Editing Pre-intuitionism

{{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>

==The introduction of natural numbers==
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The pre-intuitionists, as defined by [[L. E. J. Brouwer]], differed from the [[Formalism (philosophy of mathematics)|formalist]] standpoint in several ways,<ref name=CW/> particularly in regard to the introduction of natural numbers, or how the natural numbers are defined/denoted. For [[Henri Poincaré|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 [[Henri Poincaré|Poincaré]] to argue with [[Bertrand Russell]] over [[Giuseppe Peano| Giuseppe Peano's]] [[Peano axioms|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 [[mathematical induction|induction]] as necessary to the system. Since Peano's axiom is as [[infinity|infinite]] as the [[natural number]]s, it is difficult to prove that the property of ''P'' does belong to any ''x'' and also ''x''&nbsp;+&nbsp;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''&nbsp;+&nbsp;1, then we may infer that it will still hold to be true after ''n''&nbsp;+&nbsp;1 trials. But this is itself induction. And hence the argument [[begging the question|begs the question]].

From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of [[complete induction]] is not provable by [[logic|general logic]].

Thus arithmetic and mathematics in general is not [[analytic proposition|analytic]] but [[synthetic proposition|synthetic]]. [[Logicism]] thus rebuked and [[intuitionism|Intuition]] is held up. What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematics that is not a matter of [[language]] alone, but of [[knowledge]] itself.

==Arguments over the excluded middle==
It was for this assertion, among others, that [[Henri Poincaré|Poincaré]] was considered to be similar to the intuitionists. For [[Luitzen Egbertus Jan Brouwer|Brouwer]] though, the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they still used ''principium tertii exclusi'' (the "[[law of excluded middle]]").

The principle of the excluded middle does lead to some strange situations. For instance, statements about the future such as "There will be a naval battle tomorrow" do not seem to be either true or false, ''yet''. So there is some question whether statements must be either true or false in some [[temporal logic|situations]]. To an intuitionist this seems to rank the law of excluded middle as just as un[[rigour|rigorous]] as [[Giuseppe Peano|Peano's]] vicious circle.

Yet to the Pre-Intuitionists this is mixing apples and oranges. For them mathematics was one thing (a muddled invention of the human mind, ''i.e.'', synthetic), and logic was another (analytic).

==Other pre-intuitionists==
The above examples only include the works of [[Henri Poincaré|Poincaré]], and yet [[Luitzen Egbertus Jan Brouwer|Brouwer]] named other mathematicians as Pre-Intuitionists too; [[Émile Borel|Borel]] and [[Henri Lebesgue|Lebesgue]]. Other mathematicians such as [[Hermann Weyl]] (who eventually became disenchanted with intuitionism, feeling that it places excessive strictures on mathematical progress) and [[Leopold Kronecker]] also played a role—though they are not cited by Brouwer in his definitive speech.

In fact Kronecker might be the most famous of the Pre-Intuitionists for his singular and oft quoted phrase, "God made the natural numbers; all else is the work of man."

Kronecker goes in almost the opposite direction from Poincaré, believing in the natural numbers but not the law of the excluded middle. He was the first mathematician to express doubt on [[nonconstructive proof|non-constructive]] [[existence theorem|existence proofs]] that state that something must exist because it can be shown that it is "impossible" for it not to.

== See also ==
* [[Conventionalism]]

== Notes ==
{{Reflist}}

== References ==
*[https://web.archive.org/web/20051029080803/http://www.journalofcriticalrealism.org/archive/ALETHIAv3n2_straathof8.pdf Logical Meanderings] – a brief article by Jan Sraathof on [[Luitzen Egbertus Jan Brouwer|Brouwer]]'s various attacks on arguments of the Pre-Intuitionists about the Principle of the Excluded Third.
*[https://web.archive.org/web/20050111050713/http://www.acmsonline.org/Detlefsen87.pdf Proof And Intuition] – an article on the many varieties of knowledge as they relate to the Intuitionist and Logicist.
*[http://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm Brouwer's Cambridge Lectures on Intuitionism] – wherein [[Luitzen Egbertus Jan Brouwer|Brouwer]] talks about the Pre-Intuitionist School and addresses what he sees as its many shortcomings.

[[Category:Theories of deduction]]
[[Category:History of mathematics]]