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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 20th 2013

    Andrew Stacey: would you tell us what the word ’heuristic’ means? I’ve seen you complain in various places at MathOverflow about how people use this word (e.g., here, in the comments). Frankly, I’ve been frustrated by your not elaborating further (your justification being that MO is not suited for such discussions), and now I avoid writing the word in a place where I think you might be reading, not wishing to bring upon myself your public displeasure (even though I think I know how to use the word correctly, according to what I’ve read).

    I hope you think the nForum is suitable for such discussions!

    I find myself agreeing with QQJ: “Actually the principal definition from the Concise Oxford English dictionary is (adjective) ‘enabling a person to discover or learn something for themselves’. Plenty much broad scope there to zero your aagghhument.” And indeed, my Compact Unabridged OED (2nd edition) gives simply “serving to find out or discover” as their definition (a).

    The “rule of thumb” definition mentioned by Per Vognsen is supported and elaborated on by Wikipedia:

    Heuristic; Greek: “Εὑρίσκω”, “find” or “discover”) refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal. Where the exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts to ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense.

    In more precise terms, heuristics are strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines.

    and it seems to me Per did use the word correctly according to that range of meanings. The ways in which I would use the term would also fall within this range.

    These applications also seem supported by the OED. Under definition (b), OED cites the IBM Journal of Research and Development (1958), which is fairly precise as illustrative quotations go:

    For the moment… we shall consider that a heuristic method (or a heuristic, to use the noun form) is a procedure that may lead us by a short cut to the goal we seek or it may lead us down a blind alley.

    What do you say? Concrete examples of what you have in mind would be wonderful.

    • CommentRowNumber2.
    • CommentAuthorjim_stasheff
    • CommentTimeJul 20th 2013
    cf. ansatz?
    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 26th 2013

    I finally googled ’andrew stacey heuristic’ and found this:

    @KaneBlackburn It’s a chip on my shoulder from reading Polya’s How to solve it book where he uses the word in an extremely precise way. Using “heuristic” to mean “the rough idea” leaves us no word to describe something that is incredibly useful and is in danger of being forgotten: that proofs can be figured out and there are some very good guidelines for working out those proofs. In short, don’t take over much notice of me (but if you haven’t read Polya’s book, do take note that I heartily recommend it).

    I think this can suffice as an answer to my question: ’heuristic’ here refers to a method that enables one to discover [for oneself], very much in keeping with the Greek origins of the word. I am guessing Andrew is bothered by alternative definitions (e.g., “rule of thumb”) that water down the idea that a heuristic method, as applied for example to reconstruct a mathematical proof for oneself by following general guidelines or principles, can be as precise as one could wish – an idea that Polya’s book helps to keep alive.

    Andrew, would you please confirm this, or correct as necessary, so that the matter can be laid to rest? I’d really appreciate it.

    • CommentRowNumber4.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 26th 2013

    My apologies! Judging by the date of the original question, you asked it when I was in the middle of moving country so although I appear to have read it, I would not have been in a position to answer it and appear to have forgotten about it in the intervening time. I certainly did not do so deliberately.

    What I object to is using “heuristic” not in relation to problem-solving. Such as saying “the heuristic idea” when you’re trying to give a rough idea of some theorem or the thinking behind some theorem. Or “heuristically speaking” when you mean “if we ignore some small details”.

    I was very much impressed by Polya’s text and felt that it had ideas in it that we are in danger of losing. In particular, the idea that we do not solve problems by knowing their solution in advance! But we do it by working out what we think the solution should be like using heuristic, and other, methods.

    So my obsession with “heuristic” is not about linguistic purity but about not wanting to lose that gold. If we lose the word to describe something we lose part of our ability to communicate it.

    I do find that Wikipedia gets it right, and maybe I was a bit over-the-top in the comment on that MO discussion that you (Todd) link to. On MO I tended (when I was active there) to be more forceful and opinionated than I actually was. So what I say there should always be taken as being slightly more extreme than what I actually think. It’s hard to put myself back in what I was thinking then, but I suspect that (again influenced by How to Solve It), I was fanatical about saving “heuristic” specifically for problem solving and not for anything else.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 26th 2013

    Thank you! Your entire message comes as a relief, and makes sense to me.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2013

    Such as saying “the heuristic idea” when you’re trying to give a rough idea of some theorem or the thinking behind some theorem. Or “heuristically speaking” when you mean “if we ignore some small details”.

    I usually use the word “intuitive” in cases like those.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeOct 27th 2013

    Me too … although I'm not sure if this fits the proper definition of the proper definition of that word either.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2013

    Whether or not it fits the dictionary definition, I think it’s a widely accepted meaning of that word within mathematics.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2017
    • (edited Oct 18th 2017)

    [years later…]

    I vaguley remember the above discussion each time I type “heuristic” or “intuitive” into any entry. But this is the first time that I searched for it to come back to it.

    It makes me think that we really should have an entry heuristic and maybe in addition an entry intuition, or the like, which would make some of the points above such that they could usefully be linked to whenever one uses these words.

    • CommentRowNumber10.
    • CommentAuthormaxsnew
    • CommentTimeOct 18th 2017

    Personally, I have tried to “de-program” myself from using “inuitively” for this and always write “informally” because it can be discouraging to see something that you don’t understand at all called “intuitive”. This might be because of my field though (computer science) where many people are not famiiliar with techniques I use from category theory.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2017
    • (edited Oct 18th 2017)

    We might explain all this in the entry. In what I am currently looking at, quantum field theory, it happens to be the exact opposite: People have unbounded intuition about the “path integral” and little sensitivity that it is “informal”.

    Moreover, I come to think that the reason the non-existing “path integral” is so useful is that in fact it is a great heuristic in the Andrew’s sense above: namely thinking of an integral makes you discover all the right properties that you should put as axioms on the real thing (the axiomatic S-matrix).

    So all aspects come together.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 18th 2017

    Urs, how strongly do you mean “non-existing”? I understand no one has produced a framework in which such a creature is developed rigorously, but I get the impression you think that’s a hopeless pursuit.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2017

    There doesn’t seem to be a theorem that the path integral measure for non-toy field theory examples does not exist, but practically it seems indeed hopeless. At least there is no idea or conjecture known for how to even approach it, nobody is working on it, as far as I see. The methods that work in the toy examples all break down when one goes beyond.

    What may be worse is that even in the toy examples where the path integral measure has been constructed (“constructive field theory”), it seems a fairly sterile mathematical exercise that hasn’t opened the door to further interesting developments. It’s not been fruitful.

    Finally, all what is good about the idea of the path integral, all the good intuition about field theory and mathematics that it has provided, has been absorbed into mathematical concepts that actually work, such as causal perturbation theory.

    It seems to me that the real place of the idea of a “path integral measure” is in generalizing from integrating up quantities with values in the ring of complex numbers to integrating up quantities with values in more general ring spectra, as in “quantization via stable homotopy types”. This, too, is presently not constructed for field theories of dimension 2 or higher, but contrary to the naive measure this has a clear perspective that it should work (for instance pull-push in tmf is technically demanding but plausibly related to the quantization of 2d CFT) and it is non-sterile, following the tao of mathematics.

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