## Continuum

What is the basis for reason? And mathematics?

Moderators: AMod, iMod

wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Wed Sep 02, 2020 6:48 am R represents the set of real numbers.

To use your prior context f(R) = 6.
Ok thanks for clarifying.
Skepdick wrote: Wed Sep 02, 2020 6:16 am Exactly at 6. Since you discretised it.
How so? Does the squaring function "discretize" 25 when you input 5? And where have you ever seen the concept of a function that inputs the reals and outputs the number 6? Why are you using such a peculiar example to make such an incorrect point?
Skepdick wrote: Wed Sep 02, 2020 6:16 am We've had this discussion. Higher order functions are sometimes continuous.
Are you referring to forms of constructive math in which all functions are continuous? That's fine, in its proper context.
Skepdick wrote: Wed Sep 02, 2020 6:16 am Points are discrete entities!
Yes but the real numbers with the usual topology are not a discrete set.
Skepdick wrote: Wed Sep 02, 2020 6:16 am That's digital physics - it from bit.
That's Scott Aaronson's view on the Atomists (Democritus).
Scott Aaronson does not think the real numbers are a discrete set. Nor did John Wheeler when he was alive. That would be nonsense.
Skepdick wrote: Wed Sep 02, 2020 6:16 am Yes. That's your definition. I don't go on definitions - I go on intuitions.
LOL. Makes for one-sided conversations. Explains a lot. You make up your own technical terms then argue with people. So if I say the sky is blue, you'll say the sky is green and then when I argue with you, you'll just say, "Haha for me green is blue."
Skepdick wrote: Wed Sep 02, 2020 6:16 am Because you are talking about points.

And because you can do operators on points like >(x, y) -> Bool. where x,y are in R.

You call them points. I call them bits.
If x and y are points in the plane, how do you do >(x, y) -> Bool?

The real numbers with the usual topology are not a discrete set . This is what a discrete set is:

https://en.wikipedia.org/wiki/Discrete_space

Of course you could put the discrete topology on the real numbers, but I don't think that's what you mean.
PeteJ
Posts: 426
Joined: Fri Oct 16, 2015 1:15 pm

### Re: Continuum

wtf wrote: Wed Sep 02, 2020 3:35 am I will shorten it to two points in a moment. My response to this post is long and I hope you read it. But you need not reply to anything but two specific points: Zeno and infinity; the two specific points from my earlier post that you took great pains to avoid engaging with.

So read the rest of this ... but just respond to my challenges regarding the points you made about Zeno and infinity.
My point was that Zeno draws attention to various problems associated with our usual notion of time and space. For mathematicians this may be a minor issue but for metaphysics it is the whole issue. For instance, when do events happen? They cannot happen now since there is no duration in the 'now', and they cannot happen in the past or future. Another is the the size of space, for it has no outside and can have no external size. There would be no need to refer to Zeno or anyone else for these paradoxes. They arise whenever we closely analyse our notions of space-time and the continuum. Another would be the staccato/'legato question, since it makes no sense that space and time are either. Hence Weyl''s solution of a dual-aspect model.
Yes. And likewise I know the textbook definition of metaphysics, but that's ALL I know about it. You seem to be assuming so much background knowledge that I can't understand much of what you say.
Fair comment. This is a metaphysics forum so I assumed quite a lot.
But wouldn't you agree that the best way forward is for us to drill down on areas of common knowledge? I propose that these are Zeno and infinity. You earlier said that Zeno's paradox messes up math or messes up metaphysics or offends your metaphysical sensibilities in some way. I asked you to explain to me how any one of Zeno's several paradoxes causes you metaphysical trouble. And you didn't bother to engage with the question.
I did not say this. I said only that Zeno explores some of the paradoxes associated with space-time. How mathematicians respond to these is not my concern since they are worried only about formalities. But formalities don't solve metaphysical problems. For example, the calculus works just fine in maths for describing continuous change, but as Danziger notes this is not a metaphysical solution but just a work-around.
This is the maddening lack of specificity that I refer to in your work. Likewise you said something like "finite and infinite" are contradictory, or paradoxes, or something, I asked you to explain. You didn't.
I don't remember what I meant. I may have been referring to the problem that space cannot be infinite if BB theory is correct, yet space cannot be finite since there is no larger space to contain it. In metaphysics almost every aspect of the world is paradoxical according to out usual notion of it, and this is the problem of metaphysics.
I propose that you and I drill down on your thoughts about Zeno and about infinity, and explain to me -- and probably to yourself -- what troubles you about Zeno's paradoxes.
They don't trouble me. Our usual notion of space-time is paradoxical and I see no problem with this. The problem is with our usual notions, and in this respect metaphysics is profoundly iconoclastic. These paradoxes are easily solved by abandoning our usual notions. None of this need be of concern to mathematicians.

A case study might be Russell's (or Cantor's) paradox. There are various pragmatic mathematical solutions but they are not metaphysical solutions. For a metaphysical solution we need an idea like Brown's existential calculus, which requires a major paradigm-shift away from our usual notions of existence and the continuum.
I'm going to make a guess about something. You keep talking about paradoxes and how they're so well-known you can't name any of them.
Metaphysics is awash with paradoxes. They overwhelm most philosophers. Not many have official names. There is no need to name them. Pick any metaphysical question and you'll find yourself dealing with paradoxes.
I was thinking that you must mean something very subtle that I've never heard of. But now I just realized that if Zeno is one of your examples, you must simply be troubled by the usual laundry list of well-known mathematical paradoxes. Zeno's paradoxes, Russell's paradox, the Berry paradox, the Richard paradox, the Burali-Forti paradox, the staircase paradox, the Banach-Tarski paradox, any my personal favorite, the infinite hat paradox. That's just off the top of my head, there are lots more. You think these falsify math or falsify metaphysics or something. You won't even state a clear thesis so I don't know what your issue with them is.
I don't know most of these and have no idea of their significance for mathematicians. In metaphysics their significance is that they reveal theories that cannot be true. In metaphysics sets are emergent and the idea that set-theory is fundamental is paradoxical. The idea that form, number, set or category is fundamental is paradoxical. Mathematicians don't have to worry about such things but metaphysicians have nothing else to worry about.
The reason I never connected all these basic paradoxes with your concerns was because there is no cause for concern. None of these so-called paradoxes threatens math or metaphysics or much of anything else. They're fun to talk about. So if YOU think they break metaphysics, or whatever it is that bothers you; it's on YOU to clearly explain what you are talking about.
Well. I feel you have a responsibility not to expect me to explain everything. Much of this stuff about paradoxes is basic in metaphysics. These paradoxes do not 'threaten' metaphysics. They threaten the metaphysical views of most people. Have you noticed that almost nobody comprehends metaphysics? This is because they cannot solve the paradoxes associated with their usual views.

“Paradoxes are serious. Unlike party puzzles and teasers … paradoxes raise serious problems. Historically, they are associated with crises in thought and with revolutionary advances. To grapple with them is not merely to engage in an intellectual game, but is to come to grips with key issues."
R.M. Sainsbury - Paradoxes
Pick any paradox and explain to me how it breaks metaphysics or is metaphysically wrong. Then, clarify your statement about the finite and the infinite being contradictory.
There is a misunderstanding here. There are no paradoxes in my metaphysical view. Paradoxes arise only where one holds a view that gives rise to them. take the 'paradox' or 'antinomy' of whether the world begins with something or nothing. Neither idea works. We have a choice between seeing this as a paradox or as indicating the need for a third option. These paradoxes do not 'break' metaphysics. They are what allow us to know that certain theories are wrong. Finite/infinite are defined as contradictory, but they create no paradox unless we adopt a paradoxical view of them. What they 'break' is naive realism, materialism and and other paradoxical views. This is the value of metaphysics.
You don't know the first thing about set theory.
Perhaps. But it seems you don't know the first thing about its importance in metaphysics.
Which Western academic metaphysicians are incomprehensible? Name one; and describe exactly what it is they said that you claim is incomprehensible.
I would challenge you to name a philosopher considered 'Western' who has a comprehensible view of metaphysics. There is no such thing. Most of them would happily concede their view of metaphysics is incomprehensible, and most would argue metaphysics simply is incomprehensible, It is not a coincidence that Russell argued this and had problems with set-theory. Have you not noticed that metaphysics is incomprehensible to most people?
So you see the author of the text read by professional set theorists clearly -- CLEARLY! -- states that set theory is not about the real world.
Yes. This is exactly my point. For a fundamental analysis It is a fiction, just like what you call the real world.
Set theory says nothing about philosophical problems or a "fundamental theory," which is a piece of jargon that has meaning to you but not to me.
Set theory talks about what we want it to. I see no reason metaphysicians cannot use set-theory to explain the origin of form, as does Brown. If we equate sets with the categories of thought then it is very useful. The phrase 'fundamental theory' is not jargon, it is the entire purpose of doing metaphysics.
History perfectly well shows that set theorists have come to accept as basic the notion that set theory is not about anything in the real world.
Yes, This seems to have been Russell's view. One could equally say the same of the calculus. These are tools and models.
PeteJ wrote: Sun Aug 30, 2020 10:56 am " ...metaphysics is hopeless when we treat number and form as irreducible."

Ok. What of it? So what? Why am I supposed to regard this as important? Number is whatever mathematicians say numbers are; and form, I don't actually know what you mean. It's not a mathematical term.
You are not thinking philosophically. If form and number are not fundamental then the 'stereotypical 'Western' world-view is nonsense.
PeteJ wrote: Sun Aug 30, 2020 10:56 am It only becomes comprehensible when we transcend set-theory, form and number for a unified state.
This is incoherent buzzword bingo. You are making things up about set theory. You are using the word form in some technical sense. Mathematicians know perfectly well what numbers are and so you do. I don't know what a unified state means. More jargon.
Afaik the word 'form' is common currency. Mathematicians can define numbers any way they wish, since their concern is formality and not ontology. If you don't know what ''unified state' means then we really are in trouble,

You don't seem prepared to distinguish mathematics from metaphysics and this makes the discussion awkward. I feel we cannot get far while you talk about mathematics and I talk about metaphysics.[/quote]
PeteJ wrote: Sun Aug 30, 2020 10:56 am This is the Perennial metaphysics, if we can call it that, and Schroedinger, Weyl and Brown all endorse it. Kant comes close with his 'thing-in-itself'.
Kant died in 1804 and Cantor wasn't born till 1845; so I strongly doubt the truth of what you say here with regard to set theory.
In metaphysics Russell's paradox predates him by thousands of years. It is a well-known ontological problem, Venn diagrams are just a neat way of expressing it. In philosophy it matters not when ideas appear. Plato has more interesting things to say than most modern philosophers.
I think you can't justify or explain a thing. You're just tossing out buzzwords and avoiding the sharp questions I asked you. It does occur to me that you are re-litigating the confusions of 1900 and imagining that they're still relevant. And you utterly refuse to be specific about anything, or engage with specific questions I ask you.
I'm sorry this is your impression. The problem may be that you're trying to drag me into mathematics and I'm trying to drag you into metaphysics. My guess is that I know just a little more about the former than you know about the latter, but it may be a close run thing.
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

PeteJ wrote: Wed Sep 02, 2020 12:37 pmThe problem may be that you're trying to drag me into mathematics ...
This is the portion of the site devoted to the philosophy of math. You volunteered to be here. To be fair, you showed up in this thread talking about the mathematical continuum. If you had a deeper or different agenda, it would not have been initially possible for me to know that. I understand that now.

I agree that we're not going to make progress on metaphysics here, especially where I'm concerned. It's not my thing. You might have better luck in the Metaphysics section. viewforum.php?f=16

Just wondering, aren't your real concerns with modern physics, and not 19th century math?
PeteJ wrote: Wed Sep 02, 2020 12:37 pm For a fundamental analysis It is a fiction, just like what you call the real world.
I find deniers of the real world tedious. We're brains in vats, we're Boltzmann brains, we're programs in a cosmic computer, we're thoughts in the mind of God. These ideas may be so, but they seem nihilistic and unproductive. I think the purpose of philosophy is to account for what is, not to deny it.
PeteJ wrote: Wed Sep 02, 2020 12:37 pm Perhaps. But it seems you don't know the first thing about its importance in metaphysics.
If I'm a craftsman of fine hammers, and someone uses one of my hammers to smash open a can of tunafish, that is not my problem. I can't stop people from wrongly using my creations in applications for which they're not suited. Perhaps set theory should come with a label: "Warning: This is not about the world." But it DOES come with such a warning label. I quoted you the relevant passage in a standard professional text. I've seen set theorist Joel David Hamkins make the same point. My sense is that this is not an uncommon view among set theorists these days.

But I would be interested to know more about these (mis-) uses of set theory in metaphysics. It seems to me that any troubles you run into would be directly as a result of applying a tool to the wrong purpose; and in no way valid criticisms of set theory itself.

As an example, suppose I say, "The world is composed of sets." Then someone whips out Russell's paradox and now my metaphysics is busted. If I complain that "Set theory must be wrong," that's a category error. The problem is that I attempted to smash open a can of tuna with a finely crafted hammer. I should have read the warning label.

I agree with you that in 1900 all this was not as clearly understood as it is today. But that was an earlier point I made. Perhaps you are talking more about the history of metaphysics. People were confused about a lot of things in 1900. That was five years before Planck discovered quantum mechanics. The twentieth century is full of mathematical and physical surprises that were unimaginable in 1900.

Perhaps you'd be interested in category theory, which is an approach to mathematical foundations based on structural relationships and not the admittedly problematic notion of set membership. Using category theory we can in fact develop set theory without the notion of elements. That's more philosophically satisfying to some people and is beginning to become a mainstream view, at least in certain areas of math like geometry, logic, and algebra.

Well this is certainly an interesting discussion for two people with barely any knowledge overlap!
Skepdick
Posts: 4964
Joined: Fri Jun 14, 2019 11:16 am

### Re: Continuum

wtf wrote: Wed Sep 02, 2020 8:31 am I agree with you that thoughts are real. After all they are associated with electrochemical signals in the brain. Would you agree that a thought about a unicorn is as real as a thought about a rock; but that a unicorn is not as real as a rock? I would take that position. All thoughts are equally real, but the objects of thoughts might or might not be real.
I don't think this position is useful within the context of our conversation. Mathematical objects are real like thoughts about rocks, not real like actual rocks. And to argue over which one is "more or less" real leads directly to Gödel ontological proof.

