## √5 and Phi

What is the basis for reason? And mathematics?

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Skepdick
Posts: 3125
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### Re: √5 and Phi

wtf wrote:
Mon Feb 10, 2020 12:41 am
Yes it's the standard definition in the computer science curriculum.
And that would be dandy, but I side-stepped academia.

I am an autodidact. I understand computer science (engineering?) by having done it for 20+ years.

Now that I am reading the theory, I am joining all the dots to the practice.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

Skepdick wrote:
Mon Feb 10, 2020 12:42 am
wtf wrote:
Mon Feb 10, 2020 12:41 am
Yes it's the standard definition in the computer science curriculum.
And that would be dandy, but I side-stepped academia.

I am an autodidact. I understand computer science (engineering?) by having done it for 20+ years.

Now that I am reading the theory, I am joining all the dots to the practice.
Ok.

Skepdick
Posts: 3125
Joined: Fri Jun 14, 2019 11:16 am

### Re: √5 and Phi

wtf wrote:
Sun Feb 09, 2020 10:47 pm
I agree completely. But that's not the definition of computability in the field of computer science.
While we are talking about computation and representation (and whether Turing's definition should be THE definition), I think you may find this perspective interesting.

Sometimes all functions are continuous
You may have heard at times that there are mathematicians who think that all functions are continuous. One way of explaining this is to show that all computable functions are continuous. The point not appreciated by many (even experts) is that the truth of this claim depends on what programming language we use.

(....)

The lesson is for those “experts” who “know” that all reasonable models of computation are equivalent to Turing machines. This is true if one looks just at functions from N to N. However, at higher types, such as the type of our function m, questions of representation become important, and it does matter which model of computation is used.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

Skepdick wrote:
Mon Feb 10, 2020 3:12 am
Thanks for the article, looks good. I know a bit about constructive math. Not only from the computational viewpoint but also from higher category theory and topos theory in abstract algebra. Non-constructivism is all the rage these days.

It's good to remember that foundations aren't cage matches to the death. They're tools. You use one foundation or another to get insight into a problem. It's the math itself that's the primary object of study, not the foundation.

Can you do me a favor and point me to the quote in the Nelson article you were referring to? I don't have time to read the whole thing. But I was curious to see why you claim he was supporting your point that 1/3 is computable in base 3 but not computable in base 10; as if the representation of a number makes any difference. I couldn't believe Nelson would say such a thing but if it did, please point me to it. Thanks.

nothing
Posts: 425
Joined: Mon Oct 14, 2019 9:32 pm

### Re: √5 and Phi

wtf wrote:
Sun Feb 09, 2020 8:01 pm
nothing wrote:
Sun Feb 09, 2020 4:53 pm
However Φ is a real solution to a quadratic equation.
Yes. That's what makes it algebraic, by definition. And since pi is transcendental -- NOT the solution to any polynomial having integer coefficients -- your equality is false.
π actually can be the solution to a polynomial with integer coefficients:

The same polynomial (also a psuedo-quadratic) equals 1 and Φ³ if/when the base is squared:

Squaring the base of π thus: 2π→2π² derives
the natural '1' from transcendental π
if/when serving as the base of Φ.

Quantitatively Φ² = Φ + "2π" is not "arithmetically" equal, as
"2π" does not arithmetically equal '1', but
each 2π implicitly concerns '1' as "1" full rotation about the unit circle described by r = 1.
This is why π and 2π describe 180- and 360-degrees of rotation.
Now whatever happens to '1' happens according to the cooperation of Φ and π.

Skepdick
Posts: 3125
Joined: Fri Jun 14, 2019 11:16 am

### Re: √5 and Phi

wtf wrote:
Mon Feb 10, 2020 4:35 am
It's good to remember that foundations aren't cage matches to the death. They're tools. You use one foundation or another to get insight into a problem. It's the math itself that's the primary object of study, not the foundation.
Yeah. I have no problem with that. It's exactly how I leverage perspectivism. Different assumptions (axioms) give a different view.

