Would math make God obsolete ?

[Brian Tenneson]

There are undecidable statements (about arithmetic)... There are true statements lacking proof. There are also false statements about arithmetic the proof of whose falsehood is impossible; not just impossible for you and me but for a computer of any capacity or other forms of rational processing. We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.

[Bruno Marchal]

On 24 Jan 2014, at 08:23, Brian Tenneson wrote:

There are undecidable statements (about arithmetic)... There are true statements lacking proof. There are also false statements about arithmetic the proof of whose falsehood is impossible; not just impossible for you and me but for a computer of any capacity or other forms of rational processing.

OK. But this is not really prove for computer having the ability to transform themselves, like acquiring new axioms. But then that acquisition cannot be programmed at the start (in which case Gödel can be applied).

Without comp, there is no means to prove the existence of an absolutely undecidable proposition. We can only prove that for a fixed machine, or a fixed RE sequence of machine, there is a proposition undecidable for that machine, or for any machine in that sequence.

We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.

Yes, arithmetical omniscience needs a "god", like the Arithmetical Truth. Even a theory as powerful as ZF or NF can only scratch the arithmetical truth. OK.

Bruno

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[Gabriel Bodeen]

FWIW, under the usual definitions, the rationals are enumerable and so are a smaller set than the reals.  I'd suppose that if people can figure that out with our nifty fleshy brains, then a well-designed computer brain could, too.
-Gabe

[Brian Tenneson]

Yes, some day a computer might be able to figure out that the set of rationals is not equipollent to the set of real numbers.  I saw somewhere that using an automated theorem prover, one of Godel's incompleteness theorems was proved by a computer.

The question I raised initially was this: will there ever be a machine or human who can correctly answer all questions with a mathematical theme that have answers?  I didn't think so in my original post but now I'm starting to wonder.  It's the existence of undecidable statements that would probably lead to the machine or human not being able to do it in general.  This reminds me of the halting problem.

The good news is we will never run out of mathematical territory to think about.

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[Bruno Marchal]

On 27 Jan 2014, at 16:12, Brian Tenneson wrote:

Yes, some day a computer might be able to figure out that the set of rationals is not equipollent to the set of real numbers.  

A Lôbian machine like ZF can do that already.

I saw somewhere that using an automated theorem prover, one of Godel's incompleteness theorems was proved by a computer.

Boyer and Moore, yes, but that is not conceptuallydifferent than ZF, except that the Boyer-Moore machine uses more efficient sort of AI path.

Gödel discovered that PM already proves his own incompleteness theorem. All Lôbian machine proves their own Gödel's theorem. They all prove "If I am consistent, then I can't prove my consistency".

The question I raised initially was this: will there ever be a machine or human who can correctly answer all questions with a mathematical theme that have answers?

All? No, for any machine i in the phi_i.

But that is less clear for evolving machines, whose evolution rule is not part of the program of the machine. Of course, at each moment of her "life", she will be incomplete, but if her evolution is enough "non computable", or using some special oracle, it might be that the machine will generate the infinitely many truth of arithmetic, but not in any provable way. 

 I didn't think so in my original post but now I'm starting to wonder.  It's the existence of undecidable statements that would probably lead to the machine or human not being able to do it in general.  This reminds me of the halting problem.

Those are related. Undecidable is always relative. Consistent(PA) is not provable by PA, but is provable in two lines in the theory PA+con(PA). Of course PA+con(PA) cannot prove con(PA+con(PA)).

What about PA+con(I), with I = PA+con(I). It exists as we can eliminate the occurence of I by using the Dx = "xx" method. Well, in this case PA+con(I) can prove its own consistency, but only because it is actually inconsistent. 

The good news is we will never run out of mathematical territory to think about.

Yes indeed, even if we confine ourselves on elementary (first order) arithmetic. There is an infinity of surprises there. 

