One line, as composed of infinite lines, being greater than or less than another line necessitates one infinity being greater than another thus one set of lines being infinitely greater than another set of numbers.Skepdick wrote: ↑Sun Aug 23, 2020 1:54 pmThe ideas off "infinitely smaller" or "infinitely larger" would've sounded absurd to Archimedes too...

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

## Continuum

### Re: Continuum

### Re: Continuum

There may be a deeper issue. Weyl suggests not just that the numbers are unreal but that extension is not real. He endorses the Perennial view for which time and space are fictional.wtf wrote: ↑Mon Aug 24, 2020 10:21 pm Agreed. I never make any claims about the reality of the real numbers. On the contrary, they're a highly abstract technical construction. They're useful as heck for the physical sciences, but I doubt they are instantiated in the physical world. I don't claim they're any kind of true continuum, only that they are taken as the mathematical continuum.

The basic issue may be that a continuum has no parts. In this case there can be no such thing as an extended continuum. All there can be is the appearance of one. Wey'l's 'intuitive' continuum, which is the 'empirical' continuum, is not extended. We do not experience extension but create it as a theory. as Kant surmised.

### Re: Continuum

There is no difference between the two if you subscribe to Brouwer's intuitionism (which Weyl initially did, till he grew disillusioned with it).

Brouwer rejected infinities until they are constructed, so what do you construct infinities from?

And points are parts, which is the confusion of Mathematicians to think they are ever speaking about continuums.

Computer scientists have asked this question: Do we ever really talk about the continuum, or do we only ever talk about finite sequences of symbols that talk about the continuum?

Off-by-1 error in the tower of abstraction....

### Re: Continuum

By shrinking I mean to take the limit.

Yes, I understand what a function is.

### Re: Continuum

The same kind of crazy emerges with 1:1 corresponding between points on concentric circles.

They are just the "infinitely-many" cross-sections of a 3D cone.

Of course, points themselves are as conceptually problematic as infinities. What's the volume/diameter/circumference of a point?

Last edited by Skepdick on Tue Aug 25, 2020 1:41 pm, edited 1 time in total.

### Re: Continuum

I'd say the the diffence is independent of our subscriptions. They are simply not the same thing.

Notional parts of a notional object, yes, but not real things.And points are parts, which is the confusion of Mathematicians to think they are ever speaking about continuums.

### Re: Continuum

That's a trivial truism. No two things can ever be "the same" thing if they differ in location.

Thats the axiomatic assumption of Non-reflexive (Schrödinger logics) - it rejects the principle of identity in favour of indistinguishability.

If you are claiming a difference, you should distinguish it.

It's the upgrade of Liebnitz' law from x = y := ∀F(F(x) ⇔ F(y)) to the quantum version <Pψ l Q I Pψ> = <ψ l Q I ψ>

Irrespective of their "realness" (as you said - senses cannot determine existence) to conceptualise "parts" is to conceptualise discreteness.

Conceptual discreteness conceptually contradicts a conceptual continuum.

### Re: Continuum

Location, shape, colour, definition, time, etc. I don't understand why you're making the issue complicated. The continuum of experience is not the continuum of mathematics, and realising this would be vital for an understanding of metaphysics.

### Re: Continuum

I am talking about "the continuum" of Mathematics.

It is incoherent IF it has discrete parts. Like points.

If you don't understand that - you don't understand metaphysics.

### Re: Continuum

There have been several interesting posts, I'll try to catch up.PeteJ wrote: ↑Tue Aug 25, 2020 12:47 pm

There may be a deeper issue. Weyl suggests not just that the numbers are unreal but that extension is not real. He endorses the Perennial view for which time and space are fictional.

The basic issue may be that a continuum has no parts. In this case there can be no such thing as an extended continuum. All there can be is the appearance of one. Wey'l's 'intuitive' continuum, which is the 'empirical' continuum, is not extended. We do not experience extension but create it as a theory. as Kant surmised.

I'm familiar (generally, but not an expert) in the efforts by Weil and Brouwer to try to make some philosophical points that the standard real numbers aren't the right model. I've studied a bit of this -- more than a bit compared to the average person, but still no expert. I've grokked a bit of constructive math, read a few papers. I'm also up on what I call the neo-intuitionists, the modern computer science-influenced mathematicians who only believe in what they can compute. That accords well with the old ideas of Brouwer, without, I will say, the mysticism of Brouwer. That aspect of his work does not appeal to me. The "free choice agent" or whatever, some mystical spirit constantly at work choosing the next digit of a never ending decimal expansion. I mention this so you know that I have

*done my homework*on this subject.

But in the end,

**it makes no difference to me**. It never did. Standard math is based on ZFC set theory, which itself is exactly the study of the collection of sets you get when you start with the empty set, and build up all other possible sets using the rules of ZFC. That's all it is. It doesn't mean anything, it's not about anything in the real world. That is a given. If one wants to be Platonic and claim the empty set is sitting out there in mystic land, right next to the Flying Spaghetti Monster and the Baby Jesus, be my guest. But I myself am not making those claims.

By coincidence, in connection with a thread on another forum, I recently happened to look at my copy of Kunen's book on set theory, first edition, and he says, plain as day, on page 94, "... 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."

That's it from the horse's mouth, a prominent set theorist and the author of one of the standard graduate texts on the subject.

*Set theory is not about anything*. And neither, from a formal perspective, is the rest of math that's built on it. If it's useful to the physicists all the better, since they'll help us get funding. But math does not look beyond itself for its motivations. History bears this out. In my opinion you are putting way too much of an ontological burden on the real numbers. Most people who think about the question for long, soon come to see that the entirety of mathematics is fiction. A beautiful fiction. I'm not just making this idea up, it's a thing.

https://plato.stanford.edu/entries/fict ... thematics/

In short I'm interested in standard math as a particular logical system, as a formal game. If you think it's "the wrong model," for all I know you could be right. Set theory is still of interest for its own sake. So your point is lost on me totally. I know people want the real numbers to "mean" something. I personally don't happen to think they mean anything. I hope this is clear. I take no more interest in whether math is true, or the right model of the physical continuum, or whatever; than I do in whether the knight in chess "really" moves that way. It's a nonsense question. Math is a formal game; and even if you don't believe that, you can still gain insight by viewing it that way. Even if you don't agree you do have to respect the point of view.

Last edited by wtf on Wed Aug 26, 2020 7:16 am, edited 1 time in total.

### Re: Continuum

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 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.

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.

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 conceptualize infinity with finite strings of symbols.

Our power of abstraction is a feature, not a bug.

### Re: Continuum

Then at any stage the shrinking interval has a length greater than zero. Any two intervals of real numbers with nonzero length can have their points put into one-to-one correspondence.

There is a one-to-one correspondence of the points in a tiny tiny interval and a huge huge interval of real numbers.

Another way to see it is to just look at the two intervals on the number line. You can slide both intervals to the origin and then scale the shorter one to the longer. You could use analytic geometry or linear algebra to work out the exact bijection that does the trick.

Earlier you expressed surprise and doubt that we can match up the points on a short line to one on a longer line in one-to-one correspondence. I take it that now you agree that this is true?

### Re: Continuum

But I do understand this.I wonder why you think otherwise. The continuum of mathematics is paradoxical when taken as a real thing and this is not news.

### Re: Continuum

Probably more familiar than myself. I can agree with Weyl but know little of Brouwer. Thanks for explaining your approach so well.

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.

Thus for metaphysics we need to take account of both the arithmetical and intuitive continuum and investigate their relationship.

Does this seem to sum up the situation?

### Re: Continuum

Then you are clearly not understanding it. Leave "reality" out of it.

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.