[chezlaw]

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I'm guessing from your post chez, that you are an undergrad mathematics student. I will agree that not much geometric intuition is required for a typical first or second course in abstract algebra. However I think that when it comes to looking for new results, there has to be some other intuition to guide the algebra.

Its only my opinion but I think that mathematics in the second half of the 20th century was hampered by an obsession with abstract algebra (category theory being the worst example of this obsession). It is only recently that mathematics has started to return to its geometric roots. People are starting to think again more about geometry - Thurston's/Hamilton/Perelman proof of Poincare, Gromov's work in geometric group theory. I think that abstract algebra is a powerful tool but it is ultimately not much use unless there is some geometric or physical intuition to back it up. Of course that's only my opinion.

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I did a maths degree. Postgrad I did AI and logic which don't help this debate much.

I also studied physics and found theoretical physics no problem. Couldn't do applied electronics but that's an inability to look at circuit diagrams without nausea rather than a visualization problem.

All my experience tells me visualization whilst very useful is not vital and that those who visualise can't understand how those who dont, think. Intuitions are vital, I'd agree but I have plenty of those.

I used to get into those debates a lot. The smug group at the back (as we were fondly known) found a schism between them and me when discussing many problems. They would talk pictures and describe twisting and turning and I would talk in terms of symmetries, invariances and abstractions. Wierd looks and misunderstandings were only discounted because we all generally ended up in the same place.

I have my mathematical/reasoning limitations but I doubt visualization is an issue. I could be wrong.

chez