Am I a constructive mathematician?
It seems to me that people think I am a constructive mathematician, or worse a constructivist (a word which carries a certain amount of philosophical stigma). Let me be perfectly clear: it is not decidable whether I am a constructive mathematician.
But seriously, if anything, you may call me a mathematical relativist: there are many worlds of mathematics, and the view of the worlds is relative to which one I am in. Any attempt to bring mathematics within the scope of a single foundation necessarily limits mathematics in unacceptable ways. A mathematician who sticks to just one mathematical world (probably because of his education) is a bit like a geometer who only knows Euclidean geometry. This holds equally well for classical mathematicians, who are not willing to give up their precious law of excluded middle, and for Bishop-style mathematicians, who pursue the noble cause of not opposing anyone.
What could be more appealing to a mathematician than the idea that there is not one, but many, infinitely many worlds of mathematics? Would he not want to visit them all, understand how they are related, and see what happens to his favorite subject as he moves between them?
Let us consider an example. The real numbers are a mathematical object of fundamental importance, and have many aspects:
- The reals as a set are uncountable and in bijection with the powerset of natural numbers.
- The reals as an algebraic structure form a linearly ordered field.
- The reals as a space are locally compact, Hausdorff, and connected.
- The reals are a measurable space on which measure theory rests.
- The reals of non-standard analysis contain infinitesimals.
- The reals as understood by Leibniz contain nilpotent infinitesimals.
- The reals as Brouwerian continuum cannot be decomposed into two disjoint inhabited subsets.
- The reals are overt.
We can have some of these properties but not all at once. History has chosen for us a combination that is taught today as a dogma. Any attempt to deviate from it is met with opposition. Thus you probably consider 1, 2, 3, and 4 as true, 5 as something exotic you heard of, 6 as Leibniz’s biggest mistake, 7 as intuitionistic hallucination (because obviously the reals can be decomposed into the non-negative and negative numbers), and 8 as something you never heard of (but you should have because it is the concept dual to compactness and you have been using it all your life).
Once we break free from Cantor’s paradise that Hilbert threw us in we discover unsuspected possibilities:
- It may happen that the reals are in 1-1 correspondence with a subset of the natural numbers, while at the same time they form an uncountable set.
- It may happen that the reals form a proper class.
- It may happen that every real number has a Turing machine computing its digits.
- It may happen that the reals are not linearly ordered.
- It may happen that the reals are locally non-compact, in the sense that every interval contains a sequence without an accumulation point.
- It may happen that every subset of the reals is measurable.
- It may happen that the reals can be covered by a sequence of intervals whose cumulative length does not exceed $1$.
- It may happen that the reals contain nilpotent infinitesimals, which validate the 17th century calculations that physicists still use because, luckily, they did not subscribe entirely to the $\epsilon\delta$-dogma of analysis.
- It may happen that every real function is continuous, and consequently the reals are not decomposable into two disjoint inhabited subsets.
- It may happen that the reals are not overt, whatever that means.
It should be admitted that some of the possibilities are rather bizarre. For example, I do not know what good it is to have the reals as a subset of the naturals, but I am sure somebody could think of something. But why should a measure theorist ignore a world of mathematics in which every subset is measurable, or a computer scientist one in which all reals are computable, or a topologist one in which all functions are continuous, or an analyst one in which all functions are smooth?
I am not proposing that mathematics should be compartmentalized so that each branch sits in its world of mathematics, incompatible with others. That would be a grave mistake indeed. In fact, the unification of mathematics under the umbrella of classical set theory has been immensely successful precisely because it allowed mathematicians to discover deep and unsuspected connections between different branches of mathematics. We have learnt to look for connections between branches of mathematics, and now we must also learn to look for connections that span worlds of mathematics.
We cannot ignore the many worlds of mathematics. Therefore, mathematics must become applicable in a wide variety of worlds. Mathematicians have to be educated so that they develop multiple mathematical intuitions that help them feel how the worlds of mathematics behave.
I have so far not given you any technical definition of a mathematical world. Such a definition may be useful for showing meta-theorems, but I think it can never be exhaustive. A world of mathematics may be a forcing extension of set theory, or a topos, or a pretopos, or a model of type theory, or any other structure within which it is possible to interpret the basic language of mathematics.
At the moment I am visiting the Institute for Advanced Study as a member of the Univalent Foundations group. We are building a new foundation of mathematics whose language is type theory rather than set theory, and whose primary objects are homotopy types and not just bare sets. Do I think this is an exciting new development? Certainly! Will the Univalent foundations disrupt the monopoly of Set-theoretic foundations? I certainly hope so! Will it become the new monopoly? It must not!