The Riemann Hypothesis paradox

[attofishpi]

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.

Portuguese then.

[Logik]

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. It's called the range/precision trade-off. It's an arbitrary choice.

Do you know that in a universe where Planck-length is the smallest unit of distance you do not require infinite-precision pi ?
Not that anybody has gone and calculated pas the 31.4 trillion-th digit anyway.
https://cloud.google.com/blog/products/ ... ogle-cloud

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.

This is where I am at.

But now you have jumped from constructivism OVER mere finitism to ultrafinitism.

You labeled me as a "constructivist". I only acknowledged in so far as I understand the techniques/strategies behind constructive proofs.

Constructing models. Constructing algorithms. Constructing knowledge. It's a verb more than anything.

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.

And that's all there is to it. unification between Mathematics, Physics and Computer Science.

Theory of Thought.

You agree with my taxonomy of philosophical orientations?

Sure. In about as much as words can define anything precisely. It's easier to discuss consequences than definitions.

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

Ultra-intuitiuonism is constructivism + rejection of infinities.

I think of Mathematics as just another language for self-expression. What's the most complex thing I could ever express in Maths? The universe.

The consequences are simply that given Planck length, Shannon-Nyquist theorem and some ballpark estimates on the expected life of the universe you can describe the system perfectly (deterministically!) with a finite set of bits.

So, you are welcome to keep infinities as they are - I just have no use for them for any practical application.

For self-amusement - sure. Maths has lots of beauty and symmetry hidden in all the abstractions.

[wtf]

So, you are welcome to keep infinities as they are - I just have no use for them for any practical application.

Ok cool! Then you are not one of these dogmatic ultrafinitists, finitists, and constructivists who deny even the purely formal game of modern math. I've seen too many of those types online.

You will allow me modern math as long as I agree it's completely useless. Then we are in complete agreement! In fact I am with G.H. Hardy (played by Jeremy Irons in The Man Who Knew Infinity) when he said, in A Mathematician's Apology, that the most beautiful math is by definition that math which is the most useless. And that by that criterion, the most beautiful branch of math is Hardy's own specialty, number theory.

[If anyone has not seen The Man Who Knew Infinity, go rent/buy/stream/torrent it immediately, however people get their movies these days].

Wouldn't Hardy be surprised if he came back to life and found out that number theory, in particular the theory of factoring, has become the key concept in all Internet security and cryptocurrencies. I imagine he'd be crushed, in his eccentric British way.

Which by the way brings up a point. Today's totally useless math often becomes tomorrow's physics. Concepts like zero, negative numbers, irrational numbers, complex numbers (so shocking they were called "imaginary") and quaternions were once considered absurd and then revolutionary. Today they're commonplace. You know who uses quaternions? Game programmers. Quaternions are good for rotating things in 3-space, which is why Hamilton invented them. Did you know that they were the first noncommutative type of number discovered? They were shocking at the time just for that. xy isn't necessarily yx in the quaternions.

And what of non-Euclidean geometry? A mathematical curiosity in 1840 till Einstein came along and showed that Riemann's weird geometry was the best way to describe how the universe worked. Actually it was Einstein's friend, the mathematician Minkowski, who saw the connection.

So how do any of us know what future use the physicists will find for mathematical concepts we consider "interesting but useless" today? Like transfinite ordinals and cardinals.

For self-amusement - sure. Maths has lots of beauty and symmetry hidden in all the abstractions.

Ok then you and I have no quarrel and frankly not even a disagreement. I never ever claim math is useful for anything. I'm surprised at how people venerate math and get upset at its counterintuitive conclusions. If you just see math as chess, a formal game, you never have to ask if it's "true"; only if it's interesting.

Of course math turns out to be supremely useful in the world. It's a philosophical mystery, since math is so profoundly unreal. Math by definition is abstract. It's inspired by reality but does not aspire to be reality.

[Logik]

Ok cool! Then you are not one of these dogmatic ultrafinitists, finitists, and constructivists who deny even the purely formal game of modern math. I've seen too many of those types online. You will allow me modern math as long as I agree it's completely useless.

I am not really sure what it means to "deny" formal game or to "deny" you math in practice. e.g what are the consequences?

I only "deny" it in as much as I don't really practice pure formalism for its own sake e.g it's always part of a bigger picture.

My denial stops at my own practices and behaviors. I can't tell you what to do or how to think...

Which by the way brings up a point. Today's totally useless math often becomes tomorrow's physics. Concepts like zero, negative numbers, irrational numbers, complex numbers (so shocking they were called "imaginary") and quaternions were once considered absurd and then revolutionary.

Where I disagree with this mode of thinking is to assume the maths was incidental not coincidental.

Necessity is the mother of invention. If you conceptualise the problem you could trivially formalise it. Invent new mathematics if need be.
In fact - I often find myself deriving stuff from first principles because I don't know what to type in Google to find the relevant Mathematics.
I have no way of knowing that somebody else has already worked on this stuff and I need it NOW. And so I am much more fond of Just-in-time manufacturing of knowledge ( https://en.wikipedia.org/wiki/Just-in-t ... ufacturing ).

I mean - we can all thank Newton for his contribution to science over the last 500 years, but with modern-day technology and a reasonable dataset I can derive his life's work in a day from first principles. And so, while we are grateful and credit Newton for his achievement (and its greatness for its era) - it's really not much to brag about in 2019.

Today they're commonplace. You know who uses quaternions? Game programmers. Quaternions are good for rotating things in 3-space, which is why Hamilton invented them. Did you know that they were the first noncommutative type of number discovered? They were shocking at the time just for that. xy isn't necessarily yx in the quaternions.

See. Nomenclature and primitives

What you call "noncommutative" I call "irreducible". Whatever we call it - it is a desirable systemic property on top of which to build anything we might call an "ontology". It's Terra firma.

Yes. I know that xy is not yx. Event-ordering matters in a temporal universe. It's a key concept in reversible computing and is a direct consequence of Noether's conservation theorems. In this instance - conservation of information/energy.

And I didn't know what "quaternions" were till about 2 minutes ago, but I have been using them for 20 years.

So do I give Hamilton credit for his work or not? I certainly didn't copy his homework...

And what of non-Euclidean geometry? A mathematical curiosity in 1840 till Einstein came along and showed that Riemann's weird geometry was the best way to describe how the universe worked. Actually it was Einstein's friend, the mathematician Minkowski, who saw the connection.

Finding the connection is the hard part. When you have Mathematicians in isolation they produce a lot of isomorphic work. It creates the illusion of a fractured body of knowledge. It results in ivory towers.

Me - I like unification, so I appeal to things like principle of equifinality and functional equivalence (isomporphism).

"If it's stupid and it works it's not stupid" is mantra in my field. A mantra that offends the sensibilities of beauty and symmetry most mathematicians love.

So how do any of us know what future use the physicists will find for mathematical concepts we consider "interesting but useless" today? Like transfinite ordinals and cardinals.

In so far as my experience of physics and invention in general has been - it's never how it works.

I re-invent the wheel first. Then somebody who has theoretical grounding tells me I've done the same work as XYZ did 150 years ago.

Principle of least effort is abound. It's often easier to solve your problem however you see fit than to look for solutions others have come up.

Even if I needed "quaternions" I didn't have the word to dig up Hilbert's work.

In any case - the way this all relates to the OP: if, for some practical (e.g not formal) reason you need to compute the Zeta function then do it.
Inductively. And as the OP observes - inductive/brute-force methods are spacetime (memory!) bound.