Sam's latest posting encouraged me to submit to our forum the following
presentation by Reviel Netz about his book,
"The Shaping of Deduction in Greek Mathematics"
given at Harvard, April 1999. I must thank Reviel for his kindness in
sending me the paper appended below.
Warm regards from freezing Montevideo, Julio.
==========
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.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* *
* General View on the History of Science *
* *
* --------------------------------------------------------- *
* *
* Theory of the Emergence of the Deductive Method *
* *
* --------------------------------------------------------- *
* *
* Description of the Practices of Greek Mathematics *
* *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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,
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * *
* Historical * * Ways * * *
* Setting * _| * of Doing * _| * Achievement *
* * * Things * * *
* * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
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),
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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.)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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
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).
* * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * *
* Setting * _| * Competence * _| * Performance *
* * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * *
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.
==========