What does it mean when an author illustrates a result or proof of some result "heuristically" ? And why is that useful ?
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1From https://en.wikipedia.org/wiki/Heuristic : In mathematics, some common heuristics involve the use of visual representations, additional assumptions, forward/backward reasoning and simplification. – Martin R Dec 09 '21 at 12:40
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Can you give a specific example of this being used that you would like explained? In general I would think it means that they're showing an argument which has merit but doesn't go through rigorously, in order to establish why a result makes sense intuitively – Stephen Donovan Dec 09 '21 at 12:42
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It's an unproven, not-quite-universal algorithm that is still nonetheless effective. – Shaun Dec 09 '21 at 12:42
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I agree with Stephen. I have nothing to add to SomeCallMeTim's excellent answer except to note another place where it is very common to do this. In analytic number theory, a common heuristic is that large primes are distributed randomly. Of course this is completely false: whether a number is prime or not is deterministic. But proceeding with this assumption has had a long history of leading to "the right answers", and so people will often justify their conjectures heuristically in this sense. – Eric Nathan Stucky Dec 09 '21 at 14:06
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@Stephen Donovan: Can you give a specific example --- See Most harmful heuristic? AND Most useful heuristic? AND Examples in number theory where a heuristic argument fails AND What are some historical examples in physics of heuristic proofs of mathematical results? AND What does one mean by heuristic statistical physics arguments? FYI, I was at NCSU 89-93. – Dave L. Renfro Dec 09 '21 at 14:09
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I associate heuristical reasoning with sensible informal shortcuts via which results or correctness isn't guaranteed. @DaveL.Renfro Thanks for the links; a 'automathography'? Who knew! – ryang Dec 09 '21 at 14:33
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Here is a very common example: We define the Riemann integral as sums of rectangles approximating the area under the curve, and then take the limit as the "width" of the rectangles go towards zero. This, along with the picture, is a more or less heuristic definition of the Riemann integral, in that the formalism is not in focus, but we concentrate on visualization and understanding.
Picture is from https://isquared.digital/blog/2020-05-27-riemann-integration/
SomeCallMeTim
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