ch00beh

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@ch00beh

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ch00beh
@ch00beh
hthrflwrs
@hthrflwrs
hthrflwrs
@hthrflwrs

i love the term "non-euclidean space." you know that one math guy from 300 bc? this space would blow his tits clean off

zaratustra
@zaratustra

Maybe not him, but that's definitely what happened to every mathematician afterwards

Because what Euclid did in his Elements book, he said:

"listen, these ten books i've written about geometry, all of this only works if a triangle's angles add up to 180 degrees. that's a requirement. i don't know what happens in other cases and i don't care"

(Actually his postulate is a little more complex but it directly implies the triangle fact above)

And people for centuries afterwards were like

"what do you mean Euclid of course a triangle's angles add up to 180 degrees all you need to prove it is .... uh [scribbling grows increasingly more frantic]"

It took two millenia of mathematicians repeatedly discovering spherical geometry and going "no this is nonsense. this is satan worship" before geometry got so detached from reality that someone was willing to go "sure, if you bend space just like this you can draw a triangle with three square angles"

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AmanitaVerna
@AmanitaVerna

You could argue, given that we now know that space is curved, and can be curved a lot in the vicinity of large amounts of mass and/or energy (including things moving really fast), that non-euclidean geometry is actually less detached with reality. πŸ˜›

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mcc
@mcc

That just makes it funnier! It took millennia for anyone to work out the coherent internal rules of the surface of a sphere or even a clear statement of the idea the surface of a sphere has rules, and this entire time we've been standing on one.

It's especially weird because the moment you start trying to map a map to a globe you start noticing "wait, this is really hard actually". And the ancient world had globes? Ptolemy even devised some proper mathematical globe<->map projections in the second century CE, and globes are known to have existed for hundreds of years before that. So this makes me wonder if spherical/"noneuclidean" geometry is one of those things antiquity might have actually not found that weird after all, it's just that only a subset of their scholarship managed to survive to the present day.

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garak
@garak

I'm being un-generous here (as must any one-sentence summary of two thousand years of history), but basically it took two millennia for academic mathematics to become detached from astronomy. There was a long period where the main driver of mathematics was predicting the movements of the stars & planets, and math which didn't do that was Wrong.

Our modern concept of mathematics is that it's a toolkit of abstract and general techniques, that any problem might be solved by applying any technique. And in fact the majority of current mathematical research is about finding connection between disparate areas, and rejecting the idea that things are separate.

By contrast, antiquity treated a mathematical model as a tool for a particular situation and used that to ground their intuition of it. Most famously, (some?) Greek mathematicians viewed polynomials as ill-founded: the idea of x^2 + x^3 makes no sense because x^2 is an area and x^3 is a volume and if you're trying to add those things together you're obviously an idiot. So whereas we see polynomials as fully-abstract terms, they still saw them as physical metaphors and that grounding guided their exploration.

So given that view of mathematics, I think there's a tendency to think of geometry as being the "rules of space," and rejecting deviations from that because they don't represent space. There was much less of an inclination to play with a system, see if it's internally consistent, explore its reach, and then later on discover can model some system. The need for external consistency came first, and from that mindset it's more about finding which axiomatic system is "right" rather than exploring the consequences of axiomatic systems.

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