Thinking and Explaining

Question

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, or describe how they are connected for you.

I've been fascinated by the phenomenon the question addresses for a long time. We have complex minds evolved over many millions of years, with many modules always at work. A lot we don't habitually verbalize, and some of it is very challenging to verbalize or to communicate in any medium. Whether for this or other reasons, I'm under the impression that mathematicians often have unspoken thought processes guiding their work which may be difficult to explain, or they feel too inhibited to try. One prototypical situation is this: there's a mathematical object that's obviously (to you) invariant under a certain transformation. For instant, a linear map might conserve volume for an 'obvious' reason. But you don't have good language to explain your reason---so instead of explaining, or perhaps after trying to explain and failing, you fall back on computation. You turn the crank and without undue effort, demonstrate that the object is indeed invariant.

Here's a specific example. Once I mentioned this phenomenon to Andy Gleason; he immediately responded that when he taught algebra courses, if he was discussing cyclic subgroups of a group, he had a mental image of group elements breaking into a formation organized into circular groups. He said that 'we' never would say anything like that to the students. His words made a vivid picture in my head, because it fit with how I thought about groups. I was reminded of my long struggle as a student, trying to attach meaning to 'group', rather than just a collection of symbols, words, definitions, theorems and proofs that I read in a textbook.

Please note: I'm not advocating that we turn mathematics into a touchy-feely subject. I'm not claiming that the phenomenon I've observed is universal. I do think that paying more attention than current custom to how you and others are really thinking, to the intuitions, is helpful both in proving theorems and in explaining mathematics.

I'm very curious about the varied ways that people think, and I would like to hear.

What am I really thinking? I'm anxious about offending the guardians of the forum and being scolded (as they have every right to do) for going against clearly stated advice with a newbie mistake. But I can't help myself because I'm very curious how you will answer, and I can endure being scolded.

Answer 1

I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, because one already has a reasonable idea of what the other person's mental model of the problem is. In some ways, I find that throwing out things to a collaborator is closer to the mathematical thought process than just thinking about maths on one's own, if that makes any sense.

One specific mental image that I can communicate easily with collaborators, but not always to more general audiences, is to think of quantifiers in game theoretic terms. Do we need to show that for every epsilon there exists a delta? Then imagine that you have a bag of deltas in your hand, but you can wait until your opponent (or some malicious force of nature) produces an epsilon to bother you, at which point you can reach into your bag and find the right delta to deal with the problem. Somehow, anthropomorphising the "enemy" (as well as one's "allies") can focus one's thoughts quite well. This intuition also combines well with probabilistic methods, in which case in addition to you and the adversary, there is also a Random player who spits out mathematical quantities in a way that is neither maximally helpful nor maximally adverse to your cause, but just some randomly chosen quantity in between. The trick is then to harness this randomness to let you evade and confuse your adversary.

Is there a quantity in one's PDE or dynamical system that one can bound, but not otherwise estimate very well? Then imagine that it is controlled by an adversary or by Murphy's law, and will always push things in the most unfavorable direction for whatever you are trying to accomplish. Sometimes this will make that term "win" the game, in which case one either gives up (or starts hunting for negative results), or looks for additional ways to "tame" or "constrain" that troublesome term, for instance by exploiting some conservation law structure of the PDE.

For evolutionary PDEs in particular, I find there is a rich zoo of colourful physical analogies that one can use to get a grip on a problem. I've used the metaphor of an egg yolk frying in a pool of oil, or a jetski riding ocean waves, to understand the behaviour of a fine-scaled or high-frequency component of a wave when under the influence of a lower frequency field, and how it exchanges mass, energy, or momentum with its environment. In one extreme case, I ended up rolling around on the floor with my eyes closed in order to understand the effect of a gauge transformation that was based on this type of interaction between different frequencies. (Incidentally, that particular gauge transformation won me a Bocher prize, once I understood how it worked.) I guess this last example is one that I would have difficulty communicating to even my closest collaborators. Needless to say, none of these analogies show up in my published papers, although I did try to convey some of them in my PDE book eventually.

ADDED LATER: I think one reason why one cannot communicate most of one's internal mathematical thoughts is that one's internal mathematical model is very much a function of one's mathematical upbringing. For instance, my background is in harmonic analysis, and so I try to visualise as much as possible in terms of things like interactions between frequencies, or contests between different quantitative bounds. This is probably quite a different perspective from someone brought up from, say, an algebraic, geometric, or logical background. I can appreciate these other perspectives, but still tend to revert to the ones I am most personally comfortable with when I am thinking about these things on my own.

ADDED (MUCH) LATER: Another mode of thought that I and many others use routinely, but which I realised only recently was not as ubiquitious as I believed, is to use an "economic" mindset to prove inequalities such as $X \leq Y$ or $X \leq CY$ for various positive quantities $X, Y$, interpreting them in the form "If I can afford $Y$, can I therefore afford $X$?" or "If I can afford lots of $Y$, can I therefore afford $X$?" respectively. This frame of reference starts one thinking about what types of quantities are "cheap" and what are "expensive", and whether the use of various standard inequalities constitutes a "good deal" or not. It also helps one understand the role of weights, which make things more expensive when the weight is large, and cheaper when the weight is small.

ADDED (MUCH, MUCH) LATER: One visualisation technique that I have found very helpful is to incorporate the ambient symmetries of the problem (a la Klein) as little "wobbles" to the objects being visualised. This is most familiarly done in topology ("rubber sheet mathematics"), where every object considered is a bit "rubbery" and thus deforming all the time by infinitesimal homeomorphisms. But geometric objects in a scale-invariant problem could be thought of as being viewed through a camera with a slightly wobbly zoom lens, so that one's mental image of these objects is always varying a little in size. Similarly, if one is in a translation-invariant setting, one's mental camera should be sliding back and forth just a little to remind you of this, if one is working in a Euclidean space then the camera might be jiggling through all the rigid motions, and so forth. A more advanced example: if the problem is invariant under tensor products, as per the tensor product trick, then one's low dimensional objects should have a tiny bit of shadowing (or perhaps look like one of these 3D images when one doesn't have the polarised glasses, with the slightly separated red and blue components) that suggest that they are projections of a higher dimensional Cartesian product.

