6

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.

The Ledge
  • 179
  • 4
  • 6
    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"). – Moishe Kohan Apr 09 '21 at 00:38
  • 1
    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". – Jean Marie Becker Apr 09 '21 at 10:29

2 Answers2

10

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.

Pablo GM
  • 7
  • 3
cesaruliana
  • 728
  • 5
  • 11
  • 3
    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. – Alexandre Eremenko Apr 08 '21 at 23:52
  • 3
    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. – Alexandre Eremenko Apr 08 '21 at 23:58
  • 3
    @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. – cesaruliana Apr 09 '21 at 22:32
  • 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. – cesaruliana Apr 09 '21 at 22:34
  • 3
    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: https://mathoverflow.net/questions/19420/what-mistakes-did-the-italian-algebraic-geometers-actually-make – KCd Apr 09 '21 at 23:13
  • I voted down both the question and this answer. What does it mean "refused to be rigorous"? Rigorous foundation of Algebraic geometry simply did not exist at that time. What does it mean "some of their results were wrong"? Whose "theirs"? Some of the results of SOME 18 century mathematicians in calculus might also wrong. Does this discredit Newton, Leibniz, Taylor, Euler, Bernoullis, Maclaurin who created calculus?? – Alexandre Eremenko Apr 10 '21 at 01:39
  • @AlexandreEremenko, thank you for the time to reply. I have reworded the answer to be more clear and incorporate your critique. I could not completely reword it because both upvotes and downvotes have been cast and changing it too much would be not be fair to those who have cast their votes. I sympathize with your complaints and hope to have improved parts I was able to do so. I reiterate that I will delete the answer if people think that the whole framing of the question and answer leads to essentially bad interpretation of math's history – cesaruliana Apr 10 '21 at 08:08
  • What "unfounded postulates" do you have in mind? I do not think they argued axiomatically. They worked with algebraic subvarieties in affine and projective spaces, so there was no need for any special "postulates." – Moishe Kohan Apr 10 '21 at 13:09
  • @AlexandreEremenko I used the phrase "refused to be rigorous" as a shorthand in the title for what I think I explained in the question: that during a time in which the importance of rigor was widely accepted by mathematicians, a certain group of mathematicians chose to ignore the standards of rigor with which they should have been familiar as a result of their formal education. Of course, I think that shortening things for the title is justified, since people presumably go on to read the details of the question. If you think I could improve this, please let me know how. – The Ledge Apr 12 '21 at 13:57
  • I chose this as best answer for now because what you describe sounds fairly close to what I heard. It may be that discrepancies between your account and my professor's account are due to my professor's own infamiliarity with the details. Though if another answer comes around which is closer to what I described, I may choose it instead. – The Ledge Apr 12 '21 at 14:00
1

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.

Eugene Zhang
  • 793
  • 6
  • 15