## Continuum

What is the basis for reason? And mathematics?

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wtf
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### Re: Continuum

PeteJ wrote: Wed Aug 26, 2020 1:23 pm
Our approaches may both be reasonable and functional. It seems your interest is formal and not fundamental. So set theory for you floats free of the world and its paradoxical features are of no interest. I get this. But as a metaphysician I'm more interested in how set theory may be axiomatised for a fundamental theory. It is here that set-theory becomes philosophically interesting. We find it cannot be fundamental unless it is paradoxical, suggesting that Reality transcends set-theory and the categories of thought. Same goes for the arithmetical number line. It works just fine as long as we don't 'reify it.
Of course I realize that math is inspired by the world and "unreasonably effective" as they say. I'm objecting to trying to make the mathematical continuum apply to reality though, when it clearly doesn't. "We find it cannot be fundamental unless it is paradoxical," --- what does that mean, please? Are you referring to the paradoxes of set theory (resolved long ago) or something else?

"Same goes for the arithmetical number line. It works just fine as long as we don't 'reify it." -- We're in agreement then.
PeteJ wrote: Wed Aug 26, 2020 1:23 pm Thus for metaphysics we need to take account of both the arithmetical and intuitive continuum and investigate their relationship.
I agree that philosophers study set theory and for good reason. And I'm ok with investigations of the continuum. I'm mostly objecting to certain posters here who seem to want to adopt a nihilistic view of mathematics because it's not real enough for them. Such as ...
Skepdick wrote: Wed Aug 26, 2020 2:47 pm The "continuum" is conceptually inconsistent if it's made up of discrete objects. It's anon-starter without turning a blind eye to the contradiction.
Then just call it "the set of real numbers." I would say that when mathematicians call the real numbers the continuum, they're just using the term as a shorthand for the real numbers. They're not implying any philosophical baggage.

Are you a follower of Peirce, is that where your objection is coming from?
PeteJ
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### Re: Continuum

wtf wrote: Thu Aug 27, 2020 2:02 am Of course I realize that math is inspired by the world and "unreasonably effective" as they say. I'm objecting to trying to make the mathematical continuum apply to reality though, when it clearly doesn't. "We find it cannot be fundamental unless it is paradoxical," --- what does that mean, please? Are you referring to the paradoxes of set theory (resolved long ago) or something else?
It means that the arithmetical line works fine as a fiction, but as a fundamental phenomenon it is paradoxical. Same as saying that set-theory is fine for everyday life but doesn't work when we try to make sets fundamental. In metaphysics this means that a realistic view of space-time is fine for everyday life but paradoxical when taken to be a fundamental or truly real.
"Same goes for the arithmetical number line. It works just fine as long as we don't 'reify it." -- We're in agreement then.
We do seem to agree.
I agree that philosophers study set theory and for good reason. And I'm ok with investigations of the continuum. I'm mostly objecting to certain posters here who seem to want to adopt a nihilistic view of mathematics because it's not real enough for them. Such as ...
Skepdick wrote: Wed Aug 26, 2020 2:47 pm The "continuum" is conceptually inconsistent if it's made up of discrete objects. It's anon-starter without turning a blind eye to the contradiction.
Hmm. I'd say Skepdick is right. Mathemticians can turn a blind eye. But when they study foundations and delve into metaphysics then this conception of the continuum gives rise to contradictions and the purely conceptual nature of the arithmetical continuum becomes apparent. I thought you shared this view.
Then just call it "the set of real numbers." I would say that when mathematicians call the real numbers the continuum, they're just using the term as a shorthand for the real numbers. They're not implying any philosophical baggage.
Yes, not grasping this causes many misunderstandings.
Are you a follower of Peirce, is that where your objection is coming from?
A fan, yes. But my views originate with the Perennial philosophy, which denies the true reality of extension. It would be its unreality that explains the paradoxes that arise for views by which the continuum really is extended. It is a double-aspect view by which both of Weyl's continuums must be taken into account. (I once wrote an article on this https://philpapers.org/rec/JONTCE)
wtf
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### Re: Continuum

PeteJ wrote: Thu Aug 27, 2020 2:54 pm It means that the arithmetical line works fine as a fiction, but as a fundamental phenomenon it is paradoxical.
PeteJ wrote: Thu Aug 27, 2020 2:54 pm Same as saying that set-theory is fine for everyday life but doesn't work when we try to make sets fundamental.
Making sets fundamental works very well in math. And math works for physics. Math works very well, that's its unreasonable effectiveness. Math works very well. I don't follow your point. Math does work. And what does it mean to make sets fundamental? If by fundamental in math, that's the great 20th century foundational project. But if you mean metaphysically fundamental, who do you think is doing this? Name them so we can shame them. But name someone. One will do, so you can convince me you're not flailing at a strawman.
PeteJ wrote: Thu Aug 27, 2020 2:54 pm In metaphysics this means that a realistic view of space-time is fine for everyday life but paradoxical when taken to be a fundamental or truly real.
What paradox? And who is taking things as fundamental that should not be so taken, other than some misguided posters on message forums?
PeteJ wrote: Thu Aug 27, 2020 2:54 pm We do seem to agree.
My complaint is against various posters on message forums who rant against math because math isn't necessarily real. But you seem to think mathematicians are making that mistake. Why do you think that?

PeteJ wrote: Thu Aug 27, 2020 2:54 pm Hmm. I'd say Skepdick is right. Mathemticians can turn a blind eye.
Can you name one? "Fred Bloggs, a prominent mathematician, was briefly detained at a supermarket when he tried to buy the powerset of omega. The store manager calmly explained to him that the powerset of omega is merely an abstract mathematical construction, but as Bloggs was led away by the police he was heard to say, "It's real, it's real!""

