433 lines
22 KiB
HTML
433 lines
22 KiB
HTML
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
|
||
|
|
<HTML>
|
||
|
|
<HEAD>
|
||
|
|
<TITLE>Mathematical entities in Plato's Republic</TITLE>
|
||
|
|
<META NAME="Author" CONTENT="Bernard F. SUZANNE">
|
||
|
|
<META NAME="GENERATOR" CONTENT="Mozilla/3.01Gold (Win16; I) [Netscape]">
|
||
|
|
</HEAD>
|
||
|
|
<BODY TEXT="#000000" BGCOLOR="#FFFFFF" LINK="#0040A0" VLINK="#006060" ALINK="#800000">
|
||
|
|
|
||
|
|
<TABLE BORDER="0" WIDTH="100%">
|
||
|
|
<TR>
|
||
|
|
<TD ALIGN="LEFT">
|
||
|
|
<FONT SIZE="-1">
|
||
|
|
© 1996
|
||
|
|
<A HREF="mailto:bernard.suzanne@polytechnique.org">Bernard SUZANNE </A>
|
||
|
|
</FONT>
|
||
|
|
</TD>
|
||
|
|
|
||
|
|
<TD ALIGN="RIGHT">
|
||
|
|
<FONT SIZE="-1">
|
||
|
|
Last updated November 21, 1998
|
||
|
|
</FONT>
|
||
|
|
</TD>
|
||
|
|
</TR>
|
||
|
|
|
||
|
|
<TR>
|
||
|
|
<TD COLSPAN="2" ALIGN="CENTER"> <FONT SIZE=-1>Plato and his dialogues :
|
||
|
|
<A HREF="../plato.htm">Home</A> - <A HREF="../life.htm">Biography</A> -
|
||
|
|
<A HREF="../works.htm">Works</A> and <A HREF="../links.htm">links to them</A>
|
||
|
|
- <A HREF="../intrpret.htm">History of interpretation</A> - <A HREF="../hypotsis.htm">New
|
||
|
|
hypotheses</A> - Map of dialogues : <A HREF="../tetralog.htm#tetramap">table
|
||
|
|
version</A> or <A HREF="../tetrapre.htm#tetramap">non tabular version</A>.
|
||
|
|
Tools : <A HREF="../tools/index.htm">Index of persons and locations</A>
|
||
|
|
- <A HREF="../tools/chrono.htm">Detailed </A> and <A HREF="../tools/synopsis.htm">synoptic</A>
|
||
|
|
chronologies - <A HREF="../tools/mapindex.htm">Maps of Ancient Greek World</A>.
|
||
|
|
Site information : <A HREF="../suzanne.htm">About the author</A>. </FONT>
|
||
|
|
</TD>
|
||
|
|
</TR>
|
||
|
|
</TABLE>
|
||
|
|
|
||
|
|
<HR>
|
||
|
|
|
||
|
|
<TABLE BORDER="0" WIDTH="100%">
|
||
|
|
<TR>
|
||
|
|
<TD ALIGN=LEFT WIDTH="20%"><A HREF="../hypotsis.htm#structured"><IMG SRC="../images/logo.gif" HEIGHT=84 WIDTH=128 BORDER=0></A></TD>
|
||
|
|
|
||
|
|
<TD align="center">
|
||
|
|
<H2>E-mail Archives :<BR>
|
||
|
|
Mathematical entities in Plato's <I>Republic</I></H2>
|
||
|
|
|
||
|
|
<P><FONT SIZE=+1>August 13-15, 1996</FONT></P>
|
||
|
|
</TD>
|
||
|
|
|
||
|
|
<TD ALIGN=RIGHT WIDTH="20%"><A HREF="../plato.htm"><IMG SRC="../images/plato83.gif" HEIGHT=83 WIDTH=65 BORDER=0></A>
|
||
|
|
</TD>
|
||
|
|
</TR>
|
||
|
|
</TABLE>
|
||
|
|
|
||
|
|
<HR>
|
||
|
|
|
||
|
|
<P><FONT SIZE="-1">This page is part of the "e-mail archives" section of a site, <A HREF="../plato.htm">Plato and his dialogues</A>, dedicated to developing a new interpretation of Plato's dialogues.
