Comment by davethedevguy 2 days ago
Likewise!
In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?
Is that even a valid question, or does it just betray my inability to perceive higher dimensions?
This is fascinating and I'm in awe of the people that do this work.
> etc.
That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)?
From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially in much higher dimensions, like 24 (!). They all "visualized" the equations but in ways that gave them much more insight than you or I might imagine just by looking at the text.
My sister is a mathematican and she used to say that if you want to understand a 24-dimensional space, you start from a generalized n-dimensional space and then set n=24.
This wasn't atypical of her. She would also say that if your house is on fire then you call the firefighters, but if it is not on fire then you set it on fire, thereby reducing the problem to something that you have already solved.
> Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”
He did. You can see / hear that line in this video from his old Coursera course.
https://youtu.be/TNhgCkYDc8M?list=PLLssT5z_DsK_gyrQ_biidwvPY...
Exactly how seriously he intended this to be taken is a matter of debate, but he definitely said it.
I have dyscalculia so I'm always studying how people who have "math minds" work, especially because I have an strong spacial visual thinking style, i thought i should be good at thinking about physical math. When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones" - I gave up my journey into math.
My two cents on this: I've done a lot of math, up to graduate courses in weird stuff like operator algebra. I've also read quite a bit of maths pedagogy.
I've come to understand that the key thing that determines success in math is ability to compress concepts.
When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concept in their minds. For them, taking the next step to multiplication is almost trivial.
When a college math student learns the triangle inequality, >99.99% understand it on a superficial level. But <0.01% compress it and play around with it in their minds, and can subsequently wield it like an elegant tool in surprising contexts. These are the people with "math minds".
Shortly after graduating as an engineer, I remember receiving much help regarding mathematical thinking from a book by Keith Devlin titled "The Language of Mathematics: Making the Invisible Visible".
What stuck with me (written from memory, so might differ somewhat from the text):
In the introductory chapter, he describes mathematics as the science of patterns. E.g. number theory deals with patterns of numbers, calculus with patterns of change, statistics with patterns of uncertainty, and geometry with patterns of shapes and spaces..
Mathematical thinking involves abstraction: you identify the salient structures & quantities and describe their relationships, discarding irrelevant details. This is kind of like how, when playing chess, you can play with physical pieces or with a board on a computer screen - the pieces themselves don't matter, it's what each piece represents and the rules of the game that matters.
Now, these relationships and quantities need to be represented somehow: this could be a diagram or formulas using some notation. There are usually different options here. Different notations can highlight or obscure structures and relationships by emphasizing certain properties and de-emphasizing others. With a good notation, certain proofs that would otherwise be cumbersome might be very short. (Note also that notations typically have rules associated with them that govern how expressions can be manipulated - these rules typically correspond in some way to the things being represented and their properties.)
Now, roughly speaking, mathematicians may study various abstract structures and relationships without caring about how these correspond to the real world. They develop frameworks, notations and tools useful in dealing with these kinds of patterns. Physicists care about which patterns describe the world we live in, using the above mathematical tools to express theories that can make predictions that correspond to things we observe in the real world. As an engineer, I take a real-world problem and identify the salient features and physical theories that apply. I then convert the problem into an abstract representation, apply the mathematical tools (informed by the relevant physical theories), and develop a solution. I then translate the mathematical solution back into real-world terms.
One example of the above in action is how Riemann geometry, the geometry of curved surfaces, was created by developing a geometry where parallel lines can cross. Later, this geometry became integral in expressing the ideas of relativity.
This maps back to the idea of "making the invisible visible": Using the language of mathematics we can describe the invisible forces of aerodynamics that cause a 400 ton aircraft suspended in the air. For the latter, we can "run the numbers" on computers to visualize airflow and the subsequent forces acting on the airframe. At various stages of design, the level of abstraction might be very course (napkin calculations, discarding a lot of detail) or very fine (taking into account many different effects).
Lastly, regarding your post of 'When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones"':
Sometimes when studying relationships between physical things you notice that there are recurring patterns in the relationships themselves. For example, the same equations crop up in certain mechanical systems than does in certain electrical ones. (In the past there were mechanical computers that have now been replaced with the familiar electronic ones). With these higher order patterns, you don't necessarily care about physical things in the real world anymore. You apply the abstraction recursively: what are the salient parts of the relationships and how do they relate. This is roughly how you can generalize things from 2 dimensions to 3 and eventually n. Like learning a language, you begin to "see" the patterns as you immerse yourself in them.
I have to wonder why your last paragraph made me feel quite uncomfortable, I can't even tell why except by the time I was done an uncomfortable feeling was inside. I appreciate your time to provide me with this much context, it seems between all the comments in this thread, I'm truly out of reasons to avoid learning math, and I suppose maybe that's why your last paragraph made me uncomfortable... I wonder what I'll see... :)
Thank you again.
> If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.
Yep — and this will generally be the case, as the equation looks like: x1^2 + x2^2 + … + xn^2 = r^2. If you fix one dimension, you have a hyperplane perpendicular to that axis — and a sphere of one dimension lower in that hyperplane.
For four dimensions, you can sort of visualize that as x^2 + y^2 + z^2 + t^2 = r^2, where xyz are your normal 3D and t is time. From t=-r to t=r, you have it start as a point then spheres of growing size until you hit t=0, then the spheres shrink back to a point.
Note that the solid set, all points within a certain distance of the center, is called a ball: https://en.wikipedia.org/wiki/Ball_(mathematics)
If the boundary is included, it's a closed ball, otherwise it's an open ball.
So the sphere is the "skin", the ball is the whole thing.
A bit different than common usage.
> In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?
For such discrete geometry problems, high-dimensional spaces often behave "weirdly" - your geometric intuition from R^3 will often barely help you.
You thus typically rather rely on ideas such as symmetry, or calculations whether "there is still space inbetween that you can fill", or sometimes stochastic/averaging arguments to show the existence of some configuration.
In my PhD I did study systems in higher dimensions (including fractal dimensions) and it is not a metaphor and no, I did not visualize them, it was more like defining a mathematical representation of the system geometry and working on top of it.
It sort of like the visualizable 3D "kissing spheres" is the story that makes it interesting, captivating and accessible and therefore competitive/social which makes it interesting even more, but basically at higher dims it's a bunch of equations as it is impossible to visualise on human wetware.
You could do kissing starfish but no one cares as there is no lore. A bit like 125m world record doesn't matter. 100m is the thing.
This is not a knock ... it is interesting how social / tradition based maths is.
Another example is Fermat's Last Theorem. It had legendary status.
However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
> just a visual metaphor
It's not really a metaphor.
An n-sphere is the set of all points that are the same distance away from the same centre, in (n+1)-dimensional space. That generalises perfectly well to any number of dimensions.
In 1 dimension you get 2 points (0-sphere), in 2 dimensions you get a circle (1-sphere), in 3 dimensions you get a sphere (2-sphere), etc.
EDIT: Also, if you slice a plane through a sphere, you get a circle. If you slice a line through a circle, you get 2 points. If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.