Is this what you claim is happening? Name the mathematician you are complaining about.
PeteJ wrote: Thu Aug 27, 2020 2:54 pm But when they study foundations and delve into metaphysics then this conception of the continuum gives rise to contradictions and the purely conceptual nature of the arithmetical continuum becomes apparent.
For sake of discussion, what contradictions? Are you fighting the philosophical wars of the 1930's?
PeteJ wrote: Thu Aug 27, 2020 2:54 pm I thought you shared this view.
I share the view that forum posters shouldn't make the mistake of reifying mathematics. You seem to think mathematicians are doing that, and I wonder if you can name one. Or be more specific as to what you are talking about. A lot of mathematics IS real.
PeteJ wrote: Thu Aug 27, 2020 2:54 pm Yes, not grasping this causes many misunderstandings.
Not among actual mathematicians unless you can name one such.
PeteJ wrote: Thu Aug 27, 2020 2:54 pm A fan, yes. But my views originate with the Perennial philosophy, which denies the true reality of extension. It would be its unreality that explains the paradoxes that arise for views by which the continuum really is extended. It is a double-aspect view by which both of Weyl's continuums must be taken into account.
I looked up the relevant Wiki page. It said nothing about math or set theory. It's about a particular religious or theological doctrine to the effect that all religions are ultimately saying or pointing at the same thing. What am I missing? There's no argument against mathematical realisim there.

https://en.wikipedia.org/wiki/Perennial_philosophy
PeteJ wrote: Thu Aug 27, 2020 2:54 pm (I once wrote an article on this https://philpapers.org/rec/JONTCE)
I see that you take this seriously, but from my limited view you seem to be fighting a strawman. I don't believe any actual mathematicians hold the views you claim they do, at least when they are doing mathematics. I think it's the physicists who reify mathematics more than they should.

I understand the philosophical point that you're making, but I don't think there are many people on the other side these days. Tegmark maybe with his mathematical universe, but he doesn't even seem to believe his own idea very strongly.
PeteJ
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### Re: Continuum

wtf wrote: Thu Aug 27, 2020 4:35 pm
PeteJ wrote: Thu Aug 27, 2020 2:54 pm It means that the arithmetical line works fine as a fiction, but as a fundamental phenomenon it is paradoxical.
There are many. Zeno has a go at proving this. Are you really not aware of these paradoxes? If not I'll find a couple of references.
Making sets fundamental works very well in math. And math works for physics.
Russell didn't manage to do it so it can't be easy. What is the fundamental set?
I don't follow your point. Math does work. And what does it mean to make sets fundamental? If by fundamental in math, that's the great 20th century foundational project. But if you mean metaphysically fundamental, who do you think is doing this? Name them so we can shame them. But name someone. One will do, so you can convince me you're not flailing at a strawman.
Who do I think is doing what? Western thinkers tend to make sets (categories) fundamental and as a result cannot make sense of metaphysics. I have no opinion on mathematicians unless they are also philosophers. Afaik I agree with Weyl in all respects.
What paradox? And who is taking things as fundamental that should not be so taken, other than some misguided posters on message forums?
Surely you are aware of the paradoxes caused by realistic views of space-time. They are numerous. Having an inside but no outside would be one. Being finite or infinite would be another since both idea don't work. This is an extyensively explored area of philosophy.
My complaint is against various posters on message forums who rant against math because math isn't necessarily real. But you seem to think mathematicians are making that mistake. Why do you think that?
Lots of people make the mistake of 'reifying the reals', some of whom are mathematicians. I have no general complaints against mathematicians. They are usually well aware their discipline is formal and not existential.
Can you name one?
Bertrand Russell.
For sake of discussion, what contradictions? Are you fighting the philosophical wars of the 1930's?
If we assume the extended continuum of space-time is fundamental paradoxes immediately arise. I'm not sure it;s fair to ask me to list them when this is so well known.
I share the view that forum posters shouldn't make the mistake of reifying mathematics.

All good then, since I agree. Unfortunately philosophers regularly do just this.
I looked up the relevant Wiki page. It said nothing about math or set theory. It's about a particular religious or theological doctrine to the effect that all religions are ultimately saying or pointing at the same thing. What am I missing? There's no argument against mathematical realism there.
Lol. I fear it would takes more than a Wiki page to cover the ground, The Perennial philosophy proposes that all categories of thought are reducible, and must be reduced for a fundamental theory. Thus mathematics must be reduced. Number and form would be created phenomena and not really real. Nothing would really exist or ever really happen. Extension would be unreal. Set theory would have to be transcended if it is to be axiomatised, as explained by Weyl and by Spencer Brown in Laws of Form. If you want to follow any of this up I'll post references.
I see that you take this seriously, but from my limited view you seem to be fighting a strawman. I don't believe any actual mathematicians hold the views you claim they do, at least when they are doing mathematics. I think it's the physicists who reify mathematics more than they should.
My view also. I'm not fighting the mathematicians. Not sure why you think I am.
I understand the philosophical point that you're making,
Great. But perhaps not the philosophical one.

I don't think were disagreeing on any mathematical issue.
wtf
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### Re: Continuum

@PeteJ, I've glanced at your site and clearly you have written a number of articles and you're thoughtful and serious. But at least based on your comments here and in my brief scan of your continuum paper, I detect a maddening vagueness and lack of specificity. You make generality after generality and never anchor your claims in specifics. I get the feeling you are lacking in peer review or any kind of feedback. I have respect for your work, and I'm especially impressed that you read through Weyl's Das Kontinuum. I do not want my remarks to sound like cheap message board sniping. But you have simply not convinced me of your point of view. In fact you've convinced me that you have not done your homework. I hope you'll take my remarks as pointed but constructive criticism.

[ps] -- Writing this after my post is complete. I know I have a particular style ... I can come off as critical past the line of civility sometimes. I want you to know that if I didn't find your ideas interesting, and if I didn't desperately want to understand you, I wouldn't have bothered at all. I would like to better understand the meaning of Das Kontinuum, which I confess I've always wanted to read but never have.

So please, just accept my word that my criticism is my struggle to understand you. I want to understand the Weyl viewpoint. I know he and Brouwer were having very deep thoughts back then, and that these thoughts are now re-ascendant in the world by way of neo-intuitionism, constructive math, computer science, Curry-Howard, Homotopy type theory, computerized proof assistants, and all the rest of the startling modern developments in foundations. So believe me, I know there's something there. And I'd love to understand what it is you're talking about. You're relating intuitionism to deeper ideas in philosophy. I'd like to understand that.

You know, if you think of me as decently knowledgable in math but a total philosophical ignoramus, that's not too far from the truth. When you say that set theory has to be an extension from from deep formulation and without such a thing the world can't exist, I think you are really far out there, man. This stuff means a lot to you but it's way over my head; and the parts I do understand, seem false. Like needing a metaphysics in order for there to be existence. Surely the world is prior. Unless I misunderstand you.

I'm just trying to understand. That's where my response is coming from. Now I'll just let 'er rip.