One reason why one wants to do this is that it helps suggest useful normalisations. If one is viewing a situation with a wobbly zoom lens and there is some length that appears all over one's analysis, one is reminded that one can spend the scale invariance of the problem to zoom up or down as appropriate to normalise this scale to equal 1. Similarly for other ambient symmetries.

This sort of wobbling of symmetries is also available in less geometric settings. When viewing, say, a graph on $n$ vertices, perhaps the labels $1,\dots,n$ on the vertices have a tendency to swap with each other every so often, to emphasise the symmetry of relabeling in graph theory. Similarly, when dealing with a set $\{a,b,c,d,\dots\}$, perhaps the positions of the elements $a,b,c,d$ in one's enumeration of the set are volatile and swap places every so often. In analysis, one often only cares about the order of magnitude of some very large or very small quantity X, rather than its exact value; so one should view this quantity as being a bit squishy in size, growing or shrinking by a factor of two or so every time one looks at the problem. If there is some probability theory in one's problem, and some of your objects are random variables rather than deterministic variables, then you can imagine that every so often the "game resets", with the random variables jumping around to different values in their range (and any quantities depending on these variables changing accordingly), whereas the deterministic variables stay fixed. Similarly if one has generic points in a variety, or nonstandard objects in a space (with the point being that if something bad happens if, say, your generic point is trapped in a subvariety, you can "reset the game" in which the generic point is now outside the subvariety; similarly one can "reset" an unbounded nonstandard number to be larger than any given standard number, etc.).

Answer 2

Final addition:

Since I've produced many rambles, I thought I'd close my (anti-)contribution with a distilled version of the example I've attempted below. It's still something very standard, but, I hope, in the spirit of the original question. I'll describe it as if it were a personal thing.

Almost always, I think of an integer as a function of the primes. So for 20, say,

20(2)= 0

20(3)=2

20(5)=0

20(7)=6

. .

20(19)=1

20(23)=20

20(29)=20

20(31)=20

20(37)=20

. . .

It's quite a compelling image, I think, an integer as a function that varies in this way for a while before eventually leveling off. But, for a number of reasons, I rarely mention it to students or even to colleagues. Maybe I should.

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Original answer:

It's unclear if this is an appropriate kind of answer, in that I'm not putting forward anything very specific. But I'll take the paragraph in highlight at face value.

I find it quite hard to express publicly my vision of mathematics, and I think this is a pretty common plight. Part of the reason is the difficulty of putting into words a sense of things that ultimately stems from a view of the landscape, as may be suggested by the metaphor. But another important reason is the disapprobation of peers. To appeal to hackneyed stereotypes, each of us has in him/her a bit of Erdos, a bit of Thurston, and perhaps a bit of Grothendieck, of course in varying proportions depending on education and temperament. I think I saw somewhere on this site the sentiment that 'a bad Erdos still might be an OK mathematician, but a bad Grothendieck is really terrible,' or something to that effect. This opinion is surrounded by a pretty broad consensus, I think. If I may be allowed some cliches now from the world of finance, it's almost as though definite mathematical results are money in the bank. After you've built up some savings, you can afford to spend a bit by philosophizing. But then, you can't let the balance get too low because people will start looking at you in funny, suspicious ways. I know that on the infrequent occasions* that I get carried away and convey at any length my vision of how a certain area of mathematics should work, what should be true and why, compelling analogies, and so on, I feel rather embarrassed for a little while. It feels like I am indeed running out of money and will need to back up the highfalutin words with some theorems (or at least lemmas) relatively soon. (And then, so many basically sound ideas are initially mistaken for trivial reasons.)

Now, I wish to make it clear that unlike Grothendieck (see the beginning paragraphs of this letter to Faltings) I find this quite sensible a state of affairs. For myself, it seems to be pretty healthy that my tendency to philosophize is held in check by the demand of the community that I have something to show for it. I grant that this may well be because my own visions are so meagre in comparison to Grothendieck's. In any case, the general phenomenon itself is interesting to observe, in myself and in others.

Incidentally, I find the peer pressure in question remarkably democratic. Obviously, a well-established mathematician typically has more money than average in the bank, so to speak. But it's not a few times I've observed eminent people during periods of slowdown, being gradually ignored or just tolerated in their musings by many young people, even students.

Meanwhile, if you're an energetic youngster with some compelling vision of an area of mathematics, it may not be so bad to let loose. If you have a really good business idea, it may even make sense to take out a large loan. And provided you have the right sort of personality, the pressure to back up your philosophical bravado with results may spur you on to great things. This isn't to say you won't have to put up with perfectly reasonable looks of incredulity, even from me, possibly for years.

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*Maybe it seems frequent to my friends.

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Added:

Since I commented above on something quite general, here is an attempt at a specific contribution. It's not at all personal in that I'm referring to a well-known point of view in Diophantine geometry, whereby solutions to equations are sections of fiber bundles. Some kind of a picture of the fiber bundle in question was popularized by Mumford in his Red Book. I've discovered a reproduction on this page. The picture there is of $\operatorname{Spec}(\mathbb{Z}[x])$, but interesting equations even in two variables will conjure up a more complicated image of an arithmetic surface fibered over the 'arithmetic curve' $\operatorname{Spec}(\mathbb{Z})$. A solution to the equation will then be a section of the bundle cutting across the fibers, also in a complicated manner. Much interesting work in number theory is concerned with how the sections meet the singular fibers.