|
||
|
|
The "e-mail archives" section includes HTML edited versions of posts that I submitted on various e-mail discussion lists about Plato and ancient philosophy.
|
||
|
|
</FONT></P>
|
||
|
|
|
||
|
|
<P><I>From: Yves Bastarache <Yves_Bastarache@itr.qc.ca><BR>
|
||
|
|
To: plato <PLATO@freelance.com><BR>
|
||
|
|
Date: August 13, 1996 22:38:32<BR>
|
||
|
|
Subject: mathematical entities in the simile of the line</I></P>
|
||
|
|
|
||
|
|
<P><I>First of all I beg your comprehension for my poor english. I am a
|
||
|
|
francophone. I read well your language but I have some difficulties to
|
||
|
|
write it.</I></P>
|
||
|
|
|
||
|
|
<P><I>I would be very grateful if someone could explain me the status of
|
||
|
|
mathematical entities in the simile of the line. They puzzle me when I
|
||
|
|
try to explain them to my students. The passage from images to sensibles
|
||
|
|
objects seem to me quite clear. I can also make sense of the line when
|
||
|
|
I link sensibles objects to Forms. How should I understand the place and
|
||
|
|
role of math. entities? If possible, join exemples to your explication.</I></P>
|
||
|
|
|
||
|
|
<P><I>Thanks in advance!</I></P>
|
||
|
|
|
||
|
|
<P><I>Yves Bastarache<BR>
|
||
|
|
Trois-Rivières, Québec.</I></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR WIDTH="100%">To: plato <plato@freelance.com><BR>
|
||
|
|
Date : August 14, 1996, 09:21:18<BR>
|
||
|
|
Subject: Re: mathematical entities in the simile of the line</P>
|
||
|
|
|
||
|
|
<P>Yves,</P>
|
||
|
|
|
||
|
|
<P>To answer your request, I think there are two things you must notice:</P>
|
||
|
|
|
||
|
|
<P>1) Whereas Plato describes the two segments of the visible in termes
|
||
|
|
of <B>beings</B> (images and "originals"), he only describes
|
||
|
|
the two segments of the intelligible in terms of <B>processes</B> (from
|
||
|
|
axioms taken for granted <B>down</B> to consequences and from hypotheses
|
||
|
|
to broader hypotheses all the way <B>up</B> to the first principle);</P>
|
||
|
|
|
||
|
|
<P>2) He only uses mathematics <B>as an example</B>!</P>
|
||
|
|
|
||
|
|
<P>Plato doesn't care about mathematics and mathematical entities <I>per
|
||
|
|
se</I>. He only uses them as the most readily available example of abstract
|
||
|
|
constructs, as a "gymnastic of the mind". For instance, in the
|
||
|
|
<I>Meno</I>, they provide, for a guy who only cares about hard facts, an
|
||
|
|
experimental proof of 1) the difference between opinion and knowledge and
|
||
|
|
2) the fact that you can look for something you don't know yet, and find
|
||
|
|
it even if you don't have the words to say it yet (the slave doesn't know
|
||
|
|
the word "diagonal" until after he has found the answer, and
|
||
|
|
it is the last word of Socrates to him once he <B>knows</B> what it is).</P>
|
||
|
|
|
||
|
|
<P>In the simile of the line, they provide for an <B>example</B> of a way
|
||
|
|
of reasoning. But Plato's ultimate purpose is not to locate "mathematical
|
||
|
|
entities" on one segment, and least of all to make them one of the
|
||
|
|
segments. Cursed be Aristotle who messed all this up!...</P>
|
||
|
|
|
||
|
|
<P>So, if mathematics don't play their role of example and cloud the issue
|
||
|
|
rather than clarify it, forget about them and look for other examples.</P>
|
||
|
|
|
||
|
|
<P>Let me try with another example, much more in line (if you allow me
|
||
|
|
the pun!) with what is Plato's ultimate goal, to know what it is to be
|
||
|
|
a man ("know thyself...").</P>
|
||
|
|
|
||
|
|
<P>1) First there is my image in a mirror, a picture of me taken by somebody
|
||
|
|
else, or, if I were rich enough and famous, a painting by some future Rembrant
|
||
|
|
or Van Gogh: no problem there, we are in the first segment, looking at
|
||
|
|
"images" of one man.</P>
|
||
|
|
|
||
|
|
<P>2) Next, there is you and me and all the list members, and my wife and
|
||
|
|
kids, and my neighbours walking in the street, and... : no problem either
|
||
|
|
there, these are "real" men and women in time and space.</P>
|
||
|
|
|
||
|
|
<P>And now, <I>en route</I> for the harder part!</P>
|
||
|
|
|
||
|
|
<P>3) Then, there is the "concept" of man, the thing I refer
|
||
|
|
to when using the word "man". It is to all men what "square"
|
||
|
|
is to all squares I may reason upon. It is born from the "images"
|
||
|
|
I get through my senses of actual men and put as an unproven "hypothesis"
|
||
|
|
that I can analyse but that is "evident" for everybody (even
|
||
|
|
for Aristotle!). It is an animal with two arms, two legs, a head and so
|
||
|
|
on. It is endowed with the ability to speak and think, and has <I>logos</I>,
|
||
|
|
whatever that be. But it doesn't give me yet the <B>principle</B> of "man"!
