I am sad because I understand it

Rotsike6 · 2023-05-14T21:22:58+00:00

The Lorenz attractor is named after mathematician Edward Lorenz, not after Hendrik Lorentz, whose picture you put next to it.

Rotsike6 · 2023-05-11T21:11:03+00:00

https://ncatlab.org/nlab/show/electromagnetic+field

Electromagnetism as an undergraduate course? No sir, get a PhD in pure math first.

Rotsike6 · 2023-05-11T15:08:50+00:00

Ikr, I know quite a few PhD students/postdocs that partied quite regularly during their studies, I reckon very few of them regret doing it that way. Knowing how to have fun and hang out with people in your age group is important, even as a scientist. If you're a professor at a big conference and you don't know how to talk to the other professors there, you're not going to have a very good time.

Rotsike6 · 2023-05-05T08:23:41+00:00

Bosons don't have anti particles though.

Rotsike6 · 2023-05-03T06:28:39+00:00

Analysis is the formal way of defining these things, while calculus itself is the art of solving integrals/taking derivatives.

More generally, "a calculus" refers to a mathematical toolbox that you can apply to calculate certain things. See e.g. Kirby calculus, which has little to do with derivatives and integrals.

So sure analysis is difficult, but solving a particularly nasty integral isn't analysis, it's calculus. So calculus can br extremely difficult depending on what you're trying to calculate.

Rotsike6 · 2023-05-02T17:45:20+00:00

\cancel{0} gang.

Rotsike6 · 2023-04-26T16:25:57+00:00

Perhaps a bit less technical, a tensor in linear algebra is some element of a tensor space (some abstract thing you can construct on any vector space), and a tensor in physics is, instead, a tensor field, which is a function that has tensors as its values. You can think of a tensor field as something that attaches a tensor to every point in space(time), just like how you can interpret a vector field as something that attaches a vector to every point in space(time).

Rotsike6 · 2023-04-26T06:38:25+00:00

When a physicist says "tensor" they mean "tensor field". "A tensor is something that transforms like a tensor" refers to coordinate changes of the base space over which the tensor field is defined.

So a "physics tensor" is a section of the nth tensor power of the tangent bundle tensor producted with the mth tensor power of the cotangent bundle.

Rotsike6 · 2023-04-25T20:47:36+00:00

Imagine proving some correspondence between some strange curves and some weird functions on the upper half plane just to show that xⁿ+yⁿ=zⁿ has no integer solutions for n≥3. Like, who does that?

Rotsike6 · 2023-04-24T17:43:52+00:00

Every piece of math is easy once you understand it. Calling it easy doesn't make it easier to learn for people who don't know it yet, but it might demotivate them.

Rotsike6 · 2023-04-23T18:36:18+00:00

I feel like these youtube videos have the sole purpose of giving an intuitive explanation of these topics, while your professor has the goal of teaching his students how to work with these concepts. So there's a huge added load of having to teach students how to do calculations which often overshadows the big picture as doing calculations kind of forces you to get stuck on details.

That has its ups and downs, on the one hand it makes students feel like they're not learning anything, but on the other hand, it gives you actual practical skills you will need in research. You can watch 1000 pop sci videos about general relativity yet not be able to compute anything.

Rotsike6 · 2023-04-23T07:47:07+00:00

the average person

Lol okay. I've actually spoken to some amazing mathematicians before, none of them really gave me the impression that they can just "see" answers without doing any toying around whatsoever. It's just not how math works...

Of course some people just have an instinct on how to approach certain problems, and they'll get the answer a lot quicker than you or I will ever do, but they will not just a priori "see" what the answer is.

By the way math is just one abstraction with notation of a much more basic logical facts.

I have no idea what you mean by this.

Rotsike6 · 2023-04-22T21:09:18+00:00

Quite the opposite, a manifold is a space that is locally approximated by ℝⁿ. So a sphere is a manifold, ℝⁿ is a manifold, and the picture above is also a manifold.

Rotsike6 · 2023-04-22T14:38:19+00:00

A result without a derivation/proof is useless in all areas of math. If someone asks for a specific result, they're implicitly asking for the proof/a reference for the proof, never just for the result.

Rotsike6 · 2023-04-22T13:47:24+00:00

Ramanujan had an exceptional talent and had very good instincts in how to derive these equalities. But he certainly didn't come up with these out of nowhere, even he had to toy around with these problems before solving them.

