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The St. John’s Review
Volume XLVI, number two (2002)
Acting Editor
George Russell
Editor
Pamela Kraus
Editorial Board
Eva T. H. Brann
James Carey
Beate Ruhm von Oppen
Joe Sachs
John Van Doren
Robert B. Williamson
Elliott Zuckerman
Subscriptions and Editorial Assistant
Blakely Phillips
The St. John’s Review is published by the Office of the Dean, St.
John’s College, Annapolis: Christopher B. Nelson, President;
Harvey Flaumenhaft, Dean. For those not on the distribution
list, subscriptions are $15.00 for three issues, even though the
magazine may sometimes appear semiannually rather than three
times a year. Unsolicited essays, stories, poems, and reasoned letters are welcome. Address correspondence to the Review, St.
John’s College, P Box 2800, Annapolis, MD 21404-2800.
.O.
Back issues are available, at $5.00 per issue, from the St. John’s
College Bookstore.
©2002 St. John’s College. All rights reserved; reproduction in
whole or in part without permission is prohibited.
ISSN 0277-4720
Publishing and Printing
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Contents
Essays and Lectures
Measure, Moderation and the
Mean............................................ 5
Joe Sachs
Plato and the Measure of the Incommensurable
Part II. The Mathematical Meaning of the Indeterminite
Dyad................................................................................
................... 25
A.P. David
Moral Reform in Measure for
Measure............................................63
Laurence Berns
Book Reviews
Eva Brann’s, The Ways of
Naysaying ................................................79
Chaninah Maschler
Eva Brann’s What, Then, is
Time?.....................................................107
Torrance Kirby
The Feasting of Socrates
Peter Kalkavage’s translation of
Timaeus...................................117
Eva Brann
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5
Measure, Moderation, and the
Mean
Joe Sachs
(with particular reference to the story Odysseus tells in
the Odyssey)
Anyone who comes to love the writings and artworks that have
survived from ancient Greece ought one day to visit Olympia. In
Athens there are wonderful things to see, but also evidence everywhere of the destructive effects on buildings and statues of some of
the most polluted air anywhere in the world. But, in Olympia, in
the Peloponnese, where the most famous of the ancient athletic
games were celebrated, one can still breathe purer air, and see glorious sights. In particular, in the museum there, at the two ends of
the large main room, restored to their complete shapes, are the two
pediments of a temple of Zeus built in the decade of the 460s BC.
(Illustrations are at the end of the text.) The form of a pediment will
be familiar to you as what sits above the appropriate sort of
entrance to a temple. Picture a rectangle, wider than it is long, made
of evenly spaced vertical columns; resting on top of this row of
columns is a triangle, shorter than it is wide, with a series of sculpted figures across it. The statue at the center of the triangular pediment is the tallest figure and the focus of the whole composition.
The eastern pediment at Olympia depicts Zeus at its center, in a
monumental style that makes one think of Egypt. In fantasy, one
might see this pediment as a doorway into ancient Greece, leading in
from the east. But the truer doorway to things that are most characteristic of classical Greece is at the other end of the room. The western pediment depicts the defeat of the Centaurs, who are men in
their heads, arms, and upright chests, but horses in their legs and horizontal lower trunks. They are attempting to carry off human
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women, and one young boy, but the sculptor has captured the
moment of their defeat. They are being fought by human heroes,
including Theseus, but they are defeated by a look and a gesture. At
the center of the pediment is Apollo, ten feet tall, looking to his right
with his right arm outstretched, the hand level, the palm downward.
The look in his eyes is not angry but serious, and his face is not
clenched in threat but calm. The centaurs cannot have their way
when faced with the power radiated by such dignity. This scene, displaying in outward figures an inner topography of the human soul,
holds in it something of the spirit of classical Greece. The fact that
you or I can see these seemingly invisible qualities, just by being
patient and receptive in front of some shaped blocks of stone, is one
of the amazing achievements that has survived from that time and
place.
Zeus was, as you know, the father and ruler of the Olympian
gods, and even the name of the town Olympia was taken from its
temple of Zeus, who was the Olympian, but somehow Apollo came
to be pre-eminent among the gods imagined as living on Olympus.
At Delphi, on Mount Parnassus, above the Gulf of Corinth, there
was an ancient temple of Gaia, Mother Earth, which was considered the center of the earth. But people were kept away from it by
the Python, an inhuman monster, until Apollo killed it. The Pythia,
the priestess of the temple, then became a medium through whom
people could consult Apollo, and learn his word, or oracle. The
story of Pythian Apollo embodies the same meaning as that of the
Apollo sculpted at Olympia, a victory on behalf of humanity, won
over older and subhuman enemies. The dragons and half-humans
are not wiped out, but become subject to something shining and
beautiful. I think you will find some version of this insight present
in almost every work you read from classical Greece, though not
everyone would agree, and it may certainly at times be something
hard won and dimly seen. But even tragedy, a type of poetry discovered by certain Greeks, always displays that, even in the most
horrendous circumstances, there is a human dignity that we can still
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recognize; that when it is recognized it commands respect; and that
this respect allows all things to be seen in their true proportions.
Above the doorway of the sanctuary of Apollo at Delphi, we are
told (Plato, Protagoras 343B) that two sayings were inscribed:
Know thyself, and Nothing to excess. These may seem to be disconnected—an exhortation to self-knowledge and a platitude about
not going overboard with anything—but to think them together is
to find the meaning of each. Know thyself means know your true
limits, the greed and ambition to which no human being should
aspire and the depths to which no human being should sink. And
Nothing to excess is not just practical advice; it means that the
nature of anything, including human life, is revealed only when its
true proportions are found—that the truth of anything is its form.
The positive version of Nothing to excess is another saying—
Measure is best—and the measure of a thing is its form.
To take a simple example, what are the right proportions for
the entrance to a temple? When I described the pediments at
Olympia and asked you to picture them and the columns under
them, I’ll bet you got their proportions just about right. The rectangle formed by the columns is wider than it is high. How much
wider? Enough so that it will not look squashed together, but not so
much that it would become stringy looking. Let your imagination
squeeze and stretch it to see what goes wrong, and then notice that
to get it right again you have to bring it back to a certain very
definite shape. This is the golden rectangle. It has been produced
spontaneously by artists, architects, and carpenters of any and every
time and place. What is the ratio of its width to its height? I can tell
you exactly what it is, but not in numbers. I can also tell it to you
in numbers, but not exactly. It is approximately 61.8 units wide and
38.2 units high. That will get you in the ballpark and your eye will
then adjust it to make the ratio exact, but it can be proven that no
pair of numbers, to any finite precision, can accurately express this
ratio, which is that formed by cutting a line so that the whole has
to its larger part the same ratio that the larger part has to the smaller. If you have a calculator, you can check that 61.8 is to 38.2 in just
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about the same ratio as that of 100 to 61.8, but no matter how
many decimal places you take it to, any ratio of numbers for the
parts will fail to match that of the whole to the larger part. We know
many things by measuring, and our usual way of measuring is with
numbers, but in this case numbers are too crude an instrument by
which to know something our eyes know at a glance.
Taking the measure of something, then, does not necessarily
require quantifying it. We are always going too far in trying to quantify things. The intelligence quotient is a precise number, and no
doubt it means something, but it doesn’t capture anything worth
calling intelligence. An acquaintance of mine, who grew up in
Baltimore, once watched an old, uneducated cook in North
Carolina make biscuits. She was writing down the recipe, and at one
point asked “How much shortening did you use?” The reply was
“Enough to make it short.” This example reveals both the genuine
intelligence of the cook, which would not show up on any test
score, and the fact that she was measuring the shortening not by its
volume or weight but by its feel as she mixed it into the dough. Her
hands were performing a qualitative measurement, just as the eyes
of your imagination were measuring the rectangle by its shape,
rather than by the lengths of its sides. You should not be too quick
to agree with me about this, because if you do, you may have to give
up many other things you believe.
I am claiming, and this is something I learned from certain dead
Greeks, that the world really has qualities in it, that they are not
subjective distortions projected onto it, but the true forms of things.
I know them by my senses, and I know them better that way than
by any theoretical explanations of them. With the golden rectangle,
the discovery of the ratio of its sides reveals something that we can
never name directly—we cannot say how many times bigger one
side is than the other, or than any possible fractional part of the
other—but we can still recognize that ratio in two ways: in its sameness with another ratio, or, even more simply, in the distinctively
shaped rectangle it produces. What is quantitatively incommensurable is qualitatively harmonious. Similarly, the experienced cook
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knows that all batches of flour and shortening are not identical, and
that they may not behave the same way at different times of the
year. If you want the biscuits to turn out right, the only thing to
trust is your hands.
We need not go through all five senses, but one example of
measurement by the ear will be helpful. Clamp a guitar string at
both ends, put a bridge under it about two-fifths of the way from
either end, and pluck the two parts. You will hear something interesting. But what if the string is not of uniform thickness all the way
along? If you have measured the two lengths to make them exactly
as two to three, you might still hear something that sounds wrong,
just a little off. The interval of a fifth is produced by strings with
lengths in a perfectly commensurable ratio, all other things being
equal, but the lack of uniformity in real strings means that one tunes
an instrument best with one’s ear. It is true that musicians nowadays
sometimes use little electronic devices that read out frequencies of
vibration. But if the machine malfunctions, it will do no good for
the musician to tell the audience he got all the numbers right. Only
for the ear is there such a thing as being in tune.
Measure, proportion, and harmony are in the nature of things,
and we have a direct responsiveness to them that orients us in the
world. These are not the ratios of mathematics, but incarnate ratios.
And the words pure and applied do not fit the distinction, because
the purer instances of measure are the ones given to our senses. A
tradition preserved by a twelfth century writer (Johannes Tzetzes)
tells us that the inscription above the doorway of Plato’s school, the
Academy, read “Let no one without geometry enter under my roof.”
Does this mean that skill in mathematics was, as we would say, a
prerequisite for his classes? I don’t think so. It seems to me important that the entrant is not required to have mathematics, but geometry. Much of mathematics develops from the act of counting, a fundamental and natural power without which we could not speak or
think, but geometry starts in a different way, from a sensory recognition of the ordering of simple visible shapes. In Plato’s Gorgias
(508a), Socrates actually tells a young man that he is without geom-
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etry, but he is not criticizing Callicles for his intelligence or learning
or skill, but blaming him for a failure of moral choice. The young
man is greedy and in danger of having no friends, Socrates says,
because he does not recognize the way geometrical equality gives all
things the proportions that let them be part of larger wholes. The
loss of a sense for geometry is equated with losing one’s way in the
human world.
An example that shows both the positive and the negative side
of this is the central scene in Plato’s Meno. Meno’s “boy,” a slave
who has never been taught geometry, begins to discover it in front
of us. Relying at first solely on his ability to count, he twice goes
wrong in trying to measure the side of the double square, but counting also shows him he is wrong. With Socrates leading the way, by
drawing figures and pointing at them, the slave eventually is led to
trust his eyes, and to see the square double itself, out of itself. And
while Socrates asks all the questions, the slave has to do all the seeing himself, out of himself, just as he was led to his mistakes, but
made them himself. This is all very elementary, but the slave has
geometry in him, and he also has a little bit of courage and determination in getting it out—two qualities his master lacked when he
found some unexpected difficulty in answering other questions.
And this finally is the point of the scene, the reason Socrates
arranges it in front of us: Meno cannot see that his “boy” is a better man than he is. We can all recognize that certain people deserve
more respect than others, if we are honest, but Meno has lost that
capacity. He has lost his way. He is without geometry.
This way of understanding geometry may help explain an
apparent inconsistency in Aristotle’s Nicomachean Ethics. Near its
beginning, Aristotle says something that might at first seem to be
opposite to the inscription on Plato’s gates. He warns the reader not
to look for the precision of mathematical demonstration in the
study of ethics (1194b 19-27). Is this not equivalent to writing on
the portals of this sort of philosophy, “let no one try to enter here
with geometry”? If so, it is odd that Aristotle fills his exploration of
ethics from the beginning with references to actions that are in pro-
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portion, or in ratio, or in a right ratio. For instance, someone may
have good fortune and a steady course through life, but be knocked
out of equilibrium by some misfortune. The inability to cope with
disaster is out of proportion (1100a 23, 1101a 17) with the rest of
the life. Since some alteration is inevitable, and some grief would be
appropriate, and no rules prescribe its amount or how it should be
expressed, only a geometrical eye can judge this. The fitness of such
actions might be measured with some precision, but it can never be
demonstrated. All the circumstances and all the history of any
action can never be known, too many considerations have to be balanced, and equally good alternative ways of handling difficulties are
always possible.
Aristotle, then, does believe that human actions can be chosen
and recognized as right or wrong with precision, but he denies that
this is the same as the precision of a mathematical demonstration.
But he not only uses the language of ratio and proportion for the
kind of precision appropriate to ethics, he also speaks of all actions
that come from virtues of character as actions that hit the mean.
This is easy to misunderstand, because readers tend to ignore the
warning he gives almost as soon as he begins talking about the
mean, that this sort of mean is also an extreme (1107a 6-8, 22-3).
In fact, people rarely understand that this sort of mean is not quantitative at all. But taking it in a quantitative sense opens the way to
identifying the mean with the mediocre, the middle of the road, or
even middle-class morality, the sort of timidity that shies away from
anything that might distinguish one from the crowd. But one of the
things that Aristotle says hits the mean is courage, and he says plainly that there is no such thing as too much courage.
Now one way to see how courage both is and is not a mean
condition is to extend the mathematical language to a second
dimension, and this is both accurate and helpful. There is no such
thing as too much courage, but there is such a thing as too much
confidence, just as there can be too little of it. Courage occupies a
mean position on a scale of fearfulness and fearlessness. The sense
in which courage is an extreme is on a different axis, one on which
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the person who has just the right amount of fear puts that attitude
into action in the most excellent way. We might even liken this twodimensional scheme to the appearance of the west pediment at
Olympia, on which Apollo occupies the middle position, but also
towers over everyone else. Courage is like that. As I say, this is true
and it helps one keep hold of Aristotle’s claim that the virtues are
extremes of human character, but also stand in and aim at a mean.
But for all that, this talk of measuring along two axes seems to
me to be misleading in the most important respect. I can show how
very simply. Just ask yourself if the power of Apollo over Centaurs
and humans would be greater if he were taller. As it is, he towers
over them, but the design could have been made in such a way that
he dwarfed them, reduced them to puny insignificance. With a little bit of play in the imagination, I think you can see that this would
destroy the sculpture’s effect. The designer of the pediment (who
may have been someone named Alkamenes) wasn’t aiming at making Apollo as big as possible, but at making him extend the human
stature just a little. The Centaurs are sub-human monsters; a gigantic Apollo would also be monstrous. The sculptor has not only
placed Apollo in the middle of the horizontal array; he has also hit
a mean along the vertical axis. All the power of the ensemble
depends on getting the figures in a right relation to one another. As
with the golden rectangle (and recall that the pediment originally
sat on top of one), it is not a matter simply of adjusting Apollo’s
height, but of forming a single design.
Apollo’s height is a precise mean between a ridiculous shortness
and a monstrous tallness, but that mean is also an extreme in the
sense that it is unsurpassably right. But the way in which it is unsurpassably right is not quantitative. It is unsurpassably right in the
design to which it belongs. It fits, and nothing else would. Liddell
and Scott, the authors of the standard dictionary of ancient Greek,
will tell you that aretê, the word for virtue, comes from the name
of Ares, the god of war, but another school of thought derives it
from a humble verb that means to fit together (arariskein), or be
fitting—it may be related to a similar humble verb, from wood-
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working (harmozein), from which we get our word harmony.
Courage too, as Aristotle or any thoughtful person would explain
it, comes not from the bloodthirstiness of the war god, but from
recognizing what one’s circumstances call for and carrying it into
action. Only when the circumstances are extreme, as they are for
Patroclus or Hector, does courage call for the extreme risk, or sacrifice, of life, or perhaps, in the case of Achilles, for the sacrifice of
revenge. At the end of the Iliad, the usual ways of confronting an
enemy are no longer fitting, and Achilles recognizes that.
The recognition that Hector’s body belongs to his father and to
his city has nothing to do with anything quantitative. It is not
arrived at by adjusting any sort of dial up from too little or down
from too much. But it is a measured response to the situation that
Achilles faces. It is geometrical equality that Achilles restores, by letting the dead man be given an appropriate funeral. It is dignity that
he measures. Priam, the miserable wreck of an old man at Achilles’s
feet, dominates his action in exactly the way Apollo dominates the
Centaurs. In both cases, anger takes up a subordinate position in the
design of the human soul. It finds its right proportion to the whole.
On a list of the various meanings of the word logos preserved from
Aristotle’s school by an ancient scholar (Theon of Smyrna), one of
those meanings was the ratio of one who gives respect to the one
who is respected. By looking at Apollo in his glory, or at Priam in
his misery, we can begin to take our own measure.
This kind of qualitative measurement is appropriately represented by ratios, because a ratio is not a quantity. A ratio limits a
quantity. It is a revealing fact that we all have trouble remembering
what Euclid means by greater ratio—that it is not the span of the
interval between two magnitudes but the size of the first in relation
to the second that he is referring to. A length, or an area, or a volume, or for that matter a weight is measured by its size or amount,
but a ratio is something on a different order of things. We measure
length by cutting it up and counting the pieces, but ratios do not
admit that kind of treatment. Fractions do. Fractions are quantities
but ratios are not. The nature of quantity is that of material. There
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can always be more of it or less, arranged this way or that. And this
way of looking at quantity helps one see that ratios belong to the
realm not of material but of form.
In the Odyssey, Odysseus tells a story that goes on for four long
books. About two-thirds of the way through it he tries to stop and
go to bed, but his hosts will not let him. He claims the story is taking too long to tell, and there is too much more of it, but they are
spellbound and persuade him to go on. The king who speaks for
them tells Odysseus that there is a morphê upon his words (XI,
367), meaning a shapeliness or gracefulness. This is one of the
words that comes later to be used for “form” in an important philosophic sense. Odysseus need not measure his words by time or number, the king is telling him, because his hearers measure them by
beauty and depth. A form does not merely surround its content
with a shape. It transforms the material and makes it be what it is,
through and through. And just as Alkinous praises Odysseus for the
form of his story, Aristotle too, in his Poetics (Chap. 8, 1451a),
praises Homer for knowing where to start and end an epic poem to
make it be one story goverened by one action.
What is the form that governs the story Odysseus tells the
Phaiakians? Neither they nor we ever take that story to be a simple
report of the events that Odysseus witnessed and took part in since
the time he left Troy. It is a story formed or transformed by art. But
if all stories that reshape events were lies, fiction would simply mean
falsehood. Alkinous distinguishes Odysseus from the multitude of
liars the dark earth breeds. His criterion is not easy to translate, but
it is understandable to us because we too have heard Odysseus tell
his story, and know exactly what he means. Lattimore makes
Alkinous say that the liars make up stories from which no one could
learn anything (XI, 366). The more usual translation has it that the
lying stories are made up out of things no one could see, and this,
in turn, either in the sense that all the human witnesses are dead, or
in a deeper sense. Both translations are possible, and both capture
something of what Alkinous is talking about. Odysseus is trying to
get something out of the Phaiakians, but he is also letting them learn
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from his experience, and they count that a fair exchange. Things
that are literally false, contrary to fact, are redeemed from falsehood if they capture truth that goes beyond the merely factual. No
one can go see if the story was accurate, but no sensible person
would try to check it in that way, because its proper subject is something that cannot be seen. The story puts in front of the eyes of our
imaginations things that are invisible.
What is Odysseus’s story about? It is, first of all, full of fabulous
beings, gods and monsters and people who live in strange ways. A
question that is repeatedly asked, not with formulaic phrasing but
with constant changes in its wording, is whether the characters that
are about to be encountered are human, that is, dwelling on the
earth and eaters of bread (VI, 8; IX, 89, 191). And even among
those who are not immortal gods and monsters, some dwell under
the earth and drink blood, some dwell in mountain caves and are
cannibals, and some eat the lotus fruit and dwell in their own psyches. But these non-humans are not only a background against
which the human form is displayed, they are constant temptations
to the humans themselves.
Some of the companions of Odysseus are seduced by the lotus
into the oblivion of ignorance, but Odysseus himself is later seduced
by the Sirens, toward the oblivion produced by the love of knowledge. On either side there is a loss of connectedness to the human
community. And Odysseus’s story begins among the Kikones, where
his men get drunk and reckless with success, and then, when their
luck turns, lose six of their companions out of each of their twelve
ships; his story ends among the cattle of Helios, where the men who
are left, less than fifty of them on their one remaining ship, get hungry and reckless in misfortune, and lose their lives. In both overconfidence and despair their hungers become unmeasured by judgement. And again Odysseus too experiences the same dangers, in his
different way. His hunger for recognition, when he has saved himself and his men from the Cyclops, results in a foolhardy judgement
which brings him Poseidon’s curse, and turns victory into needless
defeat; and this is followed by another foolhardy judgement, that he
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could stay awake for ten straight days with the bag of winds, and
arrive home the single-handed savior of his men. His hunger for
glory is as deadly to his judgement as his companions’ hungers are
to theirs.
This break-down of judgement is again a loss of the connectedness of human community, since disproportionate hunger of any
kind, whether from extreme self-indulgence or extreme need,
brings isolation. After the fiasco with the bag of winds, Odysseus
twice shows himself to us in isolation on top of mountains (X, 97
and 148), and this image surrounds his explicit comparison of a
monstrous Laistrygonian to a mountain peak (X, 113), and echoes
his earlier description of the Cyclops (IX, 187-92). Here is what
Odysseus says when he narrates his first sight of the cave of
Polyphemus: “Here a monster of a man bedded down, who now
was herding his flocks alone and afar, for he did not mingle with
others, but stayed away by himself, knowing no law, for he was
formed as a wondrous monster, not like a man, an eater of bread,
but like a wooded peak of the high mountains which stands out to
view alone, apart from others.” In his outsmarting of the Cyclops,
Odysseus displays the power that lets a puny human master a gigantic brute, but in his glorying Odysseus outsmarts himself, and ends
up no better than a Cyclops.
Finally, Odysseus is measured against the gods. This is most
apparent in his verbal jousting with Athena when he awakens on
Ithaca in Book XIII. She uses superhuman knowledge and magic to
deceive and test and tease him, while he holds his own with his
merely human skills, to her delight. “That’s my boy,” she says in
effect, and he replies, in effect, “So where have you been for so
long.” But this alliance of man and goddess as friendly rivals is not
the one that is his true test. It is Kalypso who offers him the ultimate choice, to be her lover forever, while neither of them grows
old, on an island that grows everything to delight the senses and
requires no work. He chooses to go back into the sea, to work, to
fight, to take chances, and ultimately to die. He does not talk about
any of this in the story he tells the Phaiakians, though he had told
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the king and queen the bare outline of it the day before. We know
the story of Kalypso’s island from Homer’s telling of it, before we
know how to understand it. It is Odysseus who puts it in context.
From the time, early in Book X, when he comes down from the
mountain on Circe’s island, the rest of Odysseus’s story is about his
losing battle to win back the trust of his companions. “I am in no
way like the gods,” he has said to Alkinous, “but count me equal to
whomever you know among humans who bears the heaviest load of
woe.” (VII, 208-212) But unlike another man who might say that,
Odysseus had a choice, and chose human troubles. What he lost,
with his companions, was more worthy of choice to him, than what
he could gain from Kalypso’s gift.
We make much of Achilles’s choice, to live a short and glorious
life instead of a long and ordinary one, and pay less attention to
Odysseus’s choice, to live not at ease forever but for a long but
bounded time, amid troubles that will eventually come to an end.
You probably know that the first word of the Iliad is wrath; of the
Odyssey the first word is man. The shaping of the Iliad rises from
the flare-up of Achilles’s wrath, to come to completion when that
wrath itself finds its limit, not just in duration but in submission to
a higher good; the wrathful, warlike side of human life finds its
form and proportion within a larger whole. The Odyssey is formed
in a different way. It starts in three places (Olympus, Ogygia, and
Ithaca). It backs up, and proceeds for a while on parallel tracks, as
we hear a story told and watch the interaction of the teller and hearers, and finally begins moving forward in its second half. But
through and through, the form that shapes the Odyssey is the form
of the human being, as it shows us a man travelling up to all the limits of what it is to be human, coming to know them, and choosing
to remain within them. A participle in the fifth line of the poem
(arnumenos), as it is usually translated, credits Odysseus for saving
his life, but it has a richer meaning: he earned or achieved his life,
proved worthy of it by learning that it was worthy of his choice.
