What unit of measurement is required for counting the natural numbers? Metres? Litres? Hours? Bananas? Obviously, no unit of measurement is required. You can count to ten without having to determine any unit of measurement. Therefore, counting is independent of measuring. Counting is not a "form of" measuring. — Luke

@Metaphysician Undercover, What he said.

What? — Banno

I repeated what you said. "He was obligated to do the best he could while working for a lunatic." That's TDS. That's the literal definition. Fauci has been a hack, a political apparatchik for forty years. Wrong on AIDS, wrong on SARS, wrong in the hearts of his countrymen. He only became a saint in the mind of the Trump-hating public because of exactly the the attitude that you expressed.

What's he supposed to have done? — Banno

I gave an example in this thread. Or was it a different thread? He said we should all wear two masks. Rand Paul accused him of doing it for show. Fauci said it was "science." Few weeks later, Fauci admitted it was for show, for "optics," his word. This has been going on a long time.

I could enumerate many other political flipflops in the name of "science." And not to mention that the funding for the gain-of-function research that (most likely) escaped from a lab and caused the covid pandemic has now been traced to none other than Fauci. It's like a movie where the kindly old authority figure turns out to be the arch villain. Give it some more time. Facts are getting out and public opinion is beginning to turn. Give this all another few months. I'm willing to wait. I can't explain all this to you now any more than I could have a year ago, except that today some of this information is beginning to leak into the MSM.

The first day I heard about the wet markets back in early 2020 was in an article that mentioned "Oh by the way there's a bioweapons lab a mile away." The plot twist was obvious to me that day and to many others, whose voices were suppressed. The suppression is starting to lift and the truth is going to be known. Give it time. And click around. Try not to get blindsided by the lies and propaganda.

↪jgill

I'm unsurprised that you dislike CT. — sime

@jgill does complex analysis, where CT has made few if any inroads. The theory of fields doesn't make for a good category, as I've heard it put, so the analysts haven't been categorified as the algebraists have.

Fauci - not sure I can see your point about his behaviour. He was obligated to do the best he could while working for a lunatic. He didn't walk away. — Banno

It's Trumps fault?

Trump drove a certain segment of the US population insane. — fishfry

To wit.

Then it's not science. — Banno

Yes ok I do take your point. I love science and hate scientism and fake, politicized science. So if I criticize the latter, I haven't actually said anything at all about the former, which is your point.

But if the word "science" has become equated in the public mind with fake, politicized science, isn't that a problem? When Fauci and Walensky tell us they're "following the science" as they lurch from one politicized lie to the next, it's cold comfort to think about Galileo and Maxwell. Which is my point, even as I understand yours.

I don't recall seeing any of that in Europe. — baker

Quite possibly not. Trump drove a certain segment of the US population insane.

But there are distinct China haters who've been promoting the idea that the Chinese made the virus and let it out. — baker

One need not be a "China hater" to hold that opinion. Although as it turns out, it was the Americans who paid for the Chinese research.

China hater = Trump lover? — baker

Or just someone who cares about the plight of the Uighurs. You know. Concentration camps, organ harvesting, and suchlike. Perhaps you've heard there's a move afoot to boycott the 2022 Chinese Olympics. Nancy Pelosi is one of the leading proponents. Not exactly a Trump lover. Care to revise your remark?

If I'm not mistaken, Von Neumann formalized without 'element' as primitive in 1925. — TonesInDeepFreeze

Would be most interested in a reference or more context.

I don't see much here with which one might engage. — Banno

Did you miss 2020? "Science" is a political word used to silence legitimate dissent and the actual scientific method. American science is now Lysenkoism.

Well, sure, you can dismiss all of them that way. Just put on your Trump hat. — baker

Well you know it's a funny thing about that. Last year when reputable scientists suggested that covid might have escaped from a lab, they were marginalized and called Trump-lovers. Now that Trump's gone, it's suddenly ok to follow the actual evidence and consider the very real possibility that covid escaped from a lab. So the anti-Trump feelings of many people prevented them from doing actual science.

For example:

Stories published by The Washington Post ("conspiracy theory that was already debunked"), New York Times ("fringe theory"), HuffPost ("debunked fringe theory"), and The Daily Beast ("conspiracy theory") aggressively disputed Cotton's hypothesis.

But New York Magazine and other outlets have given the theory increasing attention in 2021. If anything, the official account of animal spillover is no longer blindly accepted.

https://www.foxnews.com/media/ex-new-york-times-health-reporter-lab-leak-coronavirus-theory-looking-stronger

So if people's Trump hatred caused them to overlook serious scientific information, who are the anti-science ones?

And if you don't like the Fox News link, here's a similar story from the NYT. Something for everyone. Myself, I read everything, from the right wing whackos to the left wing whackos. If you only read one side, you miss a lot.

https://www.nytimes.com/2021/05/13/science/virus-origins-lab-leak-scientists.html

I'm seeking out those who disagree with this proposition: — Banno

I'll take up that challenge for just about any proposition you could name!