You've used the vocabulary of "instantiation" before - I am happy to continue, because it's very much the vocabulary of my own ontology.physical information
wtf wrote: Wed Sep 02, 2020 8:31 am Ok. You have a function that inputs a set and outputs a real number. And when you input R, the set of real numbers, it outputs 6. What is your point?
My point is that you produced (instantiated?) the number "6" from somewhere (in your head). I am trying to understand the process/mechanism.

I am assuming (and only assuming) that you conjured "6" from "the set of real numbers" (R).
So when I say f(R) -> 6 I believe that I am literally describing what happened in your head.

If you tell me otherwise (6 didn't come from R), then I'll have to revise my belief to something else like... "6" manifested from some other function, with unknown inputs: g(?) -> 6.
wtf wrote: Wed Sep 02, 2020 8:31 am You are thinking of set difference, not a function. If I have the function f(x) = x^2 and I put in 5, the output is 25, But 25 didn't come out of the reals, did it?
First and foremost I am thinking of "where did 5 come from? Is it the same place 6 came from?"

If it came from R, is R a singleton? Can I pick another 5? And another? (I am conceptualising f(R) as sampling with replacement)
wtf wrote: Wed Sep 02, 2020 8:31 am Set difference, R \ {6}, says to take the real numbers and remove the point at 6. In this case the result is indeed two half-infinite intervals. So 6 in this case is called a cut point by topologists. What of it?
Well, it's about conceptual design again!

if f(R) -> 6 performs sampling with replacement, then nothing of it. It's not a cut point. Because "6" is not a singleton. The number "6" isn't really being "removed' from the set per se - no discontinuity is being introduced.

If f(R) -> 6 performs sampling without replacement, then yes - it's a cut-point. Because the "6" is "removed from the set" - f(R) -> 6 introduces the discontinuity.

And if you are thinking "what the fuck is he thinking?" - I am thinking "Do the elements of R behave like singletons?"
wtf wrote: Wed Sep 02, 2020 8:31 am Sure. If you have the ring axioms you have the integers. If you add a rule saying that every nonzero element has a multiplicative inverse you get the rationals. If you toss in the least upper bound axiom you get the real numbers. Mathematicians are always playing with alternative axiom systems. And logicians, as I pointed out, are always playing with inference systems. What of it?
Simply that the distinction between "logic" and "mathematics" doesn't exist from the lens of a linguist. They are both formal languages/language games played by different grammatical rules. If you don't buy in to my point of view, consider it as mere proof by abstract nonsense.

To borrow a tag line from the NuPRL project: Starting with the slogan "proofs-as-programs," we now talk about "theories-as-systems.
wtf wrote: Wed Sep 02, 2020 8:31 am Ok. One could make the same point about math. In the integers, 1 + 1 = 2. In the reals, pi + pi = 2pi. Technically the '+' symbol has been overloaded. In the former case it's addition of integers and in the second case it's addition of real numbers. What of it? In math the symbol π stands for the number pi, the prime counting function, the homotopy groups, and a projection function, depending on context. Symbols are overloaded all the time in math.
Yes they are. So the distinction in meaning, the semantics of "+" are incomplete.
They are in the Mathematician's head- it's not encoded in the notation.

So that's one way you are "fooling" yourself - you are pretending to be working with a context-sensitive language (one that supports expressing polymorphism), but you are working with a context-free language on paper so it manifests as equivocation.

When you do encode ALL semantic differences in your notation (so that a proof assistant can recognise it), you end up with the Curry-Howard isomorphism. Syntactic equivalence between proofs and programs.
wtf wrote: Wed Sep 02, 2020 8:31 am I thought your remarks on oracles were unclear. I still do. But the discussion doesn't seem to be about anything.
No shit. It's Philosophy.
wtf wrote: Wed Sep 02, 2020 8:31 am You're trying to make a dispute where there isn't one.
I am not? I am only pointing out the paradox which you are ignoring. You are free to ignore it - I am simply pointing out that you are doing it.

Abstractly, an oracle machine can solve a generally unsolvable problem. <----- PARADOX
a Turing machine with a black box, called an oracle, which is able to solve certain decision problems in a single operation. Even undecidable problems, such as the halting problem, can be used.
wtf wrote: Wed Sep 02, 2020 8:31 am If oracles don't require an explanation, would you say people are born knowing what they are? How could you defend such a proposition?
Now look who is reifying oracles! I thought we are talking about abstractions?

You are heading down unfashionable avenues. Talking about knowledge-revelations by all-knowing entities.
wtf wrote: Wed Sep 02, 2020 8:31 am You use "axiomatic" in a funny way that makes no sense to me.
I am using it in the common sense. An oracle can solve the halting problem.

Is that not an axiomatic truth?
wtf wrote: Wed Sep 02, 2020 8:31 am Your link did not apply to the conversation.
My oracle says it did.
Yours says it didn't.

Whose oracle is wrong?

Relativization everywhere
wtf wrote: Wed Sep 02, 2020 8:31 am No such conclusion is possible from available evidence. And your link was not about either of those. It was about physics. Any by the way logic and math have been different since Gödel.
Yeah, but it's 2020 now. Curry-Howard happened after Godel and Quantum Entanglement (physics) is a model of computation, so weave that thread.
wtf wrote: Wed Sep 02, 2020 8:31 am Sorry you missed the revolution of the 1930's.
Sorry you missed the last 60 years.
wtf wrote: Wed Sep 02, 2020 8:31 am You said math isn't abstract enough. I've studied enough math to conclude that it's plenty abstract. You're changing the subject.
I am not! I am just abstracting! I a generalising Mathematics - it's a language. A formal language, but still a language.

What is language and how does it relate to the universe? That's the question Philosophy has concerned itself with for ever!

From the lens of computational linguistics: Turing machines are language recognisers AND a model of computation.
Quantum Entanglement is a model of computation.

Weave that thread.
wtf wrote: Wed Sep 02, 2020 8:31 am Makes no sense to me in the context of this conversation.
Look! Part of the point I am trying to make is that "context" is a formal notion in computing. We have continuations and we have the call with current continuation function.

Mathematics pretends to (or strives to?) subscribe to denotational semantics. Mathematical symbols have specific/precise/unambiguous/universal meaning. They don't, actually! Precisely because the extra information required to interpret a symbolic Mathematical expression exists only in the Mathematician's head.

Programming languages/Computerised Proof assistants are the attempt to extract that information from the Mathematician's head and encode it in the notation/grammar. That is what "denotational semantics" looks like in practice: context-sensitive grammars. Context encoded/expressed in the language itself.
Last edited by Skepdick on Thu Sep 03, 2020 10:56 am, edited 5 times in total.
PeteJ
Posts: 426
Joined: Fri Oct 16, 2015 1:15 pm

### Re: Continuum

wtf wrote: Thu Sep 03, 2020 3:59 am This is the portion of the site devoted to the philosophy of math. You volunteered to be here. To be fair, you showed up in this thread talking about the mathematical continuum. If you had a deeper or different agenda, it would not have been initially possible for me to know that. I understand that now.
Oops. I hadn't noticed I'd strayed from metaphysics. This would account for our misaligned discussion. Apologies.
I agree that we're not going to make progress on metaphysics here, especially where I'm concerned. It's not my thing.
Let's agree it's a mismatch.
Well this is certainly an interesting discussion for two people with barely any knowledge overlap!
It could work okay but we'd have to go back and start from scratch. I suspect neither of wish to do this.

All good. I see now why we were pulling in different directions.
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

PeteJ wrote: Thu Sep 03, 2020 10:43 am All good. I see now why we were pulling in different directions.
No prob. We'll always have ...
PeteJ wrote: Sun Aug 30, 2020 10:56 amThere's so much nonsense on this site that your approach stands out.
That's the high point of my career on this site! Thank you kindly for that remark, regardless of whether you still agree!!
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Thu Sep 03, 2020 9:04 am I don't think this position is useful within the context of our conversation. Mathematical objects are real like thoughts about rocks, not real like actual rocks. And to argue over which one is "more or less" real leads directly to Gödel ontological proof.
Mathematical objects are abstract objects. Rocks are concrete objects.

https://en.wikipedia.org/wiki/Abstract_and_concrete
Skepdick wrote: Thu Sep 03, 2020 9:04 am You've used the vocabulary of "instantiation" before - I am happy to continue, because it's very much the vocabulary of my own ontology.
Just a word. Mostly from OO programming but also a more general idea, such as the fact that a carton of eggs instantiates the number 12 in the world.

[
Skepdick wrote: Thu Sep 03, 2020 9:04 am My point is that you produced (instantiated?) the number "6" from somewhere (in your head). I am trying to understand the process/mechanism.
A function is a mapping or association. If I have a set of schoolkids and I write down all their heights, I have a function where you input "Fred Bloggs" and output 5'7". Which part of that don't you understand? I put a torch under a metal plate and measure the heat at each point. That's a scalar field. A meteorologist computes the wind speed and direction at each point of city. That's a vector field. Surely these are commonplace ideas. Which part is giving you trouble?
Skepdick wrote: Thu Sep 03, 2020 9:04 am I am assuming (and only assuming) that you conjured "6" from "the set of real numbers" (R).
So when I say f(R) -> 6 I believe that I am literally describing what happened in your head.
I associated one thing with another thing. My couch is black, my sweatpants are gray, my table is brown, my carpet is red. To each thing I associate some other thing.
Skepdick wrote: Thu Sep 03, 2020 9:04 am If you tell me otherwise (6 didn't come from R), then I'll have to revise my belief to something else like... "6" manifested from some other function, with unknown inputs: g(?) -> 6.
Formally we have two sets, A and B, and with each element a ∉ A we associate an element b ∉ B; and if we call the association f, we write f(a) = b.

So perhaps we have some set of sets: The reals, the integers, the quaternions, say; and we associate the number 6 with the reals, the number 47 with the integers, and the number pi with the quaternions. So maybe the target set is also the reals. I know you learned this in high school, I'm curious as to why the concept of a function is giving you trouble.
Skepdick wrote: Thu Sep 03, 2020 9:04 am First and foremost I am thinking of "where did 5 come from? Is it the same place 6 came from?"
You have a source set and a target set, or a "domain" and "range" as they taught you in school. Each element of the source set gets associated with an element of the target set.
Skepdick wrote: Thu Sep 03, 2020 9:04 am If it came from R, is R a singleton? Can I pick another 5? And another?
I don't know, that was your example and it's a little bizarre. But, say, {R, Z, H} is the set containing the reals, the integers, and the quaternions (H for Hamilton is traditional).

If your function assigns 6 to R, you can't then assign 7 unless it's a different function. YOU KNOW ALL THIS why are you asking? They teach functions in high school math. Input/output. A soda machine. You put in a buck and out comes a soda. It's a function.

Skepdick wrote: Thu Sep 03, 2020 9:04 am (I am conceptualising f(R) as sampling with replacement)
No, that's not right. The squaring function inputs 5 and outputs 25 every time. That's the essential property of a function: You always get the same output for a given input. Of course you could have some OTHER function like the cubing function that inputs 5 and outputs 125.

To view a functional relation as sampling is a little bizarre, I wonder why you are going off in this direction. You must know what a function is. Traffic cop sits by the road with his LIDAR gun (they don't use radar anymore). Each car gets assigned a speed. That's a function.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Well, it's about conceptual design again!
I explained to you what a cut point is. It's a point of a connected topological space with the property that if you remove it, the space becomes disconnected.
Skepdick wrote: Thu Sep 03, 2020 9:04 am if f(R) -> 6 performs sampling with replacement, then nothing of it. It's not a cut point. Because "6" is not a singleton. The number "6" isn't really being "removed' from the set per se - no discontinuity is being introduced.
I'm sorry man, this is too weird. I'm sure you know what a function is. A function is not sampling with replacement. If you remove 6 from the real line you end up with a disconnected set consisting of two half-infinite intervals as you yourself pointed out.
Skepdick wrote: Thu Sep 03, 2020 9:04 am If f(R) -> 6 performs sampling without replacement, then yes - it's a cut-point. Because the "6" is "removed from the set" - f(R) -> 6 introduces the discontinuity.
I'm at a loss to follow your train of thought. Are you trolling me?
Skepdick wrote: Thu Sep 03, 2020 9:04 am And if you are thinking "what the fuck is he thinking?" - I am thinking "Do the elements of R behave like singletons?"
I do appreciate your self-awareness here. Your line of patter has indeed made me wonder what the fuck are you thinking.

Do the elements of R behave like singletons? Yes. Any set whatsoever is the union of its individual elements regarded as singletons. The real numbers R are indeed the uncountably infinite union of all the real numbers regarded as singletons.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Simply that the distinction between "logic" and "mathematics" doesn't exist from the lens of a linguist.
This hardly follows from your earlier remarks.
Skepdick wrote: Thu Sep 03, 2020 9:04 am They are both formal languages/language games played by different grammatical rules. If you don't buy in to my point of view, consider it as mere proof by abstract nonsense.
You haven't expressed a point of view; just a bunch of questions indicating you never learned what a function is (which I don't believe) followed by unrelated and unsuppported assertions about linguistics and logic.

Abstract nonsense is the ironic term sometimes used by mathematicians to refer to category theory. Is that how you're using the term?
Skepdick wrote: Thu Sep 03, 2020 9:04 am To borrow a tag line from the NuPRL project: Starting with the slogan "proofs-as-programs," we now talk about "theories-as-systems.
This has nothing to do with what's gone before, thought I recognize it as your favorite hobby horse. Nobody's disagreeing with you, I'm just wondering where your disjointed exposition is coming from and where it's going.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Yes they are. So the distinction in meaning, the semantics of "+" are incomplete.
They are in the Mathematician's head- it's not encoded in the notation.
You're wrong about this. When mathematicians define + for the natural numbers, it is given a very specific, clear and unambiguous definition. When we then define + for the integers, then for the rationals, then for the reals, in each case we very carefully define the newly-enhanced meaning of the symbol. You would be surprised to know the extreme amount of care that goes into carefully defining each different usage of a given mathematical symbol.

I think I see the point you're making. Just like the word "tunafish" is composed of these symbols "t', and "u", and "n", and so forth, and it takes a human being who's been to school to know what the symbols refer to. I hope this isn't news to you. Symbols have no inherent meaning, people have to be taught to recognize their intended meaning. When you go to calculus class you learn the meaning of the derivative and integral symbols.

