## Status of Cantor's 'Continuum Hypothesis'

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 7th, 2015, 2:27 pm wrote:Sigh. I keep telling people what I a genious I am - I even have a web blog with my IQ test scores which I've validated against two Quizilla tests - but no one listens, all 'cause I don't no do good in exams. Exams which, mind you, I'm quite certain are rigged to reinforce the liberal establishment because they're too afraid of my ideas. Personally I think that the Illuminati is involved.

LOL I totally agree with you! Most of what passes for our collective reality is an illusion driven by people with agendas. This I do believe. If we can't agree on Searle, can we at least agree on Chomsky? :-)

I get that you're smart. I'm trying to guide you to a modern understanding of abstract mathematics, so that you can use that information to support your philosophical ideas. At the very least, be able to speak the same language as everyone else.

At times I feel that you are in danger of falling into math crankery. One of my hobbies over the years has been the study of math crankery. I recognize the symptoms. The circle-squarers, the angle trisectors. And you know what? They're always engineers! I see this all the time. There is something about being extremely hands-on with numbers that makes it difficult for people to grok the abstract mathematical view. They are always the most brilliantly creative people, usually incredibly interesting people; but they just don't get math. And they're always brilliant engineers.

I am here to save you from that fate. I am trying to save you from becoming a math crank. I want you to consider learning a little mainstream abstract math, so that you can better achieve your own intellectual goals.

So you see how crazy I am. I'm trying to save your mathematical soul!

Natural ChemE » November 7th, 2015, 1:47 am wrote:Seriously though, is it truly classical math you love, or the beauty that emerges from what you can do with it?

I have zero interest in applications. I'm aware of the historical interplay between math, physics, biology, economics, and computer science. But when I see three rocks, I see the three, not the rocks. I've always had an abstract mindset.

There's a famous British mathematician of the 1920's named G.H. Hardy who wrote a book called A Mathematician's Apology; in which he states very clearly that his work is absolutely useless; and that uselessness is to be regarded as a supreme virtue in a piece of mathematics. And therefore, he needs to offer an Apology, in the sense of a justification for why he does his work. The kicker is that in the past thirty years Hardy's speciality, number theory, has become the foundation of public key cryptography, hence Internet security. I always wonder what he'd say if he found out that his beautifully useless subject had suddenly become useful. https://en.wikipedia.org/wiki/A_Mathema ... 7s_Apology

Natural ChemE » November 7th, 2015, 1:47 am wrote:
An honest question. I mean, personally, I used to say that I love math. However I had no real attachment to its particular form - I see programming, etc., as the same thing, just cooler. Is it different to you?

I studied math but earned my living as a programmer. I always regarded programming as easy. In math, if you're stuck, all you've got is pencil and paper. You have to fight through the limitations of your own mind. In programming you can always just type something. It's the dynamic aspect of typing and watching it run, over and over. Debugging. I really enjoy the mechanics and daily work of programming very much. I was born for it. But my soul is aligned with math in some way.

Natural ChemE » November 7th, 2015, 1:47 am wrote:
I guess that what I'm trying to say is that, as far as I can tell, this is math.

Right. And I'm trying to help you sort out what's important. If you are an expert in engineering math and an authority on numerical computation; and you are surfing Wiki pages on Lowenheim-Skolem and the hyperreals; I'm trying to put this stuff into historical and conceptual context for you. You can drive yourself nuts with that stuff if you don't have a context for it.

If you're interested in how math relates to the universe and physics and computation, I'm trying to point you in the right direction. And you keep wanting to go off into conceptual backwaters out of what ... opposition? Thinking you're smarter than all the professional mathematicians? You probably are! But they still know a lot more math than you. All I'm saying is learn the basic stuff before you try to improve on it.

Natural ChemE » November 7th, 2015, 1:47 am wrote:hyperreals are business-as-usual for me

Can you explain (clearly please!) what you mean by that? I know the hyperreals as a somewhat obscure construction in mathematical logic that allows you to have a model of the reals that contains infinitesimals. In which .999... = 1 is still a theorem, by the way, but let's not drive off that bridge right now.

Can you say what you are doing with infinitesimals in a practical sense? I'd be fascinated to know that there's an application. Now I do know that physicists and engineers like to think of derivatives in terms of infinitesimals. That I know about. But using actual nonstandard numbers in some application, I'd like to know about this.

Natural ChemE » November 7th, 2015, 1:47 am wrote: At least what math is now. I'm not blind to the fact that it's not what math was a hundred years ago, but.. isn't the fact that we've moved on really cool?

Yes, since 1840 we know that physics doesn't constrain math!! :-)
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

I'm super curious about why you think that physics doesn't constrain math. Like, I get that you can conduct an operation that depends on teleportation by constructing an abstraction in which teleportation is possible - e.g. a simulation - but you still fundamentally can't use teleportation in the underlying universe. Is your point that math can build up abstractions like a simulation that allows teleportation, or that math fundamentally can be based on non-physical operations, e.g. teleportation in the real world?

As for practical uses of hyperreals, first I'd suggest seeing them like I'd explained them in this reply. Note that the $N$ in that reply is the "hyperinteger rank" or whatever.
Note: I tend to use stuff like "super-big" rather than "hyper" because I think it's more descriptive and less intimidating to general readers.
Anyway, the sciences use infinity all over the place. It's basically our way of saying, "Naw, man, don't worry about how big this value is - its super-bigness is beyond compare." Or if you don't like "super-bigness", we can call it "hyperness" or "cardinality" or whatever - ultimately it's all the same thing.

