why is it that you can take 10^10^10^10^10^10^10^10^10 points from the set [0, 1]x[0, 1] and its area would still be 1, but you can only take a finite amount of anything we know (atoms, quarks) from something before you end up with nothing? how bad of a model of reality euclidean spaces are?
>>7977448
Because math and reality are only tangentially related
>>7977448
Because matter is quantized.
You can inly use macroscopic methods when you pass a certain threshold.
>>7977448
>How bad of a model of reality euclidean spaces are?
Well for one, gravity means our spacetime is not euclidean. As far as things like elementary particles go, they are not directly related to geometry.
I mean classically, gauge theories our extremely geometrical.For instance the four-vector potential (as a differential form) is a pullback of the connection form on a principal bundle.
i.e. Let M be spacetime, take the bundle [math] P\mathop \to \limits^\pi M[/math] with fibers [math] {P_m} \cong G[/math] for some lie group G. Then for some form [math] \omega \in \Gamma \left[ {\left( {P \times \mathfrak{g}} \right) \otimes {T^*}P} \right] [/math] such that it satisfies the conditions for a connection form, using the local section [math] s:U \to {\left. P \right|_U} [/math] we can pull it back [math] A = {s^*}\omega[/math] and get something equivalent to the standard four-vector potential as a 1-form.
However all this nice geometry breaks down when you quantize a theory.
>>7977448
I'm sorry, is that 10^^9 or 10^100000000?
>>7977448
>you can take points from the set
Because particles aren't points. Area is measured by composing two 1D constructs, and points are 0 dimensional constructs. Basically R is so "dense" that it can't 'collapse' even if you take absurdly large amounts of non-density from it.
>>7977448
Because the universe is finite and not enough people listen to Wildberger.
>>7977480
wat
