Preintuitionism

In the mathematical philosophy, the pre-intuitionists were a small but influential group who informally shared similar philosophies on the nature of mathematics. The term itself was used by L. E. J. Brouwer, who in his 1951 lectures at Cambridge described the differences between intuitionism and its predecessors:

Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction [...] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic.

The Pre-Intuitionists, as defined by Brouwer, differed from the Formalist standpoint in several ways, particularly in regard to the introduction of natural numbers, or how the natural numbers are defined/denoted. For Poincaré, the definition of a mathematical entity is the construction of the entity itself and not an expression of an underlying essence or existence.

This is to say that no mathematical object exists without human construction of it, both in mind and language.

This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers.

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