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60-444: 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 . The vagueness of

120-401: 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

180-472: A conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive

240-420: A created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (Georg Cantor) (G. Cantor [8, p. 252]) The numbers are

300-412: A free creation of human mind. ( R. Dedekind [3a, p. III]) One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor [3, p. 400]) Cantor distinguished two types of actual infinity,

360-423: A fundamental way. This was not an obstacle for the recognition of the correctness of the proof by the community of mathematicians. The mathematical meaning of the term "actual" in actual infinity is synonymous with definite , completed , extended or existential , but not to be mistaken for physically existing . The question of whether natural or real numbers form definite sets is therefore independent of

420-509: A more rational consideration of intuitionism in his Introduction to metamathematics (1952). Nicolas Gisin is adopting intuitionist mathematics to reinterpret quantum indeterminacy , information theory and the physics of time . Luitzen Egbertus Jan 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

480-459: A number of theorems in the emerging field of topology. The most important were his fixed point theorem , the topological invariance of degree, and the topological invariance of dimension . Among mathematicians generally, the best known is the first one, usually referred to now as the Brouwer fixed point theorem. It is a corollary to the second, concerning the topological invariance of degree, which

540-405: 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,

600-411: A potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed." With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed. "For

660-478: A regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid). He was combative as a young man. According to Mark van Atten, this pugnacity reflected his combination of independence, brilliance, high moral standards and extreme sensitivity to issues of justice. He

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720-461: A revised version of this axiom) and the set N {\displaystyle \mathbb {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. According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there

780-492: A self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II "as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics" (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908: "After completing his dissertation, Brouwer made

840-421: 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

900-598: Is a philosophy of the foundations of mathematics . It is sometimes (simplistically) characterized by saying that its adherents do not admit the law of excluded middle as a general axiom in mathematical reasoning, although it may be proven as a theorem in some special cases. Brouwer was a member of the Significs Group . It formed part of the early history of semiotics —the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably cannot be completely disentangled from

960-660: 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: 1 , 2 , . . . {\displaystyle 1,2,...} 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 , N = { 1 , 2 , . . . } {\displaystyle \mathbb {N} =\{1,2,...\}} . In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example,

1020-599: 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 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

1080-528: Is generally explicitly stated; for example finite geometry , finite field , etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic , which has been proved only more than 350 years later. The original Wiles's proof of Fermat's Last Theorem , used not only the full power of ZF with the axiom of choice , but used implicitly a further axiom that implies the existence of very large sets. The requirement of this further axiom has been later dismissed, but infinite sets remains used in

1140-402: Is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which intuitionism attempts to construct/refute/refound are taken as intuitively given. 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

1200-513: Is never complete: elements can be always added, but never infinitely many. "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different." Aristotle distinguished between infinity with respect to addition and division. But Plato has two infinities, the Great and the Small. "As an example of

1260-440: Is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. ( C.F. Gauss [in a letter to Schumacher, 12 July 1831]) Actual infinity is now commonly accepted in mathematics, although the term is no longer in use, being replaced by the concept of infinite sets . This drastic change

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1320-403: 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

1380-544: Is now commonly accepted in mathematics under the name " infinite set ". Indeed, set theory has been formalized as the Zermelo–Fraenkel set theory (ZF). One of the axioms of ZF is the axiom of infinity , that essentially says that the natural numbers form a set. All mathematics has been rewritten in terms of ZF. In particular, line , curves , all sort of spaces are defined as the set of their points. Infinite sets are so common, that when one considers finite sets, this

1440-473: Is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also." (Aristotle) The theme was brought forward by Aristotle's consideration of the apeiron—in the context of mathematics and physics (the study of nature): "Infinity turns out to be

1500-402: Is presently commonly accepted as a foundation of mathematics, contains the axiom of infinity , which means that the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accepts infinite sets, then there are different sizes ( cardinalities ) of infinite sets, and, in particular, the cardinal of the continuum of the real numbers is strictly larger than

1560-419: 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 . There is thus an asymmetry between

1620-399: Is the best known among algebraic topologists. The third theorem is perhaps the hardest. Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology , which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes , of the treatment of general continuous mappings. In 1912, at age 31, he was elected a member of

1680-496: The Brouwer–Hilbert controversy , in which Brouwer sparred with his formalist colleague David Hilbert . Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl . In addition to his mathematical work, Brouwer also published the short philosophical tract Life, Art, and Mysticism (1905). Brouwer was born to Dutch Protestant parents. Early in his career, Brouwer proved

1740-731: The Royal Netherlands Academy of Arts and Sciences . He was an Invited Speaker of the ICM in 1908 at Rome and in 1912 at Cambridge, UK. He was elected to the American Philosophical Society in 1943. Brouwer founded intuitionism , a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays , Wilhelm Ackermann , and John von Neumann (cf. Kleene (1952), p. 46–59). A variety of constructive mathematics , intuitionism

1800-544: 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

1860-590: The 'apeiron' was some sort of basic substance. Plato 's notion of the apeiron is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides and the Philebus . Aristotle sums up the views of his predecessors on infinity as follows: "Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what