Over the years, I've had many different thoughts about this perspective. For me personally, it was truly decisive, in that I hadn't been very interested in number theory until I realized, almost with a shock, that the study of solutions to equations had been 'reduced' to the study of maps between spaces of a quite rigid sort. In recent years, I think I've also reconciled myself with the more classical view, whereby numbers are some kinds of algebraic gadgets. That is, thinking about matters purely algebraically does seem to provide certain flexible modes that can be obscured by the insistence on geometry. I've also discovered that there is indeed a good deal of variation in how compelling the inner picture of a fiber bundle can be, even among seasoned experts in arithmetic geometry. Nevertheless, it's clear that the geometric approach is important, and informs a good deal of important mathematics. For example, there is an elementary but key step in Faltings' proof of the Mordell conjecture referred to as the 'Kodaira-Parshin trick,' whereby you (essentially) get a compact curve $X$ of genus at least two to parametrize a smooth family of curves $$Y\rightarrow X.$$ Then, whenever you have a rational point $$P:\operatorname{Spec}(\mathbb{Q})\rightarrow X$$ of $X$, you can look at the fiber $Y_P$ of $Y$ above $P$, which is itself a curve. The argument is that if you have too many points $P$, you get too many good curves over $\mathbb{Q}$. What is good about them? Well, they all spread out to arithmetic surfaces over the spectrum of $\mathbb{Z}$ that are singular only over a fixed set of places. This part can be made obvious by spreading out both $Y$, $X$, and the map between them over the integers as well, right at the outset. If you don't have that picture in mind, the goodness of the $Y_P$ is not at all easy to explain.

Anyways, what I wanted to say is that the picture of solutions as sections to fiber bundles is really difficult to explain to people without a certain facility in scheme theory. Because it seems so important, and because it is a crucial ingredient in my own thinking, I make an attempt every now and then in an exposition at the colloquium level, and fail miserably. I notice almost none of my colleagues even try to explain it in a general talk.

Now, I've mentioned already that this is far from a personal image of a mathematical object. But it still seems to be a good example of a very basic picture that you refrain from putting into words most of the time. If it really had been only a personal vision, it may even have been all but maddening, the schism between the clarity of the mental image and what you're able to say about it. Note that the process of putting the whole thing into words in a convincing manner in fact took thousands of pages of foundational work.

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Added again:

Professor Thurston: To be honest, I'm not sure about the significance of competing mental images in this context. If I may, I would like to suggest another possibility. It isn't too well thought out, but I don't believe it to be entirely random either.

Many people from outside the area seem to have difficulty understanding the picture I mentioned because they are intuitively suspicious of its usefulness. Consider a simpler picture of the real algebraic curve that comes up when one studies cubic equations like $$E: y^2=x^3-2.$$ There, people are easily convinced that geometry is helpful, especially when I draw the tangent line at the point $P=(3,5)$ to produce another rational point. What is the key difference from the other picture of an arithmetic surface and sections? My feeling is it has mainly to do with the suggestion that the point itself has a complicated geometry encapsulated by the arrow $$P:\operatorname{Spec}(\Bbb{Z})\rightarrow E.$$ That is, spaces like $\operatorname{Spec}(\Bbb{Q})$ and $\operatorname{Spec}(\Bbb{Z})$ are problematic and, after all, are quite radical.

In $\operatorname{Spec}(\Bbb{Q})$, one encounters the absurdity that the space $\operatorname{Spec}(\Bbb{Q})$ itself is just a point. So one has to go into the whole issue that the point is equipped with a ring of functions, which happens to be $\Bbb{Q}$, and so on. At this point, people's eyes frequently glaze over, but not, I think, because this concept is too difficult or because it competes with some other view. Rather, the typical mathematician will be unable to see the point of looking at these commonplace things in this way. The temptation arises to resort to persuasion by authority then (such and such great theorem uses this language and viewpoint, etc.), but it's obviously better if the audience can really appreciate the ideas through some first-hand experience, even of a simple sort. I do have an array of examples that might help in this regard, provided someone is kind enough to be still interested. But how helpful they really are, I'm quite unsure.

At the University of Arizona, we once had a study seminar on random matrices and number theory, to which I was called upon to contribute a brief summary of the analogous theory over finite fields. Unfortunately, this does involve some mention of sheaves, arithmetic fundamental groups, and some other strange things. Afterwards, my colleague Hermann Flaschka, an excellent mathematician with whom I felt I could speak easily about almost anything, commented that he couldn't tell if the whole language just consisted of word associations or if some actual geometry was going on. Now, I'm sure this was due in part to my poor powers of exposition. But further conversation gave me the strong impression that the question that really went through his mind was: 'How could it possibly be useful to think about these objects in this way?'

To restate my point, I think a good deal of conceptual inhibition comes from a kind of intuitive utilitarian concern. Matters are further complicated by the important fact that this kind of conceptual conservatism is perfectly sensible much of the time.

By the way, my choice of example was somewhat motivated by the fact that it is quite likely to be difficult for people outside of arithmetic geometry, including many readers of this forum. This gives it a different flavor from the situations where we all understand each other more or less well, and focus therefore on pedagogical issues referring to classroom practice.

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Yet again:

Forgive me for being a bore with these repeated additions.

The description of your approach to lectures seems to confirm the point I made, or at least had somewhat in mind: When someone can't understand what we try to explain, it's maybe in his or her best interest (real or perceived) not to. It's hard not to feel that this happens in the classroom as well oftentimes. This then brings up the obvious point that what we try to say is best informed by some understanding of who we're speaking to as well as some humility*. As a corollary, what we avoid saying might equally well be thus informed.

My own approach, by the way, is almost opposite to yours. Of course I can't absorb technical details just sitting there, but I try my best to concentrate for the whole hour or so, almost regardless of the topic. (Here in Korea, it's not uncommon for standard seminar lectures to be two hours.) If I may be forgiven a simplistic generalization, your approach strikes me as common among deeply creative people, while perennial students like me tend to follow colloquia more closely. I intend neither flattery nor modesty with this remark, but only observation. Also, I am trying to create a complex picture (there's that word again) of the problem of communication.

As to $Spec(\Bbb{Z})$, perhaps there will be occasion to bore you with that some other time. Why don't you post a question (assuming you are interested)? Then you are likely to get a great many perspectives more competent than mine. It might be an interesting experiment relevant to your original question.

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*I realize it's hardly my place to tell anyone else to be humble.

