Intuitionism
Encyclopedia
In the philosophy of mathematics
, intuitionism, or neointuitionism (opposed to preintuitionism
), is an approach to mathematics
as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs.
original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition
. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene
formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill defined. Regardless of how it is interpreted, intuitionism does not equate the truth of a mathematical statement with its provability. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he proves is in fact intuitionistically true. This gives rise to intuitionistic logic
.
----
To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind
, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable (e.g., that there is a counterexample
). There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.
Similarly, to assert that A or
B holds, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle
, "A or not
A", is not accepted as a valid principle. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of "A or not A". However, the intuitionist will accept that "A and not A" cannot be true. Thus the connectives "and" and "or" of intuitionistic logic do not satisfy de Morgan's laws as they do in classical logic.
Intuitionistic logic
substitutes constructability for abstract truth
and is associated with a transition from the proof to model theory
of abstract truth in modern mathematics. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism
of Michael Dummett
. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct/refute/refound are taken as intuitively given.
The term potential infinity refers to a mathematical procedure in which there is an unending series of steps. After each step has been completed, there is always another step to be performed. For example, consider the process of counting:
The term actual infinity
refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers,
In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers R is larger than N, because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".
Cantor's set theory led to the axiomatic system of ZFC, now the most common foundation of modern mathematics.
Intuitionism was created, in part, as a reaction to Cantor's set theory. All forms of intuitionism reject the reality of uncountable infinite sets.
Modern constructive set theory
does include the axiom of infinity from Zermelo-Fraenkel set theory (or a revised version of this axiom), and includes the set N of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin
for a counter-example).
Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.
Finitism
is an extreme version of Intuitionism that rejects the idea of potential infinity. According to Finitism, a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps.
The first of these was the invention of transfinite arithmetic
by Georg Cantor
and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker
— a confirmed finitist
.
The second of these was Gottlob Frege
's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell
, the discoverer of Russell's paradox
. Frege had planned a three volume definitive work, but shortly after the first volume had been published, Russell sent Frege a letter outlining his paradox which demonstrated that one of Frege's rules of self-reference was self-contradictory.
Frege, the story goes, plunged into depression and did not publish the second and third volumes of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and van Heijenoort's commentary.
These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.
In the early twentieth century L. E. J. Brouwer
represented the intuitionist position and David Hilbert
the formalist
position — see van Heijenoort. Kurt Gödel
offered opinions referred to as Platonist (see various sources re Gödel). Alan Turing
considers:
"non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive". (Turing 1939, reprinted in Davis 2004, p. 210) Later, Stephen Cole Kleene
brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952).
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, intuitionism, or neointuitionism (opposed to preintuitionism
Preintuitionism
In some circles of mathematical philosophy, the Pre-Intuitionists are considered to be a small but influential group who informally shared similar philosophies on the nature of mathematics. The term itself was used by L. E. J...
), is an approach to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs.
Truth and proof
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer'sLuitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition
Intuition (philosophy)
Intuition is a priori knowledge or experiential belief characterized by its immediacy. Beyond this, the nature of intuition is debated. Roughly speaking, there are two main views. They are:...
. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene
Stephen Cole Kleene
Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...
formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill defined. Regardless of how it is interpreted, intuitionism does not equate the truth of a mathematical statement with its provability. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he proves is in fact intuitionistically true. This gives rise to intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
.
----
To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind
Mind
The concept of mind is understood in many different ways by many different traditions, ranging from panpsychism and animism to traditional and organized religious views, as well as secular and materialist philosophies. Most agree that minds are constituted by conscious experience and intelligent...
, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable (e.g., that there is a counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
). There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.
Similarly, to assert that A or
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
B holds, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
, "A or not
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
A", is not accepted as a valid principle. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of "A or not A". However, the intuitionist will accept that "A and not A" cannot be true. Thus the connectives "and" and "or" of intuitionistic logic do not satisfy de Morgan's laws as they do in classical logic.
Intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
substitutes constructability for abstract truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
and is associated with a transition from the proof to model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of abstract truth in modern mathematics. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism
Anti-realism
In analytic philosophy, the term anti-realism is used to describe any position involving either the denial of an objective reality of entities of a certain type or the denial that verification-transcendent statements about a type of entity are either true or false...
of Michael Dummett
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct/refute/refound are taken as intuitively given.
Intuitionism and infinity
Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity.The term potential infinity refers to a mathematical procedure in which there is an unending series of steps. After each step has been completed, there is always another step to be performed. For example, consider the process of counting:
The term actual infinity
Actual infinity
Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural...
refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers,
In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers R is larger than N, because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".
Cantor's set theory led to the axiomatic system of ZFC, now the most common foundation of modern mathematics.