What I find way more incredible and exciting is that all these different perspectives correspond to each other (Topology, Logic, Category Theory) and so it's less about the theories themselves and more about the thing they actually describe. Which (in my own view) is the inner workings of human epistemology.
wtf wrote:
Mon Feb 10, 2020 4:35 am
Can you do me a favor and point me to the quote in the Nelson article you were referring to? I don't have time to read the whole thing. But I was curious to see why you claim he was supporting your point that 1/3 is computable in base 3 but not computable in base 10; as if the representation of a number makes any difference. I couldn't believe Nelson would say such a thing but if it did, please point me to it. Thanks.
I think we missed each other there. I referenced Nelson in terms of his view on representational/non-represenational mathematics. Which is how/why (in the context of a discussion on "computation") I thought the "Sometimes all functions are continuous" article relevant.

The relevant quote is in this post:
Skepdick wrote:
Sun Feb 09, 2020 11:55 pm
(...)

And so the implication seems to be exactly that. Representation (base, choice in precision) matters. In practice, if not conceptually or symbolically.

That 1/3 is finitely representable in base-3 is not significant (in my view). It's a coincidence because we are looking at a particular case.There will be other divisions in base-3 which would produce irrationals, so the finite representation of the result is still subject to choosing the correct numeric base.

In practice, we are robbed of this "choice" by the underlying implementation details and leaky abstractions. And so we have to hack our way around them : https://0.30000000000000004.com/

Again: I recognize that you are coming from a place of symbolic computation, which doesn't need to concern itself with the practical limits (dogma?) of digital or quantum computers.

It's the declarative/imperative distinction.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

nothing wrote:
Mon Feb 10, 2020 11:38 am
π actually can be the solution to a polynomial with integer coefficients:

I'm afraid I see no polynomial with integer coefficients here. Do you?
nothing wrote:
Mon Feb 10, 2020 11:38 am
"2π" does not arithmetically equal '1',

True. If you don't mean equality you should try to say what it is you do mean.

Now it's interesting that e^0 = e^(2 pi i) = 1. Perhaps you have a complex exponential going on somewhere. It's well known that you can wrap the real line around the unit circle this way. Is that what you're doing?

You still can't have an algebraic relationship between phi and pi since phi is algebraic and pi is transcendental.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
The relevant quote is in this post:
Skepdick wrote:
Sun Feb 09, 2020 11:55 pm
(...)
You seem to have omitted the quote.
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
And so the implication seems to be exactly that. Representation (base, choice in precision) matters. In practice, if not conceptually or symbolically.
It could never matter in the question of whether a real number is computable or algebraic. Those are adjectives that refer to numbers and not to their particular representations.
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
That 1/3 is finitely representable in base-3 is not significant (in my view). It's a coincidence because we are looking at a particular case.There will be other divisions in base-3 which would produce irrationals, so the finite representation of the result is still subject to choosing the correct numeric base.
You're equivocating. If we define pi as Turing-computable but not Skepdick-computable, I ask again: Is 1/3 Skepdick-computable or not?
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
In practice, we are robbed of this "choice" by the underlying implementation details and leaky abstractions. And so we have to hack our way around them : https://0.30000000000000004.com/
Floating point has nothing to do with any of this. The fact that you come here every day to stand on a soapbox and refuse to accept the difference between abstract and practical things strikes me as bizarre. Something about your worldview is off in some fundamental way.

Let me ask you this. Do you understand that IEEE-754 floating point is not the same as the mathematical real numbers? That they're two different things?
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
Again: I recognize that you are coming from a place of symbolic computation,
I have no idea what that means nor do I think you have a clue about where I'm coming from. But don't you often mention you're a computer programmer? Isn't computer programming pretty much the canonical example of symbolic computation? And you say you take your worldview from that. So by your own logic YOU must be a person who comes from a place of symbolic computation. I don't get your remark at all.
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
which doesn't need to concern itself with the practical limits (dogma?) of digital or quantum computers.
I'm just not catching your overall drift. Math isn't computer programming, engineering isn't physics.
Skepdick wrote:
Mon Feb 10, 2020 8:17 pm
It's the declarative/imperative distinction.
Which is what, exactly? I don't know what that means.

