Does Doing Math Feel Like Something?

This post is a bit of a fluff piece. I mainly wanted to document some things I’ve been thinking about for a while, concerning the notion that mathematics seems to have distinctive phenomenological character. My internal experience of doing mathematics and confronting mathematical concepts feels like something more than merely a composition of sensory phenomena. I should take a moment to remark here that I am not advocating for platonism. That is, I am not claiming that mathematical objects exist independently of the mind, only that my experience of doing mathematics has a distinctive character that is not reducible to sensory phenomena. (In other words, I claim that the answer to the question in the title is yes.) I reckon there is a rich literature about this already, but I am not familiar with what’s all been written, so I’ll have to defer any deep analysis until later. I may edit this post, or create a new post as I start to read about these topics. For now I wish only to document what I have personally noticed.

Dreams

What first brought my attention to the notion that there might be something it’s like to have a mathematical idea was my own dreams. I have dreams¹ on occasion where the only identifiable phenomenal content is of mathematical objects.² This usually happens when I am quite sick and enduring feverish delirium. But the interesting part is that there is usually little to no visual or auditory content to these dreams. It’s more like I am being haunted by spectres of mathematical substance.³

Many times the experience is just “of” the concepts themselves. Most recently, I was sick with a virus that made my throat so sore I couldn’t fall fully asleep.⁵ I spent an entire night in a kind of half-asleep state where, as best as I can describe, I was hallucinating 40 three-dimensional Brownian motions conditioned on their paths belonging to the unit cube in the positive orthant, and each of their paths beginning in the center of the cube. Each of the Brownian motions’ position was represented using a hexadecimal string that ended with an ellipsis (so I suppose it was infinite precision). There was a leaderboard that tracked which Brownian motions had gotten closest to the origin, and every time there was a change in the leaderboard, I briefly woke up. I recall there was also something about Schur complements. Interestingly, other than the hexadecimal strings, there was barely any visual content to these hallucinations. So there had to have been something else to my experience that granted me this kind of knowledge about the dream’s contents.

Sometimes the dreams are additionally accompanied by a feeling of understanding some mathematical truth. In one dream, I became convinced that I had “solved logistic regression”. If I was lying on my right side, then the “answer” was 0, and if I was lying on my left side, then the “answer” was 1. I don’t remember what I had interpreted lying on my back to mean; perhaps 0.5? I do remember feeling a bit anguished about this case before declaring it solved. Of course, this is all complete nonsense, but the feeling was quite palpable.

Interestingly, in some other dreams, I have the pure feeling of reasoning without any other phenomenal content. These are immensely frustrating and exhausting dreams, since I have the experience of expending lots of mental effort trying to reason through something, but that something is never available to me. I will wake up from these dreams feeling poorly rested and confused.

Mathematics in waking life

Visual accompaniment. I think the reason that I did not notice this before is that in waking life, doing mathematics is almost always accompanied by some kind of visual content⁶, and it’s hard for me to isolate what about my experience is left when stripped of any kind of sensory or quasi-sensory mediation.

Symbol manipulation. I also usually don’t need to even confront mathematical objects directly.⁷ Usually, I outsource much of my thought to symbols. For instance, if I have a linear operator $A$ and I want to take its SVD, then I don’t need to summon the phenomenal state of working with a linear operator and decomposing its action. I just write $A = U Σ V^{⊤}$ and call it done. The only “intuition” that I might feel is one of being magnetized towards symbolic manipulations that feel most fruitful.⁸

Mathematical intuition. At a certain point, raw symbol manipulation fails and I do have to resort to mathematical intuition. While this engagement, again, often involves some kind of visual imagery, I don’t think the imagery is the intuition. Eventually, if I’m lucky, I will feel a sense of synchrony and insight, even if all of my visual imagery has stayed the same. It feels like finding a block in a Jenga tower that moves easily, or perhaps a posture or hold on a rock wall that actually lets you progress. The feeling of the mathematics “giving way” is certainly something beyond sensory phenomena, and I contend that it proceeds from a kind of phenomenal state that is not purely (quasi-)sensory. This feeling happens in just about every nontrivial proof I have to write (although, come to think of it, perhaps it’s what defines a proof as being nontrivial). It was particularly strong when I was participating in SURIM in summer 2021. I was working with Santi Aranguri and Slava Naprienko on a new direct proof of Tokuyama’s formula, as documented in section 4 of our final report, and while reaching equation (23) was doable with just symbolic manipulation, factoring the sum from there seemed impossible until, after many hours of working out examples with Santi, I felt something give way in my brain as I intuited that we could simply expand the domain of the sum without changing its value, and the problem seemed to disappear. (Well, not entirely since we still had some working to do, but that particular roadblock fell.)

