The studies mentioned in this Eurekalert article seem interesting, but it was the following passage that caught my eye:

When mathematicians prove theorems in the traditional way, they present the argument in narrative form. They assume previous results, they gloss over details they think other experts will understand, they take shortcuts to make the presentation less tedious, they appeal to intuition, etc. The correctness of the arguments is determined by the scrutiny of other mathematicians, in informal discussions, in lectures, or in journals. It is sobering to realize that the means by which mathematical results are verified is essentially a social process and is thus fallible.

I couldn't help thinking that it doesn't sound so different from how the natural sciences work.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

Thanks. I'm going to read those this weekend. I don't know about the natural sciences, but that's certainly how economics works
Anyway, and excuse the cliché, but our brains aren't that bad you know...

Penrose said he was once talking to someone and something flashed in his mind (the "lightbulb" of cartoon thinkers?)--he was in the middle of the conversation and didn't pay much attention to it till that night.

That night, he tried to remember what had caught his attention momentarily...it was a thought that just came to him--and it took him several days to write up the proof to the theorem in words, that he could "see" in an instant.

One way mathematicians (all, to an extent, some more than others, Penrose probably more than most) do math is to use a "brain language" that doesn't use words, or even mathematical symbols, just concepts, idea, entities with no name, sometimes pictures or analogies (and a picture is a type of analogy), etc. Something inside says, "yeah, that HAS to be true because (random-seeming neuron firings)". Now, to translate the neuron-firings into words.....

I guess the simplest form would be like in the American Mathematical Monthly (I think that's the right one) which every month, presents a "proof without words"--using pictures. Often, the pictures make a mathematical idea "obviously true" when writing a formal proof in words and symbols might take more time. Penrose seems to have evolved beyond pictures and into a deeper "brain language" for math.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

It is true that sometimes a mathematician (even a pretty good one) will fool himself, though. A teacher of mine said he could find most errors in a paper he was to publish by searching for the phrase "it is obvious that" in the paper. That phrase usually means he thought it too obvious to write up a proof, and sometimes what seemed obvious at 2am wasn't even true!

Peer reviews (almost always) catch these mistakes.

Luckily (kind of like science), mathematicians try to find alternative proofs for things--especially if the only known proof is long and complex (e.g. Fermat's Last Theorem, Four Color Map Theorem)--kind of like "independent verification"--also, some empirical science is done too (a lot of examples of FLT and Four Color were checked by hand and by computer to verify them). But mathematics is unique in that it is set up in such a way that it is possible to provide a rigorous proof (in theory--and we know that because there ARE (lousy) automated theorem provers for computers) and we demand that the author write it up (within reason--writing it so an automated theorem prover can understand it would be very, very tedious, be effectively unreadable to humans, and be very, very long) so that peer reviewers would look at it and agree (or ask for clarification in case a statement seems true but the reviewer can't verify it himself).

Math has succeeded remarkably (including lots of applications, for areas not thought applicable even!). The first "proof" of the four-color theorem by Kempf stood for 11 years before someone noticed the mistake. That is actually rare--mistakes are almost always noted faster. It was this 11-year gap that made the theorem well-enough known to get so much attention, and ultimately a proof not that long ago (when I was still in college).

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Good grief. 11 years! I never knew that. It's completely fixed now is it?

Also, no-one uses brackets quite like a mathematician.

The article quoted is not quite correct. It is true that there is some narrative, but that it is explanatory in nature and is rigorously logical. It is true that details that will be understood by experts are somewhat "glossed over" but only in the sense that those details are well understood, and if there is a mistake is will be caught by referees or readers. Appeals to intuition are NOT part of the proof, although sometimes in the narrative, outside of the proof of the specific theorems there is discussion that is intended to provide and intuitive picture. But that appeal to intuition is most assuredly not part of the proofs that are presented, at least in serious mainstream journals. The most important point is that the process by which mathematical results are verified is NOT a social process. It is fallible, simply because it is carried out by human beings. But the process is quite rigorous and it is not likely that a mistake will stand for long.

