Friday, January 7, 2011

Catastrophe theory; geniuses and maniacs

This post is meant as a (modestly priced) receptacle for a figure that I often find myself referencing from  V.I. Arnold's book on catastrophe theory. Here Arnold is discussing, somewhat sarcastically, a catastrophe-theoretic model of the activity of a creative personality. To quote Arnold on the setup:
We shall characterize a creative personality (e.g., a scientist) by three parameters, called "technical proficiency," "enthusiasm," and "achievement." Clearly these parameters are related. This gives rise to a surface in three-dimensional space with coordinates (T,E,A). Let us project this surface onto the (T,E) plane along the A axis. 

This is the figure:
To paraphrase Arnold, what's going on here is that if you develop your technical skills at low enthusiasm, you move up along the smooth ramp at the "far end" of the figure. If you get enthusiastic once you have the skills, you move outwards along the E axis to point 2 and become a genius. On the other hand if your initial level of enthusiasm is high, you go discontinuously from having no achievements to having a lot, once you've learned a critical amount of background information.

The "catastrophe" is when you follow line 3. Let me return to quoting Arnold:

A growth of enthusiasm not supported by a corresponding growth in technical proficiency leads to a catastrophe (at point 4 of curve 3) where achievement falls by a jump and we refer to the domain denoted in Fig. 6 by the term "maniacs." It is instructive that the jumps from the state of genius to that of maniac take place along different lines, so that for sufficiently great enthusiasm a genius and maniac can possess identical enthusiasm and technical proficiency, differing only in achievement (and previous history).

The deficiencies of this model and many similar speculations in catastrophe theory are too obvious to discuss in detail. I remark only that articles in catastrophe theory are distinguished by a sharp and catastrophic lowering of the level of demands of rigor and also of novelty of published results.

Taken seriously, the catastrophe theory of the '70s and '80s was very much in the same vein as the more recent extravagances of physicists that xkcd was mocking the other day. (Although it did lead to some very beautiful results on swimming-pool patterns.) Fig. 6, however, is an awfully useful caricature of intellectual development, and various things in life keep reminding me of it.

1 comment:

Sarang said...

I should cross-post this buzz exchange as it raises an important point:

kit wallach - This seems to support an "out of madness, genius" train of thought, rather than a "descent into madness." Only the discontinuity at 4 (or all of path 3, I suppose), describes a descent, but it is a descent from mediocrity to mania. Where is the description of the erstwhile genius, now a maniac? Perhaps the upper surface should continue outward along the "Enthusiasm" axis, falling off sharply, diving down into the realm of mania, right past where the "ses" in "Geniuses" is marked. So many reside on the edge of that little cliff.

Sarang Gopalakrishnan - The way I picture this image -- admittedly this is not clear from the graphic -- the entire fold moves outwards, so for a given level of proficiency, there is always some level of enthusiasm at which you fall off the cliff, so you can draw something analogous to path 3 for any initial level of accomplishment including genius. But the more proficient you are, the more enthusiastic you have to be to fall off the cliff... (Another thing a physicist would want to do with this picture is add a "noise term" i.e. randomly directed gusts.)