What are the dangers of visual exposition of mathematics?

Question

I've heard several times (such as this one) that it's dangerous to learn/prove/teach mathematics through images. I've also read somewhere that showing mathematics through images helps one's intuition because we understand better through images, due to our long date use of the vision sense. I can conceive that this information should be half-true - there might be things that can be taught with images and things that can not. From here, I have two doubts:

1. What can and what can't be learned/proved/taught with images and why?
2. Using mathematics this way isn't the same as geometrical thinking?

I'm also open to books/articles on this topic.

Answer 1

Mathematical arguments are of a formal nature and should be independent of any visual representations. Beginners might have problems seeing the distinction between a valid and complete proof from axioms and definitions and something that "looks obvious".

But this does not mean that geometric intuition is useless or that one cannot use it. After some training, one can often judge whether some visual argument can be made into a formal argument or not.

This makes for good reading on the topic: Terry Tao, There’s more to mathematics than rigour and proofs

Answer 2

The benefits of using images to illustrate a proof are plain $-$ complicated arguments and endless strings of symbols can be very difficult to understand and pictures can help to give a picture of what's happening. With suitable illustrations you can get intuition for what's going to happen next, and so on.

However, there are a few fairly compelling reasons not to rely on pictures exclusively, a few of which I have listed below.

• By using pictures we are very much confined to the so-called real world. Sure, we can get lots of intuition for low-dimensional topology by drawing tori, but this isn't going to get us far if we want to study $3$- or $4$-manifolds, let alone manifolds in higher dimensions.

• As category theory has done such a good job of showing over the last half-century or so, there are deep-rooted connections between all sorts of mathematical objects and fields which are not at all obvious. If we limit our understanding of certain objects and results to what we see in pictures, then these connections are likely to go unnoticed.

• Mathematics is all about abstraction. By using pictures as a means of doing maths rather than merely illustrating it, we are working exclusively in the concrete. Just because a result holds in one picture I've drawn, why must it hold in all such pictures? Can it hold in other settings? Must we draw pictures of all such settings in order to prove this result? Etc.

• Using pictures to do mathematics relies heavily on the accuracy of the pictures, as illustrated in the link in your question.

It is not fair to say that using pictures is the same as geometrical thinking. It might assist geometrical thinking, and it might even form the basis of geometrical thinking, but this thinking is not the maths itself. The maths is what follows, when the intuition is translated into a formal argument.

This isn't to say that pictures shouldn't be used at all. In fact, not using illustrations at all may be almost as damaging as using them exclusively! I mentioned category theory, and I can't say how much harder it would be to understand if we didn't use commutative diagrams.

Answer 3

I will not try to pretend to be an expert.However, as a student I feel that 1) Intuition(particularly geometrical) is essential to or helpful in understanding a problem and coming up with a solution.

Yet, a formal proof is required to prove that your thinking is correct and logical.

Answer 4

One topic where pictures are very useful is in that of van Kampen diagrams in combinatorial group theory, to show one relation is a consequence of others. Here is a diagram

which shows (as I leave you to work out exactly!) that the relation $x^7$ is a consequence of the relations

$$r= x^2yxy^3,\;\; s= y^2xyx^3.$$

The $2$-dim picture is very important, and from this one can work out a linear expression of $x^7$ in terms of $r,s$: this is discussed around p. 72 of the book Nonabelian algebraic topology, from which the above (non original) diagram is borrowed. A web search on "van Kampen diagrams" gives even more elaborate examples, and more explanation.

The assumption that mathematics is truly kosher only when written on a line fails to live up to what is going on in higher dimensional algebra, where completely valid arguments often need to put in more than one dimension. One then gets the usual problem of how one writes down what is essentially a $3$-dimensional argument on a piece of paper!