|
||
|
|
It doesn't tell me what it is to be a man. The point is, I am perfectly
|
||
|
|
able to recognize a man when I see one, as I could recognize a square,
|
||
|
|
in this world of becoming, but I don't <B>know</B> yet the ultimate truth
|
||
|
|
about man.</P>
|
||
|
|
|
||
|
|
<P>And if you think <B>this</B> is the <B>idea</B> of man, then you are
|
||
|
|
open to Aristotle (and Plato's, see the <I>Parmenides</I>) argument of
|
||
|
|
the third man. It may be a "form" of man, but certainly not the
|
||
|
|
ultimate <B>idea</B> of man, for Plato at least, in my humble opinion...
|
||
|
|
This, and all like constructs of our mind, exist only as "images"
|
||
|
|
of what we see and feel with our senses. We "put" them without
|
||
|
|
futher demonstration and deem them "evident".</P>
|
||
|
|
|
||
|
|
<P>4) So, what is left? What is left is whatever might ultimately answer
|
||
|
|
the question "what is it to be a man?" And this, for Plato, I
|
||
|
|
think, is <B>the idea of justice</B> as described at length in the <I>Republic</I>,
|
||
|
|
and as "evoqued" by a summary of the <B>principles</B> of the
|
||
|
|
<I>Republic</I> at the start ot the <I>Timæus</I>, <B>before the
|
||
|
|
myth starts</B>, to better show it is outide space and time, before other
|
||
|
|
"forms" of man will be described within the myth, within time
|
||
|
|
and space: the "form" of matter he is made of, the form of his
|
||
|
|
body, the form of his soul.</P>
|
||
|
|
|
||
|
|
<P>Such an idea doesn't stem from images of "live" entities in
|
||
|
|
the visible world. It is a pure idea. A just man, be it Socrates, is no
|
||
|
|
more "justice" than a beautiful girl is <I>to kalon</I> (see
|
||
|
|
<I>Hippias Major</I>). And, if it is an "image" of justice, it
|
||
|
|
is in a very remote sense. It is rather the "participation" of
|
||
|
|
a visible being to the "idea" that will in the end give him purpose,
|
||
|
|
unity and "being", but not being in the world of becoming, in
|
||
|
|
time and space, rather "being" in eternity...</P>
|
||
|
|
|
||
|
|
<P>And the only way we may reach this "idea" is by going from
|
||
|
|
idea to idea from the "hypothesis" of man up to virtue, beauty,
|
||
|
|
courage, and the like, from the "hypothesis" of man to the new
|
||
|
|
hypothesis of "soul" to the structure of the soul and to the
|
||
|
|
larger hypothesis of <I>logos</I>, and so on, all that under the "light"
|
||
|
|
of the idea of the "good" as the leading principle, until I figure
|
||
|
|
out what the true "good" of man is, and I find it in the idea
|
||
|
|
of "justice", but a justice that is no longer limited to "social"
|
||
|
|
justice, but starts as "internal" justice between the various
|
||
|
|
parts of my being, of my soul, as a foundation for justice in my relationship
|
||
|
|
with fellow men.</P>
|
||
|
|
|
||
|
|
<P><B>This</B> is what Plato cares for! And if you stumble on some of the
|
||
|
|
"blocks" he uses to pave the way in trying to help, forget about
|
||
|
|
those blocks and keep going. Those who, like Aristotle, drawn in the river
|
||
|
|
of mathematics and never reach the "promised land" are the same
|
||
