If a mathematical proof were to be dis proven by a counter example, what would that say about the foundations of Mathematics?

[Unity+]

This is not to say that there were will be times when this happens, but what if it does happen? Would it show Mathematics to have flawed axioms? Or would it simply reveal loop holes within assumed axioms of mathematics? What other consequences would occur?

[ajb]

It would mean that the original proof was not "watertight", there must have been some loop holes or conditions the original author missed. It could show problems with the axioms used by that author, maybe some things that should be examples are not and so on. However, there are no complete axioms of all mathematics.

 

The thing to remember is that an theorem is defined by what you assume before hand. A given theorem may not apply to situations close to what is needed to build that theorem.

[Acme]

It would mean that the original proof was not "watertight", ...

I think I would go with stronger language even than that, and say the original proof was wrong if a counter example was found. In such a case the original proof would no longer be a proof or even a conjecture and if one felt the need to further classify it I would just call it a mistake. The proper technical exclamation for the originator would then be "d'oh".

 

I mean the word proof not in the sense of lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible. ~ Carl Friedrich Gauss

 

[Unity+]

I think I would go with stronger language even than that, and say the original proof was wrong if a counter example was found. In such a case the original proof would no longer be a proof or even a conjecture and if one felt the need to further classify it I would just call it a mistake. The proper technical exclamation for the originator would then be "d'oh".

 

I mean the word proof not in the sense of lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible. ~ Carl Friedrich Gauss

 

But the point is the proof would be viable by all axioms of Mathematics, but yet there would be a counter example to it which means, as abj explained, the axioms would be ill founded and would need reconsideration.

[SamBridge]

But the point is the proof would be viable by all axioms of Mathematics, but yet there would be a counter example to it which means, as abj explained, the axioms would be ill founded and would need reconsideration.

If they disagree, it implies they had different axioms somewhere down the line, or someone just made a human-mistake when writing things out. Both the proof and counter-proof would agree that 1+1=2 if they weren't talking about modular counting, but that doesn't mean 1+1 magically doesn't equal 2 anymore just because one of those conjectures is wrong.

Edited April 24, 2014 by SamBridge

[Acme]

 

I think I would go with stronger language even than that [ajb], and say the original proof was wrong if a counter example was found. In such a case the original proof would no longer be a proof or even a conjecture and if one felt the need to further classify it I would just call it a mistake. The proper technical exclamation for the originator would then be "d'oh".

 

I mean the word proof not in the sense of lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible. ~ Carl Friedrich Gauss

But the point is the proof would be viable by all axioms of Mathematics, but yet there would be a counter example to it which means, as abj explained, the axioms would be ill founded and would need reconsideration.

 

That is a spurious argument. Regardless of axioms employed, a proof is either correct or it is not. If a counter example to a purported proof is given, then the proof was no proof at all.

[Unity+]

That is a spurious argument. Regardless of axioms employed, a proof is either correct or it is not. If a counter example to a purported proof is given, then the proof was no proof at all.

How is it a poor argument? A proof may be correct within its nature of axioms but may not quite work completely because of incomplete axioms.

 

A proof is taking the axioms and combining them to make theorems that confirm a piece of logic for something that may infinitely occur and needs a set of rules in order to establish the statement for all of what it is being applied to. Therefore, axioms would be incomplete.

[SamBridge]

How is it a poor argument? A proof may be correct within its nature of axioms but may not quite work completely because of incomplete axioms.

Then it's just plain not correct in the presence of new information, just like how Newtonian physics was with special relativity.

Edited April 24, 2014 by SamBridge

[Unity+]

Then it's just plain not correct in the presence of new information, just like how Newtonian physics was with special relativity.

That is what I am getting at, yes.

[ajb]

But the point is the proof would be viable by all axioms of Mathematics, but yet there would be a counter example to it which means, as abj explained, the axioms would be ill founded and would need reconsideration.

What "all axioms"?

