That "elements" may exist without an order is the falsity I've explained to you already. And if we say that "element" indicates an abstraction, then it is a universal, not a particular, and to assume that an abstraction is a particular is a category mistake. — Metaphysician Undercover

I think you are just not cut out for mathematical abstraction and should pick another major.

Let's assume a special type of "element", created, or imagined specifically for set theory. This type of element can exist in a multitude without that multitude having any order. Each of these elements would have no spatial or temporal relation to any other element, or else there would be an order, according to that relation. We could say that they are like points, but without a spatial reference, so that we cannot draw lines between them etc., because there is no order to them. But if they were like points, without spatial relations constituting order, there would be no way to distinguish one from another — Metaphysician Undercover

Ditto.

Unlike points though, there is something which distinguishes one element from another, so that in the set of (a,b,c,), "a" does not represent the same thing as "b" does. Can I conclude, that the distinct elements are separated from one another, and distinguished one from another, by something other than space? To make them distinct and individual, they must have separation, but the separation cannot be spatial or else they would have an order, by that spatial relation.

Do you see, that from this premise alone, we cannot give any order to any set? To give a set an order would be a violation of the fundamental meaning of "element" which allows that elements can exist as particulars without any spatial temporal; relations. To be able to talk about an order within a set, would require that we transform the elements into something other than "elements", something which could have spatial or temporal relations and therefore an order. Remember, even quantity requires spatial-temporal separation between one and the other, to distinguish separate individuals. — Metaphysician Undercover

Ditto. I'm not going to bother. You are obfuscatory in the extreme. The only thing I can't figure out is why someone with zero aptitude for mathematical abstraction is so interested in it, yet so utterly unwilling to engage with it.

No, sorry, it's not clear at all. You have imagined distinct "elements" which exist without any spatial or temporal relations, thereby having no order, though they are somehow distinct individuals. — Metaphysician Undercover

Yes exactly.

Now you want to add order. You have already defined order out of the set, to add it in, is blatant contradiction. — Metaphysician Undercover

Enough. You win. You wore me out.

What I need, is a clear explanation of what an "order relation" is. — Metaphysician Undercover

I have explained this many times. I linked you to the Wiki page on mathematical order theory. An order is a binary relation; that is, a function that outputs True or False for every pair of elements in a set; that has certain properties as I've mentioned several times already.

Here is the page. Come back when you've made a sincere effort to understand the material.

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

What type of relation are you attempting to give to these elements, which gives them an order, when you've already stipulated the premise that they have no order? — Metaphysician Undercover

I explained that it's a process of abstraction, where we start from no assumptions and layer on the structure we want. If you don't get it, you don't get it.

The point is, that to give them existence without order requires a special conceptualization which I described above. Now if we want to proceed with that conceptualization, and now bring in principles of order, we must do so in a consistent way. So, we need to describe what separates one element from another, since it's clearly not space, and what makes it distinct as an element, in terms which do not give it a relationship to the others, to allow that the multitude of them do not already have any order, Then we need a principle by which order can be initiated within this non-ordered type of separation. — Metaphysician Undercover

Pick another major.

Skipping the rest. I've done what I can. I do recommend that you read the page on order theory.

I do get that you reject the mathematical concept of set. Not much anyone can do about that. It's like saying you want to learn physics and then arguing with the concepts of space, time, force, motion, energy, and temperature. You may well have a philosophical point to make, but you are preventing yourself from learning the subject. And it's learning the subject that would allow you to make more substantive rather than naive and obfuscatory objections.

I have explained that set is an undefined term entirely characterized by its behavior under the axioms. You insist on imposing your own incorrect conceptions. So there's no conversation to be had.

Thanks for the clarification fishfry, but here's a couple more things still to clear up.

To me, the following statements contradict each other.

But the set of natural number may nonetheless be ordered in many alternative ways.
— fishfry

But no set has order. That's the axiom of extensionality. Will you kindly engage with this point?
— fishfry

Which is the case, no set has order, or a set may be ordered in many different ways. Do you apprehend the contradiction? Which is it, ordered in different ways, or not ordered? — Metaphysician Undercover

The concepts are built up in layers, like a burrito.

At the bottom is the concept of set. A set is informally a collection of objects. Formally, set is an undefined term, just as point and line are undefined terms in Euclidean geometry. We all "know" what the intended meaning is, but when reasoning formally, we can only use their properties as stated by the axioms. Likewise with sets.

By the axiom of extensionality, a set is entirely characterized by its elements, without regard to order. So the set {a,b,c} is the exact same set as {b,c,a} or {c,b,a}.

That's at the lowest layer. Now we want to layer on the concept of order. To do that, we define a binary relation, which I'll call <, and we list or designate all the true pairs x < y in our set. So for example to designate the order relation a,b,c, we would take the base set {a,b,c}, and pair it with the set of ordered pairs {a < b, a < c, b < c}. Then the ordered set is designated as the PAIR ({a,b,c,}, {a < b, a < c, b < c}). I hope this is clear.

For example we have the unordered set $\mathbb N$ consisting of all the natural numbers 0, 1, 2, ... Then we layer on top of it the usual order <, so that the ordered sets is now $(\mathbb N, \lt)$. In other words a set is an unordered collection. An ordered set is a PAIR consisting of an unordered set, along with an order relation.

In the case of the usual order relation on the natural numbers, the order relation < is actually the set of all the true order statements: {0 < 1, 0 < 2, 0 < 3, ..., 1 < 2, 1 < 3, 1 < 4, ..., 2 < 3, 2 < 4, ...}.

In order to remove the apparent ambiguity of using the symbol < as both the relation and as specific instances of it, formally a relation is a set of ordered pairs; so the usual < relation on the natural numbers is actually the set {(0,1), (0,2), (0, 3), ..., (1,2), (1,3), ...}. Again I hope this is clear, it's basic stuff for a math major but is definitely a little formalism-heavy if you haven't seen it before.

The basic takeaway is that a set has no inherent order. We impose an order on a set by PAIRING the set with an order relation. That's why earlier I noted that we can start with the set $\mathbb N$ and then form two distinct ordered sets $(\mathbb N, \lt)$ and $(\mathbb N, \prec)$, where $\lt$ and $\prec$ are distinct order relations.

I'll mention in passing that this is a very common pattern in math. We start with a set $X$, which has no inherent structure at all. Then we let $\tau$ be a topology on $X$, and we call the pair $(X, \tau)$ a topological space. Or we have an unstructured set $X$ and pair it with an operation $+$, subject to some rules for how $+$ behaves, and we call the pair $(X, +)$ an Abelian group.

Pretty much everything in math is defined as some set, along with some other structures that impose whatever attributes on the set that we're interested in.

Finally, there's always some notational ambiguity, because when we say, for example, $\lt\mathbb N$, we very often mean the set of natural numbers along with its usual order. The meaning is always clear from context. If we were being precise we would always write $(\mathbb N, \lt)$ for the set $\mathbb N$ with its usual order; and we would write $(\mathbb N, \lt, + *)$ for the natural numbers with their usual order and standard arithmetic operations of addition and multiplication.

You don't have to care about the details. What is important is that any "structured set" actually consists, formally, of an unstructured set combined with whatever additional structure we care about: an order relation, a topology, arithmetic operations, and the like. I truly hope this is clear, and if not please ask.

Let me go back to my question from the last post. What exactly constitutes "the set"? Is it the description, or is it the elements which are the members of the set. — Metaphysician Undercover

Of course the description is just a representation, as 5 is a representation of the abstract number 5 (whatever that is!) and "snow" is a representation of snow.

A set is entirely characterized by its elements; but a set is more than just its elements. It's the elements along with the collecting of the objects into a set. Maybe that's a bit philosophical, I'm not sure if I can really explain it any better than that. A set is a collection of elements, regarded as an individual thing, a set. So in Peano arithmetic we have numbers 0, 1, 2, 3, 4, ... But the axiom of infinity is much stronger. It says that there is a SET that contains all the numbers. It's the difference between 0, 1, 2, 3, ... and {0, 1, 2, 3, ...}. I hope that's clear, but if you think there's a lot of philosophical mystery that I haven't adequately explained, I'd be inclined to agree. Perhaps it's the distinction between a bunch of athletes and a team, or a collection of birds and a flock. I'm sure some philosophers have found ways to describe this. A set is a collection of elements, along with the concept of their set-hood. That's the best I can do!

If it is the description, or definition, then order is excluded by the definition. — Metaphysician Undercover

Agreed. The order is imposed by PAIRING the set with a separate order relation. Hope I made that clear in my longwinded exposition a moment ago.

But if the set is the actual participants, then as I explained already they cannot exist without having an order. If the supposed participants have no existence then they cannot constitute the set. — Metaphysician Undercover

The order relation is technically a separate set, consisting of the collection of ordered pairs that define the order. Hope I made that clear already.

That's why I ask, which is it? Can a set be ordered, or is it inherently without order? Surely it cannot be both. — Metaphysician Undercover

A set is inherently without order and without any kind of structure. We impose order and other structures (topology, arithmetic, etc.) on a set by pairing the set with additional relations, which are themselves formally defined as other sets.

And as I've mentioned, we often CASUALLY say "the ordered set X," or "the topological space X", when we REALLY mean the pair (X, <) or the pair (X, $\tau)$. That's perhaps the source of some confusion, as I can sometimes mean X the unstructured set, or X the ordered set, or X the topological space, because I'm implicitly leaving out the addition structure with which X is paired.