But it does occasionally happen. Generally important results receive a great deal of scrutiny by experts around the world and are the subject of intense scrutiny by several people in concert and individually. Seminars at individual math departments may be devoted to working through the details of particularly important results. But the process is as objective as is possible given the dependence on humans. Eventually mistakes are found out.

Sometimes is takes a while for a subtle mistake to be uncovered. I am familiar with one rather striking example. There is an important open problem in mathematics known as the "invariant subspace problem". It has some implications, I believe, for quantum mechanics. The problem is to determine whether or not it is true that any bounded operator on an complex Hilbert Space of dimension greater than 1 admits a non-trivial closed invariant subspace. There was a body of work bearing on this problem called the Tomitia Decompositon Theory, and occupied a chapter in an early version of Naimark's book "Normed Rings". In the early 1960's Pasquale Porcelli was conducting a class using Naimark's book. In the class was a rather precocious junior by the name of Joe Taylor. The class was actually presented by the students who lectured on the material and worked out problems on the board. When they got to the chapter on the Tomita Decomposition Theory, Joe presented a counter-example. That chapter is no longer in the book. Joe received a Ph.D about a year later, for different work on the measure algebra of a locally compact abelian group. So the Tomita Decomposition Theory rather belatedly was invalidated, but nevertheless it was invalidated.

More commonly mistakes are discovered much earlier in the process and never see wide publication and acceptance.

I can also recall a seminar in which I participated on Kirillov's book on representation theory. So many mistakes in that book were found that the seminar nearly ended in a book burning. That served to demonstrate the unreliability of the Russian literature. But in that case no major widely known results were invalidated, we just identified a lousy book.

Major results receive enough repeated scrutiny that it is unlikely that mistakes that will have any real effect on the broad body of mathematics can stand for long. This is important as mathematics, unlike the sciences, does not advance as a series of approximations. It is important that the major theorems on which subsequent research is built be rock solid. For that reason one finds that major theorems are heavily scrutinized, checked and re-checked and quite often alternate proofs are discovered and presented, further butressing the validity of those results.

Sometimes the original proof is found to be flawed, but the result itself is correct. That was the case with Smale's original proof of the Poincare conjecture in dimensions 5 and above. His proof had flaws, but his idea of attacking the problem in higher dimensions was of great merit. In fact once the idea was put forth, other topologists, Adams in particular, were immediately able to put together proofs that got around Smale's mistake. Smale still deserved and received a Fields Medal for his work. In the sense of this example one might consider the process social. But ultimately it is rigorous logic that validates the result itself. And the process of learning mathematics requires that each succeeding generation of mathematicians work through the details of the important results that they use in their own research, providing continuing checks on those results. Humans are fallible, but with number of independent minds that review and work through the important theorems it is unlikely that all of them can be fooled for very long.

A small off-topic statement Refs:
I get the feeling from this, and a couple of other threads, that maths and sciences are regarded as significantly different. Not highly significantly perhaps, but it seems to be there.
Which I found odd really. I think of them as closely related subjects. For example, I wouldn't consider myself to be good at, say physics, unless I thought I was good at maths as well.
Hmm. Anyway, ignore that. Carry on.

This has been discussed a bit elsewhere, but it is an important side bar. To put this in perspective, my final degree was in mathematics and my first job was a research mathematician at a large university. Mathematics is not a science. It differs from science in that experiment is not a part of the process of determining the validity of mathematical "truth".

Mathematicians and physicists are in fact somewhat different breeds of cats. Mathematics does receive many good questions and research incentives from physics. Physics does make use of some mathematical results. But they are different disciplines, even if you restrict your attention in physics to the theorists. Many physicists are adept at the use of certain types of mathematics, but they are not mathematicians and in fact view mathematics a bit differently than do mathematicians. Many mathematicians are quite conversant in physics, and many have undergraduate degrees in that subject. But they approach physical problems in way that is different from the approach taken by physicists.

