Have a group of mathematicians ever gone wrong for an extended period of time?

I normally think that in mathematics individual errors are caught by the group, and that collectively mathematicians proceed forward without making any collective errors, but is this true?

Have the community of mathematicians ever gone wrong and collectively made some error - only to years later have the error identified and corrected?
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phoffricConnect With a Mentor Commented:
I recall that for a long time, Fourier Analysis was not accepted by the math community.
Every field of research has its horror stories about great ideas that were initially rejected by the research community. In mathematics, Fourier was ridiculed, Galois’ work was rejected, the Dirac delta function was laughed at, and Cantor’s work in set theory was highly controversial. All of these ideas are standard today. A classic case in control engineering is the development of the negative feedback amplifier, a spectacular technological breakthrough recounted in:

H.S. Black, “Inventing the negative feedback amplifier,” IEEE Spectrum, vol. 14,
pp. 55-60, Dec. 1977

B. Friedland, “Introduction to ‘Stabilized feed-back amplifiers,’” Proc. IEEE, vol.
87, no. 2, 1999.

Unfortunately, most of the credit went to Nyquist, who expanded on Black’s insights in his stability theory.

... marked the birth of modern mathematical signal analysis, dates back to Joseph Fourier’s (1768–1830)  investigations into the properties of heat transfer in the early 1800’s [14]. Fourier conjectured that an arbitrary periodic function, even with discontinuities, could be expressed by an infinite sum of pure harmonic terms. The idea was ridiculed among many great scholars, but a large portion of mathematical analysis in the 19th and 20th centuries was dedicated to making the statements of Fourier precise and the Fourier transformation eventually became a tool of utmost  importance in signal analysis, quantum physics and the theory of partial differential equations.
it_saigeConnect With a Mentor DeveloperCommented:
Yes and no.  The acceptance of an incorrect theorem by the collective cancels out the original thought that an individual is the only person that makes an error.  This is true in all aspects of life.  This is how we as a species learn, grow and, hopefully, advance.


One popular example:


Great examples by the experts.
To answer your question, I would have to know what you consider a mathematical error. Applying mathematics to a physical situation where it does not fit may or may not be considered an error. For example, the mathematics in the physics of Newton is correct in most situations, but not if it involves high speeds. So was that wrong mathematics?
What is the sum of the interior angles of a triangle? Is it 180 degrees (pi radians)? Yes, in Euclidean geometry it is, but not in spherical or hyperbolic geometry. But none of these is more correct than the others. It depends on the axioms that you accepted for developing the geometry. Which one applies to the "real" world the best? That depends. Build a house and Euclidean geometry is fine, but not for surveying a continent or measuring light in the vicinity of a black hole where the others are more descriptive.
Lastly, I would say "pure" mathematics can never be wrong. It is built on accepted, not proven, axioms. Now those axioms can be consistent and produce all sorts of consistent results, or they can be inconsistent are produce ambiguous results.
How any mathematical system is applied is where it can either be a correct application or a wrong application. Also, opinions as to whether a theorem is part of a particular system may turn out to be either correct, wrong or not provable either way.
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purplesoupAuthor Commented:
Thanks - great answers.

To clarify what I mean by error - I accept in applying mathematics - or applying some rule about how things are - we can never know all the circumstances required for the rule to be true. It may turn out that there was some hidden assumption that we weren't aware of, such as when Newtonian equations of motion are applied at high speeds.

But this seems to take us into the whole realm of inductive logic, and what I was interested in was getting deductive logic correct. With deductive logic there is a "right answer", but as we know individuals can make mistakes and get things wrong, so the normal procedure is to have your work checked by others, on the grounds that the more people perform the calculation the less likely it is to make an error, and I guess with computers able to do proofs the chance of error becomes less likely (although there may be a bug in the programme of course).

My question really arises from reading Descartes, who felt truths of experience were always possible to doubt, but that mathematics and logic could never be doubted. My thought was that a logical proof can't be doubted if done correctly, but how do we know it was done correctly?
Mathematicians only?
(or math errors by scientists that caused serious tragic events)
purplesoupAuthor Commented:
I'm really focussing on being certain of something. What could we say we are most certain of (apart from our own existence)? One might think we can be certain of a mathematical proof if it has been checked by a lot of top mathematicians, but can we?