But formally, every set is inherently unstructured and unordered. We impose structure, order, arithmetic operations, etc., after the fact, by associating the set with other sets that represent order relations, topologies, arithmetic operations, and so forth.

Let's look at "concept" as a noun, as if a concept is a thing. Do you agree that a concept is the product, or result of conception, which is a mental activity? There's different mental activity involved, understanding, judgement, conclusion, and effort to remember. Would you agree that the effort to remember is what maintains the concept as a static thing, So if a "concept" is used as a noun, and is said to be a thing, it is in the same sense that a memory is said to be a thing. Would you agree that if a mathematical concept is "a thing", it is a thing in the same sense that a memory is a thing? — Metaphysician Undercover

You lost me a bit here. The original question is that you object to my use of the phrase mathematical object, and I'm just asking what to call it instead. If you want me to call numbers, topological spaces, groups, rings, and fields "mathematical concepts" instead of mathematical objects, I'll do that if it makes you happy, but really, they're mathematical objects and universally recognized as such by people trained in math.

What does this have to do with the issue we are discussing? — T Clark

No longer recall the specifics, but something about support for authoritarianism along with my dislike of police thuggery in the name of conformance to mask laws that don't actually have much impact on public health in the first place.

First - No, he did not admit that Covid might have a lab origin. He became open to the possibility based on new evidence. Second - In terms of how the pandemic has been handled here, what difference does it make where it came from? — T Clark

He finally admitted it after a year denying it, and there has been no new evidence. The evidence was there all along, as were the many reputable scientists pointing that out all year. All that's changed is that the MSM can no longer keep a lid on the truth.

And what difference it makes is that in 2014, the US outlawed gain-of-function research, only to re-authorize it in 2017. And if in the end it turns out that Fauci was the one who paid the Chinese to conduct that research, that is a hell of a news story. And a case can already be made. Money is fungible. We know he gave money to EcoHealth Alliance, and they gave money to the Chinese, and they spent some of it on GOF research that may have led to a covid lab leak. That's a news story and it matters. It matters a lot if in the end, the US is paying the Chinese to do bioweapons research that either accidentally or deliberately had such a profound effect on us. I'm a little puzzled as how you can even ask the question of "what difference does it make." Isn't that what Hillary said about a dead ambassador? Is this the story you're going with?

Again, what difference does it make in terms of our pandemic response? — T Clark

In terms of the response, not much difference at all. In terms of preventing the next similar incident, it makes all the difference in the world. Every advanced country in the world is doing bioweapons research, either for offensive purposes, which nobody admits to, or for defensive purposes, which they all claim. "The other guys might do it so we have to learn about it."

I regard it as naive and childish in the extreme for anyone to be in denial about bioweapons research, and to claim we shouldn't be asking these questions about who is funding it and what the consequences might be for humanity. If THIS pandemic came from bat soup, the next one will be an accidental lab leak, and the one after that will be a deliberate lab leak. And if you don't know that, I urge you to do your homework.

Maybe start here. Chinese scientists discussed weaponising coronavirus in 2015: Media report.

BEIJING: Chinese military scientists allegedly investigated weaponising coronaviruses five years before the COVID-19 pandemic and may have predicted a World War III fought with biological weapons, according to media reports referring to documents obtained by the US State Department.

"What difference does it make," indeed.

Also, "The Federalist" is a knee-jerk right-wing rag. They've spread misinformation about Covid from the start and promoted the stolen election lie. — T Clark

Can't counter the facts, so slime the publisher. Sadly, the information they print isn't being reported by Rachel Maddow and Anderson Cooper. If it was, I'd link it. The Federalist has a conservative take on the news, but I would not call them a knee-jerk right-wing rag, unless you also admitted that by the same criterion, the NYT is a knee-jerk left-wing rag.

As I wrote previously, I've been impressed by how well the US responded to the pandemic, even given the jerky start and all the zig-zags. — T Clark

Panic and hysteria are never appropriate responses. The US government did a terrible job responding to the pandemic. I believe that my opinion will be vindicated over the next few years as people get perspective, but the jury's still out at the moment. I do believe that the sudden realization that it may well have been a lab leak, after a year of suppressing and deplatforming and smearing credible advocates of that position, supports my conclusion and not yours.

A lot of those missteps came from right-wing political sources like "The Federalist." I think you are a reverse conspiracy theorist. — T Clark

Is that someone who thinks Caesar was stabbed by a lone knifeman and that 9/11 was done by a lone planeman? I am not sure how to take that but it strikes me as funny.

It's not that people are conspiring to do bad things, it's that people are conspiring not to do good things. — T Clark

The US response to covid was driven by panic, confusion, and hysteria. Trump derangement syndrome had a lot to do with it. This is already becoming clear. as the Washington Post just admitted the other day.

John Kass wrote a piece about this. The Wuhan Story That Finally Has Legs, Now That Trump Is Gone

The NY Post reported that the Biden admin actually shut down an investigation into the lab leak hypothesis, and NOW they have been forced to start it up again. Biden shut down Wuhan inquiry out of spite — and is now forced to reverse course

Finally, here's an article documenting the US government's restoration of GOF research in 2017

US government lifts ban on risky pathogen research
The National Institutes of Health will again fund research that makes viruses more dangerous.


This is very important in a thread about the goodness of science. In the old days if you were a bright young biology postdoc, you'd go into curing cancer or heart disease or otherwise finding ways to alleviate human suffering. Now? You follow the government grant money and devote your knowledge and skill to figuring out more clever ways to weaponize diseases. @Banno, any opinion?

To sum up, the question of whether covid came from bat soup or was accidentally or perhaps deliberately released from a Chinese bioweapons lab partially funded by American taxpayer dollars controlled by Dr. Fauci, is a question that goes directly to the heart of the goodness of science. This GOF research was outlawed in the US in 2014. Scientists and others who follow these issues knew all about this stuff years ago. It didn't just come into existence because people started getting sick in 2019.

Bioweapons research goes back to the great German chemist Fritz Haber, who did brilliant work on synthesizing nitrogen in order to make fertilizers that now feed billions of people; and who then, in WWI, invented nerve gas and personally went out to the battlefields to deploy it. His wife committed suicide, in part because of her opposition to his war work.

Now that's science. It can feed you or gas you to death. It can cure your disease, or give you a disease that you otherwise wouldn't have gotten. Science is a double-edged sword.

So, you would be happy for ANTIFA to burn down your house as long as you're given a warning. Off-topic. But, ok, check. — Baden

I am unaware of what I wrote that this is in reference to. I oppose Antifa and regard them as the modern incarnation of Hitler's brownshirts, government-sanctioned thugs who did what the government wasn't able to openly do. And FWIW I attended many Occupy protests (as an observer and amateur photographer, not so much a participant) and toward the end, saw the rallies taken over by Antifa and the "black bloc." Thousands of people would demonstrate peacefully during the day, and after dark the Antifa thugs would smash store windows and set fires in the street. What's changed since then is that back then, the MSM publicized the Antifa violence to discredit the peaceful protests. Now, MSM downplays and even denies the violence. I think that's a very bad sign.

That said, I have no idea what your remark meant or what it pertained to.

It was terrible for the Germans after WWI. The Treaty of Versailles demanded huge reparations, while at the same time annexed the Saaland, which was Germany's main access to coal for industrial and domestic fuel. At the same time Germans had democracy forced on them - and it was proportional representation, which led to a proliferation of political parties, and weak, indecisive government. It's easy to see how Germany fell prey to the Nazi regime. — counterpunch

I'm not too much of a historian but I do know the Germans got screwed at Versailles and that led to the rise of Hitler etc. I didn't mean to get into the historical nuances of the phrase "good German" and I see your point.

When all this kicked off, I looked up the statistics on Arrest Related Deaths - and apparently, there are around 10 million arrests per year, and around 1000 end in the death of the suspect. That's 0.01%. Of those, 32% are black - which may immediately seem disproportionate, given that black people are only 13% of the US population. However, when you look more closely, it turns out that black people commit a lot more crime - and so make up a larger proportion of arrests than their numbers in the population would suggest. — counterpunch

You can get in a lot of trouble these days for pointing that out, but it's true. A set of facts that can be spun many ways.

I was on twitter at the time - and shared these statistics, and was banned from twitter for doing so.[/quote[

LOL. Not laughing at you, only commiserating. It's terrible what's happened to the concept of free speech lately. And, "Twitter is a private company" is not a valid response. The southern lunch counters that refused to server blacks in the 1950's were private businesses too, until the Civil Rights Act of 1964 defined them as public accommodations. Something similar has to eventually happen to the social media companies. They are either public accommodations or common carriers, as the phone companies are. At some point Congress needs to step up to the plate.

But for sure, if you point out that blacks have a lot more per capita contact with cops than whites because they commit more per capita crimes, that will definitely put you on SJW Santa's naughty list. Would have loved to see your Twitter feed. Myself, I am not on social media. Wouldn't be able to stand the aggravation.

