In the philosophy of mathematics, 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 introduction of natural numbers The pre-intuitionists, as defined by L. E. J. 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. The principle of complete induction This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers. Peano's fifth axiom states: Allow that; zero has a property P; And; if every natural number less than a number x has the property P then x also has the property P. Therefore; every natural number has the property P. This is the principle of complete induction, which establishes the property of induction as necessary to the system. Since Peano's axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x + 1. What one can do is say that, if after some number n of trials that show a property P conserved in x and x + 1, then we may infer that it will still hold to be true after n + 1 trials. But this is itself induction. And hence the argument begs