For example:
Natural ChemE » October 15th, 2015, 1:53 pm wrote:Since both the star and obstruction are very far away (in this case 1500 light years), it's almost like the obstruction was right on top of the star. Technically there's some gap between the star and its obstruction, so the obstruction can be slightly smaller while still blocking all light.
Had we written up the trignometric equations for this situation, we'd have had an isosceles triangle representing the perspective cone for someone looking out at the star system in question. The distance to that star system, $1500{\text{ly}}$, would've been called "infinity" because I'd asserted that it was so far away that the distance was effectively infinite. However I also noted that, technically, the obstruction was closer than the star.

Using elementary math, our equations would've found that the distance between the star and its obstruction is zero - because both were at the same infinity away. Or, if we didn't assert that it was the same infinity, then it'd have been a classic ${\infty}-{\infty}=???$ situation.

Using hyperreals, our numeric system would've maintained there being an infinitesimal distance between the star and its obstruction. Exactly how we described this infinitesimal distance would've been up to our intentions, but the main benefit would've been that we wouldn't come to absurd results like the distance between the star and planet being literally zero.

When humans are doing all the math, we get the weak ontology effect where we're relying on human thinkers to implicitly keep track of where the deviations from idealized infinity are. Moving to computers requires us to be more explicit, i.e. move to a stronger ontology, so modeling real world systems with infinity approximations also requires explicit infinitesimals, or we start arriving at absurd, inconsistent results.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1 » 05 Nov 2015 01:45 pm wrote:
BioWizard » November 2nd, 2015, 7:51 pm wrote:Also, what's with the relentless attitude?

@NatChemE keeps coming up with this stuff, I'm only responding. I'll admit that I may have lost my perspective when the subject turned to Cantor denialism and outright .999... crankery. These misunderstandings have been thoroughly debunked on the Internet and I'm going to push back hard on them. Especially as @NatChemE seeks to be taken seriously. He probably appreciates it when I call out ideas of his that will be roundly mocked online. It's not 1995 on Usenet anymore, which was the golden age of online math crankery. Cantor denialism and .999... crankery are soundly debunked all over the Internet. @NatChemE would want to know that.

BioWizard » November 2nd, 2015, 7:51 pm wrote: Would you please mind dropping it?

When @NatChemE makes an assertion commonly recognized as mathematical crankery, I will push back. I am sorry if you are offended or upset by anything I say. Frankly if you would simply go back and read and understand the beautiful proof of Cantor's theorem, you would be delighted and enlightened.

You misunderstood. I never said drop the argument. I said drop the attitude with which you were charging almost every point in your posts (I don't know if you're still doing it, I stopped reading).

some1guy wrote:
BioWizard » November 2nd, 2015, 7:51 pm wrote:I don't typically enjoy math discussions, and it's starting to ruin my experience ;]

Well ... if you don't typically enjoy math discussions, you probably wouldn't have enjoyed this one even if I'd never posted a word, right? :-)

Wrong.

some1guy wrote:Seriously, I apologize if something I said offended you.

Nothing you said offended me.

some1guy wrote:It's just that I'm not exactly sure what you're referring to.

Constantly telling us how you feel so dismayed, punked, how NCE is a crank and you're wasting your time on his crankery, how you can't believe the intellectual depths he has sunken to, etc etc. I guess once or twice is OK. Saying it repetitively distracts from your argument - particularly when the other party is not instigating that kind of reaction. Maybe it wasn't attitude, maybe it was genuine expression of an emotional state. I suspect that dialing it down in this kind of discussion would be entirely to your advantage either way.

some1guy wrote:@NatChemE sometimes says outlandish things that are known to be false. Yes I will push back on those.

Please do. I would be disappointed if you don't.

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### Re: Status of Cantor's 'Continuum Hypothesis'

I'm short of time today so I'll just hit the main points. I haven't read much of @NatChemE's post yet. More tomorrow.

Natural ChemE » November 7th, 2015, 9:03 pm wrote:someguy1,

I'm super curious about why you think that physics doesn't constrain math.

I'm mindful of BioWizard's excellent advice that I tone it down a bit. So I am saying this as tactfully and non-judgmentally as I can. I've already explained this forty freakin' times!

I hope you know I'm laughing while I wrote that. In fact ... it's awesome that you asked the question!

I haven't time today to compose a detailed response; but I'd respectfully ask that you go back and read what I've written going back to my very first post (after all that stuff you copied over).

I've laid out a case. The key points are Wookies and non-Euclidean geometry. I don't understand why the meaning of my story about non-Euclidean geometry isn't clear to you. How can I say it better?

ps -- I just thought of something. If I said math is not constrained by contemporary physics would that sit better with you? Because in the end, non-Euclidean geometry IS physics. It just wasn't physics in 1850. Is that helpful? If so my point would still stand. How do you know that noncomputable numbers aren't telling us something about NEXT century's physics?

BioWizard » November 8th, 2015, 6:57 am wrote:
Constantly telling us how you feel so dismayed, punked, how NCE is a crank and you're wasting your time on his crankery, how you can't believe the intellectual depths he has sunken to, etc etc. I guess once or twice is OK. Saying it repetitively distracts from your argument - particularly when the other party is not instigating that kind of reaction. Maybe it wasn't attitude, maybe it was genuine expression of an emotional state. I suspect that dialing it down in this kind of discussion would be entirely to your advantage either way.

