Intuitionism

Intuitionism, or neointuitionism (opposed to preintuitionism), is an approach in the philosophy of mathematics, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Quotes

• Actually, the difference between intuitionism and classical mathematics is really very, very small. You can turn, roughly speaking, any mathematical argument into an intuitionistic argument just by putting a double negation in front of something. Instead of saying this result is true, you say you can't prove it's not true. And that makes it intuitionistically correct. ...
  ... It's hard enough to do mathematics if you allow yourself to use infinite sets, and making things even more difficult doesn't appeal to most people. ... So I'd say most mathematicians don't bother being constructive because it just makes things unnecessarily difficult.
• Richard Borcherds, (January 23, 2023)"Richard Borchers: E8, Witten, Langlands Modular Forms". Theories of Everything with Curt Jaimungal, YouTube. (quote at 21:37 of 1:36:06 in video)

• If we compare. e.g. the systems of classical mathematics and of intuitionistic mathematics, we find that the first is much simpler and technically more efficient, while the second is more safe from surprising occurences, e.g. contradictions. At the present time, any estimation of the degree of safety of the system of classical mathematics, in other words, the degree of plausibility of its principles, is rather subjective. The majority of mathematicians seem to regard this degree as sufficiently high for all practical purposes and therefore prefer the application of classical mathematics to that of intuitionistic mathematics. The latter has not, so far as I know, been seriously applied in physics by anybody.
  • Rudolf Carnap (1939; 51), as cited in: Paul van Ulsen. Wetenschapsfilosofie, 6 november 2017.