The Riemann Hypothesis paradox

[attofishpi]

The Riemann Hypothesis can only be proven to be false, if it is false. It can never be proven to be true...there simply is not enough time in the universe.

[Logik]

There's a continuum there along which it's worth thinking about.

We either have proofs of impossibility (100% certainty): https://en.wikipedia.org/wiki/Proof_of_impossibility
Or we have proof complexity: https://en.wikipedia.org/wiki/Proof_complexity

Since we have neither right now we are sort of stuck.

Numerical methods might help us approximate it. Either way - I would be happy even with a lower bound.
Even somebody proving that it's at least O(n!) complex.

Normally I'd be willing to throw my name under a bus by making a guestimate based on intuition. I got nothing - my intuition is looking at blinking at me with befuddlement.

[wtf]

The Riemann Hypothesis can only be proven to be false, if it is false. It can never be proven to be true...there simply is not enough time in the universe.

Excellent point. Likewise we could never prove that all even numbers are divisible by 2, since there are infinitely many of them and there's not enough time in the universe. Right? Right. Wrong.

[Logik]

Excellent point. Likewise we could never prove that all even numbers are divisible by 2, since there are infinitely many of them and there's not enough time in the universe. Right? Right. Wrong.

That depends on whether you think provability and decidability mean the same thing or different things.

It's a bit of a truism really. Because in symbolic mathematics an "even number" is defined as 2n so you don't need to prove a truism...

In Curry-Howard systems provability and decidability are isomorphic and so the way you determine if any particular number (X) is, in fact even is to divide it by 2 and check for a remainder.

Or you can do stuff like, If last_digit(X) in { 0,2,4,6,8 }

[wtf]

Excellent point. Likewise we could never prove that all even numbers are divisible by 2, since there are infinitely many of them and there's not enough time in the universe. Right? Right. Wrong.

That depends on whether you think provability and decidability mean the same thing or different things.

It's a bit of a truism really. Because in symbolic mathematics an "even number" is defined as 2n so you don't need to prove a truism...

In Curry-Howard systems provability and decidability are isomorphic and so the way you determine if any particular number (X) is, in fact even is to divide it by 2 and check for a remainder.

Or you can do stuff like, If last_digit(X) in { 0,2,4,6,8 }

I'm mocking the OP's point. Is this unclear to you? Are you defending the OP's point? Then you're wrong.

In any event you are flogging a constructivist hobby horse and don't seem to understand math.

[Logik]

I'm mocking the OP's point. Is this unclear to you? Are you defending the OP's point? Then you're wrong.

In any event you are flogging a constructivist hobby horse and don't seem to understand math.

I am mocking your mockery.

Every mathematician I've spoken to so far sees provability and decidability as different criterions.

And yet.... https://en.wikipedia.org/wiki/Curry%E2% ... espondence

So while mathematicians have been playing in the abstract realm, computer scientists have been grounding symbols in physical reality for 60 or so years

[wtf]

So while mathematicians have been playing in the abstract realm, computer scientists have been grounding symbols for 60 or so years

Like I say ... you're flogging a constructivist viewpoint and don't understand mathematics. That remark applies to everything you post here.

[Logik]

Like I say ... you're flogging a constructivist viewpoint and don't understand mathematics.

Naturally. I don't understand your conception of mathematics

But I understand what an isomorphism is. As do you, surely?

Mathematical proofs are isomorphic to algorithms. The rest is divergence in nomenclature and primitives.

I am flogging a "computer science" horse. If it happens to trivialize all of mathematics because algorithms are constructive in nature. Oh well...

You won't see me crying over slaughtering sacred cows.

[wtf]

Mathematical proofs are isomorphic to algorithms. The rest is divergence in nomenclature and primitives.

So I take it you reject the concept of uncountably infinite sets, you reject Cantor's set theory and all of modern set theory. You believe the intermediate value theorem is false, and that's because the computable real line is full of holes. You reject Chaitin's Omega, a number that's definable but not computable.