Mathematical and logical intuition. Notice here that I am actually separating the phenomenal states of doing mathematics and reasoning. Sometimes I will experience this feeling of giving way, but when I try to rationally justify my intuition, it turns out to be wrong.⁹ Conversely, my logical intuition will sometimes fail where my mathematical intuition succeeds, and I’ll have a correct proof that I for some reason feel is wrong. It’s actually rather strange to me that mathematical and logical intuitions should be so well aligned, but not perfectly so. I think deciding on axioms is a great example to illustrate this sort of disconnect, since logically we have no reason to prefer (say) the inclusion or exclusion of the axiom of choice, or the continuum hypothesis, yet I feel intuitively that some axioms are just “mathematically correct”. I wonder: if we had no logical intuitions but still had mathematical intuitions, then would we simply accept or reject mathematical statements purely based upon intuition?

Different kinds of understanding

There also seem to be different ways to feel understanding in mathematics. I consider here the case of trying to understand a result and its proof, but I acknowledge that there are different ways of understanding particular mathematical concepts as well. We use the proprioceptive metaphor often: depending how well I can “grasp” the concept, I am better able to make use of it. There is perhaps much to say about this feeling but I wish to focus on results and proofs for now.

Stepwise and holistic understanding. I might be able to follow along with the proof as an “ant on the page,” as Persi Diaconis likes to say, but not holistically. That is, I understand the steps taken individually, but I somehow lose the plot when I try to take it in at once.¹⁰ Notably, sometimes the feeling of holistic understanding appears without ever being presented with new information, and perhaps without even rereading the proof. What’s changed?¹¹ Suddenly I feel that I grasp the essence of the proof, and can reproduce it without much thought, or I can even transfer the technique elsewhere.

Essential understanding. If I do understand the proof holistically, I still might not understand why the theorem is true. I understand that the theorem is true, but I feel that I have not understood the essence of the result. The classic example is the four-color theorem, which to my knowledge lacks a satisfying proof to this day. In fact I find that there are lots of examples in combinatorics and graph theory. The proof can even be slick and carry the feeling of elegance, but still be epistemically hollow. Consider the following two proofs of Euler’s partition identity (the discussion here is partly adapted from this note by George E. Andrews).

Example

Theorem: Euler's Partition Identity

The number of partitions of a positive integer $n$ into distinct parts equals the number of partitions into odd parts.

For instance, there are $6$ partitions of $8$ into distinct parts: $8$ , $7 + 1$ , $6 + 2$ , $5 + 3$ , $5 + 2 + 1$ , and $4 + 3 + 1$ . There are also $6$ partitions of $8$ into odd parts: $7 + 1$ , $5 + 3$ , $5 + 1 + 1 + 1$ , $3 + 3 + 1 + 1$ , $3 + 1 + 1 + 1 + 1 + 1$ , and $1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$ .

Generating function proof. The generating function for partitions into distinct parts is $\prod_{k = 1}^{\infty} (1 + x^{k})$ , since each part can appear at most once. The generating function for partitions into odd parts is $\prod_{k = 0}^{\infty} (1 - x^{2 k + 1})^{- 1}$ , since each odd part can appear any number of times. To show these are equal, observe

k = 1 \prod \infty (1 + x^{k}) = k = 1 \prod \infty \frac{1 - x ^{2 k}}{1 - x ^{k}} = \frac{\prod _{k = 1}^{\infty} ( 1 - x ^{2 k} )}{\prod _{k = 1}^{\infty} ( 1 - x ^{k} )} .

The numerator is the product over even integers, while the denominator is the product over all positive integers. Their quotient leaves only odd terms in the denominator.