Science is a good thing, t — Banno

Science yes. But "science," or scientism, no. We've been told for a year to "believe the science" when in fact legitimate, distinguished professional scientists have been marginalized, deplatformed, cancelled, and fired for expressing opinions that the mainstream didn't like. For a year Fauci has been regarded as a saint; now it turns out that there's some evidence showing that he was the one who directed funding (indirectly, through shell organizations of course, for deniability) into the very gain-of-function research that may (I say may, no proof either way yet) have leaked out of a Chinese bioweapons lab.

In fact just to suggest that covid came out of a lab has been grounds for firing, deplatforming, cancelling, and marginalizing. Suddenly it's acceptable to express those thoughts.

So science, hell yes. I'm a big fan. But "science" as in a weapon to suppress legitimate dissenting opinion? That's bad. And lately we are seeing way too much of "science" and not nearly enough science.

See for example https://www.npr.org/2021/03/31/983156340/theory-that-covid-came-from-a-chinese-lab-takes-on-new-life-in-wake-of-who-repor . A year ago even a hint of a suggestion of this idea got reputable scientists fired. When science is politicized, it's not a good thing. It's a bad thing. And where exactly does one find non-politicized science these days when everything is politicized? Fauci says wear two masks, Rand Paul says it's just theater, two weeks later Fauci admits his own mask wearing is for "optics," proving Rand Paul correct. Then Fauci walks back his own statement again.

Rachel Walinsky, another politicized piece of work. A few weeks ago she gave tearful testimony about how "frightened" she was. Yesterday she's all, "You can take off your masks now." What changed? The science? Or is it that with an ongoing border crisis, a gas crisis, a brandy new war in the Middle East, and exploding inflation, the Biden administration just needed to change the subject? "Asking for a friend."

Politicized science is bad science and bad, period. And lately that's the only science we've got. I for one am tired of hearing, "Follow the science" whenever someone wants to shut someone else up.

I'm interested in finding out what happened to Bertrand Russell's Type Theory? — Shawn

Coming back as various forms of neo-intuitionism via the influence of computers and automated proof systems. See for example homotopy type theory.

But the two theories that underlie it, inflation and quantum physics, have been demonstrated to be valid. If they're right, then the multiverse is an inescapable consequence of that, and we're living in it."...Ethan Siegel — Aadee

The multiverse is a purely speculative idea, as one of its leading proponents Alan Guth admits. Especially eternal inflation, which posits that time never ends and is actually infinite. There's no possible way to verify that physically. It's a speculative mathematical theory of physics. You put this right next to QM, a theory that does have experimental verification, but they're very different.

You (or the person you quoted) are entirely wrong. QM has experimental verification but eternal inflation does not. The latter is entirely speculative. If you're talking about inflation without the eternal part, I'm not too well versed in the experimental aspects but my understanding is that it is also fairly speculative.

No God needed — tim wood

The problem with this theory is that that it's less plausible than the alternative. If we just happen to have woken up in this marvelously fine-tuned universe, you can say God did it or you can say we just got lucky. I see no way to rationally choose between one and the other. Anthropic arguments are like that. We're here because if we weren't we couldn't ask the question. That explains nothing; and to people of faith, it's perfectly clear that only God could have done it.

And how do you know God didn't create the multiverse? You can never use a scientific argument to disprove God, even a speculative argument like eternal inflation. And positing an actual infinity of time is more theology than science.

You have nothing to order, and no order to offer. — Metaphysician Undercover

LOL There's a famous off-color joke with a very similar punchline, but nevermind.

To say that the natural numbers have no inherent order, is to remove "order" as a defining feature of the natural numbers. — Metaphysician Undercover

I would use the word "abstract" rather than "remove." The point is that by abstracting the concept of order from any particular meaning, we can better study order. That's the entire point of my post, which, if it didn't help, didn't help. The point of abstraction is to take away meaning such as first base, second base, so that we can study first and second abstracted from meaning. That doesn't make abstraction meaningless, it just means that we use abstraction to study concrete things by abstracting away the concreteness.

Now we are left with quantity as the defining feature. Do you agree? — Metaphysician Undercover

Well, yes and no. Von Neumann's coding of the natural numbers has the feature that the cardinality of the number n is n. But there are other codings in which this isn't true, for example 0 = {}, 1 = {{}}, etc. So we can abstract away quantity too if we like. But that wasn't the point, Even if I grant you that cardinality provides a natural way of ordering the natural numbers, it's still not the only way.

Anyway I did my best and don't think I can add anything.

I do not deny human abstractions, I just insist that they are fundamentally distinct, different from objects. — Metaphysician Undercover

I get that you don't believe in mathematical "objects." What do you call them then? What do you call numbers, sets, topological spaces, and the like?

I do not want to deny the ideas, I want to understand them. And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference. — Metaphysician Undercover

Obviously the abstract mathematical object (oops there's that phrase again) 5 is derived or inspired by the familiar concept of 5. Sort of like Moby Dick, which is entirely fictional, was inspired by an actual whaleboat, the Essex, that was sunk by a whale.

But the 5 that mathematicians study is indeed an abstract object. It's not 5 oranges or 5 planets or 5 anything. It's just 5. That's mathematical abstraction. I guess I'm all out of explanations.