Carl Jung wrote a book about this. https://en.wikipedia.org/wiki/Man_and_His_Symbols
Skepdick wrote: Thu Sep 03, 2020 9:04 am So that's one way you are "fooling" yourself - you are pretending to be working with a context-sensitive language (one that supports expressing polymorphism), but you are working with a context-free language on paper so it manifests as equivocation.
I see your point. If I wanted to, I could write 1 +_N 1 = 2 and pi +_R pi = 2pi, indicating that +_N means addition in the natural numbers, and +_R is addition in the reals. (The underscore indicates a subscript, this would look a lot better if this site supported math markup).

But I don't need to do that because any trained mathematician understands this. You're just flat out wrong about what you're saying. Mathematicians are VERY careful about this sort of thing, but they do assume their readers are trained in math and can fill in the details. Math is a language to express ideas between humans. If we want to write for a machine, we'd supply the extra notation to remove ambiguity.

Surely you can understand this. What you call "manifesting as equivocation" is nothing more than professionals using shorthand to be read by other trained professionals. And every mathematician knows exactly how to remove this kind of ambiguity or shorthand when called upon to do so.
Skepdick wrote: Thu Sep 03, 2020 9:04 am When you do encode ALL semantic differences in your notation (so that a proof assistant can recognise it), you end up with the Curry-Howard isomorphism. Syntactic equivalence between proofs and programs.
Fine, what of it? You're just riding your hobby horse again.
Skepdick wrote: Thu Sep 03, 2020 9:04 am No shit. It's Philosophy.
I don't think so.
Skepdick wrote: Thu Sep 03, 2020 9:04 am I am not? I am only pointing out the paradox which you are ignoring. You are free to ignore it - I am simply pointing out that you are doing it.
Whatever.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Abstractly, an oracle machine can solve a generally unsolvable problem. <----- PARADOX
Whatever.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Now look who is reifying oracles! I thought we are talking about abstractions?
You made the absurd claim that oracles don't require explanation, as if college students show up to CS classes already knowing what an oracle is.
Skepdick wrote: Thu Sep 03, 2020 9:04 am You are heading down unfashionable avenues. Talking about knowledge-revelations by all-knowing entities.
What is the matter with you?
Skepdick wrote: Thu Sep 03, 2020 9:04 am I am using it in the common sense. An oracle can solve the halting problem.
Whoa dude. You just figured that out?
Skepdick wrote: Thu Sep 03, 2020 9:04 am Is that not an axiomatic truth?
Not to my understanding.
Skepdick wrote: Thu Sep 03, 2020 9:04 am My oracle says it did.
Yours says it didn't.

Whose oracle is wrong?

Relativization everywhere
You're trolling.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Yeah, but it's 2020 now. Curry-Howard happened after Godel and Quantum Entanglement (physics) is a model of computation, so weave that thread.
You're just repeating your favorite buzzwords. Ooooh, Curry-Howard. Ooooh, quantum entanglement.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Sorry you missed the last 60 years.
Not true but a good line so I'll give you that one.
Skepdick wrote: Thu Sep 03, 2020 9:04 am I am not! I am just abstracting! I a generalising Mathematics - it's a language. A formal language, but still a language.
Ok, and who says it's not, and who are you arguing with, and what are you arguing about? This isn't any conversation I'm having. You just type stuff in that has nothing to do with the conversation.
Skepdick wrote: Thu Sep 03, 2020 9:04 am What is language and how does it relate to the universe? That's the question Philosophy has concerned itself with for ever!
Deep, man, deep.
Skepdick wrote: Thu Sep 03, 2020 9:04 am From the lens of computational linguistics: Turing machines are language recognisers AND a model of computation.
Quantum Entanglement is a model of computation.
Ooooh, more deepitude. This is like something out of a stoner comic from the 1960's.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Weave that thread.
Yeah baby.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Look! Part of the point I am trying to make is that "context" is a formal notion in computing. We have continuations and we have the call with current continuation function.
Gosh this is like a lesson in computer science. Donald Knuth on his best day never rose to such heights of profundity. By the way I know what context is in computing. What makes you think I don't?
Skepdick wrote: Thu Sep 03, 2020 9:04 am Mathematics pretends to (or strives to?) subscribe to denotational semantics. Mathematical symbols have specific/precise/unambiguous/universal meaning. They don't, actually! Precisely because the extra information required to interpret a symbolic Mathematical expression exists only in the Mathematician's head.
Yeah yeah. But actually you're wrong. If we use the same symbol in two different ways in the same expression, it's either perfectly clear to trained practitioners from the context; or we can apply a subscript or some other mark to each symbol to denote the difference. Mathematicians are terribly careful about this kind of thing, I don't know why you think they're not. You're right that they don't always make these subtleties explicit, but that's only because they expect their readers have the same training. If there's any chance the notation is ambiguous they disambiguate it.
Skepdick wrote: Thu Sep 03, 2020 9:04 am Programming languages/Computerised Proof assistants are the attempt to extract that information from the Mathematician's head and encode it in the notation/grammar. That is what "denotational semantics" looks like in practice: context-sensitive grammars. Context encoded/expressed in the language itself.
I get all that, I just don't understand why you're telling me about it.
Skepdick
Posts: 4964
Joined: Fri Jun 14, 2019 11:16 am

### Re: Continuum

wtf wrote: Fri Sep 04, 2020 8:15 am Mathematical objects are abstract objects. Rocks are concrete objects.
Concrete rocks are instances (of information).
The abstract concept of a rock is a pattern (of information).
The word "rock" itself is a representation (of information). It represents either an instance or a pattern (of information).
wtf wrote: Fri Sep 04, 2020 8:15 am Just a word.
Yes, but words can represent (point to?) concrete or abstract objects (as above).
wtf wrote: Fri Sep 04, 2020 8:15 am Mostly from OO programming but also a more general idea, such as the fact that a carton of eggs instantiates the number 12 in the world.
There's great overlap with the vocabulary of OO programming, but that's not my intention. I intend to be using the vocabulary of physical information which comes from physics.

From my PoV a carton of eggs doesn't instantiate the number 12.
A carton of eggs instantiates the patterns of "cartons" and "eggs".

The number 12 is "instantiated" only if you count the eggs.
wtf wrote: Fri Sep 04, 2020 8:15 am A function is a mapping or association.
"Mapping" or "associating" are verbs. Functions are verbs.
wtf wrote: Fri Sep 04, 2020 8:15 am If I have a set of schoolkids and I write down all their heights, I have a function where you input "Fred Bloggs" and output 5'7". Which part of that don't you understand? I put a torch under a metal plate and measure the heat at each point. That's a scalar field. A meteorologist computes the wind speed and direction at each point of city. That's a vector field. Surely these are commonplace ideas. Which part is giving you trouble?
I understand all of that perfectly. None are giving me any trouble.

Is just that "measuring" is the verb/function which produces numbers.

height('Fred Bloggs') -> feet:5 + inch:11
temperature(metal plate) -> matrix:X (scalar field)

You-the-observer are the "quantizer".

The problem is that speaking about "the observer" in physics is not a problem. It is in Mathematics.
Observers don't exist in the Mathematical universe.
wtf wrote: Fri Sep 04, 2020 8:15 am I associated one thing with another thing. My couch is black, my sweatpants are gray, my table is brown, my carpet is red. To each thing I associate some other thing.
color(table) -> color:brown
color(couch) -> color:black

The part about "establishing relationships" is perfectly fine. This is Peirce's theory of signs - semiotics (meaning-making).
wtf wrote: Fri Sep 04, 2020 8:15 am Formally we have two sets, A and B, and with each element a ∉ A we associate an element b ∉ B; and if we call the association f, we write f(a) = b.
That's where we are going adrift. You are only talking about f: Set -> Set, I am talking about f: Any -> Any including f: A -> A, f: B -> B.
and even f: f -> f.

So by this metric alone my formalisms are more expressive than set-theoretic formalisms.
wtf wrote: Fri Sep 04, 2020 8:15 am we associate the number 6 with the reals
This makes absolutely no sense to me! The definition of "the reals" is such that 6 is part of R.
There is no need to "associate" it. This is a theorem: 6 ∈ R
Or to make the notation more explicit. ∈(6, R) -> True

But this is where I am sensing ambiguity and I need to ask more questions: What "exists first" (does this question even make sense?) in the Mathematical universe? What is "conceptually prior" in your own head? Numbers or sets?

The reason I am asking this question is because if you ever needing to associate 6 with R, then you are implying that they are disassociated by default.
wtf wrote: Fri Sep 04, 2020 8:15 am , the number 47 with the integers, and the number pi with the quaternions. So maybe the target set is also the reals. I know you learned this in high school, I'm curious as to why the concept of a function is giving you trouble.
Question: is there such a beast as "the set of all numbers" in your head? If yes - what notation do you use to refer to it? C ?
wtf wrote: Fri Sep 04, 2020 8:15 am You have a source set and a target set, or a "domain" and "range" as they taught you in school. Each element of the source set gets associated with an element of the target set.
OK, but you are STILL (only?) talking about f: Set -> Set relations!

So where in your Mathematical universe does this relation fit: f: f-> f ?
So what Mathematical "domains" and "ranges" do you see in the above formal expression?
wtf wrote: Fri Sep 04, 2020 8:15 am If your function assigns 6 to R, you can't then assign 7 unless it's a different function.
My function aren't "assigning" anything! Assignment happens at definition time.
My function are evaluating membership because ALL of my functions are instances of the [ur=https://en.wikipedia.org/wiki/Eval]eval()[/url]

∈(pi, Integers) -> False
∈(6, Integers) -> True
∈(7.5, Reals) -> True

Perhaps this is worth unpacking too. Because I am treating ALL functions as instances of eval() I am constantly talking about pre-images, and you are constantly talking about images.

wtf wrote: Fri Sep 04, 2020 8:15 am YOU KNOW ALL THIS why are you asking? They teach functions in high school math. Input/output. A soda machine. You put in a buck and out comes a soda. It's a function.
They didn't teach us about the eval() function in school.

wtf wrote: Fri Sep 04, 2020 8:15 am No, that's not right. The squaring function inputs 5 and outputs 25 every time.
Yes, but where did you get the "5" from? Did you get it from "the set of all numbers", "the integers", "the reals" or is it just lying around in your head as "5" and not as an element of any set?
wtf wrote: Fri Sep 04, 2020 8:15 am That's the essential property of a function: You always get the same output for a given input. Of course you could have some OTHER function like the cubing function that inputs 5 and outputs 125.
I am not pointing at the function - I am pointing at the origin of the input.
wtf wrote: Fri Sep 04, 2020 8:15 am To view a functional relation as sampling is a little bizarre, I wonder why you are going off in this direction.
If you get "6" by sticking your metaphorical hand in the abstract box (set) full of numbers - that's sampling.
If you get "6" by weighing a cat - that's measurement.
wtf wrote: Fri Sep 04, 2020 8:15 am You must know what a function is. Traffic cop sits by the road with his LIDAR gun (they don't use radar anymore). Each car gets assigned a speed. That's a function.
Yes but you are reifying now! Back in the abstract Mathematical world what is a function?

Are sets outputs of functions?
Are functions found in sets?

What is the relation between the two?
wtf wrote: Fri Sep 04, 2020 8:15 am I explained to you what a cut point is. It's a point of a connected topological space with the property that if you remove it, the space becomes disconnected.
Yes. IF you remove it.

So when you "sample" 6 from R are you removing it or not?
wtf wrote: Fri Sep 04, 2020 8:15 am I'm sorry man, this is too weird. I'm sure you know what a function is. A function is not sampling with replacement. If you remove 6 from the real line you end up with a disconnected set consisting of two half-infinite intervals as you yourself pointed out.
Yes, I have a conception of a function. And I know what an "output" is. What I am asking you is really simple.

If 6 (or ANY number - it doesn't have to be 6) is the output of a function, then what is the function and what is its input?

I gave you one f(R) -> 6. You rejected it.
wtf wrote: Fri Sep 04, 2020 8:15 am I'm at a loss to follow your train of thought. Are you trolling me?
No! I am just pointing out the multiple conceptions of a function which "selects 6 from the Reals".

You could have f(R) -> 6 such that f() introduces a discontinuity in R at 6
You could have f(R) -> 6 such that f() preserves the continuity of R.

Putting on my OO programmer hat, the function which introduces a discontinuity effectively treats "6" as a singleton.

Which of the two conceptions of the R is closer to what goes on in your head?
wtf wrote: Fri Sep 04, 2020 8:15 am I do appreciate your self-awareness here. Your line of patter has indeed made me wonder what the fuck are you thinking.
Trivially: Are the elements of R singletons or not?
wtf wrote: Fri Sep 04, 2020 8:15 am Do the elements of R behave like singletons? Yes. Any set whatsoever is the union of its individual elements regarded as singletons. The real numbers R are indeed the uncountably infinite union of all the real numbers regarded as singletons.
Then how come I have a 6 here and a 6 there? How did I instantiate a "singleton" twice?
wtf wrote: Fri Sep 04, 2020 8:15 am This hardly follows from your earlier remarks.
It doesn't "follow" anything - it's axiomatic.

Mathematics is a formal language.
Logic is a formal language
English is a language

All three are languages.

type(English) == Language
type(Mathematics) == Language
type(Logic) == Language

To ask why "Mathematics is unreasonably effective" is the same as asking "why is language unreasonably effective?"

wtf wrote: Fri Sep 04, 2020 8:15 am You haven't expressed a point of view
I am expressing it! That is what I am DOING right now. I am describing my experiences of being USING language.

Mathematics, Logic, English.
wtf wrote: Fri Sep 04, 2020 8:15 am ; just a bunch of questions indicating you never learned what a function is (which I don't believe) followed by unrelated and unsuppported assertions about linguistics and logic.
A function is anything realizable. That's why I am a constructivist! Because the BHK interpretation is my religion.

https://en.wikipedia.org/wiki/Realizability
wtf wrote: Fri Sep 04, 2020 8:15 am Abstract nonsense is the ironic term sometimes used by mathematicians to refer to category theory. Is that how you're using the term?
Yes. Abstractly speaking Mathematics and English are both languages. So they are "the same" in some abstract sense of "sameness" and some universal notion of "language".
wtf wrote: Fri Sep 04, 2020 8:15 am This has nothing to do with what's gone before, thought I recognize it as your favorite hobby horse. Nobody's disagreeing with you, I'm just wondering where your disjointed exposition is coming from and where it's going.
Linguistics.

