Editing Pre-intuitionism (section)

{{Short description|Categorization of some philosophers of mathematics}}
In the [[philosophy of mathematics]], the '''pre-intuitionists''' is the name given by [[L. E. J. Brouwer]] to several influential mathematicians who shared similar opinions on the nature of mathematics. The term was introduced by Brouwer in his 1951 lectures at [[University of Cambridge|Cambridge]] where he described the differences between his philosophy of [[intuitionism]] and its predecessors:<ref name=CW>Luitzen Egbertus Jan Brouwer (edited by [[Arend Heyting]], ''Collected Works'', North-Holland, 1975, p. 509.</ref>
<blockquote>Of a totally different orientation <nowiki>[</nowiki>from the "Old Formalist School" of [[Richard Dedekind|Dedekind]], [[Georg Cantor|Cantor]], [[Giuseppe Peano|Peano]],  [[Ernst Zermelo|Zermelo]], and [[Louis Couturat|Couturat]], etc.<nowiki>]</nowiki> was the Pre-Intuitionist School, mainly led by [[Henri Poincaré|Poincaré]], [[Émile Borel|Borel]] and [[Henri Lebesgue|Lebesgue]]. These thinkers seem to have maintained a modified observational standpoint for the '''introduction of natural numbers''', for '''the principle of complete induction''' <nowiki>[</nowiki>...<nowiki>]</nowiki> 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.</blockquote>