
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.

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.



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.



*Maybe it seems frequent to my friends.



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 $Spec(\mathbb{Z}[x])$, but interesting equations even in two variables will conjure up a more complicated image of an arithmetic surface fibered over the 'arithmetic curve' $Spec(\mathbb{Z})$. 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 $$P:Spec(\mathbb{Q})\rightarrow X$$
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.



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
$$P:Spec(\Bbb{Z})\rightarrow E.$$
That is, spaces like
$Spec(\Bbb{Q})$  and $Spec(\Bbb{Z})$ are problematic and, after all, are quite radical.

In $Spec(\Bbb{Q})$,  one encounters the absurdity that the space $Spec(\Bbb{Q})$ 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.



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.



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