Mathematics ⇔ Logic ⇔ Formal language theory

Philosophically, that's relevant because Philosophy concerns itself with how language relates to reality.
wtf wrote: Fri Sep 04, 2020 8:15 am You're wrong about this. When mathematicians define + for the natural numbers, it is given a very specific, clear and unambiguous definition.
That's practically impossible. Your medium (pen and paper) does not encode context. Mine (programming languages) do.

Best intention != actual results.
wtf wrote: Fri Sep 04, 2020 8:15 am When we then define + for the integers, then for the rationals, then for the reals, in each case we very carefully define the newly-enhanced meaning of the symbol. You would be surprised to know the extreme amount of care that goes into carefully defining each different usage of a given mathematical symbol.
Yes, but it ALL falls apart when I have to read it! When I have to learn about Mathematics I am not defining symbols - I am parsing symbols.
And if there is no way to distinguish the meaning of one + from the meaning of another + then that's equivocation/ambiguity in practice!

The context required to parse meaning is either in the Mathematician's head or in some other location unavailable to the parser.

wtf wrote: Fri Sep 04, 2020 8:15 am I think I see the point you're making. Just like the word "tunafish" is composed of these symbols "t', and "u", and "n", and so forth, and it takes a human being who's been to school to know what the symbols refer to. I hope this isn't news to you. Symbols have no inherent meaning, people have to be taught to recognize their intended meaning. When you go to calculus class you learn the meaning of the derivative and integral symbols.
It's not news to me at all. What I am pointing at is that learning itself is a process/function. To "teach people the meaning of symbols" is to teach people HOW to interpret ssymbols.

So "interpretation" (of symbols) is a function. It's an eval() function
wtf wrote: Fri Sep 04, 2020 8:15 am Carl Jung wrote a book about this. https://en.wikipedia.org/wiki/Man_and_His_Symbols
Thanks. I'll have a look.

While talking re: Carl Jung's his key idea of Synchronicity (meaningful coincidences) is a key distinction for engineers - sync/async communication has different properties.

Consistency vs eventual consistency.
wtf wrote: Fri Sep 04, 2020 8:15 am But I'll play. WHY do you think I'm fooling myself? Can you find a post or quote of mine where I asserted the claims that YOU think I asserted? When did I pretend anything at all? What the hell are you talking about?
You are asserting that Mathematics and Logic are different, no ?

I am asserting that they are the same.

You are reducing them to difference.
I am abstracting them to similarity.

If Mathematics was "abstract enough" you should've been able to abstract to the same conclusion as me , no?
wtf wrote: Fri Sep 04, 2020 8:15 am Fine, what of it? You're just riding your hobby horse again.
No shit! My hobby-horse is language recognition.

Detecting regularities.
wtf wrote: Fri Sep 04, 2020 8:15 am I don't think so.
It's a Philosophy forum?
wtf wrote: Fri Sep 04, 2020 8:15 am You made the absurd claim that oracles don't require explanation, as if college students show up to CS classes already knowing what an oracle is.
I explained it by defining it! An oracle is a machine that can solve unsolvable problems in a single operation. You dismissed it with "whatever".

Question: Will this program halt?
Oracle: <provides single correct yes or no answer>

Question: Does God exist?
Oracle: <provides single correct yes or no answer>

What is it that you don't like about my explanation/definition?

If you have an oracle, you can know the correct yes/no answer to ANY well-formed question.
wtf wrote: Fri Sep 04, 2020 8:15 am What is the matter with you?
I am just expressing my observations!

I use language to EXPRESS my thoughts.

Is this not how you use it?
wtf wrote: Fri Sep 04, 2020 8:15 am Whoa dude. You just figured that out?
What is there to "figure out"? It's true by definition.
wtf wrote: Fri Sep 04, 2020 8:15 am Not to my understanding.
Then our understandings differ.

To repeat myself: An oracle can provide the correct answer to any well-formed question.
The well-formulation of questions frames the discussion right back to the semantics of query languages.
wtf wrote: Fri Sep 04, 2020 8:15 am You're trolling.
"Is Skepdick trolling?"

If we were to formalize the question using some query language

Your oracle answers "No"
My oracle answers "Yes".

Relativization.

wtf wrote: Fri Sep 04, 2020 8:15 am You're just repeating your favorite buzzwords. Ooooh, Curry-Howard. Ooooh, quantum entanglement.
What is it that you aren't understanding in the statement "Turing machines are language recognizers, as well as models of computation" ?

To successfully encode some abstract mathematical structure into a formal notation is to encode the structure of the abstraction into the structure of the grammar.

If you successfully encode the structure of the abstraction into the structure of the grammar, then a Turing machine can recognize it.

This is what the 50s and 60s left us with: structuralism. If it has no structure (regularity? pattern?) - your brain can't parse it.
wtf wrote: Fri Sep 04, 2020 8:15 am Ok, and who says it's not, and who are you arguing with, and what are you arguing about? This isn't any conversation I'm having. You just type stuff in that has nothing to do with the conversation.
I am arguing against the person who said "Mathematics is plenty abstract".

So I abstracted Mathematics.

In computer science we call this meta-linguistic abstraction
wtf wrote: Fri Sep 04, 2020 8:15 am Deep, man, deep.
It's not deep. It's abstract.

You need to make up your mind. Either you value abstraction or you don't.

Or maybe you value abstraction in the "right amount". Not too much. Not too little.
wtf wrote: Fri Sep 04, 2020 8:15 am Ooooh, more deepitude. This is like something out of a stoner comic from the 1960's.
Perhaps that's where we are disagreeing? You are looking for "depth". There's nothing at the bottom - only infinite regress.

wtf wrote: Fri Sep 04, 2020 8:15 am Gosh this is like a lesson in computer science. Donald Knuth on his best day never rose to such heights of profundity. By the way I know what context is in computing. What makes you think I don't?
Your continued insistence on doing "mathematics" with context-free grammars?

And since you mention Knuth: Science is what we understand well enough to explain to a computer; Art is everything else.

wtf wrote: Fri Sep 04, 2020 8:15 am Yeah yeah. But actually you're wrong. If we use the same symbol in two different ways in the same expression, it's either perfectly clear to trained practitioners from the context; or we can apply a subscript or some other mark to each symbol to denote the difference. Mathematicians are terribly careful about this kind of thing, I don't know why you think they're not. You're right that they don't always make these subtleties explicit, but that's only because they expect their readers have the same training. If there's any chance the notation is ambiguous they disambiguate it.
So we are talking cross-purpose here. You expect readers to have "the same training" - I expect subsequent readers to have less and less "training".

I expect Maths to become easier and more accessible. Popularisation is desirable.

I'd like to see the number of self-taught Mathematicians approach the number of self-taught programmers one day, and I think proof assistants will go a long way towards making that a reality. I dive deep on in a proof assistant (look a the source code) - I can't dive deep with a textbook.

Little did my 16 year old nephew know that when he's learning about Java Scripts, he's accidentally learning higher mathematics. He understands monads and doesn't even know it.
wtf wrote: Fri Sep 04, 2020 8:15 am I get all that, I just don't understand why you're telling me about it.
Because I am trying to get you to unambiguously express the semantics of the continuum (R) in a context-sensitive language. So far - you aren't doing it.
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Fri Sep 04, 2020 1:37 pm
Concrete rocks are instances (of information).
The abstract concept of a rock is a pattern (of information).
The word "rock" itself is a representation (of information). It represents either an instance or a pattern (of information).
Ok.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Yes, but words can represent (point to?) concrete or abstract objects (as above).
Ok.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm There's great overlap with the vocabulary of OO programming, but that's not my intention. I intend to be using the vocabulary of physical information which comes from physics.
I've heard about that but don't know much about it and I'm not likely to dive in at the moment.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm The number 12 is "instantiated" only if you count the eggs.
I always count the eggs.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm "Mapping" or "associating" are verbs. Functions are verbs.
Well yes, function can be a verb, but don't you agree it's also a noun?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm I understand all of that perfectly. None are giving me any trouble.
I didn't think it would, which is why I'm puzzled at how you're going on about the basic notions of functions. Didn't I just correctly use "functions" as a noun?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Is just that "measuring" is the verb/function which produces numbers.
Doesn't parse.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm height('Fred Bloggs') -> feet:5 + inch:11
temperature(metal plate) -> matrix:X (scalar field)
Right. No problem.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm You-the-observer are the "quantizer".
That, I don't follow. Now there are a number of levels here. The mathematical formalization of functions, functions used in everyday language, and so forth. Perhaps some of these usages require observers. But there aren't observers in math. I don't know what you mean here.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm The problem is that speaking about "the observer" in physics is not a problem. It is in Mathematics.
Observers don't exist in the Mathematical universe.
Oh cool. We are in agreement. And math isn't physics. To the extent that you're talking about things beyond math, I'm not involved in that conversation.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm color(table) -> color:brown
color(couch) -> color:black
Yes exactly.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm The part about "establishing relationships" is perfectly fine. This is Peirce's theory of signs - semiotics (meaning-making).
Mathematical functions are just a formalization of the idea of establishing relationships. But not any kind of relationships; functional relationships. So for example, "parent of" is not a function because someone can have several kids. In a functional relationship, you always get the same output for a given input.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm That's where we are going adrift. You are only talking about f: Set -> Set, I am talking about f: Any -> Any including f: A -> A, f: B -> B.
and even f: f -> f.
In set theory, functions are sets. But there are class functions. A proper class is a collection defined by a predicate, such as "is a set." So there's a proper class of all sets but (Russell's paradox) not a set of all sets. And if you like, you can have functions from a proper class to a proper class.

Why do you think f: A -> A is a problem? In high school didn't you deal with functions from the reals to the reals, for example f(x) = x^2? Surely functions from R -> R are the most commonplace functions at the elementary level.

Now f: f -> f is problematic, since the f on the left does not mean the same thing as the second and third instances. What do you mean by it? You have some kind of weird self reference going on there without a base case. It's on you to make sense of this notation. On the other hand if you wrote g: f -> f I'd have no problem. A function is a set of ordered pairs, and there's no problem mapping a set of ordered pairs to itself. But the function can't also be f if the set of ordered pairs is f, you'd have a TYPE ERROR in your own favorite terminology.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm So by this metric alone my formalisms are more expressive than set-theoretic formalisms.
Your formalisms are perfectly fine in math, except for f : f-> f which is malformed.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm This makes absolutely no sense to me! The definition of "the reals" is such that 6 is part of R.
There is no need to "associate" it. This is a theorem: 6 ∈ R
Or to make the notation more explicit. ∈(6, R) -> True
Well, YOU are the one who wants to have a function that associates the number 6 with R. It's unusual but certainly not illegal. If there's no need to do it, why are you doing it?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm But this is where I am sensing ambiguity and I need to ask more questions: What "exists first" (does this question even make sense?) in the Mathematical universe? What is "conceptually prior" in your own head? Numbers or sets?
Well that's a really good philosophical question. In set theory, sets are prior to everything. But set theory is only a model or platform to express mathematical ideas. In terms of mathematical ideas, numbers are prior. We've had numbers for thousands of years, and sets only for the past 140 years. So there's math as a formal system and math as those mathematical notions that seem hardwired into our brains. In the former, sets are prior; in the latter, numbers are prior.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm The reason I am asking this question is because if you ever needing to associate 6 with R, then you are implying that they are disassociated by default.
First, I'm not the one who proposed that bizarre example, you are. But even so, I can live with it. Suppose we have a class function that maps the entire class of sets, to the real numbers, Then R might get mapped to 6. I don't see any conceptual problem.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Question: is there such a beast as "the set of all numbers" in your head? If yes - what notation do you use to refer to it? C ?
No there isn't. First, there is no general definition of "number" in math. Number is a historically contingent idea. Pythagoras thought there were only rational numbers, then he discovered irrational numbers. People discovered and (sometimes reluctantly) came to accept negative numbers, zero, complex numbers, transcendental numbers, transfinite numbers, and so forth, as numbers. There is no formal definition of number in math so there could never be a set of them.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm OK, but you are STILL (only?) talking about f: Set -> Set relations!
I'm ok with class functions, in notions of set theory that include proper classes. See for example Morse-Kelley set theory.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm So where in your Mathematical universe does this relation fit: f: f-> f ?
So what Mathematical "domains" and "ranges" do you see in the above formal expression?
It's a type error. Let's break it down. In set theory, a function is a set of ordered pairs. For example the squaring function is the set of all pairs (x, x^2) where x ranges over whatever set the function operates on: the integers, the reals, whatever.

So a function g: f -> f inputs a set of ordered pairs and outputs ordered pairs from the same set. Not sure I can think of a concrete example off the top of my head, but we can make one up.

Let f: R -> R be the squaring function on the reals. So it's the set of all pairs (x, x^2) as x ranges over the reals. Now g must input an ordered pair (x, x^2) and output some ordered pair. How about a constant g which maps every pair (x, x^2) to the pair (5, 25). That is indeed a function g: f -> f as required. Kind of useless, but it fits the requirement.

But what can f : f -> f mean? What are the ordered pairs? You'll have an infinite regress. You have a self reference problem. It's not a well-formed formula. f has already been defined as a set of pairs (x, x^2). But by putting f to the left of the ':', you are saying that f consists of pairs of pairs.

That is. the f on the left consists of pairs ((x, x^2), (y, y^2)). You have a type error. You've already defined f as a set of pairs, and now you want to redefined it as a set of pairs of pairs. You can't redefine a symbol in same expression like that. The expression is invalid. Not a well-formed formula. Or a type error.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm My function aren't "assigning" anything! Assignment happens at definition time.
Well that makes no sense. Suppose I have a function in a programming language whose pseudocode is:

f((x) :
return x^2

Now that is a function definition. It happens at compile time. But the act of inputting a value and collecting the output happens at runtim:

y = f(5)

assigns 25 to y. Surely you agree with this.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm My function are evaluating membership because ALL of my functions are instances of the [ur=https://en.wikipedia.org/wiki/Eval]eval()[/url]
Not clear what you mean. eval() happens at runtime. I'm familiar with it in the context of interpreted languages were you can eval an expression of the language. Is that what you mean, or do you mean something else?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm ∈(pi, Integers) -> False
∈(6, Integers) -> True
∈(7.5, Reals) -> True
This is more akin to set membership. You are using eval() in a funny way that I'm not familiar with.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Perhaps this is worth unpacking too. Because I am treating ALL functions as instances of eval() I am constantly talking about pre-images, and you are constantly talking about images.
I lost you. Why don't you carefully define what you mean by eval() so we're on the same page? Like I say, I've seen eval() in interpreted languages when you have have a string like

mystring = "2 + 2"
answer = eval(mystring)

and upon execution, answer will have the value of 4. Is this what you mean, or something else?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm They didn't teach us about the eval() function in school.
It's a programming construct. Nor have you defined it in any way that makes sense to me, because your examples aren't evals, they're expressions of set membership.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm Yes, but where did you get the "5" from? Did you get it from "the set of all numbers", "the integers", "the reals" or is it just lying around in your head as "5" and not as an element of any set?
In the example of assigning 5 to R, we have, let's say, the proper class of all sets, call it V, and the set of reals, call it R, and we have a function f: V -> R, and f(R) = 5. There's no problem with this.

