Socrates Teaching Mathematics

All I know is that I know nothing
Socrates, Of course this is a contradiction as at least Socrates claims to know one thing. Similar but opposite is the statement “I don't know what I don't know”. A better statement by him is At any rate it seems that I am wiser than he is to this small extent, that I do not think that I know what I do not know.” According to Luciano De Crescenzo a method of the positive negation to approach truth (or God) with logic: “Can you provide a proof of Gods existence? No! Can you provide a proof of Gods non-existence? No! Then you say that there is something that you do not know? Yes! Then call this thing simply “God”. And what if I just call it the “Thing” that I do not know? It does not matter it is the same thing.” De Crescenzo says that the truth exists because if it does not exist then also not the fact that the truth does not exist. Additionally is seems to be a contradiction to Plato and also Socrates theory of anamnesis: I knew most of this in advance but I did not know that I knew it.

From Plato's dialogue the “Meno”

Socrates' considered that all the knowledge we are capable of possessing is already within us, and the process of reasoning something out is really just an act of recollection, i.e., remembering things we already somehow know (Anamnesis). Socrates questions an un-schooled servant boy about a geometrical problem. Socrates draws a square, and asks the boy how he would go about constructing a square twice as large.

Initially the boy says he doesn't know, but under further questioning he thinks the answer is to make the edges twice as long as the edges of the original square:

Socrates asks the boy how much larger is the area of this square in relation to the original, and the boy correctly replies that it has four time the area. Socrates re-iterates the question: how would you construct a square with just twice (not four times) the area of the original? We need a square with half the area of the one just constructed. Socrates asks the boy if we can cut each of the four squares in half by drawing a line connecting opposite corners, and the boy answers Yes! and he provides the following figure:

The red square formed by joining the midpoints of a square is called Plato's square.

They agree that the four diagonals describe a square, and its area is twice the area of the original square.

Socrates: What do you think, Meno? Has he, in his answers, expressed any opinion that was not his own?
Meno: No, they were all his own.
Socrates: And yet, as we said a short time ago, he did not know?
Meno: That is true.
Socrates: So these opinions were in him, were they not?
Meno: Yes.
Socrates: So the man who does not know has within himself true opinions about the things that he does not know?
Meno: So it appears.
Socrates: These opinions have now just been stirred up like a dream, but if he were repeatedly asked these same questions in various ways, you know that his knowledge about these things would be as accurate as anyone's.
Meno: It is likely.
Socrates: And he will know it without having been taught, but only questioned, and find the knowledge within himself?
Meno: Yes.
Socrates: And is not finding knowledge within oneself recollection?

Socrates then goes on to speculate on when or how the boy had acquired his true opinions about geometry, and suggests that it must not have been during his present life (since Meno assures Socrates that the boy has had no instruction in geometry).

We might observe that the idea using the diagonals of the fours smaller squares was not one of the boy's own opinions. He was taught this by Socrates, so one could argue that the boy has in fact been taught something which he did not know, and which he could never "recollect" simply by examining his opinions in isolation. This highlights two different kinds of knowledge, one that derives uniquely from first principles (the common notions about geometrical shapes) and the other that is accidental and arbitrary (terminology). One could argue that people learn and acquire common notions about spatial relations and proportions during their formative years, as they organize their primitive sense perceptions. On the other hand, certain very basic aspects of our experience may be "hardwired" into the biology of our brains and sense organs. At least Socrates considers that there exist things which we have to learn:
"I do not insist that my argument is right in all respects, but I would content... that we will be better men, braver and less idle, if we believe that one must search for the things one does not know..."

A comment of Carlo Celluci about the procedure of the solution as found by lets say Meno:

Seeing the solution to a given problem by means of a diagram. The prototype of this form of the analytic method is the solution to the problem of duplicating the square described in Plato’s Meno. Current uses of the method are to be found in contemporary ‘experimental geometry’ which is intended to provide an escape from the computational intractability of certain geometrical problems by deductive means. Seeing the solution to a given problem by means of a diagram is analytic insofar as one starts from the problem and, by a trial-and-error procedure, arrives at a diagram that shows its solution immediately. The solution is obtained not through a long deduction from first principles or from previously established results, but simply by looking at a diagram. While the closed world view considers the use of diagrams as redundant because it assumes that results established using diagrams can always be derived from given axioms by logical deduction only (at least in principle; in practice diagrams occur on almost every page of Hilbert’s Grundlagen der Geometrie), seeing the solution by means of a diagram is a self-contained procedure that is not part of a global axiomatic order. Carlo Cellucci, The Growth of Mathematical Knowledge: An Open World View

A continuation of Socrates' dialogue with Meno in which the boy proves that the root 2 is irrational (by an anonymous author) (http://www.gutenberg.org/etext/254)

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