By the way I found the earlier article on continuous functions very interesting. I just can't figure out what your thing is. You work with computers therefore all of math and computer science is wrong? I don't get it. If someone works as a bricklayer that doesn't mean they don't believe in architecture.

Or are you just saying that the real world is one thing and the abstract world is another? That's true. We can conceive of perfection but never attain it. Deep thoughts.

nothing
Posts: 425
Joined: Mon Oct 14, 2019 9:32 pm

### Re: √5 and Phi

wtf wrote:
Tue Feb 11, 2020 12:38 am

I'm afraid I see no polynomial with integer coefficients here. Do you?
Honestly yes but that is because I do not abide by the restrictions of what a 'polynomial' is (or can be) according to mathematical orthodoxy.
The first term is a real integer coefficient, thus bound to the equivalent geometric expression which it itself describes, if even 'unreal' elsewhere,
which is practically negated by the inclusion of π in each term whose relationships to one another revolve around the concerned '1' in relation to the base.
If this is confusing, in other words: imagining the Φ-derived '1' serving as a "universal" datum indicating 2π / 2π, it follows that any "polynomial" constructed thereupon, intrinsically concerns the unit circle at r = 1, which itself can be adapted to fit any universal geometry whence the same.

This is overall what I mean by allowing the relationship itself to determine what '1' is, rather than arbitrarily using '1' to generate Φ. Because '1' is chosen instead of π, the '1' is merely user-defined... '1' of whatever-the-user believes '1' to be. However π is transcendental: by using that instead, it is the same operandi any universal language would employ, while simultaneously binding it to the real unit circle at r = 1 generated by Φ. Now we can place literally any "variable" in the expression, in any base, to capture any geometry relating to the unit circle, keeping in mind the entire expression naturally contains both Φ and the + 1 "ground" as intrinsically related to π. Therefor if the expression is treated generally as a function of π, rather than the integer '1', the '1' is now a universal variable that fixes π ever-in-relation to Φ, like curve and line to produce desired geometry.
nothing wrote:
Mon Feb 10, 2020 11:38 am
"2π" does not arithmetically equal '1',
True. If you don't mean equality you should try to say what it is you do mean.
It was correct observation they are not equal - I added quotations in the OP to indicate expressive and addentum to clarify.
nothing wrote:
Mon Feb 10, 2020 11:38 am
Now it's interesting that e^0 = e^(2 pi i) = 1. Perhaps you have a complex exponential going on somewhere. It's well known that you can wrap the real line around the unit circle this way. Is that what you're doing?

You still can't have an algebraic relationship between phi and pi since phi is algebraic and pi is transcendental.
Equivalently yes, above should find agreement. I am going to look deeper into the equation you provided - interesting.

But you can have an algebraic relationship between π and Φ: that π is itself transcendental does not mandate it is not geometric.

The barrier is rooted in mathematical theory: though π is itself transcendental, this is a/the "artifact" of thinking of π as if a static fixed or motionless "something". In this way it is "transcendental" because alone it is not describing anything in this universe if not attached to, and co-operating with, something that is in motion. They key to π is thus motion: if no motion (and related base, such as π, 2π, 3π, 4π etc. they are all "valid") no geometry. If motion (and the same) then geometry.

Skepdick
Posts: 3125
Joined: Fri Jun 14, 2019 11:16 am

### Re: √5 and Phi

wtf wrote:
Tue Feb 11, 2020 12:48 am
You seem to have omitted the quote.
I figured you'd click on the blue up-arrow next to the quote which will take you to the relevant post (so i don't spam the thread again).
wtf wrote:
Tue Feb 11, 2020 12:48 am
It could never matter in the question of whether a real number is computable or algebraic. Those are adjectives that refer to numbers and not to their particular representations.
But every time you write down a number - whether to paper; or to memory - you are handling a representation.

