ArticlePublisher preview available

Mathematics and Experience

Authors:
Carlo Cellucci at Sapienza University of Rome
Carlo Cellucci
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the author.

Abstract

The question of whether mathematics depends on experience, including experience of the external world, is problematic because, while it is clear that natural sciences depend on experience, it is not clear that mathematics depends on experience. Indeed, several mathematicians and philosophers think that mathematics does not depend on experience, and this is also the view of mainstream philosophy of mathematics. However, this view has had a deleterious effect on the philosophy of mathematics. This article argues that, in fact, the view is not valid. Mathematics depends on experience because experience influences the making of mathematics, indeed much mathematics arises from experience and is evaluated on the basis of experience.
A preview of this full-text is provided by Springer Nature.
Learn more
Content available from Foundations of Science
This content is subject to copyright. Terms and conditions apply.
Accepted: 7 February 2024
© The Author(s), under exclusive licence to Springer Nature B.V. 2024
Carlo Cellucci
[email protected]
1 Sapienza University of Rome, Rome, Italy
Mathematics and Experience
CarloCellucci1
Foundations of Science
https://doi.org/10.1007/s10699-024-09943-2
Abstract
The question of whether mathematics depends on experience, including experience of
the external world, is problematic because, while it is clear that natural sciences depend
on experience, it is not clear that mathematics depends on experience. Indeed, several
mathematicians and philosophers think that mathematics does not depend on experience,
and this is also the view of mainstream philosophy of mathematics. However, this view
    
   -
ences the making of mathematics, indeed much mathematics arises from experience and
is evaluated on the basis of experience.
Keywords Mainstream philosophy of mathematics · Heuristic philosophy of mathematics ·
Experience and the making of mathematics · Experience and the relation of mathematics
and physics · Mathematics and a priori knowledge
1 Problematic Character of the Question
The question of whether mathematics depends on experience, including experience of the
external world, is problematic because, while it is clear that natural sciences depend on
experience, it is not clear that mathematics depends on experience.
Indeed, on the one hand, several mathematicians think that mathematics does not depend
on experience.
Thus, Mac Lane says that “the proofs of mathematics do not depend on experience, and
indeed can often be invented or carried out by young people with little experience” (Mac
Lane, 1986, p. 442).
Monk says that “there can be little doubt that” mathematics “is independent of experi-
ence” (Monk, 1976, p. 2).
1 3
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
ResearchGate has not been able to resolve any citations for this publication.
On the Tension Between Physics and Mathematics
Article
Full-text available
  • Sep 2020
  • J Gen Philos Sci
Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The paper looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists. Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics.
View
The Search for Certainty: A Philosophical Account of Foundations of Mathematics
Book
  • Oct 2023
The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all Gödel's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.
View
Proofs and Refutations: The Logic of Mathematical Discovery
Book
  • Nov 2015
Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.
View
View
Heuristic Philosophy of Mathematics
Chapter
  • Mar 2022
An alternative to mainstream philosophy of mathematics is heuristic philosophy of mathematics, according to which the philosophy of mathematics is concerned with the making of mathematics, in particular discovery, and the method of mathematics is the analytic method. Heuristic philosophy of mathematics is not to be confused with other views of mathematics that might seem similar to it, in particular with the philosophy of mathematical practice, which is not concerned with the making of mathematics, in particular discovery, but, like mainstream philosophy of mathematics, only with finished mathematics. The chapter describes characters, origin, goal, and advantages of heuristic philosophy of mathematics.KeywordsHeuristic philosophy of mathematics Practical heuristics Philosophy of mathematical practice Heuristic view of mathematicsBottom-up approach Open systems Mathematics as problem solving Problems vs. theorems
View
View
What is Experimental Philosophy of Mathematics?(External Critiques of Experimental Philosophy)
Article
  • Mar 2015
If we take "experimental philosophy" in a wider sense, it is almost the same as methodological naturalism in philosophy. Experiments of this kind would be available and also fruitful in the philosophy of mathematics. They surely will have a substantial contribution to epistemology of mathematics. At present, however, the phrase "experimental philosophy" is used in a much narrower sense. It refers exclusively to a branch of naturalized philosophy that uses the data gathered through questionnaire surveys to investigate the intuitions of ordinary people. We claim that experimental philosophy in this narrower sense will have little relevance to the philosophy of mathematics, since the philosophy of mathematics is not engaged in conceptual analysis despite of its appearance.
View
View
Grundlagen der Mathematik I
Book
  • Jan 1968
Der vorliegende Band schliesst die Darstellung der Beweistheorie ab, die ich vor einigen Jahren zusammen mit P. BERNAYS begann. Auf meinen Wunsch hat P. BERNAYS wieder die Abfassung des Textes uber nommen. Ich danke ihm fur die Sorgfalt und Treue, mit der er meine Gedanken wiedergegeben hat, an deren Entwicklung er in jahrelanger Zusammenarbeit aufs starkste beteiligt war. Ohne seine Mithilfe ware die Vollendung dieses Buches unmoglich gewesen. Den Herren W. ACKERMANN, G. GENTZEN, A. SCHMIDT, H. SCHOLZ danke ich fur ihre freundliche Mitwirkung bei den Korrekturen. Gottingen, im Marz 1939 HILBERT Zur Einfuhrung Das vorliegende Buch soll einer eingehenden Orientierung uber den gegenwartigen Stoff der HILBERTschen Beweistheorie dienen. Wenn gleich das bisher hier Erreichte gemessen an den Zielen der Theorie sehr bescheiden ist, so liegt doch ein reichlicher Stoff an pragnanten Ergebnissen, an Gesichtspunkten und Beweisgedanken vor, die zur Kenntnis zu bringen als lohnend erscheint. Fur die inhaltliche Gestaltung dieses zweiten Bandes waren durch den Zweck des Buches zwei Hauptthemata vorgezeichnet. - Es handelte sich einmal darum, die hauptsachlichen, an das e-Symbol sich knupfenden beweistheoretischen Ansatze HILBERTS und ihre Durchfuhrung zur ein gehenden Darstellung zu bringen
View
What Mathematical Truth Need Not Be
Chapter
  • Jan 1990
I might have titled this paper “In Defense of Eidophobia”, eidophobes, in Belnapese, being those who, for one reason or another, find the idea of abstract entities distasteful. Since Nuel himself is an eidophile, I occasionally found myself at philosophical odds with him. But one of the best things about being a student of Nuel’s was that he had no need for eidological clones. So in a sense perhaps the greatest tribute I could render to his pedagogy is this paper, which makes it very clear that my eidophobia has remained intact. Studying with Nuel Belnap, one never had to fear that the spirit of independent (even eidocidal) inquiry might be compromised.
View