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In the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
, 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 Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
, who in his 1951 lectures at
Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...
described the differences between
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 __NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a foot ...
, ''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._His__...
,_
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._His__...
,_Georg_Cantor">Cantor_ A_cantor_or_chanter_is_a_person_who_leads_people_in_singing_or_sometimes_in_prayer._In_formal_Jewish_worship,_a_cantor_is_a_person_who_sings_solo_verses_or_passages_to_which_the_choir_or_congregation_responds. In_Judaism,_a_cantor_sings_and_lead_...
,_
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._His__...
,_Georg_Cantor">Cantor_ A_cantor_or_chanter_is_a_person_who_leads_people_in_singing_or_sometimes_in_prayer._In_formal_Jewish_worship,_a_cantor_is_a_person_who_sings_solo_verses_or_passages_to_which_the_choir_or_congregation_responds. In_Judaism,_a_cantor_sings_and_lead_...
,_Giuseppe_Peano">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_stand_...
,__ 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._His__...
,_Georg_Cantor">Cantor_ A_cantor_or_chanter_is_a_person_who_leads_people_in_singing_or_sometimes_in_prayer._In_formal_Jewish_worship,_a_cantor_is_a_person_who_sings_solo_verses_or_passages_to_which_the_choir_or_congregation_responds. In_Judaism,_a_cantor_sings_and_lead_...
,_Giuseppe_Peano">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_stand_...
,__Ernst_Zermelo">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_Louis_Couturat.html" ;"title="Ernst_Zermelo.html" "title="Giuseppe_Peano.html" "title="Georg_Cantor.html" "title="Richard_Dedekind.html" "title="/nowiki>from the "Old Formalist School" of Richard Dedekind">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. His ...
, Georg Cantor">Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...
, Giuseppe Peano">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 stand ...
, Ernst Zermelo">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 Louis Couturat">Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré,
Borel Borel may refer to: People * Borel (author), 18th-century French playwright * Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding ...
and
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
. 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 Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
, 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 f ...
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), ...  all hold. Informal metaphors help ...
, which establishes the property of
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
as necessary to the system. Since Peano's axiom is as
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
as the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: ') is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion, instead of supporting it. For example: * "Green is ...
. 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), ...  all hold. Informal metaphors help ...
is not provable by general logic. Thus arithmetic and mathematics in general is not analytic but synthetic.
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 all ...
thus rebuked and
Intuition Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
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. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
alone, but of
knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...
itself.


Arguments over the excluded middle

It was for this assertion, among others, that Poincaré was considered to be similar to the intuitionists. For
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
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 so-called three laws of thought, along with the law of noncontradic ...
"). 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 Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
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, and yet
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
named other mathematicians as Pre-Intuitionists too;
Borel Borel may refer to: People * Borel (author), 18th-century French playwright * Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding ...
and
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
. Other mathematicians such as
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
(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, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
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 In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also ...
existence proofs that state that something must exist because it can be shown that it is "impossible" for it not to.


See also

*
Conventionalism Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicit or implicit) agreements in society, rather than on external reality. Unspoken rules play a key role in the philosophy's structure ...


Notes

{{Reflist


References


Logical Meanderings
– a brief article by Jan Sraathof on
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
'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 Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
talks about the Pre-Intuitionist School and addresses what he sees as its many shortcomings. Theories of deduction History of mathematics