Answer 3

Just to talk about something fresh for which I still have a good memory of what I actually thought and what I wrote, let's take this example

Thoughts:

1) Ivan Fesenko teased me with the puzzle without the gradient condition 10 years ago. I solved it with the standard $(1-xy)^2+x^2$, but it would be nice to tease him with the upgraded puzzle when I talk to him next. Also, it is high time to finish it off.

2) The standard polynomial is even and the only critical point is a saddle. If you think of it, this saddle is inevitable: we have low points on the landscape on both sides of the $x$-axis and high values on the $x$-axis going to $+\infty$ both ways. Thus the mountain pass lemma will ensure a saddle somewhere.

3) This works every time when the sequence of points where the polynomial goes to $0$ has two different limiting directions. Then we can separate them by a line and run the same argument. So, the limiting direction must be unique.

4) This seems impossible because the highest (even) degree homogeneous part should vanish in this direction but then it also vanishes in the opposite direction. This makes hte second highest (odd) degree homogeneous part to vanish on the entire line too. The next degree is unclear though...

5) We certainly need something to break the symmetry here. A polynomial family $P_y(x)$ of polynomials in $x$ that have roots close to $0$ when $y\to+\infty$ and no roots when $y\to-\infty$ would be nice.

6) Hey, I know this one: $yx^2-1$. Let's try $x^2+(x^2y-1)^2$.

7) Damn, it doesn't work. The origin is still a critical point.

8) Yeah, what else would you expect: the low points on the landscape are accumulating to one direction but still are separated by the line, so the mountain pass lemma is as powerful as before. To kill it, we need to shift both descends to one side.

9) Add $x$ to $P_y(x)$. That won't change the limiting direction but will shift the zeroes a bit. So, let's try $f(x,y)=x^2+(x^2y+x+1)^2$.

10) The origin is good now: $f_x=2$ there. Actually, it is $2$ everywhere where $x=0$.

11) If $x\ne 0$, then $f_y=0$ only if $x^2y+x+1=0$ but then $f_x=2x\ne 0$ by the chain rule.

12) OK, let's post the example and keep it in memory for teasing people...

That's what I actually thought for the last hour or so (interlaced with some personal thoughts that are of no interest for this thread).

What I wrote can be seen easily if you follow the link.

Why such discrepancy?

a) Some steps in the chain like 1) and 10) are too personal to be of interest to anybody. You need them to "start the engine running" and to "vent the steam", but they aren't, strictly speaking, mathematics and do not make me look any better, so why to publish them?

b) Some steps like 8) and the heuristics in 9) are actually false. To publish them would be ridiculous.

c) 4) and 7) are "failures" on the way. There is no point in telling anyone where and how I failed. I could fill volumes with my failed attempts if I started doing it.

d) 10) and 11) are trivial computations. Everyone can do those himself.

e) 2), 3), 5) are left. 9) is the counterexample. 2), 3) are steps in the direction of the affirmative answer. Once the final answer is negative, there is no point in talking of the steps in the opposite direction.

f) 5) is a nice idea but everybody knows that $y$ is not even. One can see the whole mechanics in the answer itself, so there is no need to explain it separately.

I don't know if this account of one personal affair with one relatively simple problem can really shed much light on why we do not tell/write exactly what we see/think of, but you asked and I answered.

Answer 4

One observation: People understandably hesitate telling a half-truth. When you teach a heuristic picture to someone, you also need to teach them about how fuzzy it is and when it starts to break down. A more calculational proof has the virtue of being self-contained and robustly transmissible. This is even more important when writing a textbook.

Then there are other cases where I'm puzzled why certain heuristic means of understanding and organizing knowledge don't seem to be usually taught. Take the concept of normal subgroups. In one of his books, V.I. Arnold says that a subgroup is normal when it is relativistically invariant, and he doesn't develop that line of thought any deeper. That statement is a good example of a heuristic analogy that is specific in its detail but general in its spirit. However you phrase it, certainly you should give your students the idea that a normal subgroup is something whose structure is invariant with respect to the parent group's symmetries. As a litmus test, your students should be able to tell whether these subgroups are normal at a glance, without calculation:

Let $E^2$ be the Euclidean group of the plane and let $O^2$ be the subgroup fixing some point.

Subgroups of $E^2$:

• Translations along some particular direction.
• Translations along every direction.
• Translations and glides along every direction.
• Reflections in every line.
• Rotations around some particular point.
• Symmetries of a tessellation.

Subgroups of $O^2$:

• Symmetries of a regular polygon.
• Reflections in a line.

The way I think about the non-normal cases is that there is something non-isotropic about them, some structure that the subgroup preserves that is not preserved by the parent group. For example:

• Translations along some particular direction: Rotations don't preserve the direction.
• Translations along every direction: No special directions, so it's normal.
• Translations and glides along every direction: Ditto.
• Reflections in every line: Ditto. (This combines the two previous cases.)
• Rotations around some particular point: Translations don't preserve the point.

Answer 5

This phenomenon occurs not just in advanced mathematics but also right at the very bottom in simple mental arithmetic. If I have to do a moderately complicated calculation such as adding two three-digit numbers, there's often a part of my brain that jumps ahead to the answer before another more cautious part has got there with carefully checked calculations. The first part just sort of feels the answer and then says "I told you so" to the second part, except occasionally when the first part gets it wrong and the second part says "Now you know why I bother to be careful" to the first part.

And there are also aspects of how I carry out integer addition and subtraction that I would normally be a bit embarrassed to verbalize, such as that if I subtract 48 from 135 then there's a preliminary answer, 97, that I know from experience is wrong and has to be corrected by subtracting 10. (The justification for the preliminary answer is that 13-4=9 and that the answer must end in a 7.) It's not quite what's going on in my head, but it's almost as though I say, "OK I'll subtract 58 instead so as to get the right answer." But if I were teaching this to a child then I'd tell a slightly different story, such as borrowing 1, or first subtracting 50 and then adding 2.

