√5 and Phi

[Skepdick]

Let me emphasize something I forgot to mention the other day about lazy evaluation. Lazy evaluation is an optimization technique. It's a clever way to reduce the use of resources in a given computation. We don't evaluate an expression till we need to. There's a complementary technique called memoization, which is that once we evaluate an expression we save the result so that we don't have to calculate it again. Computer science is full of these kinds of optimizations for the purpose of using real world resources (time, space, energy) more efficiently.

I understand all of that. Hence why I keep referencing the wikipedia page on space-time trade offs.

Just like you can choose to optimise a "mathematical universe" for consistency OR completeness, but not both.
Likewise, you can choose to optimise a computation for space OR time, but not both!

That is how I interpret dichotomies. They present us with forks in the road - choices. And in so far as I am choosing X over Y I say that optimising for X.

What I am further claiming is that Mathematics has chosen to optimise for constant time complexity at the cost of infinite space complexity. Perhaps implicitly rather than explicitly but you are doing it. You insist that everything is timeless.

This is 100% compatible with the language of complexity theory. All operations/functions/calculations in the Mathematical universe are constant e.g their time-complexity is O(1). Constant time. Which is zero time. But you can't choose zero time AND zero-space! If Maths is timeless then it's spaceful.
If Maths is consistent then it's incomplete etc. etc.

It's as trivial as imagining that all Mathematical functions have been pre-computed and memoised somewhere - that's what a topological space is (in my view anyway).

And so in getting the answer for f(x) = x^2 you just have to "look it up in a table (space):

x | x ^2
---------
1 | 1
2 | 4
3 | 9
...
...

In so far as I am treating a "topological space" as a "lookup table with 0(1) time complexity" - it's a perfect hash function

However, no optimisation can have the slightest effect on computability. You could implement lazy evaluation on a Turing machine.

Obviously - you have infinite memory. Infinite tape. By the very definition of a Turing Machine.

if ALL functions/operations/calculations took 0-seconds to compute then laziness/strictness distinction becomes meaningless.
The Oracle Machine knows everything right now.

It's only a meaningful distinction if you recognise time e.g if you recognise that Turing machines perform instructions/operations.
Moving the head 1000 times is not the same thing as moving the head once.

What you are talking about is complexity: how efficiently a computation runs as a function of the input length. As the inputs get longer, does the resource consumption go up as a polynomial or an exponential of the input length? That's a huge deal in computer science these days.

Yes, but that's only one side of the coin. The other side is the memoization aspect. When I want the N-th digit of pi the input length is always 1.

The table then looks like:

x | pi-digit
------------
1 | 3
2 | 1
3 | 4
4 | 1
5 | 5
......

The question becomes about the efficiency of your Turing Machine in accessing ALL of its memory/tape.
You are assuming that reading the first cell and reading the N-th cell takes "exactly the same number of operations".

The practical distinction here is between sequential vs random access memory. O(N) read time, or O(1) read time? Linked list or hash map?

If you are telling me that your Turing Machine can give you the answer to the N-th digit of pi in constant (zero) time, then you are necessarily telling me that your Turing Machine has random access memory.

But the Turing Machine doesn't have random access memory. It has sequential access memory. Move Head Left/Move Head Right. It's a linked list.

So which model of computation are you using then?

It's possible for an optimization to improve the complexity of a given computation; but never the computability.

This is where I think we keep missing each other. If your Turing Machine had finite tape (e.g you didn't pre-suppose infinite memory resources - infinite spaces) there will be direct implication on "computability".

I don't need infinite space OR infinite time to represent "ALL positive integers" here is a Python algorithm that would generate it for me.

Code: Select all

ℕ = lambda x=0: ℕ(x+1)

I have defined (represented?) the natural numbers as a recursive function. We call these generators

Now, if I run this algorithm on a real computer, I soon get a "recursion depth exceeded" error - obviously. because finite memory.
But if I run this algorithm on a Turing Machine with infinite tape, then it produces EXACTLY the natural numbers.

So whenever you say that you are doing transformations on the Natural numbers, I am hearing "you have a function which takes N as input".
But ℕ is a function that doesn't halt?

I guess my real question is one of "high order types" do you see ℕ as a space or as a function?

[wtf]

That is how I interpret dichotomies. They present us with forks in the road - choices. And in so far as I am choosing X over Y I say that optimising for X.

You're not reaching me here at all in this post.

The practical distinction here is between sequential vs random access memory. O(N) read time, or O(1) read time? Linked list or hash map?

Yes, these are practical distinctions. Absolutely nothing to do with computability. Everything to do with complexity at the theoretical level, and "just getting stuff to run faster" at the practical level.

If you are telling me that your Turing Machine can give you the answer to the N-th digit of pi in constant (zero) time, then you are necessarily telling me that your Turing Machine has random access memory.

You say you're an auto-didact, why don't you auto-learn about TMs? A TM implements memory, just incredibly inefficiently. It's a theoretical construct for the purpose of exploring the notion of computability.

