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kay bei

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- TL;DR Summary
- Maxwell, Dirac and Mikhail Gromov have been known to think Geometrically when doing their theoretical and mathematical research. What does this mean? and How is it different from the Algebraic approach to thinking.

I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned everywhere. Mikhail Gromov has said he thinks and solves problems geometrically. How is someone supposed to think of Manifolds, and abstract spaces geometrically? I don’t understand this process they use whereby they use geometric thinking to help solve a problem. Are there any basic and more advanced examples which show how someone uses this type of thinking to solve a problem? Please help me understand this. In an interview a mathematician (a fields medalist, I can find the article for you), explained that some mathematicians think algebraically, and others think geometrically and it was said that Gromov thought geometrically. Gromov has said in his interview with AMS that John Nash's thinking style influenced him and that Nash also thinks geometrically, he mentioned also Stephen Smale. I just don’t know the difference in these thinking styles (alebraic vs geometric). How does one develops this without any good examples comparing both approaches side by side on basic and difficult problems. I’ve never had a teacher show the difference. Hopefully someone can clarify this. I am asking because I would prefer to purchase a textbook which takes one approach which suits my thinking style. I may find it easier to think geometrically vs algebraically so it would be beneficial to know what is this difference. There is even a textbook on abstract algebra : a geometric approach. This is so confusing? Is it true that some mathematicians in the field of abstract algebra think in a geometric way to solve their research problems? Do you notice this also among mathematicians? Hopefully someone can help clarify this geometric vs algebraic approach mentioned everywhere.