An abstraction must be intelligible or else it is meaningless, useless. If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible. — Metaphysician Undercover

There is no space or time in math. Why can't you accept abstraction? There's space and time in physics, an application of math. There's no space or time in math itself. Is this really a point I need to explain? What am I missing that would let me get through to you on this point? A physicist cares about 5 meters or 5 seconds. The mathematician only cares about 5. It's cardinality, its ordinality. It's role as a natural number, an integer, a rational, or a real number. Its primality. What is the thing on which you and I disagree regarding this?

And so the notion of elements isn't needed for the pure purpose of constructing abstract numbers. — sime

It's certainly interesting that one can do set theory without elements.

Or, just after Davis, Hancock, Carter and Williams laid down "Thisness" - an exceptionally gorgeous, introspective, haunting modern and abstract ballad - producer Teo Macero said to the group, "Your sandwiches are here." — TonesInDeepFreeze

Immediately before JFK was assassinated on November 22, 1963, Nellie Connolly, wife of Texas governor John Connolly, sitting in the seat in front of JFK in the presidential limo, turned around and said to JFK, "You can't say that Dallas doesn't love you, Mr. President."

You might think, that "the meaning is always clear from context", but if you go back and reread TIDF's discussion of counting a quantity, you'll see the equivocation with order.

I'm taking this from the end of your post and addressing it first to get it out of the way. As I mentioned, I didn't read any posts in this thread that didn't mention my handle. I only responded to one single sentence of yours to the effect that numbers are about quantity. I simply pointed out that there is another completely distinct use of numbers, namely order. Anything else going on in this thread I have no comment on.

All I saw in you demonstration was a spatial ordering of symbols. I really do not see how to derive a purely abstract order from this. If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it.

I assure you, I am very interested to see this demonstration, because I've been looking for such a thing for a long time, because it would justify a pure form of "a priori". Of course, I'll be very harsh in my criticism because I used to believe in the pure a priori years ago, but when such a believe could not ever be justified I've since changed my mind. To persuade me back, would require what I would apprehend your demonstration as a faultless proof. — Metaphysician Undercover

I may not be fully aware of the philosophical context of your use of "a priori." Do you mean mathematical abstraction? Because I am talking about, and you seem to be objecting to, the essentially abstract nature of math. The farmer has five cows but the mathematician only cares about the five. The referent of the quantity or order is unimportant. If you don't believe in abstraction at all (a theme of yours) then there's no hope. In elementary physics problems a vector has a length of 3 meters; but the exact same problem in calculus class presents the length as 3. There are no units in math other than with reference to the arbitrarily stipulated unit of 1. There aren't grams and meters and seconds. There's no time or space, just abstract numbers. I don't know how to say it better than that, and it's frustrating to me that you either pretend to not believe in mathematical abstraction, or really don't.

Looking ahead a bit I will try to explain abstract order after I respond to your other remarks.

There is an issue though, that I'll warn you of. Any such demonstration which you can make, will be an empirical demonstration, using symbols to represent the abstract. — Metaphysician Undercover

You can't get civilization off the ground without abstract symbolic reasoning, from language to math. Even music has a notation. But the notation is not the music, I hope you'll agree. Likewise the notational explanation of ordinals won't be the ordinals. I can show you the symbols but you have to hear the music on your own. That's the tricky bit, right?

This is a good point so I'll repeat it. You would not confuse music with its notation; so I hope you'll indulge my mathematical notation in the service of communicating abstract ideas. The ideas aren't the notation. We agree on that. You seem to want to deny the ideas themselves simply because they're abstract. That's the part of your viewpoint I don't understand.

So the onus will be on you, — Metaphysician Undercover

To explain to you the theory of ordinal numbers? I don't know if I can do that but I'll give it a shot.

to demonstrate how the proposed "purely abstract order" could exist without the use of the empirical symbols, — Metaphysician Undercover

No. That can't be right. You are correct that I believe that there is a notion of purely abstract order; and that I'm constrained to talk about it using symbols that are not the actual things being talked about. But you can't reject my presentation because of that. You can't deny the quest for justice just because it's an abstract idea that is not literally present in the string "justice." If you reject all human abstractions, there's no point in my starting, because you'll just say, "Oh that's only symbolic representation of things that don't exist." I can't overcome that level of nihiism.

or else to show that the empirical symbols could exist in some sort of order which is grounded or understood neither through temporal nor spatial ideas. — Metaphysician Undercover

Well, as it happens, Cantor discovered ordinal numbers when he was studying the zeros of Fourier's trigonometric series; which were an abstract mathematical model of heat distribution. That is, one can make the case that if you heat up one end of an iron bar under lab conditions, and carefully measure how the heat travels to the rest of the bar; you will inevitably discover the transfinite ordinals. There is physics behind abstract order theory. Nevertheless, the theory stands on its own as an expression of the notion of purely abstract order.

I'll tell you something else though, I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. — Metaphysician Undercover

I see your annoyance. I want to talk about first, second, third, but I don't want to relate them to first base, second base, or third base. I want to regard ordinals as pure ideas that can be arranged in purely abstract order. If you utterly reject that then we're done. All I can do is try to explain how mathematicians view the subject of abstract order. I can't convince you that such a thing exists. But I don't need to. I can fall back on formalism and say that even if it doesn't exist, it's a fun mental pastime, like chess. There's no physical referent for chess but it's fun and educational and some practitioners take it very seriously indeed. But it's not real. I'm sure I've made this analogy before.

