Overview

In this book, philosopher Jean W. Rioux extends accounts of the Aristotelian philosophy of mathematics to what Thomas Aquinas was able to import from Aristotle’s notions of pure and applied mathematics, accompanied by his own original contributions to them. Rioux sets these accounts side-by-side modern and contemporary ones, comparing their strengths and weaknesses.

Full Product Details

Author:   Jean W. Rioux
Publisher:   Springer International Publishing AG
Imprint:   Palgrave Macmillan
Edition:   1st ed. 2023
Weight:   0.508kg
ISBN:  

9783031331275

ISBN 10:   3031331273
Pages:   280
Publication Date:   29 June 2023
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

IntroductionPart One: Mathematical Realism in Plato and AristotleChapter One: Plato on Mathematics and the Mathematicals1. Platonism as a form of mathematical realism.2. The justification for, and place of, the mathematicals in Plato’s metaphysics.3. The attraction of platonism for some contemporary philosophies of mathematics.Chapter Two: Aristotle on the Objects of Mathematics5 Where do the mathematicals exist?6 The idealization of the mathematicals.7 What does Aristotle mean when he says they exist “materially”? (Μ.3)8 Is Aristotle a realist or a non-realist?Chapter Three: Aristotle on The Speculative and Middle Sciences1. A brief, standard account of the distinction between the practical and speculative sciencesin Aristotle, along with2. the necessary and sufficient conditions for episteme.3. How pure mathematics would fulfill those conditions.4. An account of the “middle”, or applied, sciences, and5. the priority of the mathematical sciences to the applied.Chapter Four: Aristotle on Abstraction and Intelligible Matter5 The distinguishing feature of the mathematical sciences (formal abstraction), and6 Aristotle’s careful middle position between platonic realism and nominalism.7 Intelligible matter, and why it is needed.8 What each implies abouta) the existence of mathematicals andb) how we can know them (note that these are Aristotle’s primary areas of objection tothe platonic forms in general).Part Two: Mathematical Realism in AquinasChapter Five: The Objects of Mathematics, Mathematical Freedom, and the Art ofMathematics7. Where do mathematicals exist? [Commentary on the Sentences I 2 1 3]8. The idealization of the mathematicals.9. How free is mathematics?10. Are there legitimate and non-legitimate mathematical objects?11. What of systematic studies of the non-legitimate objects?12. Fictionalism?Chapter Six: To Be Virtually4. Virtual existence in Aquinas.5. Intuitionism and the Excluded Middle.6. Remote and proximate objects (the properties of all numbers are virtually contained in theunit). SCG I 69 4 & 9Chapter Seven: Mathematics and the Liberal Arts1. One of Aquinas’ key additions to the Aristotelian account.2. How the mathematical arts differ among themselves and from the speculative and appliedsciences.3. The mathematical arts and truth.Chapter Eight: The Place of the Imagination in Mathematics1. Another key addition to Aristotle.2. Hints of this in Aristotle.3. Is Aquinas speaking of the representative or the creative imagination here?4. The implications of a reduction to the imagination as regards mathematical truth.Part Three: Aristotle, Aquinas, and Modern Philosophies of MathematicsChapter Nine: Subsequent Developments in Number Theory1. What of zero, negatives, fractions, rationals, and reals?2. What of imaginary and transfinite numbers?3. What of systematic studies of real structure (e.g., topology)?Chapter Ten: Non-Euclidean Geometry1. Are Euclidean and non-euclidean geometries sciences?2. In the same sense of the word?3. What of the successful application of non-euclidean geometries?4. Representational and non-representational systems.Chapter Eleven: Cantor, Finitism, and the 20th-Century Controversies1. The finitism / infinitism issue.2. Logicism, formalism, and intuitionism.3. Where would Aristotle’s and Aquinas’ mathematical realism fall in this schema? SCG I 6911Chapter Twelve: Realism and Non-Realism in Mathematics1. Different types of mathematical non-realism.2. Contemporary mathematical realisms (Franklin, Maddy, Bernot).3. Benacerraf’s dilemma and Aristotle and Aquinas’ solution.Chapter Thirteen: This account as compared to other modern Aristotelian-Thomistic accounts1. Aristotelians: Franklin, Lear.2. Thomists: Maurer et al.ConclusionChapter Fourteen: Foundations Restored?What would taking the claims made by Aristotle and Aquinas seriously do to contemporarymathematics?

Reviews

“Jean Rioux has written an admirable synthesis of the main debates in the philosophy of mathematics between those of a largely classical perspective ... . The book weaves together three main threads … . The best quality of the book is precisely Rioux’s interweaving of the threads of his investigation.” (Timothy Kearns, Thomistica, thomistica.net, November 21, 2023)

Author Information

Jean W. Rioux is Professor and Chair of the Philosophy Department at Benedictine College, Atchison, USA. 

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