herefore [0,1] and [0,2] are of equal cardinality. — fdrake

Wait ... @tim wood was confused about THAT?? I didn't mention it because it's so obvious, at least to anyone who would jump into a discussion of bijections. Thanks for pointing out that I gave @Tim too much credit. I thought he was confused about the far more tricky bijection between [0,1] and (0,1).

Nope. Google bijection. — tim wood

What? There's no order-preserving bijection between [0,1] and (0,1). Yet there is most definitely a bijection But why would I need to Google bijection? Perhaps you can point to the paragraph on Google that you are referring to.

Can you please be specific in your objection?

* Do you agree that there's a bijection between [0,1] and (0,1), the closed and open unit intervals in the real numbers, respectively?

* Do you understand that such a bijection could not possibly be order-preserving?

* The naturals biject to the rationals and the rationals in their usual order are not well-ordered.
— fishfry
what is their usual order? What matters is that they can be well-ordered. — tim wood

What is the usual order on the rationals? You aren't sure? Come on, man. The usual order is the usual order.

Well, reading the re

st of your post, let's try this. Name any two real numbers that are next to each other in a well-ordering of the set of real numbers. — tim wood

Pi and 47. I see that you are not familiar with the notion of well-ordering the reals, or well-orderings in general. If you were you could not ask this question.

Granted, given two such numbers as a set of two numbers, the elements of that set can be well-ordered, but the set in question is the set of reals. — tim wood

Which can be well-ordered: definitely using the axiom of choice; and possibly even in the absence of choice.

The real problem with all of this stuff is not that everyone else is wrong and you alone are correct - no danger of that. Nor even that you cannot handle it, because you can. The real difficulty with these ideas is just getting used to them. So in the words of my neighbor, a retired master sergeant, who so far has always made sense, suck it up, buttercup, and get used to them — tim wood

Wow. Just wow. What is it with this forum? Tim, you're as wrong as wrong can be.

Of course I'm perfectly used to people on this forum being wrong about some mathematical topic and getting insulting when their ignorance is pointed out. The old forum was never this nasty but this place is. But on the mathematical facts, well you haven't presented any.

Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it? — TheMadFool

There is no contradiction. The definition of cardinal equivalence is that there exists at least one bijection between the two sets. I don't know why you have a psychological block against grokking that.

Guy robs a bank, gets caught. In the interrogation room the detective says, "Fred we know you're the bank robber." Fred says, "Oh you are wrong. Here is a list of all the banks in the state that I didn't rob. I even have a notarized statement to that effect from the manager of every single bank in the country that I did not rob."

Is Fred a bank robber? Yes of course. He robbed a bank! He robbed one single solitary bank and DIDN'T rob all the others. But he's a bank robber.

It's an existential quantification, "there exists," and not a universal one, "for all." Someone is a bank robber if they ever robbed a bank, even if there are many banks they didn't rob. Two sets are cardinally equivalent if there is a bijection between them, even if -- as must ALWAYS be the case -- there are maps between them that are not bijections.

Someone murders someone, they're a murderer. No use parading before the jury the seven billion human beings they DIDN"T murder. That lady cop in Dallas a few months ago who shot a guy sitting in his living room eating ice cream. She was convicted of murder. She's in prison as we speak, ten years if I recall. No use trying to point to all of her neighbors who she didn't kill. She killed one guy. That makes her a murderer in the eyes of the law.

Why is this simple point troubling you? If you're on the jury do you say, "Well, the prosecutor showed that she murdered someone. But she didn't murder EVERYONE." You find her not guilty on that basis? Of course not! Right?

Even Hitler didn't murder EVERYONE. You think he got a bad rap? LOL.

What you leave out, and what has apparently been left out, of all of this is that the sets have to first be well-ordered. Then the bijection is a two-way Hobson's choice: next rider, next horse. And you never run out of either riders or horses. The problem with irrationals, is that they cannot be put into a well-ordering. — tim wood

I'm not sure I understand your point. Some examples that come to mind:

* The unit interval [0,1] can be bijected to the interval [0,2] but neither set is well-ordered in its usual order and both are uncountable sets full of irrationals (and some rationals too).

* Ok you say but at least in that case the obvious bijections preserve order. But how about bijecting [0,1] to the open unit interval (0,1)? Then there's a bijection but it does not preserve order.

* The naturals biject to the rationals and the rationals in their usual order are not well-ordered.

* For that matter the naturals can be bijected to the integers and the integers are not well-ordered.

* In fact any set whatsoever may be well-ordered as a consequence of the axiom of choice.

