B Do you visualize equations abstracly when doing math?

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Do you visualize equations abstracly when doing math?
 

BvU

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Yes and no


:cool:
 

fresh_42

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This answer
Yes and no
to the question
Do you visualize equations abstracly when doing math?
is actually way better than it appears at first glance. What sounds ironic is indeed what takes place.
Would you care to elaborate? :biggrin:
I visualize ##c^2=a^2+b^2-2ab\,\cos(\gamma)## but do not visualize ##\Gamma^°(\mathfrak{sl}(2,\mathbb{R})) \cong \operatorname{PSL}(2,\mathbb{R})\,.##
I even have an imagination for $$\langle \,u,v\, \rangle = 0$$ but do not for $$\langle \,f,g\, \rangle =0$$
 
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BvU

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Sure. Am I free to interpret your question ? It's not very specific, you know :wink: ...

I see terms and factors and mentally visualize them moving about (the equivalent of 'doing the same thing left and right'). I also 'see' factors popping up that may or may not come in handy (e.g. a factor ##(1+x)## when there is a ##(1-x)## in the expression).

A ##\sin## elicits a ##\cos## or a ##d\cos##, a ##{1\over 1+x^2}## an ##\arctan## etc. etc.

Matter of ingrained / internalized dealing with lots of exercises in my youth.

If you are interested, things like ##\nabla\times \bf \vec E## are the things that cause me the greatest difficulty -- no associations at all. Idem vector potential, an awful lot of thermodynamic differentials ..

What did you have in mind when you posted your question ?
 

BvU

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What did you have in mind when you posted your question ?
Well, I have some difficulty understanding formulas and equations because I can't imagine what they mean: I need a 'mental picture' and have some difficulty contructing it. Math costs me energy for that reason. So I was wondering if I am just doing something wrong or that mathematics is not the game for me... Also, I'm very curious how other people and mathematicians process math :smile:
 

DrClaude

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but do not visualize ##\Gamma^°(\mathfrak{sl}(2,\mathbb{R}) \cong \operatorname{PSL}(2,\mathbb{R})\,.##
I visualise that you forgot a parenthesis in there :wink:
 

BvU

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Funny, I'm allergic and hypersensitive wrt printing errors, yet I missed that one ! Must have the same angle as @fresh_42
 

BvU

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Well, I have some difficulty understanding formulas and equations because I can't imagine what they mean: I need a 'mental picture' and have some difficulty contructing it. Math costs me energy for that reason. So I was wondering if I am just doing something wrong or that mathematics is not the game for me... Also, I'm very curious how other people and mathematicians process math :smile:
Wouldn't worry too much. It doesn't look as if you intend to be the one to solve grand unification or quantum gravity.
Try to find out what your strenghts and weaknesses are in the learning process: exploit the first as much as you can and challenge the latter now and then, just to see how it feels -- you don't have to master everything.
 

fresh_42

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Wouldn't worry too much. It doesn't look as if you intend to be the one to solve grand unification or quantum gravity.
Try to find out what your ...
Funny, I'm allergic and hypersensitive wrt printing errors, yet I missed that one ! Must have the same angle as @fresh_42
... strenghts [strengths :wink:] and weaknesses are in the learning process: exploit the first as much as you can and challenge the latter now and then, just to see how it feels -- you don't have to master everything.
This is another good advice. Math doesn't equal math. The way of thinking and visualizing can be very different depending on the specific field. Here's a list of which I think does have different points of view and therewith different ways of anticipation:
  • Geometry
  • Analysis
  • (Abstract) Algebra
  • Stochastics
  • Numeric
  • Theoretical Computer Science
  • Number Theory
  • Algebraic Topology
Sure they all have intersections and other fields are closer related to one of them than to the rest, as e.g. measure theory to analysis, or linear algebra to geometry, however, people which are good in one aren't necessarily good in all others, too - exceptional mathematicians like Tao aside.
 
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BvU

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strenghts [strengths :wink:] -- put that one down to excitement: my 2 fingers (!) can't keep up
 

fresh_42

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strenghts [strengths :wink:] -- put that one down to excitement: my 2 fingers (!) can't keep up
Lol. Karma is a ... (But to be honest, the spell checker underlined it, for otherwise I wouldn't had recognized it.)
 
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With respect to seeing math, there are some people who suffer from or enjoy having math synthesia where they visualize numbers as colors.




You can read more about it here:

https://www.theatlantic.com/science/archive/2016/07/whats-it-like-to-see-ideas-as-shapes/492032/

and more technically here:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3827548/

Personally, I like to visualize math and so I'm drawn to things like all kinds of geometry, origami, vector analysis, tensor analysis... and am less interested in non visual math unless I can find a way to visualize it.

As an aside or example of visualizing hard concepts, BetterExplained has a good discussion on ##\nabla\times \bf \vec E## using a small paddle wheel placed into the vector field.

https://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/
 
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Here's another story of a man who was brutally assaulted and came out of it with a most remarkable math skill:

 
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Well, I have some difficulty understanding formulas and equations because I can't imagine what they mean: I need a 'mental picture' and have some difficulty contructing it. Math costs me energy for that reason. So I was wondering if I am just doing something wrong or that mathematics is not the game for me... Also, I'm very curious how other people and mathematicians process math :smile:
I should refine that: I can "see" the "workings" of variables in the equation like the ones in the nominator contributing to increasing growth and the ones in de denominator contributing to decreasing it, except if the values of the variables shrink under 1, in which case we have the reverse. I have difficulty imagining the effect in an equation of a square root because most of the time it contains a whole equation in itself. And I know the deravative of ex is itself so it grows proportionally to its own value, which is relatively easy to imagine. And so with a derivative, we have the speed of growth also.

If you rework an equation I can follow that, but each reworked version of the equation, although it is really the same equation, looks slightly different and suggests to mean something entirely different altogether.

If I imagine for instance a function of two variables, I imagine a landscape of values in two dimensions. If I imagine a function of three variables, I imagine a variable landscape of two variables. If I imagine a function of a function, I imagine the inner function dictating the speed of change of the outer function.

But generally, when I try to imagine an equation as a whole, I seem to have some difficulty doing that. Although on occasions I succeed in that. So my question: do some of you recognize this sort of thing with themselves?

I have to say I haven't practiced math in a long time, and it is obviously not my metier.
 
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BvU

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some of you recognize this sort of thing
yes, and I do math daily. What helps me (visually oriented) enormously is grabbing a pencil and sketching partial dependencies. Nowadays taking excel and plotting a function is a piece of cake, so that's the next (emphasis: not the first) step.

For physics estimating orders of magnitude, sorting out which is the dominant contribution, taking extreme cases etc. is very helpful too. Not so visual, though.

Visual is making a sketch of the core of the problem. Helps to sort out main issues from details and find a workflow.

Here in PF often I have to ask 'did you make a sketch?' repeatedly to posters who can't find an inroad to an exercise.
 
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I don't know what an abstract visualisation of an equation would consist of. I sketch diagrams so I would potentially have a better idea of the problem at hand, but I'm not sure if it qualifies as abstract. I might also draw parallels between the abstract and something familiar (or look for ways to do so). For instance, a certain semi direct product of (semi)groups (read: very very abstract notion) can be visualised as the Rubik's cube.
 

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