|
|
who miss the end of Atlantis' story by Critias and never get to work on
|
||
|
|
the <I>Laws</I>... </P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR WIDTH="100%"></P>
|
||
|
|
|
||
|
|
<P><I>From: Michael Chase <GOYA@UVVM.UVIC.CA><BR>
|
||
|
|
To: plato <plato@freelance.com><BR>
|
||
|
|
Date : August 14, 1996, 11:24:48<BR>
|
||
|
|
Subject: mathematical entities </I>[original in French, my translation]</P>
|
||
|
|
|
||
|
|
<TABLE CELLSPACING=5 WIDTH="100%" VALIGN="TOP" >
|
||
|
|
<TR>
|
||
|
|
<TD WIDTH="50%"><I>Cher Yves,</I></TD>
|
||
|
|
|
||
|
|
<TD WIDTH="50%">Dear Yves,</TD>
|
||
|
|
</TR>
|
||
|
|
|
||
|
|
<TR>
|
||
|
|
<TD VALIGN=TOP WIDTH="50%"><I>Vous avez récemment posé la
|
||
|
|
question de savoir ce qu'il en était des objets mathématiques
|
||
|
|
de Platon. Vous y avez déjà reçu de très utiles
|
||
|
|
réponses, qui contiennent, sans doute, tout ce qu'il serait utile
|
||
|
|
d'enseigner à une classe d'étudiants débutants.</I></TD>
|
||
|
|
|
||
|
|
<TD VALIGN=TOP WIDTH="50%">You recently asked what was the status of Plato's
|
||
|
|
mathematical objects. You already received several quite useful answers,
|
||
|
|
containing, no doubt, all that should be taught to a class of beginners.</TD>
|
||
|
|
</TR>
|
||
|
|
|
||
|
|
<TR>
|
||
|
|
<TD VALIGN=TOP WIDTH="50%"><I>Mais il n'en reste pas moins, à mon
|
||
|
|
avis, que la démarche de M. Suzanne est totalement illégitime.
|
||
|
|
Nous ne pouvons pas ne pas prendre en considération les longs développements
|
||
|
|
qu'Aristote consacre à la théorie platonicienne des nombres
|
||
|
|
idéaux. En effet, le Stagirite nous dit, en détail et à
|
||
|
|
maintes reprises, que Platon enseignait l'existence d'un domaine de l'existence
|
||
|
|
intermédiaire entre les Idées et le monde sensible; domaine
|
||
|
|
constitue précisément par les Nombres Idéaux. Aristote
|
||
|
|
nous dit </I>expressis verbis<I> que Platon avait transmis ces doctrines
|
||
|
|
dans son enseignement oral. Or même si l'on pense, avec M. Suzanne,
|
||
|
|
qu'Aristote etait - quoi? Bête? Fou? Délirant? - il n'est
|
||
|
|
pas possible de n'accorder aucun crédit à la tradition millénaire
|
||
|
|
du Néoplatonisme, dont les représentants - depuis Jamblique,
|
||
|
|
Syrianus et Proclus jusqu'à Michel Psellus - ont développé
|
||
|
|
ces aspects de la doctrine platonicienne, notamment en ce qui concerne
|
||
|
|
le rapport étroit qui lie ces choses mathématiques au niveau
|
||
|
|
psychique de la réalité.</I></TD>
|
||
|
|
|
||
|
|
<TD VALIGN=TOP WIDTH="50%">Yet, there remains the fact that, in my opinion,
|
||
|
|
M. Suzanne's approach is totally illegitimate. We cannot not take
|
||
|
|
into account the lengthy developments that Aristotle devotes to the Platonic
|
||
|
|
theory of ideal numbers. Indeed, the Stagirian tells us, in details and
|
||
|
|
numerous times, that Plato was teaching the existence of a sphere of beings
|
||
|
|
intermediate between the forms and the visible world, a sphere of beings
|
||
|
|
precisely made up of the Ideal Numbers. Aristotle tells us <I>expressis
|
||
|
|
verbis</I> that Plato transmitted these doctrines in his oral teaching.