 

There are no all encompassing axioms of mathematics.

I think I would go with stronger language even than that, and say the original proof was wrong if a counter example was found. In such a case the original proof would no longer be a proof or even a conjecture and if one felt the need to further classify it I would just call it a mistake. The proper technical exclamation for the originator would then be "d'oh".

It could be possible that these counter examples are somewhat pathological and that further conditions on the theorem can remove them. In that sense one adjusts the axioms (meaning assumptions/conditions in the theorem). Work in progress trying to understand something difficult I would say rather than simply wrong.

[imatfaal]

What "all axioms"?

 

There are no all encompassing axioms of mathematics.

 

It could be possible that these counter examples are somewhat pathological and that further conditions on the theorem can remove them. In that sense one adjusts the axioms (meaning assumptions/conditions in the theorem). Work in progress trying to understand something difficult I would say rather than simply wrong.

 

ydoaps was threatening a few years ago to do a series of blogs on godel - I still think we need it as completeness, consistency and self-reference in maths is poorly understood here (not excluding myself).

 

I think it depends on the proof and the nature of the counter example - one simple triplet of numbers would not show Andrew Wiles' famous proof to be almost correct or a work in progress; it would show that both Fermat and Wiles were wrong.

 

However a counter-example to one of the myriad lemma and computer-assisted checking of assumptions that Wiles' proof entailed would indeed mean that the proof was still potentially salvageable and might be ok with more work.

 

I think we also need to draw a firm and heavy line between physic and maths here - mathematical theorems are proved within certain axiomatic frameworks whereas physical theories are shown to be reliable realistic models between certain observational limits within nature.

[studiot]

I started this reply before ajb and imatfaal posted further, however the comments are still valid.

 

I suggest you reread ajb's post#2.

 

However, there are no complete axioms of all mathematics.

 

 

 

This was proved in the early part of the 20th century.

 

But we need to go back a lot further.

 

A system of mathematics (there are many, there is no single one) is a logically self consistent construct that builds on axioms to create theorems and other results.

 

However it is not built on axioms alone.

Axioms in isolation cannot provide sufficient information.

 

The history of geometry is a good example.

 

The original 5 axioms (he called them propositions) of Euclid were supported by 23 definitions and 5 what he called 'common notions', without which we could not have Euclidian Geometry today.

 

Without definition 4 (a straight line) the rest is nonsense. If the analysis is not restricted to straight lines many of the results can be negated by a curved line as a counterexample. Exactly as ajb has indicated.

 

In the 18 century (I think) one of the axioms was changed and projective geometry was born.

 

(Some of) The theorems and results of projective geometry are at variance with standard Euclidian geometry, but the new system is consistent within itself and its altered axiom.

 

In the 19th century, the fifth axiom was removed altogether to found Riemanian geometries.

 

In the 20th century Geometry moved from discussion of figures and shapes as being the fundamental to discussion of sets, symmetries and groups.

[ajb]

ydoaps was threatening a few years ago to do a series of blogs on godel - I still think we need it as completeness, consistency and self-reference in maths is poorly understood here (not excluding myself).

It is not something that I am very familiar with if I am honest. I don't really care too much about foundational issues, mathematical logic and so on. My understanding is rather pedestrian here.

 

I think it depends on the proof and the nature of the counter example...

For sure and without some specific examples one cannot be any more concrete.

 

My general point is that a theorem is only as strong as what you need to create it. If a theorem does not hold for all the classes of objects that you first envisaged, then maybe it does hold if you modify your theorem or the class of objects you are discussing. Or of course it could just be wrong, but I am thinking more along the lines of how one works with mathematics here.

[Unity+]

It is not something that I am very familiar with if I am honest. I don't really care too much about foundational issues, mathematical logic and so on. My understanding is rather pedestrian here.For sure and without some specific examples one cannot be any more concrete.My general point is that a theorem is only as strong as what you need to create it. If a theorem does not hold for all the classes of objects that you first envisaged, then maybe it does hold if you modify your theorem or the class of objects you are discussing. Or of course it could just be wrong, but I am thinking more along the lines of how one works with mathematics here.