But that's formalism. Excessive formalism to model a weird function that nobody would ever bother to need or use. But if the philosophical question is where did I get the notion of 5, I got it when I happened to glance at my fingers when I was counting the eggs.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm I am not pointing at the function - I am pointing at the origin of the input.
The domain (ie, "origin of the input") is a set, or a proper class, or a type in a programming language (int, float, etc.)
Skepdick wrote: Fri Sep 04, 2020 1:37 pm If you get "6" by sticking your metaphorical hand in the abstract box (set) full of numbers - that's sampling.
If you get "6" by weighing a cat - that's measurement.
I don't see the difference. A function is an association. The method of making the association is irrelevant. All that matters is that the association is functional. We always get the same output for a given input.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Yes but you are reifying now! Back in the abstract Mathematical world what is a function?
A mathematical function is a set of ordered pairs having the functional relationship. That is, if we have two sets A and B, a function is a subset of the Cartesian product A X B such that each element of A occurs once and only once in the set of ordered pairs.

https://en.wikipedia.org/wiki/Function_(mathematics)
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Are sets outputs of functions?
They could be. More usually outputs of functions are elements of sets. But in set theory, everything is a set. So it's technically correct to say that sets are outputs of functions.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Are functions found in sets?
Sure why not. Say I'm in calculus class and I want to have the set of elementary functions; that would be the set containing all the constant functions, the polynomials, the trig functions, exp and log.

Set theory is a very general formalism. You can do pretty much anything you want that makes sense within the formalism.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm What is the relation between the two?
That you asked about them. There are functions that input numbers and output numbers. There are functions that input schoolkids and output heights. There are functions that intput sets and output other sets. There are functions that input functions and output functions, such as the derivative operator in calculus. Input x^2 and it outputs 2x, right? A function goes in, a function comes out. The function concept is very general. And as I said, if it makes you happy you can extend the idea to functions defined on proper classes, in set theories that support proper classes.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Yes. IF you remove it.
Right. If you have a connected topological space and you remove a cut point, you end up with a disconnected topological space. That's a function too! It maps topological spaces to topological spaces. Connected space in, disconnected space out.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm So when you "sample" 6 from R are you removing it or not?
You would have to tell me, since sampling is not a set theoretic notion. If by sampling you mean removing it, then you're removing it. If by sampling you only mean associating it, then you're not. You'll have to explain what you mean by sampling. You could do it either way, actually.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Yes, I have a conception of a function. And I know what an "output" is. What I am asking you is really simple.
You've asked a lot of things but I can't see what you're getting at. You would be better off reading about the definition of mathematical functions on Wik that I linked, https://en.wikipedia.org/wiki/Function_(mathematics)
Skepdick wrote: Fri Sep 04, 2020 1:37 pm If 6 (or ANY number - it doesn't have to be 6) is the output of a function, then what is the function and what is its input?
There are infinitely many functions whose output is 6 for some given input. What kind of question is this? It most definitely seems trollish. I get that you're trying to understand something but I can't figure out what.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm I gave you one f(R) -> 6. You rejected it.
I didn't reject it. I said it's unusual. But I also gave a perfectly good interpretation. Let V be the class of all sets; let R be the set of real numbers; let f: V -> R be some function such that f(R) = 6. That's perfectly acceptable to me. By the way V is traditional in set theory as the class of all sets.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm No! I am just pointing out the multiple conceptions of a function which "selects 6 from the Reals".
Absolutely. There are lots and lots of functions I can think of that input the reals and output 6. For example the function f : {R} -> {6} from the set containing the set of reals, to the set containing 6. There is exactly one such function, namely the one that assigns 6 to R. This is the minimal function that does the job. But there are lots and lots of others.

It occurs to me that there is a more natural way to express what you want, the selection of a particular real number out of the set of real numbers; and that is by using a constant function. Let f : R -> R be the function that maps everything to 6; that is, f(x) = 6. You may remember from high school math that this is a horizontal straight line six units above the x-axis. I'd see this as a more natural way of expressing the idea of picking out a particular real number. The set of all constant functions is essentially the same as the set of all real numbers. There's a constant function for each real and a real for each constant function.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm You could have f(R) -> 6 such that f() introduces a discontinuity in R at 6
You could have f(R) -> 6 such that f() preserves the continuity of R.
No function introduces a discontinuity. When I measure the heights of the kids, each kid does not disappear when measure his or her height. Functions don't delete anything from the domain. They associate with each element of the domain some element of some other set (or proper class if you prefer).
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Putting on my OO programmer hat, the function which introduces a discontinuity effectively treats "6" as a singleton.
You are very confused on this point. Assigning a value to some object does not delete the object.

Of course there is an operation in set theory, "set difference," that does delete elements from a set. Is that what you are talking about?

https://en.wikipedia.org/wiki/Complement_(set_theory)
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Which of the two conceptions of the R is closer to what goes on in your head?
Functions don't delete anything. That's not my head, that's the official definition of a function. But perhaps you are thinking more of set difference, in which we could start with R, say, and delete 6. That's a perfectly valid operation.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Trivially: Are the elements of R singletons or not?
No, the elements of R are real numbers. But in a sense, you could think of them as singletons if you like. Let's take a simpler example, the five element set {1,2,3,4,5}. That's a set of numbers. I could, if I wanted to, write down a different set: {{1},{2},{3},{4},{5}}. That is a set of singletons.

In set theory it's strictly speaking a different set. But with our philosopher hat on, perhaps with our structuralist philosopher hat on, those two sets can usually be taken as proxies for one another. They have the same number of elements and all we've really done is rename the elements. So you can say it's structurally the same set, while set-theoretically being a different set.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Then how come I have a 6 here and a 6 there? How did I instantiate a "singleton" twice?

When you were in high school and they showed you the identity function f(x) = x, whose graph is a straight line through the origin making a 45 degree angle with the positive x-axis; and you take that function and notice that when you input 47 you get 47 as the output; how do you end up with two copies of 47? Perhaps you can answer that so I can understand where you're coming from.

Every set has an identity function defined on it. Input x, output x. How do you get two copies of x?

Let's talk programming. In pseudocode again I have:

f(x) :
return x

When I input 47, what comes out? 47, right? Do you regard this as a mystery requiring explanation?

At the risk of going on too long about identity, perhaps you are asking a very subtle and clever question. In high school they showed you the coordinate plane with a copy of the real numbers as the x-axis and another copy of the reals as the y-axis. Now we know that every set is completely characterized by its elements; so where did they get two copies of the same set????

Well that turns out to be a good question. There's a trick. If we have a set, we can make another copy of the same set by coloring the elements of one copy red, and the other copy blue. That's what we do conceptually. How do we do that in set theory? If we have a set A and we need two copies of it, we take the Cartesian product A X {1} which consists of all the ordered pairs (a, 1) as a ranges over A; and then we take the Cartesian product A X {2} which is the set consisting of all the pairs (a, 2). So in this way we have two distinct sets having different elements, that can stand in for "two copies of the same set." It's a technical trick. Once we know that we can always do that to get as many copies of a set as we need, from now on we just say, "Consider 37 different copies of the reals," and there is no problem. We just do the Cartesian product trick 37 times.

Is that the answer to your question? If so, you asked an insightful question and that's the answer. This is another one of those things that mathematicians are trained on. When we need two or a million or infinitely many copies of the same set, we know that we can always do the Cartesian product trick; and instead of explicitly mentioning it and working with the different Cartesian products, we just remember that we know how to do this and we act like we can make copiesof the same set. But if someone ever challenged us to do this rigorously, we know how to do it.

ps -- Ok now I actually see where you're going with this idea of mapping R to 6. If R is the domain, you want to know where the 6 came from. How could I use it twice? If that's your question, it is indeed a good one and if so, I'll expand on it next post and fill in a detail or two. The point is that you are right, there is only one copy of every set. But we can use the Cartesian product trick to make more copies. Is this what your questions have been about?

Skepdick wrote: Fri Sep 04, 2020 1:37 pm It doesn't "follow" anything - it's axiomatic.
Forgot what's being referred to here.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Mathematics is a formal language.
Logic is a formal language
English is a language
Ok. I'd disagree about English, since it's historically contingent and constantly changing. New words and new usages are being invented all the time. There's no formal set of rules for what's a legal sentence of English. Poets are in the business of creating new meanings. The fog creeps in on little cat's feet. Formally that makes not a lick of sense; but it's a beautiful image to anyone who can read English. It expands the language even as it delights us. Natural languages are not formal languages.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm All three are languages.
Yes, but English is not a formal language. There's no single set of rules that defines the language for all time. You couldn't go back to England 500 years ago and communicate, even if they spoke something called English. You couldn't go forward too many years either. If fifteen years ago I said, "People got triggered by the president's latest tweet," what would that mean? Twitter wasn't founded till 2006; and we didn't have a president who triggered people with his tweets till 2016; and the word "triggered" did not acquire that particular meaning till a few years ago. When did that happen, exactly, do you remember? When did the trigger warnings start? English is dynamic and has no absolute rules.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm type(English) == Language
type(Mathematics) == Language
type(Logic) == Language
Not with the same meaning of Language. You're equivocating. Logic and math have formal rules. A computer can determine what's a valid sentence of logic or math; not so with English. Surely you know that natural language processing is a very difficult problem.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm To ask why "Mathematics is unreasonably effective" is the same as asking "why is language unreasonably effective?"
I don't agree with that at all, but that would lead into a huge discussion on its own. We came out of caves and started producing grunts, and then one day a primate grunted out the Gettysburg address. How the heck does that work? And then a hundred fifty years later, another primate tweeted. That's not how formal language works at all.

Skepdick wrote: Fri Sep 04, 2020 1:37 pm I am expressing it! That is what I am DOING right now. I am describing my experiences of being USING language.
You've totally changed the subject. Natural language is way too complicated to address here.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Mathematics, Logic, English.
Which one of those is not like the other?????
Skepdick wrote: Fri Sep 04, 2020 1:37 pm A function is anything realizable. That's why I am a constructivist! Because the BHK interpretation is my religion.
I learned long ago not to argue with people about their religion. This explains a lot. I'm arguing with a street corner preacher. Only fun as long as I care to play, but can never be productive.
Skepdick wrote: Fri Sep 04, 2020 1:37 pm That's practically impossible. Your medium (pen and paper) does not encode context. Mine (programming languages) do.
Now THIS is utter NONSENSE. Any program can be written down with pencil and paper and executed by hand, just as you can implement a Turing machine by hand. The most powerful supercomputer in the world can't (in principle) compute anything that you couldn't compute with pencil and paper. It can only do it faster. How on EARTH could you claim that you can do something with a programming language that you can't do with pencil and paper?
Skepdick wrote: Fri Sep 04, 2020 1:37 pm Because I am trying to get you to unambiguously express the semantics of the continuum (R) in a context-sensitive language. So far - you aren't doing it.
Ok I skipped ahead and left out a lot of stuff. If this is your objective, let's focus down on it.

I know how to define the real numbers from first-order predicate logic plus the axioms of Zermelo-Fraenkel set theory. Tell me what you find unsatisfactory about that procedure. Be aware that as part of this process, the various overloads of + are perfectly well defined and could be given different symbols if desired. But why? Don't you believe in operator overloading?

But if you believe in proof assistants, surely R can be defined in any of the usual popular proof systems. What's wrong with those definitions?

R is the unique Archimedean totally-ordered field. That definition is categorical, meaning that any two models are isomorphic. What's wrong with that definition?

See https://en.wikipedia.org/wiki/Real_number

But this is an example of why I find talking with you so frustrating. If this is the question you're really driving at, why not just say that up front? You completely wore me out before I got to it. But if that's what you're interested in, answer the questions I just put to you and read the Wiki page on the the real numbers and we can talk about that.
Skepdick
Posts: 4964
Joined: Fri Jun 14, 2019 11:16 am

### Re: Continuum

wtf wrote: Sat Sep 05, 2020 7:53 am Well yes, function can be a verb, but don't you agree it's also a noun?
If you think Mathematical operators are nouns we have some deep-seated misunderstanding.

f(x) -> x

The only noun here is the unbound variable.

wtf wrote: Sat Sep 05, 2020 7:53 am I didn't think it would, which is why I'm puzzled at how you're going on about the basic notions of functions. Didn't I just correctly use "functions" as a noun?
Functions aren't "basic" notions?!? They are the highest forms of abstractions! Operators.

Every Mathematical expression can be expressed in Combinatory logic, which means we can do away entirely with quantification over variables.

Quantifiers themselves are operators/iterators.
wtf wrote: Sat Sep 05, 2020 7:53 am Doesn't parse.
count(eggs) = 12

Counting. Verb/function.
Eggs. Noun.
wtf wrote: Sat Sep 05, 2020 7:53 am That, I don't follow. Now there are a number of levels here. The mathematical formalization of functions, functions used in everyday language, and so forth. Perhaps some of these usages require observers. But there aren't observers in math. I don't know what you mean here.
(...)
Oh cool. We are in agreement. And math isn't physics. To the extent that you're talking about things beyond math, I'm not involved in that conversation.
OK, but that is conceptually a non-starter for me. There are no "observers" in maths. Are there no Mathematicians in math? Who does Mathematics?
Who assigns meaning to Mathematical expressions?

Few posts back you mentioned "self understanding", how do you plan on achieving that if you have no meaningful notion of "self" (Mathematicians) in Maths?
Your field puts an unreasonable restriction on self-reference.

wtf wrote: Sat Sep 05, 2020 7:53 am Mathematical functions are just a formalization of the idea of establishing relationships. But not any kind of relationships; functional relationships. So for example, "parent of" is not a function because someone can have several kids. In a functional relationship, you always get the same output for a given input.
OK, but that's only a subset (used informally) of all functions. You are specifically talking about deterministic/memoisable/pure functions.