Intuitionism was created, in part, as a reaction to Cantor's set theory. All forms of intuitionism reject the reality of uncountable infinite sets.
Modern constructive set theory
Constructive set theory
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first-order language of classical set theory, and although of course the logic is constructive, there is no explicit use of constructive types...
does include the axiom of infinity from Zermelo-Fraenkel set theory (or a revised version of this axiom), and includes the set N of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin
Alexander Esenin-Volpin
Alexander Sergeyevich Esenin-Volpin is a prominent Russian-American poet and mathematician.Born on May 12, 1924 in the former Soviet Union, he was a notable dissident, political prisoner, poet, and mathematician...
for a counter-example).
Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.
- "According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence." (Kleene (1952): Introduction to Metamathematics, p. 48-49)
Finitism
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
is an extreme version of Intuitionism that rejects the idea of potential infinity. According to Finitism, a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps.
History of Intuitionism
Intuitionism's history can be traced to two controversies in nineteenth century mathematics.The first of these was the invention of transfinite arithmetic
Transfinite arithmetic
In mathematics, transfinite arithmetic is the generalization of elementary arithmetic to infinite quantities like infinite sets. It was originally discovered by the Russian-born German mathematician Georg Cantor.- See also :* transfinite number...
by Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
— a confirmed finitist
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
.
The second of these was Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, the discoverer of Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
. Frege had planned a three volume definitive work, but shortly after the first volume had been published, Russell sent Frege a letter outlining his paradox which demonstrated that one of Frege's rules of self-reference was self-contradictory.
Frege, the story goes, plunged into depression and did not publish the second and third volumes of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and van Heijenoort's commentary.
These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.
In the early twentieth century L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
represented the intuitionist position and David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
the formalist
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
position — see van Heijenoort. Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
offered opinions referred to as Platonist (see various sources re Gödel). Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
considers:
"non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive". (Turing 1939, reprinted in Davis 2004, p. 210) Later, Stephen Cole Kleene
Stephen Cole Kleene
Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...
brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952).
Contributors to intuitionism
- L. E. J. BrouwerLuitzen Egbertus Jan BrouwerLuitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
- Michael DummettMichael DummettSir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
- Arend HeytingArend HeytingArend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
- Stephen Kleene
Branches of intuitionistic mathematics
- Intuitionistic logicIntuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
- Intuitionistic arithmetic
- Intuitionistic type theoryIntuitionistic type theoryIntuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...
- Intuitionistic set theory
- Intuitionistic analysis
See also
- Anti-realismAnti-realismIn analytic philosophy, the term anti-realism is used to describe any position involving either the denial of an objective reality of entities of a certain type or the denial that verification-transcendent statements about a type of entity are either true or false...
- Benjamin PeirceBenjamin PeirceBenjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics....
- BHK interpretationBHK interpretationIn mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer, Arend Heyting and independently by Andrey Kolmogorov...
- Brouwer–Hilbert controversy
- Computability logicComputability logicIntroduced by Giorgi Japaridze in 2003, computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth...
- Constructive logic
- Curry–Howard isomorphism
- Foundations of mathematicsFoundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
- FuzzFuzzFuzz may refer to:*Vellus, a type of short, fine body hair on an animal*Tomentum, a filamentous hairlike growth on a plant*Focus , a blur effect*Fuzzbox, an electric guitar distortion effect*A derogatory slang term for the police...
(mathematics and computer science) - Game semanticsGame semanticsGame semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes. In the late 1950s Paul Lorenzen was the...
- Model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
- Intuition (knowledge)Intuition (knowledge)Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...
- Ultraintuitionism
Further reading
- "Analysis." Encyclopædia Britannica. 2006. Encyclopædia Britannica 2006 Ultimate Reference Suite DVD 15 June 2006, "Constructive analysis" (Ian StewartIan Stewart (mathematician)Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...
, author)
- W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, New York, 1994.
- In Chapter 39 Foundations, with respect to the 20th century Anglin gives very precise, short descriptions of PlatonismPlatonismPlatonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism...
(with respect to Godel), FormalismFormalism (mathematics)In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
(with respect to Hilbert), and Intuitionism (with respect to Brouwer).
- Martin DavisMartin DavisMartin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...
(ed.) (1965), The Undecidable, Raven Press, Hewlett, NY. Compilation of original papers by Gödel, Church, Kleene, Turing, Rosser, and Post. Republished as
- John W. DawsonJohn W. DawsonJohn W. Dawson was Governor of Utah Territory in 1861.Born on October 21, 1820, in Cambridge, Indiana he was a lawyer, a farmer and a newspaper editor before he entered politics, unsuccessfully running for a seat in the Indiana House of Representatives in 1854, Secretary of State of Indiana in...
Jr., Logical Dilemmas: The Life and Work of Kurt GödelKurt GödelKurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, A. K. Peters, Wellesley, MA, 1997.
