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Article  Maths, like quantum physics, has observer problems

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EXCERPTS: . . . Is there room for the first-person perspective in mathematics as well?

At first glance, the answer appears to be “no.” [...] It is the most objective of all sciences, and mathematicians take pride in the certainty and the timeless nature of mathematical truths. Indeed, if Leo Tolstoy wasn't born or died before he wrote Anna Karenina, this book wouldn’t exist; no one else would have written the exact same novel. But if Pythagoras had not lived, someone else would have discovered the same Pythagoras theorem (and in fact, many did). Moreover, his theorem means the same thing to everyone today as it meant 2,500 years ago when he discovered it, and there’s every reason to believe that it will mean the same thing to everyone 2,500 years from now, regardless of their culture, upbringing, religion, gender, or skin color.

[...] However, Pythagoras’ theorem is not true in the framework of non-Euclidean geometry ... What’s going on here? ... A theorem does not exist in a vacuum; it exists in what mathematicians call a formal system...

[...] This creates the impression that all of mathematics could be done by a computer. But that’s not the case. It’s true that once the axioms have been chosen, there’s no ambiguity in what constitutes a theorem in our formal system. This is the objective part that could indeed be programmed on a computer. But one still has to choose the axioms, and this choice is crucial.

For example, Euclidean geometry of the plane and non-Euclidean geometry of the sphere differ in just one of their five axioms; the other four are the same. But this one axiom (the famous “Euclid’s fifth postulate”) changes everything. Theorems of Euclidean geometry aren’t theorems in non-Euclidean geometry and vice versa. (Fun fact: for about 2,000 years mathematicians tried to prove the fifth axiom of Euclidean geometry from the first four — only to realize, in the 19th century, that it is impossible; on the contrary, one can replace it with a different axiom, and this leads to the non-Euclidean geometry discussed above.)

How do mathematicians choose their axioms? (MORE - missing details)

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