PeteJ wrote: Fri Aug 28, 2020 1:47 pm There are many. Zeno has a go at proving this. Are you really not aware of these paradoxes? If not I'll find a couple of references.
Zeno? Zeno?? All this is about Zeno? Ok, I'll play. Please pick any one of Zeno's several paradoxes -- pick a specific one. State your thesis ("Reifying the real numbers is bad metaphysics" or whatever -- please state a clear thesis so I do know what you are saying), and show how any one of Zeno's paradoxes supports your thesis. I must say that I find this remark of yours trivial in the extreme. I was expecting something a lot more subtle. But I'm willing to hear you state your case and clarify your thinking.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Russell didn't manage to do it so it can't be easy. What is the fundamental set?
I said that set theory is fundamental to math and math is fundamental to physics; and you ask "what is the fundamental set?" That's a disingenuous remark, playing some kind of word game instead of engaging with what I wrote. But since you ask, every set used in mathematics can be built up starting with the empty set and taking successive powersets. This procedure is the von Neumann universe of sets. It contains every set that can possibly be used in standard mathematics.

But let me restate my point so you can either grapple with it or not. Set theory is the foundation of math; and math is essential modern physics. And Russell is not a very convincing example here.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Western thinkers tend to make sets (categories) fundamental and as a result cannot make sense of metaphysics. I have no opinion on mathematicians unless they are also philosophers. Afaik I agree with Weyl in all respects.
WHICH WESTERN THINKERS? This is an example of your lack of specificity and your lack of doing your homework. Which Western thinkers said what about what? Convince me you're not just waving your hands about things you haven't really challenged yourself to verify.

Which Western thinkers can't make sense of metaphysics? Are you taking on all of modern philosophy? Maybe you're right, I don't know enough philosophy to disagree. But at least state exactly who and what you are talking about. Else you're saying nothing and convincing no one.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Surely you are aware of the paradoxes caused by realistic views of space-time.
Pretend I'm not. Tell me about some of them. Identify them by name. Tell me how they destroy Western metaphysics.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm They are numerous.
Then you should have no problem (1) naming some of them; and (2) explaining how they falsify or create problems for Western metaphysics.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Having an inside but no outside would be one.
Like a Mobius strip or a Klein bottle? Non-orientable manifolds are upsetting you? What on earth are you talking about? Give specific examples and show how they support your point.

I am not just giving you a hard time. I'm giving your work a hard review, as a teacher would. You are woefully lacking in specifics and your reader has no idea what you're talking about.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Being finite or infinite would be another since both idea don't work.
Explain what you mean. There is nothing that is both finite and infinite at the same time since those terms mean the opposite of each other. It is true that there are sets that are infinite in one definition and finite in another. For example if you don't assume the axiom of choice, there is an infinite set that's Dedekind-finite. That means it's not bijective to any finite natural number; but it's also not bijective with any of its proper subsets. Is that what you're talking about? What are you talking about?

Both don't work? You mean there are no finite sets AND no infinite sets? I'm afraid you are simply talking out of your hat.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm This is an extyensively explored area of philosophy.
By whom? You have not given me a single clue as to what you are talking about.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Lots of people make the mistake of 'reifying the reals', some of whom are mathematicians.
Name one.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm I have no general complaints against mathematicians. They are usually well aware their discipline is formal and not existential.
You haven't stopped complaining about mathematicians since you started your posts in this thread. Without managing to name a single one.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Bertrand Russell.
Laughable example. Russell did no math outside of his foundational attempts, which were totally blown up by Gödel. Surely you can get past the primitive foundational stumblings of 1900. Is that where you're stuck?

PeteJ wrote: Fri Aug 28, 2020 1:47 pm If we assume the extended continuum of space-time is fundamental paradoxes immediately arise.
What is the extended continuum of spacetime? In view of modern quantum physics, such a thing is questionable on the one hand; and inaccessible to our current theories even if such a thing did exist. What are you talking about?
PeteJ wrote: Fri Aug 28, 2020 1:47 pm I'm not sure it;s fair to ask me to list them when this is so well known.
List them or retract your remark. What paradoxes? Zeno? Are we in high school?
PeteJ wrote: Fri Aug 28, 2020 1:47 pm All good then, since I agree. Unfortunately philosophers regularly do just this.
Which philosophers?
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Lol. I fear it would takes more than a Wiki page to cover the ground, The Perennial philosophy proposes that all categories of thought are reducible, and must be reduced for a fundamental theory. Thus mathematics must be reduced.
Reduced to what? You're using jargon that has meaning to you but not to your readers.

Tell me, does chess need to be reduced? Isn't chess just a formal system? Isn't math just a formal system? Why does math need to be reduced, and to what must it be reduced, and why, and who says so, and to what effect?
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Number and form would be created phenomena and not really real.
Created by whom? Numbers aren't real? I'll certainly agree that numbers are abstract objects. That doesn't make them any less real. The law that says you go on green and stop on red is an abstract thing. A martian physicist can distinguish red from green by their wavelengths, but she can't tell you which means go and which means stop because that is a social convention. Yet if you don't treat it as real you can die. So it's real. See Searle, The Construction of Social Reality.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Nothing would really exist or ever really happen.
Without someone writing down some philosophy? That's nonsense, utter nonsense.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Extension would be unreal.
No idea what you are talking about.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Set theory would have to be transcended if it is to be axiomatised, as explained by Weyl and by Spencer Brown in Laws of Form. If you want to follow any of this up I'll post references.
I don't want references. I want you to explain yourself to me, a reasonably intelligent person who is reasonably well informed about math and set theory. Wiki calls the Laws of Form a "cult classic." It certainly was when I first heard about it fifty years ago.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm My view also. I'm not fighting the mathematicians. Not sure why you think I am.
You haven't stopped saying "mathematicians do this and mathematicians do that" since this thread started and when challenged, all you can come up with is Bertrand Russell. You have convinced me that you have not challenged yourself to to your homework and support your ideas with specifics.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm I don't think were disagreeing on any mathematical issue.
You haven't stated any mathematical issues. You don't seem to know any mathematics beyond some vague notions from the foundational wars of the Frege/Russell era.

If you went through your own work and every time you say, "Mathematicians believe ..." or "Western philosophers believe ..." and did your homework and replaced that with, "Fred Bloggs believed ..." you would have a solid piece of argumentative writing. As it is, I am unmoved. And frustrated, because you clearly have things to say. I just can't find anything solid to grab on to. You don't like finite sets, you don't like infinite sets, you don't like non-orientable manifolds, you think Zeno's paradoxes invalidate 2000 years of Western philosophy, you take Bertrand Russell as an example of a mathematician with wrong ideas.

