The Making of Mathematics: Heuristic Philosophy of Mathematics

ISBN-13: 9783030897338 / Angielski / Miękka / 2023 / 450 str.

Carlo Cellucci

Opis

This book offers an alternative to current philosophy of mathematics: heuristic philosophy of mathematics. In accordance with the heuristic approach, the philosophy of mathematics must concern itself with the making of mathematics and in particular with mathematical discovery. In the past century, mainstream philosophy of mathematics has claimed that the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics as presented in published works. On this basis, mainstream philosophy of mathematics has maintained that mathematics is theorem proving by the axiomatic method. This view has turned out to be untenable because of Gödel’s incompleteness theorems, which have shown that the view that mathematics is theorem proving by the axiomatic method does not account for a large number of basic features of mathematics. By using the heuristic approach, this book argues that mathematics is not theorem proving by the axiomatic method, but is rather problem solving by the analytic method. The author argues that this view can account for the main items of the mathematical process, those being: mathematical objects, demonstrations, definitions, diagrams, notations, explanations, applicability, beauty, and the role of mathematical knowledge.

Recenzje

"The book is extensive and circumstantial. ... The book is well written, based upon sources ... with a clear intention to advance the case of 'heuristic' philosophy of mathematics. ... the book is a valuable position which could start a debate and possible revision of modern philosophy of mathematics." (Roman Duda, zbMATH 1497.00010, 2022)

"The book is an interesting contribution to the new trend in the philosophy of mathematics, the trend in which the attention is paid mainly to the mathematical practice, to making of mathematics and not to mathematics as a finished structure of theorems and proofs." (Roman Murawski, Mathematical Reviews, November, 2022)

Spis treści

1. Introduction

Part I. Heuristic vs. Mainstream

2. Mainstream Philosophy of Mathematics

3. Heuristic Philosophy of Mathematics

Part II. Discourse on Method

4. The Question of Method

5. Analytic Method

6. Analytic-Synthetic Method and Axiomatic Method

7. Rules of Discovery

8. Theories

Part III. The Mathematical Process

9. Objects

10. Demonstrations

11. Definitions

12. Diagrams

13. Notations

Part IV. The Functionality of Mathematics

14. Explanations

15. Beauty

16. Applicability

Part V. Conclusion

17. Knowledge, Mathematics, and Naturalism

18. Concluding Remarks

Index

O autorze

Carlo Cellucci is emeritus professor of logic at Sapienza University of Rome. He is the author of seven books: Teoria della dimostrazione (Boringhieri, 1978); Le ragioni della logica (Laterza, 1998); Filosofia e matematica (Laterza, 2003); Perché ancora la filosofia (Laterza, 2008); Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method (Springer, 2013); Breve storia della logica: Dall’Umanesimo al primo Rinascimento (with Mirella Capozzi, Lulu Press, 2014); Rethinking Knowledge: The Heuristic View (Springer, 2017).

Blurb

Mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century, claims that the philosophy of mathematics cannot concern itself with the making of mathematics, in particular discovery, but only with finished mathematics, namely mathematics presented in finished form. On this basis, mainstream philosophy of mathematics argues that mathematics is theorem proving by the axiomatic method. This, however, is untenable because it is incompatible with Gödel’s incompleteness theorems, and cannot account for many features of mathematics.

This book offers an alternative approach, heuristic philosophy of mathematics, according to which the philosophy of mathematics can concern itself with the making of mathematics, in particular discovery. On this basis, the book argues that mathematics is problem solving by the analytic method, and that this can account for all the main features of mathematics: mathematical method, objects, demonstrations, definitions, diagrams, notations, explanations, beauty, applicability, and knowledge.