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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 peop ...
, intuitionism, or neointuitionism (opposed to
preintuitionism In the philosophy of mathematics, 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, who in his 1951 lectures at Cambridge ...
), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.


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's 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 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; ...
. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning.
Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
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. 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 they prove is in fact intuitionistically true. This gives rise to
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
. 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 In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
; 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''. There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement ''P'' is provable, then ''P'' certainly cannot be refutable. But even if it can be shown that ''P'' cannot be refuted, this does not constitute 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 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 noncontradi ...
, "''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 In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
as they do in classical logic.
Intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
substitutes constructability for abstract
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belie ...
and is associated with a transition from the proof of model theory to 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 In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument ...
of
Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He w ...
. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g.
Fuzzy Sets In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
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.


Infinity

Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity. The term
potential infinity In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extend ...
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 In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, exten ...
refers to a completed mathematical object which contains an infinite number of elements. An example is the set of
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 ...
s, 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 is larger than , 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
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
(ZFC), now the most common foundation of modern mathematics. Intuitionism was created, in part, as a reaction to Cantor's set theory. Modern
constructive set theory Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a con ...
includes the axiom of infinity from ZFC (or a revised version of this axiom) and the set of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see
Alexander Esenin-Volpin Alexander Sergeyevich Esenin-Volpin (also written Ésénine-Volpine and Yessenin-Volpin in his French and English publications; russian: Алекса́ндр Серге́евич Есе́нин-Во́льпин, p=ɐlʲɪˈksandr sʲɪrˈɡʲejɪ ...
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)


History

Intuitionism's history can be traced to two controversies in nineteenth century mathematics. The first of these was the invention of transfinite arithmetic by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
and its subsequent rejection by a number of prominent mathematicians including most famously his teacher
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, ...
—a confirmed finitist. The second of these was
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful
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, ...
, the discoverer of
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
. Frege had planned a three volume definitive work, but just as the second volume was going to press, Russell sent Frege a letter outlining his paradox, which demonstrated that one of Frege's rules of self-reference was self-contradictory. In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell's paradox. Frege, the story goes, plunged into depression and did not publish the third volume 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 Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
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 ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952).
Nicolas Gisin Nicolas Gisin (born 1952) is a Swiss physicist and professor at the University of Geneva working on the foundations of quantum mechanics, and quantum information and communication. His work includes both experimental and theoretical physics. He ...
is adopting intuitionist mathematics to reinterpret quantum indeterminacy, information theory and the physics of time.


Contributors

* Henri Poincaré (
preintuitionism In the philosophy of mathematics, 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, who in his 1951 lectures at Cambridge ...
/
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 structur ...
) * L. E. J. Brouwer *
Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He w ...
* Arend Heyting *
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...


Branches of intuitionistic mathematics

*
Intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
* Intuitionistic arithmetic *
Intuitionistic type theory Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician an ...
* Intuitionistic set theory * Intuitionistic analysis


See also

*
Anti-realism In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument ...
* BHK interpretation * Brouwer–Hilbert controversy * Computability logic *
Constructive logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...
* Curry–Howard isomorphism *
Foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
* Fuzzy logic *
Game semantics Game semantics (german: dialogische Logik, translated as ''dialogical logic'') is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on game theory, game-theoretic concepts, su ...
*
Intuition (knowledge) 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; ...
* Model theory *
Topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
* Ultraintuitionism


References


Further reading

*"Analysis." ''Encyclopædia Britannica''. 2006.
Encyclopædia Britannica 2006 Ultimate Reference Suite DVD An encyclopedia (American English) or encyclopædia (British English) is a reference work or compendium providing summaries of knowledge either general or special to a particular field or discipline. Encyclopedias are divided into articles ...
15 June 2006, "
Constructive analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more com ...
" ( Ian Stewart, 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
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
(with respect to Godel),
Formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scie ...
(with respect to Hilbert), and Intuitionism (with respect to Brouwer). * Martin Davis (ed.) (1965), ''The Undecidable'', Raven Press, Hewlett, NY. Compilation of original papers by Gödel, Church, Kleene, Turing, Rosser, and Post. Republished as * * John W. Dawson Jr., ''Logical Dilemmas: The Life and Work of Kurt Gödel'', 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 Goldstein, ''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 Gödel was reticent in the presence of the logical positivism of the Vienna Circle. Goldstein discusses
Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...
's impact and the impact of the formalists. Goldstein notes that the intuitionists were even more opposed to
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
than
Formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scie ...
. * van Heijenoort, J., ''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'' eprinted with commentary, p. 334, van Heijenoort:* Andrei Nikolaevich
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, 1925, ''On the principle of excluded middle'', eprinted with commentary, p. 414, van Heijenoort:* L.E.J. Brouwer, 1927, ''On the domains of definitions of functions'', eprinted 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'', eprinted with commentary, p. 490, van Heijenoort:* Jacques Herbrand, (1931b), "On the consistency of arithmetic", eprinted 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". * * Arend Heyting: * :In Chapter III ''A Critique of Mathematic Reasoning, §11. The paradoxes'', Kleene discusses Intuitionism and
Formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scie ...
in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. *
Stephen Cole Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
and Richard Eugene Vesley, ''The Foundations of Intuitionistic 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 Putnam and
Paul Benacerraf Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He ...
, ''Philosophy of Mathematics: Selected Readings'', Englewood Cliffs, N.J.: Prentice-Hall, 1964. 2nd ed., Cambridge: Cambridge University Press, 1983. : Part I. ''The foundation of mathematics'', ''Symposium on the foundations of mathematics'' :* Rudolf Carnap, ''The logicist foundations of mathematics'', p. 41 :* Arend Heyting, ''The intuitionist foundations of mathematics'', p. 52 :* Johann von Neumann, ''The formalist foundations of mathematics'', p. 61 :* Arend Heyting, ''Disputation'', p. 66 :* L. E. J. Brouwer, ''Intuitionnism and formalism'', p. 77 :* L. E. J. Brouwer, ''Consciousness, philosophy, and mathematics'', p. 90 *
Constance Reid Constance Bowman Reid (January 3, 1918 – October 14, 2010) 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 ...
, ''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 Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) * Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...
, ''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. * Jacques Hartong and Georges Reeb
''Intuitionnisme 84''
(first published in ''La Mathématique Non-standard'', éditions du C.N.R.S.) : A reevaluation of intuitionism, from the point of view (among others) of
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
and
non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
.


Secondary references

* A. A. Markov (1954) ''Theory of algorithms''. ranslated by Jacques J. Schorr-Kon and PST staffImprint Moscow, Academy of Sciences of the USSR, 1954 .e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, WashingtonDescription 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. A248.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''.... . 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. * {{Authority control Epistemology Constructivism (mathematics) Philosophy of mathematics de:Intuitionismus