Bruno

On Mon, Jan 27, 2014 at 6:58 AM, Gabriel Bodeen <gabeb...@gmail.com> wrote:

FWIW, under the usual definitions, the rationals are enumerable and so are a smaller set than the reals.  I'd suppose that if people can figure that out with our nifty fleshy brains, then a well-designed computer brain could, too.
-Gabe

On Friday, January 24, 2014 1:23:40 AM UTC-6, Brian Tenneson wrote:

There are undecidable statements (about arithmetic)... There are true statements lacking proof. There are also false statements about arithmetic the proof of whose falsehood is impossible; not just impossible for you and me but for a computer of any capacity or other forms of rational processing. We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.

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[Brian Tenneson]

Some basic.questions.  When you say PA, do you mean the set of all theorems entailed by the axioms of Peano arithmetic?  Does this include the true (relative to PA of course) wffs that are not provable from PA alone?

How can it be that PA+con(I) can prove its own consistency because it is inconsistent?  Do you mean that it is consistent relative to itself but inconsistent in the "metalanguage"?  Or else how can we have it be both consistent and inconsistent?

This is probably way off the subject (hope that's ok with you): isn't all mathematical truth relative to the formal system one is operating in?  "all mathematical truth is relative to the formal system one is operating in" is relative to the formal system I call "rational discourse" in which "mathematical discourse" and "machine-level discourse" are sub-systems.

[John Clark]

On Fri, Jan 24, 2014 at 2:23 AM, Brian Tenneson <ten...@gmail.com> wrote:

> There are undecidable statements (about arithmetic)... There are true statements lacking proof.

Yes.

> There are also false statements about arithmetic the proof of whose falsehood is impossible;

A proof is a FINITE number of statements establishing the truth or falsehood of something; if Goldbach's Conjecture is untrue then there is a FINITE even number that is NOT the sum of 2 primes. It would only take a finite number of lines to list all the prime numbers smaller than that even number and show that no two of them equal that even number, and that would be a proof that Goldbach's Conjecture is wrong.

The real problem would come if Goldbach's Conjecture is true (so we'll never find two primes to show it's wrong) but can not be proven to be true (so we will never find a finite proof to show its correct).

  John K Clark

 

not just impossible for you and me but for a computer of any capacity or other forms of rational processing. We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.

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[Brian Tenneson]

You could always just add it and its negation to the list of axioms (though not at the same time, of course) and see where that leads, if anywhere.

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[Bruno Marchal]

On 27 Jan 2014, at 17:30, Brian Tenneson wrote:

Some basic.questions.  When you say PA, do you mean the set of all theorems entailed by the axioms of Peano arithmetic?  

Yes. In some context it means only the axioms, but often I use the same expression to denote the axioms and its logical consequences (theorems).

Does this include the true (relative to PA of course) wffs that are not provable from PA alone?

No.

How can it be that PA+con(I) can prove its own consistency because it is inconsistent?  

PA+con(I) is inconsistent, because it can prove its own consistency (in one line), and that makes it inconsistent by the second incompleteness theorem (no consistent Löbian theory can prove its own consistency).

Then, being inconsistent, it can prove its consistency, like it can prove any proposition.

Do you mean that it is consistent relative to itself but inconsistent in the "metalanguage"?  

No, it is totally inconsistent. It proves f by its own axioms and the modus ponens rule.

Or else how can we have it be both consistent and inconsistent?

No. That would made us inconsistent.

This is probably way off the subject (hope that's ok with you): isn't all mathematical truth relative to the formal system one is operating in?

Why? I doubt this for some theories, like those who specify logically a Turing universal system. In all case a machine will stop or not stop, independently of the description and language used to describe the system. But I can imagine that you are partially right for richer theories.

 "all mathematical truth is relative to the formal system one is operating in" is relative to the formal system I call "rational discourse" in which "mathematical discourse" and "machine-level discourse" are sub-systems.

Hmm.... Above arithmetic, I can make some sense on this. But to just define formal system, I need some absolute part, and I use the (second order) distinction between finite and infinite to do this, at the metat-level. 

http://iridia.ulb.ac.be/~marchal/

[Bruno Marchal]

On 27 Jan 2014, at 19:55, John Clark wrote:

On Fri, Jan 24, 2014 at 2:23 AM, Brian Tenneson <ten...@gmail.com> wrote:

> There are undecidable statements (about arithmetic)... There are true statements lacking proof.

Yes.

