Poincaré’s main philosophical interest lay in the formal and logical character of theories in the physical sciences. He is especially remembered for the discussion of the scientific status of geometry, in La Science et l’hypothèse (1902, trs. as Science and Hypothesis, 1905). The axioms of geometry are not analytic, nor do they state fundamental empirical properties of space. Rather, they are conventions governing the description of space, whose adoption is governed by their utility in furthering the purpose of description. Poincaré’s conventionalism about geometry proceeded, however, against the background of a general realism about the objects of scientific enquiry, and he always insisted that there could be good reason for adopting one set of conventions rather than another. In his late Dernières pensées (1912, trs. as Mathematics and Science: Last Essays, 1963), Poincaré attacks the logicist programme of Frege and Russell, denying that mathematics can be reduced to logic, and arguing that further intuition is always needed to derive the properties of numbers.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Poincare.html A biography of Poincaré

http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20061102.shtml An audio discussion of Poincaré’s work by three experts