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The senses of vision and hearing are commonly recognized as being important for the study of mathematics, with fields like geometry and topology relying heavily on vision, while algebra and number theory can be communicated through hearing alone. There are many examples of mathematicians who have made significant contributions to the field despite being blind or visually impaired. Some examples include Leonhard Euler, who continued to work on mathematics after losing his sight. So while vision is certainly a helpful sense for studying mathematics, it is not necessarily a requirement for success in the field.

However, it is worth considering whether other senses might also play a role in understanding mathematics. For example, in an episode of the TV show The Big Bang Theory, a character (Sheldon Cooper) said "prime numbers appearing red and twin primes smelling like gasoline." While this was likely just a joke, it raises the question of whether other senses, such as smell, could potentially be used to aid in mathematical understanding.

There may be other examples of mathematicians using senses other than vision and hearing to help them understand mathematical concepts. For instance, in a response to a question on MathOverflow Thinking and Explaining, Prof. Terence Tao mentioned rolling around on the floor with his eyes closed to help him understand the effect of a gauge transformation that involved different frequencies interacting with each other.

This question was prompted by the experience of preparing a math lecture for a general audience. While it is possible to use pictures, symbols, equations, and even music to convey mathematical concepts, all of them only rely on vision and hearing. I'm curious whether other senses are useful in doing math.

Edit I'm sorry if my question might have caused any confusion or offense. Please know that I didn't mean any harm towards individuals who may have visual or hearing impairments. I was simply curious about how mathematicians engage other senses so I could gain new perspectives to learn and communicate math ideas to others. I'm very grateful for all the excellent answers and intriguing comments and it's extremely hard to decide which one I should accept.