Answer 6

I think the root of the phenomenon is that we can only communicate to others what we know, not what we understand.

Also, it is not unreasonable to think that one's mental images are not going to be of any help to others (In fact, they may well make things more complicated, or confusing for others: I have been told mental images by others---sometimes indirectly, by the choice of the word introduced in a definition---and been thereby misled; here «misled» means «led in a direction different to the one I personally would follow in order to form my own mental image of the concept».) For example, for me resolving the singularities of algebraic varieties makes a clicking (or clacking) sound: this is quite significant for me in a way, but when talking to others I doubt I'd make any mention of this, for seriously doubt it would help :)

Answer 7

I am a visual thinker and I often try to describe what I see to my students. I've been known to say things like "everyone knows that HF looks like a rectangle" as I proceed to draw a rectangle on the board. (By the way, HF is the set of all hereditarily finite sets.) I find that I naturally associate different shapes with different properties of objects. Angular shapes correspond to well-defined objects whereas rounded shapes correspond to variable objects. The number of angles or curves is a measure of how complex an object is. I don't explain my scheme to my students, but I suspect the consistency of the presentation becomes transparent over time.

I recall one instance where I deliberately concealed the true nature of my illustration to my students. I was describing a complex construction on infinite trees. I began the description by drawing five vertical lines that I promptly explained were "infinite trees viewed sideways." It so happens that the simplest case of the construction was when the trees consisted of single branches in which case the picture was completely accurate. This is the case I secretly had in mind for the entire description but I never said that since the result was utterly trivial in that case. This was a subtle way to reduce the complex construction to the trivial case.

Answer 8

I have a worse problem than having unspoken thought processes: some of my best thought processes are simply beneath the level of consciousness and I don't notice them at all until they're finished. Even then, I often get only an answer and not an explanation out of them. Surely this happens to everyone: the problem solved during sleep, the idea on a walk in the woods, the conviction that a conjecture is true on utterly minimal evidence, the argument that pops up full formed in the middle of a conversation.

My mathematical process is roughly this: consciously, I try a lot of stupid things which essentially have no chance of working but do have the benefit of exposing me to lots of examples; these examples pile up and are subconsciously masticated for days, weeks, months -- I'm not old enough mathematically to put "years" here yet -- and eventually by some inner and unobservable process I just have a feeling about what to do.

Perhaps that's an exaggeration. But I certainly do feel that way sometimes, and to the extent that it's true, it means that the whole project of trying to communicate how I thought of something is just telling stories, at least if I say anything other than "well, I just knew one day."

Answer 9

I am immediately reminded of the following phenomenon:

"Let's consider an elliptic curve $E$ over $\mathbf{Q}_p$."

~Speaker begins to draw a donut at the board~

Clearly this is wrong. The speaker may even mention how wrong it is. But there's enough of a kernel of truth to the picture where it may be helpful for the audience. Justifying why the picture is valid in the situation being considered would involve some hard and not undue model theory, but doing anything more than waving your hands and saying the magic words ("a-la-ca-Lefschetz!") is likely to derail your talk so badly as to effectively destroy it. Even if the picture is not completely valid for what you're talking about, it's a visual aid which is at least an easily described first approximation to the truth.

Answer 10

The issue seems, to me, that a lot of these mental pictures are very personal.

I am reminded of an anecdote by Richard Feynman, from "The Pleasure of Finding Things Out". He explains how counting, for him, is a verbal process (he speaks the numbers to himself as he goes along), but that a friend of his would manage visually. (Text here)

He finishes by saying:

I often think about that, especially when I'm teaching some esoteric technique such as integrating Bessel functions. When I see equations, I see the letters in colors — I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students.

Because of this, I think there might not always be a significant value in trying to pass those mental pictures over - the real aim is to provoke the student into developing his own mental pictures, that he can strongly relate to. Some words such as "homological" or "homotopical" spark up very distinctive feelings in me, in a similar way as hearing "mountain" would make me visualise various mountains, hills, cliffs, etc. But whereas the meaning of "mountain" came to me through vision (mainly, but also other senses), the origin of my mental images of mathematical ideas comes through the practice of mathematics. As such, it seems harder to convey these mathematical pictures: they must be backed up by precise mathematical understanding, which at any rate should end up conjuring these mental pictures. Of course, many mental pictures are simple enough or "canonical" enough that one might imagine everyone would come to develop very similar ones upon understanding of one particular concept; the previously mentioned example of cyclic groups comes to mind. So there might be value in passing that on, but in the end I would think that understanding accompanied by the attention to what meaning it provides already goes a long way towards developing personal mental images.

Answer 11

People have mentioned examples which are hard to share due to some kind of prerequisites. Here's one: I learned PDE from a professor who, in his mind, was always thinking about distribution theory, but officially could not talk about it until after he covered the material relevant to the exams. In distribution theory, whenever you see an integral over a domain $\int_\Omega u(x) dx$ you actually picture the characteristic function $\int \chi_\Omega(x) u(x) dx$ or $\int H(f(x)) u(x) dx$ if $f$ is a defining function for $\Omega$ and $H$ is a heaviside function. From this point of view, you imagine that all functions are smooth and compactly supported (or you can imagine their approximations), so that if you integrate by parts on $\int \chi_\Omega \nabla u(x) dx = - \int \nabla \chi_\Omega u(x) dx = \int \delta(f(x)) \nabla f(x) u(x) dx$. The boundary terms come when the derivative hits the characteristic function. Same thing for Stokes' theorem, Gauss's divergence theorem. It's pretty handy to compute this way.

For a little while this was all I understood until I later found out what was going on. The limit of difference quotients of $\chi_\Omega$ is clearly supported on the boundary of $\Omega$ and it's clear, especially if you picture an approximation, that $\nabla \chi_\Omega$ points in the direction of increase of $\chi_\Omega$ -- i.e. the inward normal. More simply: there are two points of view -- if you were to take difference quotients of $u$, you use a Lagrangian point of view in which the point at position $x$ moves in the direction $i$, and you observe a change in $u$ between those points; instead, you can take an Eulerian point of view, (where the adjoint difference quotients go on the characteristic function) and you can instead look at movement of the region with $u$ fixed.