— counterpunch

But wait, because the plot thickens. Data on arrest related deaths was collected by the Bureau of Justice Statistics from 2003-2012, whereupon the Obama administration shut it down, the year before BLM was formed in 2013. So, this kicking off in the weeks leading up to the Presidential election looks mighty suspicious. One has to ask why Obama would shut down data collection on the race of arrest related deaths if it was such a huge issue that forming BLM was necessary. About 300 black people die every year - which is plenty of fuel for a social media narrative, while statistically, there's no evidence of racism on the part of police, and every indication of extraordinary professionalism. — counterpunch

I'm in agreement. Obama was a race hustler who made race relations far worse than before he became president. I'm an old MLK-style liberal (content of character etc.) appalled by what's become of race relations. I have no idea where it's going, whether the present moment will die out or get worse.

@darthbarracuda

:up: :100: :clap:

am not going to continue playing a part in another of your political rants, conspiracy theories, and alternatives to facts. — Fooloso4

Forbes magazine is a mainstream periodical, hardly a hotbed of conspiracy theories. They explored the question. Did Covid-19 Come From A Lab? Was It Deliberate Bioterrorism? A Biodefense Expert Explores The Clues

The title of the thread is Science. If you are not aware that every advanced country in the world is engaged in bioweapons research, conducted by scientists, actual scientists, I do hope you will take the trouble to educate yourself. And if not you, perhaps other readers will. We don't know the origin of covid. Bat soup seems very unlikely at this point. It was most likely either an accidental leak from a bioweapons lab -- one partially funded indirectly by Dr. Fauci himself -- or a deliberate leak. The latter remains a possibility until it's ruled out. And you can't rule anything out scientifically by calling ideas you find unpleasant, conspiracy theories.

What exactly do you think gain of function research is? They take a naturally occurring disease, and they try to figure out how to make it easier to pass from one human to the next. Why? Because they are either planning to use it as an offensive weapon; or they need to study it to defend against the other guys doing it. In either case it's the same thing.

Here, read and learn. Science is about having an open mind. And very very sadly, science is about bioweapons research these days. When we say science, we're not talking James Clerk Maxwell anymore.

https://www.factcheck.org/2021/05/the-wuhan-lab-and-the-gain-of-function-disagreement/

For the record, there are no smoking guns. Only questions. Questions you smear as conspiracy theories, but that reputable scientists are asking.

@banno, This is a better response to your question. Can I say something bad about science? Yes, the US government spends a lot of money figuring out how to kill people with bioengineered viruses. That's science and it's evil. Perhaps a necessary evil, in the sense that our adversaries are doing the research and we have to know enough to defend ourselves. But this is what science has come to. Newton at Woolsthorpe during the plague year, it ain't.

For the majority of the past year your man Trump was in office. — Fooloso4

If I can recall that far back, my other choice was Hillary. I'd do it again. That doesn't make Trump "my man." It only means that millions of people who voted for Obama twice, couldn't stomach voting for Hillary. I was one of those millions. You could look up the numbers. Hillary doesn't call Trump supporters a basket of deplorables, she wins. She goes to Wisoconsin, she wins. She's even slightly less of a corrupt warmonger, she wins. She couldn't do or be any of that because she's Hillary. She managed to be the only person in the country who couldn't beat a guy like Trump.

Subsequently, the party I'd been a member of all my life, the Dems, went all-in for Russiagate hysteria and Muellergate and Ukrainegate and are now completely off the rails. I'm not the only former liberal who feels this way.

What evidence do you have of that? Again, you hear part of something and make up your own story or blindly believe conspiracy theories as if the are "alternative facts". Even if it came from a lab that does mean it was deliberately released as a test run of a bio-weapon. — Fooloso4

I took the trouble to admit that this is not proven. So why do you pretend to have not seen me do that? It's certainly a possibility. Every government in the world that can afford a bioweapons lab is doing the research.

I did not say she got what she deserved. I questioned your comparison to what happened to a man who was killed by having his neck kneeled on for over nine minutes. — Fooloso4

At trial it was revealed that Chauvin's knee was on Floyd's upper back and not his neck; and that Floyd was complaining about not being able to breathe before cops even laid a hand on him; and that Floyd had three times the fatal dose of fentanyl in him; and that his dope dealer, sitting next to him in the car and subject to prosecution if he admitted giving Floyd a fatal dose of drugs, took the Fifth and refused to testify.

Calling Chauvin's trial a kangaroo court is an insult to marsupials from the family Macropodidae. Not that Chauvin was officer of the year, by all accounts he was a bad cop. But if you believe in fair, impartial justice, this was a bad day all around.

I do apologize if I confused your response with someone else's. Someone said she got what she deserved. Well as Clint Eastwood said in Unforgiven to the Kid, who said of a guy he'd just shot to death, "He had it comin'": "Kid, we ALL have it comin'."

Then why make claims about what you didn't read? — Fooloso4

Don't believe I did. My claim was that people deplatformed and smeared anyone who even dared to suggest the possibility a lab origin for covid, The article claimed to delineate the process by which that happened. I confess I have a bad habit of posting links I don't read. Perhaps from now on I should include a disclaimer when I do that.

Only that finally, after a year, people are starting to admit the possibility.
— fishfry

And why do you think that is? — Fooloso4

Because Trump is no longer president, so they don't feel the need to irrationally object to everything anyone says that Trump might have agreed with. TDS killed people because it make liberals totally irrational. If Trump said the sky is blue the New York Times would deny it. That was a problem for the past several years.

In addition to not bothering to read the article you linked to it seems you have not bothered to find out the facts either. — Fooloso4

I'm painfully aware of the facts of censorship of reputable scientists who dared to oppose the MSM orthodoxy. I wonder why you're seemingly defending it.

It was the political hack who was elected President who suppressed the facts and forced Fauci to play by his rules. He is not anti-science and has the credentials to prove it. — Fooloso4

Fauci's credentials are that he's a career bureaucrat who never practiced medicine. He's no scientist and surely you know that. And say what you will about Trump, you can't call him a political hack. He's the anti-politician. One of his worst weaknesses was that he knew nothing of politics and how things get done in Washington. Trump was an anti-politician who had never run for elective office in his life. You can't call him a political hack. That's your TDS talking. You killed people with that malady.

What evidence did they have? What did Tom Cotton know? On the one hand you — Fooloso4

What's wrong with saying, "Let's keep an open mind and find out," as opposed to smearing him as a conspiracy theorist when, in the end, he's probably going to turn out to be right? Reputable scientists lost their jobs and got deplatformed for stating their opinion.

The first article I ever read about covid identified the wet market in Wuhan as the source, and then said that there just happens to be a bioweapons research facility a mile from there. At that moment I was a proponent of the lab leak hypothesis, because it makes more sense than bat soup. Every developed nation in the world is doing bioweapons research and it's inevitable for the occasional bug to escape. The most benign explanation is an accidental lab leak. In the worst case, it was a deliberate trial run for a global bioweapon. You still hanging on to the bat soup theory? Not even Fauci believes that anymore.

Next organism might be much worse. COVID-19 was a lucky test run. — frank

Nice to see you admit it was a test run. A test run of man-made bioweapon, a test run of media-induced hysteria as a means of social control.

[Former assertion not definitively proven; latter is perfectly obvious and will become more so as time goes on].

Unfortunately we learned that large swaths of America didn't evolve due to the test run, which is weird. Cultural flaw revealed, I guess. — frank

Or perhaps the globalists are being revealed as the evil madmen and women that they are, and the public is starting to wise up. Or didn't you get the memo about Bill Gates? He was hanging around with child predator Epstein in the hopes Epstein could get him a Nobel prize. The bloom is off THAT rose, it's fair to say.

"party A once used a building for some unspecified military purpose" to justify party B bombing and killing innocent civilians in party A's community who happen to be in that building. Can we get that much common ground — Baden

Not following the discussion except sporadically, and just quote-grabbing this one sentence in case I'm missing some context. But didn't the Israelis give sufficient warning for everyone to get out of that building first so that no live would be lost? And wasn't the building used for CURRENT and not just past Hamas terrorist activities? And what kind of news gather organization is AP if they don't know they're sharing a building with a terrorist organization? And finally, didn't AP say last year that reporters shouldn't get worked up over property damage? They said that when BLM and Antifa were burning down small businesses. Guess it all depends on whose building is destroyed.

Thanks, feel free to heap abuse.

That's simply th converse of treating policy as if it were science. — Banno

I am criticizing those who in the past year constantly called policy by the name of science. As in "follow the science," when they really meant, "Shut up and follow the latest contradictory policy." But surely if you say, "What do you think of apple pie," I'm entitled to say that I love apple pie but hate apple pie laced with rat poison. What we've been treated to over the past year is science laced with rat poison.

So what kind of answer are you looking for? I like science. Is that ok? And I deplore its misuse. Is that not ok to say?

Cantor, Boltzmann, Godel and Turing — Wayfarer

Cantor died of a heart attack. Boltzmann was a physicist. Turing was most likely killed by the Brits because he was blackmailable and knew too many secrets. So some say. The Beeb ain't what it used to be. Also FWIW sets can contain themselves.

https://en.wikipedia.org/wiki/Non-well-founded_set_theory

https://plato.stanford.edu/entries/nonwellfounded-set-theory/

fishfry is talking about politics and policy but calling it science. — Banno

I agree and admitted as much a couple of days ago. I'm all for science, and wholeheartedly against the WORD "science" being used as a synonym for "shut up and do what you're told, which is the opposite of what we told you yesterday." That's not science, but lately it's being PRESENTED as science, and more than one person in this thread has DEFENDED it as science. As for example claiming that Dr. Fauci has been doing science, when he does nothing but politics. And that's not a criticism, because Fauci is a career bureaucrat and not a scientist. People should understand that. If you want to say he's been a good bureaucrat, you might almost have a case. If you say he's been doing science, the facts are against you.