Shijie Gu
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    To me, good mathematics smells like chalk dust and tastes like coffee. – Carl-Fredrik Nyberg Brodda Mar 28 '23 at 04:56
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    Mathematics often feels to me the way my religious friends describe feeling during intimate faith-based moments, so if you consider faith a sense I'd say the answer is yes. – Alec Rhea Mar 28 '23 at 05:38
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    It has to be said that sometimes a purported proof just doesn't smell right, without knowing what's wrong with it. – Dave Benson Mar 28 '23 at 08:40
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    Pontryagin was a famous blind topologist. – bof Mar 28 '23 at 09:28
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    algebra and number theory can be communicated through hearing alone - Any information can be conveyed through any sense once it's encoded. My point is, many things in algebra and number theory use visual aids in understanding. While there have been blind mathematicians, I'm not convince they think about algebra and number theory in the same way. – Kimball Mar 28 '23 at 12:29
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    I would even say that hearing is not a good sense to encode mathematics at all. A long formula can usually be read at a relatively short glance even if it involves double fractions and multiple indices, but dictating something like this in an unambiguous way would be a serious exercise in patience. – mlk Mar 28 '23 at 13:07
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    What is a sense? This question has been the subject of much discussion. There's a nice article by the Sensory Trust on the topic: How many senses do we have? See also the BBC article, How many senses do you have? – Timothy Chow Mar 28 '23 at 16:47
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    This other question about aphantasia might be relevant: https://mathoverflow.net/questions/237243/mathematicians-with-aphantasia-inability-to-visualize-things-in-ones-mind – kjetil b halvorsen Mar 28 '23 at 23:52
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    @Carl-FredrikNybergBrodda Nice! It sounds like "gut feeling" or hunch or maybe the 6th sense? Inspired by your comment, there are some mathematicians using "smells" and "tastes" to determine if the math is correct. For instance, Eugenia Cheng applied math to the baking process. She explained how geometry of different baking shape may affect the final product. But I think one still need to smell and taste the bakery to judge if the mathematical analysis is truely right. However, examples like this are very restrictive. – Shijie Gu Mar 29 '23 at 01:28
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    @bof Yes. There are many mathematicians who are blind and made great contributions in math. However, I guess most of them are not born blind. So, the concepts like shape must have already established in their minds. Are there examples of geometors or topologists who are born blind? I would appreciate it if some born blind geometors/topologists could share their experience in studying math. – Shijie Gu Mar 29 '23 at 02:22
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    @ShijieGu Outside of mathematics, I have heard from bond traders that their pattern recognition can happen outside the brain. Say, they feel something in their pinky finger, they do not know why, and they know from past experience that that sensation usually predicts trouble in the bond markets. If this is off-topic, I will delete this comment – Rodrigo de Azevedo Mar 29 '23 at 09:56
  • @RodrigodeAzevedo Interesting. I guess that belongs to propriocetion? – Shijie Gu Mar 29 '23 at 11:29
  • @ShijieGu I am not qualified to comment on that – Rodrigo de Azevedo Mar 29 '23 at 11:30
  • @ShijieGu Your welcome. I guess you might have mixed him up with Ludwig Boltzmann who is known for having bad eyesight. – StefanH Mar 29 '23 at 13:36
  • @StefanH Yes, I think so. Thank you for pointing that out. – Shijie Gu Mar 30 '23 at 04:27
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    Bernard Morin, who was among the first to show that "a sphere may be inverted", had lost his sight as a small boy. Bill Thurston commented that the visual sense could even sometimes get in the way, and that apprehension of such things might be partly kinesthetic. (See his interview in More Mathematical People.) Von Neumann said his dominant sense in mathematics was hearing (and I don't think developing a fine-tuned sense of logical structure is primarily a visual affair, even if we use symbols as an aid in logic). – Todd Trimble Apr 01 '23 at 11:51
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    It's metaphorical, but sometimes smell is invoked for mathematical intuitions that are difficult to describe. "That smells like a theorem we proved about five years ago." C.F.J. Ross said Jung hypothesized that intuition may have been the remnant of the faculty of smell that was well-developed in many mammals, but underdeveloped in the human cortex compared to the other cortical lobes (Coll. Wks 18, p. 326). In an interview toward the end of his life, Jung stated ‘The intuitive is a type that doesn’t see the stumbling block before his feet, but smells a rat for ten miles’ (Evans 1976, p. 103). – Todd Trimble Apr 01 '23 at 12:07
  • @ToddTrimble, you really need to read up on the neurosciences concerning the sense of smell (and taste). If you make a sufficient effort, I think you will be surprised. – Tom Copeland Apr 26 '23 at 01:31

8 Answers8

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Anecdotally, it seems that many mathematicians use gestures to aid in understanding, not only when explaining but also when thinking privately. I believe that while this is connected to vision, it is something more general about spatial and temporal metaphor that can be associated with touch and the perception of one’s body positions. Indeed, Einstein reported using a “muscular type” of intuition.

Monroe Eskew
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    "proprioception". I think the most basic example is counting on one's fingers, but other things like the slope of a line, too. For a lecture, it might be fun to ask the audience to close their eyes and then hold up 3 fingers, 1 finger, 4 fingers...then ask if they had to think about how many 3 was in order to hold up 3 fingers, or if they have muscle memory of this gesture equals three. – user3067860 Mar 28 '23 at 15:40
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    Thank you for the nice answer. I accepted it mainly because I think it is probably the easiest for individuals to acquire in contrast to synesthesia which might only belong to the chosen ones. – Shijie Gu Mar 29 '23 at 16:42
  • Poincaré insisted that our understanding of physical laws, of space, and hence of geometry, is ultimately linked to or based on bodily sensations (we "feel" force, acceleration). Actually, what another answer calls proprioception. I'll try to dig up exact sources when I have time. – Torsten Schoeneberg Mar 30 '23 at 16:21
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Synesthesia could be an extreme case of the processing of mathematics using alternative pathways in the brain. This site on mathematical synesthesia gives some examples:

Number-form synesthesia consists of perceiving numbers as having specific and consistent spatial positions. The people who have this type often benefit from it as they can mentally manipulate the numbers they can “see” or even “touch”, finding them very easy to remember.

