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See also a
conference of Reviel Netz about Greeck diagrams
A presentation of the
book, by Gibert Arsac is available on the
Proof website
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Presentation by Reviel Netz
about his book, given at Harvard, April 1999.
I have to apologise, this is not a "paper". What I want
to do now is really to introduce the book, to lead to a
discussion. I shall describe,"very quickly, what happens in
the book, especially following the selection I have
prepared.
I begin with a couple of very simplified schemes
for the book, which I shall gradually fill in with more
detail.
First, in my introduction I write that "this book can be
read on three levels: first as a description of the
practices of Greek mathematics; second as a theory of the
emergence of the deductive method; and third as a case-study
for a general view on the history of science.". This then
can be given as a scheme, as in the transparency.
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General View on the
History of Science
Theory of the Emergence of
the Deductive Method
Description of the
Practices of Greek Mathematics
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The three levels are in a way independent: the
description of the practice stands with or without the
theory of the emergence of deduction, and the theory of the
emergence of deduction stands with or without its further
generalization. The book is like a three-tiered investment
portfolio. The investment in the description of the practice
is the most solid, in itself it is the least controversial;
the theory of the nature of Greek mathematical deduction is
more controversial and risky, while the top and most general
level is also the riskiest. As the risks rise, so do the
stakes. The book tried to keep a certain balance,
essentially to stick to the most solid level of description.
It is a very theoretical book, but the theory is driven by
the descriptive level. In such a presentation, the balance
must to some extent be transformed and the theoretical
issues must come to the fore, must become more independent.
The nature of the investment changes. I accept this, and
even as I mention this evening some of the descriptions
offered in the book, I shall always stress their theoretical
implications - I shall stress the methodology. But please
bear in mind that this does imply a certain transformation
of the book.
So, to introduce in a very simplified way the
"theory" of the book, here is another similar simple
scheme,
which may capture the structure of the argument of the
book. Put in such simple terms, I try to show this: that a
certain historical setting led to a certain kind of
practice, a certain way of doing things; which in turn made
a certain achievement possible, namely deduction. This again
can be given as in the transparency. You see that I use the
phrase "way of doing things" as a synonym for "practice",
and I shall say more about this later on.
With the aid of the lower scheme we can now
unpack a little the upper scheme. At the most basic level,
the book is a description of the main element in my
explanatory scheme - it is a description of the Greek
mathematical practices, of the Greek mathematical ways of
doing things. It also offers a specific theory, that a
certain historical setting made Greek mathematicians write
in a certain way, a way which explains the achievement of
deduction; and it implies the general argument, that
scientific achievements are made possible by certain
specific ways of doing things, which in turn are explained
by historical settings. Of course this general argument, as
just stated, sounds almost tautological, but in fact my
notion of ways of doing things is more specific; what I
shall mainly do this evening is to give some indications for
my understanding of "ways of doing things".
As can be seen even from the table of contents,
the bulk of the book is indeed concerned with the Greek
mathematical way of doing things. Of the seven chapters
(transparency),
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Table of
contents:
Introduction
1
A specimen of Greek Mathematics 9
1 The Lettered Diagram 12
2 The Pragmatics of Letters 68
3 The Mathematical Lexicon 89
4 Formulae 127
5 The Shaping of Necessity 168
6 The Shaping of Generality 240
7 The Historical Setting 271
Appendix: The Main Greek Mathematicians Cited in
the Book 313
(Plus preface,
bibliography, etc.)
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the first four are mostly an analytical description of
elements of this way of doing things. The first four
chapters are wholly compressed within the middle box.
Chapters five and six go on to show how the achievement of
deduction was made possible by the Greek mathematical
practice, so they venture out of the middle box to the
right-hand arrow and to the right-hand box, but they still
add a lot to the middle box, a lot of the description of the
mathematical practice is offered in chapters five and six.
Chapter seven is the only chapter where I say relatively
little on the middle box itself; it deals with the
historical setting of the practice, and so it looks at the
left-hand box and at the left-hand arrow. You see that I
decided, in the arrangement of the chapters, to bring the
practices to the foreground, and for this reason I have
postponed chapter seven - which is in a sense the starting
point for the argument - till the end. I really wanted to
get as quickly as possible to my analytical descriptions of
practice, which are perhaps the main methodological feature
of the book. So let me move immediately on to those
analytical descriptions of practice.
