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Quoting Gian-Carlo Rota (from the Foreword to Richard Stanley's Enumerative Combinatorics Volume I), "In mathematics, however, the burden of choice faced by the writer is so heavy as to turn off all but the most courageous. And of all mathematics, combinatorics is nowadays perhaps the hardest to write on, despite an eager audience... Shall the author yield to one of the contrary temptations of recreational math at one end, and categorical rigor at the other? or to the highly rewarding lure of the algorithm?"

Does anyone understand this last statement? What is the lure of the algorithm? Is it a reference to something particular?

I am sometimes quite happy to attain an elegant algorithm, but when it comes to writing I tend to leave the details of hard-to-formalize algorithms to the reader (e.g. "We omit the proof that this establishes the desired bijection."). And if there is a "lure", I cannot think of any example to substantiate Rota's slightly derisive tone. I admire authors who contribute algorithmic intuition to the literature. In Stanley's own book (Volume II) is an impeccable discussion of algorithms concerning the enumeration of rooted trees.

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Are you sure that his tone was intended to be derisive? I would think the lure is in the ability to exhibit constructions explicitly, and in the goal of making the construction itself as elegant as possible. –  Niel de Beaudrap Oct 7 '11 at 13:36
    
@Niel: Maybe you should make/expand that into an answer? :) –  Mike Spivey Oct 7 '11 at 16:07
    
Maybe he meant "the algorithm" is merely one of several attractive things. –  Michael Hardy Oct 7 '11 at 19:08

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I can only speculate about Rota's attitudes, but it's not obvious to me that this passage is meant to be derisive to algorithms.

I would think the lure is in the ability to exhibit constructions explicitly, and in the goal of making the construction itself as elegant and as general as possible. Doing so is a challenge to understand the construction as deeply as possible. This is, at least, why I am strongly drawn to algorithms.

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“And of all mathematics, combinatorics is nowadays perhaps the hardest to write on, despite an eager audience... Shall the author yield to one of the contrary temptations of recreational math at one end, and categorical rigor at the other? Or to the highly rewarding lure of the algorithm?”

It seems to me that Rota contrasts four things here:

  • ‘Good’ writing on combinatorics
  • Recreational math
  • Categorical rigor
  • The lure of the algorithm

and suggests that someone attempting the first may be led astray into one or more of the others.

As to the “lure of the algorithm”, an algorithm for (say) enumerating (or even displaying) combinatorial objects may provide an ‘answer’ of a sort without providing any real mathematical understanding. The algorithmic approach offers this for relatively little hard (mathematical) work, and may be a lure for those who want instant results (or dislike hard thinking). It's fun playing with Mathematica, after all! Even the most sophisticated algorithm, despite requiring hard work from a software engineering perspective, still may provide very little in terms of mathematical insight.

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Right. A spectacularly effective or ingenious algorithm may perhaps contribute nothing to understanding. –  Lubin Jan 17 at 22:35

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