Simple legacy

Benoit Mandelbrot makes the following observation in The Fractal Geometry of Nature.

Many creative minds overrate their most baroque works, and underrate the simple ones. When history reverses such judgments, prolific writers come to be best remembered as authors of “lemmas,” of propositions they had felt “too simple” in themselves and had to be published solely as preludes to forgotten theorems.

If you’re not familiar with lemmas and theorems, think of a musician who is famous for a short prelude written as an introduction to a longer piece nobody remembers. For example, Rossini’s four-minute William Tell Overture is far more famous than the four-hour William Tell opera it introduces.

Returning to famous mathematicians, I remember as an undergraduate hearing of Schwarz’s lemma and waiting for the corresponding theorem that never came. The same applies to PoincarĂ©’s lemma, Zorn’s lemma, and Fatou’s lemma.

We’re all naturally proud of things we work hard for. We expect other people to value our work in proportion to the amount of effort we put into it, but the world doesn’t work that way. It can be discouraging focus on the big, complex projects we’ve worked on that haven’t been appreciated. On the other hand, it can be very encouraging to think of the potential impact of small projects and simple ideas.

11 thoughts on “Simple legacy

  1. I think you’ve got something wrong here.
    Often “Lemma” stands for a short but important observation that leads to a big theorem and many authors follow the “definition -> lemma -> theorem” path but the term “Lemma” often stands for an single very important “theorem” itself – as is the case with the lemmas you mentioned.

    But nevertheless you’re quite right with your conclusion but I might add that most peoples work won’t get a public appreciation at all.

  2. True, most people’s work gets no appreciation. But there’s a difference between “appreciation” and “impact.” My last paragraph could imply that I equate the two ideas, but I don’t. I would say they are unrelated or even inversely related, though some things do need to be recognized before they can have impact.

    Here’s an example of a small act that could have big (anonymous) impact: picking default values in software. Our lives are influenced far more than we realize — for better and for worse — by the default values programmers choose.

  3. An even more extreme musical example is the Amen break. Astonishing!

    One of my most popular software efforts was originally just an effort to evaluate a particular automatic interface generator.

    How about the 4th Earl of Sandwich? [Aside: I thought it could not possibly be true that he invented the sandwich, but a retired CIA officer turned food historian could not find any example of a true sandwich before the famous Earl. Tacos, yes. "Open faced" sandwiches, yes. Stuffed buns, yes. But he could find no historical evidence that anyone had actually made a sandwich with two slices of bread. Probably someone did, but evidently no one noticed or bothered to write about it. The Wiki reference disproving his invention mentions a sandwich made from matzos as being the first, but matzos are not slices of bread.]

    As far as anonymous impact goes, how about the guy who defied the tanks in Tiananmen square in one of the most famous recent photographs? It turns out no one know who he was. Link in next comment to avoid spam filter.

  4. Lokks like I forgot a closing HTML tag. Anyway, the world-famous but still anonymous (even though photographed!) individual can be seen at:

    en.wikipedia.org/wiki/ Tank [underscore] Man

    [hyperlink munged to allow John's blog to continue to be read in the P R C]

    [actually, John you may want to edit out my reference to the T-word in my previous comment]

  5. Comparably, your posts on CodeProject illustrate this point. The floating point articles probably generate more traffic and comments, yet represent the least amount of code in your examples.

    You are pointing out a general tendency to focus on stuff we can wrap our brains around, yet is sufficiently puzzling to hold our interest. Seemingly seminal works published by an author or authors define their contributions and reputation, even though the publication is a rather minor part of their overall production.

  6. Another famous “mere” lemma: the Neyman-Pearson lemma, the key theorem in the theory of hypothesis testing in statistics.

  7. I am best known for my work on Slope One (see the wikipedia entry) which is maybe the simplest research result I ever got. Meanwhile people happily ignore what I spent most of my time on. I think it is really funny.

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