* And finally, and counterintutively, the reals might be well-ordered even in the absence of the axiom of choice.

Well-ordering doesn't have anything to do with bijections. I can't understand the point you're making.

ps -- One more item.

* Even two well-ordered sets may not be bijectable. There are uncountable well-ordered sets.

Baden ok but please tell fishfry to do better. — Maw

If you have a reference to the roll call for the Continuing Resolution I'd appreciate it. I had read that AOC voted for the bill and then that she tweeted against it. I can't find the actual roll call. I read that 10 Dems voted against it but can't find the actual vote. But as I say the larger point stands. Those who rail only against Trump are missing the larger picture of how our government actually works. The Dems supported the Patriot act since day one and never stopped.

it's the only thing you've talked about in the last year. — Maw

Too bad you still haven't heard it. A while back someone screeched, "Trump put kids in cages!" as evidence that Trump is uniquely bad. It's a very common refrain from the left. They're willfully ignorant on the subject but love to morally preen. I'll keep mentioning it till they get it. I have hopes the present insanity of the left is only a temporary phase.

Today AOC voted to renew the PATRIOT act. [It was cynically buried in a larger appropriations bill]. So goes what passes for the left in this country. Didn't make the MSM, or if it did the links are hard to find. Some lefty sites with their integrity and human decency still intact reported on the matter.

https://newrepublic.com/article/155793/hell-democrats-just-extend-patriot-act

My remarks are general, not specifically aimed at you. I quoted your post because it provided a convenient hook for a post I wanted to make anyway. If anything it's aimed at whoever originally made that inane "Trump caged kids!" remark as a substitute for actual critical thought about US policy at the Mexican border.

ps -- AOC reportedly tweeted against the bill. I had read that she voted for the bill. I could not find a definitive account of her vote just now. The larger point stands regarding the state of the Democratic party.

It's tragic that I'm your only source of information about the world.
— fishfry

What does this even mean? — Maw

It means that if you get your news from MSM sources you're missing a lot.

Here's a story from yesterday that caught my eye. A UN report came out on Monday saying that the US has more children in detention than any other country. Multiple news outlets reported this along with criticism of Trump's immigration policy.

Then the UN issued a correction pointing out that the numbers were from 2015, when Barack Obama's administration had more children in detention than any other country in the world.

Did these news outlets issue a correction and then mention that oh by the way, Obama had a hellacious humanitarian crisis on the border in 2014-2015 and separated families and put kids in cages?

Of course not. These media outlets all simply retracted and memory-holed the story. [Some did make the correction but others did not].

This is a problem. The MSM is committed to a particular political point of view; and if you don't read alternative sources, you become an ignorant pawn in a larger political game.

This article's from the National Review, conservative but anti-Trump. I'd love to find a more mainstream link but there isn't one. The public must not know of Obama's caged kids lest they see the cynical hypocrisy of the Dems' immigration rhetoric.

https://www.nationalreview.com/news/multiple-outlets-retract-stories-on-trump-admin-child-deportations-that-relied-on-obama-era-data/

Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things. — aletheist

Or as Bertrand Russel said, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable. — aletheist

This is not true. There's a proper class of Alephs, indexed by the proper class of ordinal numbers. After $\aleph_0, \aleph_1, \aleph_2, \dots$ come $\aleph_\omega, \aleph_{\omega + 1}$, and onward forever; too many Alephs to ever be captured in a set.

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

Cranky.
— fishfry

The author giving just one reference and that being the Wikipedia page of the diagonal argument is telling by itself.

And seems like the author is simply confused about infinite sets. And one really has to understand how different the reals are. — ssu

Are you saying I was excessively judgmental? Perhaps.

I really hope so too.Thanks. — TheMadFool

Thank you for being so gracious. Not always my experience around here and I appreciate it.

Let me see if I can respond directly to your plucked chicken remark. In that case there is an underlying reality, namely the existence, or the fact, of human beings. Our existence on earth logically and historically precedes any formal description of the matter. So when we say, "A human is a featherless biped," we are NOT actually defining what it means to be human. A human is so much more complex than any mere formalization could encapsulate. So at best when we say "A human is a featherless biped," what we are doing is laying down the rules for a formal, logical treatment of what it means to be human. And if we are wise we will recognize that. Personally I believe in science but I oppose scientism.

Now when it comes to sets, the matter is totally different.

In this case there is no underlying reality. There is no "setness" that we are trying to formalize. Rather, sets are whatever satisfies the rules we're writing down. Before the rules are written down, there are no sets! Unlike with humans, whose existence precedes any humanly contingent theory of them.

