# Clarification on constructivist’s counter-example

In the last maths post, I asked whether there was a proof that

The idea is that although trichotomy is an axiom of the reals (any two numbers are equal, or one larger than the other), what is actually used in proofs is something very different philosophically. It is distinct to say no more that some one of the three cases holds; or that a particular one of the three cases holds. The function given shows exactly the divide. While

This really is something to think hard about, because the obvious rejoinder is not so simple as it seems. As realists, we will say, ‘Yes, but it does not matter whether we can put a given real into one or other of the cases; what matters is that we know we it is in one, so the proof is without difficulty.’ In the end, I will admit to agreeing with this obvious resolution, but the struggle is in understanding why this seems so far from the way we apply mathematical intuition. Reading the proofs in the Part II Graph Theory course, it is striking just how many proceed by cases, and how uniformly the language is phrased in a constructivist way. Now, this is a lucky course to pick because thankfully most of the things proved are determinable, so this thought paradigm works, but why do we naturally gravitate to this flawed model of polychotomy in our thinking?

Constructivism fails in the end, especially Brouwer’s intuitionism, because it cannot capture our intuition that, for example, the reals exist. If we allow ourselves to let the indeterminability of

Constructivism stills leaves a challenge though, because it questions why the expression of our mathematical thought so frequently lines up with intuitionism, even if we are ready to reject it as soon as we have to to defend our idea of the reals, and indeed all higher mathematics.

For an interesting alternative approach, see Gowers’ A dialogue concerning the need for the real number system, Part IV.