A branch of mathematics which refused to be rigorous?

Question

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.

Answer 1

You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with a lack of rigour.

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 an 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 rigour 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 were disproven, foundational problems were found that shown their methods inconsistent, and the whole tradition was abandoned in the 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 discuss that some members were more careless than others, and further that standards of rigour 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.

Answer 2

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.