So you may be thinking, "This bijection business might be logically correct, but it doesn't match my beliefs about sets." But the right way to think about it is: "The knight in chess moves the way it does because that is the rule. And sets behave the way they do because those are the rules!" We must learn to adjust our intuitions to the symbology. We have to cast off our intuitions and just do exactly what the symbols say. That's the essence of learning math.

[Now if you want to talk about the "unreasonable effectiveness" of math in the real world then that's another subject. But math must be taken on its own terms! It need not conform to reality nor does it, most of the time].

In high school they tell you that sets are collections, but as we all know Russell ruined Frege's day be discovering that this cannot possibly be. Ever since then, set is an undefined term just as point and line are undefined terms in Euclidean geometry.

If you want to know what a set is, the answer is that nobody has any idea what a set is. A set is whatever satisfies the rules of set theory. In fact set theorists like to mess around with different rules to see what kinds of crazy sets they can come up with.

So when we see a plucked chicken we may fairly say, "Oh, our formal characterization was good as far as it went, but it needs to be refined. How about featherless bipedal mammals?" Then someone will shave a gorilla and we'll be off to the races again. A formal system can never completely capture every aspect of reality.

Whereas with sets, that thought process doesn't happen because sets don't exist outside of the rules we stipulate for them. There is no underlying reality to appeal to. Sets are what the rules say they are. So when we say two sets are cardinally equivalent if there's at least one single solitary lonely bijection between them; then that is the absolute truth of the matter. Because in set theory, the rules precede the reality. With humans, the reality precedes the rules.

tl;dr: It's chess not chickens.

↪fishfry
Do you agree that there exists at least one bijection from E to N?
— fishfry

So there was nothing wrong when Socrates defined humans as featherless bipeds and someone came along with a chicken plucked of all its feathers and declared "this is a human"? After all there was/is at least one human that fit the definition. — TheMadFool

I can only implore you to carefully re-read what I and others have written. Perhaps Cantor's beautiful ideas will come to you at some point. Perhaps not.

Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4 — sandman

Cranky.

Wittgenstein also criticized the diagonal argument. He was wrong but his objections were at least coherent and interesting.

If a definition leads to a contradiction? — TheMadFool

It doesn't.

Do you agree that there exists at least one bijection from E to N?

If you agree, then you must agree that E and N have the same cardinality, because the definition says that there must be at least one bijection between them, and this is manifestly the case.

ps -- It's like a guy who cheats on his wife. She says to him, "You're a cheater." He says no! Think about all the times I DIDN"T cheat on you.

But that's not the point. If you cheat once, that's the definition of a cheater. If you cheated Monday but not on Tuesday or Wednesday, you can't say you're not a cheater because you didn't cheat on Wednesday. Right? The definition is doing it once.

Likewise the definition of cardinal equivalence is that there's at least one bijection. It doesn't matter that some other function isn't a bijection.

That's where the problem is isn't it?

The definition is inadequate for the reason that, on one hand, Cantor's "preferred" bijection leads to an equivalence between the set of even numbers and natural numbers but on the other hand there exists another bijection that shows that the set of natural numbers is not equivalent to the set of natural numbers. — TheMadFool

It's a definition. It can't be right or wrong.

You do agree that there exists at least one bijection between N and E, namely f(n) = 2n. You agree with that, yes?

Then by definition the two sets have the same cardinality. That's all it means. If there exists a bijection, then the sets are said to have the same cardinality.

It's true that there are many functions between E and N that aren't bijections, but it only takes one to satisfy the definition.

Do you agree that even if you don't like the definition, E and N satisfy it by virtue of that one bijection?

there exists another bijection that shows that the set of natural numbers is not equivalent to the set of natural numbers. — TheMadFool

But no. The existence of a non-bijection doesn't prove anything. As long as there's a single bijection, the definition is satisfied. Not because it's right or wrong, but because that's the definition.

This then is used to "prove" that the set E is equivalent to set N — TheMadFool

No, that's a great source of confusion. If there exists a bijection between two sets, even a single one, even if there are plenty of functions that aren't bijections, then we DEFINE the two sets as being cardinally equivalent. It's a definition, not a proof.

Given the naturals N and the evens E, there exists a bijection. So we say, by definition, that the two sets have the same cardinality. The fact that there are many functions between the two sets that aren't bijections is irrelevant. The condition is an existential, not universal, quantification. If out of all the functions from one set to another there happens to be a single one that's a bijection, then the two sets are cardinally equivalent, by definition.