There are two main components to the Greek
mathematical practice, the visual and the linguistic. The
first two chapters deal mainly with the visual - with
diagrams - and chapters three to four deal with
the linguistic - the technical, mathematical use of
language. I gave a sample of both, and I shall begin
with diagrams.
I have chosen to concentrate, in your sample, on one
central feature of the practice: how diagrams and text are
integrated. The intuition behind the question is very
simple. A text is accompanied by a diagram - this is the
universal rule in Greek mathematics - and in some way the
two must be correlated, so that they can belong together.
How is that done? What is the practice underlying that?
There are all sorts of forms such an integration of diagram
and text can take in principle. Which depends on which? What
is involved in reading a text with a diagram, in seeing a
diagram with a text? Well, it is clear that this integration
is somehow mediated by the diagrammatic letters - the alpha
and the beta in the text, which refer to points or lines in
the diagram.
Now I offered the following test. For each
letter, as it is introduced in the text, we may ask whether
it is fully determined by the text. For instance if we have
a circle, and we are told "let its centre be alpha", alpha
is fully determined - there is only one centre, that's it
(draw). I call such letters white letters (write). But now
imagine the following: "let the radius be alpha-beta" (draw)
- not a Greek expression, but never mind that - here there
is an underdeterminacy, not because there are infinitely
many possible radii - for the purposes of the proposition
this may be immaterial, so that "any radius" is a
sufficiently determinate object. But what's underdetermined
is the relative identity of alpha and beta - we don't know
which is which, which is the centre and which lies on the
circumference. This is underdetermined by the text and of
course determined by the diagram. I call such
underdetermined letters grey. And finally there are cases
such as these: "let the diameter be alpha-beta" (draw),
"therefore alpha-gamma equals gamma-beta". Here is a case of
a letter which is completely undetermined by the text,
gamma, which as far as the text is concerned appears out of
the blue. I therefore call it a blue letter (write). Clearly
we know what gamma means by the diagram alone - aided of
course by our general understanding of the mathematics
involved. But here clearly the balance moves from text to
diagram, the information is conveyed essentially by the
diagram.
Now I have done this sort of test, classifying
letters according to their textual determinacy, for two
Greek books - Apollonius' Conics Book I, and Euclid's
Elements' book XIII - a sample, but a substantial one, with
847 letters, 847 tests. And I have found that most letters
are grey, and a substantial number are blue, that is, very
often the text does not determine the identity of the
objects it refers to, and this identification is based on
the diagram.
This then was the starting point for what I had
to say on the practice of diagrams. I went on to show how
arguments actually rely on diagrams, how diagrams are
perceived as the metonym for mathematics and for a
mathematical proposition - so that when a Greek thought of a
mathematical proposition, he could probably think of it as a
diagram introduced by a text, rather than a text accompanied
by a diagram, etc. I also went on, in chapter two, to look
in detail on the practices concerning diagrammatic letters
as such - which letters do you chose and how are they
allowed to combine.
Generally speaking, I have tried to show not just that
diagrams are more important than usually thought, but
something slightly more complicated - that instead of
thinking of diagrams alone, or of texts alone, and then of a
relation between the two, we should think of a single
vehicle of information, the diagrammatic text or the
textualised diagram, which is the vehicle of information
used in Greek mathematics. This was their way of doing
things, of conveying information - not through text alone or
through diagram alone, but through a composite unit. I shall
return to say more about this notion of ways of doing
things, of practice.
But first, I want to say a bit more about my own
practice. I have given in more detail the case of the
analysis of determination of letters, because this not only
reveals something deep, I think, about the Greek
mathematical practice, but is also a very simple example of
the kinds of things I did in this book. I have throughout
employed tests of this kind, and I want to say something
about such tests.
