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Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics and social sciences, rather than issues in the foundations of mathematics.
A primary argument in the movement 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.
Hilary Putnam stated in 1975 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 argued, necessarily be at least 'quasi'-empirical (embracing 'the scientific method' for consensus if not experiment).
Eugene Wigner had noted in 1960 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."
Related work
Additions to discussions about this concept could be several. Gregory Chaitin's and Stephen Wolfram's work, though their positions may be considered controversial, apply. Chaitin's work suggests an underlying randomness to mathematics; Wolfram's 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 influenced by computational issues. To quote Chaitin: "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!" (Limits of Mathematics). Wolfram's collection of 'undecidables' is another example. Wegner's recent paper suggests that Interactive Computation can help mathematics form a more appropriate framework (empirical) than can be founded with rationalism alone.
See also
"Quasi-empiricism in mathematics"