√5 and Phi

[wtf]

Yeah, that's intuitive to me and I would accept it without proof.

It's not intuitive to me! For example if x is algebraic that means there's some polynomial with integer coefficients that has x as a zero. If x and y are algebraic, what's are the polynomials that show that x + y, xy, x - y, and x/y are algebraic? Not obvious at all without some work IMO.

Where I am tripping up is mapping your language to mine so I can grok it e.g "closed under addition...." sounds like what I call total, halting functions etc.

A set S is closed under an operation @ if for s1, s2 elements of S, s1 @ s2 is in S.

So the integers are closed under addition since the sum of any two integers is an integer.

But the integers are not closed under division since, for example, 2/3 is not an integer.

To say the algebraic numbers are closed under addition, subtraction, multiplication, and division just means that the sum, difference, product, and quotient of two algebraic numbers is algebraic.

Hope that's clear. Computability has nothing to do with this at all. And all binary operations are total (unless specified otherwise).

[Skepdick]

It's not intuitive to me! For example if x is algebraic that means there's some polynomial with integer coefficients that has x as a zero. If x and y are algebraic, what's are the polynomials that show that x + y, xy, x - y, and x/y are algebraic? Not obvious at all without some work IMO.

Perhaps because we accept different foundational axioms somewhere deep in our heads?

Why it's intuitive to me is "Everything implementable on a classical computer is algebraic".

If it halts, it's either polinomial time, or polinomial space. I don't know what the polynomial is, but it exists.

[wtf]

Why it's intuitive to me is "Everything implementable on a classical computer is algebraic".

But of course this is false. Pi is computable, as is e, as is every well-known transcendental constant.

If it halts, it's either polinomial time, or polinomial space.

That's also of course completely false in computer science. Halting refers to computability. Poly time/space refer to complexity. There are exponential algorithms that halt. For example the classical trial division algorithm for factoring integers.

[Skepdick]

But of course this is false. Pi is computable, as is e, as is every well-known transcendental constant.

Our definitions of "computable" are incongruent.

Any algorithm you implement on a classical computer (physical machine and I will give you infinite memory for the sake of argument) to compute pi to infinite precision requires infinite time.

It will never halt. It will keep spitting out digits forever.

That's also of course completely false in computer science. Halting refers to computability. Poly time/space refer to complexity. There are exponential algorithms that halt. For example the classical algorithm for factoring integers.

I think what we are disagreeing here is the definition of "halt".

The general, human intuition for "halting" is that "it gives a final, definite answer in finite time".

The only "definite" answer possible for pi is symbolic, not numeric. Everything else is an approximation.

[wtf]

I think what we are disagreeing here is the definition of "halt".

The general, human intuition for "halting" is that "it gives a final, definite answer in finite time".

Your definition conflicts with that of Turing and the entire computer science profession. We need not go back and forth on this, you're entitled to your alternate reality.

It's ironic that you asked me to ignore our history, which frankly I didn't remember. And now you just want to pick a fight by using definitions contrary to the ones used in computer science. If you don't think pi is a computable number you're wrong. There's an algorithm that can approximate it to any desired degree of accuracy. That's the definition.

[Skepdick]

Your definition conflicts with that of Turing and the entire computer science profession. We need not go back and forth on this, you're entitled to your alternate reality.

It's entirely possible. I have no idea what definitions I am using.

I am using my knowledge having acquired it empirically - I am a practitioner, not a theoretician.

It's ironic that you asked me to ignore our history, which frankly I didn't remember. And now you just want to pick a fight by using definitions contrary to the ones used in computer science.

I am not looking for a fight.

I am trying to figure out my axioms.

In as much as it's becoming obvious to me - what I am doing is far closer to reverse mathematics.

It's how engineers think. Theorems first.

[wtf]

I am trying to figure out my axioms.

In as much as it's becoming obvious to me - what I am doing is far closer to reverse mathematics.

https://www.cs.virginia.edu/~robins/Tur ... r_1936.pdf

If you want to use a different definition, call it quasi-computable or Skepdick-computable so that people who actually know computer science aren't confused. There's nothing wrong with your making an alternate definition. But note that if you insist that a Skepdick-computable number must reach its final answer in finitely many steps, then 1/3 = .333333... is not Skepdick-computable.

