Grothendieck’s mind was indeed extremely strange. The levels of abstraction upon abstraction he achieved in algebraic geometry boggles the mind.
But I don’t think you can really make meaningful comparisons between thought processes based on self-reporting. One complication is that different fields of mathematics work differently in this regard. In things like statistics, analysis, and geometry, you rely heavily on examples. In things like algebra, examples can indeed be cumbersome and hindering, because the point of algebra is to simplify things to symbol manipulation. Of course, it might also be the case that people with more abstract-type thinking are naturally drawn to algebra.
It would be useful to look at the ‘information content’ of storing examples vs. storing symbolic representations, and see how that compares across different mathematical subjects.
I’m skeptical that the relevance of the two modes of thinking in question has much to do with the mathematical field in which they are being applied. Some of grothendiek’s most formative years were spent reconstructing parts of measure theory, specifically he wanted a rigorous definition of the concept of volume and ended up reinventing the Lebesgue measure, if memory serves, in other words, he was doing analysis and, less directly, probability theory...
I do think it’s plausible that more abstract thinkers tend towards things like algebra, but in my limited mathematical education, I was much more comfortable with geometry, and I avoid examples like the plague...
Maybe the two approaches are not all that different. When you zoom out on a growing body of concrete examples you may see something similar to the “image emerging from the mist”, that grothendiek describes.
Grothendieck’s mind was indeed extremely strange. The levels of abstraction upon abstraction he achieved in algebraic geometry boggles the mind.
But I don’t think you can really make meaningful comparisons between thought processes based on self-reporting. One complication is that different fields of mathematics work differently in this regard. In things like statistics, analysis, and geometry, you rely heavily on examples. In things like algebra, examples can indeed be cumbersome and hindering, because the point of algebra is to simplify things to symbol manipulation. Of course, it might also be the case that people with more abstract-type thinking are naturally drawn to algebra.
It would be useful to look at the ‘information content’ of storing examples vs. storing symbolic representations, and see how that compares across different mathematical subjects.
I’m skeptical that the relevance of the two modes of thinking in question has much to do with the mathematical field in which they are being applied. Some of grothendiek’s most formative years were spent reconstructing parts of measure theory, specifically he wanted a rigorous definition of the concept of volume and ended up reinventing the Lebesgue measure, if memory serves, in other words, he was doing analysis and, less directly, probability theory...
I do think it’s plausible that more abstract thinkers tend towards things like algebra, but in my limited mathematical education, I was much more comfortable with geometry, and I avoid examples like the plague...
Maybe the two approaches are not all that different. When you zoom out on a growing body of concrete examples you may see something similar to the “image emerging from the mist”, that grothendiek describes.