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).

Leopold Kronecker (1823–1891) was a German mathematician who worked on number theory and algebra. He is considered a pre-intuitionist, being only close to intuitionism because he rejected Cantor’s Set Theory. He was, in fact, more radical than the intuitionists. Unlike Poincaré, for example, Kronecker didn’t accept the transfinite numbers as valid mathematical entities.

Frege discussed Mill’s empiricist ideas and Kant’s rationalist ideas about the nature of mathematics, and employed Set Theory and logico-philosophical notions to develop a new concept for the natural numbers. All this is objectively exposed by this paper.

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)

Let’s suppose all the rules of physics will change, but, before the change, we finally figured out everything there was to be figured out about physics. This means that we achieved pragmatic completeness at that point. It’s not a universal Platonic completeness, but everything there was to be expressed about the physics at that moment was expressed.

Kant and Descartes followed an extreme clever, secure way of reasoning. For them, there must be a world of differences, or of movement, before we can extract anything (ideas, laws, concepts, etc.) from the world. For Kant, these “changes” that secure the possibility of knowledge were the ones we can measure with the categories of space and time. While, for Descartes, since there exist two things: “me” and “the world”, we can say knowledge is possible. But I think we can (...) extract a more abstract, general principle from the similarities between both Descartes and Kant’s mobilist tendency. (shrink)