I surely have not denied that "5" has conceptual meaning. To say that the numeral "5", when it is properly used, must refer to five distinct particular things, is to give it conceptual meaning. It is a universal statement, therefore conceptual. I am not saying that it must refer to one specific group of five, as a name of that group, I am saying that it could refer to any group of five, therefore it is a universal, and this indicates that the "5" in my usage refers to a concept, what you've called an abstraction, rather than any particular group of five. — Metaphysician Undercover

Yes ok. I misspoke myself when I was unclear that a SET can be given an arbitrary order, and that no one order is to be preferred above any other; but that nevertheless you are correct that the natural numbers individually are either cardinals or ordinals, referring to quantity or order. You're right about that I should have been more clear.

For example, if I said that to properly use "square", it must refer to an equilateral rectangle, or "circle" must refer to a plane round figure with a circumference which has each point equidistant from its center point, I give these terms conceptual meaning, because I do not say that the words must refer to a particular figure, I allow them to refer to a class or category of figures. — Metaphysician Undercover

You're right, 5 refers to the class of sets having 5 elements, or it refers to a canonical representation in set theory of the number 5, or in Peano arithmetic it refers to the successor of 4. Quantity and order are essential aspects of natural numbers. I was wrong not to realize earlier that I should have noted that.

Even if I said that "5" must refer only to one particular group of five, or that "square" must refer only to one particular figure, it could still be argued that this is "conceptual meaning", because to understand this phrase "must refer only to one particular", is to understand something conceptual. In reality any meaning assigned to word usage is conceptual, so this position you've thrust at me, that I deny the conceptual meaning of 5, is nonsense. What I say is that the conceptual meaning given to "5", in some situations, namely that it refers to a type of object called a number (as described by platonic realism), ought to be considered as wrong. Do you accept the fact that concepts can be wrong? For instance, your example of "justice". A group of people could have a wrong idea about what "justice" means. Likewise, a group of people could have a wrong idea about what "5" means. — Metaphysician Undercover

Have I clarified my earlier inaccuracy enough yet? 5 refers to fiveness, quantity or order. I agree.

Why would you want to make this change to "inspired" rather than "grounded"? Logic is grounded in true premises, and this is an important aspect of soundness. If your desire is to remove that requirement, and insist that the axioms of mathematics need not be true, they need only to be "inspired", like a work of fiction, the result would be unsound mathematics. Sure this unsound mathematics might be fun to play with for these people whom you call "pure mathematicians", and I call "mathemagicians", but unsound mathematics can't be said to provide acceptable principles for a discipline like physics. — Metaphysician Undercover

Does our disagreement on this point go away now that I've clarified my inaccuracy?

OK, I assume that "less than" refers to quantity. — Metaphysician Undercover

Sigh. Less than refers to whatever is on the left side of x < y if '<' denotes a strict order relation.

So we're right back to my original argument then. Numerals like "1", "2", "3", "4", refer to a quantity of objects, "3' indicating a quantity which is less than that indicated by "4", and "first" indicates a lower quantity. How do you propose to remove the quantitative reference to produce a pure order, not grounded in a physical quantity? — Metaphysician Undercover

Ok. You're right, up to a point. Natural numbers refer to quantity or order. But the set of natural number may nonetheless be ordered in many alternative ways.

If you are grounding your definition of "order" in "less than", as you have, then numbers simply indicate quantity, and your "order" is just implied. It is not the case that "2" indicates "first" in relation to "3" and "4", it is the case that "2" indicates a quantity which is less than the quantity indicated by "3" and by "4". And by your premise, that the "first "is the one which is less than the others, you conclude that "2" is first. — Metaphysician Undercover

I hope I've clarified my exposition here. I see that I caused myself trouble by not being more clear earlier.

Therefore "order" as you have presented it is not indicated by the numbers, only quantity is indicated by the numbers. Order is indicated by something other than the numbers, it's indicated by your premise that the numeral signifying a quantity less than the others, is first. — Metaphysician Undercover

Sorry that one lost me.

You clearly haven't followed what I've been saying, — Metaphysician Undercover

Not for lack of trying.

and I realize that I did not make myself clear at all. — Metaphysician Undercover

Ok let's skip all this and hopefully go forward with my admission that natural numbers individually do refer to quantity or order; but (imperative you get this) the SET of natural numbers may be reordered at will.

The point is that if we remove the reference to a quantity of individual objects, from numerals, then the ordering of numbers requires a spatial or temporal reference. You seemed to believe that we could remove the quantitative reference, and have numbers with their meanings understood in reference to order only. Clearly, "less than" does not provide this for us. And your example of the length of the word here, is a spatial reference. — Metaphysician Undercover

I agree that natural numbers have a quantitative reference. I don't know why I obfuscated that earlier. My bad.

Your other example, of the best score being first is only made relevant through a quantitative interpretation. How is 3 better than 4? Because it's less than. So you have not removed the reference to quantity as the necessary aspect of numerals, to provide a purely ordinal definition. Therefore I am still waiting for you to prove your claims. — Metaphysician Undercover

You swapped out temporal for quantity. Sneaky sneaky.

You have shown me absolutely nothing in the sense of a number not dependent on quantity for its meaning. — Metaphysician Undercover

Quaternions? Transcendentals? p-adics?

If your point is that "order" is defined by " less than", and this is supposed to be an order which is independent from quantity, then you've failed miserably at making your point. — Metaphysician Undercover

Or you've failed miserably to understand my point.

Obviously, "in the playground" is not "in the classroom", and you're clutching at straws in defense of a lost cause. — Metaphysician Undercover

You're hanging on to the other end of the straw.

Instead of addressing my argument you portray me as mathematically ignorant. It's not a matter of ignorance on my part, it's a refusal to accept a mathematical axiom which is clearly false. So I'd correct this to say that this is an instance of your denial, and willful ignorance of the truth, for the sake of supporting a false mathematical axiom. — Metaphysician Undercover

I based my statement on your general mathematical ignorance, and the way you use it as a weapon in debate.

Show me that set which has no order then. — Metaphysician Undercover

{a, b, c}.

Or $\mathbb N]/math] without any particular order being implied.<br />$

And remember, there is a difference between a thing itself, and the description of a thing. Therefore to describe a set which has no order is not to show me a set which has no order. — Metaphysician Undercover

But no set has order. That's the axiom of extensionality. Will you kindly engage with this point?

I think you need to make clear what "set" means. — Metaphysician Undercover

LOL. For purposes of our discussion, anything that satisfies the ZF or ZFC axioms. The very first of which is the axiom of extensionality which says that sets have no inherent order, being completely characterized by their elements.

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

Does it refer to a group of things, or does it refer to the category which those things are classed into? — Metaphysician Undercover

A set is a mathematical set.

The two are completely different. Take your example of "schoolkids" for instance. Does "set" refer to the actual kids, in which case there is necessarily an order which they are in, even if they are running around and changing their order? Or, does "set" refer to the concept, the category "schoolkids", in which case there are no particular individuals being referred to, and no necessary order? Which is it that "set" refers to, the particulars or the universal? Or is "set" just a clusterfuck, a massive category mistake? — Metaphysician Undercover

Schoolkids are not mathematical sets. I'm using them purely as an analogy.

You just named it for me. "Mathematical" is the word I use to refer to mathematical concepts. In ethics there are ethical concepts like justice, in biology there are biological concepts like evolution, and in physics there are physical concepts like mass. Why do you think mathematics ought to be afforded the luxury of treating their concepts as if they are objects? — Metaphysician Undercover

So I can use the phrase mathematical, but not mathematical objects? But mathematical is an adjective and mathematical object is a noun. You've still not answered the question.

But are you saying that if I call 5 a "mathematical concept" you're ok with that, but NOT with my calling it a mathematical object? Ok, I can almost live with that. Although to me, it's a mathematical object.

Fauci is anti-science.
— fishfry

No need to say anything further, that speaks volumes. — Wayfarer

Fauci is not even a scientist. He has an MD but never practiced medicine. He's been a bureaucrat all his life. Surely you can't hold him up as a scientist. And his endless politicized flipflops speak for themselves.

I read in the weekend papers that the Lab Escape theory is being re-considered. If that turns out to be the case, then so be it, although presumably it might have serious ramifications for China. — Wayfarer

Gosh without that, China would be a wonderful country. Or as I like to say: Uyghur please!

But you are making my point for me exactly. A year ago if you espoused the lab leak theory nobody said, "We should keep an open mind and wait for the evidence." THAT would be science. On the contrary, reputable scientists lost their jobs, got banned from social media, got labeled conspiracy theorists. And now that even Dr. Fauci finally admits that the lab leak might be true, all you can say is that it might be bad for China.

Tell me, a year ago when you read that it started in a wet market and "Oh by the way there's a bioweapons lab a mile away but pay no attention," what exactly was your thought process at the time? Were you unaware that every developed country in the world conducts bioweapons research? The only thing I didn't know about it was that the US actually funds China's bioweapons research. Now that's shocking, but then again, maybe not so shocking.

So - you're either pro-science, or you're relegated to pagan superstition. They're your choices. — Wayfarer

You make my point for me. Smearing, deplatforming calling people conspiracy theorists, is anti-science.