The Phaiakians understand his story, and honor his choice by making one in its image: they choose to risk their easy life by taking on
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his troubles as their own, and their journey to Ithaca is their last
carefree voyage. The first thing we hear about the Phaiakians is that
they live far away from men who earn their bread (VI, 8), but the
human form becomes visible to them in Odysseus, and draws them
out of their isolation.
In the west pediment at Olympia, human dignity is made visible in the figure of Apollo. In the toils and troubles of Odysseus at
sea, human worth becomes apparent against a background of goddesses and monsters and bad choices. The beauty of the Phaiakians’
action is set against the perversion of the human image in the young
suitors who have taken over Ithaca. The suitors are worse than the
Centaurs at Olympia, who are simply appetites that have not yet
come under control. The suitors have no respect for any man or
woman (XXII. 414-15), and so they cannot be reformed. What they
cannot recognize, they cannot take as formative. Their image, in
their feasting, reflects that of the human pigs on Circe’s island; in
their obliviousness to someone else’s home, it reflects that of the
lotus eaters; and in their reasoning that Telemachus is about to
become an obstacle to their pleasure and so, of course, should be
killed, they are no different from the cannibal Cyclops. Odysseus
knows what to do when immortality is offered to him, because he
has learned to respect the claims of human need, and wants to
redeem his loss of his companions, for which he bears not all, but
enough, of the blame. And he will have to use the same standard to
decide what to do about the suitors.
But in Ithaca and abroad, in the story that surrounds that of
Odysseus, there is a gallery of portraits of simple human dignity.
They work on us to convey the power we respect in old people
whose experience has brought them understanding. One of them is
Nestor, who responds to strangers first by feeding them and only
afterward asking whether they are pirates. (III. 69-74) Pre-eminent
among these figures is Eumaeus, the swineherd, a victim of pirates;
born the son of a king (XV 412), he was kidnapped and sold into
.
slavery, but came to accept his lot as the lowliest of servants with no
bitterness (XIV 140-147). He balances the picture of life on Ithaca:
.
SACHS
19
as the suitors have turned a palace into a pig-sty, Eumaeus, with his
courtesy and shrewd judgement, has turned a pig-sty into a place of
gracious hospitality. Homer refers to him as the godlike swineherd
(XIV 401, 413), and as the swineherd, first in the ranks of men
.
(XVII. 184). But surrounding and woven through all these portraits
of age and wisdom is the un-regarded figure of Mentor. Odysseus
had left him in charge in Ithaca (II. 225-7), but his power to rule
rested on nothing but respect. With the invasion of the suitors, the
foundation of civilized life on Ithaca collapsed, and in the resulting
chaos we hardly notice Mentor, since he cannot fight, and barely
raises his voice. He is glorified in the last line of the poem, when
Athena, in a poetic equivalent of the sculpted figure of Apollo at
Olympia, has put an end to the violent strife of people who are all
alike (XXIV 543), making herself recognizable in the voice and liv.
ing form of Mentor. These last words of the whole poem confirm
our sense that its first word, man, is what it intends to reveal to us,
and the final embodiment of that revelation is in a radiant presentation of a character so humble the poet had to compel us to notice
him at all, a character whose dignity lives only in the medium of our
respect, while that dignity, in turn, is the only foundation for shared
human life. Homer makes us err, in overlooking Mentor, and come
to ourselves in recognizing him, so that, in a small way, we mimic
Odysseus’s journey.
But if we are to take the human measure from
Mentor, that must mean that he displays human excellence, and that would be a very strange claim to make.
The poet Homer can play in a serious way by putting the
kingly soul of Eumaeus in a position in which he has only
pigs to rule over, and he can leave us with the vision of a
goddess who makes a humble man resplendent, but neither of these figures seems to display any maximum of
human possibility. Instead, what we seem to see in them
is the last shred of dignity that cannot be taken away from
any human being by any sort of mistreatment from others, but can only be lost by one’s own act. When
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THE ST. JOHN’S REVIEW
Odysseus comes out of the sea alone on the island of the
Phaiakians, he burrows under a pile of leaves. Here is the
way Homer describes this action: “As when someone
hides away a glowing ember in a black ash heap at the
end of the earth, with no countrymen anywhere near, no
others at all, saving the seed of fire in a place where there
is no other source from which he could start a fire, so did
Odysseus cover himself up with leaves.” (V 488-91)
,
Odysseus almost lost himself on his journey. And the
thing that nearly smothered the last spark of humanity in
him was his drive to excel.
We are told in the third line of the poem that many were the
people whose cities he saw and whose intellects he knew, and for
Odysseus every new experience was a test. Seeing and knowing
were never for their own sake for him. He was always taking the
measure of any new places and their inhabitants, and that, for him,
came to be for its own sake, continually to prove himself more than
the equal of any kind of skill or strength or strategem, and worthy
of respect from anything that exists that can pay respect. He wanted to go beyond anywhere others had been, to find every limit and
surpass it. This fits a conventional understanding of excellence, but
it makes no sense. It aims at nothing but beyond everything, so that
the task is infinite and formless. To achieve excellence in this way is
to measure oneself against what is measureless. Only a being of
infinite capacity could be genuinely successful. One image of human
finitude in the Odyssey is our need to sleep. The journey from
Aeolia to Ithaca is long and hard, but achievable, but also just barely longer than anyone could stay awake for. With a dangerous cargo
like the bag of winds, a sensible captain will have to admit his own
limits to himself, and take someone else into his confidence, but
Odysseus does not permit himself such weakness. That stubbornness costs him more than nine years of trouble, and eventually costs
every one of his companions his life. When we see Odysseus give
way to sleep again, the meaning is exactly the opposite of the former occasion. His sleep brings to an end his efforts to persuade his
SACHS
21
comrades, and they eat forbidden meat and die; they decide that
they are no heroes, and cannot hold out indefinitely against hunger.
Afterward, Odysseus never ceases to defend them. But it is usually not his companions themselves that he refers to, but the common lot of human beings that he discovered by paying attention to
them. No less than six times he lectures people about the cursed
belly, and the things its need can drive people to (VII. 215-21; XV
.
343-5; XVII. 286-9, 473-4; XVIII. 53-4; XIX. 71-4). The man who
once despised weakness in himself is now the fierce defender of
those whose strength fails them. His rejection of the offer of immortality is in part a gesture of solidarity with his companions, and his
disguise as a beggar on Ithaca in some way displays the truth. In
front of the Phaiakians, Odysseus could have told his story to present himself as the hero of Troy, the most important man in the
world, but he chooses instead to make his loss and his need central.
He tells one of the suitors “Nothing feebler than a human being
does the earth sustain, of all the things that breathe and crawl on
the earth” (XVIII, 130-1), using the same adjective he chose when
telling Kalypso “I know very well that thoughtful Penelope is feebler than you in both form and stature” (V 215-17). He has learned
,
to see what is fragile in us and in need of protection as having a
higher claim on his effort than any extraordinary achievements that
might extend human glory.
But the radiant dignity conferred on Mentor at the poem’s end,
and glowing from within Eumaeus in its midst, is not the whole of
the human image either. There is also heroic action that is not ambitious for glory but called forth in defence of what is dignified but
weak. In Aristotle’s ethics the word that names human dignity is
spoudê, seriousness, the quality that is apparent in certain exceptional people who know what to take seriously. But in the Odyssey
the focus is on aidôs, respect, the quality present in all of us that
enables us to recognize dignity. Respect can take the place of force,
and can bind together a community, establishing the conditions of
life under which the things that have seriousness and dignity can be
given their due. The actions that embody respect constitute what
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THE ST. JOHN’S REVIEW
Aristotle calls distributive justice, the paying of what is due not
merely in the quantitative medium of money but by reference
always to the qualitative medium of honor. In a just community,
according to Aristotle, there will never be simple equality, but rather
proportional equality, actions and titles and gestures that make evident what different people deserve. And this is what Socrates called
geometrical equality, since it requires an act of seeing rather than
one of calculating.
In the Odyssey, our seeing is put to work most vividly beyond
the world in which we live and make choices, envisioning the
Cyclops, the passage between Scylla and Charybdis, or Odysseus
lashed to the mast while the Sirens sing, but as in the west pediment
at Olympia, these figures depicted as outwardly visible display the
shape of the invisible human soul. The soul that Homer lets us recognize as unsurpassably right in its ordering is the one that we see
in the hero in rags, in his feeble old father in armor (XXIV 513-25),
.
in the boy who calls an assembly of adults, in the woman who neutralizes the strength of 108 men (XVI. 245-51) and stops time itself
for four years by unweaving every night what she wove by day (II.
94-110). It is the human balance in which strength has reason to
give way to weakness, and weakness has resources to find strength.
It is the human mean that can live only within a community. The
best human life is a topic that demands philosophic reflection, but
such reflection would not be possible if one could not, in the first
place, simply see its form.
NOTE:
The central importance in the Odyssey of the respectful attitude aidôs
that makes human communities possible is something I first learned by reading Mary Hannah Jones’s senior essay, “A First Reading of the Odyssey,”
included in the collection of St. John’s College Prize Papers, 1977-78.
SACHS
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THE ST. JOHN’S REVIEW
25
Plato and the Measure of the
Incommensurable
Part II. Plato’s New Measure:
The Mathematical Meaning of
the Indeterminate Dyad
Amirthanayagam David
I shall argue that the controversial developments—some
would say the reversals—in Plato’s later metaphysical outlook
were in fact an inspired response to some truly epochal developments in the mathematics of his day; in particular, to certain
seminal advances in the theory of the irrational. Following on
my reading of the geometry lesson at Theaetetus 147, and of its
significance for that dialogue and for the Sophist and the
Politicus, I can now shed light on one of the most obscure
notions associated with Plato, a thing known to Aristotle as the
“indeterminate dyad.” The discovery and description of this
remarkable object—remarkable, all right, yet thoroughly nonmystical and mathematically legitimate—can be seen as the
motive force behind some of the arguments and constructs in the
late dialogue Philebus. In interpreting the ancient testimony, my
reconstruction demonstrates that the mathematical meaning of
the late Platonic metaphysics was either not transmitted to, or
simply lost on, the successors of Plato and their critic Aristotle.
But where the philosophers strayed, the mathematicians found a
fruitful path: the conclusion to the work started by Theaetetus
and Plato finds a home of concision and elegance in the mathematics of Euclid’s Book X. A historian of ancient philosophy
may have to distinguish in future between the academics who
inherited Plato’s arguments, and the mathematicians who understood them.
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THE ST. JOHN’S REVIEW
Perhaps the best evidence for a revision, radical or not, in
Plato’s thought comes from Aristotle’s intellectual biography in
Metaphysics A. He there refers to a kaí flsteron, an “even
afterwards” in Plato’s career (987b1). The passage is explicit
that there was a before and an after in Plato’s thinking which
was not apparently defined by the death of Socrates. What is
more, the change was apparently of some considerable moment;
the whole force of the expression is in the kaí; Plato is said to
have accepted the premise of universal flux espoused by Cratylus
and the Heracliteans, even afterwards. The theory of sensation
we have discussed in the Theaetetus is an example of his new
approach to an old premise, an approach based on a new mathematics of measurement.
At one time during the geometry lesson in the Meno, Socrates
counsels the slave boy, who is trying to find the line from which a
square the double of a given square is generated, “if you do not care
to count it out, just point out what line it comes from (e£ m±
boúlei ¡rivmeîn, ¡llà deîxon ¡pò poía$, 84a).” This is the
vintage Socratic irony, a playful but possibly sinister half-telling:
there is in fact no straightforward way to count out such a line with
the same unit measures that count off the side of the given square.
In a passage that means to inspire confidence in our ability to learn,
Socrates hints at a shadowy impediment that lurks, even as the slave
boy triumphs. This problem of incommensurability was the bane of
measurement science—metrhtik≠, that science which assigns
number to continuous magnitude—perhaps onwards from the time
of Pythagoras. Measurement prò$ ållhla, or mutual measurement, the reciprocal subtraction (¡nvufaíresi$) of two magnitudes, came to an end or limit (péra$) at the common measure of
these magnitudes; but if the magnitudes were incommensurable, the
process of subtracting the less from the greater, and then the
remainder from the less, would continue indefinitely (i.e., it was
unlimited, åpeiron). Such everyday magnitudes as the diagonals of
squares with countable sides were årrhton, inexpressible, or
DAVID
27
ålogon, irrational, in terms of those sides, an embarrassment to
any serious measurement science.
The in-betweenness of irrational lengths with respect to rational (countable) ones—in the Meno, Socrates takes pains to show by
a narrowing process that the required length, the side of an eightfoot square, lies somewhere in between two and three feet (83ce)—may have been the clue to a new approach. Plato’s Stranger
proposes a new branch of measurement science in the Politicus
(283d ff.); alongside measurement prò$ ållhla, there is now to
be measurement prò$ t±n toû metríou génesin, measurement
toward the generation of the mean. I have suggested that
Theaetetus’ seemingly humble classification of roots (Theaetetus
147c ff.) was the ultimate inspiration for this formulation; his novel
use of the mean proportional allows number and magnitude (the
phenomena of arithmetic and geometry) to be subsumed successfully under a revitalised and heuristic measurement science.
“’Squaring’ is the finding of the mean (› tetragwnismò$
mésh$ eflresi$, De Anima 413a20),” and he who defines it this
way, says Aristotle, is showing the cause of the fact in his definition.
To square a given rectangle, one has to find the mean proportional
between the lengths of its sides. Theaetetus distinguishes between
two kinds of length as sides of squares: a mêko$ is the length of a
side of a square number (4, 9, 16, etc.), the mean proportional (or
geometric mean) between the unit and a square number; a dúnami$ is the side of a square equal to a rectangular number (2, 3, 5, 6,
etc.)—i.e., the geometric mean between the unit and a rectangular
number—which is incommensurable with the unit in length
(m≠kei) but commensurable with it in square (dunámei).
Taken by itself, this classification is hardly more than a new
way of naming the phenomena of measurement science. Even at
this stage, however, the roots of non-square numbers, formerly
irrational and intractable, have become more expressible (@htá);
they are at least commensurable in square. A third category can
now be envisioned—incommensurability in length and in square—
so that where we had a polar division of opposites (rational-irra-
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THE ST. JOHN’S REVIEW
tional), now we have an enumeration of the phenomena: rational,
expressible, irrational.
But the true mathematical utility of this re-classification lies in
the lucid quality of the geometric mean. We recall that for any interval, this mean can be approximated in length by interpolating successive pairs of arithmetic and harmonic means within the given
extremes. Since in a rational interval, like that between the unit and
a non-square number, the interpolated means are also rational, and
since they define an evanescent sequence of rational intervals
around the same geometric mean, the incommensurable roots of
non-square numbers can now be systematically approximated with
numbers of their own. Each of these lengths, which we nowadays
call √2, √3, √5, etc., is approximated as a geometric mean by one
or more series, each unique and infinite, of arithmetic and harmonic means, which give better and better rational over- and under-estimates (respectively) of each incommensurable length. Though the
geometric mean is never reached, each successive pair of interpolations reduces the interval containing it by more than half, so that
each of the approximating extremes approaches closer than any
given difference to the mean (by Euclid’s X.1). Hence the process is
unlimited in its degree of accuracy.
The uniqueness of each of these “dyadic series,” corresponding
to each of the incommensurable roots, is the key to their achievement. Numbers may now be introduced, in a mathematically useful
and rigorous way, to describe the lengths of these roots.
Measurement science can thereby fulfil its mission, once paralysed
in these cases, to number the greater and the less. Irrational roots
are no longer vaguely “in between”: each dyad of interpolated
means defines all rational lengths, whole or fractional, than which
a particular incommensurable root is greater, and all than which it
is less. Since the “dyadic interval” can be made to shrink indefinitely, these incommensurable lengths have been uniquely measured in
terms of a given unit, as uniquely as any commensurable length.
A rational length is measured by one number, a “one many,” a
single collection of so- and so-many units (and fractional parts).
DAVID
29
These lengths are therefore measured both absolutely and relatively in terms of the unit length; one can answer the question, “How
many is it?” with respect to them. An irrational but expressible
length, on the other hand, is measured by a series of pairs of numbers, a unique but “unlimited” or “indeterminate” dyad (¡óristo$
dúa$). Such lengths are only relatively measured in terms of the
unit; for them, one cannot answer the question “How many is it?”
with a definite number, but one can always answer the question, “Is
it greater or less than this many?” There are now two ways in
which number can be applied to continuous magnitude—with a
normal ¡rivmó$ measured by the unit, or an indeterminate dyad
of such ¡rivmoí—so that both the diagonal and the side of a
square can be “counted off ” in terms of the same unit length.
The original significance of the unit and the indeterminate
dyad can now be recognised in the context of the new branch of
measurement science: the former, already a principle and product
of the existing branch, measurement prò$ ållhla—for the unit
is the measure of all commensurable magnitudes, and the ultimate
result of the reciprocal subtraction of commensurable quantities—
is a measure of all rational means (including the roots of square
numbers). The latter is a way of measuring all the expressible geometric means (the roots of rectangular numbers); it is a principle
and product unique to the new branch, measurement toward the
generation of the mean, for paired interpolation represents a way
to “generate” an expressible geometric mean numerically, and the
resulting indeterminate dyad of greater and lesser values is a precise
and exhaustive way to locate an expressible length within the scale
of the rational continuum. The unit and the indeterminate dyad,
the respective measures of rational and expressible means, are
therefore rightly conceived as the two proper principles of that science which approaches measurement through the construction of
means.
*
*
*
*
*
*
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THE ST. JOHN’S REVIEW
In the Philebus (23c ff.), Socrates proposes a four-part division
of all beings. The first two segments cover the limited and the
unlimited, the once all-embracing Pythagorean pair of opposites.
The third division encompasses those beings produced by the mixture of the polar principles; this mixed category represents the distinctive late Platonic innovation in ontological thinking, outlined
also in the Sophist (see 252e). A fourth division is enumerated to
cover the cause of the mixing in the category of mixed beings.
At first glance, the mathematical subtext of this classification seems fairly straightforward. The unlimited stands for continuous magnitude, that which admits of being greater or less
(24e); the limited stands for number and measure (25a-b). The
mixed class stands, as could be expected, for continuous phenomena that admit of measurement or a scale: Socrates mentions music, weather, the seasons, and “all beautiful things
(Øsa kalà pánta, 26a-b).” The demiurge of the Timaeus,
who constructs a cosmic musical scale out of elements he has
mixed (35b ff.), could be seen as a mythical archetype of the
fourth kind of being, the cause of mixing. The mixer is also a
measurer.
Certain peculiarities in Plato’s presentation suggest, however,
that it is motivated by the developments in ancient measurement
theory that I have described. First of all, the distinction made
between the limited and the unlimited is virtually analytic. This
would not be necessary for a distinction between number and magnitude, because of the phenomenon of commensurability. But the
class of the more and the less, the pair which characterises the
unlimited, is said to disallow the existence of definite quantity; if it
were to allow quantity (posón) and the mean (tò métrion) to be
generated in the seat of its domain (‰drˆ ™ggenésvai), the moreand-less themselves (a dual subject in Plato’s Greek) would be made
to wander from the place where they properly exist (24c-d). The
class of the unlimited therefore stands for the greater-and-less qua
greater and less, those magnitudes which refuse numerical measurement of any kind, like the radically incommensurable lengths
DAVID
31
(commensurable neither in length nor in square). The class of the
limited, on the other hand, is said to cover only those things which
admit of everything opposite to the more-and-less (toútwn dè tà
™nantía pánta decómena):
prôton mèn tò ªson kaì £sóthta, metà dè tò
ªson tò diplásion kaì pân Øtiper ∂n prò$
¡rivmòn ¡rivmò$ ˚ métron ˜ prò$ métron...
(25a-b)
first the equal and equality, and after the equal the double and everything whatever which is a number in relation to a number or a measure to a measure.
The limited is therefore the class of commensurable magnitude.
Is the distinction between limited and unlimited then a descriptive
one based on that between number and magnitude, or really an analytic one between two kinds of magnitude, the commensurable and
the incommensurable?
The mixed class is also described as the class (£déa) of the equal
and the double (25d); this means it must be meant to include within it the whole class of the limited or commensurable. One could
have expected this if it corresponds to a class of scalable magnitudes. But Socrates goes on to add this curious category to its
domain:
...kaì ›pósh paúei prò$ ållhla t¡nantía
diafórw$ ®conta, súmmetra dè kaì súmfwna
™nveîsa ¡rivmòn ¡pergázetai (25d-e)
also so much of a class as stops things which are opposites, differently disposed to one another, and fashions
them into things commensurable and harmonious by
putting in number.
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THE ST. JOHN’S REVIEW
This function appears to be unique to the mixed kind of being.
Since only incommensurable things can be made commensurable,
the unlimited did indeed signify the incommensurable, as was surmised; and the class mixed from the limited and the unlimited
appears to include a new species not found in either apart, which
makes incommensurable magnitudes commensurable by “putting
in” or “inserting” (™ntívhmi) number. With somewhat uncharacteristic acuity, Protarchus understands Socrates to mean that certain constructions (or “generations,” genései$) follow from the
mixing of the Pythagorean opposites (25e). (This interchange
seems to be a single Platonic exposition split between two speakers. The author better remembers his dramatic premises when,
within less than a Stephanus page, he has Protarchus suddenly
express his unsureness about what Socrates could have meant by
the members of the third class.)
The two ways of measuring magnitude in terms of a single unit
length, by means of a number or an indeterminate dyad of numbers,
correspond to the two classes which make up Socrates’ third category. In particular, the second way of measuring corresponds to that
construction described above which is unique to the mixed category. Both take up magnitudes that were formerly irreconcilable, subsumed by an opposition of greater to less—i.e., incommensurables
belonging to the category of the unlimited—and make them concordant and commensurable by “inserting number.” But neither of
them does this in such a way as thereby to reduce these magnitudes
to the class of the limited. Rather, certain lengths turn up in the
measurement of magnitude, incommensurable as such but commensurable in square, that call forth a peculiar application of number, one which inserts greater and lesser values in such a way that
they become more and more equal. This use of numbers comes to
light only in measurement science, and hence only in the mixed category of beings; it does not suggest itself in the operations of pure
arithmetic, the science of the class of the limited (governing numerable, discrete quanta and their formal equivalents, like commensurable lengths). An indeterminate dyad is a numerical description of
DAVID
33
a peculiar kind of length, neither irrational nor rational, but belonging to a third analytic class called “expressible.”
The mathematical subtext of Socrates’ proposal therefore runs
as follows: the distinction between unlimited, limited, and mixed
is, after all, a descriptive one based on that between magnitude,
number, and measured magnitude. But when Socrates attempts to
bring unity to each category, drawing together into one (e£$ ‰n,
25a, 25d, etc.) the beings subsumed by each, he employs a threepart analytic distinction that applies properly to magnitude alone.
That is to say, he brings unity to each of the three realms—number, magnitude, and measured magnitude—by describing each of
them in terms of the particular kind of length, the particular kind
of one-dimensional magnitude, which uniquely characterises it.
Hence the class of the unlimited is not just the class of the greaterand-less, but the class which positively rejects numerical description, like that of the radically incommensurable lengths. (The analogy is strict, for recall that this class is said to reject from its own
rightful seat both definite quantity (posón) and the mean (tò
métrion); on my reconstruction, this means it rejects the only two
ways of counting lengths, either with a single number, or with an
indeterminate dyad of numbers that approximate a geometric
mean.) The class of the limited, likewise, is not just the class of
numerable things, things which can be expressed as ratios of a
number to a number, but also the class of certain kinds of magnitude, those which can be expressed as ratios of a measure to a
measure, for commensurable lengths share all the properties of
numbers. Hence the distinction between magnitude and number
(unlimited and limited) can be reduced to a distinction between
two kinds of line. And finally, the mixed class, or the class of the
scale, though it includes within it the class of the limited, comes to
be characterised by a use of numbers and a kind of magnitude
which are each unique to it. These are the indeterminate dyad and
the lengths which it measures, once incommensurable but now
made “expressible” by the insertion of number. The expressible
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THE ST. JOHN’S REVIEW
roots form a third analytic possibility within the field of onedimensional extension, alongside rational and irrational lines.
