The philosophy of mathematics attempts to describe the basic ethics, epistemics, and metaphysical assumptions that make mathematics effective in describing or predicting events in the world recognized by the human being.
It is distinct from the philosophy of science that asks what makes one theory more acceptable than another despite imperfect empirical validation and limits of the scientific method - and assumes mathematics as a neutral point of view. Scientists, in general, do not share the concerns of philosophers of mathematics except insofar as they must collaborate on reasonable method and foundation ontology of research:
Math: mind or matter?
"The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind." (Thomas Tymoczko)
In terms of which, the reasonable method is the best description we can make of the best human mind, and the foundation ontology is the best description we can make of "reality, something out there to be discovered."
As use of mathematics is not confined to scientists making predictions, but is employed also in law and economics and political science, most researchers find it appropriate that "we" be a somewhat larger (some say less disciplined) group than physical scientists, and that reasonable method be perhaps more inclusive than feasible method or scientific method.
20th century controversies
Key figures in the 20th century philosophy of mathematics include Bertrand Russell, Frege, Alfred North Whitehead, Wittgenstein, Alan Turing, Kurt Godel, Peano, David Hilbert, Paul Erdös and Eugene Wigner. These often debated the emotional and cognitive meaning of abstractions, mostly in dialogues, footnotes, and correspondences, that were "meta" to their mathematics work.
Much of this settled on whether the basic mathematics of Pythagoras to Leonhard Euler, Rene Descartes and Galois, even with the new fundamental proofs in number theory by Paul Erdös and collaborators, actually mapped onto the physicists' particle physics foundation ontology or if they were simply another sacred geometry like that of Plato - a useful but limited model that awaited understanding of some deeper ontology.
What does it rest on?
Over the last two millenia it has been assumed that logic, then set theory, then interpersonal trust between mathematicians, then the human embodied mind, actually constituted a foundation ontology. But even the habit of counting itself, and belief in collections of somehow-similar "objects", remains immune to a complete understanding despite the cognitive science and computer science devoted specifically to these fundamental subjects.
Some body philosophers argue that mathematics describes nothing but action, and seek to install a cognitive science of mathematics as a new foundation ontology. This is the most extreme version of the claim that "mathematics is the product of the human mind." The "embodied mind" thesis of Rafael E. Núñez and George Lakoff, published in 2000, is the most widely regarded version of this claim.
Lakoff makes the rather more extravagant claim that the thesis itself is not even amenable to challenge - in the history of the philosophy of mathematics such claims are not unusual! Others point out that the lack of falsifiability claimed makes any such theory unscientific, by excluding it from criticism by the scientific method.
Proofs and collaboration
However, other postmodernists see more reality in collaboration itself:
Thomas Tymoczko in 1986 edited a new anthology that went beyond traditional "foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics -- one that addresses issues of theoretical importance in terms of mathematical experience." In 1998 a new edition added essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. This had become especially important in light of the role of symbolic processing and artificial intelligence in proof work.
A singular figure in the philosophy of mathematics is Paul Erdös, a Hungarian mathematician whose work most mathematicians regard with awe. As almost all his work was performed in collaboration, and he spent the last thirty years of his life doing fundamental proofs of number theory to remove every trace of any complex analysis, it is possible to say that he drastically simplified the foundation problem in mathematics, to the point where any factor or bias common to his 507 co-authors could in fact be said to represent the bias in number theory itself.
This would be the strongest claim that could be made for the social capital (or mutual trust and cohesion) of mathematicians as the basis of numbers. In other words, that the collection of collaborators is the most fundamental collection, and counting these collaborators is the most fundamental form of counting. That "there is an external mathematical reality" and it can best be simulated by large collections of mathematically competent minds, each of which proxies that reality by auditing new proofs.
Between the posts
Between these two extreme "goal posts", many other positions seem possible - indeed a whole political spectrum of beliefs regarding epistemology.
The critique of Number, e.g. by John Zerzan and others, claims that inclusion tests, counting methods, and peer group acceptance of these may well matter more than any external reality. Such views have led in recent years to an emerging study of ethno-mathematics, e.g. by Ed Doolittle whose work emphasizes "folk mathematics" of Nearctic indigenous peoples.
Another significant contribution is the theology of Pope John Paul II whose "Fides et Ratio" (Faith and Reason), attempts to draw ethical dividing lines between the applicability of mathematical prediction, and of natural human love and faith derived from God. This, too, is not an extravangant claim given the history of this field, and is in fact likely held by more people than any of the other views above.
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