The zealotry with which the American Left pursues the Right can actually get out of hand. The FBI tried to entrap that guy into selling weapons to the far-Right, which he refused, and they later waged a full-blown military operation against him in more or less his shack. It gave birth to the American Militia Movement and, by proxy, kind of a lot of existent far-Right groups today. There was an Anarchist in Aufheben who got a lot of flak for conceptualizing some sort of "soft-policing powers". I doubt that he was taken at his word, but such measures would seem to be a lot more effective in countering the far-Right than the heavy-handed measures of Ruby Ridge or Waco, Texas. — thewonder

Yes and vice versa. Back in the day it was right wing zealots pursuing the left. Now it's reversed. Not much better. Human nature is awful.

Personally, I'm of the theory that, should law enforcement regulate the drug trade in such a manner to where it is nonviolent and not Fascist, the far-Right will just simply dissolve. Good luck convincing a single other person of that, though. — thewonder

Not entirely sure what that means. The CIA is the world's greatest drug trafficker, has been for decades.

This piece lacks credibility:

Now that it’s largely accepted that the disease escaped a Chinese laboratory, have any of those above issued a correction or so much as an update?

First of all, it may be widely accepted by readers of the Federalist, but it is not widely accepted by those who have the expertise and information to have an informed opinion. Second, there is at this point no reason to issue a correction, there is not conclusive evidence that it did come from a lab. Third, Fauci did issue an update. He said he is no longer convinced that it could not have come from a lab and thinks that more investigation is needed. — Fooloso4

Full disclosure, I didn't read the entire article. I do not agree that it's "widely accepted," nor is it known. Only that finally, after a year, people are starting to admit the possibility.

I found nothing in the article about "systematic suppression" of the origins of the virus.

Both you and whoever wrote this piece seem to not understand how science works. — Fooloso4

Science works by saying, "Let's keep an open mind and look at the facts." Not, "Let's decide on one conclusion in spite of available facts, and deplatform and smear anyone who dares to differ." That's anti-science, and that is what happened over the past year.

Did Tom Cotton have sufficient evidence to declare in February 2020 that the virus came from a lab? Without such evidence his claim was irresponsible. Fauci's response is both reasonable and responsible. Follow the evidence. — Fooloso4

Fauci is a political hack who changed his mind and flipflopped with the wind. Fauci is anti-science.

This is what Politifact has posted on its website: — Fooloso4

Extremely biased site.

That seems to be exactly what you are doing. — Fooloso4

A year ago, when people suggested a lab origin, they were deplatformed, fired from their scientific jobs, and labeled conspiracy theorists. That's politics, not science. Comrade Lysenko would be proud.

You'd have made a good racist! I think perhaps you mean Nazi. — counterpunch

I think you know what I meant. The "good Germans" we heard so much about, as in "Where were the good Germans?" Now that I've seen the past few years in the US, I understand better where they were.

George Floyd's choices created the situation. If he'd complied he wouldn't have died. The jury decided the police officers actions were disproportionate - and I accept that, but it remains, he could have got in the car, and he'd still be alive. — counterpunch

Well then we're all in agreement. Personally I don't resist cops, I comply and act polite. That's because when I was young and foolish, I sassed off to a cop and got a night in the Oakland, CA city jail for my troubles. Got my Ph.D. in the criminal justice system that night. Now that I'm old and foolish, I'm polite to cops.

And you'd have make a piss-poor philosopher. Was she tased to death? — Fooloso4

I don't understand the kind of human being who can watch a video like that and conclude that "she got what she deserved." I will agree with you that she should have just either left or put on her effing mask. I'll grant you that. But what I'm questioning is the emotional response to that video of "she got what she deserved." You're lacking in basic human decency.

And for the record, I'm a piss-poor philosopher.

deserved everything she got. — counterpunch

You'd have made a good German. And if she deserved everything she got. didn't George Floyd? Or is your violent authoritarianism one-sided?

I see that you saw the same video I did. I can't fathom the kind of human being that would see that video and say "she deserved what she got." People like you frighten me.

In 1952 Dulles was officially appointed as head of the CIA. — Apollodorus

John J McCloy (of the Rockefeller law firm Milbank, Tweed & Hope) Stimson’s Assistant Secretary, trustee of the Rockefeller Foundation, president of the World Bank, and US High Commissioner for Germany. — Apollodorus

Great summary, thanks much. Dulles and McCloy of course were also members of the Warren commission. Dulles had been fired by JFK after the Bay of Pigs, and hated JFK long before that. So his appointment to the commission was inexplicable. His role was mainly to hide the involvement of the intelligence community's connections to Oswald (full disclosure, these connections are still not proven) and their partnership with the Mafia to kill Castro (absolutely proven and well known).

Most of what we know about the workings of our government is a carefully curated myth, yet so many these days take government pronouncements at face value. Just yesterday Fauci finally admitted that the lab leak hypothesis for covid may be true. This is after a year of MSM smearing of that suggestion as a "conspiracy theory."

ps -- Almost forgot to mention these hot-off-the-presses nuggets. Story came out yesterday that three workers at the Wuhan bioweapons lab got sick in late 2019. And this morning the indefatigable Jen Psaki said we have no evidence it's true. Keep spinning, Jen.

And Maggie Haberman of the NYT said that the reason the MSM colluded to suppress all dissident speculation about the origin of covid was ... It's Trump's fault!! I am not making this up.

https://www.breitbart.com/clips/2021/05/24/nyts-haberman-blames-trump-admin-for-media-discrediting-theories-of-covid-19-origins-made-this-instantly-political/

Let's hope that it doesn't become another legend of Ruby Ridge. — thewonder

Let's hope. The government wants to label everyone who dissents a "white supremacist." It's not true, but it's an example of the Big Lie. We'll have to see how this plays out over the next few years.

It doesn't seem to me to be a precursor to category theory, but I don't opine. — TonesInDeepFreeze

Thanks much.

From what I've seen in your posts, you and I have a very different understanding of how science and science-based policy making are supposed to work. — T Clark

Shortly after my last post to you, I ran across a video of a woman being tased for refusing to put on a mask. Just yesterday, Fauci finally admitted that covid might have a lab origin. This morning The Federalist ran a long piece about how sensible independent thought regarding the origin of covid was systematically suppressed.

Most of what comes from our authorities these days is absolute bullshit. I can't understand the mindset of people who uncritically accept everything without question.

I surely have not denied that "5" has conceptual meaning. — Metaphysician Undercover

Hi, I didn't read the rest of your post yet but I realized I needed to clarify what I wrote last night.

4 is indeed the cardinal number 4 or the ordinal 4, and 5 is the cardinal or ordinal 5. So 4 and 5 do indeed represent quantities, or orders. I misspoke myself, or rather I failed to adequately address your point.

What I was saying about order is that the usual or standard order on the entire set of natural numbers is not the only possible order. So if I decide to reorder the naturals as 1, 2, 3, 5, 4, 6, ..., where 5 comes before 4, it is still the case that 5 represents a set of five elements. So each individual natural number can be seen as representing a quantity, or an order. For example in the Peano construction, 5 is the successor of 4, which is the successor of 3, etc.

So it's the SET of natural numbers that have no inherent order. But an individual natural number does represent a quantity or order.

On the other hand, rational, real, and complex numbers can't be seen as representing quantity in the same way that natural numbers do. Quaternions are little known, but it turns out that game developers use quaternions because they're the most natural tool for representing 3-D rotations. So you can add "rotation" to quantity and order. Every number represents some quality of interest, but there are more of those than just quantity and order.

Hope this clarifies a point of confusion that I didn't adequately address last night. I'll get to the rest of your post later.

The FBI claims to no longer be engaged in such operations — thewonder

No comment. Chuckles and guffaws, but no comment.

https://www.cnn.com/2021/05/03/politics/dhs-partner-private-firms-surveil-suspected-domestic-terrorists/index.html

Conspiracy theories are a different matter. They are often offered as explanation for events or situations for which there may or may not exist non-conspiratorial explanations. — Apollodorus

The phrase "conspiracy theory" is used to dismiss and marginalize any dissent from official opinion.

The CIA didn't invent the phrase, but they promoted it in order to smear critics of the Warren report.

http://www.jfklancer.com/CIA.html

But don't just take the word of some "conspiracy theory" website. Here is the exact same CIA memo, written up in the New York Times.

https://www.nytimes.com/1977/12/26/archives/cable-sought-to-discredit-critics-of-warren-report.html

And in a bit of double-reverse counterspin, someone even wrote an article trying to debunk or deflect from the implications of this very CIA memo.

https://theconversation.com/theres-a-conspiracy-theory-that-the-cia-invented-the-term-conspiracy-theory-heres-why-132117

Many people mindlessly use the phrase conspiracy theory to dismiss ideas that make them uncomfortable, so that they don't have to think too deeply or actually marshal facts and arguments in support of their own beliefs. As you noted, conspiracies are a major part of human history. The Department of Justice employs thousands of attorneys to investigate and prosecute conspiracies. They could properly be called conspiracy theorists.

Or as I like to put it: Julius Caesar was not stabbed by a lone knifeman; and 9/11 was not perpetrated by a lone planeman. They were two of the greatest conspiracies in world history. And if you don't know that powerful people are conspiring against your interests at this very moment, then you'll fall for the next Reichstag fire or Gulf of Tonkin or WMDs or War on Drugs or War on Terror that they try to foist on you.