The reductionist spirit of Socrates’ analysis is in the best traditions of ancient mathematics. To reduce one problem to another is
of course heuristic of a solution, but the process can also be useful
in definitions and classifications. An example has been given in
Aristotle’s reduction of the problem of squaring to that of finding
a mean proportional line. One effect of Euclid’s proposition II.14,
which contains a solution to Aristotle’s reduced problem, is in turn
to reduce a comparison in magnitude between any rectilinear
figures to a comparison between squares, and hence to a comparison in one dimension, between square roots. A later and particularly virtuosic example is to be found in Apollonius’ use of the
three kinds of application of area upon lines, the parabolic, hyperbolic, and elliptic, to both name and define the three kinds of conic
section. In Plato’s case, the distinctions between his ontological
realms of the unlimited, limited, and mixed—two of which, as
opposites, had had a long-standing currency in metaphysical thinking—have been reduced to the distinctions between the three kinds
of line studied in the new measurement science: irrational lines
that are incommensurable both in length and in square; rational
lines that are commensurable both in length and in square; and the
expressible lines that are incommensurable in length, but commensurable in square.
This analysis is also in the spirit of the “enumerative” method
Socrates had earlier set out (16c-17a). One is to seek out the form
(£déa) which lends unity to a field of phenomena, and then seek out
those things measured by this hypothetical unit-form (i.e., those
phenomena which are “numbers” if the original form is taken as a
unit). The method intends to be self-correcting, for one is enjoined
in turn to analyse the original unit (tò kat ¡rcà$ ‰n, 16d) in
the same way that one has analysed the enumerated phenomena, to
see “how many” it might actually be. A converse procedure is equally espoused in the case of a science like grammar (18a-d): when the
datum seems unlimited or continuous, as does the phenomenon of
DAVID
35
human vocalisation, one is first to discover the numbers into which
it naturally divides, which govern pluralities such as those marked
out by the distinction between vowels and consonants, before one
proceeds to analyse these further into their units. There may be an
analogy here with modern analyses in terms of “sets,” which also
presume that things need to be sorted before they can be counted
or related. Euclid’s definition of ratio (V requires a relation of
.3)
kind between the compared terms. Even the infinite field of number
itself is nowadays divided in such a way that unitary types may be
distinguished (“Reals” over “Rationals” and “Irrationals”) while
individual members remain both infinite and infinitely instantiatable. An “enumerative theory of forms” would seem to reflect the
ontological and epistemological implications of the interdependence of sorting, on the one hand, and counting or measuring on the
other. The new Socratic method is developed as an explicit reaction
to the Parmenidean or Pythagorean type of thinker—but also, perhaps, to the early Plato—who analyses everything in terms of
opposed principles like the one and the many or the limited and the
unlimited, and fails to articulate the crucial phenomena that are
ordered, like numbers, in between such opposites. Hasty and simplistic analysis in terms of opposites is said to characterise arguments that are made eristically, while the enumerative method, the
method that discovers the numbers of things and their ordered relations, characterises the truly dialectical approach (17a).
Socrates had earlier made it clear (14d-15c) that the familiar
paradoxes of the one and the many were no longer his concern.
Any lazy riddler could prove that an individual like Protarchus, or
a thing made up of parts, was at the same time one and many. It
was the possibility of formal unity, in the face of the sensible births
and deaths of numberless individuals, the unity that is asserted of
things in discourse—whether of “man” or of “ox” or of the beautiful or the good—that was of vital philosophical interest. Did any
such units exist? How might they persist as individuals? And how
is it that they partake of the infinite multiplicity of things that
come into being? The genuineness of these perplexities calls forth
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his enumerative approach, a philosophical pathway that Socrates
says he had ever loved, but which had often deserted him in the
past (16b). The method is hard, but the results can apparently be
astonishing; all the achievements of the arts (técnai) are said to
have been discovered on this road (16c).
The implications of this method, shot through as it is with the
influence of the burgeoning measurement science, are staggering for
the “classical” Plato. Consider that we are here hypothesizing the
existence of forms as measures, enumerating phenomena in terms
of a posited unit-form, and then examining the posited unit, presumably against the phenomena themselves, to check for its possible plurality. The method itself is therefore mixed, in such a way as
to cancel Plato’s earlier formulations. Neither is this the unhypothetical reasoning from forms to forms, whatever that may have
meant in The Republic, nor is it a reasoning from unquestioned
hypotheses, in the manner of synthetic geometry. The once eternal
forms, the objects and immutable guarantors of knowledge, have
become provisional and heuristic.
God is said to have made all beings out of the one and the many
with the limited and the unlimited as innate possessions (16c). This
would tend to insure that all phenomena will be inherently numerable, and hence to guarantee their susceptibility to an enumerative
method; we shall find the unifying form, for it is in there (eflr≠sein
gàr ™noûsan, 16d). It is as though the pairs of opposed ontological elements, once the principles of the eristic disputations, have
now been “re-packaged” in the premises, made the condition for
the possibility of an enumerable reality. Inasmuch as it was
Aristotle’s understanding (Metaphysics M.4, 1078b12) that the theory of forms was invented in the first place to account for our sense
of dependable knowledge in the face of a Heraclitean flux—and
note that the premise of a reality in flux is still accepted at Philebus
43a—it seems that this theory has now been modified to make sense
not so much of our ability to know as of our ability to count. And
this change of purpose is sparked in turn by a renewed confidence
in this sovereign ability, in light of Theaetetus’ successful attack on
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the irrational. Number had at last been restored to some of her
Pythagorean glory, as a measure of the things that are, that they are,
and the things that are not, that they are not, and what is more, of
the things in between. The victory here was sweet indeed, for the
irrational square roots were recovered from the domain of flux and
incommensurability on the very terms by which this domain is distinguished. The indeterminate dyad is both a measurement and a
process of measurement: interpolating means between means
involves a measurer and a thing measured which are continually
changing, just as in the Heraclitean or Protagorean contentions; yet
this process of itself yields a unique measure of the fixed mean proportional between the interpolated means, and makes expressible
and commensurable the once irrational root of a rectangular number.
Indeed, this process of measuring or counting in an indeterminate dyad has proved to be revelatory of form, in the sense that it
creates the class of the expressible and defines the mixed category
of being. On the one hand, things need to be sorted before they can
be counted, and hence the knowledge of form has primacy over
measurement, and the ability to count depends upon the ability to
know. But it would seem in this case that the act of measurement
can itself be disclosive of form, and hence that knowing can depend
on counting. There appears therefore to be a dialectical relationship
between sorting and counting, which is reflected in a self-correcting,
enumerative theory of forms. This methodology of the Philebus can
be seen as reincorporating certain aspects of the Pythagorean, in the
sense that once again, knowledge has become coordinated with
measurement, and to know something is in some sense to comprehend its number.
Confidence in the grounds of an enumerative approach to the
sensible world—a confidence that may once have deserted Socrates
in the face of an irrational diameter, leading him, with Meno’s
honest slave, to the abyss of irony—can allow that significant guarantees of veracity will come from the method itself. There are, for
example, different ways to “count” or measure a phenomenon,
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each of them legitimate, based on the premises and aims of the
investigator, as the several alternate divisions of the sophist and the
statesman make clear. One measure of the truth of a hypothesis,
that such-and-such a form is a genuine unit, must, under this
method, be the economy and scope of the enumeration it affords,
as a unit in fact. A criterion for a successful articulation, a guarantor that a dialectical enumeration corresponds to a real one in the
world, must therefore be the elegance of that articulation, in terms
of the economy of means and breadth of cover which problemsolving mathematicians have always striven for in the concrete
practice of their art.
Indeed, it is an informed sense of respect for developments and
concrete formulations in the arts that seems to move the older
Plato. In the spheres of grammar and music, for example, although
it appears that an abstract analysis in terms of opposites, in the manner of the sofoí, may to some extent be applied in the interpretation of phenomena, by itself such abstract analysis simply does not
make you much of a useful theorist (17b-c). An investigation into
the numbers and kinds of sounds, on the other hand, or an enumeration of the different scales and modes and the vagaries of
rhythm—these, it seems, can truly render you wiser than the common run, in music and in grammar.
Behind this sensitivity of Plato’s to the enumerative and the
concrete aspects of the arts, as against the approach through dogmatic first principles, may rest his experience of the dramatic
changes in the mathematics of his day. A distinction like that
between the rational and the irrational, which must have seemed as
basic to the science as that between odd and even numbers—an
eternal, immutable opposition, seemingly a part and principle of the
order of things—was made obsolete by the emergence into history
of a new formulation through the mind of a single, brilliant practitioner. Recall that Theaetetus’ reforms began very humbly on the
level of classification and definition: he makes the distinction
between square and non-square the basic one for number, beyond
the distinctions between, say, odd and even or prime and compos-
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ite. But of itself this suggests a new way to approach the measurement of lengths, as geometric means, and this further yields, or
reveals, a third, formally distinct category of magnitude called
expressible. Experiencing this revolutionary development, as witness or participant, must lead a thinker away from a view of tà
mavhmatiká as eternal, innate verities that can be investigated and
learned as though by recollection, towards a view of mathematics
that must acknowledge the importance and ingenuity of the problem-solver in situ, together with the power of classifications, definitions and measurements to reveal, or to obscure, the fundamental
nature of their objects. As the traditional theory of forms and the
doctrine of mávhsi$ ¡námnhsi$ can be seen as responding to the
ontology and epistemology of the earlier geometry, so can a selfcorrecting, enumerative theory of forms be seen as a response to the
ontological and epistemological implications of the new mathematics and a dynamic measurement science.
Insofar as other arts aspire to the mathematical, the new philosophical outlook must also apply to them; although, to be fair, the
provisional, enumerative approach would have long since guided
the formulations of practitioners in music and grammar, without a
felt need for a mathematical paradigm or a philosopher’s blessing.
Perhaps one should credit Plato only with waking up to the new
realities of science and art around him, much in the spirit of later
revolutions in philosophy. One need not qualify, however, one’s
estimate of the implications of this change of view for Plato’s political thought; they are as great as the differences between the
Republic and the Laws. In this vein, while Plato’s guardians had
learnt their lessons and then interpreted the world, so that nature
and politics alike would have been for them a kind of applied mathematics, Plato’s statesman is of an altogether different mould of
mathematician. He is a problem solver, in amongst it like a navigator or a physician, who must be able to adapt his laws to suit changing conditions, or improve upon his formulations to serve the present (see Politicus 295c ff., 300c). It is of course notorious that the
guardians’ inability to solve a problem—the numbering of love, and
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40
its irrational quantities—leads inexorably to the degeneration of
their regime.
*
*
*
*
*
*
In Metaphysics N, Aristotle introduces his redaction and criticism of the Platonist (or Academic) metaphysics with this statement:
“All thinkers make the principles opposites (pánte$ dè poioûsi
tà$ ¡rcà$ ™nantía$, 1087a30).” There appear to have been
various schools of thought among Academic ontologists, all of
whom posited the unit as a first principle or “element,” but each of
whom disagreed as to the nature of the opposite principle, whether
it was the “greater-and-less” or the “unequal” or “plurality”.
Aristotle makes short shrift of all these formulations, as they treat
affections and attributes and relative terms as substances (1088a16).
In N.2, he mentions a group who posit the indeterminate dyad as
the opposed element, as a way of getting around some difficulties in
the other versions; but it is still a relative principle, and in addition,
all these formulations fall to Aristotle’s argument that eternal things
simply cannot be composed of elements (1088b28-35).
Aristotle then feels, before he adumbrates his own approach to
ontology, that he must explain why these thinkers ever came up
with formulations so narrow and forced, constrained as they are by
the dogma of opposed principles (1088b35 ff.). His answer is that
they had framed the problem of ontological multiplicity in an oldfashioned way (¡rcaikô$, 1089a1-2), for they were still arguing
in response to certain paradoxes of Parmenides. The implications of
this reconstruction of recent intellectual history are decisive both
for our sense of Aristotle’s access to Plato, and for our knowledge
of Academic thought and its relation to Plato. All the Academics,
and thus Plato as well, are said to reason about existence in terms
of an opposed pair of first principles—always the unit and something else; they do this under the direct influence of Parmenides,
perhaps as part of a tradition of arguing against certain eristic dogmas of his, such as the one which Aristotle quotes:
DAVID
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o∞ gàr m≠pote toûto dam˜, e–nai m± ™ónta
For this may never be enforced, that things which are
not, are.
These thinkers are said to have felt that the possibility of multiplicity in the world would be threatened unless Parmenides were
refuted, and some other thing than unity or being were allowed to
exist. This was the origin of the “relative” principles that stood
opposite the unit. The unit and the indeterminate dyad, on this
scheme of Aristotle’s, are but one alternative among several pairs of
first principles proposed by different Academic philosophers.
The first thing to note is that the Philebus itself is Plato’s direct
and unambiguous criticism of the ontological reasoning based on
two opposed principles, in favour of a technical, empirical, enumerative approach. From the perspective of philosophical method, the
dialogue can hardly be said to have any other point. Plato conceived
of his enumerative method as a more illuminating and more useful
way of articulating phenomena, which comes to yield significant
new categories in the analysis of being (e.g., the mixed one and the
cause of mixing). No further clue seems to be necessary for the conclusion: Aristotle, somehow or another, has entirely missed the
point of Plato’s late formulations, by classing them with the type
that Plato himself characterises as eristic rather than dialectical, and
from which he most particularly wants to distinguish his own.
The next point, however, is that there must actually have been
a vigorous tradition of thought which both preceded Plato and
outlasted him in his own Academy, characterised by the use of
opposites as first principles. To believe so much is the only way to
attach any seriousness to Aristotle’s redaction. This tradition originates with Parmenides, and must once have included Plato in its
ranks, again if one is to pay any respect to Aristotle’s judgement.
But Plato came to argue against such thinkers not only in the
Philebus, but also in the Sophist, where they are called “the friends
of the forms (o‹ tôn e£dôn fíloi, 248a).” These were the lat-
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ter-day champions of eternal, immutable, unmixing forms, the
kind of weary theoretical construct that is often now taught as
Platonism. When the differences seem so clear, the question must
become: How could Plato’s new “mixed” ontology have come to
be confused with the old-fashioned approach through polar principles?
Recall that on my reading of the Philebus, there are for Plato
three ontological realms apart from the agent of cause. The first is
the realm of the limit, the realm of arithmetic, whose principle is
the unit. The second is the realm of the unlimited; its principle,
analytically opposed to the unit, is the dual greater-and-less, the
principle of irrational flux. The third realm is that of the mixed
beings, which I have interpreted as the realm of measurable things.
Its principles are two, and reflect the two ways that magnitudes
may be numbered or made commensurable, absolutely in terms of
the unit or relatively (but uniquely) by an indeterminate dyad. The
thing to note is that the unit appears as a principle twice in this
scheme, opposed in two different ways to two different things. The
distinction between the unit and the greater-and-less is strictly analytic, and belongs squarely in the Parmenidean tradition; whereas
the distinction between the unit and the indeterminate dyad is
merely descriptive, serving to recognise ways of applying numbers
inside the sphere of measurement that happen not to arise in arithmetic. The unit and the dyad are therefore not opposites; they are
simply different.
If a thinker in the Parmenidean tradition, or a historian of the
Parmenidean tradition, were to interpret Plato’s scheme in light of
their own practices, or to force it into a Parmenidean mould to flatter a historical premise, the conflation of the two distinctions would
be an inevitable result. If the Philebus could not be consulted—if it
were ågrafo$ in the sense “unpublished”—no recourse could be
had to the original reasoning; but even if there were such recourse,
Plato’s three realms of number, magnitude, and measure, and the
important differences between the distinctions unit/greater-and-less
and unit/indeterminate dyad, could only be understood in light of
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an underlying mathematical paradigm, as I have argued. Such a
thinker or such a historian would not be likely to know or to care
about the analytic possibilities in one dimension. (This is as much as
to say, he would not know what was meant by the indeterminate
dyad.) He will look for the polar principles in any ontological
scheme; at best he will see that the indeterminate dyad must connote something different from the greater-and-less, as the principle
chosen to stand opposite the unit. But he will never envision a
scheme that encompasses both oppositions.
The question next to ask is whether it was his Academic
sources, or whether it was Aristotle himself who did not understand
the mathematical meaning of the indeterminate dyad. There is
intriguing evidence in Metaphysics M and N for the latter interpretation. It would seem that his sources were in the dark about this
too; but whatever one concludes about the Academy, there is evidence that Aristotle had Plato’s accounts at hand either to quote or
to paraphrase, and that he could not make sense of them.
In N.1 (1087b7 ff.), Aristotle mentions a group of thinkers who
attempt to generate the numbers, o‹ ¡rivmoí, from the “unequal
dyad of the great and small,” taken as a material principle in relation to the formal “one,” and someone else who would generate
them from the principle of plurality. (He probably intends, respectively, the followers of Plato and Speusippus.) The generation of
numbers does not seem to have been a concern of Plato’s, however; the “problem” of multiplicity, or of how things can be both one
and many, which when posed by Parmenides might have led his successors to theorise in the abstract about the generating of numbers,
seems to be regarded in the Philebus (14c-15a, 16c-17a) as merely
a staple of the eristic paradoxes, now subsumed within the premises of Socrates’ concrete enumerative approach. Which is to say, it
appears that Plato is no longer so interested in number theory as he
is in simply counting. I am therefore inclined to think that neither
the above-mentioned group nor the ‘someone else’ represents
Plato’s line of argument, or Plato’s understanding of the unequal
dyad. Aristotle bears this out by going on immediately to mention
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an individual who speaks of the one and the unequal dyad as ontological elements (1087b9), thereby distinguishing him from the
group who had used them (afterwards, I presume) as formal and
material elements in the generation of numbers. Aristotle’s complaint about this individual is that he does not make the distinction
that the unequal dyad of great and small is one thing in formula
(lóg¨), but not in number (¡rivmõ).
Why would not Plato have made this distinction? The unequal
dyad is not one thing in formula alone: the successive pairs of interpolated numbers relate uniquely to one object as well, the side of
the square that is their single geometric mean. Further, since it consists of successively more equal sides of a single rectangular number,
the dyad can quite emphatically and strikingly be said to be of one
number, with a rationale that Aristotle might have appreciated if he
had been more familiar with the construction.
On this model of progressively “equalised” rectangular numbers, we have a transparent motivation for the original formulation
of terms like “unequal,” “indeterminate dyad,” “greater-and-lesser,” and “exceeding and exceeded,” which find their way into the
theories of Plato’s followers. In addition—and this point would
seem to be decisive for the interpretation—we should expect to
find them opposed in this context to a concept of the unit which is
associated with the square or “equal”. On no other grounds but
those of the new measurement science, as I have described them
here, would such an association be expected. Sure enough, the unit
in these theories is described as the equal (1087b5, 1092b1), in
such a way as to mystify not only Aristotle but also modern interpreters of these passages.
Neither Aristotle nor his Academic sources seem to connect
these various expressions with geometrical representations of number; the theories on the generation of numbers betray no influence
of Theaetetus’ square/oblong distinction, nor of the geometrical
interpretation of number that is settled convention by the time of
Euclid. The Academics seem to have posited “ideal” numbers which
were generated individually in succession (two, three, four, as
DAVID
45
Aristotle says in M.7 1081a23, and so without distinction as to
square or oblong) from the unit and the indeterminate dyad.
Aristotle takes some pains to make sense of this theory: if the units
(monads) of ideal numbers are all the same and addible, then they
are not ideal at all, but normal mathematical numbers (cf.
1081a19); but if the monads of each ideal number are distinct and
inaddible, they must be generated before each of their respective
numbers can be generated, as a point of logic (1081a26 ff.). This is
true no matter how these monads are generated; but Aristotle once
more quotes “he who first said it” (› prôto$ e£pµn, 1081a24)—
again distinguishing him from those who later used such phrases as
the “unequal dyad”—to allude to a possible mechanism for this generation of inaddible monads (¡súmblhtoi monáde$): they arise
out of unequals, once these are equalised (™x ¡níswn (£sasvéntwn gàr ™génonto)).
To begin with, Aristotle cannot rightly make attribution to
anyone of a theory on the generation of inaddible monads. As he
says, no one actually spoke that way (1081a36). Aristotle, perhaps
himself in reaction against the eristic movement, constructs these
arguments to save his opponents from the obvious fallacy of ideal
numbers composed of normal, identical, addible monads; yet the
alternative, unstated by them, but which he says follows reasonably from their own premises, turns out to be impossible as well,
if truth be told (1081b1). There is therefore no reason to suppose
that Plato thought or said that the generation of inaddible monads, or any monads, was connected with his notion of the unequal.
On the contrary; Plato seems to have anticipated Aristotle’s notion
of the unit as a measure, both in the intuitions of the enumerative
method and in the specifically mathematical context. At 57d-e, the
distinction is made in the Philebus between the units of the arithmetic of the many, which change as different things are counted,
and those of the arithmetic of the philosophisers, which are always
identical. It would of course have been an easy (but pointless) solution to the problem of the irrational to say that incommensurables
are simply measured by different unit lengths than commensurables.
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The enumeration of Theaetetus and Plato, on the other hand, is
predicated on the assumption of identical units. While some lengths
still remain incommensurable on these terms, all the formerly irrational square roots become expressible through an indeterminate
dyad, and the achievement of this articulation would be lost without the assumption.
What can be attributed to Plato, however, is that his notion of
the unequal involved a process of equalizing it. In neither place in
M where Aristotle mentions this idea (as above, and at 1083b24)
can he make anything of it, nor does it seem to have any intuitive
connection to the Academic number-generation theories he covers
there. The only conclusion, I suggest, is that Aristotle refers to this
conception of the unequal merely because he knows it to have been
true of Plato’s thought. The “Platonists” speak of the unequal as a
generative principle, Aristotle might have reasoned, and who knows
what they mean, as to how it generates; Plato himself also spoke of
the unequal, and the only action he attributed to it was “being
equalized”; perhaps this was somehow the “generating action,” as
obscure as that seems; one ought therefore to mention what the old
man said, in fairness to them. In N, Aristotle for the first time mentions a number-generation theory which did, perhaps, try to interpret the process; it first declares that there is no generation of odd
numbers at all, and that the even numbers are generated out of the
great and small when these are equalised. Aristotle’s criticism of the
logic of this account verges on the sarcastic: faneròn Øti o∞ toû
vewrêsai ‰neken poioûsi t±n génesin tôn ¡rivmôn.
(“Clearly, it is not on account of philosophical theorizing that they
produce their generation of the numbers.” 1091a29) Neither
Aristotle, for whom the notion seemed fatuously self-contradictory,
nor these latter theorists, for whom it was received dogma, could
have known the original mathematical context, for neither could
interpret or properly apply the notion that the unequal as an elemental principle involved a process of being equalized. We can now
restore the context, in the process of “equalizing” an unequal,
oblong number with an indeterminate dyad of more and more
DAVID
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equal rational factors. (It is particularly striking that these latter
Academics seemed to know that the notion “unequal-when-it-isequalized” served in such a way as to divide all numbers, but they
tried, with dismal consequence, to apply it to the familiar, venerable distinction between odd and even; they must have been unaware
of the division of numbers by square and oblong, which supplanted
the earlier distinction in the course of Theaetetus’ study of irrational roots, and where alone the notion of the “equalized unequal”
has any use or coherence.)
“Those who say the unequal is some one thing, making the
indeterminate dyad from great and small, say things that are far
indeed from being likely or possible,” in Aristotle’s view (M.1,
1088a15). He complains that to adopt such ideas is really to adopt
his lowly Category of the “relative” as a substantial, unitary first
principle. Something is great or small only in relation to something
else. Unlike the superior Categories of quality and quantity, which
have more substance because they involve absolute change, whether
by alteration or increase, there is no such change proper to the
Category of the relative. While a compared term may remain substantially the same, it becomes greater or less merely by quantitative
change in the other term. Aristotle is therefore at a loss as to why
such metaphysical honour should be paid to concepts that are
inherently relative.
Plato could have replied: “Consider the nature of measurement
toward the generation of the mean.” In this process, the relative
terms do not depend simply on each other, but both are related to
an unchanging third thing, a single geometric mean. Furthermore,
the pairs of relative terms are uniquely related to their proper mean,
the root of a particular oblong number. And because the greater and
lesser lengths approach closer than any given difference to the
unchanging length of the root, their status in relation to this length,
qua members of an infinite succession of approximating pairs, poses
a heady puzzle for any common-sense idea of their ontological difference from, or identity with, this single length. There is therefore
every reason to see the indeterminate dyad of great and small, a self-
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correcting binary approximation of a single geometric mean, as a
unitary and substantial thing in its proper mathematical context.
But if the context was lost, and one had access only to the words in
its name, then Aristotle’s objections might seem judicious.
That Aristotle knew about the geometry of means is clear
enough, but he must not have been familiar with the interpolation
of means in the peculiar configuration of the indeterminate dyad,
where means become extremes, which in turn beget means, which
then in turn become extremes, while each pair of harmonic and
arithmetic means serves as the extremes to the geometric mean in
the middle. The notion of relativity embodied in this configuration,
involving a process of equalising, and motion towards a fixed
object, is more subtle and peculiar than that involved in a simple
comparison, or even a static analysis expressed in terms of a mean
and extremes. I claim it is this peculiar conception of the relative
that Plato raised to the level of a principle, to stand in tandem with
the absolute measure connoted by the unit.