But the Turing Machine doesn't have random access memory. It has sequential access memory. Move Head Left/Move Head Right. It's a linked list.

So effing what? The legal system isn't justice, but you don't reject the abstraction of justice. You accept it as a conceptual ideal.

So which model of computation are you using then?

There is, up to equivalence, only one model of computation. The TM. Nobody's ever discovered a more powerful one. Nobody. Church's lambda calculus is equivalent.

This is where I think we keep missing each other. If your Turing Machine had finite tape (e.g you didn't pre-suppose infinite memory resources - infinite spaces) there will be direct implication on "computability".

The tape is unbounded. Like I say, you are the one claiming to have taught yourself. So why don't you go learn? There are MOOCs on computation theory, there are books, there are websites.

I guess my real question is one of "high order types" do you see ℕ as a space or as a function?

I have no idea what is the subject of this conversation or what you're going on about. Honestly I don't. Are you standing on a soapbox announcing that TMs aren't the same as real-world computers? Why? It's perfectly obvious to everyone who spends more than five seconds thinking about the subject.

What exactly is it you think you know that Alan Turing and every computer scientist of the following 84 years doesn't?

[Skepdick]

You say you're an auto-didact, why don't you auto-learn about TMs? A TM implements memory, just incredibly inefficiently. It's a theoretical construct for the purpose of exploring the notion of computability.

Well, precisely! Your theoretical construct has theoretical limits.
A TM has exactly the same theoretical limits as a linked list.

A linked list reads elements in linear time. O(N)
What kind of theoretical model are you using that has 0(1) read times. Show it to me!

So effing what? The legal system isn't justice, but you don't reject the abstraction of justice. You accept it as a conceptual ideal.

Of course I accept the abstraction of computation. I don't accept your definition of "computability" devoid of an interpretation in complexity theory.

To claim that your "computable functions" have O(1)-time AND O(1)-space complexity is materially equivalent to claiming that Mathematics is consistent AND complete.

Pick a horse and back it.

There is, up to equivalence, only one model of computation. The TM. Nobody's ever discovered a more powerful one. Nobody.

Up to functional equivalence, yes. That is the Turing tarpit.
There are qualitative differences that you refuse to acknowledge.

Grammatical differences. Representation. Expressive power.

The tape is unbounded. Like I say, you are the one claiming to have taught yourself. So why don't you go learn? There are MOOCs on computation theory, there are books, there are websites.

And there's my 20+ years of practical experience and intuition that no amount of textbooks can teach you about limits.
They are hard things to recognize when your head is in the abstract universe and your feet don't touch the ground.

"Unbounded tape" means exactly the same thing as "infinite space".

The Python function which I gave you generates ALL the natural numbers. It takes input of length 1.
The space-complexity of the natural numbers is O(∞).

You've practically ignored Godel.

Computation is about language recognition. Not numbers! Linguistics and cognitive science.

https://en.wikipedia.org/wiki/Chomsky_hierarchy
Type-0 grammars include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine.

[wtf]

(double post)

[wtf]

I don't accept your definition of "computability" ...

It's not my definition.

It's Turing's definition, and it's the standard definition in the field of computer science. I can't argue a point like this. There's no rational content. If you won't agree to use the same terminology everyone else uses, others have no obligation to struggle to understand you.

It's as if we're doing math and you tell me that pi is an integer. I point out that actually the number pi is a real number that is not a member of the set of integers; and you tell me, "I don't accept your definition of integer."

How the heck am I supposed to respond to that? How do I deal with such a worldview? "Argument by making up my own terminology."

[wtf]

Yes I am aware, however π is not an ordinary constant like 47: coupling π to Φ generates
the equivalent of a variable rotating base (π per half-rotation) coupled to
the creative Φ ratio, along with all geometries intrinsic to it.

I'm going to start a religion based on multiplying the definition of phi by 47/47.

[Impenitent]

I observe they are whitish in color

So, would you say that you've empirically experienced, and therefore - intuitively understand the concept of "color"?

there is a package of eggs on the counter

So would you say that you empirically experienced, and therefore - intuitively understand the concept of a "set" e.g a "collection of things"?

I understand language and how it is used... even when applied to separate and unique empirical events...

-Imp

[Skepdick]

I understand language and how it is used... even when applied to separate and unique empirical events...

You may understand HOW language is used, but do you understand what language means?

To paraphrase wtf

When you understand language sufficiently then it's perfectly clear to you on an intuitive, experiential basis that the meaning of any linguistic expression (symbol, word, sentence or paragraph) is precisely the "empirical mental event" which language causes.

Communicating with language requires shared experience.

[Skepdick]

It's Turing's definition, and it's the standard definition in the field of computer science. I can't argue a point like this. There's no rational content. If you won't agree to use the same terminology everyone else uses, others have no obligation to struggle to understand you.

I would agree to your terminology if it made any rational sense. But it doesn't. In your "immutable Mathematical universe" you have mutable (read: stateful!) Turing Machines. Lolwut?!? You feign immutability, but then you use variables. Lolwut?!? 99% of the time the rules apply every time.