Let's get this straight. I am not talking physical referents here. I am talking space and time, which are conceptual. — Metaphysician Undercover

Now that is very interesting. When you say space and time, I've understood you to be referring to those words as understood in physics. The space and time of the physical world. Which makes sense. You would be claiming that 4 comes before 5 in terms of physical space or time.

But now you are saying that space and time have "conceptual" meaning; at the same time you deny that 5 or other numbers can have conceptual meaning. I confess you've lost me and perhaps lost your own point as well. If space and time are abstract conceptual things, then why can't numbers be also?

The issue is that when we remove the physical referents (required for "counting" in the sense of determining a quantity, as the things counted), for the sake of what you might call purely abstract numbers, the meaning of the numbers is grounded in the abstract concepts of space and time. — Metaphysician Undercover

How about "inspired by" rather than grounded? As in Moby Dick being a work of fiction nevertheless inspired by a real historical event. Of course we get our concept of number from real, physical things. Nobody's denying that.

Numbers no longer refer to physical objects being counted, they refer to these abstract concepts of space and time. — Metaphysician Undercover

Ok, but now I'm confused by your claim that space and time are no longer physical things, but rather conceptual things. Why can't numbers be conceptual things inspired by physical things too?

Now, we have only deferred the need to refer to physical existence, because if our conceptions of space and time are inaccurate, and the ordering of our numbers is based in these conception of space and time, then our ordering of the numbers will be faulty as well. You seem to think that in pure mathematics, a logician is free to establish whatever one wants as "an order", but this is not true, because the logician is bound by the precepts of "logic" in order that the order be logical. — Metaphysician Undercover

You've swapped out mathematicians for logicians, and I'm not sure I can accept that. I'm talking about mathematical practice, which goes far beyond logic. Logicism's dead, right?

I am making the point that in order theory, one order is as good as another. An order is ANY relation that's reflexive, antisymmetric, and transitive. That's what an order is in order theory. I didn't make this up, it's on Wikipedia and as someone with a (little) bit of mathematical training, I can confirm that Wiki got this one right. https://en.wikipedia.org/wiki/Order_theory

Of course you are correct that the natural order of the positive integers is 1, 2, 3, ... but that is not the ONLY possible order relation on them, there are many others.

For example, a self-contradicting premise is not allowed. — Metaphysician Undercover

As long as my order is reflexive, antisymmetric, and transitive, it's allowed. And for the umpteenth time (umpteenth is an ordinal!!), a contraction is a statement P such that both P and not-P can be proved from a given set of axioms. A contradiction is NOT merely something that offends your intuition. In math we get quite accustomed to having our intuitions challenged and corrected.

So there are fundamental rules as to the criteria for "order" which cannot be broken. — Metaphysician Undercover

Yes there are. Reflexive, anti-symmetric, and transitive. Those are the rules. Or sometimes we required a strict order for convenience, and deny reflexivity (ie < rather than <=) but that's a small point.

But there ARE rules, and the rules are documented on Wikipedia, and I've pointed them out to you.

And even if you argue that the order could be a completely random ordering of numbers, the rule here is that each thing in the order must be a number. — Metaphysician Undercover

Even that's not true. I can put an order on red, green, blue. Say lex order: blue, green, red. Or length order: red, blue, green. In each case the order is reflexive, anti-symmetric, and transitive.

I'm sorry your intuition is challenged on this point, but a big part of learning mathematics is having our intuitions challenged, so that we come out the other end with better intuitions.

And every time a logician tries to escape the rule, by establishing a principle allowing oneself to go outside that rule, there must be a new rule created, or else the logician goes outside the field of logic. — Metaphysician Undercover

I have no idea why you've swapped in logicians. I'm talking about mathematicians. I'm explaining to you how mathematicians define order. I can't help your naive intuitions, I'm trying to dispel them in favor of more clarifying concepts.

And the point, is that if the rule is not grounded in empirical fact (physical existence) the logic produced is faulty, and the proposed rule ought to be rejected as a false premise. — Metaphysician Undercover

The only thing faulty is your intuition about what an order relation is.

Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it? — Metaphysician Undercover

Well the "first" element of a total order is an element that is less than any other element. Some orders have a first element, such as 1 in the positive integers. Some orders don't. There's no first positive rational number.

That's what first means.

Yes that is my point as to how counting order is different from counting a quantity. — Metaphysician Undercover

Now that's funny, as we got off onto this conversation by pointing out to you that numbers can indicate order as well as quantity. But of course ordinals are different than cardinals. Two distinct ordinals can have the same cardinal.

To count a quantity requires particular things, but to count an order requires only time. — Metaphysician Undercover

red, blue, green. Three words ordered by length. There is no time involved. You are stuck on this point through stubborness, not rational discourse. The player who finishes first in a golf tournament is the one with the lowest score, NOT the one who races around the course first.

However, time is something in the world, and that's why I don't believe in what you call "the pure concept of order". — Metaphysician Undercover

Your belief in mathematics is not required by mathematics. Mathematics can exist in the world side-by-side with your willful ignorance and obfuscation.

If order is not essential to numbers, then something else must be, because to be a concept is to be definable according to essential properties. I propose, then that quantity is essential to numbers. Do you agree? — Metaphysician Undercover

I have already given many counterexamples such as rationals, reals, complex numbers, p-adics, hyperreals, and various other exotic classes of numbers studied by mathematicians. What quantity or order does $3 + 5 i$ represent?

There is no general definition of number in math. That's kind of a curiosity, and it's kind of an interesting philosophical point, and it's also factually true.

If for example you make an order, or a category, of odd numbers, or even numbers, or prime numbers, it is something about the quantity represented by the number which makes it belong in one or more of these categories.[/quote}

What makes 6 an even number is that it's divisible by two; or equivalently, that it's residue class mod 2 is zero. That's how we recognize 6 as an even number.

Here's a more striking example that even you will have to concede. I can recognize 45385793759385938534 as an even number without knowing ANYTHING about its quantity or order. I merely have to note that the low-order digit is even, and appeal to the theorem that a number is even if and only if its low-order digit is.

— Metaphysician Undercover

If it's not quantity which is essential to numbers, as the defining feature of "number" then what do you think is? You've already ruled out order. — Metaphysician Undercover

There is no particular attributed that's a defining feature of number. The concept of number is a historically contingent opinion of mathematicians. Zero didn't used to be a number, neither did $5 + i$, and neither did $\aleph_{47}$. Today they're all regarded as numbers.

There is no general definition of number; nor is there any particular defining property by which we can say, "This thing is a number," and "That thing isn't." The concept of number is whatever the mathematicians of a given era agree is a number.

That's how it is.

No, I am saying that if order is secondary to the existence of numbers, then quantity must be primary — Metaphysician Undercover

There's no general attribute that uniquely characterizes a thing as a number. What is or is not a number is a matter of historically contingent opinion of mathematicians.

That's not true at all, it's the fallacy I referred to. The schoolkid must have height, and that height must be the height that the schoolkid has. Therefore it is impossible that the schoolkid has a height other than the height that the schoolkid has, and very obviously impossible that "it could be any height". To make such a claim is clearly fallacious, in violation of the law of identity, because you are implying that a thing could have properties other than those that it has, saying it could have any property. Obviously this is not true because a thing can only have the properties that it has, otherwise it is not the thing that it is. — Metaphysician Undercover

I think we're at a point of diminishing returns in this convo. You're flailing and not saying anything I find interesting enough to even argue with.

It gives me distress to see you describe something so obviously fallacious as "providing beautiful logical clarity". If you consider circumventing the law of identity as beautiful logical clarity, I have pity. — Metaphysician Undercover

Why don't we table this till next time. I've made my point and all you have is mathematical ignorance. That's all you ever have. You've even fatally undermined your own thesis by agreeing that space and time aren't even the space and time of physics, but are rather "conceptual," while denying the same status to numbers.

Again, you're continuing with your fallacy. A classroom full of kids must have an order, or else the kids have no spatial positions in the classroom. — Metaphysician Undercover

You haven't seen them in the playground at recess. Of course that's only when I was a kid. These days I gather they don't let the kids run around randomly at recess.

Clearly though, they are within the classroom, and whatever position they are in is the order which they have. To deny that they have an order is to deny that they have spatial existence within the room, but that contradicts your premise "a classroom full of kids". — Metaphysician Undercover

You're flailing and no longer even trying to make a coherent point.

Above, is your CLEAR example of contradiction "a classroom full of kids has no inherent order". By saying "there is a classroom full of kids", you are saying that there is an order to these kids, they exist with determinate positions, in a defined space. You contradict this by saying they have no inherent order. — Metaphysician Undercover

That's manifestly false.

So, if "a set" is like the kids in the classroom, then it must have an order to exist as a set. — Metaphysician Undercover

If you don't know that sets have no inherent order, there is no point in my arguing with your willful mathematical ignorance.

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

We can say that the order is accidental, it is not an essential feature, so that the same set could change from one order to another, just like the kids in the class, and still maintain its status as the same set. — Metaphysician Undercover

No that is not true. It's entirely contrary to the concept of set. A set has no inherent order. An order is a binary relation that's imposed on a given set. If I have a set and don't bother to supply an order relation, then the set has no order. Sets inherently have no order. That's what a set is. You can sit here all day long and make up your own definitions, but that's of no use or interest to anyone.

However, we cannot say that a set could have any order by reason of the fallacy described above, because this is to say that it has no actual order which implies that it does not exist. — Metaphysician Undercover

It can have any order as long as the order satisfies the properties of a partial or total order.