While the Academic metaphysicians may appear to have used
these very same principles, right down to the letter of their formulation, it is clear that neither they nor Aristotle grasped their proper function. They have nothing to do with accounting for multiplicity in the universe, or with the generation of numbers. They
have everything to do with the measurement of numbers. After
Theaetetus, numbers are figured as square or rectangular; they can
be compared not only in quantity, but in size, by the length of their
square roots, just as after Euclid’s II.14, any rectilinear figures can
be compared by the sides of their equivalent squares. While all
numbers have either absolutely or relatively measurable rootlengths, not all lengths have countable squares. This is one of the
odd new ways that arithmetic and geometry, number and magnitude, become interlinked after Theaetetus’ happy reformulation.
It is therefore in this context, the context of measurement, that
Plato is likely to have distinguished the absolute from the relative,
being-in-itself from relative being. Aristotle alludes to just such a
distinction, in a passage which once again exemplifies his peculiar
DAVID
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mire: he wants to review the Academic theories on the generation
of multiplicity based on certain contrary principles, including principles first conceived by Plato, but conceived in a context where in
some cases they weren’t even contraries, and where they had had
nothing to do with generating either multiplicity or numbers; he
knows the language of Plato’s own articulation of these principles,
but doesn’t have the mathematics to interpret the words. In this
case, he may even foist his own innovations in usage back on to
Plato’s original phrases, just to make sense of them.
At 1089b16, Aristotle once again invokes “he who says these
things,” claiming this time that this person had also proved for himself (prosapef≠nato) that that which was potentially a “this” and
substance (tò dunámei tóde kaì o∞sía) was not “existent in
itself ” (πn kav afltó); it was the “relative” (tò pró$ ti). What
the expression “potentially a ’this’ and substance” may have meant
for Plato is a difficult thing to determine. In particular, Aristotle
seems to take dunámei, with obvious anachronism, in his own characteristic sense of “potentially”; he had just now used the word this
way when introducing part of his own familiar solution to ontological analysis, that we must hypothesize in each case what a thing is
potentially (¡nágkh mèn oun...flpoveînai tò dunámei πn
flkást¨, 1089b15-16). Perhaps Aristotle is here weaving his own
terminology into the Platonic materials? But his next comment is a
scholium, on Plato’s appropriation of the term “relative,” that it is
just as if he had said “quality” (¸sper e£ eªpe tò poión); and
there was never a scholium without a text.
So what could the Greek text “tò dunámei tóde kaì
o∞sía” have meant to Plato? Recall Knorr’s observation that
dúnami$ and dunámei mean “square” and “in square” throughout
Greek mathematical literature. (The only exception is the very passage in the Theaetetus [148a] where the eponymous hero applies
the term dúnami$, for the first time, to a square root.) Thus in
Plato’s context, the same words may well have signified “that which
has particularity and existence in square”—i.e., that which is countable (because it is commensurable) only in square (dunámei), like
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the expressible as against the rational lines. It is these very magnitudes which one could expect to find distinguished as relative in
their being, insofar as their being depends on their measure; the
rational lengths, on the other hand, have the self-subsistent being of
definite quantity, in length and in square, while the irrational lines,
which cannot be made commensurable in either length or square,
are captive to the realm of flux and non-being. If Plato equated
“that which has being” with “that which can be counted”—and his
enumerative method suggests a move in this direction—then it is
entirely and specifically appropriate that that which has being in
square be allowed only a relative existence. It has no autonomous
number, but only a relative count. Even the phrase pró$ ti may
have had a specific connotation for Plato, which is lost in the
anachronistic aura of the Categories; for such beings are measured
by a process that is inherently pró$ ti, “towards something,” measurement toward the generation of the mean. Plato’s distinction
would have been between that which exists or is measured on its
own terms (tò πn kav afltó)—the equal, the square, and rational lengths—and that which exists or is measured toward something
else (tò πn pró$ ti), the unequal being equalized, the rectangle
approaching the square, and the indeterminate dyad approximating
the mean.
It seems clear that any such significance in these phrases could
never have been allowed to emerge through the schemata of
Aristotle’s redaction. He explains (1089b4 ff.) that in response to
the diversion caused by Parmenides, the philosophers posited the
relative and the unequal as the types of opposed principle which,
when mated with being and the unit, generated a manifold reality.
He points out, however, that neither of these posited principles is in
fact the contrary (™nantíon) or the negation (¡pófasi$) of being
and unity; each is rather another single nature among the things that
exist (mía fusi$ tôn øntwn). This is also the point of his critical scholium on Plato’s use of the phrase pró$ ti: the Category
“relative” is no more a legitimate candidate than the Category
“quality” for that contrary and negation of being and the unit which
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the Academics were supposed to be seeking; each is simply “some
one” of the beings (‰n ti tôn øntwn, 1089b20). He goes on to
complain that if Plato had meant to explain how things in general
are many, he shouldn’t have confined his investigation to things that
lie in the same Category (whether this be “substance” or “quality”
or “quantity,” let alone the insubstantial “relative”).
The sense of this reading ranges from the misguided to the wilfully obtuse. In the first instance, we cannot fault Plato for failing
either to prophesy or to apply the revolutionary insights into
ontology expressed in Aristotle’s theory of the Categories. Nor can
we fault him for not being interested any longer, as indeed he wasn’t, in the problem of how things are many. Still less can we fault
him for giving up the reasoning by opposites. He would of course
have agreed that his conception of the relative, in the configuration
of the indeterminate dyad, is in no sense the opposite of the unit
and its measure, but simply a different way of measuring, based
also on the unit, that applies to certain types of being (i.e., certain
two-dimensional numbers and one-dimensional magnitudes—
oblongs and their roots). But the full picture of Aristotle’s plight as
a redactor emerges when one throws in the fact that Plato’s complete formulation did in fact include a genuine opposition as well,
between the unit and the greater-and-less qua greater and less. One
then has a recipe for the peculiar quandary of Metaphysics M and
N towards Platonic thought, based in part on unwitting conflations, but in part also on flagrant, self-serving anachronisms, and
characterised by a haplessness in the face of Plato’s own expressions, when read in light of their borrowed use in the irrelevant
theories of the Academy.
A question remains: where did Aristotle get those “texts” of
Plato, which he seems to treat as quoted material? Although the distinction between absolute and relative being may be consistent with
the Philebus and with other ontological discussions in the later
Plato, the specific phrases which Aristotle comments on, such as tò
dunámei tóde kaì o∞sía, do not seem to occur in the dialogues.
Where, then, did Plato draw this mathematical distinction, and to
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52
what did he apply it? Was it perhaps in a Lecture on The Good—a
lecture which seemed to promise moral philosophy, but delivered
mathematics—a lecture which nobody understood?
*
*
*
*
*
*
The mathematical development of ancient measurement science will prove much easier to trace than its philosophical obfuscation at the hands of Academics and Peripatetics. As forbidding
as the structure of Euclid’s Elements X seems to be, I believe its
logic is profoundly simple, following directly in the spirit of
Plato’s enumerative method, and upon Theaetetus’ geometrical
interpretation of number.
After Theaetetus’ first efforts had rendered all the square roots
countable, he next sought to extend his classificatory net even further into the uncharted regions of the irrational. He could use his
already successful methods as a paradigm: since exploring numbers
in terms of the means between them had yielded the class of
expressible lines, he was led to explore the possibility of means
between the expressible lengths themselves, and the possibility of
irrational means. While in general such means could not be “counted off,” since the expressible lengths, treated as extremes, had not
the fixed values necessary for a computation of means, the mean
lengths could still be constructed and named with respect to rational lengths; just as at the time of the Meno, the root length of the
double square could not as yet be counted, but it could be constructed within the unit square and was named “diameter” (or the
“through-measure”) by the professors (Meno 85b). Orders of irrationals could thus be defined in terms of means, though they could
not be made commensurable.
Just such an assignment of orders is credited to Theaetetus by
Pappus, in his commentary on Elements X, on the authority of
Eudemus’ history of mathematics (now lost):
...it was...Theaetetus...who divided the more generally
known irrational lines according to the different means,
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assigning the medial line to geometry, the binomial to
arithmetic, and the apotome to harmony, as is stated by
Eudemus, the Peripatetic.32
The passage does not suggest that Theaetetus invented the three
lines and their names, but only that he first saw the essential parallelism between the structure of their relations and those of the
familiar means. The medial simply is the geometric mean between
two expressible lengths. That is why it is called méso$, the mean
proportional; the name “medial” serves only to distinguish it in
English. The binomial is a sum of two expressible lengths, and so
can be associated with the arithmetic mean, which is half the sum
of two rational lengths; but the apotome is merely a difference of
expressible lengths, and the connection with the harmonic mean is
less obvious. This also comes clear, however, as one recalls the fundamental feature of pairs of arithmetic and harmonic means which
makes possible the measurement by an indeterminate dyad: if one
applies a rectangle contained by rational extremes to the length of
their arithmetic mean, the height of the new rectangle turns out to
be the length of their harmonic mean. Euclid’s X.112-14 illustrate
a significantly parallel property of binomials and apotomes: if one
were to apply the same rational rectangle to a length that was
known to be a binomial, the height would turn out to be an apotome; further, and curiously enough, the expressible terms of such
a binomial and an apotome would be commensurable with each
other, and in the same ratio. If Theaetetus was responsible for these
propositions, he might well have been led to view the binomial and
apotome as “irrational means” between rational extremes, or as
irrational factors of an oblong number, counterparts to the rational
arithmetic and harmonic means.
It is clear, however, that Euclid’s presentation is not designed as
a theory of means. The bulk of his 115 propositions in Book X are
concerned with enumerating and constructing twelve different
kinds of binomial and apotome, making with the medial thirteen
types of irrational line; the full list is given by Euclid after Prop.
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111, before the proofs that establish the analogy between the binomials and apotomes, and the arithmetic and harmonic means. The
rationale for this enumeration becomes more apparent if one considers David Fowler’s handy grouping of the propositions:
X1-18: general properties of expressible lines and rectangles,
X19-26: medial lines and rectangles,
X27-35: constructions underlying binomials and apotomes,
X36-41, 42-7, 48-53, 54-9, 60-5, 66-70, & 71-2:
blocks of propositions dealing with each of the six types
of additive irrational lines. They are described in X3641 and also, in a different geometrical configuration, in
the Second Definitions following X47,
X73-8, 79-84, 85-90, 91-6, 97-102, 103-7, & 108-10:
blocks of propositions, parallel to the previous, dealing
with each of the six types of subtractive irrational lines.
They are described in X73-8 and also, in a different
geometrical configuration, in the Third Definitions following X84,
X111-14: the relations between binomials and
apotomes,
X115: medials of medials...
As Fowler himself observes, the propositions seem to represent an
exploration of the “simplest operations of adding, subtracting, and
squaring pairs of expressibles.” Before Theaetetus classified them in
relation to the different rational means, the binomial and apotome
may have first been distinguished and defined as part of an investigation of the “arithmetic” of expressible lengths. An investigator
might have said, if we are to understand the expressibles the way we
understand numbers—and indeed, numbers are the very paradigms
of our understanding—then we must comprehend their arithmetic;
what might the manipulations of arithmetic look like when applied
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to expressible lines?
Whereas the prospect of such an investigation might have
daunted the most optimistic of researchers, with its seeming openendedness and unlimited number of possible cases, Euclid was
able, by manipulating squares and rectangles, to organize the
infinite additions and subtractions of expressible lengths into six
types each. Thus Euclid accomplished the first ever rigorous ordering of radically incommensurable lengths, as the sums and differences of expressible ones. One cannot measure these sums and differences as such, and so one cannot “count off ” the irrational lines
that are produced; but one can number their types, and enumerate
their orders.
While the fundamental early propositions of Book X are generally credited to Theaetetus, and the propositions about mean proportionals (“medials”) seem to suit his historical and mathematical
character, the enumeration of the binomials and apotomes must
belong to Euclid. Pappus says that Euclid, following Theaetetus,
“determined...many orders of the irrationals; and brought to light,
finally, whatever of finitude (or definiteness) is to be found in
them.” This should naturally refer to his ordering of possible binomials and apotomes, and the enumeration of six corresponding
types. Though they do not depend on the proofs involved in
Euclid’s enumeration, Theaetetus’ propositions, about the relations
between binomials and apotomes, are then placed by Euclid at the
end of Book X, so that they can be expressed in terms of that enumeration, and take on a new authority: each pair belongs to one of
six sets of ordered pairs of binomials and apotomes whose terms
turn out to be commensurable and in the same ratio; each pair consists of corresponding members of one of a finite number of possible combinations of additive and subtractive expressible lengths.
It is possible, then, to trace the genesis of Book X in this way:
Theaetetus first extended the insights of measurement toward the
generation of the mean by using the three means involved in that
science as heuristic paradigms with which to interpret irrational
magnitudes. Just as an expressible length is a geometric mean
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between rational extremes, a medial length is a mean proportional
between expressible extremes; and just as arithmetic and harmonic
means are pairs of commensurable rational factors of the rectangle
contained by the extremes of their interval, binomials and apotomes
are pairs of irrational factors of the same rectangle. In his investigation of binomials and apotomes, Euclid discovered their classification, and thereby produced an ordering of irrationals in terms of
possible types of sum and difference—an arithmetic of expressible
lines. This in turn advanced the classificatory scope of Theaetetus’
propositions on the relations between binomials and apotomes,
when they were placed after Euclid’s work, at the end of Book X.
While Theaetetus could likely have proved that a rational area
applied to a binomial produces an apotome as breadth, and that the
terms of these irrational factors are commensurable and in the same
ratio, Euclid could now add, as he does in the enunciations of
Propositions 112 and 113, that such a binomial and an apotome
belong to the same order.
David Fowler approaches the book from a very different angle,
as part of his reconstruction of the ancient mathematics of ¡nvufaíresi$. He proposes an anthyphairetic theory of ratio, where
ratios between quantities are described by counting the number of
mutual subtractions which can occur between them: one counts the
number of times the lesser subtracts from the greater, then the number of times the remainder can be taken away from the lesser, then
the remainder of that transaction from the former remainder, and
so on; the list of numbers thus produced gives a unique description
of the particular ratio. He finds evidence for the historical existence
of this approach in several quarters, including a direct allusion in
Aristotle’s Topics to a definition of same ratio as same antanairesis
; and he sees the peculiar implications of this ratio theory as providing the most economical of many proposed rationales for the
total sequence and layout of Euclid’s Book II. The most surprising
fact he uncovers is a remarkable periodicity that arises in the anthyphairetic description of ratios of the form √m:√n—that is, ratios of
expressible lines.
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The achievement of Fowler’s work is to have rediscovered, and
in some measure to have resurrected in our day, the other branch of
measurement science, measurement prò$ ållhla. The periodic
repetition of the terms in the otherwise infinite mutual subtraction
of expressible quantities would have been the great discovery of this
science; as Fowler observes:
Those ratios that can be now completely understood
and described in finite terms by the arithmoi include the
ratios of the sides of commensurable squares, that is the
ratios of expressible lines √m:√n...
Note how fitly this parallels the development I have
described in the science of measurement toward the generation
of the mean: those lengths which can now be uniquely measured
in terms of the ¡rivmoí include these same expressible lines,
the sides of commensurable squares.
As far as the rationale for Euclid’s Book X is concerned,
however, Fowler’s reconstruction of the mathematics of anthyphairesis shows only why the relations between expressible lines
would have seemed a thing worth investigating. We gain no
insight into the specific form of the book as we have it, into its
method and structure in the classification of the irrationals;
these are better explained as an integral outgrowth of the new
science proposed in Plato’s Politicus, the science of measurement
toward the generation of the mean. This is not just because
Theaetetus is said to have classified the irrationals in terms of the
different means. Consider that the entire investigative strategy
of Book X, including the work I have ascribed to Euclid, is to
manipulate squares and rectangles, a manipulation in two
dimensions, in such a way as to distinguish and to enumerate the
forms of the associated lines. This approach was born with the
science of measurement toward the mean, on one fateful day. As
he lies dying off-stage, the story is told of how the young
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Theaetetus, Theodorus’ student, on the day of Socrates’ appearance in court, divided all numbers between the square and the
oblong, and distinguished two kinds of line as the sides of
squares equal to each kind of number. The “square side” of an
oblong number is the geometric mean between the sides of the
oblong. The names Theaetetus chose for these two lengths,
mêko$ and dúnami$, did not survive, for the implications of a
classification by sides of squares made the distinction itself obsolete: both kinds of length would now be called @htá, expressible. But the technique applied in his classification was to direct
the exploration of lines to its crowning achievement, in the enumerations of Euclid’s Book X.
We ought, however late, to acknowledge the dramatist who
saw the significance of such a day for history, saw it in a way that
must combine the personal and the universal, the historical and
the mathematical. Innovations in mathematics must have moved
that man in a way that made even innovation in religion seem a
distant charge, a memory of youthful import. We must come to
recognise the changes in this chronicler of the human argument,
as he took his bearings anew, and found new patterns, enumerative structures, emerging in a discourse that strains to keep
pace—paradigms of order no longer laid up in heaven, yet resonant, perhaps, with a piece of divinity. His myth of the globe’s
reversal (Politicus 268d-274e) encompasses a deteriorating
world, but also a return, through the numbering of its classes
and kinds, to the elegance of god’s tenure. Let him stand
absolved at last of the mystifications of his followers: Plato’s
own measures, his own mysteries, must finally furnish our count.
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2. Wilbur R. Knorr and Miles F. Burnyeat, “Methodology, Philology, and
Philosophy,” Isis, 1979, 70:565-70
3. Miles Burnyeat, “The Philosophical Sense of Theaetetus’ Mathematics,”
Isis, 1978, 69:489-513, on pg. 513, pg. 513
4. Knorr, Evolution, pg. 192
5. Ibid., pg. 192
6. Ibid., pg. 69 ff.
7. Ibid., pg. 96 (In full: “(a) The proofs are demonstrably valid. (b) The
treatment by special cases and the stopping at 17 are necessitated by the
methods of proof employed. (c) The proofs will be understood to apply to
an infinite number of cases. (d) No use may be made of the dichotomy of
square and oblong numbers in Theodorus’ studies, either in the demonstrations or in the choice of cases to be treated. (e) Theodorus’ proofs utilize the
special relations of the lines in the construction of the dynameis. The geometrical methods of construction are of the type characteristic of metrical
geometry as developed in Elements II and are closely associated with a certain early style of arithmetic theory. (f) But the arithmetic methods by which
Theaetetus could prove the two general theorems, on the incommensurability of lines associated with non-square and non-cubic integers, were not
available to Theodorus.”
8. Malcolm Brown, “Theaetetus: Knowledge as Continued Learning,”
Journal of the History of Philosophy, 1969, 7:359-79, on pgs. 3678
9. Knorr, Evolution, pg. 158
10. This proof is given by Knorr, Evolution, pg. 184
11. Ibid., pg. 159
12. see Euclid’s Elements X Def. 3
NOTES:
1. Wilbur R. Knorr, Evolution of the Euclidean Elements
(Dordrecht and Boston: D. Reidel Pub. Co., 1975), pgs. 65-9
13. see Plato’s Politicus, 278b-e
14. see Euclid, The Elements, 3 vols., Vol. 3, ed. Sir Thomas Heath
(Annapolis: St. John’s College Press, 1947), pg. 3
15. see Euclid II.14 and VI.13
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16. Brown, “Theaetetus,” pg. 371 ff.
17. Proclus, In Platonis Timaeum Commentaria, 3 vols., Vol. 2, ed.
Ernst Diehl (Leipzig: Teubner, 1903-6), pgs. 173-4
18. Brown, “Theaetetus,” pg. 371
19. see David H. Fowler, The Mathematics of Plato’s Academy
(Oxford: Clarendon Press, 1987), pg. 14 ff.
20. see Plato’s Timaeus 36a for this usage
21. The reading of B and T; editors usually read to )to
22. Brown, “Theaetetus,” pgs. 376-7
23. Ibid., pg. 377
24. see Theaetetus, 185c
25. Brown, “Theaetetus,” pg. 374
26. quoted in Brown, “Theaetetus,” pg. 373, note 38
27. Euclid, X.1
28. Brown, “Theaetetus,” pg. 379
29. Julia Annas, Aristotle’s Metaphysics Books M and N, Oxford:
Oxford University Press, 1976, pg. 195
30. Ibid.
31. Knorr, Evolution, pgs. 65-9
32. tr. W.Thomson and G.Junge, in Fowler, Mathematics, pg. 301
33. Fowler, Mathematics, pgs. 169-70
34. Ibid., pg. 192
35. tr. Thomson and Junge, in Fowler, Mathematics, pg. 301
36. Fowler, Mathematics, pg. 17 ff., and see Aristotle, Topics 158b
37. Ibid., pg. 192
38. see Ibid., pgs. 190-1
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63
Moral Reform in Measure for
Measure
Laurence Berns
(St. John’s College, Annapolis)
To what extent are the principles of classical political philosophy and the American polity reconcilable? The Declaration of
Independence did not mean, Lincoln tells us, that all men are equal
in all respects. The Declaration, however, presupposes that the difference between man and man is never as great as the difference
between man and beast, on the one hand, and man and God, on the
other. This “equality” by superiority to beasts and inferiority to the
divine sets limits both to human servitude and to human sovereignty.1 These principles issue in the rule of prudence that just government derives its authority from the consent of the governed. This
equality, as Locke put it, “in respect of Jurisdiction or Dominion
one over another” is not incompatible with the classical principle of
fundamental inequalities in capacities to govern. As a matter of fact
the institution of free elections (the Declaration’s “Right to
Representation”) introducing a principle of merit into the system is
predicated on the existence of such inequalities of ability, and the
capacities of electors roughly to discern them. (This does not, of
course, mean that the judgment of the electors is always correct, but
that it is sufficiently deliberate and well-informed to avoid disasters
that would unhinge the very frame of government.)
The classical position on democracy has been put, I believe,
with great clarity by Thomas Aquinas quoting St. Augustine:
If the people have a sense of moderation and responsibility and are most careful guardians of the common
weal, it is their right to enact a law allowing such a people to choose their own magistrates for the government
Delivered at the Convention of the American Political Science Association,
September 1993, The Washington Hilton Hotel, Washington, D.C.
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THE ST. JOHN’S REVIEW
of the commonwealth. But if, as time goes on, the same
people become so corrupt as to sell their votes and
entrust the government to scoundrels and criminals,
then the right of appointing their public officials is
rightly forfeit to such a people, and the choice devolves
to a few good men. [S.T., I-II, Q. 97, A. 1.]
I have no problem with this statement in principle, despite the
questionable practicality of its remedy for corruption. As Benjamin
Franklin put it, “If any form of government is capable of making a
nation happy, ours I think bids fair now for producing that effect.
But, after all, much depends on the people to be governed. We have
been guarding against an evil that old States are most liable to,
excess of power in the rulers; but our present danger seems to be
defect of obedience in the subjects. There is hope, however, from
the enlightened state of this age and country, we may guard effectually against that evil as well as the rest.” [Lett. to Ch. Carroll,
5/29/1789] What most threatens the required state of enlightenment today, it seems to me is not any paucity of economic resources
devoted to education, but rather the reigning generally accepted
opinions about what constitutes enlightenment. The AugustineThomas statement suggests, at the very least, that there is a natural
connection between the will to preserve free institutions and the
sense that those living in accordance with them are worthy of them.
How can a corrupt people be reformed? This, of course, is the
problem set for its protagonists by Shakespeare’s Measure for
Measure. Some distinctions between Duke Vincentio’s situation and
ours must be made. He has a single city and its environs to reform,
we have a huge and highly diversified nation. Our laws derive their
constitutional authority from the very people needing reform, his
do not. His polity is monarchical, ours is not. Our polity contains a
diversity of religious sects, his does not. Religious authority and
moral authority, if not united, form a well-functioning team in his
regime, in ours ... they do and they don’t. Obviously we are not
likely to find immediately applicable recipes from a study of
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Measure for Measure. We are obliged to put things in constitutional terms: “the abuse of the first Amendment”; the tendency of
lawyers and judges to ape intellectual fashions, sanctioning licentiousness with shallow-pate notions like freedom of expression, bargain-basement moral autonomy.2 We can, as teachers, try to change
the intellectual fashions. The only way I know how to do that is to
try to rise beyond the realm of intellectual fashion altogether, by
trying to understand the Duke’s problem as much as possible, as my
better, William Shakespeare, understood it.
