How can visualizing abstract spaces aid in understanding mathematical concepts?

[trees and plants]

Hello. How to think in abstract spaces?Like manifolds? Or metric spaces? Or function or Banach spaces? Are they not just generalisations of the usual 2d or 3d Euclidean spaces we know? So they could be studied by generalising or extending things we know from 2d or 3d euclidean spaces and if they have curvature or other features that we can see or know in 2 dimensions or 3 dimensions like projections, derivatives, integrals, limits and others? Generally, that is the way all abstract spaces in mathematics are studied or thought? By thinking of the things we know from 2 or 3 dimensions and generalising or extending them? Are there any other ways?Perhaps no? This is the only way? I am sorry if this seems like a trivial question but i get stuck on this. Thank you.

 

[trees and plants]

I think my question is quite obvious to be answered. Is there anyone who wants to answer?Anyone?

 

[martinbn]

I think my question is quite obvious to be answered. Is there anyone who wants to answer?Anyone?

Which question!? You have written quite a few question marks.

 

[trees and plants]

Okay, those questions i made i mean. I asked one of my professors in the past about how theorems and proofs are made and he told me if i remember correctly sometimes they are made at the same time. So, the person who is looking for the theorem he wants to make does not know sometimes the statement of the theorem but he is trying to reach to it, produce it by proving and following the proof? Is this correct?

 

[Vanadium 50]

I think my question is quite obvious to be answered. Is there anyone who wants to answer?

Treating us like servants - servants working too slow for your liking - is not a very good idea. You are not entitled to answers from volunteers on your time scale, or for that matter, you aren't entitles to answers at all. A little gratitude might even help.

Your message #4 seems to have very little overlap with your message #1. I think you need to be a lot clearer on what exactly you want to know.

Further, we were working with you on basic proofs back in November (under one of your old aliases) and you just stopped participating. This was unwise. You would have learned it then. People also are less inclined to help if last time you just walked away.

Finally, you want to study advanced topics, but you haven't got the basics down (e.g. the ability to prove numbers are either even or odd.). This seldom works.

 

[wrobel]

Are they not just generalisations of the usual 2d or 3d Euclidean spaces we know?

"just" is a dangerous word. Yes they are just generalizations. And they have a lot of features that you never meet in 2d or 3d

 

[mpresic3]

Trees and Plants, you seem to be asking whether mathematicians and physicists generalize 2 and 3 dimensions to higher dimensions in treating abstract mathematics. This sometimes useful. It is beneficial to learning relativity (differential geometry with 3 spatial dimension/1 time dimension) if one already knew differential geometry in 3 spatial dimensions. It is often said Riemann almost had general relativity and may have thought of gravity as a curvature in space, however without the theory of space-time, he missed it and Einstein developed it.
However, although experience with lower dimensions is useful, there are times when proving a theorem in general is actually easier than considering specific cases. It is also easy to mislead oneself, when reading about the lower dimension stuctures, like scalars, covectors, and vectors, we fully understand tensors as extensions, or if we totally understand Hilbert spaces, we can consider Banach space as a "trivial" extension. All told, if I had a good answer for you about how to go about learning all this, I would be a mathematician and not a physicist. We all struggle. Best of Luck

 

[ChinleShale]

Mathematical spaces are often defined as the natural domains in which phenomena can be described. For instance phase spaces are the mathematical domains of Mechanical systems. Their dimensionally derives from the number of degrees of freedom of a mechanical system e.g. the phase space of a particle with a given mass in three dimensional Euclidean space is six dimensional since it includes three position variables and three momentum variables.

Another six dimensional example in Mechanics is the phase space of three particles that are constrained to move on a straight line.

A purely mathematical example of this type of thinking is in Riemann's definition of a Riemann surface. The idea was to define a domain in which a complex function becomes single valued rather than multi valued as in the complex square root.

 

[Vanadium 50]

It appears the OP has just walked away again. Pity.

 

[trees and plants]

Well, i think it is not necessary to visualise what is happening, you know the proved things,like theorems, proofs and definitions so you can prove with those. Correct me if you want if i made any mistake.

 

[Mark44]

Well, i think it is not necessary to visualise what is happening, you know the proved things,like theorems, proofs and definitions so you can prove with those.

I disagree with this statement. Linear algebra has many problems that involve surfaces in higher dimensions for which a sketch can be very helpful, but theorems and definitions not so much.
For example, describe in words the intersection determined by this pair of equations:
##2x + y - z + 3w = 5## and ##x - y - z + 5w = 2##

 

[trees and plants]

I disagree with this statement. Linear algebra has many problems that involve surfaces in higher dimensions for which a sketch can be very helpful, but theorems and definitions not so much.
For example, describe in words the intersection determined by this pair of equations:
##2x + y - z + 3w = 5## and ##x - y - z + 5w = 2##

I thought of it geometrically. Is it a plane?

 

[Mark44]

thought of it geometrically. Is it a plane?

No. This is why being able to visualize things is important. Each of the equations I wrote is a hyperplane, the higher dimensional analog of a plane in 3 dimensions. The two hyperplanes intersect in a line in 4-dimensional space.

 

[trees and plants]

Okay, so when is it helpful to visualise things in abstract spaces?

 

[Mark44]

Okay, so when is it helpful to visualise things in abstract spaces?

I just gave you an example. There are many more,

 

[trees and plants]

You mean like having analogues of curves or surfaces or points in higher dimensions?Is there a complete set of kinds of examples regarding this topic? Thank you.

 

[Mark44]

You mean like having analogues of curves or surfaces or points in higher dimensions?