Yes thank you for pointing that out so clearly. I hope my last couple of posts show that I've processed out my thoughts and figured out why NatChemE's posts are hitting my buttons. As a longtime crankologist I felt -- and feel -- that NatChemE is right on the verge. That he can be saved and brought back to conventional mathematics; which, in the end, will further his own philosophical aims.

Now that I understand the source of my discomfort, I'm not annoyed anymore. I'm on a mission. And yes I realize how futile that is. No one in history has ever changed anyone's opinion about anything online. Credit me that much self-awareness.

But yes I am on a mission. I am on a mission to persuade NatChemE that using the concepts and terminology of mainstream modern math will benefit his own philosophical and scientific aims.

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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 7th, 2015, 9:03 pm wrote:
I'm super curious about why you think that physics doesn't constrain math.

I am aware that Wookies don't exist. Yet I enjoy watching movies about Wookies, like Wookie Wars, The Wookie Strikes Back, and Luke: I am your Wookie. What makes you think math is constrained by historically contingent ideas about physics? In 1840, didn't educated people believe that non-Euclidean geometry was logically impossible?

Natural ChemE » November 7th, 2015, 9:03 pm wrote:Like, I get that you can conduct an operation that depends on teleportation by constructing an abstraction in which teleportation is possible - e.g. a simulation - but you still fundamentally can't use teleportation in the underlying universe. Is your point that math can build up abstractions like a simulation that allows teleportation, or that math fundamentally can be based on non-physical operations, e.g. teleportation in the real world?
]

No. I'm not sure what teleportation has to do with this. A fly always ends up getting into the teleportation chamber. But now that you mention it, I don't believe teleportation is physically possible using conventional modern physics (can't say about the future); but I do enjoy watching movies about teleportation. Perhaps you too enjoy watching movies about things that could not happen in the real world.

Natural ChemE » November 7th, 2015, 9:03 pm wrote:As for practical uses of hyperreals, first I'd suggest seeing them like I'd explained them in this reply. Note that the $N$ in that reply is the "hyperinteger rank" or whatever.
Note: I tend to use stuff like "super-big" rather than "hyper" because I think it's more descriptive and less intimidating to general readers.
Anyway, the sciences use infinity all over the place. It's basically our way of saying, "Naw, man, don't worry about how big this value is - its super-bigness is beyond compare." Or if you don't like "super-bigness", we can call it "hyperness" or "cardinality" or whatever - ultimately it's all the same thing.

Are you saying that you actually don't understand what the hyperreals are, and how they differ from the standard reals? The hyperreals are a technical construction in mathematical logic. No new theorems are possible to prove in the hypperreals that are not already theorems in the standard reals. .999... = 1 is a theorem in the hypperreals.

Are you saying (I think you actually did just say this) that you don't understand the distinction between the idea of cardinality, and the construction of the hyperreals? You know, I'd be very glad to explain it to you if you were curious.

The hyperreals have been suggested as offering better pedagogy to calculus students. A calculus text based on nonstandard analysis was published in the 1970's. The idea has seen very little adoption in the past 40 years.

Natural ChemE » November 7th, 2015, 9:03 pm wrote:For example: Since both the star and obstruction are very far away (in this case 1500 light years),

Are you saying that you don't know that all physical measurement is approximate; and that the subject we are talking about here has nothing at all to do with measuring physical quantities?

Natural ChemE » November 7th, 2015, 9:03 pm wrote:

Using hyperreals, our numeric system would've maintained there being an infinitesimal distance between the star and its obstruction. Exactly how we described this infinitesimal distance would've been up to our intentions, but the main benefit would've been that we wouldn't come to absurd results like the distance between the star and planet being literally zero.

Are you saying that you think the hyperreals have something to do with physical measurement?

Natural ChemE » November 7th, 2015, 9:03 pm wrote:When humans are doing all the math, we get the weak ontology effect where we're relying on human thinkers to implicitly keep track of where the deviations from idealized infinity are. Moving to computers requires us to be more explicit, i.e. move to a stronger ontology, so modeling real world systems with infinity approximations also requires explicit infinitesimals, or we start arriving at absurd, inconsistent results.

Perhaps you can give a specific example from the scientific literature of the use of hyperreals or nonstandard analysis in any branch of the physical sciences.

I'm wondering if perhaps you are failing to distinguish among three different things:

* Physical measurement in the real world, which is always approximate and always subject to the physical limitations of our experimental apparatus;

* The modern theory of the real numbers, in which there are infinite sets, but no infinitely small quantities. In this theory, the idea of the infinitesimal has been replaced by the idea of the limit. Limits do not depend at all on "infinitely small" quantities. Nobody claims that the real numbers have physical existence. They're used to develop abstract physical theories; but the experiments are done with finite, imprecise measurements;

* Various constructions that contain infinitesimals; that is, quantities that are positive yet smaller than 1/n for any positive integer n; that are not used in any physical theory that I am aware of.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1 » November 9th, 2015, 2:05 pm wrote:.999... = 1 is a theorem in the hypperreals.

As previously linked on Wikipedia, this isn't true. Would another website or a journal article help, or do you mean something else?
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 1:12 pm wrote:
someguy1 » November 9th, 2015, 2:05 pm wrote:.999... = 1 is a theorem in the hypperreals.