There are other kinds of functions. The read_temperature() function is not that kind of function, but it's still a function!

Function application to self is a valid (and consistent) construct in Combinatorics. Why are Mathematicians being so conservative and rejecting it without nuanced consideration?

wtf wrote: Sat Sep 05, 2020 7:53 am In set theory, functions are sets. But there are class functions. A proper class is a collection defined by a predicate, such as "is a set." So there's a proper class of all sets but (Russell's paradox) not a set of all sets. And if you like, you can have functions from a proper class to a proper class.
OK, but we are still talking about f: A -> A. Instead of A being a set, A is now a proper class. Abstractly that doesn't matter. Sets and Proper classes are still abstract objects. Which is why I proposed f: Any -> Any (which includes f: f -> f) as way of generalising the discussion.

wtf wrote: Sat Sep 05, 2020 7:53 am Why do you think f: A -> A is a problem? In high school didn't you deal with functions from the reals to the reals, for example f(x) = x^2? Surely functions from R -> R are the most commonplace functions at the elementary level.
It's not a problem. f: Any -> Any addresses it. I just assumed you had a reason to use distinct placeholders (such as A and B) instead of A and A.
wtf wrote: Sat Sep 05, 2020 7:53 am Now f: f -> f is problematic, since the f on the left does not mean the same thing as the second and third instances. What do you mean by it? You have some kind of weird self reference going on there without a base case. It's on you to make sense of this notation. On the other hand if you wrote g: f -> f I'd have no problem. A function is a set of ordered pairs, and there's no problem mapping a set of ordered pairs to itself. But the function can't also be f if the set of ordered pairs is f, you'd have a TYPE ERROR in your own favorite terminology.
It's not a type error.

Code: Select all

``````assert type(type) == type
``````
I mean exactly what it says: The type of type is type.
wtf wrote: Sat Sep 05, 2020 7:53 am Your formalisms are perfectly fine in math, except for f : f-> f which is malformed.
Q.E.D you have some grammatical restrictions (that I don't care about) which is limiting my expressivity.

So Mathematicians can express one less thing than I can.

If Mathematicians care about maximising the expression of Mathematical ideas, then they should abandon contemporary Mathematics.

wtf wrote: Sat Sep 05, 2020 7:53 am Well that's a really good philosophical question. In set theory, sets are prior to everything. But set theory is only a model or platform to express mathematical ideas. In terms of mathematical ideas, numbers are prior. We've had numbers for thousands of years, and sets only for the past 140 years. So there's math as a formal system and math as those mathematical notions that seem hardwired into our brains. In the former, sets are prior; in the latter, numbers are prior.
Both of those systems exist within the same Mathematician's head.

My example re: f(R) -> 6 was aimed at the system in which sets are prior to numbers.
f(R) is how you "make" numbers from a set full of them.

But here's the paradox I am seeing. If numbers are singletons, how is it that you have numbers (prior to sets) on the one hand, and numbers (as elements of sets) at the same time. That's not a singleton - that's numbers being instantiated twice.
wtf wrote: Sat Sep 05, 2020 7:53 am First, I'm not the one who proposed that bizarre example, you are. But even so, I can live with it. Suppose we have a class function that maps the entire class of sets, to the real numbers, Then R might get mapped to 6. I don't see any conceptual problem.
OK, but you are moving the goal posts now, so I'll re-state my question given the new playing field.

In order to map Anything to Anything you need a "mapping function". Do classes exist prior to functions?

Start with an "empty" Mathematical universe, then tell me what exists "first"

wtf wrote: Sat Sep 05, 2020 7:53 am I'm ok with class functions, in notions of set theory that include proper classes. See for example Morse-Kelley set theory.
wtf wrote: Sat Sep 05, 2020 7:53 am So where in your Mathematical universe does this relation fit: f: f-> f ?
The type of all types is a member of itself. I think it means something like (but I could be wrong on the nomenclature) all types are inhabited.

I guess now would be a good time to agree on whether types are sets or not.
wtf wrote: Sat Sep 05, 2020 7:53 am So what Mathematical "domains" and "ranges" do you see in the above formal expression?
I don't know what you mean by that. If we are talking about denotational semantics, the expression means what it says.

Anything else is an interpretation. What semantics are you using?
wtf wrote: Sat Sep 05, 2020 7:53 am It's a type error. Let's break it down. In set theory, a function is a set of ordered pairs. For example the squaring function is the set of all pairs (x, x^2) where x ranges over whatever set the function operates on: the integers, the reals, whatever.
type() ranges over all abstract objects in the type system and returns their type. Whan applied to itself, it returns itself.

Code: Select all

``````In [1]: type(True)
Out[1]: bool
In [2]: type(5)
Out[2]: int
In [3]: type('hello')
Out[3]: str
In [4]: def t(test): pass
In [5]: type(t)
Out[5]: function
``````
wtf wrote: Sat Sep 05, 2020 7:53 am So a function g: f -> f inputs a set of ordered pairs and outputs ordered pairs from the same set. Not sure I can think of a concrete example off the top of my head, but we can make one up.
g: f -> f is the same thing as g: Any -> Any. We already agree that this is not problematic in Math,

wtf wrote: Sat Sep 05, 2020 7:53 am But what can f : f -> f mean?
It means exactly what it is defined to mean. A function, thatn when applied to itself - returns itself.

It's the Fix() function in Combinatorics. A fixed point.
wtf wrote: Sat Sep 05, 2020 7:53 am That is. the f on the left consists of pairs ((x, x^2), (y, y^2)). You have a type error. You've already defined f as a set of pairs, and now you want to redefined it as a set of pairs of pairs. You can't redefine a symbol in same expression like that. The expression is invalid. Not a well-formed formula. Or a type error.
And yet... it computes on a computer.
wtf wrote: Sat Sep 05, 2020 7:53 am Not clear what you mean. eval() happens at runtime. I'm familiar with it in the context of interpreted languages were you can eval an expression of the language. Is that what you mean, or do you mean something else?
Yes. It is what I mean. 2+2 is an expression in a language.

eval(2+2) = 4
wtf wrote: Sat Sep 05, 2020 7:53 am This is more akin to set membership. You are using eval() in a funny way that I'm not familiar with.
I am using eval() in the same way any interpreter uses it. The equivalent to eval() in Math is apparently an "inverse function", but intuitively that doesn't work for me.

if eval(expr) = 5 is an inverse/pre-image, then it suggests that there is an image such that f(5) = expr.

The set of all expressions which evaluate to 5 ? Is that even a set? I don't know...

wtf wrote: Sat Sep 05, 2020 7:53 am I lost you. Why don't you carefully define what you mean by eval() so we're on the same page? Like I say, I've seen eval() in interpreted languages when you have have a string like

mystring = "2 + 2"
answer = eval(mystring)

and upon execution, answer will have the value of 4. Is this what you mean, or something else?
Yes. That is what I mean: the function which evaluates (assigns value) to formal expressions.
wtf wrote: Sat Sep 05, 2020 7:53 am It's a programming construct. Nor have you defined it in any way that makes sense to me
I don't need to "define" LISP. That's the whole point of homoiconicity. eval() is a first-class citizen in LISP like sets are first-class citizens in Maths. eval() exists prior to anything else!

eval() is a model of computation.
wtf wrote: Sat Sep 05, 2020 7:53 am because your examples aren't evals, they're expressions of set membership.
They ARE expressions. That is precisely what eval() does! It evaluates expressions. In whatever language.

The expression "2 ∈ I" evaluates to True.
eval(+ 2 2) -> 4
wtf wrote: Sat Sep 05, 2020 7:53 am In the example of assigning 5 to R, we have, let's say, the proper class of all sets, call it V, and the set of reals, call it R, and we have a function f: V -> R, and f(R) = 5. There's no problem with this.
Where did you get the function from, if functions aren't first class citizens in your universe?

wtf wrote: Sat Sep 05, 2020 7:53 am But that's formalism. Excessive formalism to model a weird function that nobody would ever bother to need or use. But if the philosophical question is where did I get the notion of 5, I got it when I happened to glance at my fingers when I was counting the eggs.
So you counted you fingers? That's a verb!

f(fingers) = 5
wtf wrote: Sat Sep 05, 2020 7:53 am The domain (ie, "origin of the input") is a set, or a proper class, or a type in a programming language (int, float, etc.)
So the origin/input is a proper class. Not your fingers?

A. f(Proper Class) = 5
B. f(fingers) = 5
wtf wrote: Sat Sep 05, 2020 7:53 am I don't see the difference. A function is an association. The method of making the association is irrelevant. All that matters is that the association is functional. We always get the same output for a given input.
The method is NOT irrelevant! The method is all that matters if you take the intensional view of a function!

The intensional view of the eval() function is the grammar itself. The grammar is a first class citizen in the system.
You can re-define everything. You can re-define "define".

(define define 5)

You have to take the intensional view of eval() otherwise you have no frigging clue what it does!

And then... the same output for the same input given? That seems artificial. I certainly don't expect the same output for the input "Alexa, what is the time?". Surely you don't either?

f: Any -> Any !
wtf wrote: Sat Sep 05, 2020 7:53 am A mathematical function is a set of ordered pairs having the functional relationship. That is, if we have two sets A and B, a function is a subset of the Cartesian product A X B such that each element of A occurs once and only once in the set of ordered pairs.

https://en.wikipedia.org/wiki/Function_(mathematics)
That's a pure function. Surely there are other kinds of functions in the "set of all functions" ?

I've given you a bunch of examples already. That should suffice as demonstrating that your notion of a function is too narrow?
wtf wrote: Sat Sep 05, 2020 7:53 am They could be. More usually outputs of functions are elements of sets. But in set theory, everything is a set. So it's technically correct to say that sets are outputs of functions.
So functions are sets also?

Or do functions exist prior to sets?
wtf wrote: Sat Sep 05, 2020 7:53 am Sure why not. Say I'm in calculus class and I want to have the set of elementary functions; that would be the set containing all the constant functions, the polynomials, the trig functions, exp and log.
Does your set of functions include or exclude the eval() function?
wtf wrote: Sat Sep 05, 2020 7:53 am The function concept is very general. And as I said, if it makes you happy you can extend the idea to functions defined on proper classes, in set theories that support proper classes.
It's becoming more and more obvious to me that my notion of a "function" is more general than yours.

I don't discriminate against any functions.

No inputs, only outputs? Cool.
Iputs without outputs? Cool.
No inputs or outputs? Cool.
Self-applicable functions? Cool.
wtf wrote: Sat Sep 05, 2020 7:53 am You would have to tell me, since sampling is not a set theoretic notion. If by sampling you mean removing it, then you're removing it. If by sampling you only mean associating it, then you're not. You'll have to explain what you mean by sampling. You could do it either way, actually.
Lets not get bogged down in the word "sampling". In a conceptual framework where sets are prior to numbers, then the only way to "manufacture" numbers (as far as I can tell) is to take them from a set of which they are elements: R.

Then the process of selecting element x from set R is expressed as f(R) -> x. It's a "choice" function.

This is not "mapping" or "association", this is "reach in the jar of Real numbers and grab one". According to you this is not a valid mathematical function, because given the same input it may (statistically certain: WILL) produce a different output every time.
wtf wrote: Sat Sep 05, 2020 7:53 am You've asked a lot of things but I can't see what you're getting at. You would be better off reading about the definition of mathematical functions on Wik that I linked, https://en.wikipedia.org/wiki/Function_(mathematics)
OK. I'll make it explicit.

The set (or class, or collection; or conglomerate - which is the biggest?) of mathematical functions is smaller than the set of all functions.
eval() is not a "Mathematical function", but it is a function.
eval() can be used to implement Mathematical functions.

So eval() is more general (abstract?) than Mathematical functions.
wtf wrote: Sat Sep 05, 2020 7:53 am There are infinitely many functions whose output is 6 for some given input. What kind of question is this? It most definitely seems trollish. I get that you're trying to understand something but I can't figure out what.
This is where confusion ensues. From my POV there are infinitely many inputs for which eval(input) -> 6

There is only one function. eval() itself !!!

So why then can't I construct something crazy like....

Let Expr be the set of all mathematical expressions, then....

[ x for x in Expr if eval(x) == 6 ]

That will produce the set of all expressions which evaluate to 6.

Formally, what's wrong with this crazyness?
wtf wrote: Sat Sep 05, 2020 7:53 am By the way V is traditional in set theory as the class of all sets.
Is eval() somewhere in V?

wtf wrote: Sat Sep 05, 2020 7:53 am It occurs to me that there is a more natural way to express what you want, the selection of a particular real number out of the set of real numbers; and that is by using a constant function. Let f : R -> R be the function that maps everything to 6; that is, f(x) = 6. You may remember from high school math that this is a horizontal straight line six units above the x-axis. I'd see this as a more natural way of expressing the idea of picking out a particular real number. The set of all constant functions is essentially the same as the set of all real numbers. There's a constant function for each real and a real for each constant function.
As a particular case that's fine. If I want to "pick a real number" f(R) = 6 is sufficient to express it. But where I was going with this is expressing the general idea of "selecting an element from a collection (be it a set or a class)".

Lets call it f(V)

IF such a function exists, then the set is discrete! Because you are able to select precisely 6 from the other elements. Exactly like you can pull ping pong balls out of an infinite jar of ping pongs.
wtf wrote: Sat Sep 05, 2020 7:53 am No function introduces a discontinuity. When I measure the heights of the kids, each kid does not disappear when measure his or her height. Functions don't delete anything from the domain.
Yeah, but measuring height isn't an abstract operation. Choosing an element from a continuous set is.
wtf wrote: Sat Sep 05, 2020 7:53 am You are very confused on this point. Assigning a value to some object does not delete the object.
I am only confused by your words. Assigning value to expressions is what eval() does. eval(2+2) = 4, eval(2+2 == 4) = True.

Assigning value to variables (binding free variables) requires memory. You said you don't have observers (with memory) in Mathematics.
wtf wrote: Sat Sep 05, 2020 7:53 am Functions don't delete anything. That's not my head, that's the official definition of a function. But perhaps you are thinking more of set difference, in which we could start with R, say, and delete 6. That's a perfectly valid operation.
So set difference is an operation, but it's not a function? In my world every operation is a function.

wtf wrote: Sat Sep 05, 2020 7:53 am When you were in high school and they showed you the identity function f(x) = x, whose graph is a straight line through the origin making a 45 degree angle with the positive x-axis; and you take that function and notice that when you input 47 you get 47 as the output; how do you end up with two copies of 47? Perhaps you can answer that so I can understand where you're coming from.
You don't end up with two copies. You have two axes with f() relating them.