I want you to challenge your own vagueness and write something compelling. Take a red pencil to your own work and replace every generality with a specific supporting example.
PeteJ
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### Re: Continuum

wtf wrote: Sat Aug 29, 2020 10:22 pm @PeteJ, I've glanced at your site and clearly you have written a number of articles and you're thoughtful and serious. But at least based on your comments here and in my brief scan of your continuum paper, I detect a maddening vagueness and lack of specificity.
Thanks for your thoughtful post. There's so much nonsense on this site that your approach stands out. But it's too long for me to deal with it all in one go. As you note, I'm not a mathematician so will benefit from your comments.

I'll post a few thoughts and perhaps you could pick out an issue or two to get started on. First. please note that when I formed my view of metaphysics a decade ago I conducted a reality-check by writing a long supervised dissertation. This has never received a single objection. I say this to make it clear I don't pluck ideas out of hats.

I suspect that the main difficulty is the idea that Reality transcends the categories of thought, thus sets, categories, form and number. You bridle at my suggestion that nothing really exists, yet this is basic stuff for those who endorse a doctrine of Unity and non-duality. Without this notion my comments about the continuum and set-theory may seem quite wild. Note the word 'really' here, without which the statement would be obvious nonsense.

It would be the reification of number and from that causes Western academic metaphysics to be incomprehensible. It prevents the construction of a fundamental theory. It causes metaphysical problems to be intractable and leads to the dismissal of metaphysics as a waste of time. It means that in the Academy nobody understands metaphysics. Indeed, the idea that anyone ever could is often ridiculed.

The connection with set theory is that (for the purposes of metaphysics at least) a set is a category-of-thought. This is so even for the empty set. The emtpy set is not as basic as an absence of sets. Spencer Brown disposes of sets entirely for his 'axiomatisation' of set-theory, reducing all Venn diagrams to the blank sheet of paper on which they are drawn. This is pretty much what Weyl does for the numbers and the idea of spatial and temporal extension. Schroedinger takes the same approach, reducing the phenomena of the space-time world to 'the canvas on which they are painted'. More recently the same approach is endorsed by Ulrich Mohrhoff in his text-book on quantum mechanics.

This reduction sets to (what seems to be) nothingness allows us to construct a fundamental theory and solve philosophical problems. There would be no other way to do it, as history shows. It is very clear form the literature that metaphysics is hopeless when we treat number and form as irreducible. It only becomes comprehensible when we transcend set-theory, form and number for a unified state. 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'.

Are you okay with this so far?
Skepdick
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### Re: Continuum

wtf wrote: Wed Aug 26, 2020 7:00 am Mathematicians do not make that category error. They know that they are talking about the set of real numbers, which is a particular technical construction carried out in the framework of ZFC. They don't make claims for whether it's a good model of this or that, or whether it's real, or true, or whatever.

Those are philosophers' questions, but those questions are a category error. You might as well ask whether the knight "really" moves that way in the real world. The question's absurd. It's confusing a formal game with someone's idea of reality.
I think we are in agreement about that. I am happy to use the word "real" in the sense that the object we speak about exists in my head.
How you apply any abstract model to practical purposes is not at all what I am pointing at.

I am talking about performing abstract operations on abstract objects - if you are manipulating abstractions then that's sufficient evidence for me that they are reified.
something that was previously implicit, unexpressed, and possibly inexpressible is explicitly formulated and made available to conceptual (logical or computational) manipulation.Informally, reification is often referred to as "making something a first-class citizen" within the scope of a particular system.
From where I am looking, Mathematicians have reified sets.
wtf wrote: Wed Aug 26, 2020 7:00 am I have in fact come to understand this point of view. It came to me during an online conversation with a constructivist. The point is that when mathematicians talk about the real numbers and the universe of sets, they are not actually experiencing or instantiating these strange things. Rather, they are manipulating finite strings of symbols according to specific rules. So we are fooling ourselves about these infinities.
Mostly agreed - that is the formalist (formal language theorist) perspective. From that view-point logic itself is a formal language (so no, I am not a Peircian, because I see logic and Mathematics as the same kind of activity: symbol transformations).

But this is where my brain fires up more questions than answers. Given "the set of real numbers" (R), and function (f) such that f(R) produces a real number (x), would you say that f(R) instantiates x? If yes - is R a singleton? Is R discontinuous now that x has been instantiated?

What I am pointing at is this conceptual conundrum: if we are manipulating strings according to rules, what are rules? Are rules not abstract objects themselves? Can we manipulate the rules themselves according to other rules? In programming languages capable of introspection we can.

This is approximately the view of Jean-Yves Girard when he wrote the paper "From the rules of logic, to the logic of rules". It's one level higher in the tower of abstraction.
wtf wrote: Wed Aug 26, 2020 7:00 am However, we can find semantic meaning for uncountable sets in abstract computability theory. Oracles in computer science can be explained in terms of noncomputable numbers. So some computer scientists are interested in infinitary processes and concepts.
Oracles don't require explanations. They are axiomatic truths. Some Oracles are defined to be able to solve the halting problem. If Oracles can do the impossible, then Oracles defy the semantics of "impossibility".

A conceptual paradigm shift is necessary to reconcile that. Something along the lines of David Deutsch's constructor theory.
wtf wrote: Wed Aug 26, 2020 7:00 am But isn't this typical of what humans do? There is no perfection on earth, but we conceive of the perfection of heaven. We are creatures with the gift of abstraction. We can let variables stand for things. We can conceptualise infinity with finite strings of symbols.

Our power of abstraction is a feature, not a bug.
It is a feature! My argument (if any) is that Mathematics is not abstract enough. Conceptualizing any abstract object as a finite string is precisely the idea of Kolmogorov complexity. The construction in any abstract object is in constructing the grammatical rules which produce the object, while preserving all the properties which make the object what it is.

In this paradigm there's no difference between syntax and semantics. Syntax is all you have.
Skepdick
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### Re: Continuum

wtf wrote: Thu Aug 27, 2020 2:02 am Then just call it "the set of real numbers." I would say that when mathematicians call the real numbers the continuum, they're just using the term as a shorthand for the real numbers. They're not implying any philosophical baggage.

Are you a follower of Peirce, is that where your objection is coming from?
I think I addressed this in my previous post, but my objections are coming (trivially) from having a function that can produce elements of a continuum.

If R is the set of Real numbers and you have any function f, such that f(R) produces elements of the set, then conceptually the reals are discrete.
Or... by instantiating a real. Any real - a discontinuity is introduced.

