Maths and Philosophy #1: Socrates

I have come up with the idea to start themed weeks (where all posts during that week follow a specific theme). This week I will be talking about ancient Greek mathematicians that delved in the fields of Maths and Philosophy, starting with Socrates.

Socrates is credited as one of the founders of modern Western philosophy. His student’s, Plato, dialogues are one of the most comprehensive accounts of Socrates to survive. In today’s post I wanted to discuss Plato’s dialogue entitled “Meno” in particular, as here we can see Socrates’ teaching of mathematics.

“All I know is that I know nothing”

In this dialogue, Socrates’ considered that all the knowledge we can possess is already within us, and that the process of reasoning is merely an act of recollection.

Socrates questions an un-schooled servant boy about a geometrical problem. He begins by drawing a square, and asks the student to construct a square twice as large.

The boy initially says that he does not know, but after further questions he thinks that the answer is to make the edges twice the size of the original square, as shown below.

However, the area of this new square is greater that twice the original square; the student correctly observes that it is four times the area. Socrates reiterates the question, asking “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 the student 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, producing the final result below.

The square formed is called Plato’s 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?”

In this dialogue, Socrates joins mathematics and philosophy together by using mathematical proof to demonstrate a philosophical belief.

Stay tuned for the post on Wednesday! M x