There are basically two mainstream schools of thought, (Neo-) Platonism and Formalism, and a third somewhat heretical view, Constructivism.

According to Neo-Platonists, mathematics exist independent of human quest, so they are in fact discovered, not invented. Even the most abstract mathematical objects are real and invariable, immaterial of course and in no way related to physical existence, space and time, but anyway they do exist in a non-objective world. For a Neo-Platonist, there is an answer to Cantor's Continuum Hypothesis, only we do not have the means to obtain it, that is, we do not understand real numbers sufficiently.

A strong argument in favor of the Neo-Platonic view is the "unreasonable effectiveness of mathematics in the natural sciences" (Eugene Wigner, Nobel prize-winning physicist). Another one is given by the Russian mathematician I. Shafarevitch: 

"History of mathematics has known many occasions where a discovery made by a scientist remains unknown until somebody else makes it again later, with astonishing preciseness. In the letter that Galois wrote the day before his fatal duel, he reached some conclusions of extreme importance in the study of integrals of algebraic functions. More than twenty years later, Riemann, undoubtedly unaware of Galois' letter, re-discovered and proved the same propositions. Another example: after Lobachevski and Bolyai built the foundations of non-Euclidean geometry independent of each other, it appeared that two other mathematicians, Gauss and Schweikart, had both reached the same conclusions ten years earlier, also working independently. There is a strange feeling in reading exactly the same ideas, as coming from one mind, in the work of four scientists who studied the subject alone" (talk given to the Göttingen Academy of Sciences, 1973)
 

For Formalists on the other, mathematical objects do not exist. Mathematics consist of symbols, axioms/sentences composed of such symbols and rules to transform sentences into others (e.g. theorems), but none of these has any particular meaning. Mathematics is therefore a humanly constructed language devised by human beings for definite ends prescribed by themselves.

Formalists often speak in terms of Neo-Platonic real objects, but only for reasons of convenience. For a Formalist, the Cantor's Continuum Hypothesis is meaningless for there is no such a thing as a complete understanding of the real numbers. As long as we follow the strict rules of transforming series of symbols into other series of symbols, there is no point in asking whether we approach reality or not, because there is no reality.
 

Constructivists take the extreme view that if something cannot be constructed in a finite number of steps it does not exist. The leader was L.E.J. Brouwer who even devised a famous counter-example to show that the trichotomy law for real numbers (every real number is either negative, zero or positive) is not true. The argument involved a strictly defined but impossible calculation with Pi the result of which would define in turn the sign of a related number. Constructivists would dispose all questions about infinity on these grounds.
 
Links on the Logic and Philosophy of Mathematics can be found at the University of Waterloo/ Department of Philosophy site here
 

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