What possible future mathematical methods are not considered rigorous math right now?

Question

In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing controversy about not being "rigorous enough" but may have a shot to become an important part of the future mathematical canon?

Answer 1

In a talk called "Unclear Concepts" that I gave some years ago, I mentioned the following examples, which seem to fit your question.

(1) Feasible numbers. A natural number $n$ is said to be feasible if I can count up to it. So 1000 is feasible but $2^{100}$ is not. One expects some sort of induction principle to hold, at least when the number of "iterations" is feasible, i.e., if $P(0)$ and $(\forall n)\,(P()\to P(n+1))$, then all feasible $n$ should satisfy $P(n)$. With ordinary logic, this unfortunately lets you prove that $2^{100}$ is feasible; some version of linear or affine logic (omitting the contraction rule) seems more promising.

(2) Choice sequences. Brouwer introduced various versions of choice sequences in intuitionistic logic, but it seems difficult to provide a foundation for these in ordinary (classical) logic. Free choice sequences (where there are no restrictions on the choices) have been treated, but there seem to be problems if one is allowed to choose restrictions on future choices (including restrictions on future restrictions, etc.).

(3) The universe of all sets, i.e., the full cumulative hierarchy of sets, forming all subsets at each stage and iterating forever. Here "all" and "forever" are vague. Part of their meaning is well captured by the ZFC axioms (plus maybe some large cardinals), but the intuition seems to contain a lot more than the axioms, and it's not clear how to make that "lot more" rigorous.