Opposing the reduction of mathematics to logic

«For Wittgenstein and Carnap, mathematics is nothing more than a language indifferent to the contents that it expresses. (…)

If one tries to understand the reasons for this progressive fading of mathematical reality, one can be brought to conclude that it results from the use of the deductive method. Wanting to build all the mathematical notions starting from a small number of notions and primitive logical propositions makes one lose sight of the qualitative and integral character of the constituted theories. (…)

One thus understands Brunschvicg’s distrust with respect to all the attempts which would like to deduce the unit from mathematics starting from a small number of initial principles. Brunschvicg also opposed, in “Les Etapes de la philosophie mathématique”, the reduction of mathematics to logic (…).»

– Albert Lautman, introduction to “Essai sur les notions de structure et d’existence en mathématiques: Les schémas de structure”. Paris, Hermann & Cle Ed., 1938. p. 7-15.


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