Stanisław Leśniewski: Rethinking the Philosophy of Mathematics

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find ‘one sort of thing’ that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege’s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Leśniewski (1886–1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Leśniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology – a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Leśniewski’s project and to briefly comment on its contemporary relevance.

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Kronecker, Leopold (1823–91)

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y073-1 ◽

2018 ◽

Author(s):

Ulrich Majer

Keyword(s):

Nineteenth Century ◽

Set Theory ◽

Point Of View ◽

Academic Institution ◽

Late Nineteenth Century ◽

Foundations Of Mathematics ◽

Epistemological Perspective ◽

Classical Mathematics ◽

Late Nineteenth

Leopold Kronecker was one of the most influential German mathematicians of the late nineteenth century. He exercised a strong sociopolitical influence on the development of mathematics as an academic institution. From a philosophical point of view, his main significance lies in his anticipation of a new and rigorous epistemological perspective with regard to the foundations of mathematics: Kronecker became the father of intuitionism or constructivism, which stands in strict opposition to the methods of classical mathematics and their canonization by set theory.

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Hegel’s ‘Bad Infinity’ as a Logical Problem

Hegel Bulletin ◽

10.1017/hgl.2016.18 ◽

2016 ◽

Vol 37 (2) ◽

pp. 258-280 ◽

Cited By ~ 1

Author(s):

Vojtěch Kolman

Keyword(s):

Set Theory ◽

Philosophy Of Mathematics ◽

Main Line ◽

Foundations Of Mathematics ◽

Modern Logic ◽

Infinitesimal Calculus ◽

Logical Problem ◽

Philosophical System

AbstractThe paper analyses the concept of ‘bad infinity’ in connection with Hegel’s critique of infinitesimal calculus and with the belittling of Hegel’s mathematical notions by the representatives of modern logic and the foundations of mathematics. The main line of argument draws on the observation that Hegel’s difference is only derivatively a mathematical one and is primarily of a broadly logico-epistemological nature. Because of this, the concept of bad infinity can be fruitfully utilized, by way of inversion, in an analysis of the conceptual shortcomings of the most prominent foundational attempts at dealing with infinite quanta, such as Cantor’s set theory and Hilbert’s axiomatism. As such, the paper is an attempt at reconstructing Hegel’s philosophy of mathematics and its role in his philosophical system and, more importantly, as a contribution to logic in the more general and radical sense of the word.

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Andrzej Mostowski. Recent results in set theory. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 82–96. - G. Kreisel, A. Robinson, L. Kalmár, and A. Mostowski. Discussion. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 97–108. - Andrzej Mostowski. O niektórych nowych wynikach meta-matematycznych dotyczących teorii mnogości (On some new metamathematical results concerning set theory). Polish with Russian and English summaries. Studia logica, vol. 20 (1967), pp. 99–116.

Journal of Symbolic Logic ◽

10.2307/2272452 ◽

1972 ◽

Vol 37 (4) ◽

pp. 765-766

Author(s):

T. Jech

Keyword(s):

Philosophy Of Science ◽

Set Theory ◽

Philosophy Of Mathematics ◽

Publishing Company ◽

Foundations Of Mathematics ◽

International Colloquium ◽

North Holland Publishing Company ◽

Imre Lakatos ◽

North Holland Publishing

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Platonism and Mathematical Intuition in Kurt Gödel's Thought

Bulletin of Symbolic Logic ◽

10.2307/420946 ◽

1995 ◽

Vol 1 (1) ◽

pp. 44-74 ◽

Cited By ~ 24

Author(s):

Charles Parsons

Keyword(s):

Set Theory ◽

Real World ◽

Mathematical Knowledge ◽

Philosophy Of Mathematics ◽

Foundations Of Mathematics ◽

Mathematical Objects ◽

The Real ◽

Robust Realism ◽

Mathematical Intuition

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

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A Second Pilgrim’s Progress

10.1093/oso/9780198803478.003.0009 ◽

2017 ◽

Cited By ~ 1

Author(s):

Mark Wilson

Keyword(s):

Set Theory ◽

Applied Mathematics ◽

Philosophy Of Mathematics ◽

Pilgrim's Progress ◽

Open Questions

Influenced by Quine, self-styled naturalist projects within the philosophy of mathematics rest upon simplistic conceptions of linguistic reference and how the inferential tools of applied mathematics help us reach empirical conclusions. In truth, these two forms of descriptive enterprise must work together in a considerably more entangled manner than is generally presumed. In particular, the vital contributions of set theory to descriptive success within science have been poorly conceptualized. This essay explores how a less onerous “naturalism” can be conceived on this corrected basis. A useful distinction between “mathematical optimism” and “mathematical opportunism” is introduced, which draws our attention to some open questions with respect to the concrete representational capacities of applied mathematics.

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Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic

10.1142/12456 ◽

2022 ◽

Author(s):

Douglas Cenzer ◽

Jean Larson ◽

Christopher Porter ◽

Jindrich Zapletal

Keyword(s):

Mathematical Logic ◽

Set Theory ◽

Foundations Of Mathematics

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Type-theoretical checking and philosophy of Mathematics

Twenty Five Years of Constructive Type Theory ◽

10.1093/oso/9780198501275.003.0005 ◽

1998 ◽

Author(s):

Nicolaas Govert de Bruijn

Keyword(s):

Set Theory ◽

Subject Matter ◽

Mutual Influence ◽

Mathematical Reasoning ◽

Philosophy Of Mathematics ◽

General Information ◽

Point Of View ◽

De Bruijn ◽

Level Of Understanding ◽

The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

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Dedekind, Julius Wilhelm Richard (1831–1916)

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y072-1 ◽

2018 ◽

Author(s):

Howard Stein

Keyword(s):

Nineteenth Century ◽

Set Theory ◽

Mathematical Knowledge ◽

Mathematical Structure ◽

Georg Cantor ◽

General Notion ◽

Algebraic Numbers ◽

Integral Calculus ◽

Foundational Work ◽

First Time

Dedekind is known chiefly, among philosophers, for contributions to the foundations of the arithmetic of the real and the natural numbers. These made available for the first time a systematic and explicit way, starting from very general notions (which Dedekind himself regarded as belonging to logic), to ground the differential and integral calculus without appeal to geometric ‘intuition’. This work also forms a pioneering contribution to set theory (further advanced in Dedekind’s correspondence with Georg Cantor) and to the general notion of a ‘mathematical structure’. Dedekind’s foundational work had a close connection with his advancement of substantive mathematical knowledge, particularly in the theories of algebraic numbers and algebraic functions. His achievements in these fields make him one of the greatest mathematicians of the nineteenth century.

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