No, I don't see that at all. They are all concepts, ideas. By what principle do you say that mathematical concepts are "objects", but concepts like "justice" are not objects. I mean where is your criteria as to what constitutes a conceptual "object". I know it's not the law of identity. — Metaphysician Undercover

I'm asking you, if you don't accept the phrase mathematical object, what phrase do you use to name or label conceptual entities that are mathematical, as opposed to conceptual entities like justice that are not mathematical?

You've never heard "the object of the game"? — Metaphysician Undercover

That's a different meaning. You're being silly now, unserious.

So in this context, "first" means best. Clearly this is not how "first" is used in mathematics. In mathematics, "first" has a temporal reference of prior to, as I said, not a qualitative reference as "best". — Metaphysician Undercover

You're wrong. If you have a set, and you impose an order on the set, and in that order there's an element that's less than every other element, that order may be called the first element. That's the definition.

Your attempt at equivocation is not very good, I'm happy to say, for your sake. Ask Luke who is the master of equivocation for guidance, if you want to learn. I think you ought to stay away from that though. — Metaphysician Undercover

Ad hominems is all you've got left, I see. I hope you will not mind if I'm done here, nothing new can be said at this point.

.The problem obviously, is that you, and mathematicians in general, according to what you said above, haven't got a clue as to what a number is. — Metaphysician Undercover

Project much?

It's just an imaginary thing which you claim is an object. It appears like you can't even tell me how to distinguish the number 4 from the number 5, because you refuse to recognize the importance of quantity. — Metaphysician Undercover

I distinguish them just fine.

And if you would recognize that it is by means of quantity that we distinguish 4 from 5, then you would see that "4", and "5" cannot each represent an object, because one represents four objects, and the other five objects. Why do you take numbers for granted?
. — Metaphysician Undercover

It's been fun chatting. I'm done with this topic. Till next time.

The original paper is in Jean van Heijenoorts's 'From Frege To Godel'. — TonesInDeepFreeze

I see that I can buy a copy, but I didn't find a pdf. I'm wondering if you could summarize. Did von Neumann anticipate the categorical approach to set theory way back on 1925? Wouldn't be surprised, just curious to know, but not curious enough to buy the book. I have half a dozen physical books already stacked up to read. It's so much easer to buy books than to read them. Someday we'll just be able to upload via neural interface.

And it's even classified as Vital. There must be 10K pages on math on Wiki. I wonder how many are added each day? — jgill

What's the smallest positive integer that Wiki doesn't have an article on? That would make it deserving of its own article!

As ugly as it got, I have been impressed with how well science-based policy making worked in the US. — T Clark

I hope you would allow that people of good will could see the politicized science of the past year very differently.

Or, as Jerry Seinfeld reminds us, taking Silver in the Olympics just means you're the best of all the losers. — TonesInDeepFreeze

Or in a two-person race, the loser finished second, and the winner finished next to last.

I did? Odd, 'cause that ain't what I wrote. — Banno

Ok.

OK, so you define "order" as "having no meaning". That is your starting premise? What's the point? Any meaning you give to it will be logically invalid, as contradictory to that definition. There is nothing to study in a concept which has no meaning. — Metaphysician Undercover

Excellent job of misquoting me, attributing to me things I didn't say, and launching yourself into another irrelevant tirade against math, or abstraction, or the ordinal numbers, or whatever it is you're against.

Perhaps you're right that meaning isn't the correct word. If I said we remove a concept from its worldly or physical referent, would that be better? We care about first, second, third, and not first base, second base, third base. So how would you describe that? I'm focusing on ordinality itself and not the things ordered. So you're right, meaning was an imprecise word.

Of course it makes it meaningless, you just said you take away meaning from it. If you take away all the meaning from "first" and "second", you just have symbols without meaning. — Metaphysician Undercover

We have order, without reference to the things ordered. We still have meaning, I'll concede that meaning was the wrong word. What would would you use?

If you leave some sort of meaning as a ground, a base, you have a temporal reference, first is before, (prior to) second. — Metaphysician Undercover

There is no temporal reference.

You are using "abstract" in a way opposite to convention. We do not "take away meaning" through abstraction, abstraction is how we construct meaning. — Metaphysician Undercover

No not at all. One meter, one fish, one planet is meaning. One by itself is a mathematical abstraction. I'm not entirely sure that it means anything now that you mention it. But we can still study it, and then apply what we learn in the abstract setting to any particularities of interest.

There is a process called "abstraction", by which we remove accidental properties to give us essentials, what is necessary to the concept. We do not abstract away the meaning, we abstract what is judged as "necessary" from the concreteness, leaving behind what is unnecessary, "accidental". — Metaphysician Undercover

Ok. I agree that I'm having trouble precisely defining abstraction and I sort of see your point. But ordinal numbers are purely about order, but they're not about any particular things being ordered. How would you describe that? It's not meaningless, yet it refers to nothing in the world at all other than the pure concept of order. Which you don't seem to believe in.

Sure, cardinality is not the only possible way of ordering numbers, but if the point is, as you described, to allow for any possible order, then we have to deny the necessity of all possible orders. — Metaphysician Undercover

No, although we do deny the primacy of any particular order. That is, in order theory, the usual order 1, 2, 3, ... is no more important or special than one of the funny orders like 1 2, 4, ..., 3. Although the 1, 2, 3, ... order is important enough to give it a name, the "standard order" or "usual order" on the natural numbers. But you are correct that in order theory, the process of abstraction does put us in the position of regarding all possible orders as equally valid. Not unlike lining up the schoolkids by height, by alpha last name, by reverse alpha first name, by date of birth, by test score, etc. Each of those orders is equally valid in some particular context, and none is inherently preferred over any other. Right? Surely you'll grant me that. And then further grant me that mathematically, sometimes the usual order on the natural numbers is useful (like in most ordinary usages of math), and other times alternate orders are (like when studying or using the higher ordinals).

That is to say that there is no specific order which is necessary. — Metaphysician Undercover

Correct correct correct. Although I suspect you're about to object to that! But yes, that is exactly right. One order is as valid as another if we're studying pure order theory; although we DO honor the grade school teachers of the world by giving 1, 2, 3, ... a special name, the standard or usual order.

This removes "order" as a defining feature of numbers, — Metaphysician Undercover

Absolutely correct. I think you're trying to disprove or invalidate the idea, but actually you're understanding the process perfectly.

because no order is necessary, so numbers do not inherently have order. — Metaphysician Undercover

Correct correct correct. One order is as good as another, though the standard order has considerable mindshare among the general public and of course among mathematicians too. I'd be pushing the point too far if I denied that the standard order is special. After all in the standard order, the numbers are arranged by cardinality, which important; and the ordinals are arranged by set membership, whichis also important. So yes there IS in fact a "natural" way of characterizing the standard order is important.

Therefore order is not essential to the concept of numbers — Metaphysician Undercover

Just as being lined up by height is not essential to the concept of school children. Being orderable in one or many ways is an attributed of children and numbers, but it is not essential to the concept.

Then, we need something else to say what makes a number a number, or else we just have symbols without meaning. — Metaphysician Undercover

The question of what is the meaning of numbers is an interesting one. I'm not sure mathematicians concern themselves about it, just like biologists don't spend much time talking about the meaning of cells, or physicists (when they are doing physics) talk about the meaning of quarks. When physicists are doing philosophy, they talk about the meaning of quarks. And when mathematicians are doing philosophy, perhaps they talk about the meaning of numbers. But even on that last point, I'm not too sure.

I don't know what numbers "mean." I had dinner earlier and I don't know what my dinner meant. I know it tasted good. Is that a problem, that I don't know the meaning of dinner? What do you even mean by meaning in this context?

We could try saying that it is necessary that numbers have an order — Metaphysician Undercover

It isn't. We could consider the set $\mathbb N$ of natural numbers, which has no particular order at all, or that implicitly comes along by convention with its standard order. But order is not essential to numbers, it's imposed afterward. At least in the mathematical formalism. I get that you are drawing a distinction between the mathematical formalism, in which order is secondary to the existence of numbers; and philosophy, in which order is an essential aspect of numbers.

But in the Peano formulation, order is inherent via the successor relation. In the past you've rejected the Peano axioms, but now it seems that you should be happy with them. Because in Peano arithmetic we have 0, and we have S0, and then SS0, and then SSS0, and so forth, and there is an inherent order to the process. Happy now?

, but the specific order which they have is not necessary, like we might say a certain type of thing must have a colour, but it could be any colour. — Metaphysician Undercover

A schoolkid must have a height, but it could be any height. But with numbers it's even worse than that. A set of numbers, like a set of anything, has no inherent order. Order is a relation imposed upon a set. The set is logically prior to the order. Yes you are right about that, and I get that you're unhappy about that, but that's how it is. At least in the modern formulation of these matters.

But this will prove to be a logical quagmire — Metaphysician Undercover

You see it that way. I see it as providing beautifully logical clarity. We have the set of natural numbers, and we have the standard order and we have a lot of other orders, and we can even consider the entire collection of all possible orders, which itself turns out to be a very interesting mathematical object. It's quite a lovely intellectual structure. I'm sorry it gives you such distress.

because it's really just a way of smuggling in a contradiction. — Metaphysician Undercover

A contradiction is a proposition P such that both P and not-P may be proven from the axioms. Perhaps you would CLEALY state some proposition whose assertion and negation are provable from the concept of order as I've presented it. I don't think you can.

