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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

YCor
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Trent
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    I'm in grad school, and a fellow grad student here has aphantasia. She said it makes some things tough but that in general it hasn't affected her too much. And she does geometric topology! I can put her in touch if you want. – Kevin Casto Apr 24 '16 at 22:31
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    I can think of situations when this is in fact an advantage (you are not distracted by deceptive images, forced to formalize things more rigorously from the beginning, etc.) – მამუკა ჯიბლაძე Apr 24 '16 at 23:28
  • Interesting; never heard of aphantasia before. Is that a medical term or coined by someone with that phenomenom? – Manfred Weis Apr 25 '16 at 03:03
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    @KevinCasto: wow, geometric topology is about the last discipline i would've guessed one could find someone with aphantasia in. yes, am interested in asking her some questions about math and aphantasia (will shoot you an email on your uchicago adress titled "Aphantasia and Math"). Though, if she's comfortable (& entirely understood if she's not comfortable speaking about such a personal matter / philosophical topic in a public forum) , i imagine there are alot of people other than me who would be curious to read some sort of reply on mathoverflow. – Trent Apr 25 '16 at 05:01
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    @ManfredWeis: medical term which received a name in an article by Adam Zeman and colleagues in the journal Cortex in 2015 http://www.sciencedirect.com/science/article/pii/S0010945215001781 – Trent Apr 25 '16 at 05:01
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    I don't have aphantasia, but I find it very very hard to visualize some mathematical objects. But not the ones you'd expect: I have no problem visualizing a model of set theory, an infinite dimensional vector space, or convergence in the cofinite topology over $\Bbb Z$. I am having a very hard time imagining, however, a plane moving through a ball, or what are gradients, and what the hell are Lagrange multipliers? Luckily, all those have comfortable formal definitions for me to use as a cushion when I have to. – Asaf Karagila Apr 25 '16 at 10:38
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    @KevinCasto: Meanwhile, Antoine (necklace) and Morin (sphere eversion) were blind, though neither from birth, – Adam Epstein Apr 25 '16 at 13:03
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    Please, do not include metadiscussion about the fate of the question in the question itself. If you want to discuss the closure, the appropriate place is http://meta.mathoverflow.net (specifically, http://meta.mathoverflow.net/questions/223/requests-for-reopen-and-undelete-votes-for-on-hold-closed-and-deleted-question). FWIW, there are already two votes to reopen the question. – Emil Jeřábek Apr 25 '16 at 19:44
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    I don't think I have aphantasia, but I don't generally produce very detailed and vivid mental images. I think this can be an advantage when dealing with mathematical objects: their geometrical relationships, when they have them, tend to be rather simple, and you don't want to get distracted by unnecessary details. Also, most of mathematics is dealing with objects, e.g. those of more than three dimensions or with nontrivial topology, which nobody can really visualize. It helps to be used to not panicking when something resists visualization. – Robert Israel Apr 25 '16 at 21:58
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    Thank you for posting this question. I always thought I am the only person who cannot visualize. It was a great relief to read the article by Blake Ross and to learn that 2 percent of the population have this anomaly too. – Johann Cigler Apr 26 '16 at 15:15
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    A link in the open web would be more useful. – YCor Oct 04 '21 at 07:29
  • I recently found out that I have aphantasia, I can't picture realistic images. But... I can actually picture schematically very well, for example I can make knots in my mind and imagine a curve on a torus. In mathematics I often like to make use of geometrical intuition like imagine a t-distribution with a cone intersecting a sphere and ordinary least squares as a projection on a span vectors. – Sextus Empiricus Feb 28 '24 at 08:02

5 Answers5

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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.

MurphyKate Montee
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    Here is Bill Thurston on the example of the blind mathematician Morin and his work on sphere eversion (More Mathematical People, pp. 337-338): "It's something most people have a great deal of trouble visualizing. In fact, I think that vision is somehow distracting to the spatial sense, because we have a spatial sense that is more than just vision. People associate it with vision, but it's not the same. If I close my eyes and imagine what this room is like, I will have a sense in my mind that there's a table here and something here and there. (cont.) – Todd Trimble Apr 26 '16 at 01:53
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    It will be a sense of the room that doesn't have much to do with perspective. It's hard -- that is, it takes a lot of training -- to go from a spatial image to a picture on paper. So these things are not necessarily stored in our minds in a visual sort of way. We translate what we see into a sense of space. If you think of it, you realize that if you imagine a table with four chairs around it, it doesn't matter whether you can see the seats of the chairs. You just know that they are there. It's kinesthetic as well as visual." MurphyKate, does this at all correlate with your experience? – Todd Trimble Apr 26 '16 at 01:54
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    Todd - yes, I think that exactly what I'm using. – MurphyKate Montee Apr 26 '16 at 02:03
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    This is closely related to Todd's question, but I don't know if it is asking about exactly the same sense: When I was learning how to do a 360 front flip, I learned via manipulating my body in certain ways, then mentally simulating how I predicted changes in motion would change how my body moved, executing the changes to the best of my ability, then repeating the cycle until I learned the move. I didn't use visual perception or visual perception in my mind's eye to figure out how my body was moving, I used a kinaesthetic sense of bodily/spatial awareness. I definitely don't have aphantasia, – Trent Apr 26 '16 at 05:55
  • but - even now that I've done a 360 front flip who knows how many times - I'm not capable of visually replaying in my mind's eye what I see when I do a 360 front flip. It is however easy for me to replay in my minds eye what it kinaestheticly feels like to do a 360 front flip.