The English semantics of "element" imply Mathematical discreteness.
wtf
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### Re: Continuum

PeteJ wrote: Sun Aug 30, 2020 10:56 am Thanks for your thoughtful post. There's so much nonsense on this site that your approach stands out.

Thank you.

PeteJ wrote: Sun Aug 30, 2020 10:56 am But it's too long for me to deal with it all in one go.
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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am As you note, I'm not a mathematician so will benefit from your comments.
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.

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.

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 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.

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.

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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am I'll post a few thoughts and perhaps you could pick out an issue or two to get started on.
No prob. I'll respond to some of your points; but my main point is that I need you to be specific about two areas of overlap in our knowledge. Why do you think Zeno breaks metaphysics; and why do you think the finite and the infinite do. Those two subjects must be drilled down.
PeteJ wrote: Sun Aug 30, 2020 10:56 am First. please note that when I formed my view of metaphysics a decade ago I conducted a reality-check by writing a long supervised dissertation. This has never received a single objection.
I was not on your committee!!

PeteJ wrote: Sun Aug 30, 2020 10:56 am I say this to make it clear I don't pluck ideas out of hats.
I didn't say you did. I understand that your ideas come from your research and reading. What I do say is that your exposition is totally lacking in specific examples to connect your ideas with reality. And that when challenged on specifics, you just avoid engaging at all.

Zeno and infinity. That is where we must focus.
PeteJ wrote: Sun Aug 30, 2020 10:56 am suspect that the main difficulty is the idea that Reality transcends the categories of thought, thus sets, categories, form and number. You bridle at my suggestion that nothing really exists, yet this is basic stuff for those who endorse a doctrine of Unity and non-duality. Without this notion my comments about the continuum and set-theory may seem quite wild. Note the word 'really' here, without which the statement would be obvious nonsense.
This is frankly incoherent babble. If your words refer to prior ideas, they're not ones I'm familiar with. I'm not bridling. I'm pointing out a claim that's absurd on its face. You don't know the first thing about set theory.

PeteJ wrote: Sun Aug 30, 2020 10:56 am It would be the reification of number and from that causes Western academic metaphysics to be incomprehensible. It prevents the construction of a fundamental theory. It causes metaphysical problems to be intractable and leads to the dismissal of metaphysics as a waste of time. It means that in the Academy nobody understands metaphysics. Indeed, the idea that anyone ever could is often ridiculed.
Which Western academic metaphysicians are incomprehensible? Name one; and describe exactly what it is they said that you claim is incomprehensible.

But please don't do that. I don't care. I'm just pointing out your inability to be specific about your ideas; and that when challenged, you fall back on more generalities.

I want you to tell me about Zeno and infinity.

PeteJ wrote: Sun Aug 30, 2020 10:56 am The connection with set theory is that (for the purposes of metaphysics at least) a set is a category-of-thought. This is so even for the empty set. The emtpy set is not as basic as an absence of sets. Spencer Brown disposes of sets entirely for his 'axiomatisation' of set-theory, reducing all Venn diagrams to the blank sheet of paper on which they are drawn. This is pretty much what Weyl does for the numbers and the idea of spatial and temporal extension. Schroedinger takes the same approach, reducing the phenomena of the space-time world to 'the canvas on which they are painted'. More recently the same approach is endorsed by Ulrich Mohrhoff in his text-book on quantum mechanics.

You haven't the foggiest idea what you're talking about. Nobody thinks any of those things about set theory; and if they do, you had better name them because this is not a mainstream point of view.

There's a passage in Kunen, Set Theory, first ed. This book is one of the standard texts for graduate-level set theory. The book is highly technical but Kunen also faces the philosophical aspects. He says:
Some questions about sets are irrelevant to mathematics.

First irrelevant question. Is there anything which is not a set? Certainly there is in the "real world" of cows and pigs, but our axioms of set theory say nothing about this "real world", since we have declared that they talk only about sets - in fact, hereditary sets
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.

If you claim that some other set theorist or some philosopher says otherwise, then please name them and post a quote from their work. That's called being specific about your ideas.

PeteJ wrote: Sun Aug 30, 2020 10:56 am This reduction sets to (what seems to be) nothingness allows us to construct a fundamental theory and solve philosophical problems.
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.

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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am It is very clear form the literature
What literature? Name the book, name the author, name the page, quote the passage that supports your point of view.

PeteJ wrote: Sun Aug 30, 2020 10:56 am that 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.

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.

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.
PeteJ wrote: Sun Aug 30, 2020 10:56 am Are you okay with this so far?

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 think if you were making a historical point: "This is what people were confused a about in 1900"; your ideas would make more sense. As it is, your claims and beliefs about set theory are just flat out wrong. YOU are the one doing all the reifying here.

I summarize:

* You said Zeno's paradoxes (which ones?) break metaphysics or are evidence of SOMETHING that bothers you. At this point I ask, indeed demand, that you clearly explain (1) The point you are making; (2) Why any of Zeno's paradoxes support your point. Or if you prefer, take any other paradox.

*. You said "infinity and the finite" are inconsistent or something. Explain yourself. State your thesis, make your case.

For the record I copy the relevant passages from your earlier post.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm
There are many. Zeno has a go at proving this.
Be specific. How does (any one of) Zeno's paradoxes falsify or break math or reality or metaphysics? State a thesis, make a case.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Having an inside but no outside would be one. Being finite or infinite would be another since both idea don't work.
What on earth are you talking about? State a thesis and make a coherent argument.
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### Re: Continuum

wtf wrote: Wed Sep 02, 2020 3:35 am
PeteJ wrote: Sun Aug 30, 2020 10:56 am Thanks for your thoughtful post. There's so much nonsense on this site that your approach stands out.

Thank you.

PeteJ wrote: Sun Aug 30, 2020 10:56 am But it's too long for me to deal with it all in one go.
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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am As you note, I'm not a mathematician so will benefit from your comments.
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.

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.

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 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.

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.

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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am I'll post a few thoughts and perhaps you could pick out an issue or two to get started on.
No prob. I'll respond to some of your points; but my main point is that I need you to be specific about two areas of overlap in our knowledge. Why do you think Zeno breaks metaphysics; and why do you think the finite and the infinite do. Those two subjects must be drilled down.
PeteJ wrote: Sun Aug 30, 2020 10:56 am First. please note that when I formed my view of metaphysics a decade ago I conducted a reality-check by writing a long supervised dissertation. This has never received a single objection.
I was not on your committee!!

