World Library  
Flag as Inappropriate
Email this Article

Quasi-empiricism in mathematics

Article Id: WHEBN0000159731
Reproduction Date:

Title: Quasi-empiricism in mathematics  
Author: World Heritage Encyclopedia
Language: English
Subject: Philosophy of mathematics, Sociology of scientific knowledge, Thomas Tymoczko, Mathematical practice, Interactive computation
Collection: Empiricism, Philosophy of Mathematics, Theoretical Computer Science
Publisher: World Heritage Encyclopedia

Quasi-empiricism in mathematics

Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics. Of concern to this discussion are several topics: the relationship of empiricism (See Maddy) with mathematics, issues related to realism, the importance of culture, necessity of application, etc.


  • Primary arguments 1
    • Operational aspects 1.1
  • See also 2
  • References 3

Primary arguments

A primary argument with respect to Quasi-empiricism is that whilst mathematics and physics are more frequently being considered as closely linked fields of study, this may reflect human cognitive bias. It is claimed that, despite rigorous application of appropriate empirical methods or mathematical practice in either field, this would nonetheless be insufficient to disprove alternate approaches.

Eugene Wigner (1960)[1] noted that this culture need not be restricted to mathematics, physics, or even humans. He stated further that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." Wigner used several examples to demonstrate why 'bafflement' is an appropriate description, such as showing how mathematics adds to situational knowledge in ways that are either not possible otherwise or are so outside normal thought to be of little notice. The predictive ability, in the sense of describing potential phenomena prior to observation of such, which can be supported by a mathematical system would be another example.

Following up on Wigner, Richard Hamming (1980) [2] wrote about applications of mathematics as a central theme to this topic and suggested that successful use can trump, sometimes, proof, in the following sense: where a theorem has evident veracity through applicability, later evidence that shows the theorem's proof to be problematic would result more in trying to firm up the theorem rather than in trying to redo the applications or to deny results obtained to date. Hamming had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.

1) "We see what we look for." Why 'quasi' is apropos in reference to this discussion. 2) "We select the kind of mathematics to use." Our use and modification of mathematics is essentially situational and goal driven. 3) "Science in fact answers comparatively few problems." What still needs to be looked at is a larger set. 4) "The evolution of man provided the model." There may be limits attributable to the human element.

Hilary Putnam (1975) [3] stated that mathematics had accepted informal proofs and proof by authority, and had made and corrected errors all through its history. Also, he stated that Euclid's system of proving geometry theorems was unique to the classical Greeks and did not evolve similarly in other mathematical cultures in China, India, and Arabia. This and other evidence led many mathematicians to reject the label of Platonists, along with Plato's ontology—which, along with the methods and epistemology of Aristotle, had served as a foundation ontology for the Western world since its beginnings. A truly international culture of mathematics would, Putnam and others (1983) [4] argued, necessarily be at least 'quasi'-empirical (embracing 'the scientific method' for consensus if not experiment).

Imre Lakatos (1976 - posthumous), [5] who did his original work on this topic for his dissertation (1961, Cambridge), argued for 'Research Programs' as a means to support a basis for mathematics and considered thought experiments as appropriate to mathematical discovery. Lakatos may have been the first to use 'quasi-empiricism' in the context of this subject.

Operational aspects

Recent work that pertains to this topic are several. Chaitin's and Stephen Wolfram's work, though their positions may be considered controversial, apply. Chaitin (1997/2003) [6] suggests an underlying randomness to mathematics and Wolfram (A New Kind of Science, 2002) [7] argues that undecidability may have practical relevance, that is, be more than an abstraction.

Another relevant addition would be the discussions concerning Interactive computation, especially those related to the meaning and use of Turing's model (Church-Turing, TM, etc.).

These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003): "Now everything has gone topsy-turvy. It's gone topsy-turvy, not because of any philosophical argument, not because of Gödel's results or Turing's results or my own incompleteness results. It's gone topsy-turvy for a very simple reason — the computer!".[6]

The collection of "Undecidables" in Wolfram (A New Kind of Science, 2002) [7] is another example.

Wegner's recent paper [8] suggests that interactive computation can help mathematics form a more appropriate framework (empirical) than can be founded with rationalism alone. Related to this argument is that the function (even recursively related ad infinitum) is too simple of a construct to handle the reality of entities that resolve (via computation or some type of analog) n-dimensional (general sense of the word) systems.

See also


  1. ^ Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13:
  2. ^ R. W. Hamming, 1980, The Unreasonable Effectiveness of Mathematics, The American Mathematical Monthly Volume 87 Number 2 February 1980
  3. ^ Putnam, Hilary, 1975, Mind, Language, and Reality. Philosophical Papers, Volume 2. Cambridge University Press, Cambridge, UK. ISBN 88-459-0257-9
  4. ^ * Benacerraf, Paul, and Putnam, Hilary (eds), 1983, Philosophy of Mathematics, Selected Readings, 1st edition, Prentice–Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983
  5. ^ Lakatos, Imre 1976, Proofs and Refutations. Cambridge: Cambridge University Press. ISBN 0-521-29038-4
  6. ^ a b Chaitin, Gregory J., 1997/2003, "Limits of Mathematics", Springer-Verlag, New York, NY. ISBN 1-85233-668-4
  7. ^ a b Wolfram, Stephen, 2002, A New Kind of Science (Undecidables), Wolfram Media, Chicago, IL. ISBN 1-57955-008-8
  8. ^ Peter Wegner, Dina Goldin, 2006, Principles of Problem Solving. Communications of the ACM 49 (2006), pp.27-29
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.