He discusses two relatively simple proofs, the first being Euclid's proof of the infinity of the primes, the second being Pythagoras's proof of the irrationality of the square root of two:
> "Euclid’s theorem tells us that we have a good supply of material for the construction of a coherent arithmetic of the integers. Pythagoras’s theorem and its extensions tell us that, when we have constructed this arithmetic, it will not prove sufficient for our needs, since there will be many magnitudes which obtrude themselves upon our attention and which it will be unable to measure: the diagonal of the square is merely the most obvious example. The profound importance of this discovery was recognized at once by the Greek mathematicians. They had begun by assuming (in accordance, I suppose, with the ‘natural’ dictates of ‘common sense’) that all magnitudes of the same kind are commensurable, that any two lengths, for example, are multiples of some common unit, and they had constructed a theory of proportion based on this assumption."
> "Pythagoras’s discovery exposed the unsoundness of this foundation, and led to the construction of the much more profound theory of Eudoxus which is set out in the fifth book of the Elements, and which is regarded by many modern mathematicians as the finest achievement of Greek mathematics. The theory is astonishingly modern in spirit, and may be regarded as the beginning of the modern theory of irrational number, which has revolutionized mathematical analysis and had much influence on recent philosophy."
Another good discussion is Poincare's "Science and Hypothesis" (1902) in which he asserts that one defining feature of all mathematics is the self-consistency of arguments, which is the freedom from contradiction (a view all philosophy would hopefully adopt, vs. say, the expediency of political behavior and nation-state propaganda).
> "To sum up, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. The only limit to its power is the necessity of avoiding all contradiction; but the mind only makes use of it when experiment gives a reason for it."
https://archive.org/details/AMathematiciansApology-G.h.Hardy...
He discusses two relatively simple proofs, the first being Euclid's proof of the infinity of the primes, the second being Pythagoras's proof of the irrationality of the square root of two:
> "Euclid’s theorem tells us that we have a good supply of material for the construction of a coherent arithmetic of the integers. Pythagoras’s theorem and its extensions tell us that, when we have constructed this arithmetic, it will not prove sufficient for our needs, since there will be many magnitudes which obtrude themselves upon our attention and which it will be unable to measure: the diagonal of the square is merely the most obvious example. The profound importance of this discovery was recognized at once by the Greek mathematicians. They had begun by assuming (in accordance, I suppose, with the ‘natural’ dictates of ‘common sense’) that all magnitudes of the same kind are commensurable, that any two lengths, for example, are multiples of some common unit, and they had constructed a theory of proportion based on this assumption."
> "Pythagoras’s discovery exposed the unsoundness of this foundation, and led to the construction of the much more profound theory of Eudoxus which is set out in the fifth book of the Elements, and which is regarded by many modern mathematicians as the finest achievement of Greek mathematics. The theory is astonishingly modern in spirit, and may be regarded as the beginning of the modern theory of irrational number, which has revolutionized mathematical analysis and had much influence on recent philosophy."
Another good discussion is Poincare's "Science and Hypothesis" (1902) in which he asserts that one defining feature of all mathematics is the self-consistency of arguments, which is the freedom from contradiction (a view all philosophy would hopefully adopt, vs. say, the expediency of political behavior and nation-state propaganda).
https://archive.org/details/scienceandhypoth00poinuoft/page/...
> "To sum up, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. The only limit to its power is the necessity of avoiding all contradiction; but the mind only makes use of it when experiment gives a reason for it."