I would believe that you have some philosophical unease. That's not the same as a contradiction. Can you see that?

It is impossible, by way of contradiction, that something must be a specific colour, and at the same time is possibly any colour. — Metaphysician Undercover

But I have not asserted that a set must have any order at all. The set $\mathbb N$ has no inherent order at all. Just like a classroom full of kids has no inherent order till the teacher tells them to line up by height or by alpha firstname or reverse alpha lastname or age or test score or age. Why can't you see that?

A set has no inherent order. Order is imposed on a set afterward, and only for our own convenience in a given context. Sometimes one order, sometimes another, depending on what we're trying to achieve or express.

It is only possible that it is the colour that it is. — Metaphysician Undercover

Terrible analogy. Physical objects have color, but sets don't have an inherent order. Besides I could play the game of pointing out that while physical objects reflect light of a particular wavelength, their color is a function of the physiology of the visual system of the perceiver. So the color isn't really inherent in the object.

But I won't go there. Rather, I will just note that physical objects do have color (or at least a wavelength that gets reflected when it is hit with white light), and mass, and electric charge, and various other physical parameters. But sets do not have inherent order and this is absolutely fundamental to the nature of sets. Axiom of extensionality again: A set is completely characterized by its elements. Order has nothing to do with it, and a set by itself has no inherent order at all.

Likewise, it is impossible that numbers must have a specific order, but could possibly be any order, because the order that they currently have, would restrict the possibility of another order. — Metaphysician Undercover

Not at all, and I've shown you the example several times. The order sets $(\mathbb N, \lt)$ and $(\mathbb N, \prec)$ are the same underlying set of elements, each with a different order. Neither order is inherent to the underlying set.

The point was, that if remove all order, to say that numbers are not necessarily in any order, then we must define the essence of numbers in something other than order. — Metaphysician Undercover

Absolutely agreed. Yes. The essence of a set of numbers is NOT in their order, since we can easily impose many different orders on the same underlying set. Just as the ordering by height is not essential to the classroom of kids, since we can impose a different order; or by letting them loose in the playground at recess, we can remove all semblance of order! Surely you must take this point.

If this is cardinality, then cardinality is not an order. — Metaphysician Undercover

Of course cardinality is not an order, I thought that was abundantly clear long ago. But yes we can order a set by cardinality, if the set consists of elements of distinct cardinality. We can do that. We can order the kids by height, if in fact their heights are all distinct. If two kids have the exact same height then we can't linearly order the class by height.

They are concepts, abstractions. — Metaphysician Undercover

Ok. But that's not good enough. I asked how do you call mathematical objects like topological spaces. But justice and property are concepts and abstractions, yet they are not mathematical objects.

If you don't like the phrase, "mathematical object," what do you call them? Sure they're an abstraction, but that's way too general. You see that I'm sure.

I apprehend a difference between concepts and objects, because concepts are universals and objects are particulars. There is an incompatibility between the two, and to confuse them, or conflate them is known as a category mistake. — Metaphysician Undercover

Ok. I call numbers, sets, topological spaces, Abelian groups, etc., by the collective name mathematical objects. What do you call them? You can't say "abstractions," because justice and property are abstractions that are not mathematical objects. Consider yourself challenged to come up with a better name, if you don't like "mathematical object."

It's an idea, and ideas are not objects. — Metaphysician Undercover

Ok. this is a difference between us. I say 5 is a mathematical object, a very familiar one. Relatively few people know what a topological space is, but every child knows what 5 is. You say it's an idea and not an object. I think you're wrong about that. But we've been arguing this point for a long time.

I have an idea to post this comment, and this idea exists as a goal. Goals are "objects", or objectives, in a completely different sense of the word. So if you want to say that numbers, as ideas are "objects", we'd have to look at this sense of the word, goals. — Metaphysician Undercover

An object is not a goal. An (American) football is an object, and the goal is to get it across the goal line. You would not say the football is a goal. I think you're way off the mark with your claim that an object is a goal or objective. 5 has no object or purpose. It's just the number 5. A mathematical object. An abstract object, as all mathematical objects are.

But it doesn't make too much sense to say that they are objects in this sense, nor does it make any sense at all, to say that numbers, as ideas, are objects in the sense of particulars, because they are universals. — Metaphysician Undercover

We disagree, since I say 5 is a mathematical object. And I don't think you have a good theory to the contrary. And 5 is a PARTICULAR mathematical object. A universal is a "... class of mind-independent entities ..." (Internet Encyclopedia of Philosophy). 5 is not a class of entities, it's a single entity. A mathematical object.

Space and time are themselves abstractions, and these concepts very clearly enter into, and are fundamental to mathematics. — Metaphysician Undercover

They don't exist anywhere in mathematics. Of course "space" is a technical term very commonly used in math, as in a topological space or Euclidean space or a Banach space etc. But space as conceived in physics, as well as time, do not exist in math. If you would carefully study the axioms of set theory, you will see no references to time or space. Of this I am quite certain.

Are a circle and a square not a spatial concept, which are mathematical? — Metaphysician Undercover

They're idealized geometric mathematical objects. There are no circles or squares in the world, only approximations to the mathematical ideal.

Is the order of first, second, third, fourth, not a temporal order whish is mathematical? — Metaphysician Undercover

No, not in the least. How can you say that? That's not even the meaning of the words in everyday speech in the real world. The winner takes first place and the runner up takes second place sometimes (as in a foot race) but not always (as in a weight lifting contest) by being temporally first. You must know this, why are you using such a weak argument? First place in golf goes to the player with the lowest score, not to the player who finishes the course first. This is a TERRIBLE argument you're making here.

If you seriously think that you can separate mathematical concepts from spatial and temporal concepts, then yes, this is something you really need to explain, — Metaphysician Undercover

No, the onus is on you, as space and time play no role in the axioms and principle of mathematics.

because I've been trying to do it for many years and cannot figure out how it's possible. — Metaphysician Undercover

If you stop confusing math and physics you will be enlightened.

So please oblige me, and explain. — Metaphysician Undercover

Space and time have nothing to do with mathematics. Show me where it says they do. Look up the axioms of set theory (here for instance) and show me time and space. They're not there.

Nor are they even philosophically a part of math. A physicist has 5 meters or 5 seconds. Math just has the number 5. I hope you weren't fooled by grade school when they tell you that Sally had 5 apples and Fred had 3 apples, how many apples do they have together? Once you get past that level, they just ask you to add 5 plus 3. The apples are part of physics or biology or grocery store management. Applications of math, not math.

The problem is that "5" means nothing without a spatial or temporal reference. — Metaphysician Undercover

If I ask you if 5 is prime, can you answer? How did you do that? Did you really have to mentally imagine 5 apples?

If you think that the mathematician believes that "5" refers simply to the number 5, without any further reference to give the concept which you call the number 5 meaning, then you must believe that mathematicians think that the number 5 is a concept of nothing. — Metaphysician Undercover

No, it's the mathematical object 5. It's the third prime number. It's the order of the group of integers mod 5, and the number of sides of a pentagon. It's the largest positive integer such that all groups of order less than or equal to it are Abelian. That property uniquely characterizes the number 5. Martian mathematicians must necessarily have the same theorem. Likewise the Martians must know that there is no algebraic formula to solve a polynomial equation of degree 5 having integer coefficients. Any sentient race anywhere in the universe must necessarily discover this sooner or later.

Mathematicians have many concrete, if you'll permit that use of the word, ideas and concepts about the abstract mathematical object 5. They can represent it within set theory as a natural number, an integer, a rational, a real, a complex number, or a quaternion. Each such representation is a distinct set. We can identify them all via standard conventions. These representations might NOT be used by Martian mathematicians. The representations are contingent, but the facts about the number 5 are not. Mathematicians could write a dissertation on 5. Wikipedia has a relatively long article on the number 5. And that ain't nothing!

They find each other sexy. See? Not tricky at all. — Book273

Now THAT's action at a distance!

I'll just go along with Nature and WHO on this one; sensational articles don't help one way or the other. — Banno

This was an interesting convo. I hadn't read the rest of the thread and just grabbed on to your challenge to say something bad about science. Now I happen to be a big fan of science, so I used the opportunity to rail against the fake, politicized science of the past year. And when you said that you saw nothing to respond to or talk about, I noted that you are right, I wasn't talking about science, but rather fake, politicized science.

But then on further conversation, it turns out that you actually think the fake, politicized science of the past year is actually science. In which case my earlier remarks were perfectly on point; and it turns out that you think science is worshipping some political hack and deplatforming actual scientists who dissent. I'm afraid you are sadly in tune with the ethos of the day.

Thanks for the chat.

But the villification of Anthony Fauci is another thing altogether — Wayfarer

As has been the vilification of legitimate dissent all year long.

looks to be crap to me — Banno

If what Fauci's been doing is what you call science, then I'm against it.

https://thebulletin.org/2021/05/the-origin-of-covid-did-people-or-nature-open-pandoras-box-at-wuhan/

He said we should wear two masks? Shock! Horror! So what? — Banno

Disingenuous much? He said he did it because of the science, denied in front of Congress that it was political, then admitted that it was. You call that science? Then I'm against science. That's exactly the kind of science I'm against and now I see that you are totally for it. You don't even know what science is if you think the US response to covid has been about science.