Philosophy of Mathematics

2016 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Discourse ◽  
Mathematical Objects ◽  
Ontological Realism ◽  
Truth Value ◽  
And Mathematics

This article examines a number of issues and problems that motivate at least much of the literature in the philosophy of mathematics. It first considers how the philosophy of mathematics is related to metaphysics, epistemology, and semantics. In particular, it reviews several views that account for the metaphysical nature of mathematical objects and how they compare to other sorts of objects, including realism in ontology and nominalism. It then discusses a common claim, attributed to Georg Kreisel that the important issues in the philosophy of mathematics do not concern the nature of mathematical objects, but rather the objectivity of mathematical discourse. It also explores irrealism in truth-value, the dilemma posed by Paul Benacerraf, epistemological issues in ontological realism, ontological irrealism, and the connection between naturalism and mathematics.

scholarly journals El antiplatonismo del Tractatus de Wittgenstein

2002 ◽  
pp. 137-152
Author(s):  
Sílvio Pinto
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Discourse ◽  
Mathematical Objects ◽  
Mathematical Equations

This essay presents the framework of the early Wittgenstein’s philosophy of mathematics. The author shows how, by conceiving mathematical equations and tautologies as rules for the use of signs, the early Wittgenstein did not require postulating, or inventing, mathematical objects in order to explain the objective character of mathematical discourse.

scholarly journals PERKEMBANGAN MATEMATIKA DALAM FILSAFAT DAN ALIRAN FORMALISME YANG TERKANDUNG DALAM FILSAFAT MATEMATIKA

Sepren ◽  
2021 ◽  
Vol 2 (2) ◽  
pp. 17-22
Author(s):  
Robin Tarigan
Keyword(s):  
Learning Outcomes ◽  
Historical Perspective ◽  
Philosophy Of Mathematics ◽  
Long Time ◽  
And Mathematics ◽  
The Relationship

Philosophy of mathematics does not add a number of new mathematical theorems or theories, so a philosophy of mathematics is not mathematics. The philosophy of mathematics is an area of ​​reflection about mathematics. After studying for a long time, one needs to reflect on learning outcomes by reflecting on the philosophy of mathematics. Mathematics and philosophy are closely related, compared to other sciences. The reason is that philosophy is the base for studying science and mathematics is the mother of all sciences. There are also those who think that philosophy and mathematics are the mother of all existing knowledge. From a historical perspective, the relationship between philosophy and mathematics underwent a very striking development. This article discusses the development of mathematics in philosophy and the flow of formalism contained in the philosophy of mathematics in particular

Constructivism in mathematics

2018 ◽  
Author(s):  
David Charles McCarty
Keyword(s):  
Twentieth Century ◽  
Real Analysis ◽  
Constructive Logic ◽  
Mathematical Objects ◽  
Mathematical Ontology ◽  
The Law ◽  
And Mathematics ◽  
Excluded Middle

Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, in one way or another, that proofs of crucial existential theorems in mathematics respect constructive existence: that a crucial existential claim which is constructively admissible must afford means for constructing an instance of it which is also admissible. Allegiance to this idea often demands changes in conventional views about mathematical objects, operations and logic, and, hence, demands reworkings of ordinary mathematics along nonclassical lines. Constructive existence may be so interpreted as to require the abrogation of the law of the excluded middle and the adoption of nonstandard laws of constructive logic and mathematics in its place. There has been great variation in the forms of constructivism, each form distinguished in its interpretation of constructive existence, in its approaches to mathematical ontology and constructive logic, and in the methods chosen to prove theorems, particularly theorems of real analysis. In the twentieth century, Russian constructivism, new constructivism, Brouwerian intuitionism, finitism and predicativism have been the most influential forms of constructivism.

Semantic Realism

1998 ◽  
pp. 6-12 ◽  
Author(s):  
Elaine Landry
Keyword(s):  
Category Theory ◽  
Linguistic Analysis ◽  
Philosophical Study ◽  
Physical Sciences ◽  
Mathematical Concepts ◽  
Ontological Realism ◽  
Semantic Realism ◽  
And Mathematics

I argue that if we distinguish between ontological realism and semantic realism, then we no longer have to choose between platonism and formalism. If we take category theory as the language of mathematics, then a linguistic analysis of the content and structure of what we say in and about mathematical theories allows us to justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Insofar as this analysis relies on a distinction between ontological and semantic realism, it relies also on an implicit distinction between mathematics as a descriptive science and mathematics as a descriptive discourse. It is this latter distinction which gives rise to the tension between the mathematician qua philosopher. In conclusion, I argue that the tensions between formalism and platonism, indeed between mathematician and philosopher, arise because of an assumption that there is an analogy between mathematical talk and talk in the physical sciences.

