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.
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.”
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.
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".
I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))
I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.
The foundations for these concepts were laid by Piaget and Brissiaud, but most of their work is in french. In English, "Young children reinvent arithmetic" by Kamii is an excellent and practically oriented book based on Piaget's theories, that you may find useful. Although it is 250 pages.
This approach has become mainstream in maths teaching today, but unfortunately often misunderstood by teachers. The point of using different strategies to arrive at the same answer in arithmetics is NOT that children should memorize different strategies, but that they should be given as many tools as possible to increase the chance that they are able to play around with and compress the concept being learned.
The clearest expression of the concept of compression is maybe in this paper, I don't know if it helps or if it's too academic.
I should be able to chat with an llm about this paper, but my gut says you've given me the glimmer of where I need to go. This is something I've been deeply deeply frustrated about for 30 years now, I had really given up hope of ever being able to process mathematics (whatever they are) properly, it's a real task to figure out how to get someone to see how your brain work and then have them understand how to provide you with some framework to grasp what they know.
Once again I wanted to thank you for slowing down and taking the time to leave this thoughtful comment, if everyone took 5 minutes to try to understand what the other person is saying to see if they can help, the world would be a considerably better place. Thank you.
Just a tangent, but there's a nice trick for 8 x 12.
In algebra, you learn that (a - b)(a + b) = a^2 - b^2. It's not too hard to spot this when it's all variables with a little practice but it's easy to overlook that you can apply this to arithmetic too anywhere that you can rewrite a problem as (a-b)(a+b). This happens when the difference between the two numbers you're trying to multiply is even.
For a, take the halfway point between the two numbers, and for b, take half the difference between the numbers. So a = (8 + 12) / 2 = 10. b = (12 - 8) / 2 = 2.
Here, 8 = 10 - 2 and 12 = 10 + 2. So you can do something like (10 - 2)(10 + 2) = 10^2 - 2^2 = 100 - 4 = 96.
It's kind of a tossup if it's more useful on these smaller problems but it can be pretty fun to apply it to something like 17 x 23 which looks daunting on its own but 17 x 23 = (20-3)(20+3) = 20^2 - 3^2 = 400 - 9 = 391
Calculating 8x12 in my head relies on a trick / technique - they call it "chunking", I believe, in the Common Core maths curriculum that US parents get so angry about - that (I'm also in my 40s) was never demonstrated in schools when we were kids. (They tried to make me memorize the 12x table, which I couldn't, so I calculated it my way instead; took a little longer, but not so much that anyone caught on that I wasn't doing what the teacher said.) I'd like to think I was smart enough to work it out for myself, but I suspect my dad showed it to me.
I'll show it to you, but first: are you able to add 80 + 16 in your head? (There's another trick to learn for that.)
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.
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.