No. A point is a point in whatever kind of space we're talking about. A straight line is a straight line. A so-called "hyperplane" is a flat surface whose dimension is one less that the dimension of the space it's embedded in. It's convenient to sketch the two hyperplanes in my previous example as if they were ordinary planes in space. From this visualization, and the realization that the hyperplanes aren't parallel, it's easy to see that the two hyperplanes must intersect in a line.

Is there a complete set of kinds of examples regarding this topic?

No, of course not.

 

[trees and plants]

But i think without sketching by following the right theorems, proofs and definitions you can solve the problem you mentioned. Anything in math can be solved i think this way.

 

[Mark44]

But i think without sketching by following the right theorems, proofs and definitions you can solve the problem you mentioned.

What theorems, proofs, or definitions would you use?
IMO, what you're proposing is something like building a house without having a design plan to look at. Something that I learned a long time ago, that I believe is true, is that one side of the human brain is more analytic, and the other side more visual. To attack a problem successfully, it helps to have both sides of your brain working on the problem.

 

[ChinleShale]

What theorems, proofs, or definitions would you use?
IMO, what you're proposing is something like building a house without having a design plan to look at. Something that I learned a long time ago, that I believe is true, is that one side of the human brain is more analytic, and the other side more visual. To attack a problem successfully, it helps to have both sides of your brain working on the problem.

I also think that with long study, one gets an intuitive sense of a subject, a feeling of familiarity and a sense of what can and cannot be right. When visualization is possible it makes this much easier but it is still there without it.

 

[trees and plants]

What theorems, proofs, or definitions would you use?
IMO, what you're proposing is something like building a house without having a design plan to look at. Something that I learned a long time ago, that I believe is true, is that one side of the human brain is more analytic, and the other side more visual. To attack a problem successfully, it helps to have both sides of your brain working on the problem.

I think now you gave me the answer i wanted. Visualisation is very important in every problem and theorem and can lead and give the directions and ways one needs to solve the problem. Is this correct? Thank you.

 

[trees and plants]

I think any abstraction in math or physics can be eventually visualised and help in producing or solving or understanding problems, solutions, theorems, definitions. Is this correct? Now i feel better visualising things in math, physics generally. Thank you.

 

[Mark44]

I think any abstraction in math or physics can be eventually visualised and help in producing or solving or understanding problems, solutions, theorems, definitions. Is this correct?

I don't think so. Visualization is important, but I don't believe that every abstractions can lend itself to being visualized. It's a bit difficult to visualize an element in a Hilbert space, for example.

But it's not all or nothing, as you seem to imply in your responses in this thread. Visualization is very helpful in some areas, such as the problem I posed earlier, but other times it's not helpful. In the area of linear algebra again, are the vectors <1, 2, 0, 4>, <1, 1 -1, 0> and <1, 3, 1, 8> linearly dependent or linearly independent? Visualizing these vectors would be very difficult, if not impossible, as they are all elements of ##\mathbb R^4##. The best approach here would be to use the definitions of linear linear independence.

 

[StoneTemplePython]

No. This is why being able to visualize things is important. Each of the equations I wrote is a hyperplane, the higher dimensional analog of a plane in 3 dimensions. The two hyperplanes intersect in a line in 4-dimensional space.

I think this is unfortunate... the algebraic approach I'd suggest is collect the (coefficients of the) 2 equations into a ##2\times 4## matrix ##A##. Looking at the left-most ##2\times 2## submatrix we see it has rank 2. Therefore ##A## is surjective and a solution exists. By rank-nullity ##\dim \ker A = 4-2 = 2##, hence the solution space has dimension 2, which isn't a line.

 

[Mark44]

I think this is unfortunate... the algebraic approach I'd suggest is collect the (coefficients of the) 2 equations into a ##2\times 4## matrix ##A##. Looking at the left-most ##2\times 2## submatrix we see it has rank 2. Therefore ##A## is surjective and a solution exists. By rank-nullity ##\dim \ker A = 4-2 = 2##, hence the solution space has dimension 2, which isn't a line.

Sorry, my mistake. I was thinking in terms of planes in space, not in terms of hyperplanes in ##\mathbb R^4##.

 

[FactChecker]

I think it is not correct to think of them as generalizations or extensions of 2d or 3d euclidean spaces. One of the important questions is what properties of known spaces are no longer true in abstract spaces where certain assumptions are dropped. Abstract spaces are carefully categorized by what properties are assumed and what properties can be proven, given only those assumptions. I wouldn't call that just a generalization or extension of 2d or 3d euclidean spaces.

 

[Office_Shredder]

Sorry, my mistake. I was thinking in terms of planes in space, not in terms of hyperplanes in ##\mathbb R^4##.

One might note this means your attempt at visualizing instead of writing down definitions caused you to make a mistake.

 

[Mark44]

One might note this means your attempt at visualizing instead of writing down definitions caused you to make a mistake.

Maybe so. OTOH, I can visualize and sketch the vectors <2, 1, -1, 3> and <1, -1, -1, 5> in the plane in ##\mathbb R^4## that they determine, placing them at the approximate angle they form (about 33.95°).

My only point is that being able to visualize the situation can be helpful at times.

 

[ChinleShale]

One might note this means your attempt at visualizing instead of writing down definitions caused you to make a mistake.

Visualization is a prized exercise among mathematicians. Often visualizations brings life to definitions and serve in many cases as a way to understand them. It is worth risking a mistake.

 

[FactChecker]

Visualization is a prized exercise among mathematicians. Often visualizations brings life to definitions and serve in many cases as a way to understand them. It is worth risking a mistake.

I agree. Although any visualization and intuition gained from it would need to be followed up by some disciplined proof.