Glaisher’s bijective proof. Given a partition into distinct parts, write each part as $2^{a} m$ where $m$ is odd, and replace it with $2^{a}$ copies of $m$ . For example, $6 + 5 + 1 = (2^{1} \cdot 3) + (2^{0} \cdot 5) + (2^{0} \cdot 1)$ becomes $3 + 3 + 5 + 1$ , a partition into odd parts. Conversely, given a partition into odd parts, group equal parts together. If there are $k$ copies of an odd number $m$ , write $k$ in binary and convert each power of 2 back into a distinct part.

The generating function proof is slick, but I find it to be unsatisfying as an explanation. It tells us nothing about why distinct parts and odd parts are related structurally. Glaisher’s bijective proof, on the other hand, reveals the structural reason behind the identity: the binary representation of multiplicities mediates between distinctness and oddness.

Contextual understanding. And even if I can feel and intuit why a theorem is true and how the proof manages to demonstrate it, there can still be a lingering sense of incompleteness, a feeling that the statement’s place in the broader mathematical landscape has not been made clear. I feel this most when there are some objects or there’s a collection of results that I feel should be related, and I can understand each individually, but I don’t see any obvious kind of unifying principle. Off the top of my head, one result that sated this feeling of disconnect was Zabreiko’s lemma, which quite tidily unified the four major results one learns early in functional analysis.¹² Without going into too much detail, I currently feel this way about the state of optimization algorithms. There are lots of different kinds and classifications of optimization algorithms—first-order methods, second-order methods, population methods, stochastic methods—and it feels to me like they should all be much more closely related than we currently treat them. The book Large-Scale Convex Optimization: Algorithms & Analyses via Monotone Operators by Ernest Ryu and Wotao Yin does a phenomenal¹³ job at showing how a lot of first-order methods can be analyzed through a unified framework, that of monotone operators. I feel that much more unification can be done, perhaps not in the same way as monotone operators describe first-order methods, but in some way that realizes all of these different kinds of optimization algorithms through a natural framework.

Natural definitions. As a final point, I want to bring up the feeling of naturalness, particularly as it appears in definitions. The above feelings of holistic, essential, and contextual understanding in proofs, as well as the comfort and ease with which I can use some mathematical idea, seem to depend a fair amount on having made the most natural choice in definitions and notation. There is a feeling that we have named and signified the most salient features and affordances of the mathematical landscape, and that we have placed distinctions in the right places to capture exactly the essential qualities of an idea. We haven’t gerrymandered, doing something akin to assigning a name to some arbitrary combination like “the two easternmost legs of a chair in this room combined with the hour hand of Big Ben”. No, we’ve assigned names only to the legs, the back, the seat, and the composite object (the chair). The feeling of naturalness is that we have definitions that track genuine structure.

I won’t dwell on this point too long here. Jamie Tappenden discusses what this feeling could mean much more extensively and precisely in Mathematical Concepts and Definitions and Mathematical Concepts: Fruitfulness and Naturalness, both of which are chapters of the book The Philosophy of Mathematical Practice edited by Paolo Mancosu. There he takes seriously the idea that the feeling of “naturalness” is about something, and that we can test ideas’ naturalness quasi-empirically. He examines the idea that naturalness is about fruitfulness, and provides the Legendre symbol as a central example.

Closing thoughts

I haven’t written this to try to make any kind of commitment to or argument for or against any kind of metaphysics regarding mathematics. I’ve simply tried to document some observations about what doing mathematics feels like from the inside, that there seems to be phenomenal content beyond the sensory, that mathematical and logical intuition can come apart, that understanding comes in qualitatively different flavors which are themselves felt rather than merely known. I am not yet sure what knowledge can be drawn from these feelings to be honest. But I find it curious that these experiences are so vivid to me and yet so rarely discussed among my colleagues. Perhaps others have noticed the same things.

