prize to the person who settles the Riemann hypothesis, but only if they settle it positively, I think. Robert Katz, 1964, Axiomatic Analysis,. You talk as though the Greeks discovered an objectively existing object, the square root of two, which was not rational. Okay, so this was 1927, this was Émile Borel with his paradoxical know-it-all real number that answers every yes/no question, and it is a mathematical fantasy, not a reality. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
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How do you know there isn't a formal definition that works better for the rationals? We can look at a graph and this jump stares us in the face. Secondly, and more importantly, they provide a context in which the arguments we naturally use to justify the existence of those useful numbers are valid. He was a rationalist, and he believed that if anything is true, it must be true for a reason. You may well say, "All I care is will the program stop in a reasonable amount of time, say, a year. So you're given a program that's self-contained, and want to know what will happen. The world of mathematics is a toy world where we ask fantasy questions, which is why we have nice theories that give nice answers. And there are lots of perfect numbers. Proving this is the first half of one proof of the fundamental theorem of algebra.
Try to imagine a world without mathematics - Be a Maths Scholar
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