# What subfields of mathematics better lend themselves to visualization?

In some of my classes (e.g. graph theory, mechanics), the professors encourage the students to visualize solutions to problems; I do well in these classes. In other classes (e.g. linear algebra), we are encouraged instead to reason about abstract concepts; I usually do worse in these classes (relative to the same set of peers). My professors have corroborated my suspicion that the usefulness of visualization depends on the type of problem being solved.

Overall, it seems that I do well with visualization, but have a harder time reasoning about symbolic formulas or abstract entities.

Have people here found that certain areas of mathematics better lend themselves to specific types of mathematical reasoning? I am especially interested in identifying more subjects in math & science that emphasize visual reasoning since I very much enjoy these subjects.

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As a practicing mathematician, my opinion is that mathematical reasoning is certainly a collection of subskills. But I think your question is off-topic for MO: to get anything like a correct answer, you should rather be asking a math educator, a psychologist or a cognitive scientist. –  Pete L. Clark Jul 6 '10 at 4:59
Good point, Pete. I edited the question to put more of a specific focus on the mathematical subfields themselves. I'm more interested in the subjective experience of professional mathematicians than the cognitive science behind it, hence why I think there's value in asking this audience. I'm just looking for practical recommendations as I decide what fields to explore in mathematics. Let me know if I can edit it further to make it more appropriate. –  RexE Jul 6 '10 at 5:06
@Charles: Done. –  RexE Jul 6 '10 at 8:42
I am not at all convinced that e.g. linear algebra does not lend itself to visualization. I (and I think many others) test intuitions by drawing pictures of $\mathbb{R}^2$, and in general I tend to think of linear transformations taking the unit ball to a (possibly degenerate) ellipsoid, with axes given by eigenvectors, etc. One proof of the spectral theorem for compact self-adjoint operators on a Hilbert space proceeds via this intuition--one finds $v$ on the unit ball with $||Tv||$ maximal, and proves that that this is an eigenvector. –  Daniel Litt Jul 6 '10 at 13:48

The classes that lend themselves to visualization certainly include graph theory and mechanics, and I expect you would do well in classical geometry too. In other cases, it may depend on which book you use. For example, I am sure you would enjoy the approach to complex analysis in Tristan Needham's book Visual Complex Analysis (Oxford University Press 1997), whereas you may find other complex analysis books less friendly.

By all means use your strength in visualization to get a foothold in various areas of math (this should also be possible in linear algebra), but try to develop other strengths too -- you will need them eventually.

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+1, mainly for the 2nd para (though the suggestions in the 1st para seem eminently sensible to me) –  Yemon Choi Jul 6 '10 at 8:01

Since geometry and algebra are often discovered to be two sides to the same phenomena, I suggest that you develop your geometric intuition to understand algebraic phenomena. A typical example here is to use the abstract tensor formalism to understand Hopf algebras. From my personal experience, Hopf algebras did not come alive until I understood that the axioms could be drawn as little bits of string. When I listen to a Hopf algebra talk now, I try to envision the proof via these diagrams.

This leads me to a second point which no one yet has suggested. knot theory is an inherently visual subject that is easily entered via geometric intuition.

To truly make progress as a research mathematician, you may have to also develop tools for symbol manipulation. Geometry can always be a guide to discovering the formulas. Knot theory, abstract tensors, linear algebra, and group theory all are easily approached via geometric techniques. Many of us delight when an arcane algebraic concept is reinterpreted as a geometric one.

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Hopf algerba axioms as string sounds intriguing... any chance you've got a reference? (For some reason searching "hopf algebra string" on google isn't terribly productive :p) –  Tom Boardman Jul 6 '10 at 19:11
I am reminded of a remark made by the interlocutor Simplicio in Galileo's Dialogues concerning Two New Sciences: "Indeed I begin to understand that while logic is an excellent guide in discourse, it does not, as regards stimulation to discovery, compare with the power of sharp distinction which belongs to geometry." –  grshutt Jul 6 '10 at 21:29
@Tom This is certainly not a primary source but see <a href="arxiv.org/abs/0705.3231">; this paper </a>. The first place that <em> I </em> learned about it was in a <a href="arxiv.org/abs/math/9201301">; paper</a> by Greg Kuperberg. See also S. Majid. –  Scott Carter Jul 6 '10 at 21:34
ARRGH! I thought I used standard html to indicate those: arxiv.org/abs/0705.3231 and arxiv.org/abs/math/9201301 –  Scott Carter Jul 6 '10 at 21:36

To start with something of an anti-answer, when I took courses in algebra (not linear algebra - I mean groups, rings, modules etc.) or representation theory as an undergraduate, I found it practically impossible to get anywhere by trying to visualize what was going on. I suppose, concurrent with other answers, the important thing is that I understood the material some other way.

By contrast, and as you say, some parts of combinatorics are certainly very visual. And (obviously?) topology (not so much first-course point-set stuff, but the real stuff) and differential (at least) geometry can both be very visual subjects. It can be a lot of fun trying to find ways to use geometric inituition to attack something that is ostensibly out of reach visually (e.g. in 4 dimensions or something not embedded in $R^3$ etc.)

At the moment I'm interested in geometric analysis, where I have come across some of the most pleasingly visual things yet.

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I can see why your professors would be keen to lead you away from false intuitions based on visualization, but even false intuitions can be useful as they often suggest a possible proof (which you then find doesn't hold in general) so you can at least get started on a problem.

For me linear algebra is a `visual' subject - there are nice geometric interpretations of linear transforms (preserve lines, parallelism, intersections, etc.) that I personally find extremely useful in my work.

You asked for practical advice - here is mine: Do what works for you to solve problems, do what your professors say to pass the course.

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As you pursue this topic, I think you will discover that the distinction you make between visualization and reason is specious. Consider, on the one hand, the way in which Courant and Robbins in What is Mathematics? use sets of dots in rectangular boxes to explain the laws of the arithmetic of integers, and on the other hand, the interesting remark made by the logician Dale Jacquette: "Like many another logician, therefore, I can report that I was drawn to study logic in part by the 'beauty', or, as I prefer to say, the absorbing visual interest, of logical syntax" (Masses of Formal Philosophy 59).

You may be interested in four "experiments" Timothy Gowers conducted on his WordPress blog to gain some insight into how people think when they are doing mathematics.

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If you can excuse the flippancy, one might define category theory as algebra that benefits from two-dimensional equations.

To be more serious, sketches in category theory, or the various classes of visual inference in logic, such as Peirce's existential graphs, show that visually represented reasoning can be quite as rigorous as textually represented reasoning.

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The fact that you are good in visualizing doesn't necessary mean that you should ignore fields where visualization is less important. If you will go into areas where visualization is very important, then you'll be surrounded by other people with such skills and you might lose you relative advantage. But in a field where visualization is used less frequency, you could have an advantage in problems where visualization is required, and in most areas of mathematics visualization is useful to some extent.

In any case, as you are in the beginning of your career, you should try to obtain a broad basis rather than specialize in some direction.

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