How could antirealism about truth affect classical mathematics?

Adopting an antirealist view of truth—which restricts what is considered true to what can, in principle, be known or proved—has profound implications for classical mathematics.

The primary consequence is the challenge to the law of excluded middle, which asserts that for any proposition p, either p or its negation, not-p, must be true. Under a strongly realist conception, this law is a fundamental logical principle. However, if truth is limited to what is verifiable or provable, the law of excluded middle cannot be guaranteed, as there is no assurance that an arbitrary mathematical proposition can be proven either true or false.1

Because many significant theorems in classical mathematics rely on this principle, an antirealist stance calls large portions of the field into question. This perspective supports revisionary, constructivist approaches such as intuitionism, which rejects the use of the law of excluded middle in proofs involving infinite sets, as seen in the work of L.E.J. Brouwer.12