Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy which deals with the philosophical foundations of mathematics.

Some of the major viewpoints include: Constructivism, finitism and ultrafinitism generally entail a rejection of much of classical mathematics; so, they tend not to be popular with practising mathematicians. However, there are alternative systems of mathematics which have been worked out based on their assumptions — in particular, constructivist or intuitionistic mathematics, is a significant and continuing area of mathematical research. Some of its formulations (e.g. intuitionistic logics) have been found useful by those who don't necessarily agree with the philosophical theories which motivated their development.
 * Platonism: the position that mathematical objects (numbers, sets, fields, etc.) have a real existence independent from the physical world, that they are real existing immaterial objects
 * Formalism: the position that mathematics is nothing more than the manipulation of symbols according to formal rules; those symbols are not ascribed any ultimate existence independent of the physical universe in which they are contained
 * Constructivism or Intuitionism: the position that only mathematical objects that could be constructed, at least in principle, by human mathematicians, are real. ("Mathematics exists in the minds of mathematicians; so anything which cannot fit in the mind of a mathematician cannot exist in mathematics either.") This implies the rejection of some, but not necessarily all, infinite sets. Infinite sets are often permitted so long as we have a procedure which can successively enumerate their members, e.g. the natural numbers. But, infinite sets which cannot be enumerated, such as the (classical) reals, are rejected. (A related position is countabilism, which accepts the existence of countably infinite sets, but denies the existence of uncountable sets.)
 * Countablism: countablists reject the existence of uncountable mathematical objects; only countable mathematical objects really exist. $$\aleph_0$$ is the largest cardinal.
 * Finitism: finitists reject the existence of infinite mathematical objects; only finite mathematical objects really exist.
 * Ultrafinitism: ultrafinitists go beyond finitism to reject the existence of mathematical objects which - while finite - are so large as to be impossible for any human being to think about. According to classical mathematics, there are infinitely many natural numbers, almost all of which are so large that no mathematician will ever think of or refer to them individually. Almost all of them are so large that there is not enough space in the universe to even refer to them in any conceivable notation. Ultrafinitists deny that such finite numbers exist.
 * Paraconsistency: paraconsistent logic and mathematics rejects the principle that every proposition must be either true or false, but not both — it posits that there are propositions that are both true and false simultaneously. Thus, for example, there may be some set, which both does and does not contain a given element. (This is proposed as an alternative solution to paradoxes such as Russell's paradox and the liar paradox.)