In the book by Kleene "Introduction to Metamathematics" I have read that Poincare was intuitionist. Nevertheless, due to the fact that I am an undergraduate student in physics, I know that Poincare made significant contributions in physics theories.
Question: My confusion is about how could Poincare do theoretical physics while being intuitionist, because, as I understand it, intuitionism rejects completed infinite which is required to define real numbers, but on the other hand, theoretical physics is mainly about theorems that use real numbers.
Is it the case that he was both intuitionist and both formalist? How can that be? Maybe it is the case that most mathematicians are members of many schools of philosophy of foundations of the mathematics? But do then it makes sense to have such distinct schools at all? Did Poincare maybe do theoretical physics using constructive analysis (I don't think so because then I assume physicists would not understand him/his ideas)?
I would appreciate any ideas and comments!