However, this requires that I divorce myself from the conventional idea of time which sees time as derived from spatial change. Instead, we need to see time as required, necessary for spatial change, and this places the passing of time as prior to all spatial existence. — Metaphysician Undercover

There is no need for time or space in math. I can't talk or argue or logic you out of your disbelief in human abstraction.

This is why I said what I did about modern physics, this position is completely incompatible with the representation of time employed in physics. In conceiving of time in this way we have the means for a sort of compromised pure a priori order. It is compromised because it divides "experience" into two parts, associated with the internal and external intuitions. The internal being the intuition of time, must be separated from "experience" to maintain the status of "a priori", free from experience, for the temporal order. So it's a compromised pure a priori. — Metaphysician Undercover

And yet they get 13 decimal places of agreement between theory and experiment. That has to count for something. It's all we've got. It's helped us to crawl out of caves and build all this. For whatever that's worth.

I didn't deny the distinction between quantity and order, I emphasized it to accuse Tones of equivocation between the two in his representation of a count as bijection. — Metaphysician Undercover

I can't comment on your conversations in this thread that I didn't read. But as a technical matter, in cardinality theory we care about bijections. In order theory we care about order-preserving bijections.

That is exactly why I attack the principles of mathematics as faulty. There are empirical principles based in the law of identity, by which a physical, and sensible object is designated as an individual unit, a distinct particular, which can be counted as one discrete entity. There are no such principles for imaginary things. — Metaphysician Undercover

Doesn't Captain Kirk = Captain Kirk? Look, we're never going to get to ordinals at this rate. I don't know what you mean that the law of identity doesn't apply to fictional entities but there's a whole philosophy of fictional entities that I don't know much about.

Imaginary things have vague and fuzzy boundaries as evidenced from the sorites paradox. so the fact that "there is no mathematical difference between counting abstract or imaginary objects...and counting rocks", is evidence of faulty mathematics. — Metaphysician Undercover

You just phrase things like that to annoy me. How can you utterly deny human abstractions? Language is an abstraction. Law, property, traffic lights are abstractions. So is math.

As I said, all you've given me is a representation of a spatial ordering of symbols. If you are presenting me with something more than this you'll have to provide me with a better demonstration. — Metaphysician Undercover

This thread's already long. Do you want me to talk about ordinals or not?

I go both ways on this. Of space and time, one is continuous, the other discrete. But this is another reason why I think physics has a faulty representation of space and time, they tend to class the two together, as both either one or the other. — Metaphysician Undercover

Ok, you reject math, you reject physics. And you miss the distinction between physics and metaphysics, between a mathematical model and the thing being modeled. Whatever. Let me talk about ordinals.

Ordinals as abstract order types
========================

I'll keep this relatively brief since the rest of the post is long. I hope we can talk about ordinal numbers and not have any more endless disagreements about human abstraction. Our capacity for abstraction is one of the foundations of civilization, along with the opposable thumb. It's pointless to argue about it.

Ok first finite sets. You have a class full of school kids. You line them up by height. Or you line them up alphabetically by last name. Two distinct ways of ordering the same set. One cardinality but two distinct orders.

However, you will observe that these two distinct orderings nevertheless have the same order type. By that I mean that there is an order-preserving bijection between the set of kids in height order, and the set of kids in alpha order. You just match the first to the first, the second to the second, and so forth.

It's not hard to believe, and not hard to prove, that any two distinct orderings of a finite set have the same order type; in other words, that there is an order-preserving bijection. or order isomorphism, between the two orders. So orders on finite sets aren't very interesting.

So now, infinite sets. In fact only one infinite set is of interest to us at the moment, the natural numbers $\mathbb N = \{0, 1, 2, 3, 4, \dots \}$.

I hope you will grant me the abstract existence of this set, else there's nothing to talk about.

And I hope you won't be so tedious as to complain, "Well those dots are bullshit and they don't really stand for anything or mean anything blah blah blah." I pretty much agree with you, literally. The notation is only suggestive of a deeper abstract truth, that of the idea of an endless progression of things, one after the next, with no end, such that each thing has an immediate successor. Again if you want to stand on a soapbox and deny that, there's no point in this conversation. You have to at some level believe -- or at least accept, for purposes of playing the game -- the reality of such an endless progression. $\mathbb N$ is not the symbol or the list in brackets with the mysterious dots at the end. That's only a notation for the music. I want you to imagine the music, and form an association in your mind between the symbols, and the deeper abstract idea they represent. Surely you must be able to do this, after all you do it just fine using the English language. It's not really any different. Meaningless symbols that stand for abstract ideas. You do it every day with letters and words. It's no different in math.

Now the set of natural numbers $\mathbb N = \{0, 1, 2, 3, 4, \dots \}$ has no inherent order. You may recall that sets have no order. The set {a,b,c} and the set {b,c,a} are the exact same set. This is in fact the axiom of extensionality in set theory. It says that one of the rules of the game of set theory is that two sets are the same if and only if they have the exact same elements.

