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On A.Ya. Khinchin's paper ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ (1926): A translation with introduction and commentary
Abstract
The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.
... The target of this investigation lies at the confluence of the history of math especially the proliferation started in the 19th century as modern math gradually developed, philosophy, and education. This is a common aim for mathematicians to navigate the boundary between intuition and rigor, or in a higher hierarchy, the classic schools of intuitionism and formalism (where debates flourish between definitions of intuitionism especially the foundational debate between L.E.J. Brouwer (1881Brouwer ( -1966 and David Hilbert (1862-1943) [5]). Hilbert's program aimed to make intuitionism ideologically accepted by his audience or supporters while Brouwer believed that there is a clear 'subject matter' that makes a connection with formalism [6,7]). ...
Intuition and rigor are two inevitable components in mathematical problem-solving. Traditionally, features of education in mathematics show that those two themes are seen to be mutually exclusive rather than complementary. The famous foundational debate between L.E.J Brouwer and David Hilbert shows that intuition and rigor are often seen as being at odds. The two characteristics of mathematics play a pivotal and dynamic role in education, reasoning problem-solving, and different applications of maths. This essay delves into the intricate relationship between these seemingly contrasting approaches, underscoring their significance in the evolution of mathematical development. The research employs a blend of historical analysis and philosophical inquiry to explore the roles of intuition and rigor and how they interact. The essay argues that while intuition often sparks the initial insights in mathematical discovery, it is the rigor that ensures the robustness and validity of these insights, leading to their acceptance within the mathematical community. Through case studies of renowned mathematicians and well-known problems, this research uncovers the symbiotic nature of intuition and rigor and is demonstrating how they collectively contribute to the richness of mathematical problem-solving. Those findings suggest that a balanced integration of both features is crucial for the integrity including consistency and completeness of mathematical knowledge.
... Hilbert's roundabout way to treat mathematical contents involving infinities via finitary syntactic constructions, which they represent, Kolmogorov describes as a "brilliant art" (or Khintichin who in 1926 published a philosophical paper Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics [42], English translation in [89]. The "subject matter" in the title is English translation of Russian "predmet", which can be also translated into English as "object" and into German as "Gegenstand" -notice the same German word used by Kolmogorov in his critique of Hilbert in the above quote. ...
Kolmogorov's Calculus of Problems is an interpretation of Heyting's intuitionistic propositional calculus published by A.N. Kolmogorov in 1932. Unlike Heyting's intended interpretation of this calculus, Kolmogorov's interpretation does not comply with the philosophical principles of Mathematical Intuitionism. This philosophical difference between Kolmogorov and Heyting implies different treatments of problems and propositions: while in Heyting's view the difference between problems and propositions is merely linguistic, Kolmogorov keeps the two concepts apart and does not apply his calculus to propositions. I stress differences between Kolmogorov's and Heyting's interpretations and show how the two interpretations diverged during their development. In this context I reconstruct Kolmogorov's philosophical views on mathematics and analyse his original take on the Hilbert-Brouwer controversy. Finally, I overview some later works motivated by Kolmogorov's Calculus of Problems and propose a justification of Kolmogorov's distinction between problems and propositions in terms of Univalent Mathematics.
“Classical” set theory in the style of Cantor, Dedekind and Zermelo enjoyed dominance during the early twentieth century, playing a prominent role in many of the “modern” mathematical developments, in analysis, algebra, topology, and so on. Yet the theory was polemical, and one finds a characteristic pattern of second thoughts about set theory, after an initial enthusiasm; examples we briefly discuss are Borel, Weyl, and Rey Pastor, which contrast with Hilbert’s optimism. As a result, one can speak of a proliferation of set theories – in the plural – during the period 1910–1950. Classic examples are the (more or less coherent) predicative proposals of Russell and Weyl, the “intuitionistic set theory” developed by Brouwer, and several other systems which we cannot discuss here. We consider in this paper whether such criticism of set theory was a symptom of traditionalism, which leads to an analysis of the notion of modernism, paying especial attention to the case of L.E.J. Brouwer. I shall argue that modernism is a somewhat ambiguous notion, and that Brouwer (like Weyl) can indeed be regarded as prototypically “modern” in a sense that was characteristic of the Inter War period 1918–1939.
