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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.

I didn't want to bump the question for this, but Gil Kalai's question "Taking lecture notes in lectures" http://mathoverflow.net/questions/12638 reminded me of Thomas Körner's essay "In Praise of Lectures" http://www.dpmms.cam.ac.uk/~twk/Lecture.pdf which could be taken to be part of an answer to this question. In general, I found the answers given by others to Gil Kalai's question to be interesting in light of your question as well.

– j.c. – 2010-11-05T19:18:46.057@CamMcLeman, that's probably a description of

matricesrather than of linear transformations, right? A linear transformation isn't allowed to know which vectors you've called basis vectors, so it has to communicate with all vectors on an equal footing; whereas a matrix isforcedto respect the chain of command that you describe. – LSpice – 2017-06-09T22:47:07.9601@LSpice Sure, fair point! The memory of how it was phrased specifically has now quite faded, but it was indeed an elementary linear algebra class, so the matrix perspective was probably forefront. – Cam McLeman – 2017-06-11T01:24:07.973

146What I am really thinking is that, if you had posted this anonymously, this question would have been closed in five minutes. Back to the question: I'd not have trouble saying what I think about some math, except I often have a complicated mental picture which is impossible to convey. – Felipe Voloch – 2010-09-14T02:22:19.607

44Just for the record, many people do use the vivid internal analogies they have when conversing with students. One of my favorites from undergrad was describing a linear transformation as a commander-in-chief, who told the generals (a basis) where to go, who in turn tells all the soldiers (the rest of the vectors) where to go. The chain of command in action in a linear algebra class. – Cam McLeman – 2010-09-14T02:24:52.613

5I'm making this community wiki since this is indeed a canonical soft question. – François G. Dorais – 2010-09-14T02:31:08.207

29(By the way, I really think this is a great question!) – François G. Dorais – 2010-09-14T02:32:56.690

29@Felipe Voloch. Yes, I too realized that I could get away with more than most people. But that also means that mathematicians sometimes say less of what's on their mind to me than they might to someone less intimidating to them.

The phenomenon of the complicated mental picture is exactly what I'm asking about. I can't expect you to convey the picture except in general terms, but just to talk more of its role in your mind, so why don't you write an actual "answer". @Cam McLeman: that's exactly what I'm asking for. Why not turn it into a full answer? – Bill Thurston – 2010-09-14T02:49:02.387

5Well, when I'm really tired, I sometimes think one thing, but say another thing, often completely unrelated (i.e "homomorphism" becomes "holomorphic", you get the idea -- it can get particularly embarrassing during a talk, when what I write on the blackboard is consistent, but what I say absolutely non-sensical). I wonder if this is the outcome of working with a lot of formula's and few concepts (my field is analytic number theory). I end up thinking more visually, in the sense that I visualize formula's, but rarely pronounce many (math) words, in my head. – anon – 2010-09-14T03:59:41.103

2This is an excellent question! – 16278263789 – 2010-09-14T04:04:07.800

6Gosh, changing backticks to quote-marks gains me 23% of the question! (Bill, backticks are special characters so shouldn't be used as quotation delimiters). This is definitely borderline

on MOfor me (with the qualifier thatas a questionI think it's great), and the "real world" reputation of the questioner is one factor in my decisionnotto vote to close. Not because I don't want to offend him, but because I think that people answering will give better answers because they know who's asked the question. – Loop Space – 2010-09-14T07:24:04.29056Felipe is absolutely right: posed by somebody else, this question would have been closed within a picosecond. The rationale, ironically, being that professional mathematicians of quality would flee this site in droves because the question is "discussiony", "vague", "has no right answer", etc. This is the Matthew effect in all its glory: "For to all those who have, more will be given, and they will have an abundance; but from those who have nothing, even what they have will be taken away" (Matthew 25:29) Needless to say, I'm euphoric at the thought of Bill's participation in MathOverflow. – Georges Elencwajg – 2010-09-14T07:46:53.913

13@Georges - I agree with you to some level, but then again it is very rare that a new user will ask a soft question that a) explains why it is interesting (at least to the OP) b) is written in a clear manner with illustrations and examples, showing that effort really was put into the question. I agree that this is a borderline question, but most of the time, when I vote to close on a soft question it's because of the lack of (a) and (b). – Gjergji Zaimi – 2010-09-14T12:07:29.460

7@Georges - I don't see any irony. I see rather the swallow effect: "One swallow does not a summer make". That

oneprofessional mathematician of outstanding (irl) reputation posts a soft question does not mean that we should allowallsuch questions. The argument is thattoo manyof these questions will make MO useless for research purposes andthatis what will drive people away. Zero tolerance is unachievable, so even us hard liners sometimes let the odd soft question through. Very few are as clearly written and well-motivated and focussed as this one. – Loop Space – 2010-09-14T13:53:19.80315FWIW, I +1'd this question before I saw who asked it... – Vectornaut – 2010-09-15T16:16:18.933

3I agree with Gjergji. It is very rare for a new user to put this much effort into asking such a question. – Qiaochu Yuan – 2010-09-15T22:06:06.590

5

Have you seen Barry Mazur's article on this? http://www.math.harvard.edu/~mazur/preprints/Delphi.pdf

– Peter Arndt – 2010-09-17T14:00:14.727@Peter Arndt: No, I hadn't seen it. Thanks. – Bill Thurston – 2010-09-17T15:14:30.180

to @Bill : Could you (under the same format) asked the complementary question : about "understanding and thinking". How do we grasp a mathematical concept? What are the typical scenarii that each one has when/before hitting a point...

It would be nicer ( if you are interested in the question) to have those two questions asked in the same mood.

Note that some of the answers/comments to you question where in fact flirting with this aspect.

Note : THIS IS META !! I see it as wrong to put it in an answer. – Jérôme JEAN-CHARLES – 2010-10-01T01:11:31.973

@Jérôme, Thanks for the idea. I'll think about it and perhaps post another question.

I wish more people would write down their responses. I suspect there are people who are interested, but shy. Do you have any suggestions? – Bill Thurston – 2010-10-06T12:16:31.343