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Where is number 7? No, this is not the beginning of a spy novel, but a question philosophers have been asking for over two thousand years. Does the number 7, along with all the other natural numbers (1,2,3,4........), exist independently of rational minds, perhaps in some separate realm? Or is the number 7 a kind of concept we develop as a result of experiencing groups of seven items? Which came first - the idea of number 7, or groups of 7 items? Are numbers mind-independent or mind-dependent?
In the history of philosophy, there have been followers, in some form or other, of both views. Perhaps the most extreme view of all was that posited by Pythagoras. He not only believed that each natural number is a separate entity dwelling in a kind of heaven, but that all things are numbers, including the harmonies of music and all things of nature. However, the Pythagoreans destroyed this theory themselves when, as a consequence of Pythagoras' theorem for triangles, they discovered a number that could not be expressed by a relation between two natural numbers : the square root of 2 (which is irrational).
Plato was struck by the eternal and perfect nature of numbers. Since in our world nothing is quite that perfect, and since mathematics is about objects known through the mind, he concluded that numbers must belong to a world different from the one we experience. They must exist in another realm. Modern Logicists take a similar view to Plato, believing that numbers are real entities, which are discovered rather than created.
However, modern Nominalists and Aristotle deny the independent existence of numbers. For Aristotle, numbers are essentially numbers of things. The number 7 is not identical with any, or even all, groups of 7. Its existence lies in the fact that groups of 7 exist. Many empiricist philosophers took a similar view, including Berkeley, who believed that numbers are useful fictions that do not exist outside of a mind doing mathematics.
More recently, some philosophers have tried to answer the question "Do numbers exist" by reframing the question itself. For example, Carnap said that the answer to the question "Are there numbers?" depends on the context in which the question is put. If it is asked as a metaphysical question, Carnap thought it was meaningless. However, if the question were framed within talk about mathematics, then it did have meaning. Arguing against Nominalism, the modern philosopher Quine claimed that, by believing in modern Physics (which relies so heavily on mathematics), we are already committing ourselves to believing in the existence of numbers, whether we like it or not.
Has the success of modern Physics brought us almost full circle back to a Platonic, or even Pythagorean, kind of viewpoint? Max Tegmak, a contemporary physicist, in his Mathematical Universe Hypothesis, posits the existence of mathematical entities and even mathematical monism (that nothing exists except mathematical objects). The contemporary philosopher Badiou asserts that Number takes the form of a type of Being, and that ontology (the things which can be said to exist) is nothing other than mathematics itself.
I do not know if this question will ever be answered, but I am sure that Mathematics will continue to grow and deepen, whatever the nature of existence of numbers.
