# Is fuzzy mathematics useful in pure mathematics ?

Fuzzy sets and logic seem to be mostly used for applying to real-world situations, control-theory, game-theory, economics, statistics, data management, artificial intelligence, automated reasoning etc

Are there any proofs of theorems in pure mathematics of a non-fuzzy nature that make use of fuzzy concepts ?

Fuzzy set theory may be defined axiomatically and therefore be "pure" however here are some quotes from the Fuzzy logic article at Scholarpedia which highlight the applied nature: "Humans have a remarkable capability to reason and make decisions in an environment of uncertainty, imprecision, incompleteness of information, and partiality of knowledge, truth and class membership. The principal objective of fuzzy logic is formalization/mechanization of this capability."

"During much of its early history, fuzzy logic has been an object of skepticism and derision, in part because fuzzy is a word which is usually used in a pejorative sense. Today, fuzzy logic has an extensive literature and a wide variety of applications ranging from consumer products and fuzzy control to medical diagnostic systems and fraud detection"

If you're thinking that the idea of fuzzy proofs of nonfuzzy theorems is strange, then I would say that it doesn't, on the face of it, seem to me to be any less strange than proofs by the probabilistic method.

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As far as I can tell from the Wikipedia page, en.wikipedia.org/wiki/Fuzzy_mathematics, fuzzy math is a part of pure math. Please clarify what you mean. – Zev Chonoles Jan 17 '11 at 14:04
This question has already attracted two votes to close: it is somewhat offensive to people working in "fuzzy mathematics" (of which I don't know anybody), and also, it is very vague. You should rephrase your question (use the edit button). Try also to be explicit about the following: what would you like to learn from the answer? – André Henriques Jan 17 '11 at 14:09
I believe I mildly support the reopening of the question. Some complained about the last sentence (containing the word "madder"). I found this to be an argumentative rather than contentful question, so I have deleted this sentence. – Pete L. Clark Jan 18 '11 at 2:33
@Todd: I think I agree with you, but it is not a useful way of framing the question. (After all, there is nothing "mad" about probabilistic proofs...) – Pete L. Clark Jan 18 '11 at 7:13
@unknown: relax. Don't worry about one comment upvote. If people are at all offended, they will surely let you know. (FWIW, I didn't take what you had written earlier as disparagement.) – Todd Trimble Jan 18 '11 at 8:09

## 1 Answer

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.

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