Until I understood this point of view in a simpler way, it would not really be sensible to explain it to others. But now I know that giving a watered down version of the same proof when "proving" the fundamental theorem of calculus / Gauss's divergence for a calculus class in fact does not lose any key ideas (except for technicalities like how you need the mean value theorem to ensure the difference quotients are bounded). Of course, I would also talk about characteristic functions to any math student, since it is a nice point of view.

By the way, in the calculus of variations, when your $u(x) = L(x, \phi(x) )$ is a Lagrangian and $\phi(x)$ is a solution to the Euler-Lagrange equations, and you take difference quotients using the flow of a vector field whose flow preserves the Lagrangian (a "symmetry"), you end up with Noether's theorem through only this one variation (there are only boundary terms in what I called "the Lagrangian point of view" because you vary through a family of solutions except at the boundary). So it's also a nice way to prove conservation laws in one swoop.

My point: for a little while, distribution theory seemed like a magical theory with prerequisites that made it unexplainable in everyday talk, but once I really understood the ideas I could usually discard the vocabulary (actually, the whole theory can often be replaced by cutoffs, partitions of unity, Taylor expansion, and changes of variable -- although I still think it's great to learn). I suspect that this phenomenon is not uncommon for elementary applications of "fancy" mathematical theories. I believe that often once one has a more basic understanding, one can throw away the new words but still fully reveal the ideas (but maybe that's completely due to my own background). People here have talked about Feynman -- he was good at doing this in the context of physics. If you watch his (outstanding) lectures on Project Tuva you will see more or less the proof of Noether's theorem about which I just wrote.

A second point:

Another thing I think happens to me is that I feel some pressure not to convey just how often I rely on geometric modes of thought, especially when they go against the usual way of explaining things, or the background of a typical student, and are not completely necessary.

Example 1: When you row-reduce a matrix, you make a bunch of changes (most importantly some "transvections") in the basis of the image space until a few of your basis vectors (say $v_1 = T e_1, v_2 = T e_2$) span the image of the matrix $T$. When you picture the domain of $T$ foliated by level sets (which are parallel to the null space of $T$), you know that the remaining basis vectors $e_3, e_4, ...$ can be translated by some element in the span of $e_1, e_2$ (i.e. whichever one lies on the same level set) in order to obtain a basis for the null space. Now, this is how we visualize the situation, but is it how we compute and explain? Or do we just do the algebra, which at this point is quite easy? If the algebra is easy and the geometry takes a while to explain and is not "necessary" for the computation, why explain it? This is a dilemma because once algebra is sufficiently well-developed it's possible that the necessity of (completely equivalent) geometric thinking may become more and more rare; and algebra seems to be more "robust" in that you can explore things you can't see very well. But then, when students learn the implicit function theorem, somehow I feel like having relied on that kind of foliation much more often would help understand its geometric content. Still, even if it's in your head and very important, are you going to draw a foliation every time you do row operations? We know the geometry, know the algebra, but it would take a while to repeatedly explain how to rely on the geometry while executing computations.

Example 2: (Things that aren't graphs)

Another problem geometric thinking faces is that modern math often seems to regard pictures as not being proofs, even if they are more convincing, so there is a bias regarding how to choose to spend class time. Let's say you want to differentiate $x^3$. You can draw a cube, and a slightly larger cube, and then look at the difference of the cubes and subdivide it into a bunch of small regions, three larger slabs taking up most of the volume. Algebraically, this subdivision corresponds to multiplying out $(x+h)^3$; collecting the terms uses the commutativity, which corresponds to rotating the various identical pieces. It is no different to write this proof out algebraically, the difference is that the algebraic one is a "proof" but the geometric one is.. not? Even if it's more convincing. So it's like the picture is only there for culture.

Maybe I have the lecture time to teach both, I will. But I would like to go farther than that. When I differentiate the cube root function, the same cube appears and I go through it again if I feel like it just to convince myself of the truth. Actually, every time I ever use the product rule I always picture the same rectangle with a slightly larger rectangle. My point of view is that one important "definition" of multiplication is in terms of areas, and that a linear function is not necessarily a graph. When you think of a linear function, you should also picture things like rectangles, sectors, similar triangles like the kind that come up when "proving" basic differentiation formulas. Differentiating the integral may seem like a magical trick, but it's really just a continuation of the point of view that multiplication can look like an area/volume and differentiation means taking a small change in the input.

Now, I'd like that point of view to be absorbed, but it's not exactly in the textbook, or completely consistent with what students' other teachers taught them. It's hard to go against the idea that "you should think graphically" -- if I ever think about the sine or tangent function now, it might be the area of a triangle, it might be the length of some vertical line segment, but it's basically never using the graph, which contains basically no additional information. If I have more than one shot at it, I'll try to explain both, but is it really of service to go around saying all the time why graphs aren't the end-all-be-all?

Also, while I can express the pictures in my head one at a time, the fact that I repeatedly, repeatedly see this pictures is something that I feel is harder to express. After all, can't you just do algebra and get through this stuff more quickly? The algebra is "easier" too; it takes up less space.

Answer 12

When I talk about determinants, I generally talk about something on the spectrum between "it measures how much volume scales" and "it's the induced action on the top exterior power." But the way I think about determinants (especially in combinatorics) is the picture coming from the Lindstrom-Gessel-Viennot lemma: I imagine that the entries of the matrix describe transition amplitudes and that the determinant is an alternating sum over transition amplitudes in which "histories" of $n$ particles can constructively or destructively interfere. I have a hard time making this picture precise so I rarely talk about it, but for me it gives some intuition for why determinants should be useful in combinatorics (which the elegant basis-free definition, at least for me, does not).

Edit: Let me also mention that something I really like about this perspective is that it makes intuitive not only the multiplicativity of the determinant but even the Cauchy-Binet formula.

Answer 13

There is a huge gap.

