© 1996 Bernard SUZANNE Last updated November 21, 1998
Plato and his dialogues : Home - Biography - Works and links to them - History of interpretation - New hypotheses - Map of dialogues : table version or non tabular version. Tools : Index of persons and locations - Detailed and synoptic chronologies - Maps of Ancient Greek World. Site information : About the author.

E-mail Archives :
Mathematical entities in Plato's Republic

August 13-15, 1996

This page is part of the "e-mail archives" section of a site, Plato and his dialogues, 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.

From: Yves Bastarache <Yves_Bastarache@itr.qc.ca>
To: plato <PLATO@freelance.com>
Date: August 13, 1996 22:38:32
Subject: mathematical entities in the simile of the line

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 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.

Thanks in advance!

Yves Bastarache
Trois-Rivières, Québec.

To: plato <plato@freelance.com>
Date : August 14, 1996, 09:21:18
Subject: Re: mathematical entities in the simile of the line


To answer your request, I think there are two things you must notice:

1) Whereas Plato describes the two segments of the visible in termes of beings (images and "originals"), he only describes the two segments of the intelligible in terms of processes (from axioms taken for granted down to consequences and from hypotheses to broader hypotheses all the way up to the first principle);

2) He only uses mathematics as an example!

Plato doesn't care about mathematics and mathematical entities per se. He only uses them as the most readily available example of abstract constructs, as a "gymnastic of the mind". For instance, in the Meno, 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 knows what it is).

In the simile of the line, they provide for an example 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!...

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.

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...").

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.

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.

And now, en route for the harder part!

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 logos, whatever that be. But it doesn't give me yet the principle 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 know yet the ultimate truth about man.

And if you think this is the idea of man, then you are open to Aristotle (and Plato's, see the Parmenides) argument of the third man. It may be a "form" of man, but certainly not the ultimate idea 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".

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 the idea of justice as described at length in the Republic, and as "evoqued" by a summary of the principles of the Republic at the start ot the Timæus, before the myth starts, 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.

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 to kalon (see Hippias Major). 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...

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 logos, 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.

This 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 Laws...

From: Michael Chase <GOYA@UVVM.UVIC.CA>
To: plato <plato@freelance.com>
Date : August 14, 1996, 11:24:48
Subject: mathematical entities
[original in French, my translation]

Cher Yves, Dear Yves,
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. 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.
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 expressis verbis 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. 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 expressis verbis 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.
Recevez, cher Yves, mes meilleurs salutations depuis la côte Pacifique, de la part de Please accept, dear Yves, my best greetings from the Pacific coast, from
Michael Chase
University of Victoria.
Michael Chase
University of Victoria

To: plato <plato@freelance.com>
Date : August 15, 1996, 10:51:12
Subject: Re: mathematical entities

Dear Yves and Michael,

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".

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 Timæus, 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 processes, not entities). The Timæus may be viewed, from this standpoint, as an example of the descending approach (the one in the first segment of intelligible) starting from the smallest possible number of simple hypotheses to build from them a scientific model 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, anagkè, with which you have to make do.

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 theôria of the world is to us an example of order, of kosmos, we should use as a model to bring order to our cities and our lives through laws stemming from our logos. 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 means toward a higher end, the building of man in time and space. Mathematics, like all other sciences and technics, are neutral. They don't tell us how we should use them for good or bad. This is the whole message of the Hippias minor, and already of the Charmides. That is why they only belong to the first segment of the intelligible.

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 examples; 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 Phædo.

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 within themselves 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 malgré lui". 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...

To: plato <plato@freelance.com>
Date : August 15, 1996, 19:27:17
Subject: Re: mathematical entities

Nicholas Denyer writes, in answer to my statement that mathematics, for Plato, were "neutral", neither good nor bad:

> Aristoxenus Harm. El. 2.30 "They came [to Plato's lecture on the Good] 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" (trans. Ross).

This may simply mean that Plato, in order to lead his audience toward the idea 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 were the good.

> and also of the long passage in Gorgias 507a-508c, especially 508a: "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."

This passage is all about justice as the goal of man, and fits very well with what I have been saying. Man must put measure in his life, and geometry gives examples 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. In itself, 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.

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 Protagoras, even if it shouldn't be taken litterally as Protagoras himself might interpret it. But numbers alone are not by themselves the good...

Plato and his dialogues : Home - Biography - Works and links to them - History of interpretation - New hypotheses - Map of dialogues : table version or non tabular version. Tools : Index of persons and locations - Detailed and synoptic chronologies - Maps of Ancient Greek World. Site information : About the author.

First published December 15, 1996 ; Last updated November 21, 1998
© 1996 Bernard SUZANNE (click on name to send your comments via e-mail)
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