Basically physicists usually consider mathematicians as a picky and overly concerned with rigor, sometimes getting blocked by rigor from reaching necessary physical conclusions. For instance, physicists are relatively comfortable with the procedure in quantum field theory known as "renormalization". Mathematicians are still waiting for a rigorous definition of the process.

Mathematicians consider physicists as rather loose with mathematics and lacking in attention to detail and rigor in logic. They worry that conclusions reached by nonrigorous methods may not stand up to scrutiny and may be shown to be false. Physics proceeds by successive approximations. Mathematics proceeds in small steps but each step is checked and re-checked for absolute truth (really for absolutely correct logic based on a very small set of universally agreed-upon axioms). Mathematical "truths" are not refined. Mathematicians and physicists are very different in methods and outlook. But they need each other.




A mathematican and physicists (usually this is told with an engineer in place of the physicist) are presented with a problem. They are in a cabin in the woods near a stream. The cabin has a window overlooking the stream and curtains on the window. A bucket lies below the window. The bucket is empty. The curtains are on fire.

The mathematician and the physicist each solve the problem by taking the bucket to the stream, filling it with water, returning to the cabin and extinguishing the fire by throwing the water on the curtain.

Now the same problem is presented with one change. The bucket below the window is filled with water.

The physicist picks up the full bucket and throws the water on the curtain.

The mathematician picks up the bucket, pours the water on the floor, reducing the problem to the one previously solved.

Lol! That is funny. Explains it perfectly as well.

You have undoubtedly heard the following joke:

A mathematics professor is giving a lecture when a student stops him to ask why one of his assertions is true. The professor says "It is obvious." But then he stops and stares at the board for several minutes. Then he leaves the lecture hall, all without any further words.

Quite a bit of time passes and the normal dismissal time draws near. One of the students goes down the hall to the professor's office and sees him working at his chalkboard. The student returns to the classroom.

The professor returns to the class a couple of minutes later and announces "Yes, it is obvious."

It gets better. I once told that joke to a rather senior engineering professor from MIT. He responded, "That is not a joke. The professor was Norbert Weiner. I was in the class."



You are dating yourself (and me). That proof was published about 1976 in the Illinois Journal of Mathematics by Appel and Hakin. That issue of the journal included micrifiche with the computer output. I hate to admit it, but I must agree that it was not that long ago. :-)

I even recall seeing a short article on the proof in the weekly Science News and showing it to a graph theorist in the department. He had not heard of the proof and it generated a bit of excitement.

And I can see why!

That "appeal to intuition" was thrown in there for just that purpose, it's bogus.

Anytime that is done, in practice, it is only to motivate others to work on the actual proof, it is never accepted as a mathematical proof. That is not done when "mathematicians prove theorems in the traditional way"
I've often said the same thing since I burned it into my brain over thirty years ago.

Yes--though you get disagreements, like with anything

the proof had to be checked by computer, and that makes some nervous--after all, computers do flip bits (though less often than brain cells misfire).

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Yes, I heard the joke, from my thesis advisor (before he became such).

Ok--1976--I was thinking more like 1990s. In 1976, I was in Kindergarten, not yet freezing as it took a couple years for our school to turn down the thermostat.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

1976--that means I was lied to! In the fifth or sixth grade, 'round 1980, I went to "Math Field Day" where I first learned of the "Four Color" problem--the speaker told us he had a counterexample map but the darn cleaning lady threw it away! I wonder if he knew it was proved. He told us that there was no proof at the time.

Of course, he "could" have drawn his counterexample on a donut, as the theorem applies only to maps on a plane or sphere.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

I agree with you that they are closely related, although not the same.

Exactly, and that is what I meant.

I think you two are thinking of an idealised notion of mathematics, rather than of how mathematics has actually worked, historically.