As previously linked on Wikipedia, this isn't true. Would another website or a journal article help, or do you mean something else?

It's an irrelevant point. It doesn't help your philosophy and it certainly doesn't help us understand any math.

I don't concede that the Wiki author knows what he's talking about, since earlier he acknowledges that the transfer principle applies. If that is so, then .999... = 1 is a theorem.

However, the hyperreals are entirely irrelevant to our original discussion and of no interest to me. I'm more concerned that your earlier post about stars showed that you don't understand the distinctions among physical science, mathematical abstraction, and obscure, alternative, technical models of the real numbers that are not capable of proving any theorems that the standard reals can't already prove.

I would add that you originally invited me to this thread to discuss math. If you'd said that you are only interested in arguing about nonstandard models of the reals, I would have had no interest in the conversation. I am not sure why you think this is an important topic. It's simply not important nor does it shed any light on the philosophical topics of interest.

Finally, please note that on page 5 of the journal article you linked, it's clearly stated NOT that .999... < 1, since that is false even in the hyperreals; but rather that .999... with an underbrace under the 9's is less than 1. That's a DIFFERENT MATHEMATICAL OBJECT than the standard real .999... In fact .999... = 1 is a theorem in the hyperreals. You are Wiki-surfing without understanding, and thereby confusing yourself. I've pointed that out before. Go read page 5 again carefully and you will see that it does not say what you think it does.

Go ahead and write down equation 4.1 on page 5 exactly as written and you will see that you are missing the significance of the underbrace. The statement as written is

$\underbrace{.999...} \ < \ 1$

If you would take the trouble to understand what the paper is actually saying, you would realize that the underbrace changes the meaning of the symbol ".999..." in such a way as to make the inequality work out. But since the transfer principle applies to the hyperreals, .999... = 1 (without the underbrace) is a theorem even in the hyperreals. Every first order theorem of the standard reals is a theorem of the hyperreals. That's the transfer principle.

Again: THIS DOESN'T MATTER. It has nothing to do with the actual issues of interest here. But even on the facts, you are Wiki-surfing without taking the time and trouble to understand the technical math, which does not say what you claim it does.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

See the later notes on removing the underbrace, starting in Section 7 on Page 8.

Please forget about the transfer principle for the moment; it's throwing you off.

Also please note the end of Section 14 on page 19, "or to acknowledge the ambiguity of an ellipsis". This is immediately relevant to my earlier point about $0.{\bar{9}}{\neq}0.{\bar{9}}$ not necessarily being true.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 2:24 pm wrote:someguy1,

See the later notes on removing the underbrace, starting in Section 7 on Page 8.

Please forget about the transfer principle for the moment; it's throwing you off.

Please forget about .999... It's throwing you off. What does it have to do with anything you actually care about? It has nothing to do with anything I care about. So this has been a one-sided conversation for quite some time now. I can't see the relevance of nonstandard models of the reals to anything we've been talking about. And surely you don't think the hyperreals (constructed using a weak form of the Axiom of Choice) have anything to do with measuring stars.

The transfer principle is why the nonstandard reals have never caught on. The transfer principle entails that the nonstandard reals add nothing of interest to mathematics.

Natural ChemE » November 9th, 2015, 2:24 pm wrote:Also please note the end of Section 14 on page 19, "or to acknowledge the ambiguity of an ellipsis". This is immediately relevant to my earlier point about $0.{\bar{9}}{\neq}0.{\bar{9}}$ not necessarily being true.

A thing is equal to itself. There can be no rational discussion with someone who denies that. Of course it's certainly possible to interpret the symbolic string ".999..." in different ways. But if you use them in the same equation, they must have the same interpretation. Else you can't get rationality itself off the ground.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

The $0.{\bar{9}}$ thing is the most trivial, simplistic example of this topic that I could think of.

The stars thing is gonna have to come after you get the introductory points. Apparently I was skipping ahead.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1 » November 9th, 2015, 3:26 pm wrote:
Natural ChemE » November 9th, 2015, 2:24 pm wrote:Also please note the end of Section 14 on page 19, "or to acknowledge the ambiguity of an ellipsis". This is immediately relevant to my earlier point about $0.{\bar{9}}{\neq}0.{\bar{9}}$ not necessarily being true.

A thing is equal to itself. There can be no rational discussion with someone who denies that. Of course it's certainly possible to interpret the symbolic string ".999..." in different ways. But if you use them in the same equation, they must have the same interpretation. Else you can't get rationality itself off the ground.

So ${\infty}={\infty}$ and $\frac{{\infty}}{{\infty}}=1$?
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 2:35 pm wrote:So ${\infty}={\infty}$ and $\frac{{\infty}}{{\infty}}=1$?

I don't know, you'd have to tell me precisely how you're interpreting your symbols. In the extended real numbers your first equation is true and the LHS of your second is undefined. I discussed the extended reals in my lengthy response to you in the original Infinity thread. In the standard (nonextended) reals, both the LHS and the RHS of your first equation are undefined. On the Riemann sphere, your first equation is again true. In order to determine the truth value of a statement you have to specify the domain of interpretation, right?

Natural ChemE » November 9th, 2015, 2:31 pm wrote:someguy1,

The $0.{\bar{9}}$ thing is the most trivial, simplistic example of this topic that I could think of.

