Hmm, this author doesn’t win my confidence by stating on pg. 4 that “the mathematical induction principle … is a tautology, and as such it does not provide any new useful information to number theory, so it is superfluous and can be removed.” He goes on to suggest that we need induction as a metamathematical rule, and not an axiom (schema) as it’s normally phrased.

Of course, it’s not too difficult to show that PA (with the induction schema) is not finitely axiomatizable, and PA- (without induction) has just four axioms. So it’s obviously added something.

Nonetheless a fundamental flaw in a peer reviewed published attempted proof of Godel’s Theorem would be a significant achievement.

But this author is making an elementary error of logic: assuming that for all n S|-P(n) implies S|-(forall n)P(n) , and hence that induction is redundant. Or something like that. At least you can see roughly what the mistake is. Most crackpots simply make no sense whatsoever.

“Nonetheless a fundamental flaw in a peer reviewed published attempted proof of Godelâ€™s Theorem would be a significant achievement.”

Yes, not only an achievement, but a very unlikely one, given the immense scrutiny which Godel’s work received at the time and since. Most mathematicians found (and still find) Godel’s theorems extremely upsetting, so his proofs have had probably more scrutiny than those any other mathematician, with the exception of Cantor’s proofs for the uncountability of the reals. (Another common topic for the crackpots, attempting to disprove a result they do not like.)

far be it from me to criticize the author of this blog, however putting the title “Goedel’s Theorem is Invalid” on an article linking to an arXiv preprint (not peer reviewed) is irresponsible journalism and a bit sensationalistic. Perhaps the title, “Preprint claims Goedel’s Theorem Invalid” or “Goedel’s Theorem Challenged” would be more appropriate.

Why should you care about journalistic integrity? Well, if you are attempting to serve the mathematical community, one where facts and truth matter above all else, then you should be very careful about what you publish, lest mathematicians decide that you are no longer a trustworthy resource (and hence not worth reading). Overall I like this site very much, but please do be careful.

I disagree with you that in mathematics “facts and truth matter above all else”. I think the vast majority of pure mathematicians are completely agnostic about what is or is not true. Instead, they are interested in proof — how you establish what happens to be true, or falsify what is false. This is what interests and excites them, this is what they write about, and this is what arguments in mathematics are mostly about, not about whether particular claims are true or false. Instead, it is the scientific domains, such as physics and biology, where people argue about what is true or not.

Well, I didn’t mean that the theorems and definitions matter above all else. I mean that mathematics depends completely on each statement you use being true. For example, if a colleague tells me he may have disproven the index theorem (but the work is still a preliminary, in preprint form), and then I go around telling everyone that it is already done, and the index theorem is invalid, then I have diminished my credibility. Or let’s say I assume the falsity of the index theorem and then go prove some other stuff based on that. Well my results are basically worthless if it turns out that the index theorem is true (which is is). I agree with you that the process of proving theorems is what concerns mathematicians for the most part (although quite a bit of time is dedicated to structuring the framework of material–what connections there are between different fields, etc.) but if we go around claiming things to be true just because we read the title or abstract of a preprint, then that spells trouble. In order for mathematicians to reach new truths (or demonstrate new falsehoods), they to know what is already true or not. I don’t think that a site that purports to deliver news to the mathematics community should make such a bold statement (“Goedel’s Theorem is Invalid”) when that is just a preprint.

Whether or not it turns out that the proof presented in the preprint is valid, it is just a preprint right now. Until it has been checked and confirmed, it is a claim and should be treated as such. That’s all I’m saying.

There is no suggestion that we should argue about what is true and what is false, but we should be very careful that when we say definitively that something is true or false, we are able to back it up with a valid proof.

Hmm, this author doesn’t win my confidence by stating on pg. 4 that “the mathematical induction principle … is a tautology, and as such it does not provide any new useful information to number theory, so it is superfluous and can be removed.” He goes on to suggest that we need induction as a metamathematical rule, and not an axiom (schema) as it’s normally phrased.

