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Mathematical Intuition
Nov. 22nd, 2010 06:48 pmI've noticed when I read mathematical works, I use two different perspectives alternately. On one, I read the descriptions in English (where there are such) of what the formulae and/or proofs mean. If I remember anything from a mathematical work, it's likely to be in this category--something expressed in English. On the other view, I try to plug in terms for the symbols and follow the logical derivations behind the formulae and results.
The problem for me is forging any kind of connection between the two. When we studied Apollonius my sophomore year, I enjoyed following the manipulation of ratios he employed to do his proofs. But if you asked me what it had to do with the conic sections--that I had a much harder time understanding, even though diagrams were provided. I can manipulate formulae all day and be reasonably good at it, but unless I also have knowledge of the first kind, I won't know what I did. I don't remember Apollonius's results particularly well.
Of course we need rules (formulae) to express relations that are supposed to hold for whole classes of things--to supply the "any"s in a concise way--and to work with. But I wonder whether the ability to move back and forth between these perspectives more smoothly is what characterizes people with "mathematical intuition."
*Note: I realize this is close to the synthesis/analysis distinction, which is expressed especially clearly in the work of Viete, who himself combines tmy two "perspectives" as fluidly as I've seen.
The problem for me is forging any kind of connection between the two. When we studied Apollonius my sophomore year, I enjoyed following the manipulation of ratios he employed to do his proofs. But if you asked me what it had to do with the conic sections--that I had a much harder time understanding, even though diagrams were provided. I can manipulate formulae all day and be reasonably good at it, but unless I also have knowledge of the first kind, I won't know what I did. I don't remember Apollonius's results particularly well.
Of course we need rules (formulae) to express relations that are supposed to hold for whole classes of things--to supply the "any"s in a concise way--and to work with. But I wonder whether the ability to move back and forth between these perspectives more smoothly is what characterizes people with "mathematical intuition."
*Note: I realize this is close to the synthesis/analysis distinction, which is expressed especially clearly in the work of Viete, who himself combines tmy two "perspectives" as fluidly as I've seen.