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1920-497: The Renaissance and by early modern times the voices in favor of actual infinity were rather rare. The continuum actually consists of infinitely many indivisibles ( G. Galilei [9, p. 97]) I am so in favour of actual infinity. ( G.W. Leibniz [9, p. 97]) However, the majority of pre-modern thinkers agreed with the well-known quote of Gauss: I protest against the use of infinite magnitude as something completed, which

1980-457: The axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction . The present-day conventional finitist interpretation of ordinal and cardinal numbers is that they consist of a collection of special symbols, and an associated formal language , within which statements may be made. All such statements are necessarily finite in length. The soundness of

2040-487: The cardinal of the natural numbers. Actual infinity is to be contrasted with potential infinity , in which an endless process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an infinite series , infinite product , or limit . The ancient Greek term for

2100-411: The completed infinity of the integers. For intuitionists, infinity is described as potential ; terms synonymous with this notion are becoming or constructive . For example, Stephen Kleene describes the notion of a Turing machine tape as "a linear 'tape', (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a read head along it in finitely many steps:

2160-460: The concept of actual infinity has been a subject of debate among philosophers. Also, the question of whether the Universe is infinite is still a debate between physicists. The concept of actual infinity has been introduced in mathematics near the end of the 19th century by Georg Cantor , with his theory of infinite sets , later formalized into Zermelo–Fraenkel set theory . This theory, which

2220-437: The development of intuitionism at its source was taken up by his student Arend Heyting . Dutch mathematician and historian of mathematics Bartel Leendert van der Waerden attended lectures given by Brouwer in later years, and commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on — and only on — the foundations of his intuitionism. It seemed that he

2280-424: The fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately." Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the [actual] infinite and do not use it" ( Phys. III 2079 29). The overwhelming majority of scholastic philosophers adhered to

2340-440: The greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension . Brouwer also became a major figure in the philosophy of intuitionism , a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth . This position led to

2400-598: The heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude. Aristotle handled the topic of infinity in Physics and in Metaphysics . He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity

2460-510: The intellectual milieu of that group. In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism , which has been described by the mathematician Martin Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer then "embarked on

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2520-401: 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 . 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

2580-450: 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. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics,

2640-432: The manipulations is founded only on the basic principles of a formal language: term algebras , term rewriting , and so on. More abstractly, both (finite) model theory and proof theory offer the needed tools to work with infinities. One does not have to "believe" in infinity in order to write down algebraically valid expressions employing symbols for infinity. The philosophical problem of actual infinity concerns whether

2700-619: The motto Infinitum actu non datur . This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity . There were exceptions, however, for example in England. It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. ( G. Cantor ) Actual infinity exists in number, time and quantity. (J. Baconthorpe [9, p. 96]) During

2760-422: 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 (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 includes the axiom of infinity from ZFC (or

2820-426: The opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." (Aristotle) Belief in the existence of the infinite comes mainly from five considerations: Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than

2880-530: 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. Intuitionism's history can be traced to two controversies in nineteenth century mathematics. The first of these

2940-421: The potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon . Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite , respectively. Anaximander (610–546 BC) held that the apeiron was the principle or main element composing all things. Clearly,

3000-408: The question of whether infinite things exist physically in nature . Proponents of intuitionism , from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand, constructive analysis does accept the existence of

3060-422: The set of all real numbers R {\displaystyle \mathbb {R} } is larger than N {\displaystyle \mathbb {N} } , because any attempt 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

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3120-485: 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". Later, Stephen Cole Kleene brought forth

3180-403: The tape is therefore only "potentially" infinite, since — while there is always the ability to take another step — infinity itself is never actually reached. Mathematicians generally accept actual infinities. Georg Cantor is the most significant mathematician who defended actual infinities. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects

3240-529: 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

3300-661: The transfinite and the absolute, about which he affirmed: These concepts are to be strictly differentiated, insofar the former is, to be sure, infinite , yet capable of increase , whereas the latter is incapable of increase and is therefore indeterminable as a mathematical concept. This mistake we find, for example, in Pantheism . (G. Cantor, Über verschiedene Standpunkte in bezug auf das aktuelle Unendliche , in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts , pp. 375, 378) Actual infinity

3360-400: The transfinite numbers and sets of mathematics. A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set [of members of the kind in question] is only a part of it, I will call an infinite multitude. (B. Bolzano [2, p. 6]) Accordingly I distinguish an eternal uncreated infinity or absolutum, which is due to God and his attributes, and

3420-517: Was initialized by Bolzano and Cantor in the 19th century, and was one of the origins of the foundational crisis of mathematics . Bernard Bolzano , who introduced the notion of set (in German: Menge ), and Georg Cantor, who introduced set theory , opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3)

3480-440: Was involved in a very public and eventually demeaning controversy with Hilbert in the late 1920s over editorial policy at Mathematische Annalen , at the time a leading journal. According to Abraham Fraenkel , Brouwer espoused Germanic Aryanness and Hilbert removed him from the editorial board of Mathematische Annalen after Brouwer objected to contributions from Ostjuden . In later years Brouwer became relatively isolated;

3540-504: Was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy." About his last years, Davis (2002) remarks: Actual infinity In the philosophy of mathematics , the abstraction of actual infinity , also called completed infinity , involves infinite entities as given, actual and completed objects. Since Greek antiquity ,

3600-441: 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

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