I mean, sure, since the 19th century there has been great concern with rigor in mathematics. But if you look at the "golden age" of the 17th-18th centuries, you had Newton and Leibniz and Kepler and many others happily manipulating entities such as infinitesimals and fluxions and infinite series, which were only defined rigorously centuries later. Folks like Heaviside paved the way of great mathematical advances even though they never defined their terms rigorously. I would describe their work very much as an approximation, by modern standards. Nowadays even, mathematicians will publish articles that assume the validity of conjectures like the Riemann hypothesis, for example, which remains unproven thus far.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

Actually there are versions of the coloring theorems that apply to surfaces of various genus. The chromatic number varies with genus.

http://www.jstor.org/pss/2037902
http://mathworld.wolfram.com/ChromaticNumber.html


Wolfram, BTW puts the date for the proof as 1977, but my memory says 1976. In any case I am sure it was published in the Illinois Journal of Mathematics. http://mathworld.wolfram.com/Four-ColorTheorem.html

Can you think of a proof, that the author considers a proof, and that we (the contemporary audience then, for historical proofs) considered a proof, where the author directly made an appeal to intuition? It is true that unspoken assumptions have slipped into mathematical proofs, and later had to be "tightened" up, we're only human.

Haken.

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My point was that that tightening up you speak of hasn't always been automatic, historically.

Of course, I don't deny that in contemporary mathematics great care is taken not to let anything depend on intuition. Yet, for all that care, mistakes are still made, and it often takes input from other egos to spot them. We're all human, indeed. But the corollary to that is that we tend to get better results when we work with others.

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

What does it mean to be "automatic" in this sense? I have a feeling it's as automatic as it could be, but I'm not sure what you mean by automatic and I don't find any reference in the other posts.I'm just talking about appeals to intuition, not mistakes, nor Andrew Wiles.

I am thinking of research mathematics as I have personally practiced and witnessed it.

I consider Heaviside an electrical engineer and physicist, not a mathematician, although others may feel diferently.

Some might consider Feymnan's path integrals as mathematics, but theyt are not on solid mathematical footing. They represent good (excellent) physics but not mathematics.

It is fair game to demonstrate an implication of the Riemann hypothesis, but do not consider that as an assumption that it is true. It is rather widely recognized that the Riemann Hypothesis remains an open question, and implications of it are NOT theorems. Not until it is proved.

An appeal to intuition in a mathematical proof is immediate grounds for dismissal of the resuslt as a theorem and re-classification as a conjecture.

Correct. But I don't feel too awful bad about the mistake, given that I was working from a memory over 30 years old.

The joke is pretty old. But that fact that it is true story and not just a joke is not widly known. I heard the joke over 30 years ago. I learned the reality of the story only about 11 years ago, and don't know of any publication of it.

Ok Disinfo, I carefully read what I understood (the waffle), and skimmed through the parts I didn't (computer programming). Interesting recursive (?) problem:

1. Formal proof is tedious and time consuming.
2. Computers are ideally suited to this task.
3. Computer software is full of bugs.
4. Checking software code for bugs is tedious and time consuming...

Luckily (4) seems to be less of a problem than (1). Also, they're slowly buliding up libraries of computer-completed proofs which they hope mathematecians can now use as 'building blocks'. Formal proofs for 80% of the 'top 100 maths theorems' have already been completed. Link.

Anyway, that's some of what I could gleam from a first read...

An appeal to intuition can be a mistake. That's your point.

My point is that often in the history of mathematics -- as with fluxions and infinitesimals and Euclid's Parallel Postulate -- correct propositions were provisionally justified by an appeal to intuition, sometimes for centuries before they could be put on an impeccably logical footing. Not that this was a bad thing. It was, as you say, the best they could do at the time; but that doesn't make it any less of an appeal to intuition.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

O no, I'm saying that the appeals were not recognized as appeals to intuition. Even since Euclid, an appeal to intuition was seen as invalidating a proof.Eucild's parallel postulate, considered as an appeal to intuition, is not a mathematical error! Recognizing it as an unproven premise removes it from the proof.

I was asking for examples of appeals to intuition within a proof. But maybe I see more of your, or the OP, point: that by our choice of unproven premises, and assuming they apply to reality, we're making appeals to intuition. Still, that seems to be outside of mathematics.

