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.
Best Answer
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.).