The Making of Mathematics
Heuristic Philosophy of Mathematics
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Beschreibung
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.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.
Provides an alternative to current philosophy of mathematics; heuristic philosophy of mathematics Shows that mathematics is not merely theorem proving by the axiomatic method Of interest to both graduate students and scholars working in philosophy of mathematics
Autor*in
Carlo Cellucci
Themen in »The Making of Mathematics«
heuristic philosophy of mathematics mathematics and philosophy the irrelevance view foundationalist views on mathematics mathematics and logicism mathematics and formalism the existence of mathematical objects
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“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)
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