# Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures

# Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures

## Synopsis

• the mathematical image

• platonism

• picture-proofs

• applied mathematics

• Hilbert and Godel

• knots and nations

• definitions

• picture-proofs and Wittgenstein

• computation, proof and conjecture.

The book is ideal for courses on philosophy of mathematics and logic.

## Excerpt

What’s the greatest discovery in the history of thought? Of course, it’s a silly question—but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principle fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it. The case for Platonism, however, needs to made carefully. Let’s begin with a glance at the past.

**The Original Platonist**

We notice a similarity among various apples and casually say, There is something they have in common.’ But what could this *something* they have in common be? Should we even take such a question literally? Plato did and said the common thing is *the form of an apple*. The form is a perfect apple, or perhaps a kind of blueprint. The actual apples we encounter are copies of the form; some are better copies than others. A dog is a dog in so far as it ‘participates’ in *the form of a dog*, and an action is morally just in so far as it participates in *the form of justice*.

How do we know about the forms? Our immortal souls once resided in heaven and in this earlier life gazed directly upon the forms. But being born into this world was hard on our memories; we forgot everything. Thus, according to Plato, what we call learning is actually recollection. And so, the proper way to teach is the so-called Socratic method of questioning, which does not simply state the facts to us, but instead helps us to remember what we already know.

The example of the slave-boy in the *Meno* dramatically illustrates Plato’s point. After being assured that the slave-boy has had no mathematical training, Socrates, through a clever sequence of questions, gets him to double the square, which, for a novice, is a rather challenging geometric construction. Not only does the slave-boy do it but, after a false start, he recognizes that he has finally done it correctly. Plato’s moral is that the slave-boy *already* knew how to double the square, and Socrates, the self-described ‘mid-wife’, simply helped him in bringing out what was already there.

Plato’s theory is both wonderful and preposterous. It’s wonderful because of its tremendous scope. It explains what all apples have in common, what makes a moral act moral, how we acquire knowledge and, above all, it tells us what mathematics is. This last feature especially rings true—even if nothing else about Platonism does. When we talk about circles, for example, we don’t seem to be talking about any particular figure on the blackboard. Those are only approximations. We’re talking about a perfect circle, something . . .