    Does this type of kinaesthatic spatial perception at all correlate with the perception you are using here: "I know I don't have a picture in my head, but I sort of know the schematics, how things relate to each other."? Can you kinaesthatically (not visually) replay in your mind's eye eg what it feels like to walk?

     – Trent Apr 26 '16 at 05:56
  • @Trent: Wow, I really don't know. I've never thought about walking, for example, in that way. I guess I would say that I know how to ride a bike, and what it feels like, and I know what riding a bike looks like (I could draw a picture of me on a bike), but no, I don't really know what I see when I'm riding a bike. But if you showed me some videos, I could identify which one was a person riding a bike. I'm not sure I'm answering your question at all. – MurphyKate Montee Apr 26 '16 at 17:01
  • Maybe this is a better way of phrasing it. Say you're riding a bike with your eyes closed. Can you imagine what that would feel like, and (not via vision, via propriocreption/kinaesthesia) imagine how even when riding a bike with your eyes closed you know how your legs are moving and where they are? – Trent Apr 26 '16 at 19:09
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    @trent: yes. I have pretty solid spatial awareness/proprioception. – MurphyKate Montee Apr 26 '16 at 23:11
  • Are you also unable to remember what colors look like when you cannot see them? – Michael Hardy Feb 28 '22 at 21:36
  • Hmm. If I close my eyes and think about “green” I don’t see green. It would be hard to describe it, but I could tell you that it’s the color of grass and leaves. I recognize it when I open my eyes. It’s not really a memory problem for me; it’s that I can’t picture it in my head when my eyes are closed. – MurphyKate Montee Mar 02 '22 at 00:11
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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.

Angela Powers
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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.

Ray Butterworth
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    Like the old joke. Freshman: "How do you visualize four-dimensional space?" Professor: "Oh, it's easy, you just visualize $n$-dimensional space and then set $n=4$." – Nate Eldredge Oct 04 '21 at 02:37
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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.

Johann Cigler
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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.

Timothy Chow
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  • Is it clear that we literally only see two spatial dimensions? I would think that, literally, we see three …. – LSpice Oct 04 '21 at 21:32
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    @LSpice Depending on what you mean by "literally," we may not literally see any dimensions. Visual perception is a complicated process. But certainly, a bottleneck occurs at the retina, which is essentially two-dimensional. – Timothy Chow Oct 04 '21 at 21:38
  • I would say that a lot of people can mentally visualise in three dimensional objects by visualising "moving" relative to the object. – JP McCarthy Oct 05 '21 at 05:05
  • @JPMcCarthy Right. But even when you have your eyes open and are looking directly at the world around you, 3D space is something synthesized in the mind by some complex mental process from the 2D images that you see. (And even the 2D images are complex syntheses from the signals entering your visual system.) Your "mental image" of 3D space is even further removed from what you actually see with your eyes; it is highly abstract. So I'm not surprised that some people don't use these abstractions when reasoning mathematically about geometry. (continued) – Timothy Chow Oct 05 '21 at 11:57
  • (continued) To solve the puzzle I mentioned, even though it would seem to be the sort of thing that could be solved directly via "mental imagery," I am not able to do so, and I expect that the same is true of many others. I have to resort to other forms of abstract reasoning. So I am not sure that even those of us who do not have aphantasia are relying on "mental imagery" to do mathematics as much as we think we are. – Timothy Chow Oct 05 '21 at 12:00
  • "I have doubts about how much the ability to conjure up 'mental imagery' helps mathematical thinking." Certainly I use is frequently. However, concerning your remarks about the boundary between China and Kyrgizstan, how about the following. On the fourth page of Chapter 2 of a novel that I have before me, it says: "He continued downhill for a couple of hours. The road was hardly more than a cart track but it was easier than the maintenance road. Somewhere below, when the hills gave way to the valley where the foreigners lived," and so on. Now suppose three days after reading that,$,\ldots,$ – Michael Hardy Mar 01 '22 at 01:17
  • $\ldots,$you are asked to recall what it said. Do you remember it verbatim? Probably not. Do you remember what it said the protagonist was doing at that point? Probably you do. Similarly with the map. The visual image doesn't necessarily have all the details, but you still remember that the boundary between Manitoba and Ontario has that angle where it turns from going north to going northeast. And so on. I certainly don't remember the convoluted details of the boundary between Vermont and New York, but my mental picture emphatically shows the east-to-west breadth of Vermont getting $\ldots,$ – Michael Hardy Mar 01 '22 at 01:21
  • $\ldots,$bigger as you go further north. And lots of things like that. @TimothyChow $\qquad$ – Michael Hardy Mar 01 '22 at 01:22
  • Many things that happened when I was two or three or four years old I can remember only because the remembered visual images are vivid. – Michael Hardy Mar 01 '22 at 01:24
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    @MichaelHardy I don't think anything you say here contradicts anything I said. My point is only that there is a much bigger difference between an actual visual image and "mental imagery" than most people realize, and that what you think of as "remembering a mental image" involves a lot more complex processing than the naive analogy of loading a bitmap from disk into cache. – Timothy Chow Mar 01 '22 at 01:43