So this can be thought as Einstein and his Special and General relativity , where Special relativity only worked in certain circumstances while the General version applies to almost all circumstances(except on the quantum level as far as I know).

Edited April 24, 2014 by Unity+

[imatfaal]

So this can be thought as Einstein and his Special and General relativity , where Special relativity only worked in certain circumstances while the General version applies to almost all circumstances(except on the quantum level as far as I know).

These are physical theories - and whilst mathematical self consistent - they are not able to be proven; all you can do is say that all of the measurements we have taken so far are consistent with the theories' predictions.

 

SR is completely mathematically self consistent - but in the real world we see that it cannot be the whole story. GR is less mathematically pleasing but applies to a greater proportion of potential scenarios.

 

Mathematical theorems can be shown to be absolutely true and proven within certain axiomatic frameworks. You choose your axioms and then move on from there; it has been shown that there is no set of axiomata that are universal and incontrovertible - so the choice of axiomata is arbitrary. But once that choice has been made - you can then prove (without fear of contradiction) certain theorems; ie eulidean geometry under the parallel postulate.

[Acme]

...

My general point is that a theorem is only as strong as what you need to create it. If a theorem does not hold for all the classes of objects that you first envisaged, then maybe it does hold if you modify your theorem or the class of objects you are discussing. Or of course it could just be wrong, but I am thinking more along the lines of how one works with mathematics here.

In such a case, the original proof of a conjecture or theorem was not a proof, i.e. it was wrong. A consequent proof the same conjecture or theorem, whether it uses part of the original wrongness or not, is then a proof all of its own. As Gauss said, there is no partial proof. Not 1/2 or 3/4 or 7/8. It's all or nothing. That is exactly how one works with mathematics.

[ajb]

It's all or nothing. That is exactly how one works with mathematics.

Sure, but in practice one builds up knowledge by making mistakes, not quite understanding something, finding examples and counter examples, having "partial proofs" and so on. At the end of it one should have proper proofs that are up to the standard and style of the particular branch of mathematics. More than that, hopefully what you have done should shead light on some something and improve our overall understanding.

So this can be thought as Einstein and his Special and General relativity , where Special relativity only worked in certain circumstances while the General version applies to almost all circumstances(except on the quantum level as far as I know).

As pointed out, these are physical theories and not mathematical theorems. Within the mathematical structure of relativity you can construct mathematical theorems.

[Acme]

Sure [Acme], but in practice one builds up knowledge by making mistakes, not quite understanding something, finding examples and counter examples, having "partial proofs" and so on. At the end of it one should have proper proofs that are up to the standard and style of the particular branch of mathematics. More than that, hopefully what you have done should shead light on some something and improve our overall understanding.

 

...

I agree about the steps and find it proper that you put "partial proofs" in quotes. My point is that all of that understood and answering the OP, there are no counter examples to a proof.

[studiot]

 

I agree about the steps and find it proper that you put "partial proofs" in quotes. My point is that all of that understood and answering the OP, there are no counter examples to a proof.

 

 

Yes but the OP asked something slightly different.

 

Yes, a single counterexample discredits a proof, but proofs are about theorems and the OP asked about axioms.

 

There are no proofs involved with axioms.

 

So all you can say is that a 'counterexample' highlights an area where the writ of one or more axioms does not run.

 

This is what I described in my earlier post.

Edited April 24, 2014 by studiot

[SamBridge]

That is what I am getting at, yes.

But it's only proven wrong if other axioms of mathematics are still true, otherwise you have no mathematical justification for how its wrong.

Edited April 24, 2014 by SamBridge

[imatfaal]

A system of mathematics (there are many, there is no single one) is a logically self consistent construct that builds on axioms to create theorems and other results.

Agree entirely with your latest post but you mentioned your previous which made me go back and re-read.