So given this set $\mathbb N$, we would like to put an order on it. What is an order? Well again, we play a symbolic game. We say that an order is a binary relation on a set that's reflexive, antisymmetric, and transitive. These terms are defined in the Wiki article I linked but they're not important. What is important is that they characterize the binary relation that we usually call $\leq$, the "less than or equal" relationship.

In the present case we also require that the order be total, in the sense that given two elements $x$ and $y$, either $x \leq$ or $y \leq x$.

And finally for convenience in this context, we prefer to work with the strict order $\lt$, which works like "less than or equal" but we disallow the equal; that is, $n \lt n$ is disallowed. Again this is all common sense that you already know, the details aren't important.details aren't important.

What IS important is that we have defined order without regard to any external meaning. It's all a formal symbolic game.

Then, we can formally define the symbols 0, 1, 2, 3, ... according to von Neumann's clever idea such that $0 \in 1 \in 2 \in 3 \in \dots$

Having done this, we now have a formal definition of each natural number within the rules of set theory; and then we can make the definition: $n \lt m$ just in case $n \in m$.

The point of all this is that we can define the '<' relation without regard to quantity and without regard to time or space or anything physical or meaningful. It's just an arbitrary symbol in the formal game of set theory; in principle no different to saying how the knight moves in chess.

Having done that, we have defined what's known as the usual order on $\mathbb N$. It's the order you learned in grade school, the one everyone knows. But the point is that I have defined '<' in such a way that this order relation means nothing at all other than the formal relation of set membership; which in set theory actually has no definition at all. $\in$ is an undefined symbol, just as point and line are in Euclidean geometry.

Having now stripped the usual order of any meaning, I'm free to define alternate orders like $\prec$ that I defined earlier; which is the same order as $\lt$ except that $n \prec 3$ for all $n \neq 3$.

This is clearly an alternate order on $\mathbb N$, just like lining the kids up by height versus by alpha last name. But in this case, these two orders represent distinct order types. There is in fact no conceivable way to create an order-isomorphism between these two ordered sets $(\mathbb N, \lt)$ and $(\mathbb N, \prec)$.

We can see this because $(\mathbb N, \lt)$ has no largest element, and and $(\mathbb N, \prec)$ does: namely, 3.

Now what I have outlined is the mathematical point of view in which:

* Order is an entirely arbitrary and meaningless binary relation on a set;

* That there are multiple possible orders on a given set; and that

* In the case of infinite sets, there can be distinct orders that are also distinct order types; that is, there are distinct orders that can not be put into order-isomorphism with each other.

This is the foundation of the idea of ordinal numbers, which are just order types of sets. And in passing, I hope I have made the point that while you object to the meaninglessness of math; on the contrary, it's the very meaninglessness of math that is essential! We have stripped all notion of external meaning from the order relation; in order to be able to investigate the properties of order without regard to the things that may be ordered.

What you call a vice, math calls a virtue. Meaninglessness, or lack of reference to anything tangible, is the heart of the power of mathematical abstraction.

As Wiles said when he proved Fermat's last theorem at a conference: "I think I'll stop now."

Yes, I've apprehend this, and I respect it. I know that's why you keep on engaging me. it's not easy to understand unorthodox and unconventional ways of thinking like mine though, so I've seen your frustration. But I do appreciate the effort. I've see the same effort to understand from jgill. I don't think TonesInDeepFreeze quite has that attitude though, and Luke just seems to be always looking for the easiest ways (mostly fallacious) of making me appear to be wrong, no matter what I say. — Metaphysician Undercover

After a little lighthearted back-and-forth (/s) over on the Israel/Palestine thread it's nice to return here to the things that really matter!

I have taken your point a long time ago that I can't actually \give a logically coherent definition of a mathematical object that doesn't depend on the contingent opinions of mathematicians. But I can't agree with your apparent extrapolation from that to an apparent rejection of all abstract math.

Let me tell you something. The magnetic moment of an electron is a defining feature of how magnetism effects a massive object. Therefore it is not measured it is a stipulation based in specific assumptions such as a circular orbit. But if the electron's orbit is really not circular, then the stipulated number is incorrect. — Metaphysician Undercover

I'm not enough of a physicist to comment. My point was only that you seemed to reject QM for some reason. I noted that you can't dismiss it so trivially, since QM has a theory -- admittedly fictional in some sense -- but that nevertheless corresponds with actual physical experiment to 13 decimal places. That's impressive, and one has to account for the way in which a fictional story about electrons can so accurately correspond to reality. Of course all science consists of historically contingent approximations. But lately some of the approximations are getting really good. Your dismissal seems excessive.

FWIW I don't think anyone thinks the orbits are circular anymore. They're quantum fields, sort of everywhere at once, and in any particular place only with a calculable probability. Or even worse, we can calculate the probability of where we'll find it if we look. Where it "really" is, is a matter of metaphysics. The question lies outside of science.

Now I agree with you (if this is what you are saying) that this makes the whole enterprise metaphysically suspect in some sense. But you still have to account for the amazing agreement of theory with experiment. We might almost talk about the unreasonable effectiveness of physics in the physical sciences!

I protested it in London. — Baden

I thank you and admire you for that.