The stars thing is gonna have to come after you get the introductory points. Apparently I was skipping ahead.

I have nothing more to add. I truly did not come here to discuss the nonstandard reals or argue about .999... Perhaps you can explain why this topic is relevant to your overarching philosophy. Or why you think that interpreting the same symbol in two different ways on the two sides of an equation is going to hold much water with any reader. I can deny that x = x if x on the left side is 47 and x on the right side is a bowl of jello. Who is going to take me seriously if I try to pass that off as a debating point?
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1 » November 9th, 2015, 3:36 pm wrote:In order to determine the truth value of a statement you have to specify the domain of interpretation, right?

Yup, there needs to be a clear context. Context needs to be explicitly stated when implicit inferences can lead to ambiguous interpretations. When there's a strict generalization available, it's the implicit context because there's no ambiguity by shifting into it.

For example, $\frac{3}{2}=1$ in integer arithmetic, though in general $\frac{3}{2}=1.5$ is a better result. We needn't worry about ambiguity because integer arithmetic is a strict subset and thus not the implicit context unless specified as such in some way.

In general, ${\infty}{=}{\infty}$ isn't necessarily true because it's not in the general context. We can reduce scope to more limited contexts, e.g. on your Riemann sphere, in order to provide further definition.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 2:48 pm wrote:
someguy1 » November 9th, 2015, 3:36 pm wrote:In order to determine the truth value of a statement you have to specify the domain of interpretation, right?
Often, definitely. However when there's a strict generalization available, it's the implicit context.

For example, $\frac{3}{2}=1$ in integer arithmetic, though in general $\frac{3}{2}=1.5$ is a better result. We needn't worry about ambiguity because integer arithmetic is a strict subset and thus not the implicit context unless specified as such in some way.

What does this have to do with anything? You are trying to argue that you can sensibly interpret the same symbol in different ways in a single equation? Can you please step back and try to put this line of argument into a larger context for me?

ps -- Integer arithmetic is a "strict subset" of real number arithmetic? Of course that's not true. You just showed the counterexample!
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

This line of logic's mostly about indeterminate forms, e.g. $\frac{{\infty}}{{\infty}}$ is indeterminate in general because ${\infty}{=}{\infty}$ isn't necessarily true. However in more limited contexts in which ${\infty}={\infty}$, then $\frac{{\infty}}{{\infty}}$ is determinable and evaluates to $1$.

The relevance - like why I brought this up in the first place - is that stuff like hyperreal systems are a first step into strict generalizations of elementary math. I was trying to talk about general perspectives because they provide context for their subsets, e.g. elementary math, making them an excellent explanatory mechanism.

Also, for integer arithmetic, I probably didn't pick the best example there. It may be misleading since division maps to a floor function (obvious corrections for negatives). Non-trivial mapping probably isn't a good thing to talk about right now; I just picked it because I thought that it showing a visually different result would help reinforce the point.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 3:00 pm wrote:someguy1,

This line of logic's mostly about indeterminate forms, e.g. $\frac{{\infty}}{{\infty}}$ is indeterminate in general because ${\infty}{=}{\infty}$ isn't necessarily true. However in more limited contexts in which ${\infty}={\infty}$, then $\frac{{\infty}}{{\infty}}$ is determinable and evaluates to $1$.

Name one single specific context in which that's true. In every context that I know about in which the symbol $\infty$ is defined, the quotient $\frac{{\infty}}{{\infty}}$ is undefined. But I'm always open to learning new things. In what context do you think that's true?

Natural ChemE » November 9th, 2015, 3:00 pm wrote:The relevance - like why I brought this up in the first place - is that stuff like hyperreal systems are a first step into strict generalizations of elementary math. I was trying to talk about general perspectives because they provide context for their subsets, e.g. elementary math, making them an excellent explanatory mechanism.

You haven't succeeded in convincing me that you understand enough basic math to "generalize" it.

That's not to say that you haven't got all this worked out in your head. Just that you haven't been able to communicate it to me. I'm not seeing any generalizations of anything here. Nor do I understand the relevance to your overall philosophy of mind, physics, computability theory, and math. I just don't see where you're going with this. You're free to work in the hyperreals. Like I say, a guy wrote a whole calculus textook based on it. Keisler I think. But what's this got to do with anything?

Natural ChemE » November 9th, 2015, 3:00 pm wrote:Also, for integer arithmetic, that's a misleading example

Natural ChemE » November 9th, 2015, 3:00 pm wrote: since division maps to a floor function (obvious corrections for negatives). Non-trivial mapping probably isn't a good thing to talk about right now.

Well your claim that integer arithmetic is a strict subset of real number arithmetic certainly "floored" me! :-)
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1 » November 9th, 2015, 4:07 pm wrote:In every context that I know about in which the symbol $\infty$ is defined, the quotient $\frac{{\infty}}{{\infty}}$ is undefined. But I'm always open to learning new things. In what context do you think that's true?

Two stars are infinitely far away from Earth in either direction. The ratio of the magnitudes of their distances is $1$.

This is to get back to the earlier example of hyperreals in the star system. Specifically, we can have different scopes of infinity.