PeteJ wrote: Sun Aug 30, 2020 10:56 am I say this to make it clear I don't pluck ideas out of hats.
I didn't say you did. I understand that your ideas come from your research and reading. What I do say is that your exposition is totally lacking in specific examples to connect your ideas with reality. And that when challenged on specifics, you just avoid engaging at all.

Zeno and infinity. That is where we must focus.
PeteJ wrote: Sun Aug 30, 2020 10:56 am suspect that the main difficulty is the idea that Reality transcends the categories of thought, thus sets, categories, form and number. You bridle at my suggestion that nothing really exists, yet this is basic stuff for those who endorse a doctrine of Unity and non-duality. Without this notion my comments about the continuum and set-theory may seem quite wild. Note the word 'really' here, without which the statement would be obvious nonsense.
This is frankly incoherent babble. If your words refer to prior ideas, they're not ones I'm familiar with. I'm not bridling. I'm pointing out a claim that's absurd on its face. You don't know the first thing about set theory.

PeteJ wrote: Sun Aug 30, 2020 10:56 am It would be the reification of number and from that causes Western academic metaphysics to be incomprehensible. It prevents the construction of a fundamental theory. It causes metaphysical problems to be intractable and leads to the dismissal of metaphysics as a waste of time. It means that in the Academy nobody understands metaphysics. Indeed, the idea that anyone ever could is often ridiculed.
Which Western academic metaphysicians are incomprehensible? Name one; and describe exactly what it is they said that you claim is incomprehensible.

But please don't do that. I don't care. I'm just pointing out your inability to be specific about your ideas; and that when challenged, you fall back on more generalities.

I want you to tell me about Zeno and infinity.

PeteJ wrote: Sun Aug 30, 2020 10:56 am The connection with set theory is that (for the purposes of metaphysics at least) a set is a category-of-thought. This is so even for the empty set. The emtpy set is not as basic as an absence of sets. Spencer Brown disposes of sets entirely for his 'axiomatisation' of set-theory, reducing all Venn diagrams to the blank sheet of paper on which they are drawn. This is pretty much what Weyl does for the numbers and the idea of spatial and temporal extension. Schroedinger takes the same approach, reducing the phenomena of the space-time world to 'the canvas on which they are painted'. More recently the same approach is endorsed by Ulrich Mohrhoff in his text-book on quantum mechanics.

You haven't the foggiest idea what you're talking about. Nobody thinks any of those things about set theory; and if they do, you had better name them because this is not a mainstream point of view.

There's a passage in Kunen, Set Theory, first ed. This book is one of the standard texts for graduate-level set theory. The book is highly technical but Kunen also faces the philosophical aspects. He says:
Some questions about sets are irrelevant to mathematics.

First irrelevant question. Is there anything which is not a set? Certainly there is in the "real world" of cows and pigs, but our axioms of set theory say nothing about this "real world", since we have declared that they talk only about sets - in fact, hereditary sets
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.

If you claim that some other set theorist or some philosopher says otherwise, then please name them and post a quote from their work. That's called being specific about your ideas.

PeteJ wrote: Sun Aug 30, 2020 10:56 am This reduction sets to (what seems to be) nothingness allows us to construct a fundamental theory and solve philosophical problems.
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.

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.

PeteJ wrote: Sun Aug 30, 2020 10:56 am It is very clear form the literature
What literature? Name the book, name the author, name the page, quote the passage that supports your point of view.

PeteJ wrote: Sun Aug 30, 2020 10:56 am that 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.

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.

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.
PeteJ wrote: Sun Aug 30, 2020 10:56 am Are you okay with this so far?

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 think if you were making a historical point: "This is what people were confused a about in 1900"; your ideas would make more sense. As it is, your claims and beliefs about set theory are just flat out wrong. YOU are the one doing all the reifying here.

I summarize:

* You said Zeno's paradoxes (which ones?) break metaphysics or are evidence of SOMETHING that bothers you. At this point I ask, indeed demand, that you clearly explain (1) The point you are making; (2) Why any of Zeno's paradoxes support your point. Or if you prefer, take any other paradox.

*. You said "infinity and the finite" are inconsistent or something. Explain yourself. State your thesis, make your case.

For the record I copy the relevant passages from your earlier post.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm
There are many. Zeno has a go at proving this.
Be specific. How does (any one of) Zeno's paradoxes falsify or break math or reality or metaphysics? State a thesis, make a case.
PeteJ wrote: Fri Aug 28, 2020 1:47 pm Having an inside but no outside would be one. Being finite or infinite would be another since both idea don't work.
What on earth are you talking about? State a thesis and make a coherent argument.
This response is to PeteJ. Finiteness can be observed as multiple infinities. Using the line as an example it is infinite as composed of infinite lines. However when broken down to 2,3,4, etc. number of lines it results in multiple infinities.

This can be observed under the regressive analysis of any set of multiple phenomena as well. Upon continual analysis the phenomena can be observed as continually broken down to an infinity of parts thus necessitating infinite regress occurs for any specific phenomena. In observing multiple phenomena we are observing multiple infinities.

Finiteness is multiple infinities.
wtf
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### Re: Continuum

Skepdick wrote: Sun Aug 30, 2020 1:45 pm I think we are in agreement about that.
@Skepdick, it frightens me to hear that you agree with anything I say. Isn't that the first time ever?

Skepdick wrote: Sun Aug 30, 2020 1:45 pm I am happy to use the word "real" in the sense that the object we speak about exists in my head.
How you apply any abstract model to practical purposes is not at all what I am pointing at.

I am talking about performing abstract operations on abstract objects - if you are manipulating abstractions then that's sufficient evidence for me that they are reified.

Ok. Just placating you, not agreeing. I don't know exactly what point you're making.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm From where I am looking, Mathematicians have reified sets.
As abstract objects that do not refer to anything in the real world. I hope you read the Kunen quote I described in my reply to @PeteJ. Set theory is not about anything in the real world and that is official.

Skepdick wrote: Sun Aug 30, 2020 1:45 pm Mostly agreed - that is the formalist (formal language theorist) perspective. From that view-point logic itself is a formal language (so no, I am not a Peircian, because I see logic and Mathematics as the same kind of activity: symbol transformations).
Please don't take my lack of objections as agreement. I'm worn out from replying to @PeteJ and I'm not sure exactly what point you are making to me.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm But this is where my brain fires up more questions than answers. Given "the set of real numbers" (R), and function (f) such that f(R) produces a real number (x), would you say that f(R) instantiates x? If yes - is R a singleton? Is R discontinuous now that x has been instantiated?
f is a function that inputs the entire set of real numbers and outputs a number, like 6? What is the relevance of that somewhat bizarre example?