Paul Benacerraf and Hilary Putnam. Introduction. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, 1964, pp. 1–27. - Rudolf Carnap. The logicist foundations of mathematics. English translation of 3528 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 31–41. - Arend Heyting. The intuitionist foundations of mathematics. English translation of 3856 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 42–49. - Johann von Neumann. The formalist foundations of mathematics. English translation of 2998 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 50–54. - Arend Heyting. Disputation. A reprint of pages 1-12 (the first chapter) and parts of the bibliography of XXI 367. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 55–65. - L. E. J. Brouwer. Intuitionism and formalism. A reprint of 1557. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 66–77. - L. E. J. Brouwer. Consciousness, philosophy, and mathematics. A reprint of pages 1243-1249 of XIV 132. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 78–84. - Gottlob Frege. The concept of number. English translation of pages 67-104, 115-119, of 495 (1884 edn.) by Michael S. Mahoney. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 85–112. - Bertrand Russell. Selections from Introduction to mathematical philosophy. A reprint of pages 1-19, 194-206, of 11126 (1st edn., 1919). Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 113–133. - David Hilbert. On the infinite. English translation of 10813 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 134–151.

1969 ◽  
Vol 34 (1) ◽  
pp. 107-110
Author(s):  
Alec Fisher
Keyword(s):  
New Jersey ◽  
English Translation ◽  
Philosophy Of Mathematics ◽  
Bertrand Russell ◽  
Hilary Putnam ◽  
Foundations Of Mathematics ◽  
Rudolf Carnap ◽  
Von Neumann ◽  
David Hilbert ◽  
And Mathematics

Zen, Mathematics, and Rāmānujan: Uncommon Links

2016 ◽  
Vol 4 (1) ◽  
pp. 25-47
Author(s):  
Masato Mitsuda
Keyword(s):  
Scientific Revolution ◽  
Technological Progress ◽  
18Th Century ◽  
Philosophy Of Mathematics ◽  
Zen Buddhism ◽  
Fundamental Difference ◽  
The Third ◽  
Late 18Th Century ◽  
And Mathematics ◽  
Celestial Sphere

For centuries, religion has been the main impulse for moral and humanistic advancement, and ever since the rise of the Scientific Revolution (from 1543, the year Copernicus published De revolutioni bus orbium coelestium [On the revolution of the celestial sphere] – to the late 18th century), mathematics has been the cardinal element for scientific and technological progress. Mathematics requires a logical mind, but religion demands a receptive and compassionate mind. Even though there is a fundamental difference between the two subjects, the aim of this essay is to explore the relationships between Zen, mathematics, and Rāmānujan. The first section expounds on Bodhidharma’s and Hui neng’s notions of “no mind” and the “essence of mind,” as they are deemed an important bridge between Zen and mathematics. The second section presents how mathematics and Zen Buddhism relate to each other. Accordingly, the views on intuition, imagination, freedom, and language based on Einstein, Cantor, Brouwer, Poincare, et al. are discussed. The third section discusses the work of the most renowned mathematician in modern India in relation to Zen Buddhism. Rāmānujan’s unparalleled accomplishment in the field of number theory is well known among mathematicians. However, he is not well presented in the philosophy of mathematics, because of his unusual approach to mathematics.

Descartes on the Eternal Truths and Essences of Mathematics: An Alternative Reading

Vivarium ◽  
2016 ◽  
Vol 54 (2-3) ◽  
pp. 204-249
Author(s):  
Helen Hattab
Keyword(s):  
Rene Descartes ◽  
Philosophy Of Mathematics ◽  
Ontological Status ◽  
Divine Simplicity ◽  
Historical Evidence ◽  
Eternal Truths ◽  
Mathematical Objects ◽  
Divine Essence ◽  
René Descartes ◽  
Alternative Reading

René Descartes is neither a Conceptualist nor a Platonist when it comes to the ontological status of the eternal truths and essences of mathematics but articulates a view derived from Proclus. There are several advantages to interpreting Descartes’ texts in light of Proclus’ view of universals and philosophy of mathematics. Key passages that, on standard readings, are in conflict are reconciled if we read Descartes as appropriating Proclus’ threefold distinction among universals. Specifically, passages that appear to commit Descartes to a Platonist view of mathematical objects and the truths that follow from them are no longer in tension with the Conceptualist view of universals implied by his treatment of the eternal truths in the Principles of Philosophy. This interpretation also fits the historical evidence and explains why Descartes ends up with seemingly inconsistent commitments to divine simplicity and God’s efficient creation of truths that are not merely conceptually distinct from the divine essence.

scholarly journals Theological Underpinnings of the Modern Philosophy of Mathematics.

2016 ◽  
Vol 44 (1) ◽  
pp. 147-168
Author(s):  
Vladislav Shaposhnikov
Keyword(s):  
Twentieth Century ◽  
Modern Philosophy ◽  
Philosophy Of Mathematics ◽  
Main Concern ◽  
Early Twentieth Century ◽  
Pure Mathematics ◽  
And Mathematics

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

scholarly journals The Normative Pluriverse

2020 ◽  
Vol 18 (2) ◽  
Author(s):  
Matti Eklund
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Objects ◽  
Normative Realism ◽  
Important Form ◽  
The Way

According to a certain pluralist view in philosophy of mathematics, there are as many mathematical objects as there can coherently be. Recently, Justin Clarke-Doane has explored what consequences the analogous view on normative properties would have. What if there is a normative pluriverse? Here I address this same question. The challenge is best seen as a challenge to an important form of normative realism. I criticize the way Clarke-Doane presents the challenge. An improved challenge is presented, and the role of pluralism in this challenge is assessed.

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