How do you think? Logical vs visual?

[MadRocketSci2]

There is often a dichotomy posed between verbal thinking and visual thinking. I'm definitely very very visual. I hardly think in terms of words at all.

However, within the context of math, there are two different mental mechanisms that come into play when I'm solving a math problem. There is the visual understanding where I am at my strongest. Then there is the logical aspect of math - the formal application of operations on mathematical objects.

I am very very strongly inclined to visual understanding of math. In fact, I don't even consider myself as understanding something until I can visualize what the operators are "doing" to the objects. I can apply operators to objects according to the rules without coming up with visualizations of the concept, but then I have no real idea of what I'm doing or what the result means. I can keep going ad nauseum, barring mistakes, but finding my way back to seeing what it means, what I'm doing, is difficult after two or three operations.

The "visualizations" don't even have to be real 3 dimensional pictures (for example, I can be operating on 4+ dimensional objects, but I can tell I'm using that part of my brain, and not the part where I logicaly operate on symbol-strings)


The logical approach seems to be favored by modern formal mathematics. It isn't considered rigourous until there is some logical definition to the concept. It seems similar to me to what you can do with computers functionally operating on strings, and reducible to some Turing problem. I think there was work done on searching spaces of "theorems" derivable from "axioms" using computers.

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How do you approach mathematics? Is your visual reasoning primary to your understanding of the problems, or are you a primarily logical thinker? Do you understand the problem when you can visualize an answer? Or when you have an algorithm that operates on the mathematical objects?

Were any of you extraordinarily precocious in mathematics? Did you achieve this via visual or symbolic reasoning?

How do you think various historical mathematicians and physicists reasoned? Do you suspect that Einstein was a visual thinker, or a symbolic thinker? How about Reimann? Euler? Gauss? Newton?, ect.


Also, if you are strong in one style of thinking, have you ever managed to build the other up to par, and how did you go about it?

 

[MadRocketSci2]

Two examples from my personal experience - I used to struggle with how to deal with random distributions. If you had some general function operating on two random variables z=f(x,y), how do you get z. Then I managed to visualize what I was doing in terms of integrating across contours in the configuration space of the two input variables, the contours being defined by the value of the output variable - and all of a sudden, I got it. Now I know where all the integrals, ect, come from and why they are over what they are over.

Right now, it occurs to me that there is probably an operator that generalizes the cross product for an arbitrary number of dimensions.

The cross product in two dimensions can take two vectors and return a scalar. It can also take a single vector and return a perpindicular vector.

In three dimensions the cross product typically is understood as taking two vectors and returning a third.

However, in 4 dimensions, for that to make sense, it would need to take three vectors to return a vector. If it takes two vectors, it returns a rank-2 tensor standing for the normal plane. (Is it a general rank-2 tensor, or is it restricted to being a bi-vector?). Would 2 rank-2 tensors return a scalar? (Again, do the inputs have to be restricted to multilinear n-vectors?)

This suggests to me a general operator in N dimensions, taking tensors that sum to rank M, and returning a tensor or tensors of rank N-M.

But I'll have to break out formal logic to work out the details of how this operator behaves. PS - I know it has probably already been explored and thouroughly mapped out already. It's just new to me.

 

[Gerenuk]

I'm not entirely sure what you mean by the difference between "visual" and "logic".

In any case at the beginning you mention "verbal". I think verbal is definitely the wrong way to go. The human mind cannot multitask and I believe one can either do verbal or maths. Many student think in terms of words and then I'm not surprised that they cannot grasp Maths since they constantly keep distracting themselves with words.
Actually doing symbolic rearrangements is visual thinking as one pictures how to juggle around the symbols!

You might be referring to the difference between visually understanding an expression and purely taking an expression as a set of symbols with rules.
Visual thinking only works where you have some concept of space (vectors, planes, integrals in planes). But my opinion is that even then one should use visualizations only to translate a problem into symbolic maths. From then on it is much easier to blindly apply the mathematical rules for transformations and in the end translate the result back into pictures. It is very tough to picture statements from logic or 3D space at each step.

I would be very careful with applying concepts from visualizations to higher dimensional space. I haven't looked at the precise definition of the cross product generalization you mention, but I would feel most comfortable with a definition that says something like "the dot product of the cross product with its components is always zero" (plus orientation). With symbols I can be sure not to misuse analogies.