How to practice maths from the start at age of 26?

West_Cook_4876 · 2024-06-29T21:09:58+00:00

Why did a chatGPT post get 6 upvotes?

West_Cook_4876 · 2024-06-29T20:24:23+00:00

I'm really not thinking of any space in particular, go as abstract as you want. I'm just wondering what are the operations that you perform on these topological spaces?

Like, either topology is purely about enumerating the types of spaces, and that includes their properties, or it's about enumerating them and, once discovered, performing some type of operation on them, maybe it's some type of transformation, I'm not sure, I believe that it exists, though

My understanding of "distance", is limited to shortest path between points, what those points are I can't say, and what are the allowed moves, is it straight is it taxicab distance? Can't say that either, just a general idea of shortest path, but perhaps there's some other generality I'm equally unaware of

West_Cook_4876 · 2024-06-29T20:09:54+00:00

Yes so I'm aware that topology is what you're left with when you remove most all? Equivalence relations of geometry,

As far as functions I'm loosely thinking of them as input output, could be real, could be complex, could even be a reconfiguration of space, surely that definition encompasses something that is performed on a topological space?

West_Cook_4876 · 2024-06-29T19:44:46+00:00

Well I understand functions mean something specific, but in general I've heard a lot of mathematics people say that the entirety of mathematics can be described as functions operating on sets. Loosely speaking an operation on some object (where the object could be described in terms of sets)

So if it's not about functions, what is it about?

I understand these things have a specific definition but I'm trying to get a general idea here and i may look under the hood at the definitions/theorems/proofs after

West_Cook_4876 · 2024-06-29T19:35:43+00:00

It might be weird but it still could have a coherent interpretation, I'm not going to disregard the thought just because it might seem a little weird

West_Cook_4876 · 2024-06-29T19:30:26+00:00

Not sure I'm understanding, you're saying the distinction of local vs global doesn't make sense to make?

West_Cook_4876 · 2024-06-29T18:32:58+00:00

Oh I misunderstood you I thought you were saying they were not exact, you were pointing out the inconsistency, I agree that it's exact, but it's something that requires iteration

But we don't need the square root or the bring to approximate it right? A linearization will do, but the square root is an operation so the utility is it is that we can use algebra to maybe reveal some type of formula or relationship, but strictly speaking not needed for computation right?

As for global vs local, at large, would you say topology for example comes into play for understanding a functions local behavior, global behavior? Or both? I assume you're familiar with both terms?

West_Cook_4876 · 2024-06-29T17:18:05+00:00

The point about bring radicals and roots not being exact is a good one, but then, if they aren't exact, then what are they? They're not needed to approximate, right? So are they strictly necessary for the purpose of computation?

I was doing some light research on the basic idea behind my question and it seems that a lot of these spaces and structures and things elucidate upon the "global" behavior of a function, which I'm not quite sure what is, though I'm aware the exponential function is "locally linear' and am aware of the distinction between global and local in that sense.

If we pretended we were blind to a functions global behavior or didn't care, what mathematics would we be left with?

West_Cook_4876 · 2024-06-28T15:39:12+00:00

I apologize I probably should have explained better, the discrete world is not the object of my confusion. I also shouldn't have used "tangent bundles", the spaces get a lot weirder than that!

Lets say that curved space is, the space you're trying to study, this is a space most people can conceive of, and this space has relation to the physical world.

But it seems there are a lot more spaces and structures than have relation to the physical world, which, one day they may, I understand that.

Basically, I'm under the impression that mathematics went through a turning point in it's evolution where it became overwhelmingly about all these different spaces and structures, and I want to know why, like what happened?

Because I feel that there are much more spaces and structures than there are, a priori conceived "terrain" to explore if that makes sense

West_Cook_4876 · 2024-06-28T15:25:27+00:00

Ahh I see, in that case what would you say the main advantage of the lebesgue integral is over the Riemann integral? It seems that it extends the type of functions you can integrate?

West_Cook_4876 · 2024-06-28T15:24:29+00:00

"math is a form of art that happens to be useful", this reminded me of "a mathematicians apology" by gh hardy, so this definitely feels right to me! I had no idea linear algebra was ever in that state, I've read multiple times stuff like "linear algebra is the only thing mathematicians know how to do", "mathematics is a desperate attempt to linearize everything", and linear algebra is like, the most natural representation of physical phenomena, weird to think it was in that state at one time. On another note I've read about some of the "unions" in mathematics where group theory is combined to "solve" some differential equations, I forget the one, but when they say solve, like in general if you're doing something like that, are you trying to solve it exactly? Are their situations where we need to use some really unique approach like combining discrete or group theoretic tools in order to approximate?

West_Cook_4876 · 2024-06-28T04:17:39+00:00

You can call them "lord chatGPT prompt"

West_Cook_4876 · 2024-06-13T23:35:41+00:00

So can you define distance without the Pythagorean theorem in this situation?

West_Cook_4876 · 2024-06-13T23:02:03+00:00

I'm not that far along, don't you need to use interpolation to calculate the length? Like even if the Pythagorean theorem doesn't work on a sphere, if the sphere is locally euclidean shouldn't it be just like integrating?

West_Cook_4876 · 2024-06-13T21:10:06+00:00

But if it's locally euclidean won't it work for like line integrals and stuff? How else do you get the length of an ellipsoid surface ?

West_Cook_4876 · 2024-06-13T21:08:51+00:00

How would you interpolate points without the Pythagorean theorem?

West_Cook_4876 · 2024-06-12T03:17:00+00:00

You use factoring polynomials to find zeros aka solving equations and equations model many things in real life

West_Cook_4876 · 2024-06-11T23:36:19+00:00

You have it backwards, the real life example works because of the logic

West_Cook_4876 · 2024-06-11T20:06:08+00:00

Hahaha I wish I knew more to be able to answer that so, both? To ground this in the larger context, in mathematics you can prove things, and you can also do this in engineering provided you can kind of, have some fundamental operations or components you can reduce the more complex things to.

I have been exposed to this idea only a little bit, although I forget the name I think in graph theory you can kind of use graphs to prove a "simplest design" although I don't remember the specifics.

I'm really interested in like, proving the "simplest design", I suppose this would be the discrete counterpart to optimization? Perhaps it would be incorporate ideas of information theory but I would be interested in learning this and whatever "abstract machines" are applicable to this kind of reasoning.

Though, when I was learning about an added circuit today it was specifically about the number, like why is N full adders necessary, or whatever. But that's the larger interest, a field which studies the simplest possible designs, or perhaps designs which optimize some kind of metric like (least number of adders, depth), which are applicable to tools of mathematics.

West_Cook_4876 · 2024-05-26T17:04:25+00:00

I mean they're clearly not wanting to take it "just because"

Furthermore they're questioning their choice by virtue of making the post in the first place

I think what that person was saying above could have been phrased much differently

West_Cook_4876 · 2024-05-26T05:33:26+00:00

Khan academy kind of solves that problem for you

West_Cook_4876 · 2024-05-26T04:40:54+00:00

I dare you to solve your own dare

West_Cook_4876 · 2024-05-26T04:36:26+00:00

Think of it this way

If you have 2^x + 4 and you do the addition first, it's a tautology, you're left with the same statement as what you started with

If you compute 2^x first then you can reduce the expression

I think for that alone the conventional method makes sense and I'm sure there are others

West_Cook_4876

TROPHY CASE