As has been noted, this is not the only way to compare the relative sizes of sets. There's the subset relation. Clearly E is a proper subset of N so E is "smaller" than N with respect to the proper subset relation

Yet another way that was mentioned a few posts back is natural density, also known as asymptotic density. Using this method we see that the limit as n gets large of the proportion of even numbers among the first n numbers goes to 1/2. So the natural density of E in N is 1/2. Likewise the natural density of the multiples of 3 in N is 1/3; and the natural density of the primes is 0. [Proof by the interested reader for that last assertion].

So there are lots of ways to compare the relative size of sets. Cardinality happens to be one way. It's not a very fine-grained one. For example the intervals [0,1] and [0,2] of real numbers have the same cardinality, since there's a bijection between them: namely f(x) = 2x. But [0,1] has length 1 and [0,2] has length 2. The generalization of the idea of length is called measure, and it's yet another way to compare the relative size of sets. In fact measure theory underlies modern probability theory, so it's very important. But as we saw with the intervals, cardinality does not respect measure. In that sense cardinality is a very crude way of thinking about the relative sizes of sets.

Just as we use a hammer for one thing and a spatula for another and a torque wrench for yet something else, mathematicians have a lot of tools in the toolbox. Cardinality, as defined by bijection, is just one tool, to be used as appropriate.

Basically two different bijections are possible. One agrees with Cantor's "proof" but the other contradicts Cantor. You'll have to show that Cantor's bijection is the correct one and the alternative is nonsensical. — TheMadFool

Cantor's definition of cardinal equivalence via bijection is not right or wrong. It's a definition. In the past 140 years it has proven to be both interesting and useful, which is why it's stuck around. But it's not the only way of comparing sets; and it's neither right nor wrong. It's just one of the tools in the mathematical toolbox. E and N have the same cardinality. The natural density of E in N is 1/2. And E is indeed a proper subset of N. All three points of view are equally valid. One or the other viewpoint might be more useful in a particular context.

Perhaps you could be a bit more specific. What immigration crap from white liberals? Do you mean things like Daca, for instance? — praxis

The people who came in legally and worked to establish themselves resent those who jump the line and are rewarded for it. Isn't that perfectly sensible?

But this conflation is itself anti-immigrant, because it refused to recognize the difference between those who subvert the laws of the country with those who spend the time and effort to become American. — NOS4A2

Yes and no, mostly no. The problem is that the US demand for illegal labor far (far!) exceeds the supply of legal slots available by law.

One might think ok, pass a law raising the legal immigration quotas to accommodate all the demand for farm laborers and chicken pluckers and suburban lawn cutters, nannies, and housekeepers.

But that would defeat the purpose. The entire purpose of keeping immigration illegal is so that the workers can't organize, can't complain, can't report being underpaid or abused, can't ask for any of the normal labor protections that American workers take for granted. This is the entire point of the sick, depraved, immoral system.

The fruits and veggies need to be picked now, and the laborers need the work now. Nobody can wait for the legal queue. The incentives force economically desperate migrants to sneak across the desert to work to feed their families. And the employers are only too happy to have such a supply of below-market labor that can't complain no matter what.

That is the system that's evolved over the decades. Sure, in a rational world we'd balance immigration quotas with labor demand. But then the workers wouldn't be so cheap anymore.

The hypocrisy is the entire point of the system. It's a lie to say that illegal crossers are bad people subverting the system. They're responding responding rationally to the incentives presented by a hypocritical and immoral system.

My original response to this was deleted, but I do want you to know this is exceptionally pitiful. — Maw

It's tragic that I'm your only source of information about the world. I regret that I haven't the time to take on such a responsibility.

So please explain to me where this substantive block of ethnic minorities are that believe Trump has been good for them. — Maw

Like I say I find argumentum at linkum, or argument by flinging links at each other, tedious.There are sources out there to support pretty much everything. I have in fact read several credible articles supporting the idea that Trump has African-American and Hispanic support. I'd ask you to stipulate that I'm making that statement in good faith and good will. I don't feel like going out on Google and curating the links for you, which you could just dismiss anyway as being not from approved sources, or outright lies or whatever. I'm just choosing to not even start that game.

Oh and also, you know who are the most anti-illegal-immigrant people around? The Mexican-American citizens who came here legally and established themselves. The Hispanic middle class doesn't buy the immigration crap from white liberals.

Not sure that minorities helped him that much in 2016. I recall that only 1% of black women voted for him. — praxis

That's still perfectly consistent with my point. In every situation there is the seen and the unseen. We can SEE the percentage of blacks who came out to vote against him. We can NOT see the millions who simply stayed home because they couldn't pull the lever for Ms. Superpredators.

