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This is a soft question concerning the mental processes at work when doing maths. I hope this question is not too vague, and I believe I haven't found a similar one previously posted on MathSE.

In some of the answers I received on previous posts, I've been confronted to very abstract concepts which I find difficult to grab, because I don't see any physical meaning to them. In the same way, Grothendieck's programmes in his biography, which I read once out of curiosity, leave me completely puzzled.

My question is : do mathematicians have a concrete/visual mental picture of the (sometimes very abstract) concepts they manipulate, which can help them finding new leads/paths/theorems ?

(For example, in this interview (in French) of Benoit Mandelbrot, he says that during his math education, he realized he could turn any math problem (even algebra or arithmetics) into a geometrical problem, and that was his way of solving it.)

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Some do, some don't. I don't think you'll get an accurate answer that applies to all mathematicians. –  Qiaochu Yuan Sep 7 '11 at 8:18
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Depends very much on the mathematician. I don’t visualize in any literal sense, but I do have mental models of some kind for some things. –  Brian M. Scott Sep 7 '11 at 8:20
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Personally I don't think mathematics would be anywhere near as far or large as it is today without mathematicians generally having a minimal mental "picture" of what it is they're doing. Rearranging symbols like mere puzzle pieces (what I think some people view math as close to being) is outpaced by creative conceptual insight. Of course, if you're asking if all of mathematics can be understood pictorially in 2 or 3-dimensions as we normally "see" things in our visual consciousness, I say no: this is much too narrow to capture the form of deeper or more sophisticated math. –  anon Sep 7 '11 at 8:37
    
@AlexPof I had a real analysis lecturer saying that visualisation has its limitation in mathematics, e.g. maybe you can draw open balls and visualise the meaning of an open set but then what happens in higher dimensions? –  fpqc Sep 7 '11 at 8:40
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2 Answers

I am not a professional mathematician, but there are some things from experience that I can say. Obviously these will be open for debate.

For example say you would like to prove the second isomorphism theorem. You go about doing some exercises like that if $H,N \leq G$ and $N \unlhd G$, then:

$(1) H \cap N \unlhd H$,

$ (2)HN \leq G$, so on and so forth. $HN = \{ hn : h \in H, n \in N\}$.

Finally you want to prove that $(HN) / N \cong H / (H \cap N)$. The way I tried was to draw some diagrams, trying to see things like

"If $X$ is isomorphic to $Y$, and $\phi$ some surjective map from $Y$ to some other group then the quotient..."

So you play around with this for a while, but maybe you need to construct an example of a surjective homomorphism from one group to some quotient. Then you may need to move to more "concrete" concepts of cosets, how multiplying elements works, checking if maps are well defined, etc.

The point is that you need to be able to work with both the abstract and concrete. For me sometimes when things get too abstract, I try to do as the above, construct some example that perhaps involves some computation to illustrate concepts.

Lastly, try proving the following statement "concretely" or "abstractly" (I will explain below) :

"Given a matrix $A$, the dimension of its row space is equal to the dimension of its column space."

By "concretely" I mean considering a matrix and its row-reduced echelon form, taking into account things like pivot variables, etc.

By "abstractly" I mean considering orthogonal complements and rank-nullity.

I hope the examples I gave above helped!

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@AlexPof Now what I meant to say actually was that you work on the specifics (particular example) to see how say the first isomorphism theorem works. It does not mean you should just prove a specific case of it! Pure math by the way is very hard, the level of abstraction that one has to get through first before trying to solve a problem (I am referring to Algebra) is a lot. I even doubt many people in my class know what is going on, or the full implication of the theorems they are proving. –  fpqc Sep 7 '11 at 22:24
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My question is : do mathematicians have a concrete/visual mental picture of the (sometimes very abstract) concepts they manipulate, which can help them finding new leads/paths/theorems ?

Speaking from my experience, I think most have.

Whenever I doing differentiation, I always imagine "zooming" into the graph, to see it's locally a plane... With this intuition I discovered many laws about differentiation (and other geometric stuff) before I learned about them. The latest example was the formula for the total derivative, so when you have a function $f(t) = g(x(t), y(t))$ and you want to get the $df/dt$.

When working in 4 higher dimensions or higher in the simplest cases making Schlegel diagrams (so in 4 dimensions simply dividing dividing $x$, $y$ and $z$ with $w$ ans imagine the resulting picture) or simply projecting into 3D, by omitting coordinates often work. But more often it's not working. In this case I try to find dimensional analogies. For example no one can directly imagine 4 dimensions. But still we can measure distances, because the formula in 1D is $\sqrt{\Delta x}$, in 2D it's $\sqrt{\Delta x + \Delta y}$, in 3D it's $\sqrt{\Delta x + \Delta y + \Delta z}$. One can easily continue it in 4D and have the formula $\sqrt{\Delta x + \Delta y + \Delta z + \Delta w}$.

When you become accustomed with the concepts, you will need to imagine less, and simply use the laws you learned. Then you will develop an intuition when you can simply "feel" where is the right path, or whether the results you got makes sense or not.

In some of the answers I received on previous posts, I've been confronted to very abstract concepts which I find difficult to grab, because I don't see any physical meaning to them.

It depends on the concept. Sometimes you simply cannot connect it to a physical meaning for example: prime numbers.

Also the problem I see is that writing about math in understandable way requires good writing skills, that only a few have.

So when learning something new I also struggle a lot with papers/textbooks where definitions and theorems work with whole fields, whole sets, $n$s and $m$s. The definition and theorem is impenetrable, no one can nitpick. But I think accompanying it into a plain English sentence would greatly increase the chances to understand and see the concept, even if it's not precise enough.

For example: matrix determinant has a difficult and artifical definition, but telling something about parallelogram areas and parallelepiped volumes would help a lot.

Or another example: a the graph of a differentiable function generally don't have cusps or edges.

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