- Less readable than Goldstein but, in Chapter III Excursis, Dawson gives an excellent "A Capsule History of the Development of Logic to 1928".
- Rebecca GoldsteinRebecca GoldsteinRebecca Goldstein is an American novelist and professor of philosophy. She has written five novels, a number of short stories and essays, and biographical studies of mathematician Kurt Gödel and philosopher Baruch Spinoza....
, Incompleteness: The Proof and Paradox of Kurt Godel, Atlas Books, W.W. Norton, New York, 2005.
- In Chapter II Hilbert and the Formalists Goldstein gives further historical context. As a Platonist GödelGodelGodel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
was reticent in the presence of the logical positivismLogical positivismLogical positivism is a philosophy that combines empiricism—the idea that observational evidence is indispensable for knowledge—with a version of rationalism incorporating mathematical and logico-linguistic constructs and deductions of epistemology.It may be considered as a type of analytic...
of the Vienna Circle. She discusses Wittgenstein's impact and the impact of the formalists. Goldstein notes that the intuitionists were even more opposed to PlatonismPlatonismPlatonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism...
than FormalismFormalism (mathematics)In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
.
- van Heijenoort, J.Jean Van HeijenoortJean Louis Maxime van Heijenoort was a pioneer historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and from then until 1947, an American Trotskyist activist.-Life:Van Heijenoort was born in Creil, France...
, From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. The following papers appear in van Heijenoort: - L.E.J. Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]
- Andrei Nikolaevich Kolmogorov, 1925, On the principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
- L.E.J. Brouwer, 1927, On the domains of definitions of functions, [reprinted with commentary, p. 446, van Heijenoort]
-
- Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
- L.E.J. Brouwer, 1927(2), Intuitionistic reflections on formalism, [reprinted with commentary, p. 490, van Heijenoort]
- Jacques Herbrand, (1931b), "On the consistency of arithmetic", [reprinted with commentary, p. 618ff, van Heijenoort]
- From van Heijenoort's commentary it is unclear whether or not Herbrand was a true "intuitionist"; Gödel (1963) asserted that indeed "...Herbrand was an intuitionist". But van Heijenoort says Herbrand's conception was "on the whole much closer to that of Hilbert's word 'finitary' ('finit') that to "intuitionistic" as applied to Brouwer's doctrine".
- Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
- Arend HeytingArend HeytingArend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
:
- In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and FormalismFormalism (mathematics)In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.
- Stephen Cole KleeneStephen Cole KleeneStephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...
and Richard Eugene Vesley, The Foundations of Intuistionistic Mathematics, North-Holland Publishing Co. Amsterdam, 1965. The lead sentence tells it all "The constructive tendency in mathematics...". A text for specialists, but written in Kleene's wonderfully-clear style.
- Hilary PutnamHilary PutnamHilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
and Paul BenacerrafPaul BenacerrafPaul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....
, Philosophy of Mathematics: Selected Readings, Englewood Cliffs, N.J.: Prentice-Hall, 1964. 2nd ed., Cambridge: Cambridge University Press, 1983. ISBN 0-521-29648-X
- Part I. The foundation of mathematics, Symposium on the foundations of mathematics
- Rudolph CarnapRudolf CarnapRudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
, The logicist foundations of mathematics, p. 41 - Arend HeytingArend HeytingArend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
, The intuitionist foundations of mathematics, p. 52 - Johann von NeumannJohn von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, The formalist foundations of mathematics, p. 61 - Arendt Heyting, Disputation, p. 66
- L.E.J. Brouwer, Intuitionnism and formalism, p. 77
- L.E.J. Brouwer, Consciousness, philosophy, and mathematics, p. 90
- Rudolph Carnap
- Constance ReidConstance ReidConstance Bowman Reid was the author of several biographies of mathematicians and popular books about mathematics. She received several awards for mathematical exposition. She was not a mathematician but comes from a mathematical family: Her sister is Julia Robinson and her brother-in-law is...
, Hilbert, Copernicus - Springer-Verlag, 1st edition 1970, 2nd edition 1996.
- Definitive biography of Hilbert places his "Program" in historical context together with the subsequent fighting, sometimes rancorous, between the Intuitionists and the Formalists.
- Paul Rosenbloom, The Elements of Mathematical Logic, Dover Publications Inc, Mineola, New York, 1950.
- In a style more of Principia Mathematica – many symbols, some antique, some from German script. Very good discussions of intuitionism in the following locations: pages 51-58 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 69-73 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146-151 Section 7 the Axiom of Choice.
Secondary references
- A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]
- A secondary reference for specialists: Markov opined that "The entire significance for mathematics of rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3, italics added.] Markov believed that further applications of his work "merit a special book, which the author hopes to write in the future" (p. 3). Sadly, said work apparently never appeared.