Political scientists did sift through the numbers, and it's clear that Obama's black turnout in his two elections was far higher than Hillary's in 2016.

Speaking of a reality disconnect, Trump hasn’t been able to bring American manufacturing out of its recession, but this doesn’t seem to be a dealbreaker for his loyal supporters in the rust-belt. It should be. — praxis

The heartland's hurting for sure. Would not be surprised if Liz peels off some of Trump's support with her talk of economic unfairness. Even if Trump's right to squeeze China on trade (I happen to agree) there's no question that the soybean farmers are unhappy.

The truth is that the manufacturing jobs are gone for good and they're not coming back. Obama was right about that. Trump's economic populism sounds good but the tide of history's going the other way.

4% of Black Americans think Trumps been good for them and 19% of Hispanic Americans think Trump has been good for them fishfry is just making stuff up — Maw

Black an Hispanic employment under Trump is way up. I'm not providing a link because I don't think a philosophy forum should be a link/counterlink battle as if we were on Craigslist or Reddit. Google around.

Trump doesn't have to get the entire minority vote. He only has to peel off a few votes form the traditional Dem majority among Blacks and Hispanics. And please remember a point I find myself repeating, that millions of black Obama voters in 2008 and 2012 didn't bother to go to the polls for Hillary. You could look that up too.

I do object to your statement that I'm making things up. That's a negative personal characterization and it's quite false. If there's one thing I do it's read and research obsessively across a wide spectrum of news and opinion. I never knowingly post anything that's untrue, if it's a matter of fact; or at least arguable, if it's opinion.

@Maw I've never personally attacked your integrity and I'd like the same respect from you.

The bottom line is whether this is to be a political conversation on a (relatively) high toned philosophy discussion forum; or whether it's just Craigslist or Twitter with pretensions.

I assume then, that you still do not understand the distinction I made between what a symbol means, and what it refers to, or stands for. Perhaps if you read up on the kind/token distinction, that will help you. — Metaphysician Undercover

You can assume what you like. What's true is that I've given up interacting with you. Your persistent rudeness reflects badly on you. I'm fully aware of the subject matter that you claim I'm ignorant of. You persistently avoid engaging with anything I write. You're a nasty piece of work. Unpleasant.

Isolationism has come upon me lately for the very reasons laid out by P. Buchanan in 2013: — frank

I love the Patster. He gets a bad rap (racist, Nazi, etc.) I don't mind even if there's something to it. He has many redeeming qualities. He's brilliant. He's a true populist. "Pitchfork Pat" when he ran for the GOP nomination in 2000. You remember some guy named Bush got nominated instead. Pat was a strong and I mean STRONG opponent of the Iraq war in real time. He was a conservative speaking out against Bush's war. You can call Pat an isolationist if you want, but he's an America first guy. What good does invading Iraq do us? None. So there's that. He's often right about things.

He's been writing about the death of the west for a long time. He's very prescient and his analysis is spot on.

Plus I like his style. I always loved watching him on the McLaughlin Group. Rachel Maddow said he was helpful to her when she was starting out at MSNBC and he was on his way to getting fired from the network in a purge of original thought.

So I like Pat. You may now reply: "ARE YOU SAYING that you like a guy who's been called a Nazi? You must be a Nazi."

Sorry haven't read the rest of the thread, just wanted to express my appreciation of Pat Buchanan.

I'm cool with that. — frank

Thank you. I'm grateful for any understanding and agreement. About anything. But I have the title now. "Confessions of a non-deplorable Trump supporter." There's an essay in there around these ideas. Flesh out my idiosyncratic point of view. Get branded a racist for my troubles. Such is the state of public dialog in 2019. Not exactly what Plato had in mind.

Changing the subject, how about old She Who Must Not Be Indicted, or "She Who Etc" for short? She's running sure as the planets follow their spacetime geodesics around the sun. What do y'all liberals think about that? You cool with She Who Etc wrecking yet another election for you?

And now that you mention it: Go Tulsi!!!!!

No, you're not paying attention fishfry — Metaphysician Undercover

I came here tonight to append a note to my previous post, which I composed in my mind before I saw this ... ahem ... remark of yours. It changes nothing. Here is what I wanted to say:

* I apologize if I sound strident. And I don't want to be strident. But in truth I've been trying to leave this conversation for a while. I stated my intention twice already and weakened.

On my part I'd be so happy to simply agree to disagree. Over time I'll go over your posts with an open mind. I would like to understand your point of view and in the process I might well learn something. I'm much more of an open-minded fellow than I sometimes appear. Make an argument I can understand, and I may well agree with you.

