“Young man, in mathematics you don’t understand things. You just get used to them.” John von Neumann.
Continuing the discussion on what does it mean to understand a proof, and following up on the discussion on a recent post by Gowers, I would like to bring up the question of what does it mean for a mathematical proof to be difficult.
First of all, I think it is important, again, to distinguish between the perspective of the person conceiving the proof and the person reading (or trying to understand) the proof.
Proofs that are hard to read
From the perspective of the reader, a proof may seem hard (or at least harder than it needs to be) simply because of the style of exposition, as already discussed.
Even some flawlessly presented papers, however, can be hard to follow, because of the complexity of the technical machinery used by the author. In many areas that have developed very sophisticated tools (PCP constructions, for example), every paper might look very complicated to the non-expert, because of the need to “interface” with the very complicated context, or the need to re-prove “standard” results because they were never stated exactly in the way that is needed and so on. To the expert, however, many such paper look relatively simple, in the sense that given the basic ideas, which can often be summarized in a few sentences, it is “routine” to reconstruct the rest of the paper. (More later on what this means for the value of such papers.)
Finally, there are well-presented papers which baffle the experts (sometimes, this includes the authors). I am thinking, for example, of the Khot-Vishnoi paper on sparsest cut. Those are papers that introduce new techniques that are not yet fully understood. (Meaning, for example, that the full extent of their applicability and generality is not clear yet.) They contain the genuinely difficult proofs, and they are the ones that are most useful to study. Note that, here, I am considering difficulty as a transient, not an intrinsic property: at some point the techniques become used over and over, a theory is built around them, we know how to make use of them, and they become “understood,” or “simple” to the experts. Needless to say, this process of clarification and simplification is extremely useful, and should be well regarded and rewarded, as it usually is. (Most of my research is of this type, and I cannot complain for the way it is received.)
Proofs that are hard to conceive
What about difficulty from the perspective of authors? Clearly, papers like the ones I just described are as difficult (or more) to come up with than to understand for the experts. One is devising new techniques and exploiting them on the fly, and it is a miracle when it all works out.
What about papers that seem difficult to non-experts and are easy (when properly presented) to experts? Here it depends. Sometime, given the right tools, the result is just routine application of the technical machinery, no harder to conceive than to verify. Sometimes, however, even if the main idea of the paper can be expressed in one sentence, that one idea can be the result of endless attempts, blind alleys, complicated first attempts, later simplifications, and so on. Of course, papers of the first type are nearly (but not completely) useless, and papers of the second type are very good.
The value of technical work
There was recently a discussion in the theory community about the value of “conceptual” papers, which, perhaps unpredictably, turned in part into a discussion on the value of “technical” papers. At the time, I completely agreed (as I still do) with statements such as “all other things being equal, a simpler proof is better than a harder one,” but I had a problem with some of the conclusions that were being drawn.
Leaving aside the issue of papers whose main contribution is a new definition or a new model (which are, in fact, the “conceptual” papers that the original statement dealt with), when we look at a paper proving a new result and introducing a new idea, then, all other things being equal, we want the authors the explain their idea and distill their proof in the simplest possible way. However, another point is also true: that given two well presented papers, both simplified as much as they can be, the more complicated one is the one that uses the more complex technical tools, and a new ideas about complex machinery is more valuable than a new ideas about simpler machinery, because the former makes powerful tools even more powerful. And, as argued above, the technical papers which are going to be more useful (by stimulating the construction of new theoretical thinking around their tools) are the mystifying ones.
- writing papers just because you can, employing difficult techniques in routine ways: BAD
(but even such papers may be somewhat useful, I may return to this point in a later post.)
- presenting a needlessly complicated version of a potentially simple argument: VERY BAD
- having a new idea on how to use difficult techniques, and explaining it as transparently as possible: GOOD
(but such papers would still be hard to read for the non-experts)
- making breakthroughs by conceiving new ways of doing things: VERY GOOD
- finding the “right” way to understand a previously mystifying argument: GOOD
(And, of course, coming up with a new definition or model that extends the reach of theoretical work: VERY EXCELLENT; but this was not the subject of this post.)