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In the
philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
, the pre-intuitionists is the name given by
L. E. J. Brouwer Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the ...
to several influential mathematicians who shared similar opinions on the nature of mathematics. The term was introduced by Brouwer in his 1951 lectures at
Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...
where he described the differences between his philosophy of
intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
and its predecessors:Luitzen Egbertus Jan Brouwer (edited by Arend Heyting, ''Collected Works'', North-Holland, 1975, p. 509.
Of a totally different orientation
Dedekind Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
,
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. ...
,
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stan ...
, Zermelo Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
, and Couturat, etc.">Louis Couturat">Couturat, etc./nowiki> was the Pre-Intuitionist School, mainly led by
Poincaré Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
, 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 /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.


The introduction of natural numbers

The pre-intuitionists, as defined by
L. E. J. Brouwer Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the ...
, 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 Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
, 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 Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
to argue with
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
over Giuseppe Peano's axiomatic theory of natural numbers. Peano's fifth
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
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 Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a simple case, then ...
, which establishes the property of induction as necessary to the system. Since Peano's axiom is as infinite as the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, 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 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 Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a simple case, then ...
is not provable by general logic. Thus arithmetic and mathematics in general is not analytic but
synthetic Synthetic may refer to: Science * Synthetic biology * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic elements, chemical elements that are not naturally found on Earth and therefore have to be created in ...
.
Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or al ...
thus rebuked and
Intuition Intuition is the ability to acquire knowledge without recourse to conscious reasoning or needing an explanation. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledg ...
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 Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
alone, but of
knowledge Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is oft ...
itself.


Arguments over the excluded middle

It was for this assertion, among others, that
Poincaré Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
was considered to be similar to the intuitionists. For 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 In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
"). 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 situations. To an intuitionist this seems to rank the law of excluded middle as just as un rigorous as 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
Poincaré Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
, and yet Brouwer named other mathematicians as Pre-Intuitionists too; Borel and Lebesgue. Other mathematicians such as
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
(who eventually became disenchanted with intuitionism, feeling that it places excessive strictures on mathematical progress) and
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
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 non-constructive 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


Logical Meanderings
– a brief article by Jan Sraathof on Brouwer's various attacks on arguments of the Pre-Intuitionists about the Principle of the Excluded Third.
Proof And Intuition
– an article on the many varieties of knowledge as they relate to the Intuitionist and Logicist.

– wherein Brouwer talks about the Pre-Intuitionist School and addresses what he sees as its many shortcomings. Theories of deduction History of mathematics