But we're talking past each other. I hereby agree to disagree with your point of view as expressed in this thread. We're talking past each other and no productive dialog is occuring.

* Ok that's pretty much what I wanted to write. But then I saw the quoted text ... and whether it's true or not, it comes off as having a bit of an edge to it. So neither of us is attaining our highest selves here. Let's let this go and meet again perhaps in some other thread.

But you are right in fact. I am not paying the slightest attention to your argument. That's another sign I should depart the thread. I'm making my arguments and you are making yours but nobody has gotten any more enlightened in many a post. So I'm out, unilaterally. I do not concede the point but I'm all outta ammo.

And yeah when I show up to apologize for sounding strident, I click on your post and I see your little snark and of course I don't read another thing. Becoming a self-fulfilling prophecy.

I do promise and commit to reading through your posts in much more detail at my leisure. I will also pick up the conversation between you and @Zuhair, which I haven't read but can see that it's an exploration of your ideas by someone who seems to at least know what you are talking about.

Bottom line I have no idea what you're talking about. But it's cool. Peace.

the fact that he won't take a clear stand against racism — frank

I don't agree with that claim. During the 2016 campaign he went to the NAACP meeting and asked for their votes. "What have you got to lose?" he asked. He has a way of getting at the heart of the matter. Millions of blacks and Hispanics are doing much better in Trump's economy than they did in Obama's. You think they don't know that? He doesn't have to win all the ethnic minorities. He just has to peel enough of them away from the Dems. He did that in 2916 and he'll do that again in 2020. The Dems are no longer connected to reality.

I guess then people think you wear a MAGA hat. :wink: — ssu

Never did, never would.

I have this explanation for my politics:

Hillary was 100% correct when she said that half of Trump's supporters are a basket of deplorables. Racist, misogynistic, homophobic, Islamaphobic, xenophobic. It's a fact, I totally agree. About 30% of the American electorate falls into that category.

Now what the Dems and the left have NEVER been willing to ask themselves is: Who are the half of Trump supporters who are NOT in that basket of deplorables? Who are the lifelong social liberals, lifelong registered Democrats, who can no longer support what the Democratic party and the left have become?

I put myself firmly in that category. I stand for peace. The left now supports war. I stand for free speech. The left now stands for no-platforming and spitting in the face (literally, if you caught that news last week) of anyone who dares to disagree with them. I stand opposed to the illiberal, corrupt, warmongering left and the Democratic party they've taken over.

Ask yourself: If half of Trump's supporters are deplorable, who are the other half? The Dems won't ask themselves that question because to ask the question requires looking in the mirror at what they've become.

I could go on. Just to take one demographic example, millions of African-Americans who voted for Obama in 2008 and 2012 didn't bother to turn out for Hillary? Why is that? Did they suddenly turn racist? Or could it be that they know that Hillary and Bill (and Biden) were behind the punitive crime bill of the 90's that destroyed the black community? And that Hillary and Bill (and Biden) were behind the punitive bankruptcy bill of the same era? Or that Hillary called young black men "super predators who should be brought to heel."

Brought to heel. Like dogs. Hillary said that. Do you need me to provide the clip? Ok.

https://www.realclearpolitics.com/video/2016/08/22/trump_tweets_video_of_hillary_clinton_referring_to_blacks_as_super-predators.html

You think black people don't know those things? The votes show otherwise. Blacks who supported Obama didn't turn out for Hillary. One of the factors that cost her the election. Even law-abiding blacks who hate gangbangers will have a negative reaction to that kind of rhetoric from a privileged white woman.

So is the country really racist for electing Trump? Or did a lot of blacks reject Hillary's implicit racism? Another question the Dems won't ask themselves.

Who are all the people who would never dream of wearing a MAGA hat yet can no longer support what's become of the Democrats? That's the question to ask if you seek to lead and unify the country.

Until it doesn't.

You see, the two ruling parties that are in symbiosis can rule only so long that people think they "waste their votes if they don't vote for one or the other". — ssu

If California were in play in a presidential election I wouldn't cast a protest vote. And I would burn in the fiery pits of hell before I'd ever vote for Hillary Clinton. It was the DNC that rigged their own process to nominate a corrupt warmonger so incompetent at politics that she managed to lose to Trump by failing to lock down the rust belt states.

So imagine there are four chairs, and we represent those four chairs with the symbol "4". — Metaphysician Undercover

We're not talking about chairs. Four chairs over here are different than the four chairs over there.