 

I cannot agree - there is no such sufficiently complicated system of axiomata that does not produce theorems that cannot be shown to be true and which must produce theorem which are not true but cannot shown to be false. This is Godels answer to Hilbert's Second (?) Question; no there is not a self consistent set of axiomata that produce all possible correct theorem but only produce true theorem.

 

ie Anything complex enough to be interesting and useful will have self-contradictions, anything simple enough to not have problems is so boring to be of academic use only

[studiot]

 

studiot, on 24 Apr 2014 - 11:09 AM, said:

A system of mathematics (there are many, there is no single one) is a logically self consistent construct that builds on axioms to create theorems and other results.

 

 

imatfaal

Agree entirely with your latest post but you mentioned your previous which made me go back and re-read.

 

I cannot agree - there is no such sufficiently complicated system of axiomata that does not produce theorems that cannot be shown to be true and which must produce theorem which are not true but cannot shown to be false. This is Godels answer to Hilbert's Second (?) Question; no there is not a self consistent set of axiomata that produce all possible correct theorem but only produce true theorem.

 

ie Anything complex enough to be interesting and useful will have self-contradictions, anything simple enough to not have problems is so boring to be of academic use only

 

 

Sorry, I really cannot follow this or what you are getting at.

 

Please explain differently.

[Acme]

 

I agree about the steps and find it proper that you [ajb] put "partial proofs" in quotes. My point is that all of that understood and answering the OP, there are no counter examples to a proof.

Yes but the OP asked something slightly different.

 

D'oh! Agreed. I should have specified the title rather than the OP, as the title makes no mention of axioms.

If a mathematical proof were to be disproven by a counter example, what would that say about the foundations of Mathematics?

The question is not even wrong. What it, the question, says is that the questioner has some confusion regarding mathematical proofs.

 

I suppose we could start mixing it up yet more by invoking proof theory, but that's a horse of a different kettle of fish.

 

proof theory @ Wilted Petal: >> http://en.wikipedia.org/wiki/Proof_theory

Edited April 24, 2014 by Acme

[Unity+]

D'oh! Agreed. I should have specified the title rather than the OP, as the title makes no mention of axioms.The question is not even wrong. What it, the question, says is that the questioner has some confusion regarding mathematical proofs.I suppose we could start mixing it up yet more by invoking proof theory, but that's a horse of a different kettle of fish.proof theory @ Wilted Petal: >> http://en.wikipedia.org/wiki/Proof_theory

That not even wrong statement does not apply here because there is no fallacy within the question. The idea of a proof is that it gives a foundation stating that something is always true and most likely will remain that way with current axioms of mathematics. However, if the theorem supported by a proof working off of he axioms of mathematics is disproved by a counterexample it would mean the axioms used are I'll founded, as others have stated.

[John]

If a theorem is known to be valid under some set of axioms, and then a valid counterexample is also given under the same set of axioms, then this would simply indicate that the set of axioms is logically inconsistent.

As noted earlier in the thread, there is no complete axiomatization of mathematics. However, we do have axiomatizations for large parts of mathematics, and some of these are strong enough to be subject to Gödel's incompleteness theorems. In this case, the second theorem is of interest. It says that no consistent sufficiently strong axiomatic system can prove its own consistency. Here "sufficiently strong" relates to the ability to make certain statements about arithmetic (more technically, the ability to interpret Robinson arithmetic).

ZFC, which is probably the closest thing we have to an axiomatization of mathematics in general, falls under this umbrella. While we believe ZFC is consistent, we cannot prove it, and thus it's not inconceivable that some inconsistency may eventually be found. If such an inconsistency is found, then we'll be in some trouble, since an inconsistent axiomatic system can be used to derive any statement, thus rendering theorems built on ZFC rather suspect. In that case, we'll need to find some new system on which to rigorously build mathematics, but again we won't be certain the new system is consistent either.

Edited April 24, 2014 by John