Also, yeah, sorry about the floor-function mapping stuff. The mapping is on Wikipedia, and it's this mapping that makes integer arithmetic a strict subset. But the specifics of the mapping are tangential; I just wanted to use it as an example with a visibly different result, apparently failing to consider that non-trivial mappings would introduce more confusion than the visual difference would dissipate.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 3:11 pm wrote:
someguy1 » November 9th, 2015, 4:07 pm wrote:In every context that I know about in which the symbol $\infty$ is defined, the quotient $\frac{{\infty}}{{\infty}}$ is undefined. But I'm always open to learning new things. In what context do you think that's true?

Two stars are infinitely far away from Earth in either direction. The ratio of the magnitudes of their distances is $1$.

This is to get back to the earlier example of hyperreals in the star system. Specifically, we can have different scopes of infinity.

You have a link to some contemporary theory of physics that posits an infinite distance between some pair of objects within the physical universe? And that the hyperreals, depending as they do on the Axiom of Choice, has relevance to physical measurement?
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

Old response - I'd suggest skipping for now
Sure, I think that the most classic example's probably energy from Coulomb's law. A more modern example is how the quantum correspondence principle reduces to classical mechanics at infinity. Statistical mechanics is also a good example. Stokes-Einstein is a good example from Kinetics in which we consider differing levels of infinity, i.e. the boundaries are infinitely far away 'til we plug the calculated diffusion constant and then consider the at-infinity level in our fluid continuity equations, e.g. Navier–Stokes.

We basically use this type of logic whenever we derive something in the physical sciences.

New response
Wait, I'm being stupid.. the Dirac delta function is the true classic example:
.
Its definition struggles in elementary math, which is often used in class to help professors talk about how elementary math is silly.
Dirac delta function, Wikipedia wrote:This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.
While reals are insufficient, hyperreals do work.

To help understand the issue, it's that the Dirac delta function is infinitely dense but still finite under integration. This works because its treated consistently at the infinitesimal level, but writing this up in classical mathematical notation bites 'cause elementary math was stupid about infinitely small/big stuff.

For example, ${\delta}{\left(1\right)}$ has $99.{\bar{9}}$% of its area ${\in}{\left(0.{\bar{9}},2-0.{\bar{9}}\right)}$ - just to have fun with $0.{\bar{9}}$'s. But in elementary math, you just can't write this in any meaningful way, being why one of the founders of Quantum Mechanics described his own equations as merely being heuristic characterizations.

You can fix this problem by injecting a hyperinteger rank $H$ into the definition, just as we did in the $0.{\bar{9}}+{10}^{-H}=1$ bit, noting that I'm omitting the underbrace with the $H$ for the infinite string of $9$'s, per usual notation and as is consistent with the the-implicit-context-is-a-strict-generalization thing we discussed a bit ago. This allows for physical correspondence while also making the math work.

Or you can use an ensemble of hyperinteger ranks $H_i$ if there're various scopes of infinitesimals. For example, this can occur if the point particles are all infinite small in the current context, yet some of them are infinitely small in the finite scope of others. For a concrete example, we can say that molecules and electrons are both infinitely small at the human scale, however molecules are finite at the molecular scale while electrons remain effectively infinitesimal. This may motivate $\left{{H}_{\text{molecular}},{H}_{\text{electron}}\right}$.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

In general please let me know whenever I go off the deep end. I mean the stuff I'm writing is correct, but that doesn't amount to much if it's incomprehensible.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

Here's paper on the Dirac delta with hyperreals:
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 3:53 pm wrote:In general please let me know whenever I go off the deep end.

I always knew I'd find my role in life.
Last edited by someguy1 on November 9th, 2015, 6:48 pm, edited 1 time in total.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 3:57 pm wrote:someguy1,

Here's paper on the Dirac delta with hyperreals:

I will read that with great interest. I have no doubt it can be done. That isn't the point. You keep saying it's the point but it is not the point. In my opinion, of course. In your opinion, it's the point. Of what, I can't figure out.

ps -- Oh it's the same author as the other paper. This confirms my point. I agree that this is a logically sound approach. It's just a distinctly minority approach that is of high interest to a handful of specialists; and of no interest whatsoever to anyone else. As I say these ideas have been around for years, I think Hewitt invented the hyperreals in 1948. That would be what, 70 years ago almost. Then in the 1970's Keisler wrote his calculus text based on it. And now there's a small subculture of mathematicians re-doing a tiny subset of standard math in hyperreal symbology. There is nothing wrong with that, if they are happy I'm happy. It's just such a tremendous distraction that I don't understand your interest in it. You didn't link me to standard physics, you linked to the same guy that wrote the other paper. And Hewitt didn't think much of the hyperreals either. He invented them and then got back to his regular work, which was in standard analysis. You could look it up.
Last edited by someguy1 on November 9th, 2015, 6:54 pm, edited 2 times in total.
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 3:20 pm wrote:someguy1,

which is often used in class to help professors talk about how elementary math is silly.

Oh! I gotcha! You mean professors in physics and engineering classes said that. Yes, I believe you!! This is the age-old split between how mathematicians think of math, and how the physicists/engineers think of math. Einstein said that after he saw Minkowski's mathematical description of relativity, he (Einstein) no longer understood it!

That's a joke with a large grain of truth. I'm told that physicists routinely think of calculus in terms of infinitesimals, and use infinitesimals in their work all the time. Not mathematically rigorous infinitesimals. But rather "informal physics-major infinitesimals!"

When mathematicians rigorously formalize the infinitesimals used by physicists, the mathematicians use the standard real numbers to do so.