Once again you are avoiding the question. We are talking about 4 + 4 = 8. You claim the two instances of '4' represent or stand for or refer to or mean two different things. I categorically deny that. I have repeatedly challenged you to explain that remark and you deflect by talking about chairs. You have no argument. You got confused by your grade school teacher and you can't get out of that psychological box.

You have claimed that in ZFC things are claimed to be equal that are not identical. I have categorically denied that and challenged you to provide an example. You have repeatedly failed to do so.

You have claimed that mathematicians use the word equality when they really mean congruence, equivalence, or isomorphism. I have categorically denied that (with certain well-understood casual figures of speech in particular contexts) and challenged you to provide a specific example. You have repeatedly failed to do so.

You have no argument but you have your misunderstandings and a lot of words and handwaving.

I ask you to introspect on the point that if you can't come up with specific examples, perhaps you don't understand your own ideas as well as you think you do.

So you're saying Trump is racist. And you voted for him. — frank

Playing the old "so you're saying that ..." game. A low form of argument practiced by people who think "gotcha" is clever.

Trump often says things I wish he wouldn't say. But in fact he got rid of NAFTA and implemented USMCA (still awaiting Congressional approval) which keeps the good parts of NAFTA and fixes the bad parts. So I would say that Trump has a nuanced and productive understanding of US-Mexican trade issues.

I will add that I vote in California, so that I could vote for Hillary or for Trump or for Jill Stein or Gary Johnson or Bozo the Clown and my vote would not matter. That gives me the liberty of casting protest votes in presidential elections with no downside. California goes for the Dem regardless of who I vote for.

It's not a matter of giving you a reference, it's just a matter of whether you understand the reason or not. Do you know how to count? Say you have "1", and that 1 signifies something. And, you have another "1", and that 1 signifies something. In order that these two 1s, when they are put together (1+1), can add up to two, they must each signify something distinct from the other. If each of the two 1s signified the very same thing, there would not be two things, only one. — Metaphysician Undercover

You should give me a reference. If this is from some branch of philosophy or some philosopher's idea, let me know what that is. As it stands I think you were just warped by your grade school teacher.

One thing I see on discussion forums is that sometimes someone is arguing a point of view but not being up front about it. Someone claims uncountable sets are incoherent and twenty posts later it turns out they're a diehard ultrafinitist of the crank variety. I don't care if they're a crank but if they'd just start by saying, "From an ultrafinitist point of view ..." then I could engage with them. But without that information, their point of view is not comprehensible.

If your ideas are original, say that. If they follow from someone's work, say that. Give me something to hang on to. Because as it stands you're just saying flat out incorrect things, and waving your hands instead of giving hard facts, evidence, and examples to support your point.

Do you understand the reasoning here? — Metaphysician Undercover

To the extent I do, you're wrong. And to the extent I don't, I really wish you'd give me a reference so that I have some idea where you're coming from.

I want specifics.

You made the outrageous and on its face absurd claim that "4" means something different each time it occurs in "4 + 4 = 8". Tell me what different things they mean. If you can't articulate the difference, perhaps your ideas aren't as clear as you think they are. Surely that's fair.

There is a difference between what a symbol "means" (as said in your fist paragraph above), and what a symbol "refers to", (as said in your second paragraph above). — Metaphysician Undercover

That is very funny. As in ironic When I wrote those two paragraphs I used "means" and "refers to" as synonyms. You take me to be saying two different things. I take meaning and reference to be the same in this context. But even "same" is a loaded word for you so I hope you're not going to go down another rabbit hole here.

I read the rest of your post and some of the interaction between you and @Zuhair. I see that you're sincere and knowledgeable about ... something. I can't figure out what because you won't supply a reference. You said that in "4 + 4 = 8" the two occurrences of the symbol "4' do not refer to (or mean?) the same thing. From my point of view there is simply no further conversation to be had. You're clearly serious, you're not trolling me. But when I try to take you seriously, I can't understand what you're saying. There is only one referent (or meaning) of the symbol "4" in the context of elementary arithmetic. [There could of course be other contexts, such as modular arithmetic in which "4' means some equivalence class mod some other integer]. But in the context of elementary arithmetic, "4" means the number 4.

I am simply not prepared, either by philosophical erudition or even the slightest interest, in debating this proposition. If you didn't seem so learned I'd honestly think you're trolling me.

What would help would be a simply clear example of WTF you are talking about. If the two occurrences of "4" in "4 + 4 = 8" refer to (or mean) something different, TELL ME WHAT THEY MEAN. Don't just toss out more paragraphs of obfuscation. Show me what you are talking about.

And -- secondly -- why won't you engage on the specific disagreement we're having about mathematical equality? I claim it is logical identity. You claim it's what's normally called equivalence, congruence, or isomorphism. This is a point we could engage on but you won't engage.

