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.
And that would be dandy, but I side-stepped academia.
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.
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.
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.
π actually can be the solution to a polynomial with integer coefficients:
Yeah. I have no problem with that. It's exactly how I leverage perspectivism. Different assumptions (axioms) give a different view.
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.wtf wrote: ↑Mon Feb 10, 2020 4:35 amCan 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'm afraid I see no polynomial with integer coefficients here. Do you?
You seem to have omitted the quote.
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.
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 pmThat 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.
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.Skepdick wrote: ↑Mon Feb 10, 2020 8:17 pmIn 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/
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.
I'm just not catching your overall drift. Math isn't computer programming, engineering isn't physics.
Which is what, exactly? I don't know what that means.
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.
It was correct observation they are not equal - I added quotations in the OP to indicate expressive and addentum to clarify.
Equivalently yes, above should find agreement. I am going to look deeper into the equation you provided - interesting.nothing wrote: ↑Mon Feb 10, 2020 11:38 amNow 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.
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).
But every time you write down a number - whether to paper; or to memory - you are handling a representation.
It doesn't matter! It's just a label. Definitions are irrelevant. Empirical measurements matter.
It's called model-dependent realism. https://en.wikipedia.org/wiki/Model-dependent_realismwtf wrote: ↑Tue Feb 11, 2020 12:48 amFloating 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.
Yes! Representation! Floating points are approximations of reals. They are not reals.
Not according to Programming Language Theory. Symbolic is just one of the many paradigms for expressing computations.
I come from a place where computation is broader than that.
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.
Imperative programming: telling the “machine” how to do something, and as a result what you want to happen will happen.
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)?
I have no use for the language of "believing in". I don't care ice hockey, but I believe in it.
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".
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.
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?
Impossible to understand what you mean.
Pure word salad. Devoid of meaning.nothing wrote: ↑Tue Feb 11, 2020 5:44 pmIf 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.
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 pmThis 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 seems very clear to me.
You might find this helpful.
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?
Pi doesn't move. It's a fixed real number sitting at a particular location on the number line.nothing wrote: ↑Tue Feb 11, 2020 5:44 pmThe 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.
Duh, got it!! Will check it out.
Of course. But the representation is not the number.
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.
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?
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.
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?
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.
Of course. Is this not obvious to you? Is it not obvious to you that it's obvious to me?
To compute the circumference of the universe? About 6 decimal digits. Or 30. I forget the number but it's relatively small.
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.
Namedropping some philosophical shit while acting like a crank is a waste of time.Skepdick wrote: ↑Tue Feb 11, 2020 10:52 pmIt's called model-dependent realism. https://en.wikipedia.org/wiki/Model-dependent_realism
Wow deep man.
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.
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.
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.
You "keep telling me" things that are perfectly obvious. Why do you do that?
Than what, you lost me on that.
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?
Then we'll all be as smart as you? You're being a crank again. You're right and everyone else is wrong.
Ok. What of it?
Oh I see. So what?Skepdick wrote: ↑Tue Feb 11, 2020 10:52 pmImperative 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.
Haskell is declarative.
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.
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?Skepdick wrote: ↑Tue Feb 11, 2020 10:52 pmMy 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 reality, man? I hear ya.
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.
Meaning what? I can't connect your personal philosophy with your technical points.
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.
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".
you asked what language is describing...Skepdick wrote: ↑Wed Feb 12, 2020 9:35 amIf 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.
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