Research topic ideas truth in mathematics

[BobSanchez]

Research topic ideas "truth in mathematics"

I need a research topic for a school seminar I've been attending called truth in mathematics. It can be about a certain person who challenged the norm or discovered something revolutionary (such as Cantor) or even a specific idea dealing with certainty (like truth tables) Any suggestions are appreciated.

 

[mathman]

Godel - undecidable proof.

 

[micromass]

What about Tarski's definition of truth?

 

[SteveL27]

Take any standard independence result and discuss whether it makes sense for this result to be true, and in what sense. For example, the Continuum Hypothesis (CH) is known to be independent of ZFC. Nevertheless, ZFC is intended to formalize the real numbers; and in the real numbers, CH must either be true or false. This is a fruitful starting point for research.

Here are a couple of starting points.

http://en.wikipedia.org/wiki/Continuum_hypothesis

http://consc.net/notes/continuum.html

 

[gb7nash]

Fermat's Last Theorem? This might be tough to present though.

 

[mathman]

ZFC is intended to formalize the real numbers; and in the real numbers, CH must either be true or false.

Why?? It is unprovable one way or the other.

 

[SteveL27]

Why?? It is unprovable one way or the other.

Provability and truth are not the same. The OP asked about truth, so this is a good area for research.

If you believe in the real numbers, then CH is either true or false in the reals. If CH is independent of ZFC, then we need to find better systems of axioms. This is a very active area in contemporary set theory.

On the other hand perhaps you don't believe in the real numbers, only in axioms. In that case there is no such thing as truth, only provability. And if there's no such thing as truth, then why do math at all, except as a meaningless logic game?

If one is asking about truth in mathematics, these are questions you have to think about.

 

[mathman]

How do you distinguish "truth" from "provability"? During my math education and ever since, something (in mathematics) was "true" if and only if it was provable.

 

[micromass]

How do you distinguish "truth" from "provability"? During my math education and ever since, something (in mathematics) was "true" if and only if it was provable.

Not necessarily, for example, if Goldbach's conjecture was shown to be unprovable, then it would necessarily be true. Sadly, I have no better reference of this than this link: http://mathforum.org/kb/thread.jspa?forumID=193&threadID=435233&messageID=1376382

 

[Hurkyl]

Not necessarily, for example, if Goldbach's conjecture was shown to be unprovable, then it would necessarily be true. Sadly, I have no better reference of this than this link: http://mathforum.org/kb/thread.jspa?forumID=193&threadID=435233&messageID=1376382

This is either a level slipping fallacy, or you're implicitly restricting yourself to a limited subset of possible semantics.

A statement of first-order logic is undecidable in some theory if and only if there exists a truth valuation that makes it true and a truth valuation that makes it false. (Both truth valuations, of course, are true for every theorem of the theory)

 

[micromass]

This is either a level slipping fallacy, or you're implicitly restricting yourself to a limited subset of possible semantics.

A statement of first-order logic is undecidable in some theory if and only if there exists a truth valuation that makes it true and a truth valuation that makes it false. (Both truth valuations, of course, are true for every theorem of the theory)

Hmmm, I knew I shouldn't have trusted that site...
Would you care to explain what's wrong with the logic on the link I posted??

 

[Hurkyl]

If you believe in the real numbers, then CH is either true or false in the reals.

I can "believe" in ZFC and be entirely agnostic about the CH, thanks to its undecidability. There simply isn't any chain of first-order logic that that can transfer my belief of ZFC to a belief or disbelief in CH.

If CH is independent of ZFC, then we need to find better systems of axioms.

Only if we care about the CH being provable or disprovable. We already know that there must be some such statements...

On the other hand perhaps you don't believe in the real numbers, only in axioms. In that case there is no such thing as truth, only provability.

Formal logic isn't limited to syntax. There's a whole subfield of semantics that covers truth valuations, interpretations, models, and so forth.

And if there's no such thing as truth, then why do math at all, except as a meaningless logic game?

There's an entirely different conclusion one can draw (and I do) -- the naive notion of truth is too naive to stand up to serious analysis.

 

[Hurkyl]

Hmmm, I knew I shouldn't have trusted that site...
Would you care to explain what's wrong with the logic on the link I posted??

I couldn't find a reference on sigma-1 completeness so I can't explain what it actually implies.


My statement follows from Gödel's completeness theorem for first-order logic:

• If P is true in every model of some axioms, then P is provable from those axioms

I suppose I should also mention

• Soundness: If P is provable from some axioms, then P is true in every model of those axioms
• Every truth valuation on a set of axioms is the truth valuation of some model of those axioms

 

[micromass]

I couldn't find a reference on sigma-1 completeness so I can't explain what it actually implies.


