Logical difficulties in modern mathematics (2012)

Logical difficulties in modern mathematics (2012)
njwildberger.com

That said, there are philosophical questions about truth, infinity, unknowable statements and the like. Mathematicians have by and large settled on a set of answers to these. Every statement is true or false, regardless of whether we know the answer, or even whether we can know the answer. Infinite sets exist, and are described by a known set of axioms called ZFC. Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?)

All of these statements are part of classical mathematics. Almost every elementary exposition will implicitly assume that they are try. Yet they can all be questioned, and their truth can never be settled in any absolute sense. However woe betide the student who dares question these in a math class.

I don't think many people see that as a matter of philosophy.

This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy.

To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics.

More precisely any contradiction found in Constructivism necessarily will lead to a contradiction in classical mathematics. The converse is only partially true. Gödel did prove that a logical contradiction in the classical handling of infinity will lead to a contradiction in Constructivism. But a flaw in a specific set of classical axioms, such as ZFC, need not lead to a flaw in usual Constructivism.

His real complaint as far as I've ever been able to determine is that he is a highly symbolic thinker; and because of that won't accept certain assumptions that everyone else takes as a given - usually what the Reals are. I'm very happy to accept than any length in geometry is a number by definition (hence sqrt(2), constructed by a 1-1-sqrt(2) triangle, is clearly a number corresponding to that length). He won't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation. This isn't really a logical issue, he just won't use everyone else's definitions.

He's worth listening too because he is good at maths despite that handicap and his perspective is interesting to provoke a bit of reflection on what your assumptions are and what does infinity really mean anyway. His complaints are otherwise unlikely to catch on.