One thing about this test is that it somehow
finds a fact. When you say "Archimedes was the greatest
mathematician in antiquity" you're probably saying something
true, yet this is not quite a fact, not quite something
solid, out there, transcending our own subjective
interpretations. But when you say that "of the 630 letters
in Apollonius' Conics Book I, only 268 are fully determined
by the text", this is solid. This is out there. I do not
deny that there is a lot which can be negotiated - there is
a lot to be cleared concerning the concept of
"determination", and of course I also make mistakes in my
tests. But then this is the way facts are: made by
negotiations and clarifications, liable to mistake. My main
pride really is that I have brought to the world, in this
book, a whole new set of facts, a new order of facts.
Further, and related to this, note that with
such tests I make the sources say things they did not intend
to say. Apollonius did not mean to say that he uses letters
in such and such a way. He meant to say something about
Conic sections. Perhaps this explains why I had to build all
those facts in ancient science. The corpus is limited, and
it is very unreflective. It is just obvious that there is
not a lot to get out of the characters themselves. Ask
Apollonius how he uses diagrams, and he is silent, he won't
say a word. The usual methods of interrogation fail. So we
need to devise new methods of interrogation, to make the
texts speak even against their will. We catch them unawares,
so to speak.
What does it mean? That we look at reality, at a level
underlying the conscious level of the practitioners, and
underlying the conscious level of the audience. For a
comparison, think, say, of prosody. Suppose we want to know
about Shakespeare's prosody. He does not say anything about
prosody, but we have got substantial evidence for his
prosody, namely his entire writing. Everything says
something about prosody. The texts may speak about kings,
their loves and violent deaths, and simultaneously they say
something, reveal something about Shakespeare's prosodic
practice - just as Apollonius' text may speak about conic
sections, and simultaneously reveal something about his ways
of conveying information. So we can find, for instance, in
Shakespeare, that inversions of stress - stress occurring at
the 'wrong' syllable - tends to occur much more often at the
beginning of the line than at its end. There is no reason to
think Shakespeare was aware of this, no more than Apollonius
was aware of the way in which he attached letters to points.
But both are facts, underlying what Shakespeare and
Apollonius did. Shakespeare, constructing effective iambic
pantameters, used stresses in a certain way, and not
another; Apollonius, constructing effective vehicles of
information, used letters in a certain way, and not another.
This is the description of the practice, then.
Now I add the following. It is clear, I hope, that it is
something about the arrangement of stresses which makes the
Shakespearean pentameter so effective, which leads to the
cognitive impact we know as the appreciation of meter. It
should be equally clear that it is something about the way
in which information is conveyed, which leads to the
cognitive impact we know as the appreciation of necessity,
of the deductive force of a claim. Over and above the
logical validity of an argument, we feel it as
deductive, as compelling, in precisely the same way in which
we feel that a line is a pentameter. We read a text, and as
we read it we are led to a perception, that it could not
be otherwise, a perception which is the essence of
deduction. Underlying all proof, no matter how logically
complicated, there is some such perception. Please notice
that I do not say that deduction is merely psychological. It
is a logical fact, that a proof proves its result. And it is
also a psychological fact, that when reading a proof, you
are convinced. As you read, you are led to what I shall now
informally call the AHA feeling, when you recognise the
compelling force of the argument. My question is, what goes
into this AHA feeling. My answer, roughly speaking, is that
this involves the fact that the information you need to
process is somehow simple and manageable, so that you are
able to see that it could not have been
otherwise. I shall clarify this a both later on, but I
stress once again: I do not at all deny the objective
validity of mathematics, I do note attempt to
reduce it to psychology. Mathematics is
logically valid. But the perception of logical validity,
like all other perception, is a psychological fact.
It should therefore be obvious why I describe
the practice in the way I do. My question is precisely, what
is the unreflective practice of deductive texts - what goes
into this special kind of perception, which is implicit in a
deductive text. And my claim is that by understanding this
reading, this practice, we shall explain the product -
deduction. This is the sense in which a way of doing things
is supposed to explain an achievement - this is the nature
of the right-hand arrow in my explanatory scheme (show).
It is of course impossible to compress the
account of this right-hand arrow into a brief presentation.