Calling each of these instances a "dream" is perhaps a little disingenuous. I am not fully comfortable calling them as such because they are often hypnagogic or hypnopompic rather than REM. Perhaps in another post I'll discuss my taxonomy of dreams, since I've put a lot of thought towards classifying the kinds of dreams that I and some others have. ↩
I am not alone in this. My roommate, for instance, has reported such dreams. When we were living on campus and slept in the barracks—I mean, dorms—I would sometimes hear him mumble in his sleep. I recall one time he said something incoherent about rational lattices. My advisor, Stephen Boyd, also has dreams like this often, except his tend to be more fruitful. ↩
Unfortunately, my oneiric mathematical visitors are kind of dumbasses. Instead of showing me infinite series evaluations of elliptic integrals or mock theta functions like they did for Ramanujan, they taunt and frustrate me with utter nonsense, or they act all coy and refuse to show anything at all.⁴ ↩
Actually, maybe I'm the dumbass. Maybe they're being perfectly clear and I am completely missing the point. That sounds more likely... ↩
See my most recent now post for more reflections on that! ↩
Just to be clear: by visual content I mean both what you might call perceptual and imagistic phenomenal content. The former is what I experience when I write a symbol or plot a graph or draw a diagram. The content of this experience is static, cohesive, and immediately and effortlessly accessible to me; I just have to look at it. Imagistic phenomenal content, on the other hand, is the quasi-visual content of imagined symbols or plots or diagrams. Holding onto this content is more effortful and ethereal, since I am not the best visualizer in the world. These images tend to be hazy and constantly in flux, and if I relax for too long or try to add too much complexity, then they completely fall apart. It's not clear to me "where" this imagistic content is, since it's never superimposed onto my perceptual content, and I have never had the experience of confusing the two. ↩
I am reminded here of the writings of Alfred North Whitehead in chapter 5 An Introduction to Mathematics, "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems ... It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them." I may make a post in the future about the design of mathematical notation, since it is a very curious thing to me. At the very least, I would love to write a rant about the notational baggage and sprawl associated with the derivative. I know it's a bit overdone, but it is truly terrible how confused our notation is, particularly on the applied side, where the kind of thing a derivative is seems to be whatever's convenient at the moment, and you end up with mismatched shapes and transposes and it's not even clear with respect to what you're differentiating... it's a nightmare trying to read some CS papers. ↩
I find it actually rather mysterious that you can get so far with pure symbolic manipulation. What are you actually doing when you simply pattern-match from one statement to another, devoid of any kind of mathematical intuition? You can do this at home! Find a mathematics textbook for a sufficiently advanced subject that you understand close to nothing about the vocabulary and notation. Flip to some early exercise, and solve it without reading the text. Just look for the relevant vocabulary and property sets, and do your standard kinds of deductions without ever trying to understand what the constituents of your argument mean. I know I've certainly done this in some of my classes when I was short on time! It reminds me of the Chinese room argument, though I haven't quite teased out what lesson can be drawn here (hence why this is a footnote). ↩
This is a very common experience, of course. Actually, the Italian school of algebraic geometry, particularly under Federigo Enriques, is a great case study for what happens when mathematicians rely much more on intuition than rigor. Despite the heavy use of informal techniques, the school made many significant contributions to the field that later turned out to withstand rigor. Of course, it eventually collapsed because this style of argumentation also produced a fair number of incorrect results. See also this paper by Silvia de Toffoli and Claudio Fontanari which contains some relevant discussion about the sort of phenomenology I've described above, as experienced by Enriques et al. ↩
I remember feeling this way when I was taking introductory analysis and I wrote my final report about the Weierstrass function. The proof of its nowhere differentiability seemed to me totally ad hoc and unmotivated. The high-level strategy is poorly documented. Of course, lots of results feel this way when first learning about a subject. I think a lot of group theory felt this way to me, particularly when learning about Sylow theory, since we examined so many specific cases and it didn't really give me a sense of what was going on, as if I should instead just "shut up and calculate". ↩
I suppose it's a bit like the spinning dancer illusion or perhaps like autostereograms (maybe you know of the Magic Eye series). The visual information entering your eye hasn't changed, yet somehow you see something new. Or perhaps you could liken it to all those instances where you've seen something your whole life, and then someone points out some detail or visual comparison, and then you can't unsee it. ↩
I mean, of course, they can all be viewed as downstream effects of the Baire category theorem, but I feel that isn't so explanatory. ↩
Pun intended? ↩