The association of numbers with sensory perceptions is an effective mnemonic tool. Daniel Tammet could recite $\pi$ to 22,000 decimal places by using the visual forms of numbers to create highly memorable mental landscapes:

"The number 1, for example, is a brilliant and bright white, like someone shining a flashlight into my eyes. Five is a clap of thunder or the sound of waves crashing against rocks. Thirty-seven is lumpy like porridge, while 89 reminds me of falling snow."

An fMRI study of these sensory pathways is at
Seeing mathematics: Perceptual experience and brain activity in acquired synesthesia

Carlo Beenakker
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    I was going to say that the Big Bang Theory quote is likely evidence that Sheldon is synesthetic rather than just a joke. I've known other people to whom different mathematical objects had different colors. I believe abelian groups were green to them, etc. – Greg Friedman Mar 29 '23 at 01:49
  • Thank you for the nice answer. I was wondering if people with synesthesia can share their feelings with those who don't have it. For instance, is it possible to feel number 1 is a brilliant and bright white? Can it be transmitted or conveyed to others? – Shijie Gu Mar 29 '23 at 11:48
  • @ShijieGu, re, surely it can be shared with someone else in the same way that any other feeling can—namely, by my (or a synæsthete's) attempting to evoke the feeling in you, whether by exposing you to the same stimuli that produced it for me or by any other stimuli; and with the same catch, that, no matter what the feeling and how common it is, there's no way for me to be sure I have evoked the same feeling in you that I myself experienced—or even what "the same" means here. – LSpice Mar 29 '23 at 20:30
  • @GregFriedman: It’s clearly meant as a depiction of synaesthesia, but it’s clearly also meant a joke. – Peter LeFanu Lumsdaine Mar 31 '23 at 08:47
  • I have the something like the color-grapheme type, which is similar to Daniel Tammet's, but not remotely as potent as his. However it does make it easier (or more fun) to remember strings of digits and to remember formulas. I think this may be a reason I was attracted to number theory, since the numbers themselves and interrelationships take on a more intrinsic meaningfulness. – PrimeRibeyeDeal Apr 01 '23 at 11:17
  • @ShijieGu in context of this answer (and to reply your question about transmitting the feelings to others) I think it is worth mentioning that there is a vast field of mnemonic tricks; in particular associating numbers with stuff is a well known way to remember large combinations. See e.g. PAO (Person-Action-Object) system. Mnemonic usage is very personalized, and may utilize different senses. One memory athlete was even using emotions as associations. To expand on how mnemonic systems work I can recommend a book "Moonwalking with Einstein". I remember some math concepts using mnemonics. – Hi-Angel Apr 01 '23 at 13:18
  • @ShijieGu, oh, just to clarify: while synesthesia is basically an automatic subconscious way of applying mnemonics, but you don't have to be a synesthete to use mnemonics to remember math concepts. I do that and am not a synesthete. I am sure that even different types of calculations may be aided by mnemonics, although I didn't dig into it because I am not a professional mathematician, so just don't have a practical need. – Hi-Angel Apr 01 '23 at 13:33
  • @Hi-Angel Aha, yes! Thank you for the comments. I think the most widely used mnemonics in math is name mnemonics, i.e., the first letter of each word is combined into a new word. Now, I believe synesthesia could be "contagious". – Shijie Gu Apr 02 '23 at 04:43
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Mathematicians use proprioception all the time.

Have you ever seen a mathematician close their eyes and wave their hands in the air while explaining some mathematics?

Lee Mosher
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I guess you can consider also the sense of touch. Consider the story of Bernard Morin, who was a blind mathematician who works in the eversion of the sphere. I suppose everybody can achieve more mathematical intuition by touching surfaces, not only watching or drawing them. Maybe 3d printers can be a good tool to develope this kind of intuition.