Of course, it’s not too difficult to show that PA (with the induction schema) is not finitely axiomatizable, and PA- (without induction) has just four axioms. So it’s obviously added something.

Even by only reading the abstract, one can identify a flaw in the author’s argument. The error is the following invalid inference:

Premise: There is an error in proof X of Theorem Y.

Conclusion: Therefore, Theorem Y is invalid.

It seems to be a common mistake of naive mathematicians to assume that an error in a specific proof disproves a claim.

Nonetheless a fundamental flaw in a peer reviewed published attempted proof of Godel’s Theorem would be a significant achievement.

But this author is making an elementary error of logic: assuming that for all n S|-P(n) implies S|-(forall n)P(n) , and hence that induction is redundant. Or something like that. At least you can see roughly what the mistake is. Most crackpots simply make no sense whatsoever.

Sigfpe –

“Nonetheless a fundamental flaw in a peer reviewed published attempted proof of Godelâ€™s Theorem would be a significant achievement.”Yes, not only an achievement, but a very unlikely one, given the immense scrutiny which Godel’s work received at the time and since. Most mathematicians found (and still find) Godel’s theorems extremely upsetting, so his proofs have had probably more scrutiny than those any other mathematician, with the exception of Cantor’s proofs for the uncountability of the reals. (Another common topic for the crackpots, attempting to disprove a result they do not like.)

far be it from me to criticize the author of this blog, however putting the title “Goedel’s Theorem is Invalid” on an article linking to an arXiv preprint (not peer reviewed) is irresponsible journalism and a bit sensationalistic. Perhaps the title, “Preprint claims Goedel’s Theorem Invalid” or “Goedel’s Theorem Challenged” would be more appropriate.

Why should you care about journalistic integrity? Well, if you are attempting to serve the mathematical community, one where facts and truth matter above all else, then you should be very careful about what you publish, lest mathematicians decide that you are no longer a trustworthy resource (and hence not worth reading). Overall I like this site very much, but please do be careful.

Thanks.

To TheDude:

I disagree with you that in mathematics “facts and truth matter above all else”. I think the vast majority of pure mathematicians are completely agnostic about what is or is not true. Instead, they are interested in proof — how you establish what happens to be true, or falsify what is false. This is what interests and excites them, this is what they write about, and this is what arguments in mathematics are mostly about, not about whether particular claims are true or false. Instead, it is the scientific domains, such as physics and biology, where people argue about what is true or not.

Well, I didn’t mean that the theorems and definitions matter above all else. I mean that mathematics depends completely on each statement you use being true. For example, if a colleague tells me he may have disproven the index theorem (but the work is still a preliminary, in preprint form), and then I go around telling everyone that it is already done, and the index theorem is invalid, then I have diminished my credibility. Or let’s say I assume the falsity of the index theorem and then go prove some other stuff based on that. Well my results are basically worthless if it turns out that the index theorem is true (which is is). I agree with you that the process of proving theorems is what concerns mathematicians for the most part (although quite a bit of time is dedicated to structuring the framework of material–what connections there are between different fields, etc.) but if we go around claiming things to be true just because we read the title or abstract of a preprint, then that spells trouble. In order for mathematicians to reach new truths (or demonstrate new falsehoods), they to know what is already true or not. I don’t think that a site that purports to deliver news to the mathematics community should make such a bold statement (“Goedel’s Theorem is Invalid”) when that is just a preprint.

Whether or not it turns out that the proof presented in the preprint is valid, it is just a preprint right now. Until it has been checked and confirmed, it is a claim and should be treated as such. That’s all I’m saying.

There is no suggestion that we should argue about what is true and what is false, but we should be very careful that when we say definitively that something is true or false, we are able to back it up with a valid proof.

The post headline does seem slightly irresponsible, although after reading the paper and your post I supposed that it was meant to be lightly ironic.

So to theDude: consider if you read the following on a physics site

Newton’s Third Law Disproved

According to a new paper, Newton’s Third Law can be avoided. There you have it.

If I read that I would probably have been expecting to read something extremely funny on the other end of a link.

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