If one thinks that certain mathematical are mere constructions, one might also reject certain mathematical methods. Take intuitionism as a case: Brouwer thought that mathematics was the product of the human mind, and restricted maths to whatever is constructible in a very strict sense. Finitism adds even further constraints, disallowing infinitary methods altogether. In these cases, the position seems to be that we cannot hold on to the usual commitment to mathematical entities.
I want to also point out that his philosophical attitude turned out to also be very productive in the end and lead to significant contributions in our understanding of type theory which are applicable today in theorem provers and functional programming languages. He also gave new proofs of various theorems independent of various other results and made room for a lot of advances in logic. So, I think even the side-effects of pursuing this kind of project can be rather valuable and have obvious practical impact.