|
||
|
|
Yet, even if one thinks with M. Suzanne that Aristotle was - what? Dumb?
|
||
|
|
Fool? Delirious? - it is impossible not to give creance to the millenary
|
||
|
|
tradition of Neoplatonism, whose followers - starting with Jamblicus, Syrianus
|
||
|
|
and Proclus down to Michel Psellus - developed those aspects of platonician
|
||
|
|
doctrine, especially as regards the close relationship between those mathematical
|
||
|
|
beings and the psychical dimension of reality.</TD>
|
||
|
|
</TR>
|
||
|
|
|
||
|
|
<TR>
|
||
|
|
<TD VALIGN=TOP WIDTH="50%"><I>Recevez, cher Yves, mes meilleurs salutations
|
||
|
|
depuis la côte Pacifique, de la part de</I></TD>
|
||
|
|
|
||
|
|
<TD VALIGN=TOP WIDTH="50%">Please accept, dear Yves, my best greetings
|
||
|
|
from the Pacific coast, from</TD>
|
||
|
|
</TR>
|
||
|
|
|
||
|
|
<TR>
|
||
|
|
<TD WIDTH="50%"><I>Michael Chase<BR>
|
||
|
|
University of Victoria. </I></TD>
|
||
|
|
|
||
|
|
<TD WIDTH="50%">Michael Chase<BR>
|
||
|
|
University of Victoria</TD>
|
||
|
|
</TR>
|
||
|
|
</TABLE>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR WIDTH="100%"></P>
|
||
|
|
|
||
|
|
<P>To: plato <plato@freelance.com><BR>
|
||
|
|
Date : August 15, 1996, 10:51:12<BR>
|
||
|
|
Subject: Re: mathematical entities</P>
|
||
|
|
|
||
|
|
<P>Dear Yves and Michael,</P>
|
||
|
|
|
||
|
|
<P>my purpose, in the answer to Yves, was not to dismiss the constant and
|
||
|
|
sustained interest Plato always had in mathematics, but only to put it
|
||
|
|
in perspective with respect to what was foremost to him, the "know
|
||
|
|
thyself".</P>
|
||
|
|
|
||
|
|
<P>Obviously Plato spent a large share of his time dealing with mathematics
|
||
|
|
and "sciences" in general, and everybody knows the Academy hosted
|
||
|
|
renowed mathematicians during his lifetime. In the <I>Timæus</I>,
|
||
|
|
Plato is the first one (at least the first whose works are extant) to offer
|
||
|
|
a mathematical model of matter (based on triangles). And if the model is
|
||
|
|
largely outdated by now, the process that leads to it, the way of approaching
|
||
|
|
things, is more than ever relevant today (and I was stressing in my answer
|
||
|
|
to Yves that Plato defines the two segments of the intelligible in terms
|
||
|
|
of <B>processes</B>, not <B>entities</B>). The <B>Timæus</B> may
|
||
|
|
be viewed, from this standpoint, as an example of the <B>descending</B>
|
||
|
|
approach (the one in the first segment of intelligible) starting from the
|
||
|
|
smallest possible number of simple hypotheses to build from them a <B>scientific
|
||
|
|
model</B> of the universe. Yet, it doesn't move upward toward the anhypotetical
|
||
|
|
principle, because triangles are neither good nor bad, neither just nor
|
||
|
|
injust. They are, that's all. That's one more way to say that matter as
|
||
|
|
such is neither good nor bad, but simply subject to its own laws; it is
|
||
|
|
necessity, <I>anagkè</I>, with which you have to make do.</P>
|
||
|
|
|
||
|
|
<P>For Plato, mathematics are an example of abstraction proving, as others
|
||
|
|
with whom I fully agree said in their answers to Yves, that the whole of
|
||
|
|
being is not limited to material, sensible beings, to what is within time
|
||
|
|
and space. And the <I>theôria</I> of the world is to us an example
|
||
|
|
of order, of <I>kosmos</I>, we should use as a <B>model</B> to bring order
|
||
|
|
to our cities and our lives through laws stemming from our <I>logos</I>.
|
||
|
|
For that reason, it all deserves our consideration and the time Plato and
|
||
|
|
his pupils spend on it. But, and that was my point, this is still no more
|
||
|
|
that <B>means</B> toward a higher end, the building of man in time and
|
||
|
|
space. Mathematics, like all other sciences and technics, are <B>neutral</B>.