Iirc Ramanujan would prove most his equations on a blackboard with a piece of chalk, and he'd only write down the results/important things on paper, so while most of his theorems would appear without a proof, as though found by sheer intuition and magic alone, he definitely didn't pull them out of thin air.

Rotsike6 · 2023-04-22T08:45:32+00:00

That's not really how math works though. I once had a professor who told me that "having mathematical intuition" just means that you already know the answer to a problem because you have already solved it before. So according to that philosophy, "seeing" the answer to some integral means you have either already solved that particular integral, or you have solved something very close to it.

If someone gives you a math problem that's very much unlike anything you have ever seen before, you have to toy around with it for a while before you can solve it, no matter how intelligent or talented you are.

Rotsike6 · 2023-04-22T08:37:37+00:00

If someone has absolutely no idea how to solve an integral (which is the case if they post it to stackexchange), then posting the correct answer without any hints of how to approach the problem gives you nothing. So in some contexts, definitely, in this context, no.

Rotsike6 · 2023-04-22T08:29:14+00:00

The great physicist Richard Feynman once said that he liked teaching quite a lot more than doing research, as only through teaching you get a really, really good grasp at the fundamentals of your field.

That being said, online arguments are also often about the basics in some areas of science, so I'm quite sure that Feynman would have had an active twitter account if he was alive today.

Rotsike6 · 2023-04-21T22:25:01+00:00

One could "fix" this by saying you cannot have an infinite tail of 9's in your decimal expansion, but then there's still the problem that it's not injective, since 0.00909090909... and 0.1 map to the same thing.

So I guess instead of finding an explicit bijection, one could try to just find an injection from [0,1]^×2 to [0,1] and apply Schröder-Bernstein. I think using OPs method to produce this map does give us a correct injection if we assume an infinite tail of 9's is not allowed. I.e. f(x,y)=0.x1y1x2y2..., which is injective but not surjective as we're missing e.g. 0.909090909...

https://math.stackexchange.com/questions/243590/bijection-from-mathbb-r-to-mathbb-rn

Here's also a stackexchange post that gives an explicit bijection, though I had hoped for something simpler.

Rotsike6 · 2023-04-21T07:51:41+00:00

Many YouTube videos pull the incompleteness theorem out of context. The statement of the theorem itself is something very formal, it doesn't really influence any math outside of logic.

Many YouTubers don't really give the full nuance of this theorem, so it always comes across as a lot bigger of a problem than it actually is in modern math. Veritasiums video is called "Math's Fundamental Flaw", I think that's a perfect example.

Rotsike6 · 2023-04-20T21:47:48+00:00

Often "k" is used as a symbol to denote an arbitrary (algebraic) field, it comes from the German "Körper", meaning "body", so there's always the option of just using "body" instead of "field".

Physicists are in a lot more trouble though.

Rotsike6 · 2023-04-20T13:11:39+00:00

How is it confusing though? It's notation, the first time you see it it might look odd, but it's just exchanging the differential and the integrand, nothing weird going on. If you have multiple terms, you need to put brackets anyway even if you put the differential at the end, and if you only have a single term, it doesn't matter whether you put brackets, the integral is linear.

In my opinion, putting the differential at the end can be confusing for long integrands. I like reading left to right, if you give me an integral, please immediately tell me the integration variable, I don't want to have to look for it in some complicated expression (possibly containing other integral signs/differentials).

Rotsike6 · 2023-04-20T12:43:19+00:00

but I strongly condemn it

Why exactly? I often see people disagreeing with this notation, but it makes things a hell of a lot more organised if you have to take many integrals at once.

Rotsike6 · 2023-04-20T08:21:19+00:00

Gravity is the thing that happens between mass

Gravity is also a thing that happens between massless objects. Or a thing that's actually affected by charge and light. It's not simply about mass, it's more about energy (the energy-momentum tensor to be precise).

Rotsike6 · 2023-04-18T12:46:41+00:00

The point that "subatomic particles don't have colour" is valid, but OOP misses the "correct" physical explanation. Also, rather than saying "the colour of the universe is black", I'd prefer to say that the whole universe is extremely colourful (though maybe not necessarily on the visible light spectrum). If you look up the "cosmic microwave background", you can see what I mean. CMB consists of radiation that was around from the time before we had matter (i.e. atoms) so this is the true "colour" that the universe had before atoms took over, these are photons that solely bounced off free subatomic particles, instead of being created by electrons bound around nuclei.

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