You are mentioning the number. In the language of C - you have a pointer.

So you could (metaphorically) think of a number as "an address on the ticker tape of a Turing machine", but it gets stupidly recursive - how do you represent the address? And yes - I am blurring lines between countable/uncountable infinities so my metaphor works.
wtf wrote:
Tue Feb 11, 2020 12:48 am
You're equivocating. If we define pi as Turing-computable but not Skepdick-computable, I ask again: Is 1/3 Skepdick-computable or not?
It doesn't matter! It's just a label. Definitions are irrelevant. Empirical measurements matter.
In 2020 we have arbitrary-precision software libraries. Having an algorithm that produces the N-th digit of an irrational number is still an issue of tractability if the value of N pushes you beyond the limits of physics.

It took Google 12 days to calculate 31 trillion digits. How many do you want/need?

If you need more than the computational resources of the universe - then it's not computable.
If you need less than the computational resources of the universe - then it's computable.

https://en.wikipedia.org/wiki/Bremermann%27s_limit
https://en.wikipedia.org/wiki/Landauer%27s_principle
wtf wrote:
Tue Feb 11, 2020 12:48 am
Floating point has nothing to do with any of this. The fact that you come here every day to stand on a soapbox and refuse to accept the difference between abstract and practical things strikes me as bizarre. Something about your worldview is off in some fundamental way.
It's called model-dependent realism. https://en.wikipedia.org/wiki/Model-dependent_realism

Abstract models are useful lenses for understanding patterns in the real world. How else do we "understand?" if we don't have a model for understanding.
wtf wrote:
Tue Feb 11, 2020 12:48 am
Let me ask you this. Do you understand that IEEE-754 floating point is not the same as the mathematical real numbers? That they're two different things?
Yes! Representation! Floating points are approximations of reals. They are not reals.

But numbers don't exist. So who cares?
wtf wrote:
Tue Feb 11, 2020 12:48 am
Isn't computer programming pretty much the canonical example of symbolic computation?
Not according to Programming Language Theory. Symbolic is just one of the many paradigms for expressing computations.

That's why I keep telling you that Mathematics is just a language. Like Python, or Haskell, or C - it's notation.
wtf wrote:
Tue Feb 11, 2020 12:48 am
And you say you take your worldview from that. So by your own logic YOU must be a person who comes from a place of symbolic computation. I don't get your remark at all.
I come from a place where computation is broader than that.

It's empirical/practical. Space and Time complexity are practical concern.
wtf wrote:
Tue Feb 11, 2020 12:48 am
I'm just not catching your overall drift. Math isn't computer programming, engineering isn't physics.
Yeah. It is. When Mathematics eventually catches on to proving theorems on computers, you'll figure out it's all just knowledge representation and organization.

Only thing different is your problem-space concerns itself with abstract problems.
wtf wrote:
Tue Feb 11, 2020 12:48 am
Which is what, exactly? I don't know what that means.
Imperative programming: telling the “machine” how to do something, and as a result what you want to happen will happen.
Declarative programming: telling the “machine” what you would like to happen, and let the computer figure out how to do it.

Python is imperative.
Mathematics is declarative. You are happy to write f(x,y) = z and skip the 10 pages in between - the procedure/algorithm to obtain the result.
wtf wrote:
Tue Feb 11, 2020 12:48 am
By the way I found the earlier article on continuous functions very interesting. I just can't figure out what your thing is. You work with computers therefore all of math and computer science is wrong? I don't get it.
My thing is straddling the invented/discovered fence. How is it, that different parts of society go to their own corners, develop their own formal theories about their own specialised interests and all of them come out with the same results (alas - different notations)?