Vienna, the seat of the Holy Roman Empire, is ruled by a
Duke, who “above all other strifes contended especially to know
himself,... a gentleman of all temperance.” Like those two political defectives, Prospero and Socrates, he has no taste whatsoever
for the theatrical pomposity endemic to political life. His apolitical temperament has caused him wrongly (“t’was my fault”) to
allow Vienna’s strict and biting laws to become toothless and contemptible; licentiousness thrives, and “Liberty plucks Justice by
the nose.”
His keen sense of justice prevents him from punishing in his
own name evil deeds bred by his own permissiveness. But purification there must be. He appoints a Lord Angelo (soon to prove a
Fallen Angelo), a man of “stricture and firm abstinence”, who
“scarce confesses that his blood flows” to stand in for himself, that
is (unlike American executives) to “enforce or qualify the laws.” But
first something should be said about why someone like puritanical
Angelo was needed.
The Vienna presented at first in the play seems to consist primarily of nunneries, monasteries and whorehouses, with almost
nothing in between: the only family man presented is the absurd
comic figure Elbow; austere celibacy, on the one hand, and saucy
profligacy, on the other, again almost nothing in between. As sexuality is debased, celibacy, for some, gains in attractiveness.
Something seems to be radically wrong with the way most Viennese
think, feel and behave in regard to their sexuality. Immediately fol-
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lowing Angelo’s appointment the Duke pretends that affairs of state
require his hasty removal to foreign parts; Angelo is on his own.
Political scientists (Bloom and Jaffa) quite properly refer to
Machiavelli’s The Prince, chapter VII, as the locus classicus for the
Duke’s mode of procedure with Angelo.3 Cesare Borgia on taking
over Romagna found that because it had been very badly governed
it was full of robberies, quarrels and insolence. To reduce it to peace
and obedience he appointed a very cruel man, Remirro de Orco, as
his deputy with full powers. Remirro soon reduced it to peace and
unity. The reform being accomplished, in order to deflect the hatred
it had generated from himself Cesare had the cruel Remirro placed
one morning in the piazza at Cesena in two pieces, a piece of wood
and a bloody knife beside him. The ferocity of the spectacle left the
people both satisfied and stupified. Bacon speaks of this way of proceeding both in his Wisdom of the Ancients [III], and his Essays
[XIII], but both seem to have been published after this play was first
presented. One is tempted to go along with our scientific fashions
and play at being “more hard-nosed than Thou,” but the differences
between Shakespeare and Machiavelli at least deserve listing. The
Duke does not kill Angelo, though he had full warrant to do so;
unlike Cesare with Remirro, the Duke is not interested merely in
using Angelo, but also as with everyone else, including himself,
making him better, reforming him; above all, since he is not omniscient, he is interested in understanding Angelo: “Hence shall we see,
/If power change purpose, what our seemers be.” It is not simply
because he courts popularity, that he doesn’t institute the reform
himself, it is rather because he is not the right man for the job, and
it would not be, or at least not seem, just for him to do so. There is
another work of Machiavelli’s that bears close comparison with
Measure for Measure, that is Mandragola4; the Duke seems to combine characteristics of both Ligurio and Frate Timoteo, but here
again the differences should prove instructive.
The Duke does not leave Vienna, he goes underground in the
guise of a “holy friar” both to observe and invisibly to correct the
course of his reform. Angelo evidently goes to work immediately:
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the houses of prostitution are put down, and a young gentleman
named Claudio is sentenced to death for fornication; for the
woman he is engaged to marry (the marriage delayed by dowry
problems), Juliet, is big with child. Angelo rejects the urgings of his
second in command and his Provost that here the punishment is
way out of proportion to the crime. Claudio has a high-spirited sister, Isabella, who has entered the austere order of St. Clare -“When
you have vow’d, you must not speak with men /But in the presence
of the prioress; /Then, if you speak, you must not show your face;
/Or if you show your face, you must not speak”- as a novice. She
wishes for an even “more strict restraint.” We are, I suppose, to
imagine her quite beautiful; her moral beauty at least engages the
affections of the play’s two main protagonists. She is urged by the
dissolute gentleman Lucio to plead with Angelo for her brother’s
life. Despite her choosing to renounce family life, her’s is the only
powerful display of family feeling in the play. While hearing her
plea the transforming event of the play takes place, Angelo finds
himself possessed by an overwhelming passion, which, both to himself and to her, he calls love for Isabella. He, on second interview,
proposes that she yield her body to him for one night in exchange
for her brother’s life.
The critique of Angelo would seem to be a critique of
Puritanism in general. Licentious Lucio thinks Angelo “a man
whose blood /Is very snow-broth; one who never feels /The wanton
stings and motions of the sense...” This is surely wrong. The Duke
had made a similar, but more penetrating, observation: “Lord
Angelo is precise; /Stands at a guard with Envy; scarce confesses
/That his blood flows...” If he must guard against envy, he feels the
desires whose indulgences he must not be envious of. With old
Escalus, before he has fallen, Angelo admits that he too has had the
desires that lead to the actions he is punishing with death, acting
upon them makes the difference. He is too good, at least too strict
and too proud to consort with the dissolute; he proves to be not
good enough to be celibate. He wants to be associated with the
highly virtuous, is attracted by Isabella’s purity; he wants to pre-
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serve the image of his gravity; and he wants the joys of what he calls
love: your brother shall not die “Isabel, if you give me love.”
He seems to be altogether confused about the difference
between “yielding up thy body” and “give me love.”5 It was
Isabella’s moving persuasiveness that led him to give more attention
to the erotic side of his soul than he could handle: “Go to your
bosom, /Knock there, and ask your heart what it doth know /That’s
like my brother’s fault. If it confess /A natural guiltiness, such as
his....” He replies to himself: “She speaks, and ’tis such sense /That
my sense breeds with it.” [2.2.137 ff.] He did warn the Duke: “Let
there be some more test made of my metal, /Before so noble and so
great a figure /Be stamp’d upon it.” The Duke knows that: “He doth
with holy abstinence subdue /That in himself which he spurs on his
power /To qualify in others.” [4.2.79] The immoderate Puritan
allows the bitterness from his own frustrated desires with perhaps a
touch of envy to spur him on to punish those who will not abstain.
The fear of falling into temptation increases the severity. The intensity of purifying zeal seems to be directly proportional to the
difficulty one has in keeping one’s own illicit desires under control.
The judgment is warped in the direction of severity by what one
feels is required to frighten oneself into abstinence. Isabella’s loveliness and what he sees when he follows her advice and looks into
his own soul push him over the edge.
And now I give my sensual race the rein:
Fit thy consent to my sharp appetite;
Lay by all nicety and prolixious blushes
That banish what they sue for. [2.4.159 ff.]
On reflection it might not seem so strange that modesty should
provoke desire.
Any competent political scientist can figure out why Angelo
never intends to fulfill his side of the bargain. Isabella can find no
“charity in sin.” “More than our brother is our chastity.” The Duke
disguised as Friar Lodowick prepares Claudio for death with a ser-
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mon crammed with Stoic commonplaces on the worthlessness of
life. From here on the Duke uses the holy privileges associated with
his disguise to inform himself of everyone else’s secrets. He overhears Isabella’s account to Claudio of Angelo’s proposal. In another remarkable scene Claudio begins by sharing Isabella’s righteous,
honorable and Christian indignation at the impossibility of Angelo’s
plan. But he has been brought to face the fear of death in a very feeling way.
Death is a fearful thing.
...to die, and go we know not where;
To lie in cold obstruction, and to rot;
This sensible warm motion to become
A kneaded clod; and the delighted spirit
To bath in fiery floods, or to reside
In thrilling region of thick-ribbed ice;
To be imprison’d in the viewless winds
And blown with restless violence round about
The pendent world: or to be worse than worst
Of those that lawless and incertain thought
Imagine howling, -’tis too horrible.
The weariest and most loathed worldly life
That age, ache, penury and imprisonment
Can lay on nature, is a paradise
To what we fear of death. [3.1.115 ff.]
Claudio’s speech is a beautiful illustration of that “very illusion
of the imagination” beautifully described by Adam Smith: the way
a man or woman’s sympathetic imagination attributes to the dead
what he or she would feel being alive, if he or she were housed in
the dead person’s body. And thus “the foresight of our own dissolution is so terrible to us, and ... the idea of those circumstances,
which undoubtedly can give us no pain when we are dead, makes
us miserable while we are alive.” The Duke certainly does not
explain anything like this to Claudio or to anyone else in this play,
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but I don’t think he would disagree with the way Smith closes this
chapter. “And from thence arises one of the most important principles in human nature, the dread of death, the great poison to the
happiness, but the great restraint upon the injustice of mankind,
which, while it afflicts and mortifies the individual, guards and protects the society.” [The Theory of Moral Sentiments, I.i.1.13] Bloom
quite properly refers to Lucretius [III. 417 ff.] in his discussion of
this passage, but Smith, it seems to me, is more balanced, even more
“classical”.
Claudio goes on to plead:
Sweet sister, let me live.
What sin you do to save a brother’s life,
Nature dispenses with the deed so far
That it becomes a virtue. [3.1.132 ff.]
This is not the first time Nature has been invoked to oppose
chastity law. The licentious but eloquent Lucio puts it in a way that
comes close to generally accepted opinion among our intellectuals.
Your brother and his lover have embrac’d;
As those that feed grow full, as blossoming time
That from the seedness the bare fallow brings
To teeming foison, even so her plenteous womb
Expresseth his full tilth and husbandry. [1.4.40 ff.]
The licentious have their say in this play. But Shakespeare has quite
naturally, but not altogether explicitly, built Nature’s answer to
promiscuity into their very speech: it is full of the imagery and fear
of venereal disease. The Duke seems to have come to the realization
that Nature in human society requires law for its fulfillment.6
Isabella is moved by Claudio’s speech, but in exactly the opposite direction. “O, you beast. . . faithless coward...dishonest wretch,”
she replies.
Wilt thou be made a man out of my vice?
Is’t not a kind of incest, to take life [i.e., to be born
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again]
From thine own sister’s shame?
...Take my defiance,
Die, perish! [3.1.135 ff.]
This is not the first time sexual imagery enters Isabella’s speech
in moments of great passion. Answering Angelo she is primarily
thinking of stripping herself for whipping:
Th’impression of keen whips I’d wear as rubies,
And strip myself to death as to a bed
That longing have been sick for, ere I’d yield
My body up to shame. [2.4.101 ff.]
The Duke’s first task is to avert the great impending injustice
brought on by his scheme, but he does it in a way that also seems to
be perfectly calculated to bring Isabella to face her sexuality, and
human sexuality in general, more temperately. His reform will turn
out to be a comprehensive reform; all the representative characters,
Pompey, Lucio, Claudio, Angelo and Isabella are in different ways
reformed. The Duke uses the religious authority he has assumed to
engage Isabella in a plot that will right all wrongs. Angelo, it turns
out, had been engaged to marry a lady, Mariana. When her brother carrying her dowry was wrecked at sea, Angelo “pretending in
her disoveries of dishonour” called off the marriage. This wronged
lady, the “forenamed maid” has unreasonably been driven by his
unkindnesses to a more violent and unruly love for Angelo. She still
regards him as her “husband.” Isabella is to agree to Angelo’s terms,
arrange for a short meeting in a very dark place; Mariana is to be
substituted for Isabella. If the encounter is acknowledged afterwards, it may compel him “to her recompense.” By this, the Duke
argues, “is your brother saved, your honour untainted, the poor
Mariana advantaged, and the corrupt deputy scaled.” The Duke as
friar will frame and make Mariana fit for the attempt. The fact that
this seems to pose no special difficulty suggests that Mariana may
indeed be as right as one can be to mate with Angelo. But it is too
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easy these days to berate Puritanism, Mariana may just see some
nobility in Angelo’s austerity, a nobility that manifests itself to us as
well even in his guilty self-condemnations. It can hardly escape
Isabella’s later reflection that what takes place between Angelo and
Mariana is in many respects parallel to what took place between
Claudio and Juliet. But this conjunction is sanctioned by a holy
man, who declares that “the doubleness of the benefit defends the
deceit from reproof.” Isabella is happy to go along. Even the pleasure of revenge on Angelo seems to be sanctioned by this holy man.
Isabella’s imagination is invited with no impiety to receive scenes of
her enemy coupled with his affianced lover, thinking he is violating
herself. If her soul is puritanical, it will have to become Puritanism
with a certain sense of humor.7
The Duke’s plan for deceiving Angelo succeeds, but Angelo
sends no reprieve for Claudio. On the contrary, he advances the
time for his beheading. He has no interest in preserving the life of
a man privy to his crime, and who, if he has the least grain of honor,
would be bound to think of little else than revenge. The Duke, again
using his assumed religious authority, attempts to get the Provost of
the prison to substitute the head of a convicted murderer,
Barnardine, for Claudio’s, to fool the wicked Angelo again. The
Duke had invoked “the vow of my order.” The “gentle Provost” is
the only one who refuses to bow to religious authority, “Pardon me,
good father, it is against my oath.” When the Duke, without fully
revealing himself, is forced to prove he is acting not only by religious authority but by the authority of the Duke himself, the
Provost goes along. He who refused to subordinate political authority to religious authority for his “care and secrecy” will be rewarded by the Duke with “worthier place.” But Barnardine has been
drinking and is not prepared for death today. He simply will not
consent to die today. This is an amazing prison. They all agree that
to take him in this condition is damnable. Luckily, the captive
pirate, Ragozine, who resembles Claudio, has died that morning:
the perfect head to substitute for Claudio’s. Besides provision of
some fine comedy, the prison scenes are essential for understanding
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the Duke’s basic strategy of reform. Pompey the procurer now put
out of work becomes the prison’s executioner’s assistant. The servant of false love, venereal disease and the unlawful begetting of life
quite easily becomes the true servant of its lawful taking. Pompey’s
coarseness is re-formed to serve the rule of law. The taking and the
begetting of life have been connected before. Angelo declares:
Ha? Fie, these filthy vices! It were as good
To pardon him that hath from nature stolen
A man already made [a murderer], as to remit
Their saucy sweetness that do coin heaven’s image
In stamps that are forbid. ’Tis all as easy
Falsely to take away a life true made,
As to put mettle [metal] in restrained means
To make a false one. [2.4.42 ff.]
As Jaffa put it, “Fornication, as a kind of false coinage of citizens, becomes more than a private action.”8 The regulation of
coinage, society’s circulating medium, is usually a rather unquestioned prerogative of sovereignty. The penalties for counterfeiting
have never been light. To “coin heaven’s image” joins biblical sanctity of begetting to the need for sovereign political regulation of that
private behavior which is the source of life for society as a whole.9
(The coining image occurs at least three more times in the play, in
speeches by the Duke [1.1.35-36], Isabella [2.4.128-29] and Angelo
[1.1.48-50].)
Threats of death color the whole atmosphere of the play. Fear
of death in potential malefactors seems to be indispensable for the
restoration and maintenance of law-abidingness. But absolutely no
one ever gets killed in this prison. It is the genius of this Duke to be
able to employ the fear without ever having to follow through with
the act. The ploy would never work, if it became generally known.
The great final act and scene of the play pulls all strands together, the return of the Duke and resumption of his authority in a
grand public ceremony, where the Duke “like power divine” reveals
all hidden iniquities and resolves all difficulties with perfect justice.
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This justice, both legal and natural justice, is primarily justice in
marriage. For the chaste sexuality of marriage, the raising and nourishing of families under the law, is the solution for the sexual corruption of Vienna.10 The dissolute gentleman, Lucio, is forced to
marry the prostitute who is the mother of his child. Angelo, who
sought Isabella, who was too good for him, is ordered to marry and
love the less scrupulous Mariana. But the Duke knows that the institution of marriage, upon which the health of his polity depends,
will not be on a firm foundation unless it shines forth at the paradigmatic center of society. He too must marry, and marry well. The
high-minded Duke asks the high-minded Isabella to be his wife.
How that works out, we never learn. As part of the apocalypse the
Duke staged for his triumphal return, Isabella was made to believe
that her brother had indeed been executed. It may be that the Duke
wanted her to weigh the events leading to that result more carefully, or merely, as he said, “to keep her ignorant of her good,/To make
her heavenly comforts of despair/When it is least expected.” He
might be made to pay for those hours of despair. These reservations
aside, it seems to be a near perfect marriage. If it should be that the
Duke comes short of perfection by contemplative leniency and
Isabella by spirited severity, it would be by the blending of their
virtues and the mitigating of their defects in their shared lives or in
their offspring that Vienna could hope to receive its perfect Lord.
NOTES:
1. Cf. H.V. Jaffa, The Conditions of Freedom: Essays in Political
Philosophy (Baltimore: The Johns Hopkins University Press, 1975), pp.
152-53; G. Anastaplo, St. Louis University Law Journal (Spring, 1965), 390.
2. G. Anastaplo, “Censorship”, The Encyclopedia Britannica, 15th
Edition, 1986 printing, Volume 15, pp. 634-641; The Amendments to
the Constitution: a Commentary, (Baltimore: Johns Hopkins
University Press, 1995), pp. 52-56: R.A. Licht, “Respect is not a Right”,
Crisis, Vol. 11, No. 7, July-August, 1993, pp. 41-47; “Communal
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Democracy, Modernity, and the Jewish Political Tradition”,
Jewish
Political Studies Review, 5:1-2 (Spring, 1993).
3. A. Bloom, Love and Friendship (New York: Simon and Schuster,
1993), p. 330; H.V Jaffa, “Chastity as a Political Principle: Measure for
.
Measure”, Shakespeare as Political Thinker, eds. J. Alvis and T.G.
West (Wilmington, DE: ISI Books, 2000), pp. 211-12.
4. The most reliable and literal translation known to me is by M.
Flaumenhaft (Prospect Heights, Ill.: Waveland Press, 1981).
5. Cf. W. Shakespeare, Sonnets, Nos. 129 and 116.
6. L. Berns, “Gratitude, Nature and Piety in King Lear”, Interpretation,
Vol. 3/1 (Autumn, 1972), Sections V and IX; “Rational Animal-Political
Animal: Nature and Convention in Human Speech and Politics”, Essays in
Honor of Jacob Klein (Annapolis: St. John’s College Press, 1976), pp.
29-35, esp. section III; [uncorrected version in The Review of Politics,
Vol. 40, No. 2 (April, 1978), pp. 231-54.]
7. L. Berns, “Transcendence and Equivocation: Some Political, Theological
and Philosophic Themes in Shakespeare”,
Shakespeare as Political
Thinker, cited n. 3, pp. 402-4.
8. “Chastity as a Political Principle: Measure for Measure”, citation
n. 3, p. 221.
9. The locus classicus for the relation between private and public, polity and family is Aeschylus’s trilogy Oresteia. The trilogy begins with a
world where family feeling, the spirit of revenge and cycles of blood feuding dominate and characterize political and social life. Agamemnon, the triumphant leader of the Trojan expedition, is killed on his return to Argos
from Troy by his wife Clytaemestra for the sake of “my child’s Justice”, that
is , to avenge the death of their daughter sacrificed to propitiate the gods
holding up the expedition to Troy. The ruling deities are the Old
Goddesses, the Daughters of Night, the “ingrown, vengeful Furies.” In the
second play, Orestes, Agamemnon’s and Clytaemestra’s son, following the
charge of Apollo’s oracle, avenges his father’s death by killing his mother.
The Furies, “the bloodhounds of my mother’s hate,” pursue him. The third
play, The Eumenides, the well-meaning ones, celebrates the founding of the
Court of the Areiopagus at Athens. Orestes seeks sanctuary at Delphi. The
Pythian oracle is overwhelmed by the pursuing Furies. Apollo himself inter-
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venes and stills the Furies long enough for Hermes to guide Orestes to
Athens for a final resolution of his case. At Athens Pallas Athene takes
charge. She who was not born of woman (born from the head of Zeus), a
most man-like female, almost a mean between male and female, turns the
trial over to an open court of Athenian citizens. The Furies argue against,
Apollo argues for Orestes; gods as advocates, before human beings as
judges and jury. The jury of twelve human beings is given the authority to
decide. The sovereignty of hitherto untamable family feeling is brought
under the supervening authority of the polis, the political community.
Although they have the authority, the human beings by themselves are incapable of deciding between conflicting rights of mother and father. The jury
splits evenly. The deciding vote is given to the goddess Athene. Divine help
is required for settling such questions. She decides for Orestes. It seems that
reasonable procedures for settling and dispensing with problems may
sometimes be more important than assurance that the solutions are correct.
These questions are no longer to be dealt with violently behind closed
doors but deliberately before public and open spectacles of law court,
assembly and theater. The Furies are unwilling to accept these dispensations of the younger gods. By a combination of threats and persuasion
Athene cajoles the Furies to integrate their authority over family feeling and
the household into the service of the greater good of the political community. They shall “win first fruits in offerings for children and the marriage
rite.” The Furies finally agree and are transformed into Eumenides. The
feelings they preside over which are capable of tearing the political community apart cannot be extinguished: they are to be redirected against the
despotically minded consumed by “a terrible love of high renown” and
external enemies; they will bolster the mutual love of fellow citizens. “This
is a cure for much that is wrong among mortals.” Cf. M. Flaumenhaft,
“Seeing Justice Done: Aeschylus’ Oresteia”, Interpretation, Vol. 17/1,
(Fall, 1989), pp. 69-109, reprinted in The Civic Spectacle: Essays on
Drama and Community, (Lanham: Rowman and Littlefield, 1994),
Chapter 1; and David K. Nichols, “Aeschylus’ Oresteia and the Origins of
Political Life”, Interpretation, Vol. 9/1, (August, 1980), pp. 83-91.
10. L. Berns, “Gratitude, Nature and Piety in King Lear”, citation n. 6, p.
50: “... love and passion ... need to be controlled by law and authority. Being
conceived outside the ’order of law’, Edmund was banished from the family circle. He is not altogether ’unnaturally’ devoid of family feeling.”
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79
A Review of Eva Brann’s
The Ways of Naysaying
Chaninah Maschler
“The first impetus” for this study,1 Eva Brann tells us in her
Preface, was the desire to deepen her understanding of the two
“capacities of our inwardness” that had been the themes of two of
her previous books, The World of the Imagination and What, Then,
is Time? “As the imagination ...makes present what is not before us
by reason of nonexistence or withdrawal, so memory ...holds what
is not with us by reason of having gone by....Therefore... to understand something of imagination, memory, and time, we must mount
an inquiry into what it means to say that something is not what it
claims to be or is not there or is nonexistent or is affected by
Nonbeing. And that is what I am after in Ways of Naysaying” (pp.
xiif, my italics).2
Addressing, I presume, readers of the first two volumes of her
trilogy, Brann explains that and why there will be less reliance on
introspection and more reliance on logic and language in the present volume: ”We could, it is thinkable, be aware of our internal
images...without having language for them....But whether we
could know about negation—that we are capable of it and how—
without speech is doubtful to me. Hence within my scheme, no,
not, non- are deeper than imagination and time, in the sense that
the former underlie the latter and are revealed in their analysis” (p.
xiii). Earlier in this paragraph, and in more detail later, when summarizing Freud’s essay “On denial” in her Chapter One, she allows
that there is a pre-linguistic “nay-saying of instinct and gesture.”
Since this paragraph is rather condensed, and much hangs by it,
let me try to say in my own words what I believe it to hold: Doing
no, for instance, spitting out or pushing away or averting the gaze,
occurs (ontogenetically and, according to Freud, also phylogenetically) before speech. And a sort of prereflective reacting to heard
Eva Brann’s The Ways of Naysaying: No, Not, Nothing, and Nonbeing. Lanham, Md: Rowman
and Littlefield, 2001. Chaninah Maschler is tutor emeritus at St. John’s College.
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“no,” or even the uttering of the syllable “no” (as mere substitute
for the gesture of rejection) may well be a phase of children’s intellectual history. Further, having images and reacting to them, or to
having had them, can occur without the one who reacts being aware
that what he or she is reacting to is an image. But to peg an image
as an image, which means, to take it as a likeness of its original, that
requires, according to Brann, the thought “This is and is not that.”
Therefore there can be no sizing up of a mirror image, memory
image, dream image, perceptual presentation as “merely” an image
until after negation has entered upon the mental scene. Now knowing about negativity, which is different from prereflectively reacting in a rejecting or separating manner, that could not occur sans
speech. While images are, therefore, existentially “prior to” (earlier
than) speech, in involving recognized negativity they show themselves conceptually “posterior to” speech.3
Brann seems to be employing some version of the
Aristotelian contrast between “first to us” and “first in nature.”
