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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 provingby 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.
List of contents
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.
About the author
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).
Report
"Carlo Cellucci ... published what can be understood to be one of the most significant books in the philosophy and pedagogy of mathematics in the past century. His erudition and explication are evident on every page, but what makes it most valuable for an understanding and presentation of mathematics is the clarity he brings to the argument ... . Such an understanding could transform the way mathematics is presented in journals, textbooks, and classrooms." (Marshall Gordon, Philosophy of Mathematics Education Journal, Issue 1, February, 2024)
"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)