What is "IT" that we all keep discovering/observing?
wtf wrote:
Tue Feb 11, 2020 12:48 am
If someone works as a bricklayer that doesn't mean they don't believe in architecture.
I have no use for the language of "believing in". I don't care ice hockey, but I believe in it.
wtf wrote:
Tue Feb 11, 2020 12:48 am
Or are you just saying that the real world is one thing and the abstract world is another? That's true. We can conceive of perfection but never attain it. Deep thoughts.
I am saying that if the abstract world is some place "elsewhere" yet we keep finding the same patterns, that sure meets the empirical bar for "reproducibility".

The question that nags me is: what is "IT" that we are observing/describing with formal languages?

Impenitent
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### Re: √5 and Phi

Skepdick wrote:
Tue Feb 11, 2020 10:52 pm

The question that nags me is: what is "IT" that we are observing/describing with formal languages?
the noumena

-Imp

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

nothing wrote:
Tue Feb 11, 2020 5:44 pm
wtf wrote:
Tue Feb 11, 2020 12:38 am

I'm afraid I see no polynomial with integer coefficients here. Do you?
Honestly yes but that is because I do not abide by the restrictions of what a 'polynomial' is (or can be) according to mathematical orthodoxy.
Ok that's fine. But then do this. Say, "A standard polynomial is such and so. But for my purposes I want to define a nothing-polynomial as ..." then make your definition. In math you can make up any definition you want.

But when you change the definition of a standard term without first announcing that fact, you simply appear divorced from reality, claiming to have a polynomial with integer coefficients then presenting an expression that is nothing of the kind.

If you wish to communicate with others, then whenever you re-define standard terminology, announce that fact by clearly giving your own definition. Otherwise you can't be taken seriously. Doesn't that make sense?

nothing wrote:
Tue Feb 11, 2020 5:44 pm
The first term is a real integer coefficient, thus bound to the equivalent geometric expression which it itself describes, if even 'unreal' elsewhere,
Please define "real integer coefficient" if it means something other than a standard integer. Else how can your reader know what you're talking about?

nothing wrote:
Tue Feb 11, 2020 5:44 pm
which is practically negated by the inclusion of π in each term whose relationships to one another revolve around the concerned '1' in relation to the base.
Impossible to understand what you mean.

nothing wrote:
Tue Feb 11, 2020 5:44 pm
If this is confusing, in other words: imagining the Φ-derived '1' serving as a "universal" datum indicating 2π / 2π, it follows that any "polynomial" constructed thereupon, intrinsically concerns the unit circle at r = 1, which itself can be adapted to fit any universal geometry whence the same.
Pure word salad. Devoid of meaning.
nothing wrote:
Tue Feb 11, 2020 5:44 pm
This is overall what I mean by allowing the relationship itself to determine what '1' is, rather than arbitrarily using '1' to generate Φ. Because '1' is chosen instead of π, the '1' is merely user-defined... '1' of whatever-the-user believes '1' to be. However π is transcendental: by using that instead, it is the same operandi any universal language would employ, while simultaneously binding it to the real unit circle at r = 1 generated by Φ. Now we can place literally any "variable" in the expression, in any base, to capture any geometry relating to the unit circle, keeping in mind the entire expression naturally contains both Φ and the + 1 "ground" as intrinsically related to π. Therefor if the expression is treated generally as a function of π, rather than the integer '1', the '1' is now a universal variable that fixes π ever-in-relation to Φ, like curve and line to produce desired geometry.
More of same. Don't mean to be unduly critical. But parts of your posts are lucid and parts aren't, and I hope I'm giving you useful feedback by pointing out the areas where you need to work on clarifying your exposition so that people can understand you.
nothing wrote:
Tue Feb 11, 2020 5:44 pm
It was correct observation they are not equal - I added quotations in the OP to indicate expressive and addentum to clarify.
Nothing seems very clear to me.
nothing wrote:
Tue Feb 11, 2020 5:44 pm
Equivalently yes, above should find agreement. I am going to look deeper into the equation you provided - interesting.

