Editing 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 science]]s, and [[computational mathematics]], rather than solely to issues in the [[foundations of mathematics]]. Of concern to this discussion are several topics: the relationship of [[Philosophy of mathematics#Empiricism|empiricism]] (see [[Penelope Maddy]]) with [[mathematics]], issues related to [[Philosophy of mathematics#Mathematical realism|realism]], the importance of [[Philosophy of mathematics#Social constructivism or social realism|culture]], necessity of [[Philosophy of mathematics#Constructivism|application]], etc.

==Primary arguments==
A primary argument with respect to [[quasi-empiricism]] is that whilst mathematics and physics are frequently considered to be 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)<ref>[[Eugene Wigner]], 1960, "[http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html The Unreasonable Effectiveness of Mathematics in the Natural Sciences]," ''Communications on Pure and Applied Mathematics 13'':</ref> [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|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 [[Eugene Wigner|Wigner]], [[Richard Hamming]] (1980)<ref>[[Richard Hamming|R. W. Hamming]], 1980, [http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html The Unreasonable Effectiveness of Mathematics], ''The [[American Mathematical Monthly]]'' Volume 87 Number 2 February 1980</ref> wrote about [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences#Hamming.27s follow-on to Wigner|applications of mathematics]] as a central theme to this topic and suggested that successful use can sometimes trump 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. [[Richard Hamming|Hamming]] had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.

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

For [[Willard Van Orman Quine]] (1960),<ref>Willard Van Orman Quine (1960), ''[[Word and Object]]'', MIT Press, p. 22.</ref> existence is only existence in a structure. This position is relevant to quasi-empiricism because Quine believes that the same evidence that supports theorizing about the structure of the world is the same as the evidence supporting theorizing about mathematical structures.<ref>Paul Ernest (ed.), ''Mathematics Education and Philosophy: An International Perspective'', Routledge, 2003, p. 45.</ref>

[[Hilary Putnam]] (1975)<ref>[[Hilary Putnam|Putnam, Hilary]], 1975, ''Mind, Language, and Reality. Philosophical Papers, Volume 2''. Cambridge University Press, Cambridge, UK. {{ISBN|88-459-0257-9}}</ref> 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 [[Ancient Greece|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 [[Philosophy of mathematics|Platonists]], along with [[Plato's ontology]]{{snd}} 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)<ref>[[Paul Benacerraf|Benacerraf, Paul]], and [[Hilary Putnam|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</ref> argued, necessarily be at least 'quasi'-empirical (embracing 'the scientific method' for consensus if not experiment).

[[Imre Lakatos]] (1976),<ref>[[Imre Lakatos|Lakatos, Imre]] (1976), ''[[Proofs and Refutations]]''. Cambridge: Cambridge University Press. {{ISBN|0-521-29038-4}}</ref> who did his original work on this topic for [[Proofs and Refutations|his dissertation]] (1961, [[Cambridge]]), argued for '[[Imre Lakatos#Research programmes|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===
Several recent works pertain to this topic. [[Gregory Chaitin]]'s and [[Stephen Wolfram]]'s work, though their positions may be considered controversial, apply. Chaitin (1997/2003)<ref name="limits">[[Gregory Chaitin|Chaitin, Gregory J.]], 1997/2003, ''[https://books.google.com/books?id=kwBpDQAAQBAJ&pg=PA96 Limits of Mathematics]'' {{webarchive |url=https://web.archive.org/web/20060101134749/http://cs.umaine.edu/~chaitin/lm.html |date=January 1, 2006 }}, Springer-Verlag, New York, NY. {{ISBN|1-85233-668-4}}</ref> suggests an underlying randomness to mathematics and Wolfram (''[[A New Kind of Science]]'', 2002)<ref name="new">[[Stephen Wolfram|Wolfram, Stephen]], 2002, ''A New Kind of Science'' ([http://www.wolframscience.com/nksonline/notes-section-12.8 Undecidables]), Wolfram Media, Chicago, IL. {{ISBN|1-57955-008-8}}</ref> 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 [[Alan Turing|Turing]]'s model ([[Church-Turing thesis]], [[Turing machines]], etc.).

These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003): {{Quote|Now everything has gone topsy-turvy. It's gone topsy-turvy, not because of any philosophical argument, not because of [[Kurt Gödel|Gödel]]'s results or [[Alan Turing|Turing]]'s results or my own incompleteness results. It's gone topsy-turvy for a very simple reason—the computer!<ref name="limits"/>{{rp|96}}}}

The collection of "Undecidables" in Wolfram (''[[A New Kind of Science]]'', 2002)<ref name="new"/> is another example.

[[Peter Wegner (computer scientist)|Wegner's]] 2006 paper "Principles of Problem Solving"<ref>[http://www.cs.brown.edu/people/pw/home.html Peter Wegner], Dina Goldin, 2006, "[http://portal.acm.org/citation.cfm?doid=1139922.1139942 Principles of Problem Solving]". ''Communications of the ACM'' 49 (2006), pp. 27–29</ref> 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 (mathematics)|function]] (even recursively related ad infinitum) is too simple 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==
{{Div col|colwidth=20em}}
*[[Entscheidungsproblem]]
*[[Charles Sanders Peirce]]
*[[Karl Popper]]
*{{slink|Philosophy of mathematics#Beyond the traditional schools}}
*[[Postmodern mathematics]]
*[[Thomas Tymoczko]]
*[[Unreasonable ineffectiveness of mathematics]]
{{Div col end}}

==References==
{{Reflist}}

[[Category:Philosophy of mathematics]]
[[Category:Theoretical computer science]]
[[Category:Empiricism]]