Is Fuzzy Set/Measure Theory an Active Area for Research?

Question

I came across the notion of a fuzzy set the other day and since then, I've been reading about fuzzy measures and the Sugeno/Choquet integrals. While I certainly do not claim to have fully wrapped my mind around it by any means, there is something that I find appealing about this area of study. This whole subject seems to be fairly obscure as Amazon has only a few books on the subject and none of them look like bestsellers (though I find these sorts of texts to be hidden gems more often than not). Basically, I am just interested in learning more about the direction that research in this subject is headed (if anywhere) and if it is perhaps a promising area to carve out a niche (admittedly in the distant future) one day. Any insight or reference would be very much appreciated. Also if Fuzzy Sets/Measure are thought to be a dead end or a "rabbit hole" of sorts, that knowledge would be equally valuable.

Answer 1

The appeal you speak of, which I believe is/was shared by many, is that the fuzzy variants of the traditional mathematical objects are, intuitively, more realistic. After all, in real life we never know anything with certainty, so why not replace all of logic and set theory by their fuzzy cousins - surely the mathematics we'll get will be better.

Well, mathematics is about providing formal models for real life phenomena (very roughly speaking). Good mathematics is about providing good models for real life phenomenon. What makes a model good is how well it describes whatever it is we are modeling, and how powerful the axioms are, namely can we prove interesting things. Whether or not the model agrees with our intuitive understanding is a psychological issue, inherent to us humans.

So, while traditional mathematics provides fantastic models for physics and other real life situations, fuzzy mathematics, in comparison, never took off the ground. Sure, there are plenty of theorems, and much current fuzzy research is about transferring known results to the fuzzy setting, but when it comes to applications there is very little in favour of the fuzzy realm. That is not to say none at all, but, again, comparing to classical maths it is really very very little.

It makes sense that introducing fuzziness into the fabric of your mathematical models will severely weaken the formalism. After all, all those uncertainties in life are the reason for much of the difficulty in analyzing situations with any degree of accuracy. It is precisely the elimination of any doubt that makes our mathematical reasoning so powerful. The wonderful thing is that the models we then get, while not completely realistic, are fantastically good (good enough to put a man on the moon). Now, of course, sometimes uncertainties are unavoidable, but we already have probability theory, which in a sense is a sort of fuzzy mathematics - certainly a respectable one.

Finally, a general word of advise is that it is probably not a good idea to get involved in what you already recognize to be an area of mathematics with few best sellers. Broaden your horizons as you like, but ground yourself in the mainstream - at least if you seek to be employed as a mathematician.