|
||
|
|
They don't tell us how we should use them for good or bad. This is the
|
||
|
|
whole message of the <I>Hippias minor</I>, and already of the <I>Charmides</I>.
|
||
|
|
That is why they only belong to the first segment of the intelligible.</P>
|
||
|
|
|
||
|
|
<P>That Plato spent a lot of time trying to categorize the various orders
|
||
|
|
of "reality"; that, while doing so, he gave a key role to mathematical
|
||
|
|
constructs he used, as I already said, as <B>examples</B>; this is more
|
||
|
|
than likely. That, in this work, discussion with students and colleagues,
|
||
|
|
including Aristotle, lead him to overstate his case, might not be surprising.
|
||
|
|
But I doubt very much that he held mathematical entities as the ultimate
|
||
|
|
reality, as the highest order of "ideas". This would seem to
|
||
|
|
me in complete contradiction with everything he says elsewhere and with
|
||
|
|
Socrates influence on him and what he tells us in his "intellectual
|
||
|
|
autobiography" in the <I>Phædo</I>. </P>
|
||
|
|
|
||
|
|
<P>About Aristotle now. I don't consider Aristotle as a dumb, or a fool,
|
||
|
|
or a mad person, but as one of Plato's brightest students, and, I might
|
||
|
|
add, as a "good student" with all the drawback this implies.
|
||
|
|
I believe that, despite all his gifts, Aristotle was unable to follow Plato
|
||
|
|
all the way "upward" where Plato reached. Aristotle was full
|
||
|
|
of "common sense" and most likely received a solid "scientific"
|
||
|
|
education owing to his familial origins. But he was also too much of a
|
||
|
|
"materialist", too keen of "hard facts" to follow Plato
|
||
|
|
to the level of abstraction he wanted to lead him to. Besides, he was too
|
||
|
|
fond of "showing up", too anxious to give the right answers,
|
||
|
|
to show that he understood. Plato spent his life trying to help students
|
||
|
|
find <B>within themselves</B> answers to the ultimate questions. Aristotle
|
||
|
|
on the other hand spent his life trying to give others the right answers.
|
||
|
|
Aristotle, as Taylor said somewhere (using the French expression) was a
|
||
|
|
"platonist <I>malgré lui</I>". He tried to folow Plato's
|
||
|
|
approach of "modeling" the universe, but, as often arrives still
|
||
|
|
nowadays to scientists, it took the model for the real thing, he was unable
|
||
|
|
to make the difference between images and what they were images of, a difference
|
||
|
|
that Plato fully understood. Aristotle, true to his physician's ascent,
|
||
|
|
didn't understand that the "idea of man" was not a DNA molecule,
|
||
|
|
that he would call "entelechy", or an "image" describing
|
||
|
|
the "form", the "structure" of man as it might be found
|
||
|
|
under the scalpel of a surgeon, but the "ideal" man must "participate"
|
||
|
|
into in order to become what he is meant to be, an ideal of justice capable
|
||
|
|
of "building" man because it alone can bring to the lump of matter
|
||
|
|
he is made of a unity that doesn't disappear in death, as proves Socrates
|
||
|
|
still well and "alive" in the dialogues... </P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR></P>
|
||
|
|
|
||
|
|
<P>To: plato <plato@freelance.com><BR>
|
||
|
|
Date : August 15, 1996, 19:27:17<BR>
|
||
|
|
Subject: Re: mathematical entities</P>
|
||
|
|
|
||
|
|
<P>Nicholas Denyer writes, in answer to my statement that mathematics,
|
||
|
|
for Plato, were "neutral", neither good nor bad:</P>
|
||
|
|
|
||
|
|
<P><I>> Aristoxenus </I>Harm. El.<I> 2.30 "</I>They came<I> [to
|
||
|
|
Plato's lecture on the Good] </I>in the conviction that they would get
|
||
|
|
some one or other of the things that the world calls good: riches, or health,
|
||
|
|
or strength. But when they found that Plato's reasonings were of mathematics
|
||
|
|
their disenchantment was complete<I>" (trans. Ross).</I></P>