I'll just presume this is a sardonic barb. Riiiight? — Baden

Doing my best to break through on this thread. "Civilians. They're what's for dinner," didn't even get a mention.

Apparently if a bad guy comes to my house the powers that be are justified in killing us both and all my children too. No questions asked. Yes, that's just the way it works everywhere. Nothing to see here. — Baden

American liberals signed off on the invasion of Iraq, which killed 1 million Iraqis who had absolutely nothing to do with 9/11. You opposed the invasion in 2002 I hope. The evangelicals lost their soul when they signed on to Bush's war, and so did the war-supporting Democrats and liberals. A very morally clarifying moment all around.

181 dead, 52 children. — StreetlightX

52 kids who won't grow up to be Palestinian terrorists.

Wonder who will write paragraphs of thought experiments justifying this? Probably a moral black hole of a human being - or three. — StreetlightX

How'd I do?

Do you know of any way to approach this problem? I'm assuming you will reference information theory, which isn't what I think is appropriate to ascertain "complexity", or is it? — Shawn

I've referred you to Turing degree, complexity theory, and proof systems like Agda and Coq. I'm all out of ideas. A good resource for complexity theory is everything on Scott Aaronson's website. https://www.scottaaronson.com/

But, you do acknowledge that the complexity class of a theorem's proof can be indeterminate, as long as the machine engages in non-brute force methods to determine "complexity"? — Shawn

I have no idea, I know very little of these matters.

How would you define it? Is it even possible to define this? And, may I ask how is this distinct from estimating Kolmogorov complexity? — Shawn

It's defined in CS, you can Google around. It's a fairly technical subject, nothing I know much about.

Of course it's possible to define, the definition is in the Wiki article, although that article's not too informative. I don't know any more about it than you'd learn from Googling and reading some papers. None that I've seen are particularly accessible.

OK, I sat down, read it, and read it more, and seemingly you would have the have a powerful oracle machine to make the decision to do this with least decidedly complexity.

Am I getting that much right? — Shawn

I'm not an authority on CS theory. The idea seems to be that we write A < B for sets of natural numbers if A can be decided with an oracle for B. What's interesting is that this is not a linear order. There are sets A and B such that neither A < B or B < A.

Do you think complexity class can determine "complexity" quantifiably for a Turing computable algorithm? — Shawn

Perhaps Turing degree is what you're looking for. I'm not sure what you mean by quantifiable. Complexity theory puts computational problems into equivalence classes, That's perhaps more qualitative than quantitative.

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

Yes, sir. — Shawn

No, I addressed exactly the sentence he wrote. What he wrote in that sentence was correct. I am not responsible for addressing other confusions he has that might conflict with the sentence he posted and to which I responded. He wrote something correct, and I affirmed it. — TonesInDeepFreeze

I think I'll go argue with the partisans on the Israel-Palestine war thread. It seems more peaceful.

I was talking about type theory in general avoiding the issue of the set of all sets by assigning hierarchy classes as I last read about the issue. — Shawn

You understand that there is no set of all sets, correct? Even in type theory. Yes?

I was talking about type theory in general avoiding the issue of the set of all sets by assigning hierarchy types as I last read about the issue. — Shawn

In context, your remark served to amplify the OP's confusion rather than correct it. I straightened the situation out. You're welcome. I myself am singularly unconfused about this issue, which is why I challenged your technically correct but misleading-in-context claim.

It is correct that PM is one approach to avoiding the paradox. With PM there is no set of all sets. — TonesInDeepFreeze

That last sentence makes all the difference, especially in the context of @Shawn referencing the set of all sets, my telling him that there is no such thing, Shawn saying he'd look up PM, and you saying that he didn't need to, the implication being that PM confirmed his misunderstanding, when in fact it doesn't. You added to OP's confusion on this issue and I don't know why.

Why would I want to supply references to a claim that PM allows a set of all sets when I agree that PM does not allow a set of all sets? — TonesInDeepFreeze

Do you have a cat? Perhaps your cat wrote this using your handle:

You are correct that PM is one approach to avoiding the paradox. — TonesInDeepFreeze

You don't need to search. You are correct that PM is one approach to avoiding the paradox. — TonesInDeepFreeze

I do not believe there is a set of all sets either in Russell's type theory or in any version of modern type theory. I'd be grateful if you could supply references and/or context to the contrary. At best I've read that type theory "avoids the paradox," but I've seen no claim that there is a set of all sets.

resolving entailment of the set of all sets, — Shawn

There is no set of all sets, I hope we're not going down this road.

So, when I say that a proof of a theorem is subject to not being able to determine its complexity does that mean anything? — Shawn

What do you mean by the complexity of a proof? I've already suggested that it could be defined as the length of a proof in some formal proof system. What do you mean by it?

What I'm trying to determine is whether there is any possibility to determine the complexity of proofs by reasoning that a Q.E.D. would occur at the least exhaustive method of determining it. — Shawn

What is a QED? Do you mean a proof?

Does that make sense? Following with this logic, if you don't have a method of doing this, then how can you determine complexity in mathematical proofs? — Shawn

I've already suggested a way. I don't know that it's regarded as particularly important but I could be wrong about that. For sure it's important in computer science, and I pointed you to complexity theory.

