It's bizarre to say that you can be a great mathematician "even if you calculate a lot." Most great mathematicians have been extremely avid calculators. Even Riemann, generally thought a paragon of geometric insight, calculated maniacally; his nachlass is full of calculations.
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Replying to @MathPrinceps @BarbaraFantechi
For the curious non mathematicians.. what does “to calculate” or “not calculate” mean/infer?
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Replying to @rca619700 @MathPrinceps
This is an extremely good point! Most mathematicians calculate, but in very different senses of the word. I know very few who, like Don does, enjoy doing computations involving integers with dozens of digits. Most of us , myself included, wouldn't be able to.
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Replying to @BarbaraFantechi @rca619700
Calculation is a form of play. It observes certain rules, like those of tennis, and may be deadly serious. But it need not have a precisely defined goal; one does not always play "to win." Sometimes one just goes out and whacks the ball around, for the sheer fun of it.
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Even when whacking the ball around for the sheer fun of it, though, one still tries to avoid hitting the ball into the net, or outside the lines. And in this sense calculation is always constrained and structured, even if its purpose is not always perfectly clear.
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Often the play seems magically to organize itself, in some strange and surprising way. As one fiddles around, obeying the rules of calculation, one often starts to feel as though the rules themselves are somehow pushing one in a certain direction. As though they subtly bias play.
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The great thing about calculation is that one can do it even when one is stuck and frustrated -- which, for most mathematicians, is the usual state of affairs. Calculation can then be a form of active meditation. It keeps the mind turning, often in the right general direction.
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And when calculation leads to this feeling that the rules of the game itself are somehow biasing play in a certain way, rewarding some choices and punishing others, then the activity takes on a tantalizing fascination. One struggles to hear the subtle guidance it is offering.
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According to the eminent mathematician-historian Jeremy Gray, "[Gauss] himself said that many of his best discoveries were made at the end of lengthy calculations, and much of his work on elliptic functions moves in a sea of formulae with an uncanny sense of direction."
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Gray adds that many great mathematicians had an almost eerie knack of thinking in formulae. "One is struck as often by the technical power of great mathematicians as by their profundity. Nonetheless, some mathematicians have the ability more than others."
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Gray cites Euler, Gauss, and Kummer as paragons of raw technical power -- as master calculators whose profound insights were rooted in extensive play of just the sort I've been describing. (Klein and Poincare, though, he acknowledges were noticeably less virtuosic in this way.)
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Among contemporary mathematicians, Don Zaiger has long seemed to me perhaps the most temperamentally similar to Euler. For him, calculation is a delight and an indispensable source of inspiration. (Atiyah says this was also true of Zagier's great adviser, Friedrich Hirzebruch.)
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Replying to @MathPrinceps @BarbaraFantechi
Thank you! I am more in the neuro science realm, but there are intersections .. Years ago Edward DeBono coined the term/process “lateral thinking” as a way of developing non-linear innovation (paraphrase “digging lots of holes, instead of one hole deeper”) ..
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