To amplify your point, unproven premises, axioms and postulates, are quite well recognized for precisely what they are in mathematics. Euclid's parallel postulate, is precisely that, a postulate. It is consistent with the other axioms of Euclidean geometry and produces a consistent theory of geometry. It is neither true nor false in absolute terms. It does accurately describe the ordinary geometry of the plane (rather by definition) and seems to describe the real world very well locally (and this latter observation is a statement regarding science and not mathematics). But there are other perfectly consistent geometries that do not assume Euclid's parallel postulate, and in fact violate it. They too are solid mathematical theories, and seem to be useful in physics -- for instance in general relativity.

You may recall that there was for many many years an attempt to prove the parallel postulate from the other axioms. It was only with the discovery of non-Euclidean geometries that it was recognized that the parallel postulate is in fact independent of the other axioms of geometry.

Axioms and postulates are a necessary part of mathematics, and by definition they are not amenable to proof. Rather they are accepted as a basis from which other ramifications are logically deduced. All of mathematics is nothing more than the uncovering of statements that follow logically and rigorously from a set of axioms. One assumes that they are true for the purpose of further investigation, but they have no absolute truth. If you accept them, then that which is derived from them must also be accepted. You can choose not accept them, but there seems to be little utility in doing so from a more practical point of view -- the acceptance of the usual set of axioms has proven quite fruitful in the attempt to devise mathematical models for the natural world. The assumption that the axioms reflect "reality" is an assumption of science, primarily physics, and not a part of mathematics itself.

That said, I am not aware of any school or any individual who has chosen to reject the Peano Axioms. If you reject what is essentially a axiomatic formulation of the natural numbers, then you really are limited.

I am aware of the constructivist school that has chosen to pursue mathematics in the absence of the axiom of choice. They have been able to make some surprising progress without that tool. But most mathematicians work within a system that does accept it. Paul Cohen showed years ago that it is independent of the other axioms of set theory.

The basis for theorems is usually explicitly stated or at least well understood in the mathematical community. Use of the usual set of axioms is accepted and it is known that the theorems are dependent on those axioms. This is not an appeal to intuition, but rather a general agreement on the axioms from which the remainder of mathematics is derived -- if nothing is said it is safe to assume that a theorem is based on the Peano Axioms (or Zermelo Frankel set theory to be more rigorous) and the axiom of choice The axioms have themselves been studied and the pitfalls and foibles are known (Godel's incompleteness theorems for instance).

An appeal to intuition, not a postulate or axiom, in the course of a mathematical proof, is, as you noted, grounds for immediate rejection of the proof. It simply isn't done, old boy. At least not knowlingly by mathematicians.

Kempe's false proof was in the 19th century. It took until 1976 to prove it properly, by a different route, although Kempe's idea's remained crucial to the proof. Though because of the computer assistance to the proof, I think it took time to get it accepted. I seem to recall it was in about 1983, when I was studying Maths at university, that there was an announcement of proof of the Four Colo[u]r Theorem. My maths tutor was proud of a letter he received with a postmark announcing "Four colo[u]rs suffice!"

Thanks for that. I've been doing some reading about the FCT and it's history. Just basic wiki stuff. Appel and Haken. The 2nd article mentioned in the OP also deals with the FCT, although I must admit I found it tough going.

Every field has problems accepting new techniques I think. I seem to recall that you're an economist? We had our econometric squabbles. Maths and stats to describe human behaviour? How preposterous...

And colour is definitely spelt with an 'u'. The jury's still out on American English.

But they were!
Even Euclid was unsatisfied with the Parallel Postulate, according to Proclus.

And as for infinitesimals/fluxions, there's Bishop Berkeley's famous polemic against them, which however did not stop mathematicians from using them cavalierly in their arguments (some still do even today).

Right.

I don't see how that could be reasonably argued. Sure, these are questions with some philosophical flavour. But to say that they lie completely outside mathematics...!

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.