I shall very briefly mention some points along the way. One
aspect of this, to repeat, is the diagram: so after I show
that it is a real vehicle of information, not a mere
appendage, I go on to show what sort of information is taken
out of it - essentially, the information involving discrete
relations of objects in a plane, such as inclusion and
intersection; so the complexity of continuous space is
reduced to a small and manageable system of relations.
I move on to say more about this kind of
explanation, in the context of formulaic language - the next
item sampled in your reading.
Greek mathematical texts are written within a very
limited language, which is also very repetitive. It is not
only that you have a small vocabulary, but you repeatedly
use the very same expressions, the same formulae - this is
what I mean by the word formula, in this context. So the
texts are a system of formulaic expressions.
For instance, the typical expression for
proportion is roughly like "as the line AB to the line CD,
so the line EF to the line GH". This expression for
proportion is incidentally perhaps the most important
expression in Greek mathematics. This is a formula -
whenever you want to refer to proportions, you use an
expression like this. And notice that it is made of
formulae. The formula, as it were, governs other,
constituent formulae, as may be perceived in a tree
structure. There is a constituent structure to the formulaic
language of Greek mathematics. I offer here a simple way of
thinking of this proportion formula as a three-tier
constituent structure (I simplify a bit now: note that the
ultimate objects governed by the formulaic structure are
generally speaking formulae in turn, which may consist of
further simpler formulae).
The significance of this structure is that now
we may look again at a mathematical proof, and see it as a
sequence of operations upon such constituent structures,
upon such trees. Underlying the process of deduction, what
really happens, in some sense of "really", is that
constituent formulaic expressions are substituted inside
other, higher formulaic expressions. Take a simple and
typical derivation:
"Since it is: as the <square> on MY [Y
= psi] to the <square> on YI, so the
<rectangle contained> by APB to the <rectangle
contained> by DPE, but as the <rectangle
contained> by APB to the <rectangle contained>
by DPE, so the <square> on LT to the <square>
on TI, therefore also: as the <square> on MY to the
<square> on YI, so is the <square> on LT to
the <square> on TI.".
This may seem confusing, but once you are used to the
formulaic system, you are immediately led to read this off
as the transitivity of proportion. We moderns have a certain
visual typographic symbolism which brings out this
underlying structure. Greek writing was not visual and
typographic in the same way, it was just an uninterrupted
stream of letters, it looked like this: (transparency). So
clearly the underlying structure was not read of the text,
visually. What happened instead is that you interiorise a
system of formulaic expressions. It was through a certain
linguistic perception, of the expression as having a certain
constitutent structure, that you analysed this as a simple
derivation, where all you have to do is to substitute one
constituent by another.
Now, where did I get those trees from, those constituent
representations? I did not invent them, they are standard
tools in linguistic analysis. It is somehow a fact about
human language, that it is organised by constituent
structures. We do not just say one word after another, we
utter sentences which have a structure - a noun phrase
followed by a verb phrase, say, and each of the phrases has
its own structure, etc. This is how we utter expressions and
this is how we interpret them. The human linguistic
apparatus is based upon the perception of constituent
structures. Thus, we may make the following comparison
between modern and ancient mathematical representations.
Modern mathematical, visual typographic symbolism uses our
ability to recognize visual patterns, while the ancient
formulaic system of repeated phrases used the human ability
to perceive constituent structures. In both cases, then, a
certain cognitive resource is used in a systematic way, to
accomplish a specific task. In both cases, the cognitive
tool makes it possible to reduce a complex situation into a
discrete, manageable system, upon which you operate.
So this is another way of unpacking what I mean
by "ways of doing things". I mean specific ways of
mobilising human cognitive resources in systematic ways.
And, very simply, my thesis is that by mobilising
systematically the basic human cognitive resources, in
various specific ways, people were able to achieve various
specific tasks. This explains why I need to look at a level
below the conscious level - because the cognitive level
simply isn't conscious. It is precisely by taking your
resources for granted, that you use them. It is extremely
difficult to walk, consciously, deliberately to calculate
the motions of your feet and body. To explain senso-motoric
motion, you need to take it apart, to look at the level
which is no longer accessible to the practitioner. What I
tried to do is to take apart, in a similar way, the mental
motion of deduction.