[Skepdick]

https://www.cs.virginia.edu/~robins/Tur ... r_1936.pdf

I have read it.

Turing's view misses out on tractability, so it falls short of my needs.

Engineer here. Tractability matters.

[wtf]

Turing's view misses out on tractability, so it falls short of my needs.

Engineer here. Tractability matters.

Ok. Then call it tracticably computable and we have no difference of opinion. But then what of 1/3? How do you handle that objection? Or do you regard 1/3 as tracticably noncomputable?

And how about computations involving numbers whose decimal notation istoo big to be represented in the physical universe, like 2^2^2^2^2^2^2^2^2^2^2? Are finite numbers like that regarded as tracticably non-computable too? Ok. You are entitled to your definition. If you're an ultrafinitist I have no problem with that, it's an interesting philosophy though too limited in its power to serve as the foundation of physical science. It's also not currently axiomatizable, which some people consider a problem.

[Skepdick]

Ok. Then call it tracticably computable and we have no difference of opinion. But then what of 1/3? How do you handle that objection? Or do you regard 1/3 as tracticably noncomputable?

We don't have a difference of opinion once we assume infinities as intractable.

An algorithm that takes infinite time cannot be said to halt in any way that's sensible to me.

[wtf]

Ok. Then call it tracticably computable and we have no difference of opinion. But then what of 1/3? How do you handle that objection? Or do you regard 1/3 as tracticably noncomputable?

We don't have a difference of opinion once we assume infinities as intractable.

An algorithm that takes infinite time cannot be said to halt in any way that's sensible to me.

No algorithm takes infinite time. I think you are confused about basic computer science. And clearly you don't understand the definition of computability since the algorithms involved all halt in finite time.

[Skepdick]

No algorithm takes infinite time. I think you are confused about basic computer science.

Pi has no "last digit"

An algorithm that outputs "all the digits of pi one by one" cannot come to an "end".

There is no such point on the time line that you can label "after the algorithm halted"

[wtf]

No algorithm takes infinite time. I think you are confused about basic computer science.

Pi has no "last digit"

An algorithm that outputs "all the digits of pi one by one" cannot come to an "end".

There is no such point on the time line that you can label "after the algorithm halted"

I can't talk you out of your beliefs. And you're entirely within your rights to make whatever alternative definitions you like. Let me know if you have any substantive mathematical questions, which I'll do my best to answer. If you think you're right and Turing was wrong, I'll still put my money on Turing.

An algorithm that outputs "all the digits of pi one by one" cannot come to an "end".

That's not the definition of computability. All that's required is to approximate a number with arbitrary precision. You ask me to approximate pi within 1/10000, the algo halts in finite time. You ask me to approximate pi within 1/10000000000000, the algo halts in finite time. That's the definition. You want to make a different definition and argue with me about it. Homey don't play dat.

[Skepdick]

I can't talk you out of your beliefs. And you're entirely within your rights to make whatever alternative definitions you like. Let me know if you have any substantive mathematical questions, which I'll do my best to answer. If you think you're right and Turing was wrong, I'll still put my money on Turing.

I am not talking about "beliefs" here. I am trying to describe what human experience and empirical observations of a "halting algorithm" might look like in practice, to a real human observer. I am trying to turn "halting" into something empirically testable, not merely definitional.

Turing was right in as much as time doesn't matter in the Mathematical universe - the symbolic universe.

It matters in the Human universe - the numeric universe where time is a scientific quantity.

[Skepdick]

That's not the definition of computability. All that's required is to approximate a number with arbitrary precision. You ask me to approximate pi within 1/10000, the algo halts in finite time. You ask me to approximate pi within 1/10000000000000, the algo halts in finite time. That's the definition. You want to make a different definition and argue with me about it. Homey don't play dat.

Guy, I am not trying to ruffle your feathers - I am not trying to start a fight,

I am merely pointing out that an approximation of pi is not pi. Pi is Pi.

It ca only ever exist as a symbol.