You could just as well interpret f() = x to represent a generator function which produces the X-axis only. which is why I thought f: A -> A (one domain) is different to f: A -> B (two domains)
wtf wrote: Sat Sep 05, 2020 7:53 am Every set has an identity function defined on it. Input x, output x. How do you get two copies of x?

Let's talk programming. In pseudocode again I have:

f(x) :
return x

When I input 47, what comes out? 47, right? Do you regard this as a mystery requiring explanation?
Look, there's no mystery when I have memory. I know how to make copies of data, I have a bunch of strategies to pass data around: pass by value/pass by reference. The input and output of the function are not in the same memory location, so what the function does under the hood is it configures another memory address to be "like" the input memory address.

x = 47 is assignment (memory)
f(x) = 47 is evaluation.

You don't have memory in Mathematics because you don't have observers/Mathematicians. So I am way more curious about the origins of the first 47.
wtf wrote: Sat Sep 05, 2020 7:53 am At the risk of going on too long about identity, perhaps you are asking a very subtle and clever question. In high school they showed you the coordinate plane with a copy of the real numbers as the x-axis and another copy of the reals as the y-axis. Now we know that every set is completely characterized by its elements; so where did they get two copies of the same set????

Well that turns out to be a good question. There's a trick. If we have a set, we can make another copy of the same set by coloring the elements of one copy red, and the other copy blue. That's what we do conceptually. How do we do that in set theory? If we have a set A and we need two copies of it, we take the Cartesian product A X {1} which consists of all the ordered pairs (a, 1) as a ranges over A; and then we take the Cartesian product A X {2} which is the set consisting of all the pairs (a, 2). So in this way we have two distinct sets having different elements, that can stand in for "two copies of the same set." It's a technical trick. Once we know that we can always do that to get as many copies of a set as we need, from now on we just say, "Consider 37 different copies of the reals," and there is no problem. We just do the Cartesian product trick 37 times.
So, conceptually none of this works for me.

In order for you to do cartesian_product(R, {1}) so that you can get ordered pairs (r, 1) you need a function which decomposes R into each of its elements! You NEED my "peculiar function" f(R) -> r

And if you put on the Combinatory logic, you don't have quantifiers either. You need iterators.

Show me an iterator for a continuous set.

wtf wrote: Sat Sep 05, 2020 7:53 am Is that the answer to your question? If so, you asked an insightful question and that's the answer. This is another one of those things that mathematicians are trained on. When we need two or a million or infinitely many copies of the same set, we know that we can always do the Cartesian product trick; and instead of explicitly mentioning it and working with the different Cartesian products, we just remember that we know how to do this and we act like we can make copiesof the same set. But if someone ever challenged us to do this rigorously, we know how to do it.
This seems like a pretty stupid trick to me. Why can't you just define X as precisely the thing you need?
You can have ANYTHING you want, all you have to do is define it. Even infintismals aren't immune to being defined. Abraham Robinson did it in his non-standard analysis.

As far as I am concerned rigour starts with definition. Show me the intensional view of cartesian_product(A, B): .....

Like I said, the part I am most interested in is how you obtain EACH ordered pair from a continuous (uncountable!) set.
wtf wrote: Sat Sep 05, 2020 7:53 am ps -- Ok now I actually see where you're going with this idea of mapping R to 6. If R is the domain, you want to know where the 6 came from. How could I use it twice? If that's your question, it is indeed a good one and if so, I'll expand on it next post and fill in a detail or two. The point is that you are right, there is only one copy of every set. But we can use the Cartesian product trick to make more copies. Is this what your questions have been about?
My question is trivially about locating/selecting/sampling (discretising!) the elements of (sets|classes).

R is uncountable/continuous.
Cartesian product talks about ordered pairs. Those are discrete!

How do mapping functions iterate over uncountable sets?
wtf wrote: Sat Sep 05, 2020 7:53 am Ok. I'd disagree about English, since it's historically contingent and constantly changing. New words and new usages are being invented all the time.
Apparently the same goes for Maths? You only got sets 140 years ago. Categories - in the last 60.

The concept of Mathematics may be static in your head, it's not so static on the ground.
wtf wrote: Sat Sep 05, 2020 7:53 am There's no formal set of rules for what's a legal sentence of English. Poets are in the business of creating new meanings. The fog creeps in on little cat's feet. Formally that makes not a lick of sense; but it's a beautiful image to anyone who can read English. It expands the language even as it delights us. Natural languages are not formal languages.
There's no formal definition for what a number is - it doesn't bother you as much as it bothers you in English?
wtf wrote: Sat Sep 05, 2020 7:53 am Yes, but English is not a formal language. There's no single set of rules that defines the language for all time.
Exactly like Maths.

Set theory was the lingua franca. Topoligists/type theorists are getting pissed off with the orthodoxy.
wtf wrote: Sat Sep 05, 2020 7:53 am You couldn't go back to England 500 years ago and communicate, even if they spoke something called English. You couldn't go forward too many years either. If fifteen years ago I said, "People got triggered by the president's latest tweet," what would that mean? Twitter wasn't founded till 2006; and we didn't have a president who triggered people with his tweets till 2016; and the word "triggered" did not acquire that particular meaning till a few years ago. When did that happen, exactly, do you remember? When did the trigger warnings start? English is dynamic and has no absolute rules.
You couldn't go back 500 years and speak of "homomorphisms" to a mathematician either. You are still confusing grammar with semantics. Grammar is the set of rules which governs the formation of sentences.

It's the (unstated) rules which dictate that you can't say 7.5 ∈ I, but you can say 7.5 ∈ R.

In formal languages grammar is only about form/structure and so it happens to coincide with meaning.
wtf wrote: Sat Sep 05, 2020 7:53 am Not with the same meaning of Language. You're equivocating. Logic and math have formal rules. A computer can determine what's a valid sentence of logic or math; not so with English. Surely you know that natural language processing is a very difficult problem.
With a sufficiently abstract meaning of language.

All languages have rules. That's how we recognize them as being "languages" rather than just random characters strung together. That their universal mapping property cannot be strictly formalised doesn't mean it doesn't exist. This is Chomsky's universal grammar argument.

NLP is difficult, not impossible. We have GPT-3 now. It does what it does without formal axioms.

wtf wrote: Sat Sep 05, 2020 7:53 am I don't agree with that at all, but that would lead into a huge discussion on its own. We came out of caves and started producing grunts, and then one day a primate grunted out the Gettysburg address. How the heck does that work? And then a hundred fifty years later, another primate tweeted. That's not how formal language works at all.
We engineered formal languages for a purpose. Obviously. But we copied everything about them from natural languages. Grammar. Subjects, Verbs (functions). Nouns (inputs).

I kicked the ball.

self.kick(ball)
wtf wrote: Sat Sep 05, 2020 7:53 am You've totally changed the subject. Natural language is way too complicated to address here.
Sure. GPT-3 is a a complex beast. But I doubt you'll deny it's non-formal.
wtf wrote: Sat Sep 05, 2020 7:53 am Which one of those is not like the other?????
If you see a difference, you aren't abstracting away enough detail...
wtf wrote: Sat Sep 05, 2020 7:53 am Now THIS is utter NONSENSE. Any program can be written down with pencil and paper and executed by hand, just as you can implement a Turing machine by hand.
It's only nonsense in theory. In practice your "nonsense" is nonsense.

Qualitatatively "up to isomorphism" doesn't cut it.
wtf wrote: Sat Sep 05, 2020 7:53 am The most powerful supercomputer in the world can't (in principle) compute anything that you couldn't compute with pencil and paper. It can only do it faster. How on EARTH could you claim that you can do something with a programming language that you can't do with pencil and paper?
In principle, that's total horseshit. Languages are human interfaces! There are such things as more and less expressive ones. By claiming equivalence you've got both your feet stuck in the Turing tarpit

Courtesy of http://math.andrej.com/2006/03/27/somet ... ontinuous/
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.
You are saying what you are saying, but you don't actually believe it.
Else you'd (in principle) not object to using Brainfuck as your prefered Mathematical notation.
wtf wrote: Sat Sep 05, 2020 7:53 am I know how to define the real numbers from first-order predicate logic plus the axioms of Zermelo-Fraenkel set theory. Tell me what you find unsatisfactory about that procedure. Be aware that as part of this process, the various overloads of + are perfectly well defined and could be given different symbols if desired. But why? Don't you believe in operator overloading?
What I'll probably find unsatisfactory is your appeal to yet another formal language (first order logic) without ever showing me the implementation of any of the functions/operators it depends on. Such as the ∀ iterator/quantifier when applied to "continuous" sets.

And I'll probably also object to your appeal to predicate logic which rests upon propositional logic, because propositional logic is isomorphic to type theory.

It's merely a foundational objection.
wtf wrote: Sat Sep 05, 2020 7:53 am But if you believe in proof assistants, surely R can be defined in any of the usual popular proof systems. What's wrong with those definitions?
The fictitious ∀ operator which accepts uncountable sets as inputs.
wtf wrote: Sat Sep 05, 2020 7:53 am R is the unique Archimedean totally-ordered field. That definition is categorical, meaning that any two models are isomorphic. What's wrong with that definition?
For starters, how do you test if any given model is actually a model of R? The Archimedean property is defined as an inequality. The > operators do not halt when comparing infinite precision reals which are really close to each other in any model of computation. The closer they are - the longer comparison takes. There is "no time" in Mathematics.

But a human could never do such comparison in theory or in practice.

wtf wrote: Sat Sep 05, 2020 7:53 am But this is an example of why I find talking with you so frustrating. If this is the question you're really driving at, why not just say that up front? You completely wore me out before I got to it. But if that's what you're interested in, answer the questions I just put to you and read the Wiki page on the the real numbers and we can talk about that.
I am really really driving at Finite Model Theory ( https://en.wikipedia.org/wiki/Finite_model_theory ).

FMT is mainly about discrimination of structures. The usual motivating question is whether a given class of structures can be described (up to isomorphism) in a given language. For instance, can all cyclic graphs be discriminated (from the non-cyclic ones) by a sentence of the first-order logic of graphs? This can also be phrased as: is the property "cyclic" FO expressible?

Is the property of "self reference" expressible?

Not in any Mathematical formalism!
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Fri Sep 18, 2020 1:34 pm If you think Mathematical operators are nouns we have some deep-seated misunderstanding.

f(x) -> x

The only noun here is the unbound variable.
"A function is a mapping from one set to another."

Please identify the subject of that sentence and confirm whether you think it's a noun or not.

I honestly can't see your remark as anything other than disingenuous. Am I mistaken? You never heard the word function used as a noun? "

Are you being disingenuous? In the US, students are introduced to functions in the 11th grade, when they are 16 years old. So your math education must be at the 15 year old level if you never heard the word function used as a noun, and don't understand that a function is a particular thing, having definite properties.

Skepdick wrote: Fri Sep 18, 2020 1:34 pm Functions aren't "basic" notions?!? They are the highest forms of abstractions! Operators.
I don't know what you mean by that. Functions are definitely mathematical primitives conceptually. In the formalism of set theory, functions are particular sets, so sets are more fundamental. But I agree that functions are primary conceptual entities in our intuition. Functions. Mappings. Those are nouns my friend, I don't understand why a seemingly intelligent person is speaking nonsense to me.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Every Mathematical expression can be expressed in Combinatory logic, which means we can do away entirely with quantification over variables.
That's cool. What of it?
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Quantifiers themselves are operators/iterators.
That's cool. Why are you telling me this?

I don't think interacting with you is productive. You have an axe to grind or an obsession to obsess over, and anyone you interact with is just a foil for your monomania.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Who assigns meaning to Mathematical expressions?
Humans give meaning to mathematical expressions. Written math is a historically contingent artifact of people. Whether math itself has prior existence or is also an artifact of the human mind, is a philosophical question. I can't imaging we have any disagreement here; except for the fact that you think this trivia is important, and something other than a function of your own monomania and mathematical ignorance.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Few posts back you mentioned "self understanding", how do you plan on achieving that if you have no meaningful notion of "self" (Mathematicians) in Maths?
Your field puts an unreasonable restriction on self-reference.
I know that in your mind, the above sentence corresponds to some meaning in your mind. I assure you that I have no idea what the heck you're talking about.

Mathematicians study non-wellfounded set theory, which is self-referential set theory if you like. I have no problem with it, neither does math.

https://plato.stanford.edu/entries/nonw ... et-theory/

Perhaps you can show me examples of such "unreasonable restrictions," when in fact I just gave you the counterexample. And besides, computer science doesn't give you infinite self-reference. There's always a base case to every inductive procedure; and every inductive function can be expressed and implemented as a simple loop.

Skepdick wrote: Fri Sep 18, 2020 1:34 pm OK, but that's only a subset (used informally) of all functions. You are specifically talking about deterministic/memoisable/pure functions.
I would not use the word deterministic, but it's part of the definition of a mathematical function that it gives the same output whenever you put in the same input.

If you have some more general notion of function in mind, I'm sure math (and I) can accommodate it. For example in high school we say that f(x) = sqrt(x) is a function because we define the sqrt() operator to mean the positive of the two real numbers whose square is x.

But if you let z range over the complex numbers (as is traditional), then w = sqrt(z) is a multivalued function whose output is BOTH of the possible square roots. Just as the complex logarithm function has infinitely many outputs for each input. In fact mathematicians study the topological space you get when you consider the geometry of the space of all possible outputs of a function.

Check out the groovy pictures of multivalued complex functions here. They're called Riemann surfaces. They're huge in mathematics.

Perhaps it's not so much that math is limited, it's that your own KNOWLEDGE of math is limited. You say math doesn't do self-reference, but it does. You say math doesn't do multi-valued functions, but it does. You get your ideas of math from your high-school sophomore level of mathematical sophistication; and then you decide that it's math, not your own knowledge, that's deficient.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm There are other kinds of functions. The read_temperature() function is not that kind of function, but it's still a function!
I see what you're getting at, but I disagree with your interpretation. For a given ambient temperature, any read_temp() function MUST always return the same value, else it's broken. Of course I take your point that the ambient temp can fluctuate. I could even go so far as to say I understand your point of view; but that you're making way too much of it. read_temp() is implicitly a function of time; that is, read_temp(now) or read_temp(some_particular_timestamp) is perfectly deterministic.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Function application to self is a valid (and consistent) construct in Combinatorics. Why are Mathematicians being so conservative and rejecting it without nuanced consideration?
Which mathematicians? Can you name any of them and point me to articles or books or web pages where they did what you claim they're doing? I have no problem with the concept of function as envisioned in computer science. If that's slightly different than functions in math, so be it. If they're the same but only at an abstract or formal level, so be it. You're trying to make mountains out of the molehills of your own lack of knowledge and understanding. The word "function" can have many meanings. There's the meaning in formal set theory; there's the meaning in Riemann surfaces; there's the meaning in various programming languages; there's the meaning in formal CS; there's the meaning in combinatorial logic. I have no problem with this nor does anyone else.