Or do you mean f: R -> R, in other words f inputs a real number and outputs one, like f(x{ = x + 7? If the latter, it's just a function. It's a set of ordered pairs of real numbers with the property that each real occurs exactly once as the first coordinate of a pair. [That's the functional property, that f(47) is always the same thing, never something different from day to day].

Skepdick wrote: Sun Aug 30, 2020 1:45 pm What I am pointing at is this conceptual conundrum: if we are manipulating strings according to rules, what are rules? Are rules not abstract objects themselves? Can we manipulate the rules themselves according to other rules? In programming languages capable of introspection we can.
Yes, formal logic does this all day long. What of it? We have classical logic. first-order predicate logic, modal logic, second-order logic, fuzzy logic. Of course we can play with the rules. I know this isn't news to you, why do you ask?
Skepdick wrote: Sun Aug 30, 2020 1:45 pm This is approximately the view of Jean-Yves Girard when he wrote the paper "From the rules of logic, to the logic of rules". It's one level higher in the tower of abstraction.
How nice for Mr. Girard.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm Oracles don't require explanations. They are axiomatic truths.
What? Axioms aren't true at all. They're accepted without proof in order to get an axiomatic system off the ground. But oracles require a hell of a lot of explanation, since they are by definition thingies that do something that we just proved can't be done. They require quite a bit of study and explanation, which Turing did.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm Some Oracles are defined to be able to solve the halting problem. If Oracles can do the impossible, then Oracles defy the semantics of "impossibility".
Hence they require explanation. You just refuted your own point.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm A conceptual paradigm shift is necessary to reconcile that. Something along the lines of David Deutsch's constructor theory.
Oh please. Just study some CS. I know you think you have, but you are curiously obtuse at the moment in your discussion of oracles. And why exactly?

And as you often do, you supplied a link in the hopes that I wouldn't look at it and find out that it has not a freaking thing to do with the subject under discussion. Sorry dude I called your bluff and you've got nothing. Your link is on a completely different subject.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm It is a feature! My argument (if any) is that Mathematics is not abstract enough. Conceptualizing any abstract object as a finite string is precisely the idea of Kolmogorov complexity. The construction in any abstract object is in constructing the grammatical rules which produce the object, while preserving all the properties which make the object what it is.
Uh yeah, ok. Uh ... math is not abstract enough? Study some infinity categories and you'll change your mind.

Skepdick wrote: Sun Aug 30, 2020 1:45 pm In this paradigm there's no difference between syntax and semantics. Syntax is all you have.
Whatever dude. If a paradigm denies meaning, it's useless.

I don't mean to sound even more crabby than usual. Replying to @PeteJ put me in a bad mood. Like going down in warm maple syrup. Nothing to hold on to.
Last edited by wtf on Wed Sep 02, 2020 4:26 am, edited 5 times in total.
wtf
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### Re: Continuum

Skepdick wrote: Sun Aug 30, 2020 2:00 pm I think I addressed this in my previous post, but my objections are coming (trivially) from having a function that can produce elements of a continuum.

If R is the set of Real numbers and you have any function f, such that f(R) produces elements of the set, then conceptually the reals are discrete.
Or... by instantiating a real. Any real - a discontinuity is introduced.

The English semantics of "element" imply Mathematical discreteness.
Your notation f(R) is confusing. Do you mean that f is a function that inputs a real number and outputs another real number? If so, what of it? You may be aware that the great project of the 19th century was the arithmetization of analysis. In other words we DID found the continuous on the discrete. I'm aware of the philosophical problems. But you lost me on the bit about discontinuity. If f(x) = x^2 [x squared] where is the discontinuity? It looks continuous to me. You better go back to freshman calculus if you think functions on the real numbers are inherently discontinuous.

It's true that the reals are made of points, but they are NOT discrete. The integers are discrete because you can draw a little circle around each integer on the number line and each circle contains only one integer and no others. You can't do that with the reals. The real numbers are not discrete. I don't know why you think they are, unless you are confused about the technical meaning of discrete and/or making up your own definition.

https://en.wikipedia.org/wiki/Arithmeti ... f_analysis
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### Re: Continuum

wtf wrote: Wed Sep 02, 2020 4:11 am @Skepdick, it frightens me to hear that you agree with anything I say. Isn't that the first time ever?
Don't be too surprised. Disagreement is tiresome business.
wtf wrote: Wed Sep 02, 2020 4:11 am As abstract objects that do not refer to anything in the real world. I hope you read the Kunen quote I described in my reply to @PeteJ. Set theory is not about anything in the real world and that is official.
Right at the bottom you say "if it denies meaning - it's useless". Abstract objects "exist" in your head and have meaning. That meaning is real.

To me, everything in the universe is "real". My thoughts are "real". Your thoughts are "real". The conventional use of the word "real" (which implies that thoughts aren't "real") stems from philosophical dualism and it's a form of special pleading: I don't subscribe to it.
wtf wrote: Wed Sep 02, 2020 4:11 am f is a function that inputs the entire set of real numbers and outputs a number, like 6? What is the relevance of that somewhat bizarre example?
Yes that. A many-to-one function. The relevance is that you treat the set of numbers as an abstract mathematical objects. And you also treat 6 as an abstract mathematical object. Where did you conjure "6" from if not from R?
wtf wrote: Wed Sep 02, 2020 4:11 am Your comment about discontinuity didn't make any sense to me
When you take "6" out of R, You are left with a two ranges (-∞, 6) and (6, ∞). A discontinuity.
wtf wrote: Wed Sep 02, 2020 4:11 am Yes, formal logic does this all day long. What of it? We have classical logic. first-order predicate logic, modal logic, second-order logic, fuzzy logic. Of course we can play with the rules. I know this isn't news to you, why do you ask?
Yes it does. That's what programming languages do - we play with the (grammatical) rules. I am asking whether the notion of "contextual rules" means anything in Maths.

Like the + operator is polymorphic. it means "addition" when inputs are number-types "concatenation" when inputs are string-types.