This is how I construe her claims that, while imagining and recollecting are more manifest, negativity lies “deeper” than do
these “capacities of our inwardness”; and that, furthermore,
whatever is condition for the possibility of negativity lies more
deeply still. Her book as a whole will argue that the Platonic discovery that Being “holds” Nonbeing may well be the ultimate
answer to the question “was die Welt im Innersten zusammen
halt” (“what it is that most intimately holds the world together,”
Goethe, Faust Pt. 1, li 383).
The Introduction of Brann’s book is given over to etymology. It draws attention to the fact that in English, German,
French, Latin, and Greek (the languages in which the Western
philosophic tradition is expressed), most of the basic words for
naysaying—no, not,non-,nothing, and negativity—start with a
nasal sound. Jesperson, and before him Darwin, remarked on
this fact and entertained the thought that, conceivably, our signs
for negation are transitional between naturally expressive gesture and conventionally learned word. The n-word would then
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81
have originated as a substitute for the snort of aversion or refusal
and in the course of linguistic and cultural history have proliferated into a multiplicity, the last but not least of which would be
the abstract general negation word “not” we use for contradiction.
In Chapter One, Brann, like many of us today, seems to
share Darwin’s impulse to give direction to speculation about
human archai by studying child development. Accordingly, the
title of her book’s first official chapter—“Chapter One,
Aboriginal Naysaying: Willfull No”—refers initially to “the primordial ‘no’ to everything” of the toddler (p. 9), but eventually
to other respondings (Goethe’s Mephistopheles serving, seriocomically, as paradigm) that reject, disobey, or spit out what is
given. Thus some of the discussion of nihilism in Chapter Six
looks as though it were continuous with Chapter One’s analysis
of childish “no.” The toddler rejects the breast, the command or
prohibition, the saying “so it is” of the grown-up; the nihilist
turns down shared traditions, institutions, and even intersubjectively acknowledged matter of fact.
I loved the affectionate and knowing description of “the terrible twos” in this chapter. I share Brann’s admiration for Freud’s
astonishingly potent brief essay on negation, which she summarizes,
pretty much in Freud’s own words, on pp. 10-12. But it looks to me
as though Freud’s quasi-Nietzschean “genealogy” of the intellectual function of judgment out of the interplay of biologically “primary impulses” has, when Brann is through with it, become tinted with
Augustinian surmises of original sin. In evidence I cite the fact that
it is rather late in the chapter (footnote 28, p. 22) that the “healthy
naysaying” of resistance to temptation and of rebellion against
tyranny are mentioned; also, that the emphasis on self-awareness’
emerging from deeds and words of “arbitrary willfullness” (cf. p.
18) does not seem to be balanced by reflections on the child’s need
to exercise, so as to perfect, skill at matching expectation with outcome, and vice versa. What I have in mind is well-explained by
Jerome Kagan. In brief, Kagan holds that much of what we
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observe in the not quite two-year-old is made intelligible if we
view it as due to the emergence of “three related competencies”:
ability to notice that some happening or action is at odds with
what is right and regular; absorption by the idea of standards as
standards, both those set by others and those set by oneself; awareness of one’s own and the world’s ability or inability to meet standards. I am confident that Brann would agree that these competencies involve the child’s increasingly better memory: Isn’t much
of the toddler’s “testing” of the world, commandeering of adults,
and “first Adam”-like rage at the world’s or the grown-ups’ not
coming through connected with practicing the ability to match
outcome with forecast and plan, remembrance with presentation?
It is my impression that Brann writes more nearly in this spirit in
What, Then, is Time? (See p. 165).
The chief questions asked and answered in Chapter Two, where
the not of logic is taken up, are as follows: 1.What is negation?
2.Where is the sentence negated? 3.Is the positive prior to the negative? 4. How is negation related to falsity?7
Following Aristotle, Brann assigns negating (the act) and negation (the act’s sentential consequence) to the genus of opposition.
An admirable overview of types of opposition, as described and
classified by Aristotle, is provided, while opposition in general is
recognized to be indefinable.8 Plainly, the idea of not is clarified
when, through insertion into its genus, it is made evident that not
must be discriminated from fellow-contenders for naysaying primacy, for example, speaking linguistically, the particles non- or un- or
a- and, speaking semantically (?), the polar relations of contrariety
and privation. The not of contradiction is declared the winner, on
Aristotle’s authority (p. 27). But, Brann hastens to add, contradiction, which is “sheer, unintermediated opposition” (studied as such
under the heading of question 4), belongs to thinking and speaking,
not to things.
We seem, perhaps contrary to expectation, to have ferreted out
something like an answer to the question of what negation consists
in. A summing up of the interim upshot of the inquiry into the
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83
nature of not is provided on p. 29f. and concludes with the sentence: “Negation arises from the human desire and ability to make
distinctions; it is (most likely) grounded in the oppositions and
polarities that belong to beings....”
Where in the declarative negative sentence is the particle that
accomplishes negation located? is the second question. What
motivates the question? One could imagine a linguist who is trying to learn an exotic language asking it. He would, I suppose,
have tried to obtain a corpus of utterances sufficiently rich to
hold instances of all the elementary affirmative sentence patterns
of that language (supposing this possible); next, he’d have consulted with a native informant as to how one would, in his language, “say the opposite(s)” of these. Assume the native informant is a speaker of English and the linguist’s native tongue is some
non-Indo-European language, say Chinese or Hebrew. If I understand Brann correctly, she believes that the Chinese linguist
would somehow find out that all the elementary affirmative sentence patterns of spoken English are reducible to the triadic pattern S is P How could he have found this out? The best I can
.
come up with is that, in learning English, he relies on the same
logical truth on which he relied when he acquired his mother
tongue—that whatever is said is interpretable as making some
comment on a declared or otherwise manifest topic: The topic is
named by one part of the sentence; the sentence attaches the
comment to the name; and in so doing comments on, that is,
predicates the sentence’s predicate of, the thing or things in the
world that is or are the sentence’s topic.12 The question now
becomes how and why this insight into the logically dyadic T-C
structure of simple 13 affirmative sentences issues in the triadic S
is P structure. The reason for my selecting a Chinese-speaking linguist was, of course, that (as Brann reports in the long and important footnote 22 on p. 64), Chinese sentences do not require a
copula to accomplish the job of commenting. Hebrew doesn’t
either: Joseph holech “merely juxtaposes” what is, strictly speaking,
a participial (thus adjectival) form of the verb to the proper name
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“Joseph” to say what in English would be said by the sentence
Joseph is walking. But nothing stands in the way of a Hebrewspeaking linguist’s learning that in English sentences an “is” must be
inserted between “Joseph” and “walking” for the predicating job to
be accomplished.
What all this fussing is about is the issue how logical and grammatical distinctions differ and mesh. Brann’s fourfold answer to the
question where the negation particle is located in a sentence proceeds, not on the linguist’s basis of studying a corpus of English negative sentences, nor on the logician’s basis of reflecting on the negating jobs that would have to be accomplishable if the tasks of describing and reasoning rightly are to be carried out. Rather, she works
with the S is P pattern of “traditional” logic and negates, first the
“is” or copula, next the “P” or predicate, third the sentence “S is P”
as a whole, and finally, although not whole-heartedly, even the “S”
or subject. Having done this, she points out the jobs done by the
patterns which thus emerge.14
Why does she proceed in this manner? She is, usually, not at all
friendly to mere algebraic patterning. More important, she knows
that Frege, whose “deep critique of the classical view [of negation]”
was taken up appreciatively in the concluding section of the treatment of question 1 (pp. 30-32), endorses something like what I
tried to say through my fable of the Chinese or Hebrew-speaking
linguist, that what chiefly matters is the irreducible logical contrast
between naming and predicating and their complementarity,15
whereas the presence or absence of some form of the verb “to be”
is a linguistic accident.
On first reading I thought that her manner of proceeding in
Chapter Two is due to her not being as convinced as was Frege of
the need for a principled distinction between logic, as a normative
science, and psychology and linguistics as empirical sciences which
acknowledge logical norms in practice (as we all do when we think),
but which do not study them.16 Brann reports and up to a point
explains that Frege distinguishes T-C structures qua what he calls
Gedanken (“thoughts”) from “assertions” (what Kant called “judg-
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ments”). And she appreciates that “thoughts,” including negations,
are for Frege objective and atemporal whereas he regards “assertions” as acts of a speaker or thinker who at some time or other
asserts an assertable or its contradictry. She even quotes a sentence
of Frege’s which brings this contrast to bear on the issue of negation.17 But she refuses to let go of inquiry into what it is in human
beings and the world that leads to nay-saying.18
On second reading I found an outright answer to my question,
why Brann distances herself not only from Frege but also from Plato
and the Aristotle of On Interpretation, in footnote 22 (p. 64). She
writes: “I accept...[‘S is P’] as the fundamental sentence form
because people whose thought is congenial to me19 have built on it
structures that are of great interest, and because I have corroborated by introspection that it is my most basic declarative mode of
internal speech, closer to thinking than the bipartite sentence consisting of a subject and a predicative verb.”20
Postponing till her penultimate chapter, Chapter Six, inquiry
into what she calls the “greatest question,” namely, whether
Something or Nothing is ultimate, the issue in section 3 of the present chapter is whether “in human speaking denial is always derivative and in human speech negation is always secondary” (p. 36).
Boethius, ancient authors in the Aristotelian tradition, and modern
cognitive science are reported to endorse the opinion that the affirmative is prior to the negative, as at first blush it would seem to be,
since any negating particle is an “addendum.” Bosanquet and
Bradley are described as having answered the question in a more
nuanced way: “Negation is not as such a denial of affirmative judgment; it does not presuppose a particular affirmative judgment to be
denied. But it does presuppose some general affirmation, namely,
that of a world having a positive content judged to be real....The
positive judgment itself cannot take place before the distinction
between a mere idea and a fact of reality is recognized. ‘And with
this distinction the idea of negation is given’ ” (p. 40).21 Still, the
over-all conclusion of the inquiry in section 3 is that negation is
“secondary.”
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More minute examination of Bosanquet’s and Bradley’s
remarks on negation might have yielded a scheme for differentiating the diverse senses of the prior/posterior relation; causal priority
might have become differentiated from conceptual priority; priority in dignity or rank from temporal priority.22 But as it stands, section 3 seems to favor a temporal sense of prior/posterior. This bothers me because I am inclined to believe that logicians qua logicians
have no business asking about temporal priority and that conceptually the positive comes or rather is on the scene along with the negative. Thus neither is prior to either.23 As an illustration, consider
the following: At the beginning of the Prior Analytics, Aristotle
defines argument or deduction (syllogismos) as follows: “A deduction is a discourse in which, certain things being stated, something
other than what is stated follows of necessity from their being so.”
I believe that anyone who grasps the type of necessity here spoken
of grasps along with it the impossibility of the contradictory. Upon
reflection I recognize that I base this apparently psychological
observation on the conceptual (i.e. logical) truth that must-bes are
the contradictories of cannot bes and cannot be apart from them.
Indeed, it dawns on me that my seemingly psychological claim may
be nothing but the conceptual truth itelf in another form of words.
The treatment of question 4 (how negation is related to falsity)
shows a respect for Wittgenstein that was, I believe, absent from
Brann’s previous writings.24 His Tractatus is praised both for asking
and for answering the following questions: (1)“How do Truth and
Falsity come to be obverses” (i.e. opposites)? (2)“How is negation
related to them and to truth-values?” (3)“Why are propositions
bipolar?” (4)Can we justify the logic textbooks’ assumption “that
each proposition has only one negative?”(p. 47).25 The two paragraphs immediately preceding the enunciation of these questions
seem to report the answers that Brann found in Wittgenstein. I
quote them in full:
“It all begins with a discrimination exercised by us over a logical space wherein things are seated within their place in their proper relation configurations, a discrimination of the otherness of what
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is false. So prototruth26 is in the world of fact. Now comes a proposition. In its negative and positive sense it is like a solid body that
restricts all movement into a certain place; in its positive sense it has
an empty place where the object can fit in (Tractatus 4.463). These
[comparisons ]are pictures of the ... inherent bipolarity of every
proposition. It shows negation from the beginning related to the
negated proposition, for it is that hole which the negating proposition is blocking (Tractatus 4.0641). So to understand a proposition
is to see the logical space (Tractatus 3.4) and to discriminate what
the facts would have to be like to make a proposition...[i.e. a logical picture] true or false.”
“Truth, then, or falsity, is the consonance or correlation of a
propositional picture with reality (Tractatus 2.21), where reality
(Wirklichkeit) is the existence or non-existence of facts (Tractatus 2;
2.06)—a non-existent fact being one that is pushed out of the world
picture by the fact that exists. In this correspondence is truth in the
primary sense, and it comes in the duality true-false because of the
way logical space divides and we discriminate the facts. In the sense
of propositions lies the polarity positive-negative, the latter of
which is expressed in the sign not- when the facts fail to correspond
to p. Truth values, T and F, are secondary to and derived from negation: ’The sense of a truth function of p is a function of the sense of
p’ (Tractatus 5.2341). Thus T and F are not properties of propositions (Tractatus 6.111) any more than are positive and negative.
The truth values of the truth tables capture the relations of T and F
to p and not-p more than they define the latter.”
Section 4 concludes with the following remarkable observations:
(1)The examination of the Tractatus has revealed that for
Wittgenstein and other moderns “truth comes from the world, and
negation is in propositions. For traditional philosophers it is just the
other way around: Negation is in the world of appearances and in
the beings of the intellect, and truth is in the propositions” (p. 48).
(2)What Aristotle says about the true and the false in
Metaphysics Bk. IV 1011b25 and Bk VI, 1027b19ff tends to show
,
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that Heidegger was quite right when, in his Logic, he denied that
the Aristotelian texts hold a “correspondence theory of truth” (p.
48).
(3)Aristotle speaks “for a world very different from the one in
which the propositional calculus of Russell and Wittgenstein is at
home. For Aristotle negation (I mean negation in an objective form,
contrariety interpreted as Nonbeing and its effects) is in the world
and falsity (I mean the not always unintentional failure of speech to
reveal being is in statements....Whether negation is in the world or
in speech is one of the numerous but interrelated marks by which a
classical world...is distinguished from a modern world. For a world
that has negation built in responds to receptive thought since it
reveals its own distinctions, while a solidly positive one demands
constructive reason since oppositions need to be made” (p. 49).
As the just-reported grand conclusions of section 4 of Chapter
2 tend to confirm, negation became thematic for Brann by virtue of
her interest in the psychological and ontological topics that were
mentioned in the opening paragraph of this review; whereas logicians—from Aristotle through the Stoic logicians and Frege, Peirce,
Russell, Quine—attend to negation chiefly because of how it affects
what is and what is not a valid pattern of argument. Patterns of reasoning or deduction rather than patterns of judgment or of propositions may well be their primary concern.27 This difference
between herself and the logicians might also explain the otherwise
rather puzzling remark, on p. 25, that “by and large the negations
of logic28 take place in symbols and are found in books. They are
not so much naysayings as naywritings.” For the purposes of reasoning the idea of contradiction, that is, of an opposition which is
not only exclusive but also exhaustive, is indispensable: Illiterate
Athenians have no trouble grasping the sense of arguments by contraposition such as, “If virtue were teachable, there’d be teachers of
virtue, yet there are none. Therefore, virtue is not teacheable.” And
I surmise that the pattern of Euclid’s reductios (which likewise
involve the stark negativity of contradiction) was first discovered as
a debating gambit and passed on by teachers of rhetoric. I say this
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partly because it seems to me that even Euclid’s Elements still retain
a viva voce dialogic rhetorical mode.
“When we refer to a nonexistent object, what are we thinking
of and what are we talking about?” (p. 76). Chapter Three begins
by pointing out that this is a distinctively modern question,29 different from the ancient one taken up in Plato’s Sophist, how nonbeing can be, to which Chapter Four will be devoted. Four types of
non-existents are mentioned for purposes of illustration—“members of extinct species [dodos, for instance]...deceased human
beings [for example, Socrates]...artifacts no longer extant,30 but
also all the entities that never did exist in the ordinary sensible
sense, such as unicorns” (p. 79).
Roughly speaking, four types of answers are sketched in
Chapter Three: Bertrand Russell’s “theory of definite description,”
Alexius Meinong’s “theory of objects [and objectives],” the recent
version of Meinong worked out by Terence Parsons in his 1980
book Nonexistent Objects (New Haven: Yale University Press), and
any one of a number of theories according to which “pretense and
make-believe are the chief explanatory principles,...[not of the
behavior of nonexistent objects], but [of] how they manage to
come on the scene to begin with, [and] what we cognitively do to
cooperate in fiction making” (p. 99). From the way these theories
are elaborated it becomes apparent that, although —as the two earlier volumes of Brann’s trilogy argued—our ability to think and
speak truly or falsely of bygone things is testimony to the powers
of the human imagination, in that the feat of “re-calling” depends
on or consists in the imagination’s having succeeded at making
temporally absent things present, it is the “saving” of fictional entities that chiefly matters for the purposes of the present book’s
Chapter Three.
Before she turns to a fairly detailed examination of Russell’s
treatment of proper names and definite descriptions, Brann lets us
know that “the theory that is the winner in the world of logic
[namely, Russell’s], will turn out to be something of loser in the
world of fiction” (p. 76). Russell’s theory, as she tells us Parson too
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observed, pays too high a price for its clarity: “The theory commits
us to treating the sentences of fiction as false, while most of us think
they have at least a sort of truth, and some of us even believe that
they often have more truth than mere fact does” (p. 86). Meinong,
contrarywise, “comes near to saving the phenomena of that intentional experience of central interest to this trilogy...the experience
of imagining (p. 91). Russell’s excision of nonexistents from reality31 is false to the power that some non-existent beings and places
have, moving us “as models and attractors,” and “outliving us by
millennia, and in a word impinging on us as if existence were home
to them as well [as to ourselves?]” (p. 102).
Instead of recapitulating what Brann says about the technical
aspects of Russell’s theory of description and Meinongian rival theories, I want to dwell a little on Brann’s question how we are to
account for the fact that the Natasha, Pierre, and Andrey of
Tolstoy’s War and Peace or the Hari Kumar and Ronald Merrick of
Scott’s Raj Quintet32 have become our companions.
A familiar answer begins by reminding us of our unabating
curiosity about our fellow human beings, whether met in the flesh
or encountered vicariously through what our friends, our children,
our journalists report and our television news programs show. “But
the characters who people novels are immensely more memorable
than the Tom or Dick or Harry that our neighbors tell us about.”
Well, that does somewhat depend on what a particular neighbor is
capable of telling us about a particular Tom (or Jane for that matter), not to mention the particular Tom or Jane spoken about. But
to the extent that it is true, may it not in large part be the result of
novelistic characters’ (at least those that dwell in novels of substance) becoming so much better known to us than any persons not
our “real life” intimates?33 Novelists are much better at noticing
things than most of us are, and better at imparting what they’ve
noticed too. Also, our acquaintance with novelistic characters is a
shared acquaintance, shared with other make-believe characters in
the novel, with the novel’s author, and with fellow-readers of the
novel. It is hardly news that sharing (comparing notes and impress-
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sions) is immensely pleasurable, greatly contributes to a feeling of
solidarity, and is constitutive of our sense of reality.
Add to what’s been said our relish for just about all human skills
or powers, our own as readers and the novel-making skills of the
author. Most important, count in the special joys of play and makebelieve: Aren’t we well launched on the beginnings of some sort of
answer to the question “Why and how do fictional characters
become real to us?”
Brann does not think so. At least, she rejects the idea that what
we relish, in ourselves and novelists, is the exercise of the human
power of make-believe: “Being absorbed into a fiction, living in its
landscapes and with its people, is not well described as a form of
pretense—not on the reader’s or viewer’s part and so much less on
the poet’s or painter’s part....Children, to be sure, play ’Let’s pretend,’ but that is usually when the game requires that roles be
assigned , and I’d bet that the mover of the pretense doen’t often
assign, say, the submissive role to herself; in participating in a novel,
on the other hand, we may well surrender ourselves to the experiences of the underdog ”(p. 99f).34
I wonder whether childish “dramatic play” (as the child psychologists call it) and make-believe of every sort is here conceived
of in all its richness. Think of the infinite variety of solitary and collaborative pretending and letting be we catch our children at! Sure,
sometimes there is one kid in charge (“I’ll be mommy and you’ll be
baby”) but by no means always. Two games of make-believe I
remember watching were: spreading out newspapers on the floor to
be islands and going island-finding, island-hopping, and islandworking; arranging marbles in rows and letting them be children at
school. Neither of these games called for leadership. Older children
would sometimes join the younger ones at play, humbly grateful and
gratified to be allowed “in” on the game. Improvisational theatre
has some of these qualities, I believe, though I cannot be sure since
I have never participated, either as actor or as audience.35 I went on
like this because I want to make concrete that there might be ways
of “taking fiction seriously” and trying to understand why and how
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make-believe matters that don’t proceed by way of ontology but by
way of psychology. The British pediatrician and child psychologist
D.W Winnicott may have something to teach us here.36 And as for
.
grown-ups making believe, I have begun to read Kendall L.
Walton’s Mimesis as Make-Believe, On the Foundations of the
Representational Arts (Cambridge: Harvard University Press, 1990.
Part Four of this book tries to show why it is all right to do without
fictional entities. I should, however, also mention that in Austria, at
the University of Graz, much is currently being written about the
logic and ontology of fictional objects.
A reader of an earlier version of this review advised me that I
need to report where I stand on the issue of the being and non-being
of fictional characters. I am undecided, because I have insufficiently considered (to give just one example) whether my belief that one
can be as mistaken in one’s “reading” of a fictional character as one
can in one’s “reading” of a violin sonata does or does not have
ontological implications. My laziness about ontology may have
something to do with the fact that I lean toward believing that it is
more illuminating to ask questions about how imagined persons
and places are and are not like historical individuals and geographic regions, or how what one learns about good and evil from living
hooks on to what one learns about them from literature, than it is
to delve into ontology.
The rest of Chapter Three is devoted to reflections on lies and
lying37 and to Anselm’s so-called ontological proof of the impossibility of God’s non-existence.38 The setting out of Anselm’s argument is very pretty!
Chapter Four: When we begin to read Chapter Four’s first
paragraph, we are already in possession of the guidepost furnished
in the Preface (p. xiv): “Here [in section 2] comes on the scene the
Non of philosophy (my italics), a prefix signifying not the brusquely rejecting denial of fact in words but the more forgiving opposition of two elements in the same world. The thought of Nonbeing
comes among us as the unbidden effect of Parmenides’ injunction
against it, and Plato will domesticate that same Nonbeing, bringing
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it into philosophy as the relational principle of diversity, the Other.”
But to reach section 2 we must traverse section 1. It begins:
“Parmenides learned from the goddess who dwells in the house of
truth that ’Being is’ and that he must not embark on the way of
Nonbeing. As far as I know, Nonbeing had not established itself in
anyone’s thought—at least in the West—before Parmenides’ deity
warned him off this path of inquiry; nor has it ever vacated its place
in thought since. Her [i.e. the goddess’s] repeated prohibitions and
injunctions against this Unthinkable and Unsayable seem to have
done for this philosophical offense what inveighings against sin
have so often accomplished in the moral sphere—they have
launched it on its career as a well-formulated and ever attractive
presence” (p. 123).
Among the titillating suggestions of Brann’s commentary on
Parmenides’ poem there is this, that this “heroic epic” (in dactylic
hexameter) is “unmistakably [intended as?] a rival to Homer’s
Odyssey,” so that “the ancient difference between philosophy and
poetry” of which the Republic speaks (607b)39 first comes on the
world scene when the journeying of young Parmenides displaces
that of middle-aged Odysseus.
I find myself incapable of paraphrasing what Brann says about
Parmenides. Here are some more quotations: “We often use phrases like ‘sing a song,’ where the object is the action of the verb made
into a thing accomplished. Parmenides sometimes does something
symmetrical with the verb ‘to be’ at the front end of a sentence. He
turns the verbal sense into a subject. But I don’t think that Being or
its negation is thereby established as a thing....On the contrary,
mere verbal ‘Is’ remains the truest kind of showing forth, and the
nounlike forms merely display the inability, or rather unwillingness,
of the goddess’s speech to get outside the meaning of that little
word which courses through human speech surrounded by subject
and predicate. Parmenides’ poem is a rebuff before the fact to those
who will claim that Indo-European languages are indefeasibly subject-and-predicate-ridden. For this is what Parmenides is bidden [by
his goddess] to convey: the sheer Isness of which we always get hold
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when we think beyond multiplicity....The common declarative tripartite sentence...is an implicit expression of three distinctions:
between the thinker and the thought (since some thinking person is
having and uttering a thought); between the thought and what it is
about (since the sentence states a thought-proposition about an
object); and between the object and its properties (since the sentence predicates a property of its subject). At the very beginning,
before these elements have ever been formally established, the goddess wants to prevent them from being distinguished....My main
purpose in this section has been to enter just enough intothe meaning of ‘Is’ to make sense of the ’Is not’ that trails it as its unwelcome
but unshakable doppelganger” (p. 130ff.).