I've always been interested in this question from the point of not just the "hermeneutics" of mathematics, but also from the standpoint of motivation for mathematicians. I wonder to what degree doing mathematics is constructing a mental model for a mathematical object, comparing the properties of that model to the facts associated to the object and then trying to reconcile the model with the facts? Some personal examples: I often read a definition, and then try to write the definition in an equivalent form in my own words...based on whatever vague impression/model the definition inspired. After this, I compare my definition with the original and then try to see what in my conception/mental picture needs altering. Another thing I commonly do is to read the statement of a theorem and try to prove it for myself...however I am not simply trying to prove the theorem formally, but am trying to build a conceptual construct or point of view that will make the proof of the theorem evident in light of that construct. Personally, this (often failing) attempt to build a sharp intuitive mental model of a mathematical object is the primary motivation for doing mathematics at all.

This all reminds me of what was (purportedly) written on Richard Feynman's blackboard at the time of his death: If you cannot create it, you don't understand it. One is forced to wonder what constitutes "creation" here.

EDIT/ADDENDUM: In the interview with Alain Connes here there is a great description of internal mental process. This has inspired me to distill that, for me, mathematics is the triumph of concept over brute computation. The uncommunicated mental models that allow us to organize and complete computations and proofs that seem impossible is, the central source of joy and surprise in mathematics.

Paraphrasing Gowers, philosophy of mathematics is useful in that it affects the practice of mathematics. The above viewpoint is liberating as it points at what to spend time trying to do. It is frustrating to watch introductory analysis students stumble around with formalism with no apparent "picture" (not necessarily geometric) of what is going on in a proof. The typical undergraduate seems to spend very little time trying to find a mental model that generates a vivid proof. This may be due to the fact that a fig leaf covers the essential part of what we do!

Answer 14

One facet of the problem is when you replace the word others by yourself. Did it ever happen to you that you find a nice construction, a nice concept, a nice property, giving you a research paper of which you are proud ? Once the exciting period of intensive research ends, you come back and ask yourselves How did I come to find this trick ? Why did I look in this direction ? And sometimes, you just cannot remember How and When. But you did it !! If you can't explain yourself this process, do you expect to explain others we way you think ?

Answer 15

A lot of the discussion going on above is about the fact that we do not understand our mind's working. I do not even know if mathematical thinking is language-bound. I have the impression that it is not: mathematical discussions are more tedious in foreign languages but not more difficult. An analogy is perhaps a walk on a barely visible path. Sometimes you lose it and have to search for it. Glimpsing it again you tell yourself: "Ah, here it is again" but language is completely irrelevant even if it can be used to describe it to a friend. How do other people feel about this? Perhaps we are all different and do not see not the "same" reality. (Is your color "red" the same as my color "red"). Perhaps we are making a huge mistake assuming that different people think very similarly: For a dog, reality is made of a lot of scents, for a horse it is perhaps prairies and vast avenues where it can gallop. Some of us are perhaps "dogs" and other "horses" of mathematics and since most scents and prairies have not been named, we have to communicate through a limited common denominator, even among people of the same "mathematical species".

Answer 16

The well known situation of language translation is I believe akin to the tension between thinking and explaining.

I am French, I can understand, write and explained myself in English yet I am a bit at a loss when required to translate some piece of English into French so I call this translating ability a third language. My guess or feeling or personal view is that the thinking is more semantic and - for weird (and sad) cultural reasons- mathematical explaining is too often required to be on a syntactic level.

Answer 17

I'd like to take the occasion and sketch my view on reconstruction problems in graph theory: I see a graph as a set of subjects with a relation between (some of) them. Each node (= subject) has a limited knowledge of the whole graph. The question is how many of those subjects have to put their knowledge together to know the whole graph. Of course, this depends on the kind of limited knowledge each subject (= node) has. In case it knows everything but its own relations to the rest of the graph, we have Ulam's reconstruction problem. Another natural kind of limited knowledge would be: all relations except those between the most distant nodes. Or: all relations within a neighbourhood of fixed size and nothing else.

I find it enlightening to compare this situation to the case of reconstructing a 3D object from its 2D projections (another kind of limited knowledge).

Answer 18

Somehow my experience, at least these days, is quite different. I do not see much difference between the way I think about things and the way I talk or write about them. Part of the reason is being involved in semi-formal Internet activities like blogs, polymath projects, MO, etc. Another reason is that I think more about conceptual and meta questions (like this one). Yet another reason is that I am quite interested in the process (for me and even more for others) leading to ideas and answers. Also, getting older may have something to do with it: On the one hand, you are less surprised from your own partial-thoughts and unexplained intuition, and, on the other hand, you are less surprised from your half-thoughts and unexplained intuitions.

Answer 19

Professor Thurston,

In my own study, I've struggled with giving meaning to mathematical objects for many years. It's likely I've learned more slowly than most, because I find it difficult to move on unless I've found something I can rely on. Usually, when I try to explain mathematics, I will give the person I'm speaking to a simple problem to give him or her the experience of doing mathematics. Almost always, I will ask to show that $\pi > 3.$ The experience is usually familiar. One person has asked me, "how is this different from finding the answer to any question?"

Another example I like to give people is one of the several graphical proofs of the Pythagorean theorem. The one I present is #9 from this site, since it seems to be the simplest:

http://www.cut-the-knot.org/pythagoras/index.shtml

I've always had trouble explaining things like group without giving simple geometric examples, like the dihedral groups. That the turns of the Rubik's cube is offered as an example of a group is well known. Conjugation, for instance, immediately makes sense. Elohemahab Solomon offered this video

Does A "Connections" Blog/Podcast exist for Math?

where Mr. du Sautoy, (at about 12:00) in the context of groups of symmetry, compares the founding of the concept of group with that of number. To summarize Mr. du Sautoy, the symmetry a group measures is analogous to the quantity number measures. For example, we have chairs and tables, which are different, but if we have 3 of each, the quantity is the same. If we have two walls of the Alhambra with different looking designs, but have the same symmetry, then one group acts on the designs. He discusses $S_3$ and $\mathbb Z_6.$

I accepted complex numbers for a long time, but after experience with math past calculus, and the requirement for rigor slowly pervaded my thinking, I couldn't accept them as more than ad hoc constructions. Finally, I was satisfied with their construction as elements of the field $\mathbb R[x]/(x^2+1).$ I talked about this in my answer to the question:

Demystifying complex numbers

I now think of complex numbers as an abstract construction from things I have intuition about, that is, real numbers and polynomials. The usual interpretations, as vectors in the complex plane, vectors in $\mathbb R^2$ that one can multiply, are still of course, extremely useful in understanding them.

