The Making of Mathematics

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.

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).

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.