This paper objectively defines the three main contemporary philosophies of mathematics: formalism, logicism, and intuitionism. Being the three leading scientists of each: Hilbert (formalist), Frege (logicist), and Poincaré (intuitionist).

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the philosophy of mathematics by (...) writing short essays and letters. (shrink)

In 1956, a few writings of Wittgenstein that he didn't publish in his lifetime were revealed to the public. These writings were gathered in the book Remarks on the Foundations of Mathematics (1956). There, we can see that Wittgenstein had some discontentment with the way philosophers, logicians, and mathematicians were thinking about paradoxes, and he even registered a few polemic reasons to not accept Gödel’s incompleteness theorems.

Husserl came over to philosophy from mathematics and he devoted many years to the formulation of a firm foundation for Philosophy that could even secure the status of "science" for it. But unlike some of his contemporaries (like Frege and Russell), he did not seek salvation for philosophy in the mathematical method. He argued philosophy (like any other field of study) should pay attention to uninterpreted basic experience and this would lead the way to understanding the essence of things. Essence, (...) then, would become conscious and this process was addressed by him as “phenomenological reduction.”. (shrink)