My statement follows from Gödel's completeness theorem for first-order logic:

• If P is true in every model of some axioms, then P is provable from those axioms

I suppose I should also mention

• Soundness: If P is provable from some axioms, then P is true in every model of those axioms
• Every truth valuation on a set of axioms is the truth valuation of some model of those axioms

Yes, it's obvious now that it violates the completeness theorem. Still I'm quite interested in what actually goes wrong in his explanations. I'll do some more research, but I couldn't find any good reference on sigma1-completeness either

 

[SteveL27]

How do you distinguish "truth" from "provability"? During my math education and ever since, something (in mathematics) was "true" if and only if it was provable.

The OP asked about truth in mathematics, and this is the exact point I'm attempting to bring out.

Truth and provability are very different things.

In ZFC I can prove that the real numbers can be well-ordered. In ZF I can NOT prove that the real numbers are well-ordered.

Now, if truth is the same as provability, how do you account for the fact that I can prove well-ordering in one axiom system but not in another?

Provability is more about the system of axioms you choose, than about saying what's true.

So, what does it mean to ask if the reals can be well-ordered? If you believe only in provability, you would say, "Well, it depends only on the axioms you choose."

But which axiom system is the true one? Or do you think that the notion of truth should be thrown out, and we should only care about what we can prove from a given axiom system, and we don't care about the actual truth of the things we're proving.

Do you see the distinction I'm making? The OP asked about truth in mathematics, and this is the heart of the issue.

Does such a thing as mathematical truth exist, and our axioms are imperfect attempts to model it? Or is truth an illusion, and the ONLY thing we have is arbitrary axioms and their logical consequences, with the notion of truth becoming a quaint relic?

 

[mathman]

To Stevel 27

I am an old retired mathematician. The questions you raised never came up in what I did. I believe there is a "standard" set of axioms which is used by all mathematicians, working outside of symbolic logic. Much of the fundamental background that was taught to me was based on the Bourbaki approach. We didn't go any further back.

 

[SteveL27]

To Stevel 27

I believe there is a "standard" set of axioms which is used by all mathematicians, working outside of symbolic logic.

I have an interesting counterexample for your consideration.

We are all familiar with Wiles's celebrated proof of Fermat's Last Theorem.

As it happens, Wiles's proof is based on the modern formulation of algebraic geometry using a Grothendieck universe. A Grothendieck universe is a device developed by Alexander Grothendieck to avoid the use of proper classes, which are inevitable if one sticks to ZFC.

http://en.wikipedia.org/wiki/Grothendieck_universe

Now, it turns out that the existence of a Grothendieck universe is logically equivalent to the existence of a strongly inaccessible cardinal -- an object that is not only independent of ZFC, but whose consistency with ZFC can not even be proven within ZFC.

Now, does this cast doubt on Wiles's proof? Most people don't believe that. However, the question is valid enough to have generated this fascinating discussion on Math Overflow.

http://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof

There are links within that page that give quite a bit of background information on this subject.

The point is that in the search for good theorems and good formalizations, mathematicians these days freely use whatever axiom system is most convenient to the problem at hand. Nobody doubts that a complete translation of Wiles's proof from the context of Grothiendick universes to the context of ZFC could be done ... but nobody's done it, and evidently parts of Grothiendick's framework have proven difficult to translate to pure ZFC.

This would seem to be a counterexample to your claim that there is one standard axiom system that everyone uses.

So, I would reiterate my question: Do you still believe that provability and truth are exactly the same thing in modern mathematics?

Others have mentioned Godel's incompleteness theorem, which is an immediate refutation of the notion that provability = truth, but there are many other arguments that cast light on the question.

Surely you are aware that the notion of mathematical proof has evolved considerably since Euclid. If provability is a historical, evolving notion, do you think that truth is too?

 

[JSuarez]

It seems to me that this thread veered a little off-course, regarding the OP's question. To him/her, let me say that it would help if you provide more details about what you discussed in class; something related to "Truth in Mathematics" doesn't necessarily have to be a Logic-related topic. Given the said lack of details, I would point you to this video by IAS's Prof. Enrico Bombieri:

http://video.ias.edu/bombieri-2010"

 

[mathman]

To Stevel 27

I looked at the references in your last note and, as far as I can see, Wiles's use of inaccessible cardinals is controversial. I will admit it is completely far from my area of expertize (probability theory) so I can't make any judgment. However I would be very surprised that a theorem in algebra needs any such idea (inaccessible cardinals).