A. J. Pan-Collantes
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Mathematicians frequently rely on the perception of the passage of time as part of reasoning. While we could imagine all of mathematics as taking place in a "static" universe, where a real number is always interpreted spatially, we often imagine one of our real number variables as representing time. This completely changes our mental picture associated with a function.

For instance we might think of a function $H: [0,1] \times [0,1] \to \mathbb{R}^2$ as a family of parameterized curves, where the first input is the "parameter" and the second input is giving the "time". We would then visualize this by fixing the parameter and playing a movie of the the path $H(\alpha, t)$ as $t$ increases from $0$ to $1$ over time.

This is not needed. We could instead think of it as a "static" deformation of the square. However, the intuition afforded by our sense of change through time is invaluable.

In a different direction, we also rely on passage of time to understand periodic phenomena. When I think about the Chinese Remainder Theorem, this is connected intimately with situations where multiple distinct rhythms are coming together: tapping on every third beat with my right hand while tapping every other beat with my left for instance.

LSpice
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Steven Gubkin
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I cannot tell what opinion the neuroscience holds here, i.e., if you do a brain scan of a person trying to read a mathematical text or solve a problem, and then compare it with the scan of a person trying to determine the shape of an object by touch, smelling a flower, or tasting some food or drink, which would be the common areas to light up. The experiment like that is not too hard to carry out on a few subjects from laymen to professional mathematicians and the results may be quite interesting, so I'm inclined to believe that somebody has carried it out already, but, alas, I have no idea who and when. What I describe below is just my personal experience.

The answer is "all and none". Let's talk about vision, even. I mainly think in pictures (as opposed to linguistic processing). But that doesn't mean that I draw a lot of relevant things when trying to figure something out or reading a paper. Occasionally I do it, but my usual way of thinking of mathematics is with eyes closed and I just (try to) imagine the corresponding images, which aren't even images in the painter's sense of the word, because, if asked to draw what I "see", I wouldn't be able to do it. I cannot tell what exactly they are, but certainly they are closer to the idea of the visual perception than to anything else despite the fact that the visual perception itself is totally shut down and it is rather about projecting, than about perceiving. I'm inclined to believe that this kind of "perception" would be impossible without having the regular vision machinery in the brain, but again, one needs to run a series of experiments to confirm or to reject this belief.

Also even vision and hearing as such are mainly used for communication rather than for doing mathematics per se. Their only role in mathematical processing is that of a cable connecting the "external hard drive" (usually a piece of paper or a blackboard) with the brain CPU, which doesn't seem to have much memory of its own, just 5-10 registers (my brain definitely works using at most 5: when looking at a non-structured algebraic expression with 6 variables, I cannot remember and reproduce it without compression (a.k.a. finding patterns). This role is very important but, evidently, it can be delegated somewhere else and one can learn to use the internal hard drive too, though I have never managed to do it for random information, just for storage, cataloging, and access to something elusive I call "general ideas", which, by the way, are (next to) impossible to store and transmit using vision or hearing in the form they are really stored (IMHO, those are big "association tables" though, perhaps, "labeled and colored hypergraphs" would be a better choice of words).

I do not roll on the floor or punch walls to do mathematics, but I can easily do either one to vent off my frustration about my ultimate stupidity and total inability to figure the most trivial things out. Those actions are purely emotional.

Finally, I'm as much of a Platonist as far as mathematics is concerned as one can be. So, I believe that all those mathematical objects and constructs are more fundamental or "real" than the so called "real world". When working on a problem, you are just navigating that parallel universe. The words like "I have made half-way to the solution" and "I cannot find the path from A to B" are not metaphors for me. But if you ask me to tell what exactly are the means of landscape recognition, navigation, and propulsion in that parallel Universe, the only thing I'll be able to do (which I have done a few times in front of my students) will be to say "OK, throw a problem at me and I'll show you my thinking". Gowers tried to make a video like that (about computing some determinant), but after first 5 minutes that was a disaster: the first two steps essentially lead to a full solution and then something strange was happening for the next half an hour or so. I still cannot understand what went wrong with him at that moment. My experiments were occasionally more interesting but I doubt they conveyed much: we've all seen a snake effortlessly slithering on the grass. We can easily find the descriptions of the corresponding mechanics and read them. Now lie on the floor and repeat the feat. The same with the senses used. Suppose Terry Tao tells you what he uses exactly and you see it on a brain scan. Will you be ready to prove even some of his already published theorems, not mentioning the ones he hasn't done yet? I won't.

