What are the historic stances on the epistemological status of mathematics?

Question

I know that Plato and Kant thought it was synthetic a priori (although Plato would not have phrased it in that way). What other major thinkers have weighed in on this issue, on both sides of both the analytic/synthetic distinction and the a priori/a posteriori distinctions? Citations are best if you can provide them.

Answer 1

Here Marx-Engels' view on Mathematics.

Anti-Duering in 1877.

http://www.marxists.org/archive/marx/works/1877/anti-duhring/ch01.htm (emphasis mine)

That pure mathematics has a validity which is independent of the particular experience of each individual is, for that matter, correct, and this is true of all established facts in every science, and indeed of all facts whatsoever. The magnetic poles, the fact that water is composed of hydrogen and oxygen, the fact that Hegel is dead and Herr Dühring alive, hold good independently of my own experience or that of any other individual, and even independently of Herr Dühring’s experience, when he begins to sleep the sleep of the just. But it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is, to perform the first arithmetical operation, are anything but a free creation of the mind. Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered except their number — and this ability is the product of a long historical development based on experience. Like the idea of number, so the idea of figure is borrowed exclusively from the external world, and does not arise in the mind out of pure thought. There must have been things which had shape and whose shapes were compared before anyone could arrive at the idea of figure. Pure mathematics deals with the space forms and quantity relations of the real world — that is, with material which is very real indeed. The fact that this material appears in an extremely abstract form can only superficially conceal its origin from the external world. But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to separate them entirely from their content, to put the content aside as irrelevant; thus we get points without dimensions, lines without breadth and thickness, a and b and x and y, constants and variables; and only at the very end do we reach the free creations and imaginations of the mind itself, that is to say, imaginary magnitudes. Even the apparent derivation of mathematical magnitudes from each other does not prove their a priori origin, but only their rational connection. Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men: from the measurement of land and the content of vessels, from the computation of time and from mechanics. But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform. That is how things happened in society and in the state, and in this way, and not otherwise, pure mathematics was subsequently applied to the world, although it is borrowed from this same world and represents only one part of its forms of interconnection — and it is only just because of this that it can be applied at all.

So, in short, Engels thought Mathematics is historical and at the same time empirical born out of human beings' neccessities, although it appears it is standing alone itself, the origin came from the existence from the outside world, even though the numbers look like a priori, it is not, but just human being needed the numbers to count the existings outside and it became systematic itself like a very tall building and thus it only looks like as if it is a priori.

Let me edit so that the Engel's latter part of the speech gets clearer. His latter emphasized part of the speech is related with the concept or the idea of the alienation, which is, human products or what first was in human mind become(s) to behave as if it is ( they are ) an independent, outside existence. So that Engels wrote "But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform.". He meant by this the mathematics became a law despite of the nature of its origin, ( which is empirical, first only a little chunk of conception in his/her mind ) a law ( a whole independent architecture ) that stands independently against us which we have to conform to. This idea of the alienation, is penetrating throughout almost the entire works of Marx-Engels team, here only Engels adopting itself to the all the so-called science, while Marx rather more tried to analyze by it the capitalistic mode of the production.

Answer 2

Wolfgang Pauli has written on this extensively. He said

...a mathematical formula can never tell us what a thing is, but only how it behaves; it can only specify an object through its properties. And these are unlikely to coincide in toto with the properties of any single microscopic object of our everyday life.

And Arthur Eddington:

For example, we may admire the triumph of patience of the mathematician in predicting so closely the positions of the moon, but aesthetically the lunar theory is atrocious; it is obvious that the moon and the mathematician use different methods of finding the lunar orbit...But now we realise that science has nothing to say as to the intrinsic nature of the atom. The physical atom is, like everything else in physics, a schedule of pointer readings...

...matter is something that Mr. X knows. Let us see how it goes: This is the potential that was derived from the interval that was measured by the scale that was made from the matter that Mr. X knows. Next question: What is Mr. X? Well it happens that physics is not at all anxious to pursue the question: What is Mr. X? It is not disposed to admit that its elaborate structure of a physical universe is "The House that Mr. X built."...matter, in some indirect way, comes within the purview of Mr. X's mind is not a fact of any utility for a theoretical scheme of physics. We cannot embody it in a differential equation. It is ignored, and the physical properties of matter and other entities are expressed in their linkages in the cycle. And you can see how by the ingenious device of the cycle physics secures for itself a self-contained domain for study with no loose ends projecting into the unknown. All other physical definitions have the same kind of interlocking. Electrical force is defined as something which causes motion of an electric charge; an electric charge is something that exerts something that produces motion of something that exerts something that produces...ad infinitum.

Both quotes are from Quantum Physics and Ultimate Reality: Mystical Writings of Great Physicists by Michael Green.

Answer 3

Intuitionism and Formalism are the two modern positions that tend to be contrasted with the synthetic, a priori characterization of Platonism.

Formalism sees mathematics as just another science, meant to map onto outside world. It is a reaction to the discovery that our internal notions of negation and universality are very hard to reconcile, as pointed out by Russel's paradox. This makes the idea of an ideal world very hard to stomach. Either the ideals have a basic contradiction baked-in, or they are really something other than ideal. The focus is on finding a new internal basis that supports all useful mathematics, and feels more accountable internally, and discarding mathematics too far from testable application as places where internal contradictions might hide. It presumes mathematics, like all other sciences must be a-posteriori.

Intuitionism supposes that mathematics is neither an ideal nor an approximation to external reality, but part of the structure of human psychology. It finds application to external reality in that humans are evolved to survive in the world, but it is only internally consistent to the degree that human language is. It does not accept the notions of universality or negation as clean and ideal constructions, but requires constructions in verbal forms to stay away from overdependence upon them. So, for instance, reductio-ad-absurdum is not acceptable as a way of establishing existence or universality, only as a way to forestall wasted searching for proofs of the contradicted concepts applications. Thus mathematics has a-priori components, but is not entirely a-priori, as it must be unfolded through contact with other humans' language structures. It is also partly analytic and partly synthetic to the degree that the bases of language are inborn but evolved.