“The next step in the ancient story of Nonbeing is...the reversal of its outlaw status and its integration into the community of
Beings. It is taken in Athens, the city of reconciliations” (p. 138).
What follows the exquisite paragraph whose two opening sentences were just quoted40 is a fresh setting out of reflections on
Plato’s dialogue the Sophist.41
I describe a few of these.
Seasoned readers of Platonic dialogues agree in noticing that
the conversation in the Sophist begins with the question whether
corresponding to the three names or titles “sophist”, “statesman”,
“philosopher” there are three beings or three types of being. Given
the fact that there is a dialogue called Sophist and also one called
Statesman, the non-being of a dialogue called Philosopher is a glaring fact. Some Plato commentators have argued that the Philebus is
the “missing” dialogue. But Brann believes that there are indications
in the Sophist that Plato means us to understand that “sophists and
philosophers are identical,” though differing in three respects: First,
the dialectical skill which is shared by sophists and philosophers is,
in the philosopher, accompanied by a kind of professional ethics.
Dialectic is, for him, a sacred trust. For the sophist it is a moneymaking techne, for sale to the highest bidder. Second, unlike the
traveling sophist, who is detached from civic loyalties, “the philosopher never forgets his human circumstances” (p. 139). Third, the
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philosopher is “that rare sophist who acknowledges Nonbeing
without taking cover in it” (p. 139).
To catch the “vulgar” sophist, the philosopher-sophist—in this
dialogue represented by an unnamed stranger-guest from
Parmenides’ city, Elea—must somehow show that contrary to what
Parmenides’ goddess taught him, Nonbeing is.
But it is not only to catch the sophist; nor just to defend the
possibility of false speech, negative speech, and error. Rather, to
save philosophy itself (to save speech itself?), Nonbeing must be
allowed to be! (Sophist 260A). The stranger therefore, Theseus-like,
or again, Athena-like, bestows citizenship on Nonbeing by declaring
it a form among the koinonia of forms (p.141). It is the diversifying
relational principle or form Otherness, not to medamoos on,
absolute nothing. “It is its ...[being identified] as the Other that
saves it from the utter inability—which Parmenides does indeed
assert—to become sayable....Nonbeing both bonds and negates
among beings, but its negation is not annihilation” (p. 142).
The chapter’s last paragraph makes the transition to Hegel: “In
Nonbeing naysaying has found its enabling principle in the realm of
Being. Now comes a view of speech and thought [namely, Hegel’s]
as themselves having inherent negativity. As Nonbeing was a source
of ontic diversity, so this [Hegelian] negativity will be the source of
mental motion” (p. 144, my italics).
Concerning Chapter Five I merely report that it employs the
trinity Spirit, Understanding, Reason to display and classify the
kinds of negativity encountered in Hegel’s Phenomenology, Kant’s
First Critique, and Hegel’s Logic. Devotees of Hegel will find much
to admire here. The chapter concludes with a paragraph announcing that, though the earlier chapter concerning Parmenides and
Plato and the present one concerning Hegel conspire to reaffirm
that Being is prior to Nothing, this is not as yet fully established:
Therefore Chapter Six42 jousts with the “greatest question”—which
is ultimate, Something or Nothing?
Most winning, witty, and sometimes even wise of all the sections of Ways of Naysaying are the concluding pages of this chap-
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ter, Chapter Six, about Nothing, offered under the seemingly bleak
heading, “Nothing as Inescapable End: Death” (pp. 188-198)!
However much of the time I was rather lost in this chapter. The
reason, I imagine, is that Brann’s question, whether Something or
Nothing is ultimate, never jelled into being a question for me. Yet as
best I understand the chapter, the various items it gathers together—“modern nominalism” (p. 170), Epicureanism and the void (p.
171ff), the “blithe nihilism” of some of the Buddhist schools (p.
173), the political “nihilism” of the mid-nineteenth century Russian
revolutionaries portrayed in Turgeniev’s novel Fathers and Sons (p.
179), and Heidegger’s teachings concerning the nihilating nihil (das
nichtende Nichts, p. 184ff) —are thought to deserve to stand side
by side because they all affirm, albeit in different ways, that Nothing
is more C primordial, more really real, than Something. This is the
sense in which they are all of them “nihilisms.”43 Another thing that
they may have in common is an ontology in which will is prior to
understanding.
It is possible that my failure to understand the chapter and its
leading question is due to incomprehension of Heidegger: I tend to
become so overwhelmed with irritation at his preachy incantational tone, his haughtiness, his tricks of inverting grounds and their
consequents, his abuse of the scholarly riches deposited in etymological dictionaries, that I become incapable of paying attention to
what he says.
Conclusion: (1)Is all human “opposing” (in will, word, or deed)
reactive to, thus parasitic on, a “posing”? (2)Might negating
responses constitute evidence for the being of Nonbeing,
Nonexistents, or even of Nothing? (3)Supposing there are
Nonexistents and Nonbeings, by what powers of the soul do we
encounter them?
In her final chapter, Brann recapitulates the affirmative answers
she earlier gave to questions (1) and (2). But she now expands on
what was said about Nonbeing in her pivotal Chapter Four:
“Besides the nonexistents that respond to our sense of what is missing...there are also declines and falls from existence, right in the
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world around us, that we experience as a sort of nonexistence.
Take, for example, the reflection of a willow tree that appears in a
pond. Take the numerous things and people in the world without
that are not what they appear to be....This last group, fallen existences [my italics], particularly raises the question whether it is our
way of experience or the nature of things that provides the not or
non here” (p. 215). As the past participle “fallen” which I underlined just now goes to show, Brann is introducing a principle of hierarchy into the realm of being. “Nonbeing as otherness is the universal relativity....But there is also ...a vertical Nonbeing....This
Nonbeing...has in it something of absolute inferiority, of defective
or deficient Being” (p. 216). Brann has brought us back to the central books of Plato’s Republic, I mean, books VI and VII, with their
image of the sun, diagram of the unequally divided line, and story
of the prisoners confined to life in a cave.44 It is in this context that
she reaffirms the answer to question (3) that’s been with us since her
book’s opening sentence: It’s neither sensing nor thinking that give
us access to nonbeings and nonexistents but imagination and memory.45
Obviously, then, this review cannot have done justice to the
book it tried to summarize and (in some measure) appraise, since
that book is one third of a three thirds whole. I hope, however, to
have conveyed something of its extraordinary scope, writing style,
intellectual daring and imagination.
NOTES:
1. The Ways of Naysaying: No, Not, Nothing, and Nonbeing (
New York: Rowman & Littlefield, 2001)
2. Cf. What, Then, Is Time?, p. 165:“...we are able to have and interpret
images, to live consciously in the phases of time, and to think and speak negatively. My guess is that these three capacities are really triune, three-in-one.
They may be the root of our humanity, and perhaps the subject of another
book.”
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3. See also World of the Imagination, pp. 405 and 783, where Brann
expresses her agreement with Freud and Wittgenstein that one can “speak of
what is not, but not depict it.”
4. His self-identification, from Faust pt. 1, lines 1336-8, is quoted on p. 14:
“I am a part of the force that constantly wills evil and constantly effects
good....I am the spirit that constantly denies.” Omitted from the quotation,
though surely Nietzscheans would hold that they are, if not the, an arche of
“nihilism,” are the lines: “und das mit Recht; denn alles, was entste-
ht/ Ist wert das es zugrunde geht;/Drum besser ware es dass
nichts enstunde...” (“and rightly so, because everything that originates
deserves to perish. Wherefore it would have been better if nothing had originated.”)
5. Brann’s use of the Freud essay is filtered through Rene Spitz’s The First
Year of Life and No and Yes. I have not read these books. Therefore I
cannot tell whether her complaint that Freud’s speculations— about what it
was that first prompted the human race’s invention of a “symbol” for negation— fail to include reflection on not as accomplishing “denial of truth or
untruth” is also Spitz’s. “Psychoanalytic theory does not tell whence comes
mature negation and possible truth telling; these may not have a naturalistic
genesis” is the concluding sentence of her account of Freud. What a non-naturalistic account of origins might consist in is not explained.
6. The Second Year: The Emergence of Selfawareness (Cambridge:
Harvard University Press, 1981).
7. This list slights her treatment of double negation, of the logical paradoxes that are generated when negation and self-reference are allowed to combine, of the stretching of the concept of number through the introduction of
negative numbers and zero, and of Kant’s discovery or invention of “directed quantities” (vectors) in the pre-critical essay “An Attempt to Introduce the
Concept of Negative Numbers into Philosophy.” Since these topics are listed
in the well-prepared index, I omit page references.
8. Cf. Metaphysics ix, 1048b1-10
9. The quoted sentence ends with the bracketed remark “...of which the first,
the opposition of oppositions, is surely that of thinking itself to its object.”
This claim makes me uneasy, given the remark, on p. xiii of the Preface, that
“the mysteries and conuncrums of intention—denotation and reference,
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sense and meaning...are happily not within the task of this book.” In my estimation, Frege’s insistence on the need for a Sinn/Bedeutung contrast and late
Russell’s attempt to dispense with it must be discussed by anyone who investigates thinking and speaking and their “objects.” Observe also that conversational exchange is given no role in the account. A quick way of making this
manifest is that, throughout the book, saying “no” is classified or explained
in terms of exercising the will, although it surely figures when answering
what linguists indeed peg as “yes/no questions.”
10. My hunch is that Anscombe’s remarks about “internal” and “external”
negation in her Introduction to Wittgenstein’s Tractatus (see in Anscombe
pp. 31, 34, 35, 46, 47, 51), and her question (p. 53) “...Is the property of
being true or false, which belongs to the truth-functions, the very sam property as the property of being true or false that belongs to the propositions
whose internal structure does not interest us?” is what first prompted Brann
to make the question about the “location” of the negation particle thematic.
11. If we are both looking at the ocean and you say “Majestic!” my guess
that it is the ocean that is said to be majestic is pretty safe. That’s how I mean
“otherwise manifest.”
12. For the somewhat ampler statement of this Fregean type of analysis of
“simple” sentences which is the source of my remarks, see pp.132f,
Anscombe and Geach, Three Philosophers: Aristotle, Aquinas, Frege. Please
observe that although English, which has pretty nearly dropped the use of
case endings, tends to place the name of the topic early in the sentence,classical Greek and other languages that use case endings to express syntactic
structure may, for rhetorical purposes, place it late in the sentence. Note also
that nothing prevents a simple sentence’s having a “complex” topic, for
instance the ordered triple {Athena, Athens, this olive tree}, which is, on one
analysis, the topic of the sentence “Athena gave Athens this olive tree.” When
the topic is so identified, the predicate is “—gave—to—” When the item that
would, in Greek, be in the nominative case is singled out as the name of the
sentence’s topic, the predicate would be “—gave Athens this olive tree.”
What chiefly matters, from a Fregean logical point of view, is the contrast
between proper names (e.g. “Theaetetus”) and concept words (e.g “flies” or
“sits”) as in the sentences “Theaetetus flies” and “Theaetetus sits.” A person
who is unaware that the word “give” is trivalent and the word “fly” or “sit”
is monovalent hasn’t got the hang of the semantics of these concept words.
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Cf footnote 11 below. See further Anthony Kenny’s Penguin volume about
Wittgenstein, pp. 121f.
question (p. 63) “Is Being at the true center of every sentence even if it is
obscured by a predicative verb?”
13. How “simple” is to be understood in this context is, of course, much in
14. In the spirit of Kantian “architectonic,” these ways of negating a sentence
are later (p. 95) brought to bear on lying, so as to yield a classificatory scheme
for lies.
21. I note that there’s a large dose of such “idealist” thinking in Freud’s essay
on negation: “the performance of the function of judgement is not made possible until the creation of the symbol of negation has endowed thinking with
a first measure of freedom from the consequences of repression and, with it,
from the compulsion of the pleasure principle.”
15. In the “dream theory” of Theaetetus 202 the mistake is to suppose that
22. Cf. Aristotle, Metaphysics Bk V, Ch.ll.
sentences consist of nothing but names; earlier, at 190, it looks as though sentences are being spoken of as consisting of nothing but predicate words. For
explicit correction of such “homogenizing” treatment of the constituents of
sentences, see Sophist 262.
23. Peter Geach’s essay “The Law of Exclude Middle” (p. 79, Logic
need of saying.
16. Does “doing logic”/“doing empirical science” exhaust the genus “investigation”? Brann would certainly question this bipartition.
17. “Perhaps the act of negating, which maintains a questionable existence
as the polar opposite of [affirmative] judging, is a chimerical construction,
formed by a fusing of the act of judging with the negation.” (p. 128 of Geach
and Black’s Translations from the Philosophical Writings of Gottlob
Frege, Oxford: Blackwell, 1952).
18. In footnote 54, on p. 69, Brann calls on Anscombe to testify that, as
Brann puts it, the Wittgenstein of the Tractatus, in “rejecting inquiry into
the way world, pictured fact, language, and thought are related” and “pretending that epistemology has nothing to do with the foundations of logic
and the theory of meaning,” made claims that are “fantastically untrue”
(Anscombe, Introduction to Wittgenstein’s Tractatus, (London: Hutchinson
University Library, p. 28).
19. For example, and especially, Kant and Hegel.
20. This sentence continues, after a colon, as follows:”The briefest way to
put the reason why is that thinking speech brings its objects to a standstill
even as it goes about discerning them through their properties. The declarative is expresses at once that transfixing done by thought and the expansion
with which the object of thought responds.” The just cited explication of
Brann’s “introspective” report is tantamount to an affirmative answer to the
Matters, Oxford: Blackwell, 1972) contains a nice exposition of this thesis.
Geach, like Brann herself (e.g., p. 28), exploits Wittgenstein’s metaphor of
“logical space” and the notion of boundary for this purpose. Note, by the
way, that it would be a mistake to assimilate Wittgenstein’s logical space to
Brann’s psychic space, as she describes it on the opening pages of her Preface.
Studying Brann’s, Wittgenstein’s, and the cognitive scientist Gilles
Fauconnier’s uses of metaphors of space would be a delicate but worthwhile
undertaking.
24. See, e.g., What, Then Is Time?, p. 112ff. In other sections of Ways of
Naysaying Wittgenstein continues to be treated as the or a bad guy: He
would, as Brann reads him,want to prevent her and fellow philosophers from
investigating whether there is “some one truth behind [the] many appearances” of, in this instance, negativity (p. xiv and note 11 on p. xvii). In the
chapter on nihilism, Brann approvingly reports that Stanley Rosen has
“shown” that “‘Wittgenstein and his progeny are nihilists because they cannot distinguish speech from silence.’” After the brief quote from Rosen, she
goes on to say: “For [according to Wittgenstein] it makes no difference what
we say. It makes no difference because if, as the later Wittgenstein
says...speech becomes meaningful only in a context of gamelike rules and
conventions and as a ‘form of life,’ then we can never get beyond these and
never receive a sensible answer when we query a conventional usage or conventionalism itself ” (p. 183).
25. I was helped by Anscombe’s version of this last question, which runs as
follows: What right do logicians have to define “not” by telling us that “not
p” is “the proposition that is true when p is false and false when p is true”?
The phrase “the so- and- so” is, after all, legitimate only when there is a soand-so and there is only one such.
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26. Does this word (or its German equivalent), occur in Wittgenstein’s text?
27. It is a striking fact that only Aristotle’s treatment of “immediate inference” is taken up (footnote 7, p. 61) and “syllogizing” omitted. Note also that
in Chapter Three, when dealing with Russell’s account of Definite
Description, nothing is said about the need, in mathematical reasoning, for
the principle of “substitutivity of identicals” or the “principle of existential
generalization.” See Ausonio Marras’ Introduction to his anthology,
Intentionaliy, Mind, and Language (Urbana: University of Illinois
Press, 1972) for some brief remarks about the latter two. When all is said and
done, Brann does not seem to be really interested in formal logic. This is how
I account for her not catching the slip in claiming that “In symbolic logic we
do not enter the propositions as we did in section 2, but take them as primitive, symbolized by p or q, etc.” (p. 43; cf p. 212). She certainly knows that
Frege’s treatment of quantification (analysis of the sense and use of such little words as “all”, “some”, “one,” which is needed for doing predicate calculus) is what is usually singled out as the true “advance” beyond premodern
logic; Stoic logic, though “pre-modern,” had already dealt with the definitions of the logical constants of propositional logic and with its basic argument patterns.
I look as though I’m being a pedant about the history of logic. But that’s really not what I care about. Rather, ever since the days that I heard the World
War II German soldiers who were entering Amsterdam, Holland, sing
“Denn wir alle lieben nur ein Madelein, Annemarie” I have wondered, “Should I feel sorry for that girl, Annemarie, burdened with being
loved by this whole troop of men? Or are there as many Annemarie’s as there
are men in this troop, and each of the girls gets one of the singing men? For
a fine essay on this topic, see Peter Geach’s “History of a Fallacy” in Logic
Matters (Oxford: Blackwell, 1972).
28. I believe this means the not of contradiction.
29. I am not sure how “modern” is meant here: post-Occamist, that is postrealist (in the scholastic sense of that word)? I ask for clarification of the
adjective because I am not certain what, exactly, the systemic import of the
observation is. See footnote 23 on pp. 111ff. See also the remark about the
“inherent nihilism of an absolute nominalism” in her commentary on Wallace
Stevens’ poem “The Snow Man” and the continuation of this thought in her
interpretation of “The Course of a Particular,” p. 170. FootnoteA 3 on p. 199
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claims that “nominalism is one of the philosophical positions adopted by
those for whom disillusionment is a warrant of truth” and concludes with a
remark about the “fanatically honest.” These are, says Brann, the folk who
“take pride in shivering in the metaphysical cold.” The quoted passages
sound—what shall I call it?— dismissive to me. I wish there had been something more nearly like an explanation of what the nominalism/realism issue
is and why Brann favors the realists. Cf pp. 4-6 of W Stace, The
.T.
Philosophy of Hegel (New York: Dover, 1955)?
30. Artifacts no longer in use, like sliderules, or tools for living about which
we learn through literary remains but examplars of which have not been
encountered by archaeologists? I try more nearly to specify the question
because I am confused whether the general question of how we can speak or
think truly or falsely of kinds that are“bygones” is being raised or rather the
question how bygone individuals can be referred to? Cf Wittgenstein’s
Philosophical Investigations ¶79 about the many senses of “Moses did not
exist.” See also G.E.M. Anscombe and P Geach, Three Philosophers
.T.
(Oxford: Blackwell, 1973) pp. 135f about the importance of Frege’s reviving
the scholastic contrast between singular and universal propositions.
“Traditional” logic rides roughshod over the distinction. Geach’s essay “Perils
of Pauline” in Logic Matters is refreshingly lucid and unstuffy on the subject of names and descriptioons (and much else besides).
31. Cf. p. 100: ”What Russell says he means, flatly and irremediably, and
therefore he must be flatly and irremediably wrong: It cannot be the case that
what is said about and within fictions is false—unless one maintains that logically accurate speech has no correspondence with humanly normal speech.
For we say both that it is true and that it is true to life that Natasha Rostov
marries Pierre Bezuhov, and we want to keep on saying just that.”
32. See the splendid appreciation of the Raj Quintet in Brann’s contribution
to the anthology Poets, Princes, and Private Citizens edited by Joseph M.
Knippenberg and Peter Augustine Lawler (Lanham: Rowman and Littlefield,
1996).
33. The special pleasure we take in our own children is not solely due to
their being ours; it has much to do with our knowing them better than most
other people’s children.
34. I worry a little about the rhetorical effects of using the words “pretend”
and “pretense” in lieu of “make believe.” But let that pass.
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35. The novelist Jorge Luis Borges writes somewhere, “[The actor] on stage
plays at being another before a gathering of people who play at taking him
for that other person.” I acknowledge, however, that novels differ from stage
plays, involve (in addition to the things mentioned) some special a deux intimacy between the reader and the book.
36. See for example Playing and Reality, London: Tavistock Publications,
1971 and perhaps also some of the essays about Winnicottt included in the
collection edited by Grolnick and Barkin, Between Reality and Fantasy
(New York: Jason Arons, 1978). I particularly recommend Rosemary
Dinnage’s “A Bit of Light.”
37. As best I recall, Brann does not, when treating of “the lie in the soul” (p.
94), worry about what Freud called repression.
38. I was puzzled that Brann did not reserve space in her book to discuss the
important topic of children’s and grown-ups’ often being uncertain whether
this or that “really happened” and whether this or that named individual
(Satan, Cerberus) or species of entities (witches) “really exists” or not.
Helping children sort out the dreamt from what’s in the public world of the
awake is among our parental responsibilities. Thus “...does not exist” seems
to me to hold as important a story as is that about the being of non-beings.
39. Cf Epinomis 990 on that mere farmer’s almanac, Hesiod’s Works and
Days?. Parmenides reputedly was the first to propose that the moon shines
by the sun’s reflected light and that the earth is a sphere; also, that the
evening and morning stars are one and the same. I therefore keep hoping for
a reading of his poem that will show that its episteme/doxa contrast has
astronomical meaning. But no such reading is endorsed by Brann.
40. These sentences allude, of course, to Aeschylus’ Oresteia and Sophocles’
Oedipus at Colonus. This well illustrates the dramatizing vividness of Brann’s
ontological discourse.
41. Cf The World of the Imagination p. 389ff. and Jacob Klein, Greek
Mathematical Thought and the Origin of Algebra (English version,
Cambridge: MIT Press, 1966), p. 82 and A Commentary on Plato’s Meno
(Chapel Hill: University of North Carolina Press, 1965), p. 114f.
42. Corresponding to the afternoon of the day on which Man was created,
male and female, in God’s image? Yes, of course I am joking in playing with
the numbers. But I am not just joking: The chapters in Genesis that tell in
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detail how man became man (chapters 2, 3, 4) hold a plethora of negation
words, whereas the opening chapter lacks all negativity.
43. If there is an explicitly stated definition of the word “nihilism” in
Chapter Six, I need to have it pointed out to me.
44. Cf. Eva Brann, “The Music of the Republic,” St. John’s Review, volume xxxix, numbers 1 and 2. Se especially pp. 75,6.
45. Cf. the discussion of “opinion” on pp. 38ff of “The Music of the
Republic.”
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107
The Potent Nonentity: A
review of Eva Brann’s What,
Then, is Time?
Torrance Kirby
Time, Augustine claims, is so ordinary as to be impossibly
difficult (Conf. XI.14). This is the paradoxical theme to which Eva
Brann returns often (one is tempted to say “time and again”) in her
remarkable, recently published volume What, Then, is Time? Time,
the “potent nonentity,” proves to be as elusive a quarry as the
Sophist himself. The inquiry begins with a high sense of wonder
peculiarly fitting in this of all philosophical quests. The inner experience of time and its foundation or ultimate ground, constitute the
heart of this investigation. Brann employs an extended, highly elaborated aporetic approach to the search for a definition. So numerous and complex are the poriai encountered that this Protean
beast is not pinned down with a definition until well into the closing chapter of the book. The investigation as a whole is composed
in the form of a diptych with one larger panel devoted to the study
of various selected texts or “presentations” by philosophers who, in
Brann’s estimation, “have written most deeply and most engagingly about time.” A second smaller panel contains the author’s own
“reflections” on the matter. She is careful to point out, “study and
thought, though not of necessity incompatible, are by no means the
same” (159). This book is worthy of the most careful reading with
both ends in view.
The predominance of the prolegomena in this investigation is
consistent with the spirit of much contemporary, postmodern
inquiry. Brann’s approach is underscored by the splendidly postrevolutionary claim that her purpose is “not to change the world
but to interpret it!” Viewed in another light, however, the methodology of this book is resonant with the very best ancient authors,
Eva Brann. What, Then, is Time? Lanham, MD. Rowman and Littlefield, 1999.
Torrance Kirby is an assistant professor at McGill University.
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and its hermeneutical approach reminiscent of Aristotelian science.
The first part of the book, a study of earlier philosophical “presentations” of time, constitutes a “history” such as one finds at the
outset of many of Aristotle’s treatises. Brann’s study of the
attempts of her predecessors to define time is thus by no means any
ordinary history. Her extensive review of the preeminent contributions to the hermeneutics of time clarifies wonderfully the question concerning time and enables the reader to make the great
ascent from mere study to thought. In the “reflections” of the second part, Brann proceeds intrepidly to face the question “what,
then, is time?” head on.