Two wrongs make a right? — jorndoe

I didn't say that and didn't bother to read further.

why not add a few misiles and kill civilians. — Manuel

What else are civilians for then? Surely a glance at 20th and 21st century war should disabuse one of the notion that civilians are to be protected. When US sanctions killed half a million Iraqi children, Secretary of State Madeleine Albright famously said, "We think the price was worth it."

Nonetheless, we've got posters in this thread who have little interest in doing anything except feigning moral indignation, virtue signalling and just trying to drag others through the mud. — Judaka

Thought I'd go the opposite way. Civilians. They're what's for dinner.

Please forgive my lack of knowledge on the matter; but, what I wanted to say that a theorems proof for an axiomatic system in math is unable to be ascertained in complexity with regards to being neither complete and consistent. — Shawn

This sentence repeats the same misunderstanding. Individual theorems do not have the consistency or completeness attributes, in the same sense that automobile tires don't have the horsepower attribute.

With this fact being true, — Shawn

As stated, what you said is not sufficiently clear to be true or false; but if I had to guess at your meaning, it's false because consistency and completeness don't apply to individual theorems.

then how does one ever hope to gauge a mathematical theorem's proof as measurably complex or less complex. — Shawn

For computational problems, computational complexity theory is the standard approach, as you already know. For theorems, I suppose if anyone cared, we can pick any one of the contemporary proof assistants like Coq or Agda, and define the complexity of a theorem as the shortest proof of that theorem in such a system.

But completeness and consistency of axiomatic systems seem to have little or nothing to do with your question. If a theorem has a proof and you want to find the simplest one, that has nothing to do with the axiomatic system being incomplete. The axiom of choice has no proof in ZF so it's meaningless to ask about the simplest proof. And if a system is inconsistent, then everything has a one-line proof following directly from the inconsistency. So completeness and consistency are irrelevant to your question.

a theorem cannot both be consistent and complete, according to Gödel's Incompleteness theorems? — Shawn

A theorem can neither be consistent nor complete not by virtue of Gödel, but rather by virtue of the fact that the terms consistency and completeness apply to axiomatic systems and NOT to theorems. You seem unclear on this point.

Let me elaborate in an effort to be useful. Standard set theory, ZF, is incomplete by Gödel's first incompleteness theorem. We can identify a particular proposition that can neither be proved nor disproved, say the axiom of choice, That's incompleteness. Consistency just says that there's no proposition P such that both P and not-P can be proven from ZF. So consistency and completeness are properties of axiomatic systems.

A particular theorem, like Fermat's last theorem, or the fundamental theorem of finitely generated Abelian groups, or the fundamental theorem of calculus, do not have the properties of consistency and completeness. Those properties don't apply to individual theorems.

What do others think? Does this make things sounds somewhat disorganized in how proofs are written? — Shawn

Your OP and subsequent posts are a mishmash of several things and I can't figure out what you're saying. You could be talking about an extension of Turing machines that accept languages over an uncountable alphabet. You could be talking about Turing degree, the level of algorithmic unsolvability sets of natural numbers. Interestingly this is not a linear order, so that there are sets with incomparable Turing degree. You could be talking about extending an incomplete axiomatic system by adding either a proposition or its negation in order to form a larger system, which is still necessarily incomplete; and doing that over and over, hoping to eventually generate a complete system. That can't be done, and was in fact the subject of Turing's doctoral dissertation. You also tossed in Kolmogorov complexity and even everyone's favorite buzzphrase, quantum computing.

Is it possible for you to focus your subject? Can you perhaps give a specific example of what you're getting at?

By refusing to say "black lives matter" and instead just keep saying "all lives matter" you haven't really said much, except to deny or ignore something that needs addressing. — jorndoe

For sake of discussion inviting people to throw rocks at me, I'll toss out a counterargument that I already mentioned a little earlier.

There's a lot of bad police behavior in the US. Some of it is racial and some of it isn't. If you define it as a racial problem, you can't solve the problem because you've misdiagnosed it. If you tell black people that "You're being hunted by cops," you don't solve any problems that actually exist; and you encourage a certain percentage of black people to resist arrest, resulting in a self-fulfilling prophecy. Dr. King dreamed of a society where people were judged on the content of their character and not the color of their skin; in effect, "all lives matter." If you now characterize such a position as racist, or not sufficiently anti-racist, you simply polarize people and make society's problems worse.

Secondly, while it's undeniably true that black lives matter, Black Lives Matter is a Marxist organization wholly dedicated to the destruction of the American way of life. This equivocation puts lifelong nonracist people like myself in the position of saying that black lives matter but that I oppose Black Lives Matter; subjecting me to to the charge of racism. And this equivocation is no accident, it's deliberate.

If saying that "All lives matter" is racist, then the word racist has been distorted beyond all meaning and is simply used as a political epithet; as has in fact happened in the present sociopolitical moment.

Oh dear. Did you not read that section of the thread, where I described the difference between quantity and order? — Metaphysician Undercover

Quite possibly I didn't. I only read the posts that generated my mentions. I did not read anything else in the thread. And as I believe I admitted, my cardinal/ordinal remark was only in reference to a single sentence you wrote, and not at all in reference to the larger context of the discussion, of which I was and still remain ignorant.