It's likely you know this already: it looks like Hilbert coined the term "ring" from Mr. Gleason's interpretation of cyclic groups.

http://en.wikipedia.org/wiki/Ring_(mathematics)#History

A distribution is something, with whose definition I've worked successfully, but still have little sense of. I know they are generalized functions, but my lack of understanding might just be lack of experience.

Answer 20

The ring ${\mathbb Z}/N{\mathbb Z}$ is usually defined in a rather cumbersome way, and it takes some time (infinite in most cases) before students realize that you can think of it as ${\bf Z}$ with just one added relation $N=0$ to do the computations (and the problem that $xy=0$ does not necessarily imply that $x=0$ or $y=0$), so that it is indeed a very simple object and not some horribly abstract invention.

Answer 21

Thank you for starting this discussion. I think any kind of pedagogical tool should be shared with students and collegues, especially in writing form. When I read "the classics", i.e., works of famous mathematicians, I always wondered the process they went through to reach those conclusions, what imagery went through their heads while they proved a theorem, maybe that would be useful to me, or not, but I always wanted to know. I think during a lecture saying something like "here is how I do it, imagine group elements breaking into a formation organized into circular groups" is no discomfort to anyone. Maybe this explanation can help one student, or two (or all) students to grok the topic just a tad more, and that's still important. Richard Feynman used to say (paraphrasing) that he never really knew in advance how his students would understand quantum mechanics, he did not have any single method, he'd only try to explain the topic from many different angles hoping that one of those angles would provide an entry point for a student, into the subject.

Answer 22

I realize that this question is very old, is arguably off-topic (see Felipe Voloch's comment, though, I do think this question was appropriate as it's part of doing research), and that the OP has sadly passed away. However, I want to point out that there was a project devoted to precisely the OP's original question. I refer to the User's Guide Project, which was launched in 2015 and reassessed in 2020.

The project was the brain child of Luke Wolcott. Over three years, 13 mathematicians wrote a "user's guide" to go with one of their papers. Each user's guide consisted of four sections:

1. Key insights and organizing principles of the paper.
2. Metaphors and imagery the author used while carrying out the research.
3. Story of the development of the research.
4. Colloquial summary aimed at a general (non-math) audience.

The user's guides can be found here. Section 2 is highly related to what Bill Thurston was asking, and indeed this very MO post was cited in the 2015 paper. So, that provides 13 concrete examples right there.

Answer 23

I once read an anecdote of a mathematician who was teaching topology in a class. At some point he got stuck and he quickly drew a diagram in the corner of the blackboard and after studying this for a few seconds, rubbed it out and continued with his formal exposition.

It might have been an idea to explain to his students what he saw in his diagram that he couldn't see in his formal exposition!

This underlines to me, that we actually use many languages rather than just the officially sanctioned language to communicate, whether to others or to ourselves.

Another example is Richard Feynman. He wrote the tan of something as a large T with the horizontal part of T stretched out to overhang the argument. He felt that his own notation was just as good as the traditional - until he realised the importance of public languages over private languages - they enable public communication and to far greater audiences than private languages ever can.

My own Feynman like adventures with notation, having once learned the de Rham calculus was to call the operations of vector analysis - grad, curl and div by cograd, cocurl and codiv because they operate not on vectors but covectors; or rather, cofields, rather than fields; and which led me to ask - should it be a cofield of covectors, a field of covectors or a cofield of covectors!

On a more personal level, I struggled with understanding a universal definition when I first came across it in Category Theory. It simply looked gnomic. The name didn't help, since it is about characterising some mathematical object by its properties and hence might have been better named a characterising definition (though of course one can always argue that any definition ought to be characterising!)

I also had trouble with tangent bundle and functor. Eventually I understood it geometrically exactly as a functor rather than how it's built up though stages. I find that intuitively much more plausible. As an additional benefit, it made me appreciate category theory since all this can be easily encapsulated though t: T->1, a natural bundle according to Michor, Kolar and Slovak.

My own trouble I had with group theory was not understanding what a group was or is; but what it was for; here, the enlightening moment came from gauge theory when I could see the point of Lie Groups and also Lie Algebras, the tangent space at the origin. Given that originally I was interested in physics, rather than mathematics - this was perhaps only natural. This , moreover shows that meaning doesn't arise solely from a topic per se, that is in its own universe of meaning, so to speak; but also when it bridges topics, illuminating both and more. This ought not to be a surprise since knowledge, know-how and the understanding are a unitary phenomenon - no matter how we might artificially divide fields.

My personal theory of what constitutes thinking on the mathematical sphere - as well as others - derives from Plato and his notion of divine madness, which has nothing to do with madness per se, but that it's a kind of gift from the spirit (pragmatic thinking is thinking on the horizontal plane - as in walking along the ground; and creative thinking, is one on the vertical scale - that is ascending towards the sky of ideas. I'd point out here, that Parmenides, who is often credited with beginning the scientific revolution in the West began his philosophical/mystical poem with an ascent to the heavens, so to speak; in our own predominantly scientific, technological and secular culture this aspect is ignored compared to what he was saying discursively about the way of truth, being and motion); and also from Simone Weil, the sister of Andre Weil, who thought of thinking as knowing how to pay attention to things. She said that this is one of the key things that geometry teaches us, even if we are - in the end - no longer interested in geometry - that is, have moved on. Personally, I suspect that she derived this from Plato as well, and his theory of dialectical thinking.