however, make no mistake the pythagoreans were a deeply weird cult who hated beans more than they loved life. but they should be mocked for that, not the irrational number thing.

of course, the theory of the real numbers hand countless named and nameless contributors besides these. it is the bedrock of most modern mathematics, and it is from a firm position on that bedrock that we look back, see people raising hell because the ground beneath their feet is giving way, and laugh.

the other piece is the development of abstract algebra, which provided the tools to understand how the behavior of the positive and negative whole numbers is a certain self-consistent piece of the behavior of the real numbers as a whole. for this we have to credit dedikind for originating the definition of an algebraic ring. he also is responsible for giving one of the two conventional definitions of the real numbers in terms of the rational numbers, which really does solemnize their marriage.

the bridging of this conceptual gulf had what seem to me to be two vital steps. one is the development of algebraic geometry, the critical insight here being the possibility of a magnitude being a "solution" of a polynomial equation. as far as i know, this was the innovation of abu kamil, an early algebraist in the arabic tradition, though the form of the idea we are most familiar with is the coordinate plane of descartes

returning to the body of the matter, this crisis in geometric reasoning needed an entirely new theory of proportion to be developed and integrated into the system of plane geometry. that took time but wasnt a deal breaker. what was a deal breaker is that for the next millennium, more or less, (positive whole) numbers and magnitudes were conceptually different species and, though they had some similar properties, could not be made to reveal important facts about the other

as a historical aside, a number of school geometry texts quietly only proved important propositions of the theory of proportional triangles for triangles where the ratio between the sides can be written as one of whole numbers and then proceeded to use them as though they'd been proven for all triangles

but the fact of the matter is that for the pythagoreans, this result seemed as devastating as if logic itself had been shown to be faulty. they had built an entire system on representing the relations between magnitudes with relations between (positive whole) numbers and here comes a proof that demonstrates the existence of magnitudes that clearly stand in some fixed relation to one another that categorically cannot be equivalent to a relation between numbers

people laugh at how the pythagoreans are said to have murdered the guy who told people about the incommensurability of a square's side and diagonal (read: irrationality of the square root of two) but that's just because we take for granted a tremendous amount of work done on the real number system; i even had to translate the statement itself into that system because i imagine most people have never heard of incomensurables

recoiling in horror when my genealogy reveals one of my ancestors to have come from britain, relaxing only slightly when i learn that he was a scott

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