When we were kids, we used to joke that 1+1 is 11... Kids distinguished between number-types and string-types.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm This is approximately the view of Jean-Yves Girard when he wrote the paper "From the rules of logic, to the logic of rules". It's one level higher in the tower of abstraction.
How nice for Mr. Girard.
wtf wrote: Wed Sep 02, 2020 4:11 am What? Axioms aren't true at all. They're accepted without proof in order to get an axiomatic system off the ground.
You are arguing over nomenclature. So I am putting it to rest with agreement. True/agreed/axiomatic/Garbage in - garbage out.

Same thing.
wtf wrote: Wed Sep 02, 2020 4:11 am Hence they require explanation. You just refuted your own point.
I think you are applying a double standard to arrive at this conclusion.

If Oracles are axiomatic, and Oracles require explanations then axioms require explanations.

Your explanation of axioms were "we need them in order to get formal systems off the ground".
Why isn't that sufficient for Oracles? Why do Oracles "require explanations and studying" but axioms don't?
wtf wrote: Wed Sep 02, 2020 4:11 am Oh please. Just study some CS. I know you think you have, but you are curiously obtuse at the moment in your discussion of oracles. And why exactly?
because there's nothing to be said about them! They are axiomatic. They are bizarre things that give you the correct answer.
wtf wrote: Wed Sep 02, 2020 4:11 am And as you often do, you supplied a link in the hopes that I wouldn't look at it and find out that it has not a freaking thing to do with the subject under discussion. Sorry dude I called your bluff and you've got nothing. Your link is on a completely different subject.
That depends on your subjective "sameness" and "difference" functions/heuristics!

To you Logic and Mathematics are "different subjects".

To me they are EXACTLY the same thing. Formal languages.

Rule-following games.
Skepdick wrote: Sun Aug 30, 2020 1:45 pm Uh yeah, ok. Uh ... math is not abstract enough? Study some infinity categories and you'll change your mind.
Sure! But since you are studying those things, and you are observing them you can always take the "outside" view. So if infinity categories is the "internal language" of the system (its grammar). What's its external language (its semantics)?

Or if you prefer Tarski's nomenclature: object language (grammar) vs metalanguage (semantics).
Skepdick wrote: Sun Aug 30, 2020 1:45 pm Whatever dude. If a paradigm denies meaning, it's useless.

In computer science (or software engineering if you will) we just call them contexts. Same thing formalists call them. Context-free vs context-sensitive languages etc...
Skepdick wrote: Sun Aug 30, 2020 1:45 pm I don't mean to sound even more crabby than usual. Replying to @PeteJ put me in a bad mood. Like going down in warm maple syrup. Nothing to hold on to.
It doesn't bother me - I don't take anything on this site personally (as a safety precaution for my own sanity). But as a general warning: philosophy is maple syrup all the way down with no "foundations".
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### Re: Continuum

wtf wrote: Wed Sep 02, 2020 4:16 am Your notation f(R) is confusing. Do you mean that f is a function that inputs a real number and outputs another real number?
R represents the set of real numbers.

To use your prior context f(R) = 6.
wtf wrote: Wed Sep 02, 2020 4:16 am where is the discontinuity? It looks continuous to me.
Exactly at 6. Since you discretised it.
wtf wrote: Wed Sep 02, 2020 4:16 am You better go back to freshman calculus if you think functions on the real numbers are inherently discontinuous.
We've had this discussion. Higher order functions are sometimes continuous.

sometimes all functions are continuous
wtf wrote: Wed Sep 02, 2020 4:16 am It's true that the reals are made of points, but they are NOT discrete.
Points are discrete entities!

That's digital physics - it from bit.
That's Scott Aaronson's view on the Atomists (Democritus).
wtf wrote: Wed Sep 02, 2020 4:16 am The integers are discrete because you can draw a little circle around each integer on the number line and each circle contains only one integer and no others. You can't do that with the reals.
Yes. That's your definition. I don't go on definitions - I go on intuitions.
wtf wrote: Wed Sep 02, 2020 4:16 am The real numbers are not discrete. I don't know why you think they are, unless you are confused about the technical meaning of discrete and/or making up your own definition.
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.
wtf
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### Re: Continuum

Skepdick wrote: Wed Sep 02, 2020 6:16 am To me, everything in the universe is "real". My thoughts are "real". Your thoughts are "real". The conventional use of the word "real" (which implies that thoughts aren't "real") stems from philosophical dualism and it's a form of special pleading: I don't subscribe to it.
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
Skepdick wrote: Wed Sep 02, 2020 6:16 am Yes that. A many-to-one function. The relevance is that you treat the set of numbers as an abstract mathematical objects. And you also treat 6 as an abstract mathematical object. Where did you conjure "6" from if not from R?
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?
Skepdick wrote: Wed Sep 02, 2020 6:16 am When you take "6" out of R, You are left with a two ranges (-∞, 6) and (6, ∞). A discontinuity.
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?

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?
Skepdick wrote: Wed Sep 02, 2020 6:16 am Yes it does. That's what programming languages do - we play with the (grammatical) rules. I am asking whether the notion of "contextual rules" means anything in Maths.
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?

Skepdick wrote: Wed Sep 02, 2020 6:16 am Like the + operator is polymorphic. it means "addition" when inputs are number-types "concatenation" when inputs are string-types.
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.
Skepdick wrote: Wed Sep 02, 2020 6:16 am When we were kids, we used to joke that 1+1 is 11... Kids distinguished between number-types and string-types.
Ok.

Skepdick wrote: Wed Sep 02, 2020 6:16 am You are arguing over nomenclature. So I am putting it to rest with agreement.
Ok.

Same thing.
Skepdick wrote: Wed Sep 02, 2020 6:16 am If Oracles are axiomatic, and Oracles require explanations then axioms require explanations.
I thought your remarks on oracles were unclear. I still do. But the discussion doesn't seem to be about anything.
Skepdick wrote: Wed Sep 02, 2020 6:16 am Your explanation of axioms were "we need them in order to get formal systems off the ground".
Why isn't that sufficient for Oracles? Why do Oracles "require explanations and studying" but axioms don't?
You're trying to make a dispute where there isn't one. You seem confused about oracles. And the company that makes database software is Oracle. The thing in CS is an oracle. If oracles don't require an explanation, would you say people are born knowing what they are? How could you defend such a proposition?
Skepdick wrote: Wed Sep 02, 2020 6:16 am because there's nothing to be said about them! They are axiomatic. They are bizarre things that give you the correct answer.
You use "axiomatic" in a funny way that makes no sense to me.
Skepdick wrote: Wed Sep 02, 2020 6:16 am That depends on your subjective "sameness" and "difference" functions/heuristics!