I have a different strategy... practicing the axiom of unrestricted comprehension. Arriving at axioms rather than defining axioms.

Definitions don't matter - two rational agents can arrive at consensus and derive their functional definitions as needed. This is Auman's agreement theorem in game theory.

It's as if we're doing math and you tell me that pi is an integer. I point out that actually the number pi is a real number that is not a member of the set of integers; and you tell me, "I don't accept your definition of integer."

And I would tell you: It's just lip service.

How would I empirically determine one or the other? You are arguing over nomenclature, not any logical/empirical/testable properties or implications.

In so far as you are telling me that "pi is computable" - I am hearing it's a lambda function pi(n). Great! I'll give you the interface...

pi = lambda n: magic(n)

You give me the syntax of the magic() function.

I showed you the syntax of a computable function which seeds/generates the set of ALL integers. Trivially: You needed to accept the axiom that "0" is the first number and then allow for induction to deduce that ℕ is a computable function.

Can you show me the syntax of a function which can seed/generate pi? What do you start with? What axioms do I need to accept so that I can deduce the computability of pi?

Or you could simply say: Typechecking for the 2nd order Lambda calculus (System F) is undecidable, therefore 2nd order functions of the "Computable" type are undecidable.

In practical/empirical/testable terms that means EXACTLY: Given an infinite stream of digits, could you tell me if that is the number pi?

How the heck am I supposed to respond to that? How do I deal with such a worldview?

You stop arguing. And start communicating. By practicing unrestricted comprehension.

Argument by making up my own terminology."

No. By calibrating terminology. Language synchronisation.

You play a cooperative, rather than an adversarial game.

[wtf]

I would agree to your terminology if it made any rational sense. But it doesn't.

It's not my terminology. It's the terminology of Turing (1936) and universally adopted by the entire field of computer science ever since.

I didn't read any farther.

[Skepdick]

It's not my terminology. It's the terminology of Turing (1936) and universally adopted by the entire field of computer science ever since.

I didn't read any farther.

Well, I'll give you the bit that you were supposed to read.

Typechecking for the 2nd order Lambda calculus (System F) is undecidable, therefore 2nd order functions of the "Computable" type are undecidable (this is the Halting problem recursively re-stated).

In practical/empirical/testable/falsifiable (e.g scientific) terms that means EXACTLY: Given an infinite set of symbols, you can't actually tell me if that is the number pi. You can't even tell me if it's a number!

Turing machines are language recognisers. If you can't teach a Turing Machine to recognise your symbols as numbers, then your syntax (representation!) is infinitely incomprehensible.

The dwelling place of meaning is syntax; semantics is the home of illusion. --Ed Nelson

Ed nelson is not quite right, but he's less wrong than Mathematicians. The dwelling place of meaning is a world where the syntax/semantics distinction is dissolved.

Homoiconicity

[Impenitent]

I understand language and how it is used... even when applied to separate and unique empirical events...

You may understand HOW language is used, but do you understand what language means?

To paraphrase wtf

When you understand language sufficiently then it's perfectly clear to you on an intuitive, experiential basis that the meaning of any linguistic expression (symbol, word, sentence or paragraph) is precisely the "empirical mental event" which language causes.

Communicating with language requires shared experience.

that's exactly why I shared the definition of empirical

https://www.merriam-webster.com/dictionary/empirical

https://dictionary.cambridge.org/dictio ... /empirical

it is not a mental event.

-Imp

[Skepdick]

that's exactly why I shared the definition of empirical

https://www.merriam-webster.com/dictionary/empirical

https://dictionary.cambridge.org/dictio ... /empirical

And that's why I keep telling you definitions don't matter. The definition of 'experience' is a posteriori experience itself.

I know that I am experiencing my thoughts/intuitions, and you are trying to tell me that what I am experiencing is not "experience".

I guess you have me convinced then. I'll stop experiencing on account of your definition.

it is not a mental event.

-Imp

You are ignoring the cases which disagree with you.

Thought experiments/private languages are mental events.
Reading poetry is a mental event - if there were no empirical consequences you wouldn't be reading.

[nothing]

I'm going to start a religion based on multiplying the definition of phi by 47/47.

Comparing 47/47 to π is clearly a false comparison: 47 is not transcendental.

You can use the geometry that is intrinsic to Φ in relation to 2π
to generate any geometry that exists in 3D.


Now every 2π concerns each: Φ, Φ² and Φ³ in a single geometry.

Regarding the religion: normally these are "belief"-based, whereas
the geometry of Φ need not be believed, as it can be known
to some degree. Mathematicians who must "count" things for a living
are way too easily dismissing the qualitative aspect of numbers.

[wtf]

Comparing 47/47 to π is clearly a false comparison: 47 is not transcendental.

You're multiplying the numerator and denominator of the definition of phi by pi. That is, you're multiplying phi by pi/pi; which is also not transcendental. It's 1.