Perhaps you're reading too much into what some physics professors think of math. I'm sure Newton, Gauss, and Witten don't think math is silly. Einstein probably does! In the end, math and physics have a terribly mysterious interplay that we don't fully understand. Is that something we can agree on?

Thanks for the nice graphic. As it happens there's a perfectly sound mathematical definition of the Dirac delta function. It's a linear functional on the space of $C^\infty$ functions on the reals with compact support, defined by $\delta(\phi) = \phi(0)$. I'm pretty sure you know this because it's plainly described on Wiki. I won't bore you with the details but I'd be glad to walk through the meaning of the definition in detail if anyone cares.

https://en.wikipedia.org/wiki/Dirac_del ... stribution

It's perfectly natural for physicsists to informally think about infinities and for mathematicians to "clean up the mess" by grounding the physicists' tools in rigorous logic. That doesn't make math silly. It means physics is doing physics's job and math is doing math's job.

Another nice example is Fourier series. In math they define a Hilbert space as a complete inner product space; then they prove that under such and so circumstances there's a countable orthonormal basis; then you can define the Fourier coefficients and prove all the usual theorems. A special case is the real numbers with trigonometric series. So you get undergrad Fourier series and a good chunk of modern physics for free, just by defining abstract Hilbert space.

Another physics-y thing I once had an ah-ha moment about is the mysterious bra-ket notation. I always imagined that was some deep mystery I'd never understand in my lifetime. But then I found out it's just a linear functional acting on a vector, written with inner product notation. It's a standard math gadget, not mysterious at all.

In a lot of ways, one could study math and conclude that physics is trivial. It's just a special case, barely worth mentioning except in the footnotes!

Of course that wouldn't really be true; and it would be silly for someone to think that. Math is math, and physics is physics. There is a mysterious interplay between them. Math drives physics and physics drives math. Physics is about the world we live in. Math is about ... well, it's really not clear what math is about. Structure, I think. Math is the study of structure. Of course that's a very modern position. Modern math is very structural. Structure and relationship. If you want to entertain yourself read up on Category theory. That's structuralist math at its best. Or at its worst, depending on your opinion of structural mathematics. https://en.wikipedia.org/wiki/Category_theory

I really appreciate your telling me where you heard that math is silly. In physics class!! I'm sure your prof meant that as an in-joke and doesn't actually think math is silly. Or maybe he does. In any event it helps me to put your thoughts into perspective.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

Hah I'm glad that this works for ya. I agree with the focus on structuralism!

It's cool how category theory corresponds to computer type systems, right?
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### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 9th, 2015, 4:54 pm wrote:
Hah I'm glad that this works for ya. I agree with the focus on structuralism!

It's cool how category theory corresponds to computer type systems, right?

Yes, I don't know anything about type theory but I hear it's the hot stuff these days. I saw Category theory when I studied math, then I didn't hear about it for a long time, and now it's everywhere. CS, economics, logic, even physics.

I would like to sum up some points. Would you agree that

* Physics is about the structure of the actual universe -- the one we live in.

* Math is about structure in general. Therefore there are structures in math that may not necessarily be represented in the universe.

* And that therefore, math is not constrained by physics.

* Nevertheless, non-physical math often finds application in physics in the future. So we should keep an open mind; take math on its own terms; and try to understand what math's most counterintuitive theories are trying to tell us.

Is this a more clear statement of my thesis?
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### Re: Status of Cantor's 'Continuum Hypothesis'

excuse me both:
may I ask what difference is supposed to exist between an hyperreal (as I hope I have understood it to be) and a limit in classical maths?

Doesn't Dirac function represent the limit of 1/dx as dx-->0, a function whose area in dx is dx/dx, which in turn would be indefinite for dx=0 but exactly equals 1 in the limit for dx-->0 ?

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### Re: Status of Cantor's 'Continuum Hypothesis'

neuro » November 10th, 2015, 7:34 am wrote:excuse me both:
may I ask what difference is supposed to exist between an hyperreal (as I hope I have understood it to be) and a limit in classical maths?

Good question. Briefly:

* The hyperreals contain a copy of the standard real numbers and also contain infinitesimals, constructed in a logically rigorous way.

* The standard reals don't contain any infinitesimals at all. Limits are defined in terms of standard reals. There is no talk of infinitesimals in standard math. Even the phrase "infinitesimal calculus" is a historical misnomer, since the great achievement of modern calculus is to banish infinitesimals from math. This was an intellectual project that spanned over 200 years, from Newton to set theory. No more infinitesimals.

* Physicists use informal, nonrigorous infinitesimals as Newton did. Which is perfectly fine. Physicists study the universe. They value results over rigor. Mathematicians must be driven by rigor. Two sides of the same coin, perhaps, but with different areas of concern.

Here's a little more detail.

What do we mean by an infinitesimal? Wiki says it's something so small there's no way to measure it. This is a useless definition. A single point on the real number line has measure zero, but it's not an infinitesimal. The set of integers has measure zero in the real numbers, even though there are infinitely many of them. And there are nonmeasurable sets of reals, which can't be said to have any size at all. In fact when you say something is so "small" you can't measure it ... if you can't measure it, what does it mean to call it small? The Wiki definition is of historical and philosophical interest, but isn't precise enough to do math with.

In modern math a number $x$ is infinitesimal if:

* $x \ > \ 0$, and

* $\forall n \ \in \ \mathbb{N}, \ x \ < \ \frac{1}{n}$. That's a bit of notation which just says that $x$ is smaller than each of $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, etc.

