Bill, Your comment is very close to what I was trying to say. > I have been a musician for seventy years, and I was a serious mathematician until age twenty. (I graduated with a double degree.) I can assure you that I dont think only in terms of the patterns . . . In fact, in my most treasured musical experiencesand Id venture to say the same for mathematicsI barely think at all. Its an embodied understanding: I feel what I apprehendand only after the fact, with a great sense of loss, do I think about it. And when I do think I mostly struggle to find some faint simulacrum of my experience. Sometimes that might involve patterns; sometimes I might draw pictures or notes or words; sometimes I simply get up from the desk and pace, wave my arms, sing a little. (Except for the singing, the same definitely goes for mathematics.) The phrase "embodied experience" is excellent. Peirce, Einstein, Archimedes, Whitehead, and many others would agree. In fact, the way you describe your experience and the difficulty of putting it into words is very close to what Peirce said about his "left handed brain" (he was left handed). And he admitted that he had considerable difficulty in expressing himself in words -- that is one reason why he preferred diagrams. He also hoped to generalize his diagrams to "stereoscopic moving images" or even physical models. He would have loved to work with a virtual reality system. When I mentioned "structural patterns", I chose that term because I needed a noun phrase to insert in the sentence. Your description is very close to the way that most professional mathematicians describe their way of thinking. For examples, and references see the first ten slides in "Peirce, Polya, and Euclid: Integrating logic, heuristics, and geometry" http://jfsowa.com/talks/ppe.pdf . The reason why I used the analogy of math to music is that I wanted to emphasize the non-verbal way of thinking in those fields. Any description in words cannot capture the essence of what goes on when a mathematician or a musician is deep in the creative experience. I'm thinking of the musician who was asked what his composition meant. As an answer, he played it again. Fundamental principle: For mathematicians and musicians, words are not just secondary, they're almost irrelevant. To understand them, look at what they do, not at what they say. John
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