Is Mathematics Problem Solving or Theorem Proving?

Foundations of Science 22 (1):183-199 (2017)
The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable.
Keywords Problem solving  Theorem proving  Analytic method  Axiomatic method  David Hilbert  Incompleteness theorems
Categories (categorize this paper)
DOI 10.1007/s10699-015-9475-2
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 35,471
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

Critique of Pure Reason.I. Kant - 1787/1998 - Philosophy 59 (230):555-557.
Mathematical Logic.Joseph R. Shoenfield - 1967 - Reading, Mass., Addison-Wesley Pub. Co..
Collected Works.Kurt Gödel - 1986 - Oxford University Press.
What is Mathematics, Really?Reuben Hersh - 1997 - Oxford University Press.

View all 14 references / Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Mathematical Discourse Vs. Mathematical Intuition.Carlo Cellucci - 2005 - In Carlo Cellucci & Donald Gillies (eds.), Mathematical Reasoning and Heuristics. College Publications. pp. 137-165..
Mathematical Instrumentalism, Gödel’s Theorem, and Inductive Evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
From Closed to Open Systems.Carlo Cellucci - 1993 - In J. Czermak (ed.), Philosophy of Mathematics, pp. 206-220. Hölder-Pichler-Tempsky.
Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 236--317.
Non-Language Thinking in Mathematics.Dieter Lohmar - 2012 - Axiomathes 22 (1):109-120.
Arithmetic, Mathematical Intuition, and Evidence.Richard Tieszen - 2015 - Inquiry: An Interdisciplinary Journal of Philosophy 58 (1):28-56.
Identity in Modal Logic Theorem Proving.Francis J. Pelletier - 1993 - Studia Logica 52 (2):291 - 308.
Issues Around Reflective Abstraction in Mathematics Education.C. Ulrich - 2014 - Constructivist Foundations 9 (3):370-371.


Added to PP index

Total downloads
22 ( #279,690 of 2,285,717 )

Recent downloads (6 months)
1 ( #390,440 of 2,285,717 )

How can I increase my downloads?

Monthly downloads

My notes

Sign in to use this feature