 

[SteveL27]

To Stevel 27

I looked at the references in your last note and, as far as I can see, Wiles's use of inaccessible cardinals is controversial. I will admit it is completely far from my area of expertize (probability theory) so I can't make any judgment. However I would be very surprised that a theorem in algebra needs any such idea (inaccessible cardinals).

I confess I'm at a loss to respond.

Earlier you wrote:

I believe there is a "standard" set of axioms which is used by all mathematicians, working outside of symbolic logic.

I pointed out that there is a striking and reasonably well-known counterexample to that statement; namely, the modern framework of algebraic geometry developed by Grothiendick, which takes place in the axiom system ZFC + "there exists an inaccessible cardinal," a system that is stronger than ZFC.

Nobody disputes this well-documented fact. The situation is very well laid out here ...

http://www.cwru.edu/artsci/phil/Proving_FLT.pdf

Since algebraic geometry is what we'd all consider "real math" and not symbolic logic or foundations, I would think that you would concede the point. However, all you are willing to do is say that you would be "very surprised."

Does that mean that

a) You are indeed very surprised, but now that you have considered the matter, you do realize that modern mathematicians are far from agreed on a standard set of axioms, but rather choose whatever set of axioms they need to do their work and prove their theorems, providing they can get a consensus from the mathematical community that their work is valid. [This would be my personal belief. Also note the many efforts to find alternate foundational systems, such as new axioms of set theory such as assuming the existence of inaccessibles; or category theory; or Vovoedsky's belief that PA may well be inconsistent, and his current research into developing new foundations based on homotopy theory.]

or

b) You disagree that Grothiendick uses inaccessible cardinals. This would be a completely untenable position. Grothiendick universes are logically equivalent to inaccessibles, as outlined in the Wikipedia article I linked earlier: http://en.wikipedia.org/wiki/Grothendieck_universe

or

c) You have some other specific disagreement with the references I've provided, but you have not yet specified them.

I do understand that as a practicing mathematician, you may well have spent your time proving theorems, attending faculty meetings, and grading freshman calculus papers, and did not spend much time thinking about foundational or historical or philosophical issues.

But what do you think NOW, based on what I've written? If truth = provability, then truth is a function of history and choice of axiom systems. We know today that Euclid's Elements is not logically rigorous; and that Gauss's 1799 proof of the Fundamental Theorem of Algebra is logically flawed. Our notions of proof change throughout history. Whereas we would like to think of truth as something that is eternal: FTA has always been true, even long before anyone proved it. Or did it become true when Gauss proved it, then false as the logical flaws in Gauss's proof were understood, and then true again now that the average grad student can probably scratch out a proof of FTA given a hint or two?

See

http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

for an overview of the history of attempts to prove this famous theorem.

In any event, you said you spent your career in probability theory. Then you must be intimately familiar with the notion of nonmeasurable sets; and their logical relationship to the Axiom of Choice. I would think that of all the mathematical disciplines, probability theorists would be the most inclined to realize that provability is a function of the axioms you choose, and not at all necessarily related to any absolute notion of truth. Surely you are aware that AC was controversial for a long time; and the reason that mainstream mathematicians freely accept it today is because it is useful, not because we have any way at present to know if it is true, or if even calling AC true or false is meaningful.

I do apologize if all this seems like a thread hijack. But the OP asked about truth in mathematics. And a professional mathematician has claimed that truth = provability, a notion that was forever destroyed by Godel in 1931 -- as Mathman himself noted earlier in this thread! So I just don't understand where Mathman is coming from.

Mathman, YOU are the one who wrote:

Godel - undecidable proof.

And now you are saying that truth = provability, when Godel showed the exact opposite!

 

[mathman]

To get back to the beginning, "truth" to me is a philosophical concept, not a mathematical one, once you divorce it from provable.

 

[Hurkyl]

To get back to the beginning, "truth" to me is a philosophical concept, not a mathematical one, once you divorce it from provable.

Have you seen the notion of a truth valuation? Or an interpretation in the sense of formal logic?

 

[BobSanchez]

Thanks so much...there is some cool stuff to look at in there. I got plenty of ideas to work with and even shared some with other people in my seminar!

 

[mathman]

Have you seen the notion of a truth valuation? Or an interpretation in the sense of formal logic?

For better or worse, my math education did not include formal logic. From what I have encountered in this thread, I'm glad I missed it.