So far I gave some very compressed accounts of
two isolated bits of my description of the practice - major
bits, but still only bits. For reasons of time, I shall not
say a lot more about the actual practice - I shall very
quickly say something on the following chunk of text in your
reading, on the structure of proofs. Once again, you can see
my fondness of diagrams - so I use tree-diagrams to
represent the flow of arguments in proofs. Some of you are
familiar with this from my presentation on the Aristotelian
paragraph. So I represent an argument of the form (draw)
1, therefore 2,
by the line
1 and 2, therefore 3, by the triangle
and 1, because 2, by the line
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And most of the system can be composed from those
components. I shall not go into any details of my analysis
of such trees, only mention my main conclusion: that the
Greek tree of proof is generally speaking a directly flowing
north-east sequence, with few complications, trying to keep
as closely as possible to the situation where every claim is
directly based on the immediately preceding claim. This is
far from being the only possible structure, and I argue that
it represents a certain ideal of proof, what may be called
the on-line proof, the proof where you are convinced on the
spot - not by some global understanding of the structure of
the argument, but by what's directly in front of you, that
is, I finally claim, this is the ideal of oral persuasion.
But I only suggest this now and must refer you to the text
for the actual argument.
So I give you a few snippets of the description of the
practice, and of course this is unsatisfactory. If what I do
in the book is to take apart the mental motion of deduction,
then naturally the book is in fact difficult to follow in
bits. It's like getting a description of walking, with, say,
only the raising of the feet. So far, with what I gave this
evening, my Greek mathematicians are still suspended in
mid-air. I have attempted in this book something like a
total description of the Greek mathematical practice, and
this totality is I think crucial. So I hope some of you go
on to read the book. But this will have to do now for the
right-hand arrow.
Even more quickly, let me say something about the
historical setting and the left-hand arrow.
What do I need to show here? I need to show,
what in the Greek historical setting made people write and
communicate in a certain way. I need to describe the nature
of mathematical communication and the expectations
surrounding it. Once again, I look at something like the
underlying material reality: who communicated with whom, in
what contexts, in what way?
Once again, I am trying to get
facts - the solid reality underlying history. So
I look first at the demography of Greek mathematics. I
discuss for instance the question of how many Greek
mathematicians there were. The point is, there really was a
certain number of them. There are of course problems of
definition, let alone problems of evidence, but the fact
remains that there was only a certain number, and I think it
is obvious that the nature of communication is very strongly
influenced by the numbers involved. I discuss the evidence
and come up with a certain guess, namely that throughout
antiquity there were about one thousand Greek
mathematicians, i.e. no more than a few dozens active
simultaneously. My point right now, once again, is
methodological. Perhaps I am wrong about the number. Perhaps
there were ten thousands or more, or perhaps only a few
hundreds. This is now a possible field of debate, and I look
forward to such a debate. But what I hope is less
controversial is that such demographic facts make great
difference; what Greek mathematics was, as a practice, would
have been very different with different numbers of
practitioners. I argue that this practice was essentially in
the form of written communication within a very small
network of practitioners spread around the Mediterranean; in
a sense, those practitioners were mostly auto-didacts.
I then go on to situate Greek mathematics in terms of the
cultural expectations surrounding it. I shall mention one
detail of this part of the chapter. In fact this may be a
contribution to the general question of the cultural
position of Greek intellectual life.
Following the work of Geoffrey Lloyd, we now
understand very well the position of Greek intellectual life
inside the public culture of the democratic polis, with the
stress on the ideal of oral persuasion. This picture is
especially valid for the fifth century BC, and is most
important as a background for the Pre-Socratics and for the
Hippocratic corpus. Since the mid-eighties, however, Greek
historiography has rediscovered aristocracy. It is now more
and more recognised that, however democratic the Greek polis
may have become, the aristocracy did not disappear. It
continued to have its own separate identity, which however
was partly defined in reaction to the polis. The Greek elite
was involved simultaneously in two systems of relations:
vertically, inside the polis, the oral culture of more or
less democratic debate; and horizontally, throughout the
Greek world, the written culture of aristocratic networks.