https://en.wikipedia.org/wiki/Euler%27s_formula
nothing wrote:
Tue Feb 11, 2020 5:44 pm
But you can have an algebraic relationship between π and Φ: that π is itself transcendental does not mandate it is not geometric.
Please define "geometric" in this context. Of course pi has many geometric relations. But in terms of standard math, with the standard definitions, do you understand that phi is algebraic, pi is transcendental, and that these are mutually exclusive classes of real numbers?
nothing wrote:
Tue Feb 11, 2020 5:44 pm
The barrier is rooted in mathematical theory: though π is itself transcendental, this is a/the "artifact" of thinking of π as if a static fixed or motionless "something". In this way it is "transcendental" because alone it is not describing anything in this universe if not attached to, and co-operating with, something that is in motion. They key to π is thus motion: if no motion (and related base, such as π, 2π, 3π, 4π etc. they are all "valid") no geometry. If motion (and the same) then geometry.
Pi doesn't move. It's a fixed real number sitting at a particular location on the number line.

All in all, as we wrap the real number line around the unit circle, it keeps cycling over the same points with a period of 2pi. That's the period of the sin and cosine functions, and also of the complex exponential. The point (1,0) in the plane, or the complex number 1, gets hit by rotating around the circle 0 radians, 2pi radians, 4pi radians, and in general every 2pi radians you come back to the same spot. This is true.

Your relation to phi is much less clear and I can't understand what you're doing.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

### Re: √5 and Phi

Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
I figured you'd click on the blue up-arrow next to the quote which will take you to the relevant post (so i don't spam the thread again).
Duh, got it!! Will check it out.

Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
But every time you write down a number - whether to paper; or to memory - you are handling a representation.
Of course. But the representation is not the number.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
You are mentioning the number. In the language of C - you have a pointer.
Yes exactly. A representation is a pointer to the abstract concept it points to. 2 + 2 and 4 and 3.999... are distinct representations that point to the abstract number we call 4. Just as "justice" is a word that points to the abstract idea of justice, a thing we call care about and that is highly imperfectly implemented in the real world.

It's not only in math that we use abstractions as perfected examples of the imperfect things of our world, right? To be human is to abstract.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
So you could (metaphorically) think of a number as "an address on the ticker tape of a Turing machine", but it gets stupidly recursive - how do you represent the address?
Of course TMs can represent numbers. I don't see what your problem is. In computer memory if I have an address that points to a bit pattern that represents a number, so what? I hope this isn't confusing you in any way. What is your concern with a chain of a dozen pointers that end up pointing to a memory location that holds a bit pattern that represents a number? So what?
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
And yes - I am blurring lines between countable/uncountable infinities so my metaphor works.
This remark doesn't make any sense at all. What do countable and uncountable infinities have to do with anything? Except to note that TMs can only represent at most countably many numbers. Most real numbers are not computable.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
It doesn't matter! It's just a label. Definitions are irrelevant. Empirical measurements matter.
See I don't know what you mean here. Of course measurement matters. When an engineer is building a bridge I don't care if she knows abstract set theory, only that she knows how to build bridges. But that doesn't invalidate set theory. It's two different things. Engineering is about the physical world and abstract math is not. Surely you are not confused on this point either. So what is your point? That's what I don't understand? So what if bridge building isn't set theory? SO WHAT?? Why do you go on about it?
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
In 2020 we have arbitrary-precision software libraries.
But of course they are actually no such thing, even if they're called that. All physical machines are limited by time, space, and energy. So we can never arbitrarily represent all the real numbers, or even, in the age of the universe, all the positive integers. Computers are finite.

All arbitrary precision means is that you can have as many decimal places of accuracy as you like, subject to the physical limitations of the computer. You can NOT approximate every real number for the simple reason that your computer is finite. Again, surely you know that.

Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Having an algorithm that produces the N-th digit of an irrational number is still an issue of tractability if the value of N pushes you beyond the limits of physics.
Of course. Is this not obvious to you? Is it not obvious to you that it's obvious to me?

