Is there anything deep about Fuzzy sets/Fuzzy logic? I keep coming across the term "fuzzy". I browsed around a bit and read a few articles.. It seems to me that there's absolutely nothing deep or foundational about fuzzy set theory or fuzzy logic. I know they are practically useful in robotics, ..etc. but since there's a lot of buzz around them, I am led to think that there might be something deep that I just don't see.
I realize that my question is perhaps vague or "not constructive", I would understand if it were closed for that reason.
 A: I know quite a lot of mathematicians who claim that there is nothing deep about "formal logic" (in the sense of mathematical logic): they believe it is just a waste of time to spend so much time in formalizing a problem. What these people do not realize is that the interesting part of logic is not the process of formalization, what it is really interesting is the metamathematical reasoning we can do once a problem is formalized. Particular examples of useful results concerning metamathematics are: compactness theorem, Löwenheim-Skolem theorems, completeness theorem, incompleteness theorems, etc.
My interpretation about your question is that you share the same prejudices I have explained above, but towards fuzzy logic insted of logic. So, let me try to point out that what is interesting about fuzzy logic for me, at least, is not the process of formalizing using fuzzy sets (I agree that this is not worth spending time), it is the metamathematics that can be developed around this alternative semantics. To emphasize this distinction some people has coined the term "mathematical fuzzy logic" (you can find much more details in the link Mathematical Fuzzy Logic and the references there in).
Let me just summarize some statements about fuzzy logic in this metamathematical sense:

*

*First-order logic using an infinite-valued Lukasiewicz semantics (this is a natural choice of connectives) is not recursively axiomatizable, indeed it is $\Pi_2^0$-complete. The first part was proved by Bruno Scarpellini, and the second by Mathias Ragaz. Other choices of interpreting connectives give different situations (e.g., using what is called "product logic" you get a non-arithmetical set of validities, etc.).


*The comprehension axiom is consistent in case you restrict your logical rules from classical logic to Lukasiewicz fuzzy logic. Thus, an alternative approach to restricting comprehension axiom (this is method used in ZFC) consists on restricting the logical rules. To explore a bit more on this topic look at https://mathoverflow.net/questions/86151/boolean-valued-models-vs-the-infinite-valued-logic-of-lukasiewicz-and-set-theory


*Finally, let me point out that while it is known that Peano arithmetic cannot be consistenly expanded with a truth predicate, this is not so clear in case you allow the truth predicate to be fuzzy. Unfortunately, the situation is the same than in classical logic, but the proof is more tricky than the one given for classical logics. You can find the details of this in the paper http://www.jstor.org/stable/2586541 published at Journal of Symbolic Logic (see also http://www.logicmatters.net/2010/05/curry-lukasiewicz-and-field ).
A: This is what yours truly answered on a related question on MathOverflow:

Fuzzy measure theory has applications in pure measure theory. The
  Choquet capacity theorem is a standard tool for showing the universal
  measurability of analytic sets. The   theory of capacities (or
  fuzzy measures) is fairly well developed and strongly related to
  "normal" analysis. 
The theory of capacities was not created in the context of fuzzy
  mathematics, but M. Sugeno developed a form of fuzzy integration in
  his PhD thesis that shares many formal similarities with the Choquet
  integral and some work on the Sugeno integral carried over to the
  Choquet integral.
A rather extensive introduction to these topics is given in the book
  Generalized Measure Theory by Wang and Klir.

A: I'm not sure what you mean by "deep".  However, I believe you will think it reasonable to postulate that if something leads to the creation or expansion of new mathematics, then it seems reasonable to think of it as deep in some sense.  Well, fuzzy set theory can get said to have lead to the creation of new branches of group theory (and other branches of abstract algebra), topology, differential calculus, arithmetic, geometry, trigonometry, graph theory, etc., see here and here for potential sources.  On top of this, there exists possibility theory, and generalized measure theory where fuzzy sets usually come up at some point one way or another.  This, of course, is not an argument that you should like fuzzy set theory, nor an argument that you should dislike fuzzy set theory, just as if you read enough philosophers you'll almost surely find someone extremely deep who you intensely dislike. 
A: When I think of "Fuzzy logic" the first thing that comes to mind is engineering, where is it used in control systems, and information processing. For example, "fuzzy logic" is used in control systems for washing machines. I think it was a hot topic in 1980s and 1990s, and one article from that time is "Designing with Fuzzy Logic" by Kevin Self from the IEEE Spectrum magazine in 1990. 
My opinion as a logician is that "fuzzy logic" per se has not developed significant interest in mathematical logic for the study of foundations of mathematics. I have not seen any examples of "deep" applications of fuzzy logic to classical foundational issues. One reason why it will be difficult to develop deep foundational implications of fuzzy logic is that much work in foundations is motivated by a few standard structures (ignoring any philosophical objections): the natural numbers, the real numbers, the cumulative hierarchy of sets. These are not ordinarily conceived as "fuzzy" structures. 
As a personal aside, I would not use the term "fuzzy logic" to refer to multivalued logic in general, and I think it is somewhat ahistorical to refer to work of Łukasiewicz as "fuzzy logic". However, others do use "fuzzy logic" more broadly, and there is a book by Hájek entitled "Metamathematics of Fuzzy Logic", so my opinion is far from universal. 
None of this means there is a lack of value in fuzzy logic, of course. It is obviously of great interest for information processing, control theory, and engineering, where it may indeed have deep aspects. 
The introduction to the Stanford Encyclopedia of Philosophy article on fuzzy logic also remarks on some of these issues.  In that terminology, I am apparently speaking about "fuzzy logic in the broad sense".  
