The comments at that link are beautiful. If you're the sort who likes to skip comments, I encourage you to view them in this case.
I'd like to highlight a point of dissonance between the title ("How mathematicians think...") and the actual request ("anything that makes it easier to see, for example, the linking of spheres") -- emphasis mine.
I find the identification between visualization and intuition revealing. As a rule, mathematicians must be able to reason about things they cannot even begin to visualize -- non-measurable sets, infinities so large they need special names, infinite linear combinations of orthogonal functions.
That's not to devalue attempts at visualization. They're useful for developing intuition. But the original joke works because the mathematician is perfectly happy reasoning in hyperspace even though he cannot see it. The fourth dimension is not particularly harder to describe than the nth.
> infinite linear combinations of orthogonal functions
I'm not sure it's quite accurate to say that these can't even begin to be visualised. The theory of Fourier series means that someone picturing a 'reasonable' function on the circle has already begun the endeavour.
You're right. I thought, even while I was writing that, that it was a weak example. Though I had Banach spaces in mind, not fourier series, and I have always had trouble with them.
Still. I felt I needed three examples. "Pick something not from set theory," I said to myself. I couldn't come up with a strong example. Maybe that's telling.
Unmeasurable sets, though. When I try to visualize one, I see the letter E because that's what we called it in the constructive proof.
You're in good company. Supposedly, someone once asked J G Thompson (a very eminent group theorist, winner of the Fields Medal, the Wolf Prize and the Abel Prize) what mental picture he had in his head when thinking about a difficult group theory problem. "A big black letter G", he said.
At James Arthur's 60th birthday conference, he said something like the following (roughly paraphrased): "First …" and, unfortunately, I don't remember what first was (but I agree with Dove (http://news.ycombinator.com/item?id=1385842) that there should be 3 :-) ).
"Next, I studied the work of Langlands, and the group was called GL_2. Then, I studied the work of Harish-Chandra, and the group was called G."
Exotic differential structures (brief explanation: consider 4-dimensional Euclidean space, R^4; you can think of its topological structure as being determined by the way you calculate distances between points; this actually gives you more than just topological structure because you can do things like differentiate functions on the space. Well, there are different distance functions that are equivalent topologically but not differentially. The same is true for lots of other spaces, in dimensions much higher than 4.)
Huge finite groups like the "Monster".
The Galois group of Qbar over Q. (Brief explanation: Q is the rational numbers. Qbar is the set of all "algebraic numbers", i.e. roots of polynomials with integer coefficients. The Galois group consists of all "field automorphisms of Qbar", which means all functions from Qbar to itself that don't mess with arithmetic: f(x+y)=f(x)+f(y), f(xy)=f(x)f(y), etc. It's an important object in number theory.)
The very infinite-dimensional Hilbert space in which the wavefunction of the universe lives.
"For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers"
As usual Terry Tao's comment is wonderfully illuminating (at least for a non-mathematician like me). It's common knowledge that the ratio of the volume of a n-sphere to its circumscribed n-cube goes to zero as n->0 (http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_are...), but I never thought of this as another result of the law of large numbers.
That's actually what I considered to be the cause for the law of large numbers, when I tried to trace the normal distribution to axioms of three-dimensional geometry. I failed to get much beyond that, for lack of ability to reason about exponential functions, and abandoned the pursuit.
Correct but unhelpful answer: only through inordinate amount of practice with solving problems.
A possibly helpful answer to the question "Why is it probably a good idea for most people to give up on this?":
A dimension R^n is only a collection of points each of which is specified by an n-tuple over R. This isn't hard to hold in your head, but it tells you very little about how objects behave in any given space. By "think in", presumably the inquirer wants to be able to predict the behaviors of objects. The trouble is that there are many more different kinds of objects and behaviors than one is naturally inclined to assume, because one doesn't normally think of his ordinary 3D intuition as an insanely complex piece of specialized hardware that it is. You can certainly learn to do in software small parts of it, one by one. You would take those n-tuples, and do some particular bit of math on them to get you where you want to go. This will be difficult, like all math. Gather a big pile of these small pieces, train them until they're fast enough, and eventually you've got something like a crude emulator. I would hazard that unless you are interested in deep abstract (as in, non-visual) problems of the relevant math branches, you won't have the discipline to carry all this out.
I liked part two about how 2D creatures would see and maybe visualize 3D objects, that's the explanation that helped me most with how to visualize 4D objects.
There were two great quotes from the math book i used last semester.
The first was the most mathematicians equate geometry with understanding. In otherwords, if you can describe (draw) a picture that represents what you are talking about, then you really understand it.
The second was that not all math is something you can visualize. In that some of the beauty of math is that it can describe things beyond humans abilities of perception, and that it is precisely in these cases that math is its most useful.
The best part of the that joke (at least in my mind) is that generally engineers are pretty good, if not the best, at visualising multiple dimensions > 3.
Isn't it a little too convenient that the Universe seems to prefer 3d for physical processes, and that almost any kind of phenomena we need to understand can be framed in a 3d model? Or is it us the whole time, unable to see beyond our own limitations? When we see grandiose contrivances in a film plot, we naturally suspend disbelief in order to enjoy the story. I think those of us who have learned university math and who think we understand higher dimension are doing likewise—telling ourselves a story so we don't have to walk out of the theater.
