Constructivism & Intuitionism

Constructivism

The primary position of constructivism is simple. The constructivist "believes that we have no conception of mathematical truth apart from the idea of proof".

Simply, 

proof = truth

Or:

mathematical truth = proof

It follows that if truth = proof, then truth and proof are (despite Platonism) inventions of the human mind. Proof is all there is. More specifically, we can say the same about numbers. Numbers "do not exist until constructed, by operations which generate them in a finite number of steps". Mathematical operations don't just use numbers: they also construct them.

This leads us to a question: 

What did these mathematical constructions use before they constructed the numbers? 

What constituted the mathematical constructions before the numbers were actually created? Were numbers there from the beginning? In that case, who or what created them? Or, if they were there from the start, perhaps they weren't constructed (or created) at all and Plato was right after all.

The stark conclusion of constructivism is the ‘anti-realist’ idea that "all existing numbers are contained in the books and papers of the mathematicians" (384). Numbers aren't discovered or intuited by mathematicians. They're constructed or created. Thus if a number hasn't been constructed or proved, then it quite simply doesn't exist to be discovered or intuited. In addition, only numerals (not numbers) really exist. And to say "that numbers exist is to say that there are valid proofs involving numerals" (384). (This appears to be very like Hartry Field’s position.)

This position is very similar to that endorsed by Kant over a hundred and forty years earlier. Kant believed that mathematical propositions "are known a priori since we ourselves are the authors of them" (385). Is this mathematical idealism? Their a priori status is guaranteed simply because we don't need to look outside of our own minds to the empirical world (or even to a platonic realm) to discover numbers and their nature.

Intuitionism

Now we arrive at intuitionism, a variant on constructivism.

Here too proof is everything. However, there's a surprising conclusion to this emphasis on mathematical proof. We've already said that a 

"mathematical proposition is true only if there is a proof of it; similarly, it is false only if there is a proof of its negation" (385). 

But what if there is proof of neither? Does that mean that the proposition is neither true nor false? Perhaps it simply means that the proposition is "meaningless" or that it's not a genuine example of a mathematical proposition.

However, the intuitionists accepted one of these conclusions. The proposition may well be neither true nor false. It's still, however, a bona fide proposition. We must, therefore, deny the law of the excluded middle for such mathematical propositions. That is, we must deny the principle: either p or not-p. This means that such mathematical propositions must have a "third value". This third (truth?) value is often called "indeterminate".

There are more surprising conclusions one must accept if one is an intuitionist. For example,

"as Heyting demonstrated, we shall need an entirely new system of logic – which he called intuitionistic logic – in order to accommodate the constructivist vision of mathematical truth" (385).

The logicists tried to reduce mathematics to logic. Now we find that a discovery in mathematics will have a profound effect on logic itself. If mathematics requires a third truth-value (indeterminate), then so too will logic (which, of course, also deals with truth). Indeed a logical vision or system must ‘accommodate’ the new findings of constructivism or intuitionism. Does this in itself show us that logic is part of mathematics, rather than that mathematics is part of logic? Perhaps not in all cases.