So my final answer is "Who cares?". Use whatever you can and want, play it with no hands barred, feel free to experiment in both work and communication anywhere within the boundaries of common sense and slightly beyond (I once showed a cartoon to my class where the left half was the map with a moving observer and his field of vision and the right half was what he saw, when we studied "linear algebra with application" and I wanted to talk a bit about perspective and computer animation) and just see what works and what doesn't when you do it. As to "what mathematics is, really?" (it looks like we'll slide at least a bit in that direction in this or any other similar soft discussion), as far as I'm concerned, believe in anything you want to believe from Platonism to the Social Construct theory, but remember that the only way you can convert other intelligent people into your faith is by showing that you are a better craftsman than they, and with some of the most interesting characters holding the most outlandish beliefs it may turn out somewhat difficult :-)

fedja
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    Thanks for the nice answer. Comparing brain activity to the way computers work is intriguing. However, I disagree a little with the "Who cares?" part. For instance, by studying how bats "visualize", scientists have made progress in helping blind people to see. With the development of computers, we are able to generate concrete geometric pictures that more people can appreciate and understand. In the old times, those pictures probably only existed in the minds of a few mathematicians. So, I think develop senses other than the common ones might be another direction to consider. – Shijie Gu Mar 30 '23 at 10:28
  • Why the abysmal neglect of neuroscience--human and comparative--even at the level of what Scientific American used to be? Incredible. – Tom Copeland Apr 26 '23 at 01:27
  • @TomCopeland "Why the abysmal neglect of neuroscience" If you refer to the opening phrase of my answer, it is not "neglect", just sheer ignorance of mine: I merely do not know of a study that would directly relate to the question and would hardly be able to distinguish between a decent work and BS even if I had. If you can fill the gap, by all means do it :-) – fedja Apr 26 '23 at 02:01
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... and of course, lots of common sense, if possible.

AccidentalFourierTransform
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I was typing this as a comment to an answer which got deleted as I was typing, so I'll post here. (Rewritten to fit the new context)

Another user mentioned that proprioceptiom is the most basic form of counting (e.g. on the fingers), but there's actually another: subitizing, which is the ability to recognize quantities of 2, 3, 4 (sometimes more) objects without counting them. It's present in humans and a number of animals, and it's thought to be the brain's natural "proto-counting" ability.

You can try this out by placing (or having someone place) 3 or 4 objects (of the same type) in any arrangement on the table. You'll automatically recognize the quantity. Some people can subitize 5, some 6, but the it drops off. (However for written numbers like 100000 & 100000000 you probably subitize the digits in groups.)

On a similar note, the brain uses color to group disparate objects into a unit. This is why sports teams wear a single color, and so we feel we can follow the action. Otherwise it would be a mess of individual players.

PrimeRibeyeDeal
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  • My understanding is that chimps are far better at this than humans. – JP McCarthy Apr 01 '23 at 13:17
  • @JPMcCarthy I have seen the videos where chimps are handily defeating humans in short term memory tasks involving arrangements of numerical digits, but I have not seen one involving subitizing. However I would not be surprised. – PrimeRibeyeDeal Apr 02 '23 at 23:30
  • I have had a Google and it appears I have misremembered. I am fairly sure you are talking about the videos I have seen, and no they are not subitising. I find mentions of chimps subitising but nothing as concrete as I thought I remembered. – JP McCarthy Apr 03 '23 at 04:06