That also explains why I ran out of steam for engaging in further convo. I made a small point, that there are ordinals as well as cardinals. I was not intending to engage at any deeper level.

I think it's rather pointless because you do not seem at all inclined to make any effort toward understanding. — Metaphysician Undercover

I would say that I've made a considerable effort the past several years to understand your point of view. But I agree that I prefer to make the effort in small doses; and on this thread, I reached my local limit. I'm sure we'll do this again in some other thread. But in truth you made so many strange statements here that I saw no basis to continue. When you pooh-poohed the 13-digit accuracy of the measurement of the magnetic moment of the electron, you indicated a dismissal of all experimental science. This is perfectly consistent with your 2200 year old view of math. You have a 2200 year old view of physics as well. I don't wish to argue that point with you.

Actually, I'm starting to get a real feel for the problem now, and I sincerely want to thank TIDF and fishfry for helping me come to this realization. — Metaphysician Undercover

Thank you for the kind words. I did not see this, as it did not contain an @ before my handle. As I said, I've only looked at posts that contain a mention of my handle.

I now see that there is a fundamental difference between using numerals to signify quantities, and using them to signify orders. The former requires distinct entities, objects counted, for truth in the usage, while the truth or falsity of the latter is dependent on spatial-temporal relations. — Metaphysician Undercover

But since I'm here, let me note that this could not be more false. I already gave the counterexample One can order the natural numbers $\mathbb N$ with the linear order $\lt$, the usual order; or $\prec$, the "funny order" in which everything is the same as the standard order except that $n \prec 3$ for all $n \neq 3$. This is a purely abstract order relation on the natural numbers. There is no spacial or temporal referent involved. One abstract set, two distinct abstract orders. Absolutely no referents in the physical world (that we yet know of) but of critical importance in mathematical logic, proof theory, and various other abstract branches of math.

You can't claim ignorance of this illustration of the distinction between quantity and order, since I already showed it to you in this thread. So whence comes your claim, which is false on its face, and falls on its face as well?

So the truth of a determined quantity depends on the criteria for what qualifies as an object to be counted, — Metaphysician Undercover

This also is wrong, since there is no mathematical difference between counting abstract or imaginary objects (sheep, for example, as someone noted) and counting rocks.

while the truth of a determined order is dependent only on our concepts of space and time. — Metaphysician Undercover

Please show me space or time in the $\prec$ order on the natural numbers.

So, in the case of quantity, truth or falsity is dependent on the truth of our concept of distinct, individual objects, — Metaphysician Undercover

Those harpooneers don't exist, yet there are three of them; four if you count Ahab's personal harpooneer Fedallah. They can be counted sure as the planets. Even more surely, since there's no committee removing harpooneer-hood from Queequeg, as there is for removing planet-hood from poor Pluto. I remember being on vacation, sitting in the Portland airport reading the New Yorker, and discovering that Pluto was no longer a planet. Counting is not as sure a thing as you'd think. The "truth of our concept of distinct, individual objects," as you put it, turns out to be subject to the vote of a committee. Such is reality these days. A lot more tenuous than you'd think.

but in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time. — Metaphysician Undercover

I ask again for you to please show me space and time in $\prec$. It's an alternate ordering of the natural numbers, an alternate order type in fact, but there is no space or time involved. It's purely abstract.

Since we think of space and time as continuous, non-discrete, — Metaphysician Undercover

Who is this "we?" Surely there are many who can argue the opposite. Planck scale and all that. Simulation theory and all that. Of course we "think" of space and time as continuous if we are Newtonians, but that worldview's been paradigm-shifted as you know.

we have two very different, and incompatible uses of the same numerals. — Metaphysician Undercover

Of course quantity and order are two distinct aspects of the "same numerals" in the finite case. In the transfinite case we use different numerals; $\aleph_0$ for the cardinal representing the natural numbers; and $\omega$ to represent the exact same set with its usual order $\lt$.

But I don't see your point. Cardinals refer to quantity and ordinals to order. The number 5 may be the cardinal 5 or the ordinal 5. The symbology is overloaded but the meaning is always clear from context; and in any event, the order type of a finite set never changes even if its order does. The distinction between cardinals and ordinals only gets interesting in the transfinite case.

I really do not think there is any type of order which is not based in a spatial or temporal relation. — Metaphysician Undercover

You have just been shown one, namely $\prec$. But you haven't actually "just" been shown one. I showed you this example several days ago.

You know, to sum this all up, I understand that it's difficult to give a coherent account of abstract objects. But that doesn't mean they're not important. You use the former to utterly reject the latter, and that forces you into positions that are impossible to defend.

I'd say it's ordering, not counting. — Metaphysician Undercover

Wait, NOW you believe in ordinals?

I'm going to skip responding to your points. Earlier you said numbers were for quantity and I pointed out that there are other kinds of numbers (in the finite case, the exact same numbers viewed differently) for order. I made my point. Then we got off onto other things. You say you don't think there are pure mathematicians, that you can't count abstract things, that you don't believe in experimental science (since you apparently reject the example I gave) and so forth. I'm out of enthusiasm to continue. Till next time.