E-mail Archives : Non-Euclidian geometries and theories on Plato's dialogues November 12, 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.

To: plato <plato@freelance.com>
Date : November 12, 1996, 20:09:26
Subject : re: plato's geometry and intellectual pursuits

Vassilios Tsiantos writes:

> Could you please give me a definition for what is a Non-Euclidean geometry?

Bill Ball writes about my comments on the Apology:

> What evidence do you have to challenge "what Taylor and most other scholars say"?

Let me try to pull these two threads together.

Here is an example of non-Euclidian geometry:

Definition 1: a "straight" line is the shortest path between two points
Definition 2: a parallel to a given straight line (D) from a given point A outside this straight line is another straight line (D') including point A and having no intersection with line (D)

Let's assume that we stay in two-dimensional geometry for this example. The above definitions are fully consistent with Euclidian geometry so long as our two dimensional geometry is defined on a plane. But now, let's see what happens if we take our two dimensional "plane" to be the surface of a sphere (which, by the way, happens to be our situation when we draw maps of the earth).

The "straight" line of this "plane", using the same definition (shortest path between two points), as any intercontinental pilot knows, is the "great circle" including these two points, that is, the circle at the intersection of the sphere and the plane defined by the two points and the center of the sphere (in other words a circle on the sphere whose center is the center of the sphere).

Now, let's see what happens to our "parallels": let there be a "straight line" (C), that is, a great circle on the sphere. Let's now take a point A on the sphere outisde that "straight line" (that is, outside the given great circle, or else, outside the plan including the great circle and the center of the sphere). From this point, I can draw an infinity of "straight lines" on the sphere (any great circle defined by a plan including A and the center of the sphere, and there is an infinity of plans containing two points). But the point is all these "straight lines" intersect with (C) somewhere (because any two great circles on a sphere intersect, the planes containing them having always a point in common, the center of the circle). In other words, in our newly defined two-dimensionnal geometry, parallels don't exist.

By the way, if I remember correctly, this is Riemanian geometry. In another geometry, Lobatchevskian geometry (unless it is the other way around), using the same definitions as above and using an hyperbolic surface in place of a sphere, you end up with an infinity of parallels to a given "straight line" from a given point.

The point is, any scientific theory works from axioms and definitions, and looks for consistency in the consequences of these starting propositions, and if applied to the real world, for consistency between what the theory "predicts" and what experience shows. Non Euclidian geometries try to use other sets of axioms and/or definitions and see whether they can derive from them consistent consequences (consequences that are not contradictory with one another or with the axioms). Then, people look for practical applications of these sets of axioms and definitions. But, by definition, the initial axioms cannot be proven. You may change the set so that what was an axiom in one approach becomes a theorem in another, but you have to have a set of undemonstrable propositions to start with.

Well, it turns out that the situation is pretty much the same with the interpretation of Plato's writings. If you think about it, the "theories" that are proposed by scholars are just that, theories, even if many of them lose sight of this fact. We must admit that we know nothing for sure about the date of composition of the dialogues and the purpose of Plato in writing them. We have some hard facts, chief among them the dialogues themselves, and bits and pieces of informations that require themselves a lot of "decoding": Aristotle quotations and criticism, but did he really understood all what Plato said? Diogenes Lærtius, but that was hundreds of years later, and so on.

So there is currently, and there has been for the past hundred years or so, may be a little more, a "darwinian" theory that tries to explain the dialogues by a "theory of evolution" in Plato's mind. But this is just one possible theory, that relies on supposed "inconsistencies" between dialogues that have themselves to be proven, because they rely on "interpretations" of their meaning which are themselves "hypotheses" (this is a chance for scholars, because otherwise, how would they make a living?...).

As far as I am concerned, I have been trying to develop a "non-darwinian" interpretation of platonism. I am perfectly aware of the fact that it is an "interpretation", that it is "undemonstrable", because it has to rely on "axioms" that are undemonstrable in any scientific way, but so is the case of the mainstream "darwinian" interpretation(s). The only thing I am looking for is consistency in the results and with the "experience" of the dialogues. I hold that the "darwinian" interpretation leaves many questions in doubt or unanswered (that's why there are several "darwinian" variants of platonism): what, for instance, about the Menexenus and Parmenides? I hope that my alternative "theory" provides more consistency in the explanations and leaves less "holes" in the interpretation. But I know not everybody would agree with me. My own method is that I prefer to look for an explanation that assumes Plato's consistency until I can find no way to reconcile different texts (and I didn't so far), but that requires that we abandon some preconceived notions, that we accept that Plato might have been way ahead of his time, much smarter than Aristotle, even though Aristotle was later (thus an "evolution" from Plato), and maybe much smarter than many of us, even though we are 2500 years later.

Anyway, I only ask the holders of other "theories" to accept that theirs are as much theories as mine, even though I am alone and they are thousands, and to judge each theory by its own axioms (as Plato so well knew how to do and Aristitle so little!...), and the axioms by their "fruits". But we must also realize that these "fruits", the "interpretation" of the dialogues, is istelf part of the problem: a text keeps saying the same thing and we have to "interpret" it to bring meaning to sequences of letters...

Thus, my position about the Apology, comes from the role I see for that dialogue in the overall scheme I propose, from its "symmetrical" position compared to the Sophist (the visible killing of Socrates by his fellow Athenians on the side of the visible, the "parricide" in thought of Parmenides by one of his fellow Eleans on the side of the intelligible), from the detailed analysis of its plan and its relationship with the structuring principles of the whole set, and so on... On the other hand, I see nothing that is irrealistic or materially or psychologically impossible in having Plato write the Apology thus conceived as part of a grand design late in his life, and I am not aware of any hard evidence that would compel us to date it earlier; Besides, my proposal cares much less with dates and order of composition and may accomodate various schemes in that respect. So, as far as I am concerned, Taylor's hypothesis is but one possible hypothesis, even if it is shared by thousands of scholars (was it not Socrates, or Plato through the mouth of Socrates, who said that truth is not measured by ballots?...) and should be judged only as part of a whole set of assumptions and resulting interpretations. Mine is another hypothesis, and should be judged as part of a whole set and based on the overall interpretation it is a part of.