|
||
|
|
|
||
|
|
<P>This may simply mean that Plato, in order to lead his audience toward
|
||
|
|
the <B>idea</B> of the good had to use the detour via mathematics as a
|
||
|
|
path toward abstraction, which is what we have been saying all along. It
|
||
|
|
doesn't necessarily mean that mathematics <B>were</B> the good.</P>
|
||
|
|
|
||
|
|
<P><I>> and also of the long passage in </I><A HREF="http://www.perseus.tufts.edu/hopper/text?doc=plat.+gorg.+507a">Gorgias<I>
|
||
|
|
507a-508c</I></A><I>, especially 508a: "</I>You do not realise that
|
||
|
|
geometrical equality has great force amongst both men and gods. Instead,
|
||
|
|
you think you should go in for greedy acquisition, for you neglect geometry.<I>"</I></P>
|
||
|
|
|
||
|
|
<P>This passage is all about justice as the goal of man, and fits very
|
||
|
|
well with what I have been saying. Man must put <B>measure</B> in his life,
|
||
|
|
and geometry gives <B>examples</B> of measure and mean. But it is because
|
||
|
|
I know in the first place what justice is and how important to man it is
|
||
|
|
that I may take the example of the geometrical equality to illustrate it,
|
||
|
|
not the other way around. In other words, it is not the geometrical equality
|
||
|
|
which tells us it is better, it is us who use it as a means of measuring
|
||
|
|
a good we have found elsewhere. <B>In itself</B>, the geometrical equality
|
||
|
|
is neither better nor worse than the arithmetical. It only help express
|
||
|
|
and illustrate a good that I must have found elsewhere. So, granted, it
|
||
|
|
may "participate" to the good this way *in our minds*, but it
|
||
|
|
doesn't lead to the good if we don't know it yet.</P>
|
||
|
|
|
||
|
|
<P>Besides, it is very possible that Socrates is here making fun of Callicles
|
||
|
|
who despises philosophy in playing the "scientist" in front of
|
||
|
|
him, even if he believes what he says in a certain way. There is a share
|
||
|
|
of truth even in the theory of measurement of pleasures and pains Socrates
|
||
|
|
develops at the end of the <I>Protagoras</I>, even if it shouldn't be taken
|
||
|
|
litterally as Protagoras himself might interpret it. But numbers alone
|
||
|
|
are not by themselves the good... </P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR WIDTH="100%"></P>
|
||
|
|
|
||
|
|
<CENTER>
|
||
|
|
<P> <FONT SIZE=-1>Plato and his dialogues : <A HREF="../plato.htm">Home</A>
|
||
|
|
- <A HREF="../life.htm">Biography</A> - <A HREF="../works.htm">Works</A> and
|
||
|
|
<A HREF="../links.htm">links to them</A> - <A HREF="../intrpret.htm">History
|
||
|
|
of interpretation</A> - <A HREF="../hypotsis.htm">New hypotheses</A> - Map
|
||
|
|
of dialogues : <A HREF="../tetralog.htm#tetramap">table version</A> or <A HREF="../tetrapre.htm#tetramap">non
|
||
|
|
tabular version</A>. Tools : <A HREF="../tools/index.htm">Index of persons
|
||
|
|
and locations</A> - <A HREF="../tools/chrono.htm">Detailed </A> and <A HREF="../tools/synopsis.htm">synoptic</A>
|
||
|
|
chronologies - <A HREF="../tools/mapindex.htm">Maps of Ancient Greek World</A>.
|
||
|
|
Site information : <A HREF="../suzanne.htm">About the author</A>. </FONT>
|
||
|
|
</P>
|
||
|
|
</CENTER>
|
||
|
|
|
||
|
|
<CENTER><P><FONT SIZE=-1>First published December 15, 1996 ;
|
||
|
|
Last updated November 21, 1998<BR>
|
||
|
|
© 1996 <A HREF="mailto:bernard.suzanne@polytechnique.org">Bernard SUZANNE </A>
|
||
|
|
(click on name to send your comments via e-mail) <BR>
|
||
|
|
<I>Quotations from theses pages are authorized provided they mention the
|
||
|
|
author's name and source of quotation (including date of last update).
|
||
|
|
Copies of these pages must not alter the text and must leave this copyright
|
||
|
|
mention visible in full.</I> </FONT></P></CENTER>
|
||
|
|
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|