Our external physical reality is a mathematical structure. That is, the physical universe is mathematics in a well-defined sense, and in those [worlds] complex enough to contain self-aware substructures SAS [they] will subjectively perceive themselves as existing in a physically 'real' world (- Max Tegmark's mathematical universe).
Is mathematics a human invention or does it have an independent existence?
Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: ‘why is mathematics useful in describing nature?’, ‘in which sense, if any, do mathematical entities such as numbers exist?’ and ‘why and how are mathematical statements true?’
This reasoning comes about when we realize (through thought and experimentation) how the behavior of Nature follows mathematics to an extremely high degree of accuracy. The deeper we probe the laws of Nature, the more the physical world disappears and becomes a world of pure math.
Physicist and Nobel laureate Eugene Wigner noted in his classic article on the philosophy of physics: ‘The enormous usefulness of mathematics is something bordering on the mysterious…There is no rational explanation for it…The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve’.
In later life Wigner became interested in the Vedanta philosophy of Hinduism, particularly its ideas of the universe as an all pervading consciousness. He commented "It was not possible to formulate the laws (of quantum theory) in a fully consistent way without reference to consciousness."
The Pythagorean philosophy was dominated by the idea that numbers were not only symbols of reality, but also were the final substance of real things... leading to the famous saying that ‘all things are numbers.’ Pythagoras himself spoke of square numbers and cubic numbers, and we still use these terms, but he also spoke of oblong, triangular, and spherical numbers. He associated numbers with form, relating arithmetic to geometry. According to Aristotle – ‘They (Pythagoreans) said too that the whole universe is constructed according to a musical scale... and that the whole universe is a number, because it is both composed of numbers and organized numerically and musically… the principles of mathematics were also the principles of all things that be…
Theoretical physicist Max Tegmark’s MUH (Mathematical Universe Hypothesis ) states that: 'All structures that exist mathematically also exist physically'. This is in the sense that "in those ‘worlds’ complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically 'real' world".
Mathematical realism holds that mathematical entities exist independently of the human mind. We do not invent mathematics, but rather discover it. Triangles, for example, are real entities that have an existence. The major problem of mathematical Platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities?
This is not an abstract question for Mathematical Platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades. See also the chapter on mathematical realism in the book Plato's Cave (ebook).