From this viewpoint it is possible to answer an earlier question -- is mathematics an experimental subject? Indeed it is, but many of the experiments are rudimentary and are repeated every generation by children using fingers or counters to discover and verify the elementary rules of arithmetic (e.g. 3 + 4 = 7, , etc.). As the mathematician G. H. Hardy put it
The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations.It is plausible that the basic laws of logic, geometry and probability are also developed, very early in a child's development, in a similar, experimental way.
Nevertheless, there is a contrast between mathematics and natural science. In the former the most elementary laws may be obtained on the basis of experiment, but an enormous edifice is built up from here and it is very rare for experiments to be performed at the higher level. With natural science, typically, a whole theory develops, often using mathematics in the exposition of the theory, but then predictions are made and experiments are done to test that theory. It is the testing in practice that ensures that science unveils an objective understanding even if, as we have seen already, that is only a partial understanding. Indeed in the Marxist view the whole notion of objective truth can be defined as the correspondence between theory and practice.
So in order for higher mathematics to claim objectivity there must be some correspondence between theory and practice. Now the practice of mathematics can be in its applications to other sciences, its applications to some other part of mathematics itself or to a range of human activities. The theory of the differential calculus, for example, has innumerable applications to almost every science and, indeed, was developed at least partly in response to problems in other fields. Group theory is applied more often to subjects within mathematics: number theory, geometry, topology, etc., but also to sub-atomic physics. Cryptography developed directly from a problem facing society: the military need to send and intercept secret messages during the second world war [#!Hod83!#].
At its most abstract level mathematics deals with objects and structures which cannot correspond directly with any real practice. Infinite sets, for example, are unlikely to refer to any real objects or events. (If we assume a bounded universe and a quantisation of all the dimensions then infinite sets cannot correspond to any set of physical objects or events.) However the study of infinite sets is useful to mathematics, firstly because the language of infinite set theory allows us to handle finite objects quite neatly (e.g. when we talk of the set of all even numbers, or when we deal with irrational numbers like by means of an infinite series of approximations 3.141...) and secondly because it clarifies the deductive method in an area where intuition is of very limited value. It is at this highly abstract level that the most profound contradictions have been found and where the understanding of truth and meaning are most problematic
As a final example of this consider Cantor's so-called continuum hypothesis which says that there is no set whose cardinality lies strictly between that of the integers and the real numbers. It has been shown that the continuum hypothesis cannot be proved from the other axioms, nor can it be proved false -- it is independent of the axioms of set theory. Some mathematicians (e.g. Gödel) hoped that a new, intuitively obvious axiom would come to light and solve this type of problem, but this has not happened and most mathematicians now think that it never will. The Platonist has no problem with the continuum hypothesis: (s)he believes that it is definitely true (or definitely false). The human brain may be incapable of ever discovering the truth, but there is a truth. Intuitionists, too, are not troubled because they do not believe in the existence of the set of real numbers as a completed totality, as this cannot be constructed. Therefore, the continuum hypothesis is without meaning for them, or more precisely, the continuum hypothesis is trivially true for intuitionists since they do not accept the existence of the set of real numbers. The formalists could argue that the continuum hypothesis could be made true or false at will. To them, mathematics is a process of making formal definitions and conventions -- there is no meaning attached to it.
The materialist should pause before saying anything rash, but then perhaps put the following case.
Elementary mathematics, and indeed the vast bulk of mathematics, derives more or less directly from experience. It is tested by its logical consistency moreover by its applications to a number of other subjects and activities. Therefore, when we say, for example, that 5 + 5 = 10 or that there are infinitely many prime numbers, we refer to an objective truth. It does not depend on the individual who says it, nor on the type of society that made the discovery. At the most abstract level, though, mathematics does not correspond to any real process and so propositions like the continuum hypothesis cannot be construed as objective truths, but are indeed a matter of definition or convention. As Putnam put it
Urging this relativism is not advocating unbridled relativism; I do not doubt that there are some objective (if evolving) canons of rationality; I simply doubt that we would regard them as settling this sort of question let alone as singling out one unique `rationally acceptable set theory'. ([#!Put83!#] page 430.)
If it were the case that the majority of mathematics were of this very abstract type, and if mathematics were principally an internal subject with few connections to other sciences, then it would be very hard to argue that mathematics were scientific. However, neither of these conditions are met: a large part of mathematics deals directly with applications and even at the most abstract level the mathematician is always delighted if their work links in with some distantly related area.