How does a mathematican picture what he's working on?

[Li(n)]

In mathematics, what the expert
sees and does with an image is not what the novice sees, even with the same diagrams. What the teacher
sees is not what the students see. What one student sees is not what their neighbor sees. All of these
differences impact our classroom work with diagrams and visuals.

It is an illusion that mathematical reasoning is done in the brain with language. Standard
presentations of mathematics foster this illusion, but this formal public appearance does not represent
the problem solving, the thinking, the reasoning of many mathematicians. A ballet performance does
not embody the way this performer walks around their home, or the way they practice. Analogously,
what you observe in a mathematics paper or lecture does not embody what a mathematician does while
solving the problem, or when talking with a colleague. It does not match the cognitive processes of the
mathematician, the teacher, or the learner of mathematics.
What students commonly see in a mathematics classroom is also an illusion.

Visual reasoning is not restricted to geometry or spatially represented mathematics.
As an example, combinatorics is very rich in visual patterns and associated reasoning
Even the algebra, and symbolic logic, rely on visual form and appearance to evoke appropriate
steps and comparisons. All fields of mathematics contain processes and properties that afford
visual patterns and visually structured reasoning .

Mathematicians have not developed clear, consistent ways of working with visuals, as we have
with algebra and other symbolic forms. While the larger community has the discipline to agree on
shared definitions and algebraic forms, we continually develop new diagrammatic representations, in
undisciplined ways. This mixes sustainable visuals with good cognitive fit with local eccentricities.
This gap between individual or local practices and shared conventions is an obstacle to effective
sharing and learning.

So the question is : how do you visualize mathematics in your head?

If your answer is no you just don't - then what's the point of not seeing beauty with your own eyes?

If the answer is yes I visualize mathematics in my head - could you describe your experiences and how you do it?Thanks

 

[Jamma]

This is something that I have always felt- even if I'm dealing with a non-geometric problem, I always have "pictures" in my head of what's going on. These pictures are extremely rough, but I think the brain has a way of trying to "geometerise" everything. It's almost impossible to describe this process though, and I will always do so in different ways, even for the same problem but, for example, if I'm thinking about a quotient group, I may visualise an abstract group as some "blob" partitioned into other "blobs" which interact with each other in a "group-like" way.

That probably makes me sound a little mental, but that's because I'm trying to describe a process that's practically impossible to describe.

 

[Obis]

I create mental images for the most important, fundamental concepts. For example, I visualize a bijective function similarly to this :

(....)

||||||||||||

(....)Where the top (....) is some arbitrary domain, the | shows the mapping and the bottom (....) is the range of the function. I find it very convenient to visualize bijective functions from R^k to R, for example, similarly to this :

(....)
(....)
(....)
(....)
(....)

|||||||||||||

(....)

In this case, an element from the domain is a broken line, that goes through the all the top
(....).

This mental image is modified at the |||||||||| part if the function is not bijective, for example.

Similarly, I try to create mental images for various fundamental processes and actions in mathematics. Let's say a permutation of 1, 2, 3. The mental image of a permutation of these three symbols is simply a feeling of shuffling these three symbols in some arbitrary way.

Then, visualizing mathematics is an interplay of visualizing the concepts, applying various actions to them while feeling those actions and seeing what is happening.

 

[homeomorphic]

Being a visual-thinking fanatic, I guess I should give an answer.

How do I visualize it?

I don't know. I just do.

Pretty lame answer, but that's about it. It's hard to say what it's like. It's a lot like what I would draw on a chalkboard. Just figures, often with moving parts. Maybe also feeling pushes and pulls. Sometimes, a sequence might seem like a little animal hopping away.

It often takes a lot of time for me to be able to see the pictures clearly. I sometimes have to keep practicing it, until it becomes easy to see. I think I usually have to practice less these days after so much experience with it. The end result of the process is that many initially non-obvious results become obvious.

 

[chiro]

I agree with homeomorphic.

To me math provides us with a gateway to another kind of sensory perception. With our eyes we are able to represent things more or less in three spatial dimensions, along with a time dimension when you consider spatiotemporal changes.

With mathematics we have a language and a set of tools to allow us to make sense of four, eight, twelve, or even infinitely many dimensions. The tools are still being built, but the ability to have a sensory perception of these kinds of objects is getting a lot more intuitive as we build up results.

After a while you start to think in terms of the language, representation, and usefulness of mathematics because it makes many forms of abstraction found in many other kinds of senses a lot easier.

The best way to get from the non-mathematical to the highly abstract mathematical form of understanding is to use a sensory bridge. Usually we do this by relating a mathematical object, structure, and so on to something that is physical. The physical world is fairly easy for most people to comprehend because we are immersed in it every hour, every day, of every week.

After a while though, the math does have a habit of becoming intuitive which means that at some point you can stop using physical analogs and instead use mathematical analogs, even if they are relatively simple ones.

 

[Stephen Tashi]

I recall that a book about John von Neumann said that some mathematicians are "acoustic" - they think in terms of verbal statements rather than geometric pictures. For example, when von Neumann thought about a problem he would often speak softly to himself. He also could apparently remember almost every document that he every read, but he did not have a general "photographic memory" for non-verbal material.

 

[Oriako]

Obis, I literally do that exact same thing for how I visualize bijective functions or linear combinations.

An example I'd like to share for when I am thinking of a linear hull of a set of vectors.
I imagine two sides separated by a line. On the left region I visualize what the n-tuples of the basis literally look like when you write it out on paper $$(x_{1},x_{2},x_{3},...)$$ floating around sort of with $$(y_{1},y_{2},y_{3},...)$$. I can imagine expending that vector, say x_{3} into its actual coordinates (1,5,7) and then draw lines between the numbers in the ordered pair for the vector x_{3} to the vector y_{3}, while on the right side of "the line separating the two halves" I imagine those vectors in an n-dimensional space. I'm not actually able to visualize higher than 3-dimensional space well, but I just sort of trick myself by thinking of another axis orthogonal to the previous one and get this weird structure that looks a little bit like an alexander horned except with straight lines. On the right side, when I draw a line on the left side between a vector x_{3} to another vector y_{3}, I think of the linear combination of those two vectors on the right side.

To visualize the linear hull of the basis of vectors I sort of run through drawing lines between each of the vectors (taking linear combinations of all of them) and I get on the right hand side a picture of vectors quickly adding together (it seems sort of like a cycle, rotating through additions of vectors). Whenever an addition happens I imagine a certain area of a plane drawn out, and then whenever I've gone to infinity with the vector additions I know that the plane I've drawn out is the linear hull of the vectors. If I take the linear hull of three vectors in R^3 I imagine this as the vectors adding to each other with different scaling and translations in a cyclic manner which resembles a sphere of increasing radius, so I know it covers all of R^3.

I made a figure in like 2 minutes, which doesn't make a lot of sense, but maybe you might get an idea...

http://imageshack.us/photo/my-images/580/visualization.png/

The yellow part is like the plane that is being sweeped out.
-
Here is another method I use for visualizing vector spaces or how linear combinations can "fill a space", I'll use my linear combination plane sweeping method, I made a nice figure here.
http://imageshack.us/photo/my-images/507/planeconstruction.png/
You imagine how to construct an n-1 dimensional hyperplane out of your basis, and then rotate those hyperplanes around whatever dimension they are in. In the figure I was trying to get at how you could visualize R^4 but it didn't really come out right... think about rotating those planes around 4 orthogonal axes, I can't draw it correctly it works a lot better in my head.