Perhaps there is some link between the two? In the case of for example getting the units of some calculation wrong, there were probably quite a few experts who saw the calculations and didn't spot the error.
The only thing that we can say for certain is, things change.  Change is required for learning and advancing.  People make mistakes and, often times, these mistakes are not apparent...  Sometimes, it takes for a problem to surface before the mistake is found.  While other times, the mistake is on purpose because the current evidence supports the mistake.

Unfortunately the top airline industry specialist in fatal crashes say that despite the unbelievable efforts in technology and every possible mathematical calculation on weight, shape and design of parts, etc............there are still numerous unknown ways a plane can have a catastrophic failure.

And the industry doesn't know what they are or how to fix them until the next fatal crash caused by one of the unknown problems. And Boeing and Airbus have top of the line experts to make all the calculations. Yet their unknown math errors will lead to a fatal event between now and later.
phoffricConnect With a Mentor Commented:
Sir Andrew John Wiles has provided a proof of Fermat's Last Theorem. I wonder if it is even conceivable that there is an error in it. You can watch the documentary of Wiles' struggle here:

Or read about it here:

After the announcement, Katz was appointed as one of the referees to review Wiles' manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap.
Had the questions been asked differently, perhaps Wiles would not have seen the error, and perhaps once published, other mathematicians would not have seen the error for awhile.
I don't know of any mathematical system that has been wrong for any period of time. For example, in Euclidean Geometry, there are Undefined Terms, Axioms (accepted without proofs), and Theorems, Corollaries, Lemmas, etc. So following the rules of logic, the sum of the interior angles of a triangle is 180 degrees in Euclidean Geometry. And that will never change unless you change the Undefined Terms or Axioms.
But again, applying that to the "real" world depends on whether a "line" in the real world is a  "Euclidean Line". But that goes beyond a mathematical system and gets into applied mathematics.
purplesoupAuthor Commented:
I did want mathematical proofs and the detail of mathematics that had been accepted by mathematicians as being rigorous and correct and then later discovered to be in error.

This wasn't designed to be a discussion on human error or the difficulty of being certain about the behaviour complex systems.

Can we keep it in mathematics, and I'll close it tomorrow in case there any any further suggestions?

I did a paper in college 40 years ago on mathematical rigor (glad to see someone else use the word). Any errors that persisted were more in the area of opinion as to whether a theorem could be proven or not. Any errors like the supposed proof of Fermat's Last Theorem were the result of lack of rigor which is usually a gap where something was assumed that had not been proven.
purplesoupAuthor Commented:
Thanks for all the information. It sounds as if there haven't really been any examples of something that was mathematically wrong being accepted as something true - defining error in terms of mathematical rigor - although there have been short intervals in which a proof was accepted when not all assumptions had been proven. I'm not sure about the four colour theorem - that seems to have had a proof which the community was accepting as true but which wasn't? Was that an exception?

However there have been plenty of examples the other way around - in which something which is now accepted as mathematically true was thought by the community to be false. It sounds as if this wasn't to do with rigor, but with the introduction of new areas of mathematics.

I expect my question wasn't clear enough, but I would conclude that in terms of mathematical rigor, where we are talking about the correct application of an agreed procedure to achieve some proof, there has never been a case where the mathematical community has thought something was right when it was not.
Never say never. Principia Mathematica
was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,[1] being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.
Ref: http://en.wikipedia.org/wiki/Principia_Mathematica
purplesoupAuthor Commented:
Ok I take Godel's incompleteness theorems to illustrate the hierarchical nature of logic, and therefore that you are going to have some assumptions that can't be proven.

I'm not sure it is evidence of mathematicians collectively believing something had been rigorously proven only for many years later someone coming along and showing there was an error in their work.

I guess you might say - perhaps - that mathematicians for many years believed the sort of project Russell and Whitehead worked on was possible and Godel proved that assumption wrong, but I don't know if there is any evidence for that, and in any case it was just an assumption not some accepted proof.
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