In my previous blogpost (What Is Mathematics?) we saw that platonists believe that what mathematicians do is discovering things about a world which exist independently of themselves. Intuitionists, on the other hand, believe that mathematicians do not discover but create mathematical theorems*. For many mathematicians Platonism is the obvious choice here. Doesn’t nature just show us how mathematics works? We need only look at triangles drawn in the sand to see that Pythagoras’ theorem is true, don’t we? Isn’t that discovery? Let’s take a look at an example of a mathematical theorem to see whether Platonism is really that obvious.
Suppose that some brilliant mathematician, burning the midnight-oil in her lonely attic, finds some very complex geometrical theorem (say, about the volume of an exotic geometrical shape). It may be that this theorem is not instantiated; it may be that there is nothing in nature which actually has this shape (maybe the shape has a large number of dimensions). It is not so obvious that the geometrical theorem that our lonely mathematician has come up with exists anywhere outside her mind (and the paper she wrote it down on). Why should we believe that the mathematician has discovered anything? That would imply that it was already there before she came along. Might we not say that she just ‘made it up’? Such questions show that it is far from obvious that Platonism is self-evident while Intuitionism is not.
Why should we care?
‘Ok, fine.’ I hear you think, ‘discovering and creating are two different things. Platonists and intuitionists have different ideas about what mathematics is. They disagree on whether mathematics necessarily exists outside the mind of the mathematician. But why is that important? Does it matter for how we do mathematics?‘
Yes, it does. One of the rules that is often used in mathematics is ‘the exclusion of the middle ground’. It is the simple rule that any mathematical proposition is either true or it is false – there is no middle ground. Mathematicians use this rule whenever they prove a proposition by contradiction. That works as follows. Suppose we want to prove a certain proposition. If we can show that the negation of our proposition (the assumption that it is false) leads to a contradiction, then ‘the exclusion of the middle ground’ tells us “assuming the falseness of the proposition leads to a contradiction, so the proposition must be true.”
Intuitionists argue that proof by contradiction isn’t really proof at all. They believe that real proof is constructive. The only thing that is constructed in a case of proof by contradiction is that the negated proposition is not true, not that it is false. Mathematical proof, according to the intuitionist, shouldn’t be based on the metaphysical assumption that the truth is in a sense ‘binary’ (that propositions must be either true or false). In cases where ‘the exclusion of the middle ground’ seems like a useful rule, it must be possible to find real, constructive proofs. Intuitionists, therefore, believe that much of what we call mathematical knowledge is actually based on conjecture (because it is based on ‘the exclusion of the middle ground’-rule).
Can you think of a reason why the intuitionists are skeptical of the ‘exclusion of the middle ground’-rule? (please leave your answer as a comment to this blogpost).
Intuitionism & P vs NP
Some computer scientists believe that the skepticism of the intuitionists regarding ‘the exclusion of the middle ground’ is connected to the infamous P vs NP problem. In one sentence, the P vs NP problem is finding an answer to the question whether any mathematical problem whose solution can be quickly verified (to be an actual solution), can also be quickly solved. Some say that the difference between constructive and non-constructive proofs as well as the difference between solving problems and verifying solutions can be understood as a difference between bottom-up and top-down mathematical reasoning.
If there is indeed this connection (between the intuitionism debate and the P vs NP problem) then the interpretation of mathematics may be of great importance to, for example, cryptography. Much of the security of financial transactions today relies on certain mathematical problems being very difficult to solve (such as breaking very large numbers down into prime factors). What if whole swathes of mathematical knowledge turn out to be based on conjecture? Of course, these conjectures are not necessarily false, but intuitionist philosophy casts doubt on them. But how will the world’s economy fare, if so much is based on trust in financial systems? Trust which is often based on mathematical proof?
What is a Mathematician?
Back to the philosophy of mathematics. Can we answer the question from my earlier blogpost ‘What is a Mathematician’? We might say that the platonist believes that all physical things can be a mathematician, as long as they follow (or embody?) the rules of mathematics (whether it be some conscious being, a lifeless calculating machine, or even a falling stone). The intuitionist, on the other hand, believes that a mathematician must be able to apply the rules of mathematics.
I’ll illustrate this difference by comparing mathematics to a game of chess. Let’s say that every possible state of affairs on the chessboard represents a mathematical proposition and that the moves to get to that state of affairs are limited by the rules of mathematics.
The platonist would say that any state of affairs on the chessboard that can be reached by following the rules of mathematics is a mathematical proposition – whether it actually has been reached or not. They just exist. Or they don’t. The intuitionist, on the other hand, would say that mathematical propositions are constructed by the mathematician by following the rules of mathematics in moving about the pieces on the chessboard. Before the mathematician moved the pieces, the proposition simply didn’t exist yet.
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More about the philosophy of mathematics:
*) Wait a minute! What happened to Formalism? Why aren’t we discussing that anymore? To see how Formalism differs from Platonism and Intuitionism it is important to know that formalists believe that mathematics in itself is ’empty’ – consisting in tautologies whose truth is independent of anything material. Understanding this is at the basis of what I wrote about numbers at the end of the post What Is Mathematics?. I’ll discuss the emptiness of mathematics in a later blogpost and here focus on the difference between Platonism and Intuitionism.