> There are also false statements about arithmetic the proof of whose falsehood is impossible;

A proof is a FINITE number of statements establishing the truth or falsehood of something;

Not establishing the truth, but establishing the theoremhood. 

At the metalevel, if the theiry is formalized in first order logic, you will have that the proved proposition will be satisfied in all models of the theory, but this cannot, in general, be shown *in* the theory, unless the theory is Löbian (but that is not easy to prove---I don't use that).

if Goldbach's Conjecture is untrue then there is a FINITE even number that is NOT the sum of 2 primes.

Yes. The negation of Goldbach is Sigma_1. Goldbach is Pi_1. Like Riemann Hypothesis.

But Syracuse conjecture is above Pi_1. You cannot decide it, nor his negation, by a simple mechanical procedure.

It would only take a finite number of lines to list all the prime numbers smaller than that even number and show that no two of them equal that even number, and that would be a proof that Goldbach's Conjecture is wrong.

The real problem would come if Goldbach's Conjecture is true (so we'll never find two primes to show it's wrong) but can not be proven to be true (so we will never find a finite proof to show its correct).

OK.

Bruno

  John K Clark

 

not just impossible for you and me but for a computer of any capacity or other forms of rational processing. We'll never have a computer, then, that will work as a mathematically-omniscient device. By that I mean a computer such that every question that has a mathematically-oriented theme having an answer truthfully can be answered by such a device. Calculators demonstrate the concept but are clearly not mathematically-omniscient: you ask the calculator what is 2+2 and press a button and "presto" you get an answer. What I'm talking about would be questions like "is the set of rational numbers equal in size to the set of real numbers", and get the correct answer. So we will never have such a computer no matter what its capacities are, even if computer encompasses the entire human brain. Unfortunately, that means that even for humans, we will never know everything about math. Unless something weird would happen and we suddenly had infinite capacities; that might change the conclusions.

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[Gabriel Bodeen]

On Tuesday, January 28, 2014 2:14:40 AM UTC-6, Bruno Marchal wrote:

PA+con(I) is inconsistent, because it can prove its own consistency (in one line), and that makes it inconsistent by the second incompleteness theorem (no consistent Löbian theory can prove its own consistency).

Then, being inconsistent, it can prove its consistency, like it can prove any proposition.

Um, so an inconsistent theory can prove its consistency, and a consistent one can't?  What then stops the consistent theory from trying to use that principle to prove itself consistent?

-Gabe

[John Clark]

On Mon, Jan 27, 2014 at 2:35 PM, Brian Tenneson <ten...@gmail.com> wrote:

> You could always just add it and its negation to the list of axioms (though not at the same time, of course) and see where that leads,

Axioms should be simple things that are self evidently true, neither Goldbach's Conjecture nor its negation is that.

 John K Clark

[John Clark]

On Tue, Jan 28, 2014 at 3:20 AM, Bruno Marchal <mar...@ulb.ac.be> wrote:

>> A proof is a FINITE number of statements establishing the truth or falsehood of something;

> Not establishing the truth, but establishing the theoremhood. 

I stand corrected; although it would be true if the axioms in the logical system you're using are true, and that's why you need to be very very careful before adding new axioms to your system.  

 John K Clark

[Bruno Marchal]

A case like that can be made for arithmetic, but despite I do believe that indeed 17 is prime, in serious theology, it is wiser and simpler to put even that as an hypothesis. Sure, it is provable in PA, and in RA, but I take those axioms of PA and RA as hypotheses too. That is what we assume. Better to keep our personal conviction for ourselves, or for the pause-café.

That helps also for the model theory, which theorizes on semantic and truth, in relation with theories, as a subject matter.

Bruno

 John K Clark

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[LizR]

"Would math make God obsolete?"

If so, that remainds me of something...

"I refuse to prove that I exist,'" says God, "for proof denies faith, and without faith I am nothing."
"But," says Man, "The Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED."
"Oh dear," says God, "I hadn't thought of that," and promptly vanishes in a puff of logic.
"Oh, that was easy," says Man, and for an encore goes on to prove that black is white, and gets himself killed on the next zebra crossing.

[John Mikes]

Liz, that was enjoyable. In the back of it lurks the incompatibility of 'GOD" with logics. 

John

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