Discussion of the “lisping” efforts of predecessors (Metaph.
A.1) in this chase turns out to be a daunting task. The relevant texts
range “from the hard to the hellishly hard,” as Brann puts it. As in
an Aristotelian “history,” the texts are selected with a view to clarification of certain key facets of the problem of definition. Four crucial theories about the nature of time are addressed through the
study of four pairs of philosophers. The originality of Brann’s
approach is striking. The unexpected pairings - Plato and Einstein,
Aristotle and Kant, Plotinus and Heidegger, Augustine and Husserl
- prove to be both inspired and illuminating. An important element
of Brann’s purpose in this approach is to demonstrate that the larger questions about the nature of time are themselves by no means
“time-bound.” By pairing the authors in this way Brann ensures that
the problem of definition predominates over less important considerations. The first approach to the theory of time, as exemplified by
the arguments of Plato’s Timaeus and Einstein’s Special Theory of
Relativity, proposes that time is “external,” namely that time refers
to external motions of which it is the measure, as in the case of a
clock’s measurement of the diurnal rotation of the sun. (The consideration of time as the “externality” of history and its movements
is mercifully ruled outside of the present inquiry.) In the cosmos of
Timaeus, time is the very intelligibility or “numbering” of the external motion of the visible heaven. As Brann puts it, this identification
of time with phenomenal motion continues to “bedevil” the dis-
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course of physics. Einstein displays little interest in the essential
nature of time, but is absorbed rather by the question of quantifying time owing to complications arising from the implication of
temporality in locomotion. After the fashion of the hunt for the
wily Sophist in the Platonic dialogue of that name, the consequence
of this initial “presentation” of a definition of time is to introduce a
dichotomous division - namely between time in the world and time
in the soul - which is of considerable use to Brann in advancing her
own quest for an acceptable formulation. The boundaries have been
narrowed considerably by the exclusion of merely “external” time
as a fallacy.
Before proceeding to the presentations of internal time, Brann
examines a pair who propose highly speculative accounts of the
generation of time out of space. Hegel’s dialectical exposition of the
genesis of time out of space is put forward by Brann as possibly the
most profound of all treatments of “external” time. For Hegel, time
from its first genesis as a pure Becoming, behaves like incipient spirit (Geist): “Time is the Concept itself that is there and which presents itself to consciousness as empty intuition. For this reason Spirit
necessarily appears in time, and it appears in time just as long as it
has not grasped its pure Concept, that is, has not annulled Time”
(Phenomenology ¶ 801). Through a discussion of Bergson’s mission
to suppress “extensive space” in favour of “intensive time” Brann
effects a transition to the second principal stem, viz. internal time
or “time in the soul,” which is the general focus of the remaining
three pairs of texts in the series of presentations.
With her examination of the theories of Aristotle and Kant,
Brann arrives at the second crucial stem of the dichotomous division of time into the categories “external” and “internal.” Although
for Aristotle motion is properly the “substrate” of time, while conversely for Kant time is itself the ground of motion, both philosophers are “driven” to relate the notion of time to a “psychic counting.” As Aristotle says, “time is the number of motion” where
motion is understood as disclosing continuous magnitude. The
“truth” of time resides in the numbering or counting soul that meas-
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ures the before and after of this magnitude. Time, according to this
presentation, is no longer viewed as an independent, “substantive”
reality or but is rather reduced to the status of an accident or predicate which exists “for thought.” For Brann, Kant’s treatment of
time displays a deep affinity with Aristotle’s on this more general
level. The internal sense of time, however, represents much more in
the Kantian metaphysics than ever dreamed of by Aristotle. For
Kant this psychic counting is perhaps the most intimate characteristic of humanity. Indeed Brann shows that Kant’s treatment of time
is most accessible when the Critique of Pure Reason is viewed as “a
new founding of human nature whose centre is time” (55).
Appearances may be removed from time but not the reverse, which
reveals that time, for Kant, is prior in the order of knowing; the
apprehension of change is understood to depend upon the a priori
intuition of time. In one of numerous penetrating aperçus scattered
throughout the discussion, Brann draws attention to Kant’s
nonetheless restricted view of our ability to know ourselves as temporal beings by reminding us of his low opinion of music. This, in
turn, is contrasted with Leibniz’s opposing exaltation of the unconscious counting of the soul in music as “a pleasure given to us by
God so that we may know of him; in music soul is revealed to itself
and God to it” (Principles of Nature and Grace ¶ 14).
In the subsequent paired “presentations” of Plotinus and
Heidegger, the inquiry proceeds to consider the “ground” of temporality—that is, of some higher, possibly transcendent source of
this inner sense of time. Thus the dichotomous division of the
“hunt” advances to a new level of precision. For both Plotinus and
Heidegger, as Brann shows, time constitutes the “deepest condition” for humanity. Plotinus identifies time with specifically
“human” being in its manifestations of a peculiarly ecstatic nature,
by the human’s attempt to escape the element of its temporal fallenness. The Soul’s very “appetite for things to come” (Enneads III.
7.4, 34) keeps her in her fallen state. Temporal being strives for salvation, viz. the overcoming of temporal “dispersion,” through
union with the eternal hypostasis above. Happiness, understood as
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“the flight of the alone to the Alone,” is thus altogether outside
time, for it is no mere mood or emotion, but rather a fundamental
possibility for the soul, that of an undispersed present even beyond
being (Enneads I. 5.7, 15). Time is made explicable through eternity, its original ground. Although radically distinct from Plotinus
with respect to virtually the entire substantive content of his
thought, Martin Heidegger at least shares with Plotinus the supposition that temporality is the key to understanding human existence.
As a being whose essence is its existence, this ultimate ground is for
Heidegger not the transcendent eternity of the Plotinian Primal
Hypostasis, but rather the temporality of human being itself,
Dasein. The discussion stemming from this remarkable dialectical
pairing of Heidegger and Plotinus is particularly illuminating.
In chapter four Brann arrives at her final pairing of Augustine
and Husserl with the observation that no two philosophers are both
further apart and closer together. Through an examination of their
discourse on time as a temporal “stretching” of the soul (distensio,
as Augustine puts it), the argument—for it is indeed an argument—
acquires a distinctly sharper dialectical edge. The coincidence of
identity and difference in their thinking about time is uncannily
appropriate to their strongly dialectical approaches to the quest to
define time. According to Brann, while Augustine sifts through the
phenomena in search of existence and while Husserl neutralizes
existence in order to find the phenomena, both look within themselves for the phases of time, that is to say, for past, present, and
future. For both philosophers, Brann argues, the problem of “internal” time is not to be referred to a higher ontological ground for
resolution, as is the case with Plotinus, for example, but rather time
is to be understood as arising out of and discerned within the soul
or consciousness. Brann’s argument on this point is open to some
dispute, at least with reference to Augustine if not to Husserl.
Perhaps the device of pairing the presentations has led to a downplaying of Augustine’s affinity with Plotinus. It is common among
contemporary existential readings of Augustine to de-emphasise his
dependence upon Neoplatonic metaphysics. He begins his presen-
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tation on time with the “in principio” of Genesis 1, the revelation
of the divine creative activity understood as totally beyond the temporal, narrative realm of human existence. In making his transition
in Confessions from Book X on memory to Book XI on time,
Augustine shifts gears as it were from looking within at the phenomena of consciousness to looking above at the higher ground of
the life of the soul, ab interiora ad superiora. The Creator, who is
altogether above the flux of becoming, is understood nevertheless
by Augustine to be present, knowing, and active within the temporal realm.
While temporal human existence, dispersed or “distended” as
it is through phases of past, present, and future, is to be contrasted absolutely with the undivided existence of “the One,” Augustine
finds nonetheless within the soul as imago dei a positive image of
the activity of God in creation. The enigma of the human experience of time is thus referred by Augustine to the exemplar of the
Trinity for resolution. In the psychological image of the Trinity—
memoria, intellectus, et voluntas—Augustine finds a model for his
reflection upon the experience of time as at once continuous and
without extension. He points to the chanting of a psalm as a potent
revelation concerning time. He reflects upon the recitation of a
song that he knows, Ambrose’s hymn Deus Creator Omnium. The
song is stored in memory, an already completed whole which the
soul intends to sing. Before singing, the soul’s expectation possesses the complete song. As the soul sings, the relation of expectation
to memory shifts syllable by syllable until the entirety of expectation has finally become a memory of the song as completed, as having been sung. Memory, presence, and expectation are united in
the song. Through the singing of praise, itself a mode of confession, Augustine begins to see how the timeless and the temporal
become one. Through song the soul is enabled to think the divine
object in the image, and this, Plotinus certainly would regard as the
most extreme absurdity. Thus, by “collecting” ourselves, we can
escape from our temporal constitution into God’s “standing Now,”
as Brann puts it, into eternity.
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Brann concludes the part devoted to presentations of time with
an extensive and complex analysis of Edmund Husserl’s phenomenological treatment of internal time-consciousness. The text of
Husserl’s Zur Phänomenologie des inneren Zeitbewußtseins we
owe, Brann tells us, to Edith Stein’s supererogatory editing of various manuscripts and notations. By way of a background sketch,
Brann offers a helpful introduction to Phenomenology itself and
looks at the influence of Augustine, William James, and Franz
Brentano on Husserl’s reflections upon time. Husserl is particularly
engaged with the problem of integrating the phases of time. Brann
claims that he in fact “solves the problem of relating the present, the
moment of primary perception to its immediate retentional past
and protentional future by giving a model for the orderly sinking
away of perceptions and their intertwining with present consciousness” (160). With Husserl, the presentations have in a certain sense
come around full circle. Husserl brings his account of time to completion by reconstituting “external” time in the form of an absolute
temporal flux which transcends the temporal phenomena of internal time-consciousness and which is, moreover, the underlying principle which sustains human subjectivity. As Brann concludes,
Husserl’s ultimate temporal flux is “a very nearly inarticulable final
fact” (156).
In the Second Part of the book, titled “Reflections,” Brann purports to finally face the question “What, then, is time?” (The claim
that the “Presentations” are a mere exercise in “study” and that
only now, in the final pages is she going to roll up her sleeves and
get down to the serious business of “thought” seems not entirely
ingenuous. Already a good deal of hard thinking has gone into
both the pairing itself and the ordering of the pairs, all of which
serves to advance the quest for a definition.) The reflections proceed with a consideration of certain formal similarities between
time and the faculty of imagination - here, once again, is the
Sophist and the wedding of Being and Nonbeing. Brann shows that
images present a relatively constant picture, viz. Being and
Nonbeing in fusion, while time, on the other hand, is a flux of
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THE ST. JOHN’S REVIEW
“Being as passing over into Nothing and Nothing as passing over
into Being” (Hegel, Phil. of Nature ¶ 259). Time and imagination
are thus connected with one another through the way Being is
related to Nonbeing in both temporal process of becoming and in
images. As might well be expected, Brann offers a fascinating comparison of these concepts by building upon her previous exploration of the faculty of imagination.2
There follows Brann’s own interpretation of the phases of time
together with their appropriate faculties: past and memory, future
and expectation, present and perception. Throughout, she draws
upon the foregoing presentations of time by the philosophers which
provide both the categories and a vocabulary which enable Brann to
penetrate the question deeply and swiftly. This section of the book
is a wonderful demonstration of the dictum of Bernard of Chartres
who claimed to be able to see things far off by virtue of “standing
on the shoulders of giants.” In an interesting and frequently amusing section Brann proceeds to analyse various “time pathologies” as
forms of “phase-fixation.” Here we have an opportunity to reflect
on aspects of time’s “brutal tyranny,” e.g. the contemporary idolatry of novelty, a fixation on the “just now,” the trivialising of the
past in nostalgia or the future obsession of the IT phenomenon.
Brann even reviews cures for these time-induced pathologies such
as that offered by Nietzsche in his teaching on the Eternal
Recurrence of the Identical. Brann counters this frantic cycle of
reincarnation with another, much more attractive option, namely
the concept of Aevum, as manifest in the sempiternity of the angels
in heaven or, alternatively, in the fictional temporality of the novel.
All of this is delightful. Brann recommends the cultivation of “aeveternity” as at least “a partial relief for our temporal ills.”
In the last chapter of the book Brann moves closer to the final
struggle with the definition of time by way of a via negativa. Here
time is finally unveiled as the potent, indeed tyrannical, non-entity.
The revealing is apophatic. Time is not external motion, nor is it an
abstraction from process. It is not a power or force, nor a “fungible
substance” (i.e. time is not even money!). Time is certainly not a
KIRBY
115
mere linguistic usage. As Brann succinctly puts this point,
“Language can guide thought but it cannot constrain it.” (Brann
notes in passing how neatly the distinctions of philosophical inquiry
concerning time seem to turn up in the problems of linguistics.)
Time is not Dasein. Whereas Heidegger regards human finitude as
ultimately expressed in the fact our mortality, that our existence is
“destined” to end, Brann counters optimistically that human finitude is better sought in the fact that we begin, “we do not temporalize ourselves; we are born temporal.” Time is no determinate
being; it is not perceived by the senses, it is without external effects,
and elusive to insight. Time is therefore a non-entity. Though
apparently nothing, time’s “not-being” is nonetheless very powerful
(although, be it noted, not “a power”). “What, then, is time?” Here
the argument finally shifts from marked apophasis to a more kataphatic note. The affirmative definition comes in nine-fold form (a
touch which no doubt would have pleased Pythagoras). It is not this
reviewer’s intent, however, to spill the beans. In order to reap the
full benefit of Brann’s final, dramatic unmasking of Time—to be
altogether “present” as it were at the capture of this elusive beast—
readers are well advised to follow the leader of the hunt herself
along the trail through all its intricate twists and turns. And who
indeed are the intended participants in this quest? Brann recommends her book to anyone who longs to learn about time by pursuing the quest described above, to aficionados, to students who
seek to come to grips with some of the primary texts on time, and
finally to teachers who might be on the look-out for some tips on
selections for a syllabus on the interpretation of time. This is an
unusually difficult book whose author challenges the reader to “take
note” and whose rewards are proportionate to the investment of
careful, punctuated attention.
NOTE:
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THE ST. JOHN’S REVIEW
1. See Eva Brann, The World of the Imagination: Sum and Substance
(Lanham, MD: Rowman and Littlefield, 1991).
117
The Feasting of Socrates
Eva Brann
Before reviewing Peter Kalkavage’s Focus Press translation of
the Timaeus for the St. John’s community, I must, in all fairness,
confess my partiality. He, Eric Salem, and myself were the cotranslators of Plato’s Phaedo and his Sophist for the same publisher.
Together, over several years, we worked out some principles of
translation which are discernible in this Timaeus version. In fact, I
think the three of us would welcome with some glee the notion of
a St. John’s school of translation. For we wanted to be working very
much with the spirit of the Program and a possible use by our students in mind. We thought that translations of Plato should render
word for word, even particle for particle, with the greatest exactitude, what the Greek said, avoiding all interpretative paraphrase,
craven omissions, and latter-day terminology. But we also stipulated that they should catch the idiomatic expressiveness and the
changing moods of the original. These principles are clearly at work
in this rendering of the Timaeus.
We learned as well, however, that each dialogue is a unique universe of discourse, the artful representation of an inquiry with its
own approaches, terms, settings, and above all its own participants,
each of whom is in a mood specific to this never-to-be-repeated, yet
ever-to-be-continued conversation. Thus it follows that the Timaeus
made its own particular demands on the translator. It is, after all,
less a dialogue than a short tale of antiquity by Critias followed by
an account of the cosmos by Timaeus—a long one. The familiar
voice of Socrates falls almost silent as these speeches are made to be
a feast for his enjoyment—or, perhaps, amusement. Timaeus’s cosmology is full of the sort of technical matter Socrates does not scruple to spoof in the Republic—the very dialogue which establishes
the sort of ideal city that his companions agree to bring to moving
life for him by giving it its historical and cosmological setting.
Peter Kalkavage, Plato’s Timaeus. Newburyport, MA: The Focus Philosophical Library (2001).
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THE ST. JOHN’S REVIEW
Timaean cosmology involves not only the moving spheres and circles that bear the astronomical bodies and the geometric elements
from which they are constituted, but also the musical “harmonies”
(scales) that ensoul the heavens. Three beautifully clear appendices
provide the reader—and this edition is meant for the “adventuresome beginner”—with the fairly elementary knowledge needed to
enjoy this heavenly entertainment. It should be said, though, that
the cosmological astronomy of the Timaeus together with its sober
mathematical exposition in Ptolemy’s Almagest was the serious science that stood behind the New Astronomy of the dawn of modernity. (There is a story—I cannot vouch for its truth—that in the
early days of the St. John’s Program books of astronomy and
physics were to be found in the library ranged under “Music,” courtesy of the Timaeus.) The dialogue is so full of Greek science that
there is a danger of regarding it as a source of antiquarian problems.
But, the translator observes in his Preface, that is the very danger,
the one of reducing the cosmos to a collection of mummified facts
and recondite puzzles, to which the Egyptian priests are said to fall
prey. So less is more by way of learned exegesis, and the well-illustrated appendices give just enough to make the dialogue intelligible
to an amateur.
Since I’ve started at the back, let me say that here too you will
find an English to Greek glossary. The entries tell not only how a
Greek word is translated and, if more than one translation has to be
used, why that is necessary, it also gives the root or central meaning
and others that flow from it. In sum, the entries are a pretty interesting lesson in philosophic Greek.
To go to the front end of the book, there is, besides the Preface,
the Introductory Essay. The Timaeus, the only Platonic dialogue
known in medieval times and in all epochs the most influential one
among those philosophers to whom the constitution of the cosmos
was of central interest, is also, in Peter Kalkavage’s words, “the
strangest of Plato’s dialogues. It is so strange that one wonders
whether anything can be taken seriously . . . . [It] is strange not only
to us but also in itself.” The Introduction is intended to illuminate
BRANN
119
that strangeness without dispelling it. The odd but necessary question is pursued: What is the Timaeus about? Socrates is all dressed
up (kekosmenos) and in a strange mood. He gives a truncated, philosophy-free version of his Republic and asks to be told about this
stripped-down political blueprint mobilized to go to war. The
resulting verbal feast prepared for him among the three eminent
men who are present (one mysterious fourth is absent) has an oddly
skewed relation to the truth and the love of wisdom that are
Socrates’ normal preoccupation, for it is presented as a “likely
story,” and a story of likenesses, the way of being that is so dubious
for Socrates.
The festivity begins with Critias’s retelling of an antiquarian
tale about archaic Athens as told by the Egyptian priests to the visiting lawgiver of Athens, Solon. We hear that this old Athens,
ancient even to the ancients, once defeated a huge and sinister
island empire called Atlantis.* Critias thus presents a pseudo-historical Athens as the embodiment of a “pale image” of the Republic.
There is plenty to puzzle about in this beginning.
For this city Timaeus supplies the cosmic setting; we are invited to wonder how fitting it is. A divine craftsman appears out of
nowhere and makes the cosmos, the well-ordered beautiful world,
in the image of an original model. Hence the cosmos has two wonderful features. It is a copy and thus, while imperfect in its being,
capable of being in turn a model, as it indeed is in the dialogue. And
second, it is intelligible, interpretable, not only as an intentionally
made work of art, but as en-, or rather, circum-souled. For whereas the human animal has its soul within, the cosmos is encompassed
by bands of soul matter. All these wonderful and significant doings
can be read in the dialogue, but the Introduction brings out their
thought-provoking strangeness and their relevance to our humanity.
Thus after the cosmic construction there is a harsher “Second
Founding.” It has an elusive “wandering cause,” the “source of
power as opposed to goodness”—an intra-cosmic, semi-intractable
cause called “necessity” acting in a scarcely intelligible theater of
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THE ST. JOHN’S REVIEW
operation, space. Within it arise body and the human animal: “The
making of man for Timaeus is a pious desecration,” says the
Introduction. It is delegated by the Craftsman-father to his starsons.
This part of the Timaeus, the coming-to-be of organic life within the cosmos, is so weird that our undergraduates aren’t even asked
to read it, yet Peter Kalkavage shows how to begin to make humanly applicable sense of it.
Finally he returns to the question: “Why is the greatest philosophical work on the cosmos framed by politics?” An answer is suggested: The frame signals Plato’s reflection on what happens when
the Socratic search for truth is replaced by a Timaean will to order.
But this shift to the constructive will might well stand for the revolution that initiated our modernity. The means to this new age are
also adumbrated in this miraculous dialogue; in his final assessment
of the Timaeus Peter Kalkavage says that “the likely story presents
the paradigm of what it would mean to use mathematical structures
to make flux intelligible—at least as intelligible as possible.”
Twenty-one centuries later the calculus will perfect these structures,
and so the science by which we live and which Plato has prefigured
will really take off. Read this introduction to get a sense of what it
means for a work to be great, to see deep into things and far into
time.
But better yet, read the splendid translation framed by the valuable apparatus. It is trustworthy; it sticks close to the text, word for
word. But it is also readable—not translaterese but good, lively, and
flexibly intoned English, since faithfulness in translation includes
preserving something of the literary quality of the original. This dialogue in particular is, for all the wild exuberance of its philosophical imagination, written in fresh, plain Greek, though plain terms
are often put to novel uses.—Would you expect to find Being,
Becoming, Same, Other, ordinary words with a gloss of high philosophy, in a cosmological context? Perhaps the best example is the
divine Craftsman. As the translator points out in the glossary, the
Greek word, which has passed into English as “demiurge,” merely
BRANN
121
means a skilled worker available for orders from the public, so it
was just right to preserve that sense with the plain English word. To
help with background knowledge, there are lots of footnotes right
on the page.
Here’s my recommendation, then: We have all these wonderful
alumni seminars around the country. Why not devote one here and
there to a reading of the Timaeus?—And perhaps some participants
might take advantage of Peter Kalkavage’s translation (which is,
incidentally, purposely inexpensive). I’d love to come and help, and
so, I imagine, would he.
*I can’t resist a footnote.
In our own last century, there have been droves of people,
many of them now active, who have fallen into Plato’s antiquarian
trap and gone in search of this lost continent. The description of the
island, which enormous geometrically planned public works have
transformed into something formidably awful, is set out in the dialogue Critias. Its Speer-like architecture (Speer was Hitler’s architect) appealed to the Nazis, whose mythmakers represented Atlantis
as an early Nordic utopia, to be rediscovered by state-sponsored
archaeologists. These people had at least got it right with respect to
the scariness of the drawn-and-quartered, brass-walled locale. Most
modern representations, be they in books, songs, or movies (of
which Disney’s “Atlantis” is the latest) are governed by the mistaken notion that Atlantis was meant to be a lost place of marvels and
beauties, a sort of mid-ocean Shangri-la. It’s actually a totalitarian
topography, the triumph of the will over nature.
�
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<em>The St. John's Review</em><span> is published by the Office of the Dean, St. John's College. All manuscripts are subject to blind review. Address correspondence to </span><em>The St. John's Review</em><span>, St. John's College, 60 College Avenue, Annapolis, MD 21401 or via e-mail at </span><a class="obfuscated_link" href="mailto:review@sjc.edu"><span class="obfuscated_link_text">review@sjc.edu</span></a><span>.</span><br /><br /><em>The St. John's Review</em> exemplifies, encourages, and enhances the disciplined reflection that is nurtured by the St. John's Program. It does so both through the character most in common among its contributors — their familiarity with the Program and their respect for it — and through the style and content of their contributions. As it represents the St. John's Program, The St. John's Review espouses no philosophical, religious, or political doctrine beyond a dedication to liberal learning, and its readers may expect to find diversity of thought represented in its pages.<br /><br /><em>The St. John's Review</em> was first published in 1974. It merged with <em>The College </em>beginning with the July 1980 issue. From that date forward, the numbering of <em>The St. John's Review</em> continues that of <em>The College</em>. <br /><br />Click on <a title="The St. John's Review" href="http://digitalarchives.sjc.edu/items/browse?collection=13"><strong>Items in the The St. John's Review Collection</strong></a> to view and sort all items in the collection.
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The St. John's Review, 2002/2
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Russell, George
Kraus, Pamela
Brann, Eva T. H.
Carey, James
Ruhm von Oppen, Beate
Sachs, Joe
Van Doren, John
Williamson, Robert B.
Zuckerman, Elliott
Phillips, Blakely
Sachs, Joe
David, Amirthanayagam
Berns, Lawrence
Maschler, Chaninah
Kirby, Torrance
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Volume XLVI, number two of The St. John's Review. Published in 2002.
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Annapolis, MD
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St. John's Review
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