Excellent post.

In my experience, algebraic thinkers absorb information much faster than visual thinkers, but they more often make silly conceptual errors that visual thinkers don't make. For example, an algebraic thinker might accidentally add a vector to a scalar, since their symbols look identical on paper. But a visual thinker would be much less likely to do this, since their visual representations for scalars and vectors would likely be so distinct.

A rule of thumb: when short-term speed is crucial, think algebraically. When long-term understanding is crucial, think visually.

One problem with algebraic intuition is that it leave ideas "unhooked" in your mind. I mean this in the following sense:

> While you are leaning things you need to think about them and examine them from many sides. By connecting them in many ways with what you already know.... you can later retrieve them in unusual situations. It took me a long time to realize that each time I learned something I should put "hooks" on it. This is another face of the extra effort, the studying more deeply, the going the extra mile, that seems to be characteristic of great scientists. -- Richard Hamming

Algebraic proofs are stored as symbolic/syntactic movies in your head. But syntactic movies resemble other syntactic movies, causing algebraic proofs to blend together with all of the other symbolic/syntactic theorems. Visualizing proofs, on the other hand, makes each theorem significantly more distinguished from each other. You are much more likely to recall and understand important facts this way, in my opinion. You are therefore more likely to apply them in novel ways to solve new problems.

Here Einstein famously describes visual vs. syntactic thinking in a letter to Jacques S. Hadamard:

> (A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined.

> There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought — before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.

> (B) The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will.

> (C) According to what has been said, the play with the mentioned elements is aimed to be analogous to certain logical connections one is searching for.

> (D) Visual and motor. In a stage when words intervene at all, they are, in my case, purely auditive, but they interfere only in a secondary stage, as already mentioned.

A good friend of mine has a PhD in physics, and is a classic algebraic thinker. He is so, so much faster than me. But he once remarked that he forgets the proofs of almost everything he learned in grad school, and has to get back into the concrete exercises to regain his algebraic intuition. Visual thinkers may be slow, but they never forget.