The point is not to put one system of relations above
another, but to see that both had acted simultaneously. The
Greek aristocrats were subject, simultaneously, to both the
centripetal and the centrifugal forces of the polis. The
significance of this for us, is that we may interpret the
communication of Greek mathematics as representing a
transformation, of the ideal of oral persuasion, into a much
more regimented, written form of an inward-looking group. In
a nutshell, this is the most important background I invoke
in explaining the Greek mathematical practice - so, for
instance, I point to the use of the visual, of apparently
transparent and public diagrams - which are inscribed by
writing, that is by letters, the tool of the literate elite;
or to the aural nature of the mathematical language, based
on speech and nothing else - but transformed into a rigid
form consisting of a limited number of formulaic
expressions, known to the experts and implicitly defining
them. Of course there is much else I offer in the book, both
in the practice and in the historical setting, but I do not
try to give now a full account. What I try to do is to give
a flavor, of how I hope to move from historical setting and
cultural expectations, to explain the practice.
So we've come back to "practice", to "ways of doing
things". I want now finally to clarify this a bit. So what
do I mean by "practice"?
Once again, what I try to do tonight is to
introduce a discussion and to try to clarify my position, so
I shall try to do this for the general question of practice.
for the discussion, I think the following distinction might
be useful, it might clarify my sense of practice which is
perhaps not so obvious.
So I suggest we use the Saussurian distinction
of parole and langue. This works like
this. In explaining what linguistics was about, Saussure
made a distinction between two orders of reality, both of
which may be conceived as "language". One is
parole - the set of particular linguistic
events, the sentences uttered by people or written by them,
all the speech acts. Right now what I produce is a piece of
parole. So this is one thing we may refer to when
thinking about language, this is one of orders of reality of
language. But beyond that, there is another order of
reality, that of langue. Langue is the set of
principles which govern my parole, the grammar of the
language in which I speak and which is shared by my
community. My last sentence had 26 words - this is a fact
about parole. And it was a grammatical sentence of
English - this is a fact about langue. Or, for
instance, the English word order is Subject-Verb-Object,
it's an SVO language; this is a fact about English as a
langue, and it is a fact over and above the many SVO
sentences in English parole, let alone the many non-SVO
sentences in English parole. It is simply a fact not in the
order of reality of parole, but in the order of
reality of langue. Chomsky adapts this distinction in
his competence versus performance distinction - competence
is the same as langue, performance is the same as
parole. The main difference is that Chomsky is much
more explicit about the location of the order of reality of
langue or of competence: for Chomsky, competence
resides in the mind, it is a fact about human knowledge. You
have a certain piece of knowledge in your mind, which is
knowledge of the English language, a certain competence; and
this competence may then be manifested by your performance.
So on the one hand, there are the products themselves, the
production of utterances, which is parole or
performance; on the other hand, is what underlies such
production, a certain langue or competence - a
certain set of principles, a grammar, a knowledge. Not of
course explicit knowledge - no one knows the grammar of
one's own language explicitly - but an implicit knowledge,
an unconscious knowledge.
It is obvious how this kind of distinction can
be extended beyond language itself. It is somehow something
about people, that they do not just react in blind and
chaotic ways to what's outside them. People, in general, do
things in certain ways and not in other ways, because that's
how they know to do them; because they have interiorised a
certain grammar. So, in this metaphorical sense, underlying
human activity, in whatever field, underlying each set of
parole, we may think of its own langue. This
is the conviction informing my study. So you can see why I
prefer the cumbersome phrase, "ways of doing things", to the
word "practice". Because I study practice in the sense of
langue, not parole. I try to get at the
principles underlying what people do - not the things they
do, then, but their ways of doing things.
To conclude, then, I can now present this explanatory
scheme very simply. The deduction effect is a feature of the
Greek mathematical text, a feature of the product, of the
performance - explained by the competence described through
the book. In other words, the explanatory scheme is:
(transparency).
A certain historical setting, leads to a certain
combination of practices, to a certain competence, which
then of course explains the performance and its features. I
have now got a truly simple scheme; and if it appears
obvious to the point of tautology, this, I hope, is because
it captures a simple and obvious truth about human
achievements and their explanation.
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