Thomas Aquinas' Mathematical Realism

ISBN-13: 9783031331275 / Angielski

Jean W. Rioux

Opis

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.

Spis treści

Introduction

Part One: Mathematical Realism in Plato and Aristotle

Chapter One: Plato on Mathematics and the Mathematicals

1. 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 Mathematics

5 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 Sciences

1. A brief, standard account of the distinction between the practical and speculative sciences

in Aristotle, along with

2. the necessary and sufficient conditions for episteme.

3. How pure mathematics would fulfill those conditions.

4. An account of the “middle”, or applied, sciences, and

5. the priority of the mathematical sciences to the applied.

Chapter Four: Aristotle on Abstraction and Intelligible Matter

5 The distinguishing feature of the mathematical sciences (formal abstraction), and

6 Aristotle’s careful middle position between platonic realism and nominalism.

7 Intelligible matter, and why it is needed.

8 What each implies about

a) the existence of mathematicals and

b) how we can know them (note that these are Aristotle’s primary areas of objection to

the platonic forms in general).

Part Two: Mathematical Realism in Aquinas

Chapter Five: The Objects of Mathematics, Mathematical Freedom, and the Art of

Mathematics

7. 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 Virtually

4. Virtual existence in Aquinas.

5. Intuitionism and the Excluded Middle.

6. Remote and proximate objects (the properties of all numbers are virtually contained in the

unit). SCG I 69 4 & 9

Chapter Seven: Mathematics and the Liberal Arts

1. One of Aquinas’ key additions to the Aristotelian account.

2. How the mathematical arts differ among themselves and from the speculative and applied

sciences.

3. The mathematical arts and truth.

Chapter Eight: The Place of the Imagination in Mathematics

1. 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 Mathematics

Chapter Nine: Subsequent Developments in Number Theory

1. 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 Geometry

1. 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 Controversies

1. The finitism / infinitism issue.

2. Logicism, formalism, and intuitionism.

3. Where would Aristotle’s and Aquinas’ mathematical realism fall in this schema? SCG I 69

11

Chapter Twelve: Realism and Non-Realism in Mathematics

1. 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 accounts

1. Aristotelians: Franklin, Lear.

2. Thomists: Maurer et al.

Conclusion

Chapter Fourteen: Foundations Restored?

What would taking the claims made by Aristotle and Aquinas seriously do to contemporary

mathematics?

O autorze

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