I don't think we consciously "think" in any dimensions above 3. Like all life, we are neurologically hardwired by default to think in 3d, since that's what has worked for the evolution of all life on Earth. As for formal math, I am no longer even sure there is a connection between the logical, axiomatic and cultural edifice we've built up and call "mathematics", and our day-to-day "neural navigation software" that allows each of us to get from point A to B on the surface of a rotating oblate rock tethered to a nuclear fireball that is hurtling through the infinite, continuous space we call the Universe. The map is not the territory. The mathematical deductions of our neural experience are not the same thing as the experience itself.
From all of the suggestions on Mathoverflow, almost all of them are variants on projecting higher dimensional objects onto 3d and 2d objects, then comparing all the different projections in a clever way to get a "feel" for how the higher dimensional object changes. Even this is completely non-intuitive, as our brain's visual apparatus is optimized to take in a total picture and immediately spot the biggest changes. i.e. there's a predator running at us from over there! That is afterall what eyeballs and visual perception evolved for! If we can't even see the whole visual field at once, but just slices of projections of it, then our finely tuned visual hardware is thwarted and unable to detect and piece together the "shape" of objects. This is why is say nobody can "think" higher than 3d—even if you are doing it, your brain is still imperceptibly and behind the scenes translating your logical construct into a 3d "sensory" construct.
With that said the best way I've come across for visualizing 4d objects is from complex analysis, where you can use color gradients to represent a dimension. It doesn't work so well going beyond 4d, but it's a great set of training wheels.
http://www.mai.liu.se/~halun/complex/http://www.nucalc.com/ComplexFunctions.html
Oh ye gods, not that steaming pile of garbage AGAIN. Yes, if you want to make sure you never understand the true nature of higher dimensions and instead fill your brain with psuedoscientific garbage pulled straight from the cesspool of Star Trek, then by all means, watch that video a few dozen times. If you want to actually understand dimensions, pick up a topology textbook and start reading. (Or read the linked article.)
I can not anti-recommend that video enough. This video is anti-knowledge, convincing you that you know something when you actually understand it less well after viewing than before.
The actual truth is that there is nothing special whatsoever about "the fourth spatial dimension" (let alone the eighth or ninth), what's special is us. We're the ones that have trouble because we live in three spatial dimensions and our visual cortex is hardwired to work very well in 3. Four, five, six, ten, a hundred, or my personal favorite "n-dimensional" just means that you need four, five, six, ten, a hundred, or an arbitrary number of values to determine the location of something in a space. Nothing more, nothing less. Visualizing it a challenge, but mathematically it's nothing special.
I've seen that video a few times over the years, and it has always struck me as... how do I say it... not correct. Something in there just seems wrong. I do like some of the ideas in there, though.
Hopefully someone here can elaborate because it's been on my mind since I first saw it.
Well, I'm no physicist, but I agree - it's bogus. The problem, I think, is that it assumes that time is the fourth dimension. Instead it should be the last dimension. So if you're talking about 6-space, then time should be the 7th dimension. If you always call time the fourth dimension, then you get these wacky results where in 5-space and up you can see and exist in all time simultaneously.
This is the problem with calling time a dimension. It's useful, but it makes it natural to talk moving forwards and backwards in time. Error.
The root of the problem is that most people can't see the difference between 4D space and 3D spacetime.
3D spacetime behaves mathematically exactly the same as 4D space, it is a construct to make it easier to analyze and do physics calculations of 3D objects over time, by setting time as the 4th dimension, thereby giving every object a position in time as well.
But the visualization is completely different, and the things you can do are completely different. If I have a 3D left-hand glove in a 4D space, I can "turn" it in the 4th dimension and end up with a right-hand glove. But if I have the same glove in spacetime, it doesn't matter how much I wait around (i.e. move in the 4th dimension), my left-hand glove will never turn into a right-hand glove.
There were a couple of things that I noticed that I don't think are correct. I thought it was fine up through the first 3, more or less. One issue that kind of bugged me was that it seemed to imply that spaces cannot be curved. The video I think would suggest that the surface of a sphere is 3-dimensional when it's actually only 2. Calling infinity a point in the seventh dimension also bugged me.
Overall, it seems like the creator of it knew a little bit of math, and probably some physics, and extrapolated from there. More probable though is that their only knowledge of the subject came from philosophy, or epistemology.
Overall, it seems like the creator of it knew a little bit of math, and probably some physics, and extrapolated from there. More probable though is that their only knowledge of the subject came from philosophy, or epistemology.
Overall, it seems like the creator of it knew a little bit of math, and probably some physics, philosophy, or epistemology. More probable though is that their only knowledge of the subject came from late night sessions passing the bong, listening to Floyd.
I'd like to highlight a point of dissonance between the title ("How mathematicians think...") and the actual request ("anything that makes it easier to see, for example, the linking of spheres") -- emphasis mine.
I find the identification between visualization and intuition revealing. As a rule, mathematicians must be able to reason about things they cannot even begin to visualize -- non-measurable sets, infinities so large they need special names, infinite linear combinations of orthogonal functions.
That's not to devalue attempts at visualization. They're useful for developing intuition. But the original joke works because the mathematician is perfectly happy reasoning in hyperspace even though he cannot see it. The fourth dimension is not particularly harder to describe than the nth.