Skepdick wrote: Fri Sep 18, 2020 1:34 pm OK, but we are still talking about f: A -> A. Instead of A being a set, A is now a proper class. Abstractly that doesn't matter. Sets and Proper classes are still abstract objects. Which is why I proposed f: Any -> Any (which includes f: f -> f) as way of generalising the discussion.
Sorry I don't remember what this refers to. If you want to notate some function as f: Any -> Any I have no problem with that. Input a giraffe and it outputs some noncomputable real number. Input the time of day and it outputs the temperature in Celsius, a small town just outside of Duluth. Input a cheeseburger and it outputs fries. What of it? One, I'm perfectly fine with your notation; and two? who gives an eff? What on earth are you going on about?

Skepdick wrote: Fri Sep 18, 2020 1:34 pm It's not a problem. f: Any -> Any addresses it. I just assumed you had a reason to use distinct placeholders (such as A and B) instead of A and A.
You're just making up problems where there aren't any. I can tell you why f : A -> A can sometimes be better than f : Any -> Any. It's because if A is a particular set or proper class, that gives us information as to what are the legal inputs and outputs of the function f. If I tell you A is the integers, or the reals, or the complex numbers, then f : A -> A says that f inputs and outputs an integer. Or a real, or a complex number.

Whereas f : Any -> Any is a perfectly good function, it just doesn't tell me much. Like if ask you how to get to the store and you tell me "Go north then turn east," it's not as useful as, "Go north 3 blocks then east 4 blocks. f : A -> A is more specific; it conveys more information. Why is this something you're trying to say is some kind of problem?

Skepdick wrote: Fri Sep 18, 2020 1:34 pm I mean exactly what it says: The type of type is type.
Whatever. Your lack of mathematical understanding, combined with your monomania about combinatorial logic, makes it pointless to continue this discussion.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Q.E.D you have some grammatical restrictions (that I don't care about) which is limiting my expressivity.
You are free to express yourself anyway you like. Nobody has any idea what you're talking about.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm So Mathematicians can express one less thing than I can.
Don't be silly. I can take 0 = 1 as an axiom and thereby prove ANY proposition. I can express anything.

Your f : f -> f notation is malformed. It means nothing. You're free to spout nonsense. If I'm in abstract algebra class and the professor is talking about the theory of groups, and I raise my hand and say, "Mary had a little lamb, the doctor was surprised," it's certainly true that I can spout some crap that annoys my professor. I don't think anyone would claim I now have more expressive power than the prof.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm If Mathematicians care about maximising the expression of Mathematical ideas, then they should abandon contemporary Mathematics.
In the end, crankery always comes down to bad toilet training or a screechy third grad math teacher. I'm sorry you have some kind of bug up your ass about math. Besides, mathematicians are ALWAYS abandoning contemporary math. Just look at the history. Math is one radical new paradigm after another for thousands of years, continuing to today.

Skepdick wrote: Fri Sep 18, 2020 1:34 pm
Both of those systems exist within the same Mathematician's head.
So you agree.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm But here's the paradox I am seeing. If numbers are singletons, how is it that you have numbers (prior to sets) on the one hand, and numbers (as elements of sets) at the same time. That's not a singleton - that's numbers being instantiated twice.
You know what the problem is? This isn't a bad question, but by now I'm sick of typing and determined not to respond to you anymore. I could answer your question but I no longer care. Go read an undergrad book on set theory. I have no idea why you think that "numbers are singletons." The number 6 and the set {6} containing the number 6 are distinct mathematical objects. They're not the same, they're two different things entirely.

Skepdick wrote: Fri Sep 18, 2020 1:34 pm In order to map Anything to Anything you need a "mapping function". Do classes exist prior to functions?
In set theory? Most definitely. In our intuition? Everyone's intuition is their own. So I can't say.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm Start with an "empty" Mathematical universe, then tell me what exists "first"
All sets are built from the empty set. In order to get the process going you can EITHER

1) Adopt an axiom that says the empty set exists: ∅ = {x : x ≠ x}. Note the beautiful leveraging of the law of identity, which I hope you appreciate. Or:

2) Extend the logical law of identity from: "For all x, x = x"; to: "For all x, x = x, AND there exists an x." In other words some philosophers regard the universe as being nonempty.

You can build it up either way.

wtf wrote: Sat Sep 05, 2020 7:53 am I'm ok with class functions, in notions of set theory that include proper classes. See for example Morse-Kelley set theory.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm The type of all types is a member of itself. I think it means something like (but I could be wrong on the nomenclature) all types are inhabited.
I appreciate that you acknowledge that you could be wrong about something. If in your logical system "type" is a type, I'm fine with that. After all in an object-oriented language when I say Foo: x, I am implicitly creating TWO objects in the runtime: an object that represents the class Foo; AND an object that represents the instance x.

You seem to think that because set theory isn't type theory, that makes set theory wrong, or math wrong, or me wrong, or contemporary math wrong. Why do you think that? Surely you can hold two definitions of a thing in your head. There's a vector as in vector calculus, and there's a vector as a means of disease transmission, right? Why can't you breed a mosquito with a mountain climber? Because you can't cross a vector with a scalar! That's an old engineering joke.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm I guess now would be a good time to agree on whether types are sets or not.
I don't know enough type theory to answer that. Does it really matter? To whom? I'm sure you can formalize type theory in set theory if you want. And I'm sure you can formalize type theory without set theory if you want. What difference does it make?
Skepdick wrote: Fri Sep 18, 2020 1:34 pm So what Mathematical "domains" and "ranges" do you see in the above formal expression?
I don't know what you mean by that. If we are talking about denotational semantics, the expression means what it says.

Anything else is an interpretation. What semantics are you using?[/quote]

Sanity and knowledge. How about you?
Skepdick wrote: Fri Sep 18, 2020 1:34 pm type() ranges over all abstract objects in the type system and returns their type. Whan applied to itself, it returns itself.
So you've said. And I'm perfectly willing to believe you. But so what?

Here's a Python3 program I just typed in:

Code: Select all

``````% python3
Python 3.8.5 (default, Aug  8 2020, 21:47:12)
[Clang 8.0.0 (clang-800.0.42.1)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> def f() :
...     f()
...
>>> f
<function f at 0x1052300d0>
``````
So I perfectly well take your point. I just can't for the life of me imagine why you think this invalidates mathematics.
Skepdick wrote: Fri Sep 18, 2020 1:34 pm It means exactly what it is defined to mean. A function, thatn when applied to itself - returns itself.
Ok fine. So effing what? Therefore math is wrong. Ok according to you. You haven't made a rational case to me.

Ok I deleted the rest of this. I'm done typing. Have a nice evening. Thanks for the chat.
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Thu Sep 03, 2020 9:04 am I agree with you that thoughts are real.
Just using this random quote so your handle will be tagged.

I think I had an insight and can sum things up.

When I typed in the Python program

Code: Select all

``````def f() :
f()``````
I realized that what you are talking about by self-reference or "nondeterministic functions" are what I would call late binding in programming languages. The way Python treats that block of code is to allocate a memory location for the thing 'f'; and in the future, when someone happens to invoke it, we'll see what value it has THEN. "Runtime binding" is another name for it.

So now the question is, How is late binding mathematically modeled in programming theory?

I don't know but I'm sure it can be done. For example when they teach Turing machines they give the intuitive idea about an "unbounded tape and a read/write head," but shortly afterward they give a formal, set-theoretic treatment by saying that a TM is a 7-tuple M = <Q, Γ, b, ∑, δ, q_0, F>; where

* Q is a nonempty set of states.

* blah blah blah. Full details here. https://en.wikipedia.org/wiki/Turing_ma ... definition

Point being that computer scientists DO go to some trouble to mathematically defined their structures using math, in particular set theory.

So I assume that the professors who study formal language theory have mathematically modeled late binding. They would do it so they can prove things about it, develop a theory, study tradeoffs among performance, memory, etc. That's what they do.

Is this little post of mine in accord with your ideas? That we're just wondering how they model late binding mathematically. But I do agree with your point that late binding pushes the bounds of basic mathematical functions a lot farther than high school. Not that the CS professors haven't figured out how to do it.

So now I think I understand what you mean about self-reference and "nondeterministic functions." In math, the squaring function can only square a number. But in a programming language, we don't have to decide what a function does until someone calls it. We can wait till it's called to see how it's defined THEN.

I think we are in perfect agreement on all this; and the only question is how the programming language theorists model it using math. But I'm sure they've figured that out.

What do you think?
Skepdick
Posts: 4964
Joined: Fri Jun 14, 2019 11:16 am

### Re: Continuum

wtf wrote: Sun Sep 20, 2020 4:49 am I realized that what you are talking about by self-reference or "nondeterministic functions" are what I would call late binding in programming languages. The way Python treats that block of code is to allocate a memory location for the thing 'f'; and in the future, when someone happens to invoke it, we'll see what value it has THEN. "Runtime binding" is another name for it.
This is where we are tripping up over Python's grammar (so I am going to switch gears to LISP/eval()) )

It goes by many names in programming language theory. Dynamic typing. Dynamic dispatch. Parametric polymorphism. But recursion is **NOT** late binding. They are grammatically different.

https://repl.it/repls/EnchantedOtherTransversal
;;This is late binding
(declare ^:dynamic g)
(defn h [x] (g x))
(defn g [x] ( * x 2))

(= (h 5) 10) ;; true
(defn g [x] ( * x 10))
(= (h 5) 50) ;; true

;; This is recursion
(defn f [x] (f x))
(f 0) ;; StackOverflow -> would be an infinite loop if we had "infinite tape"
wtf wrote: Sun Sep 20, 2020 4:49 am So now the question is, How is late binding mathematically modeled in programming theory?

I don't know but I'm sure it can be done. For example when they teach Turing machines they give the intuitive idea about an "unbounded tape and a read/write head," but shortly afterward they give a formal, set-theoretic treatment by saying that a TM is a 7-tuple M = <Q, Γ, b, ∑, δ, q_0, F>; where

* Q is a nonempty set of states.

* blah blah blah. Full details here. https://en.wikipedia.org/wiki/Turing_ma ... definition
Yes. IF you are a set theorist, that is a fine way to define a Turing machine. But conceptually that definition requires interpretation in a set-theoretic framework. And a whole lot of my point is that eval() is a peculiar and particular type of function of great interest to Computer Scientists. It's a meta-circular evaluator.

And also lets not forget that the Church-Turing thesis only ever spoke about Nat->Nat computations and makes no mention of higher order types,

eval() is f: Any -> Any
wtf wrote: Sun Sep 20, 2020 4:49 am Point being that computer scientists DO go to some trouble to mathematically defined their structures using math, in particular set theory.
There was a paper by Djikstra 1968 or thereabout on structured programming (called "GOTO considered harmful"). Any language which has a GOTO function (or any equivalent construct which allows non-deterministic control-flow) is basically unstructured. It cannot be "proven correct" in any current Mathematical sense.

That is to say, you get computation, but you don't get computation that you can reason about analytically. It's humanly unwieldy - too much power. Shotgun-to-foot kinda thing.
wtf wrote: Sun Sep 20, 2020 4:49 am So I assume that the professors who study formal language theory have mathematically modeled late binding. They would do it so they can prove things about it, develop a theory, study tradeoffs among performance, memory, etc. That's what they do.
If they are trying to prove stuff about it, they are necessarily talking about structured expressions. But that precisely what complexity theory is all about from the lens of a Formalist.

Complexity classes correspond to the "power" of the automaton (computer?) required to parse a grammar.
wtf wrote: Sun Sep 20, 2020 4:49 am Is this little post of mine in accord with your ideas? That we're just wondering how they model late binding mathematically. But I do agree with your point that late binding pushes the bounds of basic mathematical functions a lot farther than high school. Not that the CS professors haven't figured out how to do it.
Mathematically, late binding is just polymorphism. eval() is the epitome of parametric polymorphism.

Different expressions produce different behaviour. Some expressions produce self-modifying behaviour. The notion of "self modification" is impossible to model in Maths.
wtf wrote: Sun Sep 20, 2020 4:49 am So now I think I understand what you mean about self-reference and "nondeterministic functions." In math, the squaring function can only square a number.

But in a programming language, we don't have to decide what a function does until someone calls it. We can wait till it's called to see how it's defined THEN.

I think we are in perfect agreement on all this; and the only question is how the programming language theorists model it using math. But I'm sure they've figured that out.

What do you think?
Ironically, if the squaring function doesn't discriminate between Natural and Real number-types as its inputs, then you are treating it as being polymorphic.

f(x) = x^2 is dynamically typed because x is untyped.
Skepdick
Posts: 4964
Joined: Fri Jun 14, 2019 11:16 am

### Re: Continuum

wtf wrote: Sun Sep 20, 2020 4:49 am What do you think?
What I think can be simplified down to 2 lines of LISP.

https://repl.it/repls/SillyMixedGnudebugger#main.scm

Code: Select all

``````(define x (list `(+ 2 2)))
(apply eval x)``````
This is a peculiar formalism.
first we assign x the literal expression "((+ 2 2))".

Then we apply the eval function to x which produces 4.

From a Mathematical viewpoint what is the domain of x? What is the co-domain of eval() ?

Because in my head that's f: Any -> Any
wtf
Posts: 969
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Continuum

Skepdick wrote: Sun Sep 27, 2020 12:17 am But recursion is **NOT** late binding. They are grammatically different.
I don't recall saying anything to the contrary. But clearly you went out of your way to entirely miss the point of what I was saying to you, just so that you could continue to grind your many axes.

It would be better if I stop responding to this thread since not only have I long since forgotten what it was about; there was NEVER a point when I had any idea what it was about.

I do know that whatever your core point is, you feel it with great passion. Programming is right and math is wrong. Or something. I wish you well with your crusade.