All of that's perfectly fine. You can say, "From a constructivist or computable point of view, such and so." But you are ignoring all of modern math and all the physics based on it. And by not making your viewpoint clear, you might be confusing readers who don't know what constructivist mathematics is, and that it's a fringe belief among professional mathematicians.

https://en.wikipedia.org/wiki/Construct ... thematics)

[Logik]

So I take it you reject the concept of uncountably infinite sets, you reject Cantor's set theory and all of modern set theory.

Correct. Sets are types. I consider lambda calculus/type theory foundational. I root for ultrafinitism/ultraintuitionism.

All of that's perfectly fine. You can say, "From a constructivist or computable point of view, such and so."
But you are ignoring all of modern math and all the physics based on it.

Pull the other one. In a quantum-computational conception of the universe all infinities are errors.

https://en.wikipedia.org/wiki/Renormalization

That the equations work is neither here nor there. All models are wrong - some are useful.

[attofishpi]

I'm mocking the OP's point.

HOW DARE YOU!!

...hang on, I think i've solved it...ah shit, where's my rubber.

[wtf]

Correct. Sets are types. I consider type theory foundational. I root for ultrafinitism/ultraintuitionism.

You do realize that ultrafinitism is far more restrictive than the computablism you espouse in your threads. For example the real number pi is computable, because its digits can be cranked out by a Turing machine using any of many known closed-form formulas. But pi can not exist in an ultrafinitist setting because you can't have a number with infinitely many decimal digits.

Yes? Just want to make sure you understand the difference between ultrafinitism. In fact let's just bullet-point the various ideas.

* Ultrafinitism -- Sufficiently large finite sets don't exist. Exponentiation of the positive integers is not a total function.

* Finitism -- Each of 0, 1, 2, 3, ... exists. There are infinitely many natural numbers. But there is no SET of them. This is simply ZF set theory with the negation of the axiom of infinity. It's also the same as Peano arithmetic

* Constructivism -- You believe in infinite sets, as long as there's an algorithm or Turing machine that can crank out its elements. So for example you believe in the set of even numbers, but you don't believe in the full uncountable powerset of the naturals. And you DEFINITELY don't believe in sets given by the axiom of choice, such as a choice function on the set of equivalence classes of the reals mod the rationals. Therefore you cannot do modern probability theory, which depends crucially on the axiom of choice.

* Standard math -- One believes in the ZFC axioms and all their logical consequences. This is basically the math taught to undergrad math majors.

* Modern set theory -- One believes in ZFC plus as many large cardinals as one needs for a given application, or enjoys studying for their own sake. By the way large cardinals are of interest to logicians, since they measure the proof strength of various theories.

Now you have all along been arguing from a constructivist point of view. I have no problem with that as long as you identify that orientation so that readers are not confused.

But now you have jumped from constructivism OVER mere finitism to ultrafinitism. There is currently no known axiomatic foundation for ultrafinitism. Its only virtue -- a strong one, in my opinion -- is that it's the only philosophy of math that might actually be consistent with the universe we live in. So three cheers for ultrafinitism.

You agree with my taxonomy of philosophical orientations?

I'm also curious about your reference to ultra-intuitionism. I can accept constructivism because I understand what it's about. Intuitionism is constructivism plus mysticism, and I've never understood it even though from time to time I've taken a run at it. And ultra-intuitionism I've never heard of. What's it about?

[wtf]

...hang on, I think i've solved it...ah shit, where's my rubber.

Uh ... is that a Britishism for something other than what it means in Yank?

[attofishpi]

...hang on, I think i've solved it...ah shit, where's my rubber.

Uh ... is that a Britishism for something other than what it means in Yank?

Yes...though in one respect it means 'eraser' and in the other, well, one could have a Yank.

[wtf]

Yes...though in one respect it means 'eraser' and in the other, well, one could have a Yank.

LOL!! Eraser. Thanks for the info. I'm learning to speak English. Currently I only speak American. Completely different language.