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I'm currently in a class on formal mathematics/formal logic/axiomatic set theory. Someone asked, "At the end of the day, as mathematicians, why do we care about rigor?" My professor gave an example of a mathematician who did not believe in the importance of rigor (implicitly refering to a time in which mathematics was rigorous by modern standards) and developed a number of results which others took up, following in his tradition of non-rigor, which eventually formed and became recognized as a branch or subfield of mathematics on its own. He said that the whole field was non-rigorous, and when outside mathematicians eventually bothered to look at their results rigorously it was found that the whole field was unfounded due to inconsistencies in some founding axioms. I asked what the name of the field or of the mathematician were, but he did not remember details. He only said that he thought it was a group of french mathematicians in the early 1900's studying a subfield of topology.

My question is this: Does anyone else recognize this example, and if so, could you provide me with details/references? I have already dome some searching using a variety of search terms, as well as poked around on various mathematics-related SE sites and didn't find anything that looked like what my professor was talking about.

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    $\begingroup$ Most likely, it is garbled version of the early development of geometric and algebraic topology (by Poincare, Dehn, Kneser and others). Garbled, since lack of rigor was not because the early practitioners of topology did not believe in such (or had wrong axioms), but because they were lacking adequate definitions and tools. While they made some mistakes, most of these early results were eventually justified (in some cases, it took 50 years, like with "Dehn Lemma"). $\endgroup$ – Moishe Kohan Apr 9 at 0:38
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    $\begingroup$ Henri Poincaré is hopefuly the french mathematician (and leader of a certain group) your teacher was thinking to.with the (at that time) emerging domain of topology, called by him "analysis situ". $\endgroup$ – Jean Marie Becker Apr 9 at 10:29
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You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with lack of rigor.

The short summary is that the school started with some unfounded postulates, that they used to derive a wide number of results. This must be understood in the context of a incipient field, where no rigorous foundation was available, and occurred concurrently with the struggle to give analysis a rigorous foundation (late 19th century) so it is not even clear what debate about rigor could be had at this point.

In any case, the first results could be proven by different methods in a much more laborious way, so that confidence grew in their methods. Eventually, some results of the italian school we're disproven, foundational problems we're found that shown their methods inconsistent, and the whole tradition was abandoned in early twentieth century.

This summary is likely unjust, in the sense that it does not credit the members of the italian school for advancing the field at its naissance, and does not discusses that some memebers were more careless than others, and further that standards of rigor have evolved during that time, so it is not fair to judge a whole mathematical scene on some wrong results from some of the involved.

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    $\begingroup$ This description of Italian school of algebraic geometry is inadequate. They made many important contributions, and since that time most of their results were put on a rigorous basis. $\endgroup$ – Alexandre Eremenko Apr 8 at 23:52
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    $\begingroup$ From the modern point of view all Analysis of 18-19 centuries was "non-rigorous". But all these discoveries constitute the core of modern mathematics. When the demands of rigor changes, most of them were justified in the modern framework. Same happened with Algebraic geometry. $\endgroup$ – Alexandre Eremenko Apr 8 at 23:58
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    $\begingroup$ @AlexandreEremenko, I fully agree with your comments. My answer did not intent to pass judgement, only to describe. the Italian school (maybe unfairly) has become the go-to tale regarding the importance of rigor. I tried to point out that they indeed arrived at correct results and furthered the field, but some later results were shown wrong, and eventually the school as a program withered. I do not think it is fair to judge the work done, as you said many standards of rigor change with time. I do think the OP question may have been about the italian school. $\endgroup$ – cesaruliana Apr 9 at 22:32
  • $\begingroup$ Also, if my answer seems to imply something wrong about the italian school I'll be happy to reword it, or delete it if people think that the answer is irremediably misleading. $\endgroup$ – cesaruliana Apr 9 at 22:34
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    $\begingroup$ I think this is a good answer and was the first one I thought of when reading the question. Here is a link about examples of specific wrong results by the Italian school of algebraic geometry: mathoverflow.net/questions/19420/… $\endgroup$ – KCd Apr 9 at 23:13
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Fuzzy set theory that was first introduced in 1965 as an extension of the classical notion of sets could be such a field you are looking for. In general, fuzzy logic resembles multi-valued logic but lacks a rigorous mathematical foundation. As the result, its conclusions are plausible but inexact (or fallacious).

However, the uncertainty issues tackled by fuzzy set theory are often important real-world problems beyond the realm of probability. For example, one of the most active applications of fuzzy sets is to model linguistic expressions involving adjectives and adverbs because there are no rigorous mathematical models for them so far. As the result, a lot of researchers love working in fuzzy sets because they are attracted by its plausible appeal and choose to ignore (completely) the non-rigorous fact of it.

As a result, fuzzy set theory has become a fast growing field (of mathematics) since it is introduced and boasts the most number of research papers among all fields of mathematics nowadays. Many people in practical fields such as electrical engineering, artificial intelligence and so on, like to work in fuzzy set theory.

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