What subfields of mathematics better lend themselves to visualization? - MathOverflow

What subfields of mathematics better lend themselves to visualization?

2010-07-06T04:36:02Z

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.

Answer by Bob Durrant for What subfields of mathematics better lend themselves to visualization?

2010-07-06T07:45:18Z

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.

Answer by John Stillwell for What subfields of mathematics better lend themselves to visualization?

2010-07-06T07:52:52Z

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.

Answer by Charles Stewart for What subfields of mathematics better lend themselves to visualization?

2010-07-06T08:22:32Z

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.

Answer by grshutt for What subfields of mathematics better lend themselves to visualization?

2010-07-06T08:47:40Z

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.

Answer by Spencer for What subfields of mathematics better lend themselves to visualization?

2010-07-06T10:17:18Z

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.

Answer by Scott Carter for What subfields of mathematics better lend themselves to visualization?

2010-07-06T13:45:27Z

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.

Answer by Yiftach Barnea for What subfields of mathematics better lend themselves to visualization?

2010-07-06T15:29:03Z

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.