Turing made the definition he did because he understood all this. He wrote down an idealized, abstract representation of a machine that computes but is not bound by physical limitations.

Is this not perfectly obvious? Why do you keep belaboring the point? That's what I don't get.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
It took Google 12 days to calculate 31 trillion digits. How many do you want/need?
To compute the circumference of the universe? About 6 decimal digits. Or 30. I forget the number but it's relatively small.

To express the entire decimal representation of pi? You need all of them.

But pi is not a good example, because there are many finite-symbol closed-form expressions for pi.

http://mathworld.wolfram.com/PiFormulas.html

But why are you going on about this? It's perfectly clear that we can crank out as many digits of pi as we want, ignoring resource constraints. That's Turing's definition of computability.

And since the universe is finite and computations take time, space, and energy, we can only crank out finitely many digits of pi in the age of the universe.

WHO THE HECK THINKS ANYONE IS DISPUTING THIS, or not understanding it?

It's two different things. What we can do in the physical world, and what we can do with abstract computations having unbounded resources. Do you not understand this? Do you have to explain it to me, to yourself? I'm constantly confused by where you're coming from. You are stating the most obvious, commonplace, well-known simplicities, as if they were ... what? Revelations? Talking points? What? Tell me.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
If you need more than the computational resources of the universe - then it's not computable.
If you need less than the computational resources of the universe - then it's computable.
No. Turing-computable is Turing computable, and Skepdick-computable is Skepdick-computable. You do yourself no favors by insisting that everyone else in the entire world is using the wrong definition. You can call coffee tea, but if you ask for tea at the diner they're going to give you coffee. You can't start arguing with them that YOU use the word tea to mean coffee.

In computer science, the word computable means exactly what Turing said it meant in 1936. Nobody's changed it since then and nobody's found a better idea; that is, a definition that encompasses computations that Turing would not recognize as such.

Even quantum computing has the SAME power as classical computation. It runs more efficiently on some problems (as far as we know, but not proven); but in terms of computability, it's the same.

Why are you calling coffee, tea? Computable has a standard meaning and by insisting that YOUR definition is right any the standard one is wrong, you embarrass yourself. A definition can't be wrong. If you want to talk about the limits of practical computation, call it "practically computable" or Skepdick-computable and we can have a conversation. Otherwise you're tedious.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
It's called model-dependent realism. https://en.wikipedia.org/wiki/Model-dependent_realism
Namedropping some philosophical shit while acting like a crank is a waste of time.

As long as you insist on changing a common definition and then arguing with people about it, you're a crank. Your basic ideas are fine but the way you come across is very cranky.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Abstract models are useful lenses for understanding patterns in the real world. How else do we "understand?" if we don't have a model for understanding.
Wow deep man.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Yes! Representation! Floating points are approximations of reals. They are not reals.
Floats represent reals in the sense that all floats are rational numbers. That's as far as it goes. And you can't represent ALL rationals, because your computing equipment has bounded resources. There are real numbers you can NOT approximate with floating point arithmetic running on physical hardware. It would be helpful for you to understand this point.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
But numbers don't exist. So who cares?
As long as you don't want to do physics or biology, I suppose we don't care. How much civilization are you willing to abandon to stick to your position, whatever it is? Abstractions DO exist in the world. Justice, law, traffic lights, property. We could not run our lives without abstractions. Math consists of abstractions that often do have impact in our lives.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
wtf wrote:
Tue Feb 11, 2020 12:48 am
Isn't computer programming pretty much the canonical example of symbolic computation?
Not according to Programming Language Theory. Symbolic is just one of the many paradigms for expressing computations.
Very odd response. Computer programming is not an example of symbolic computation? You would deny this? Either by deliberately mis-parsing it, or by stating an obvious falsehood? You say programming ISN"T symbolic computation? Wow. News to me.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
That's why I keep telling you that Mathematics is just a language. Like Python, or Haskell, or C - it's notation.
You "keep telling me" things that are perfectly obvious. Why do you do that?

Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
I come from a place where computation is broader than that.
Than what, you lost me on that.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
It's empirical/practical. Space and Time complexity are practical concern.
Of course. All you're doing in all of this is talking about the different between computer science and software engineering. Why do you think this is news to anyone?
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Yeah. It is. When Mathematics eventually catches on to proving theorems on computers, you'll figure out it's all just knowledge representation and organization.
Then we'll all be as smart as you? You're being a crank again. You're right and everyone else is wrong.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Only thing different is your problem-space concerns itself with abstract problems.
Ok. What of it?
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
Imperative programming: telling the “machine” how to do something, and as a result what you want to happen will happen.
Declarative programming: telling the “machine” what you would like to happen, and let the computer figure out how to do it.

Python is imperative.
Mathematics is declarative. You are happy to write f(x,y) = z and skip the 10 pages in between - the procedure/algorithm to obtain the result.
Oh I see. So what?
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
My thing is straddling the invented/discovered fence. How is it, that different parts of society go to their own corners, develop their own formal theories about their own specialised interests and all of them come out with the same results (alas - different notations)?
Ok cool. When you're explaining this stuff to me, could you do me a favor and simply use standard terminology for things that have standard definitions? Like computable, which has a particular universally-understood meaning in computer science and math. If you want to define Skepdick-computability that would be much more clear in terms of communication. You can see that, right?

Besides, Skepdick-computability keeps changing. It's a function of how fast our hardware is and how clever our algorithms are. Whereas Turing-computability is eternal. It doesn't change over time.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
What is "IT" that we all keep discovering/observing?
What is reality, man? I hear ya.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
I have no use for the language of "believing in". I don't care ice hockey, but I believe in it.
You don't care for computer science, but you also have some kind of psychological block on believing in it or using its standard terminology.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
I am saying that if the abstract world is some place "elsewhere" yet we keep finding the same patterns, that sure meets the empirical bar for "reproducibility".
Meaning what? I can't connect your personal philosophy with your technical points.
Skepdick wrote:
Tue Feb 11, 2020 10:52 pm
The question that nags me is: what is "IT" that we are observing/describing with formal languages?
Abstract mathematical objects. Just as "larceny" refers to an abstract class of behaviors deemed against the law by the legal profession. Just as genes were an abstraction developed to explain the inheritability of various biological traits. Just as quarks are an abstraction in physics.

Have you notice that humans have the power of abstraction? And that our entire civilization depends on it? Math and computer science are highly formalized examples. But when you stop at a red light and go at the green, you are reifying abstractions. Assigning meaning to colors that inherently have no meaning. Yet your life depends on doing it the same way everyone else does. So it's real. Abstractions are real things. What sort of things? Abstract things.

https://plato.stanford.edu/entries/abstract-objects/

Skepdick
Posts: 3125
Joined: Fri Jun 14, 2019 11:16 am

### Re: √5 and Phi

Impenitent wrote:
Wed Feb 12, 2020 12:19 am
the noumena
If Mathematics is introspective and it's starting at "the noumena" - then we are doing Quantum Physics (or whatever the lowest level of abstraction is when interacting with reality). If Mathematics is about abstraction then this sure seems "backwards".

It's not that! We are staring at epistemology. How the mind works.

Impenitent
Posts: 2635
Joined: Wed Feb 10, 2010 2:04 pm

### Re: √5 and Phi

Skepdick wrote:
Wed Feb 12, 2020 9:35 am
Impenitent wrote:
Wed Feb 12, 2020 12:19 am
the noumena
If Mathematics is introspective and it's starting at "the noumena" - then we are doing Quantum Physics (or whatever the lowest level of abstraction is when interacting with reality). If Mathematics is about abstraction then this sure seems "backwards".

It's not that! We are staring at epistemology. How the mind works.
you asked what language is describing...

-Imp

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