The paper examines Yankov’s contribution to the history of mathematical logic and the foundations of mathematics. It concerns the public communication of Markov’s critical attitude towards Brouwer’s intuitionistic mathematics from the point of view of his constructive mathematics and the commentary on A.S. Esenin-Vol’pin program of ultra-intuitionistic foundations of mathematics.KeywordsV.A. YankovA.A. MarkovA.S. Esenin-Vol’pinL.E.J. BrouwerA. HeytingE.A. BishopB.A. KushnerMarkov’s constructive mathematicsIntuitionistic mathematicsPhilosophy of constructive mathematicsMathematics Subject Classification:03-0303F5501A6001A72
In this paper, we focus on the launch of the Great Soviet Encyclopedia, which was first published in 1925. We present the context of the launch and explain why it was closely connected to the period of the New Economic Policy. In the last section, we examine four articles about randomness and probability included in the first volumes of the encyclopedia in order to illustrate some debates from within the scientific scene in the USSR during the 1920s.
In this paper we focus on the beginning of publication of the Large Soviet Encyclopedia, launched in 1925. We present the context of this launching and explain why it was tightly connected to the period of the New Economical Policy. In a last section, we examine four articles included in the first volumes of the encyclopedia and relative to randomness and probability, in order to illustrate some debates of the scientific scene in USSR during the 1920s.
The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.
L. E. J. Brouwer and David Hilbert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notorious Grundlagenstreit centered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his "Intuitionism" did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics. Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a nonstandard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy. Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework - not from Kant's philosophy of mathematics but from his general metaphysics - which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy.
This chapter describes the importance of consciousness, philosophy, and mathematics. Consciousness in its deepest home seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation. It seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. By a move of time, a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind. Mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding, creating new mathematical entities in the shape of predeterminately, or more or less freely proceeding infinite sequences of mathematical entities acquired, and in the shape of mathematical species.
Brouwer's dissertation contains both the germs of his topological and his foundational work. We concentrate here on the latter. The extraordinary rich thesis contains comments and critique on his contemporaries, and a novel approach to many foundational issues. Between the lines one finds the genesis of the continuum and the natural numbers via the ur-intuition, the constructive interpretation of logic, choice sequences, and a precise discussion of the language, logic, and mathematics levels. The present paper provides a survey of the material and comments on Brouwer's innovations.
Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian.1 Brouwer’s theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or “realistic”) view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is determinately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles.
The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s
This chapter describes a formal system for intuitionistic analysis and the intuitive justification of this system. The system discussed is called “Choice Sequence” (CS). CS contains numerical variables, two kinds of variables for constructive functions, and variables for choice sequences. The subsystem of CS obtained by restricting the language, axioms, and the rules of CS to expressions not involving variables for choice sequences is called IDK, because the main feature of the subsystem is an inductive definition of a class of constructive functions called K. The formal work on the system CS was started by Kreisel (which was privately circulated only) and was recently improved and rounded off to a certain extent by a joint effort of Kreisel and the author.
From the text (B. V. Gnedenko 1961): The paper presented here was written as early as sometime between 1939 and 1944 by the eminent mathematician Aleksandr Yakovlevich Khinchin [Vopr. Filos. Nauchn.-Tekhn. Zh. 15, No. 1, 91–92], who is well known for his contributions to the theory of probability, statistical physics, theory of numbers, and theory of functions. For reasons unknown to me, it remained unpublished, although I remember that Khinchin had submitted it to the periodical Usp. Mat. Nauk. After he died, while I was putting in order the scientific and literary heritage of Khinchin, I remembered this work and began looking for it. Regrettably, I was unable to find any copies of a final version and the editorial office of Uspekhi did not have any record of the article. So I decided to make use of a copy that had been retyped in 1946 by my students, E. L. Rvacheva and D. G. Meyzler, even though it had some lacunae. I am convinced that even in this state, Khinchin’s work is of considerable interest.