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We can now counter-pose two views. At one extreme we have the Platonists who believe in the objectivity of mathematics. They regard mathematics as being independent of human consciousness, but go further and consider it to be independent even of our physical universe. To them the laws of mathematics and logic are valid not just in our world, but are necessarily true in any possible universe. At the other extreme we can place the intuitionists who argue that mathematics is a process of human invention. Maths is to do with the way humans think, it has nothing to do with objective truth or real properties of the world. An intuitionist might make a contrast between the discovery of the law of gravitation and the invention of the theory of calculus. All of these views deny a link between the meaning of mathematics and the objective properties of the universe we inhabit. So the problem facing all these approaches is to explain why mathematics is such a useful subject, why highly abstract theories often lead to powerful applications in the rest of science.