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By Reuben Hersh

Number of the main fascinating contemporary writings at the philosophy of arithmetic written by means of hugely revered researchers from philosophy, arithmetic, physics, and chemistry Interdisciplinary e-book that might be valuable in different fields—with a cross-disciplinary topic region, and contributions from researchers of assorted disciplines

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This is true, for example, of number theory and of much of the theory of partial differential equations. The idea that the axiomatic method provides a strategy both for finding and remembering proofs also contrasts with mathematical experience, which shows that proofs based on the axiomatic method often appear to be found only by a stroke of luck, and seem artificial and difficult to understand. Showing only the final outcome of the investigation, established in a way that is completely different from how it was first obtained, such proofs hide the actual mathematical process, thus contributing to make mathematics a difficult subject.

I never heard of two poets writing the same poem. HIPPOCRATES Nor have I heard of such a thing. But I remember Theaitetos telling me about a very interesting theorem he discovered on incommensurable 10 Alfréd Rényi distances. He showed his results to his master, Theodoros, who produced a letter by Archytas in which the same theorem was contained almost word for word. SOCRATES In poetry that would be impossible. Now you see that there is a problem. But let us continue. How do you explain that the mathematicians of different countries can usually agree about the truth, while about questions concerning the state, for example, the Persians and the Spartans have quite opposite views from ours in Athens, and, moreover, we here do not often agree with each other?

HIPPOCRATES I do not see how to start. SOCRATES If you still have patience, let us try it together. Tell me, what is the difference between the sailor who finds an uninhabited island and the painter who finds a new color, one which no other painter has used before him? HIPPOCRATES I think that the sailor may be called a discoverer, and the painter an inventor. The sailor discovers an island which existed before him, only it was unknown, while the painter invents a new color which before that did not exist at all.

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