A moment's thought shows that there are no infinitesimals in the real numbers. For example if say $x = \frac{1}{1000}$, clearly I can just pick $n = 2000$ and then $0 \ < \ \frac{1}{2000} \ < \ \frac{1}{1000}$, so that $\frac{1}{1000}$ isn't an infinitesimal after all. We can repeat the same idea for any real number you care to choose. So there are no infinitesimals in the reals.

In physics, evidently the rigorous math approach hasn't made it across the great math/physics divide; and the physicists still think in terms of non-rigorous infinitesimals from the days of Newton. But that's ok. Their job is to study the universe, not to worry about logical purity. In fact excessive logical purity is generally what the physicists accuse the mathematicians of. That's probably what NatChemE's professor meant.

This state of affairs -- mathematicians using rigorous limits that finesse and banish infinitesimals; and physicists using nonrigorous, casual 17th century infinitesimals; is pretty much how it is these days.

In 1948, a guy named Ed Hewitt realized that you could create a mathematical system that behaved exactly like the real numbers; and that also contained genuine infinitesimals constructed in a logically rigorous manner. These are the hyperreals. The intuition, I'm told, is that surrounding each real number is a little cloud of hyperreal numbers that don't add any magnitude to the number, but are nevertheless distinct from the number.

Now you would think that mathematicians would say hey, now that we have real, logically rigorous infinitesimals, why don't we just use them instead of limits? Instead, Hewitt didn't bother to follow up and went back to his other work. In 1978 Keisler wrote a calculus textbook. Even if working mathematians weren't using hyperreals, maybe students would find that approach easier. Now it's forty years later and Keisler is still the only hyperreal-based calculus textbook. It's just never caught on, not with research mathematicians and not with teaching. We don't know what the future holds, but today the hyperreals are a mathematical backwater.

There are some fundamental reasons the hyperreals haven't caught on.

* There is a "theorem about theorems" called the Transfer principle that says that anything you can prove about the hyperreals is already true about the standard reals. If mathematicians can't use it to prove new theorems, they aren't interested. (This applies only to first-order properties and I'm not familiar with what happens when you allow second-order statements. Perhaps that's why NatChemE says I'm reading too much into the Transfer principle. In any event, the hyperreals are now almost 70 years old and haven't attracted much interest.)

* The set-theoretic construction of the hyperreals requires a form of the Axiom of Choice. Therefore, the hyperreals are marginally less ontologically solid than the standard reals, which do not depend on Choice. Some philosophers and mathematicians care about this. Their point would be that the standard reals require fewer assumptions than the hyperreals do, and therefore the standard reals are to be preferred.

* We have an intuition about the reals that if two points are not the same point, then there is some nonzero distance between them. This is summed up in the Archimedean property. The standard reals are Archimedean, the hyperreals not.

To sum up:

* Physics uses informal, nonrigorous infinitesimals in the spirit of Newton.

* Math has abolished infinitesimals entirely, by insisting that the reals are Archimedean, and using set theory to build up the Archimedean reals without infinitesimals. Limits are defined in terms of the standard reals. Infinitesimals are neither needed nor allowed. This approach has been working well for a long time and nobody seems inclined to change it.

* There are a handful of systems that act like the reals and have infinitesimals. The hyperreals are one. The surreals are another. There may be others. These are all interesting systems but are regarded as curiosities in mainstream math. This is not to say people won't think differently in a hundred years. No way to know.

I mention in closing that I never actually said what a modern limit is, just that they're defined in terms of standard real numbers. For sake of exposition I did not want to descend into technicalities. This wiki page is halfway decent but frankly if anyone is interested, just ask and I'll write up a couple of paragraphs with epsilons and such. The Wiki article also mentions the definition of a limit in the hyperreals, which is based on taking the "standard part" of a hyperreal number to get from the little cloud around a real number to the real number itself.
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### Re: Status of Cantor's 'Continuum Hypothesis'

someguy1,

I'm curious about your outlook on what's interesting/used/etc. Basically I feel like you're dismissing the tools that we currently use in applied fields based on their obscurity in non-applied fields, which seems very strange to me.

Hypothetically, say that engineers were using mathematics effectively unreproducible in terms of the systems that you're talking about. Say that human progress is advanced by progress in these tools unknown to non-applied mathematicians. In this hypothetical case, would you be interested in such tools, or would their obscurity within non-applied fields continue to cause them to be disinteresting to you?
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### Re: Status of Cantor's 'Continuum Hypothesis'

ps -- It occurs to me that I can describe how modern math uses the real numbers to define limits without infinitesimals, without getting too technical.

Consider the sequence $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, ...

We'd like to say that it "goes to $0$" or as a limit of $0$.

The way that's defined is to note that the sequence gets arbitrarily close to $0$. So for example suppose you say, well, can it get within $\frac{1}{1000}$ of $0$? Yes, just go far enough out in the sequence, past $\frac{1}{1024}$, $\frac{1}{2048}$, $\frac{1}{4096}$, ...

So instead of having a vague notion of "infinitely close," we replace that with a precise definition of arbitrarily close.

In the hyperreals, we can actually make a precise definition of "infinitely close." In standard math, we replace that with "arbitrarily close."

That's the difference. I probably should have just written this and skipped the rest of it :-)
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