Mathematicians with aphantasia (inability to visualize things in one's mind) Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like?
I realize that this question probably falls outside of the scope of Mathoverflow, but it's so shocking that such a fundamental mental difference exists that I think the question is worth asking here anyways. Even if it gets closed, which I suspect it will, if even one mathematician with aphantasia sees this and has the startling revelation that they have aphantasia, I'll be 1000000% glad I posted the question.
*inability to visualize things in one's mind. see this note that went viral recently for a more detailed explanation: https://www.facebook.com/notes/blake-ross/aphantasia-how-it-feels-to-be-blind-in-your-mind/10156834777480504
 A: I'm the grad student mentioned above by Kevin Costo. I should give a little disclaimer, which is that I am self-diagnosed based on the VVIT questionnaire. I was reading an article that described aphantasia, and instantly recognized myself in the description. 
As Kevin mentioned, I do geometric topology. I find that it doesn't present much of a problem for me, pretty much because I can still keep track of relationships between, say, points - I just don't have a mental image. As an example, in the opening of Thurstons book on 3- manifolds, he talks about what one would see when sitting in a 3-torus and looking around. This, while "visual," made perfect sense to me. 
Another example is the game Skribble, in which one person decribes a drawing of a shape (say, a train) using simple geometric shapes (rectangles, circles, etc). The idea is to guess what the shape being described is. I can play this game in my head. How? I don't know, to be honest. I know I don't have a picture in my head, but I sort of know the schematics, how things relate to each other. 
I'd say the biggest problem I've run into is when something is sufficiently complicated and I need a sequence of steps to simplify it. If I need to, I just draw a picture. In some ways, I think it may be easier for me to be a geometer than, say, a category theorist, because I've already found ways to think about actual objects and their relationships. I don't have a way to visualize abstract objects. 
A: I do not think I have aphantasia, but I want to point out that "mental imagery" is very different from an image that you are directly looking at.  Consider a map of the world.  If you ask me to mentally visualize a map of the world, I believe I can do so, but if you then ask me, "Does China share a border with Kyrgyzstan?" I can't answer that question by just "looking at that part of the map" and getting the answer, which I certainly could do if I were looking at an actual (political) map of the world.  My geographical knowledge of that part of the world is somewhat fuzzy and so my mental image is fuzzy, but it's not fuzzy in the way that an actual blurred visual image is fuzzy.  Furthermore, even when I fancy that I can mentally "see" a border, such as the border between the U.S.A. and Canada, you can easily ask me difficult questions that I can't answer using my "mental image" that I could trivially answer by looking at an actual map.
I have also noticed many people talking about "visualizing" 3-dimensional space, but surely this is another example of a clear difference between mental imagery and actual visual images.  At best, we literally see only two spatial dimensions.  Three-dimensional space is something we conceptually construct, not something we see directly. I'd go further and say that even our mental "images" of two-dimensional projections of three-dimensional objects are quite different from actual two-dimensional images. Take for example the old puzzle where we take a square-based pyramid with equilateral triangles as faces and we glue a regular tetrahedron to one of the faces of the pyramid.  I think I can form a mental image of this object, but suppose you ask me whether it has 7 faces or 5 faces.  With the actual object, if you were unsure, you could just pick it up, look at it from an appropriate angle, and see that the faces are flush.  But with a mental image, if you are unsure, you probably cannot solve the problem just by "looking."  You probably have to think about it and work it out.
All this is to say that I have doubts about how much the ability to conjure up "mental imagery" helps mathematical thinking.  For those with the ability, I am sure it helps to some extent, but I suspect that it helps less than it might seem, and that much of what we attribute to our ability to form mental images is actually a far more complex process than (ahem) meets the eye.
A: Disclaimer: I don't have aphantasia, so this may be irrelevant to your question. 
I graduated university with a BS in math and am starting grad school this fall. I am not an aphantasiac, but when I learned about aphantasia I immediately related. My thought process almost never defaults to mental imagery, although I can make wispy, vague mental images with much effort which are gone almost the second they arrive. My thinking is almost completely semantic - dialogue, facts, data, concepts, etc. with no accompanying mental sensation. I actually sense that this is a strength for me. Visual representation, to me, is a huge distraction, especially in mathematics. As the geometric topologist pointed out, the way things (whether that be actual points in space or concepts) are related to each other, factually, is usually the meat of what you're looking for, and having a mental process that cuts right to that without any middle-man step in between "feels" more efficient. But maybe that's just because it's all I've ever known. I suspect that it's the reason I'm able to mentally handle large amounts of data at a time - because in my head it very rarely comes with any cumbersome (visual or otherwise) attachments. 
A more practical example of this is the way I remember driving directions. I don't remember what the road looked like or possess any spatial feel or awareness of what my surroundings were, but I remember a series of facts. Example: pass two stop signs, then turn right... Instead of simply approaching the road and visually remembering, ah, this is where I turn right. I guess in that sense, any time I drive somewhere is like the first time again. I'm giving myself instructions in my head as if I'm giving them to someone else who needs directions. 
Ironically, one of my strongest areas in my undergrad was graph theory. I know the factual characteristics of any certain graph, but it comes to my mind unencumbered by the visual representation, and I quite like that. It's more streamlined and "clean" to me. My weakest area was vector calculus; the professor constantly told us to just "see" the dimensions of whatever problem we were working on in the mind's eye. Since the discovery that the strength of one's ability to visualize may exist on a spectrum, I understand why that seemed nearly impossible and so frustrating to me. So it seems that I am more inclined toward anything dealing with relationship (or fact and pattern, like number theory) as opposed to problems dealing with measurement and real-world dimension. 
A: Though I surely am aphantasmic I can only guess how my mathematical experience differs from that of visualisers since I do not know what visualisers “see” when doing mathematics. I had no problems in studying mathematics but I had to do it in my own way. I learned most things from books or articles since I have difficulties in absorbing information through listening to lectures. To understand an abstract theory I must do many concrete examples or find analogies to already familiar theories in order to get a “feeling” of the theory. I think my bad memory or my inability to play chess must also partly be due to my inability to generate images in my head. But perhaps some visualisers also have similar problems.
A: I didn't realize that aphantasia wasn't the way almost everyone experienced the world until I was 67.
In retrospect, it explains my career choice (which wasn't really a choice, it was simply what I was).
Computer science, and in particular programming, is entirely verbal, with all thoughts in nothing but words (including artificial languages).
It perhaps also explains my aversion to GUI user interfaces.
I remember decades ago reading things like Einstein's ability to think in 4 dimensions, and not understanding what was so special about that; it's just like 3 dimensions, but one with an additional orthogonal direction.
It simply never occurred to me that people could actually close their eyes and "see" three dimensional images.