You said a while back that a grade school teacher once told you that an equality is between two different things. I'm afraid that experience imprinted an incorrect idea in your mind. You must let it go. With a handful of well-understood exceptions, mathematical equality is logical identity. You have not presented any specific examples to the contrary.

I say Captain Ahab was already there before he was thought into existence. For the same reason I can invent a random word at will, like fragalagadingdong and although it's never been heard in its entirety before, all the component parts were there, which I jumbled up together. The first words spoken by humans were a bridge between sounds they could already make and some action or object of reference.

Free will or original thought are illusions. I feel like I have free will but logically I know that cannot be. If I invent something new, which looks and seems new to everyone, it is only as a response to the need for said things invention to begin with. — Razorback kitten

Well ok. I understand. We can say for example that, if we agree on an alphabet of symbols, such as the English alphabet, in some sense every possible finite-length string already exists. From a, aa, aaa, ... to z, zz, zzz... The conceptual or abstract space of all possible finite-length strings of letters and punctuation marks already exists. Just as the sculptor extracts the statue that was already in the stone; the novelist simply selects one long finite string out of all the ones that could possibly exist.

You COULD look at it this way, if you chose to deny human creativity. And as you say that free will and original thought are illusions, I see that I'm reading you correctly.

I disagree, but I can't justify it! For all I know you're depressingly correct.

But wait. Can't we accept your point yet save creativity? Out of ALL the possible configurations present within the stone, the sculptor chooses one. Out of all the possible finite strings of symbols, the novelist chooses one finite string that tells a story about a tragic whaling expedition.

So even if we accept that every possibility is potential; doesn't it still take an act of human creativity to select one of those potentials and make it actual?

And when we do creative work, we discover that it's damned difficult to turn those potentials to actuals. We have a great idea for an essay or article in our minds but the thing doesn't write itself, we must struggle. The mathematician knows what's true but struggles for years with the proof. In theory the proof was already inherent in the axioms; but in practice, only the long struggle brings it out.

It's the struggle of the artist in bringing the potential into the actual that matters. So I still believe in creativity.

I understand what you're saying. Trump played his part by not filling vacancies for judges at the border, and then Honduras went into crisis.

And then with the ling history of the US absorbing Latin America into itself. Not much of this is about racism. — frank

Well it's definitely about the racism of the US against our neighbors to the south. Very long history. Wars fought. Not the thread to get into it. I think there's always been racism involved in US policy towards Mexico and central America.

Countries typically have policies and procedures that transcend party lines. It's simply a myth that in a democracy government day-to-day operations would differ so much depending on what party is in power. — ssu

Interesting point. In recent years Congress has become weaker and the presidency more autocratic. You may recall a certain recent Democratic president who boasted that he'd govern with a "pen and a phone." Remember when Obama said that? It's the same deal. I for one would love to see Congress step up to its responsibilities. You know if the Dems really wanted to abolish ICE or open the borders they'd pass a law to do so. Far easier to criticize Trump for enforcing the very laws Dems implemented, and putting kids in the very same cages Obama built.

Please don't misunderstand me. I don't support Trump's border policies. I didn't support Obama's. Obama deported far more people than Trump. I'm simply appalled at the massive hypocrisy of the left. Is Trump an autocrat? Yes. Was Obama? Yes. Do Congresspersons like to give speeches but are often nowhere to be found when it comes to passing bills? Yes. Imperial presidency, weak Congress. The historians are already writing about it, it's been decades in the making.

Am I taking your point correctly? I agree that the US should be governed by laws, not autocrats. Congress should step up. We're constantly at war but haven't declared war since the day after Pearl Harbor. That's where it starts. Congress won't do its job so presidents rule by fiat. Signing statements, executive orders and the like.

Separating thousands of children from their parents is not a fine point, my unsympathetic friend. — praxis

Your argument, as I understand it, is that when Obama separated and caged kids, he was doing it for Just and Wise reasons; whereas when Trump does it, he's Adolf Hitler running concentration camps. That is an ignorant and disingenuous position.

I read and follow a lot of sources on both the left and right. That's why in the summer of 2014 I read dozens of articles and as many cable news reports of the awful humanitarian crisis Obama had on the southern border that overwhelmed our immigration system. But very little of it was reported in the MSM or liberal-leaning cable networks. The result is that many liberals honestly don't know about the summer of 2014. In fact many times liberals have tweeted out photos of "Trump's kids in cages" that turned out to be photos from Obama's cages in 2014.

Now it is true that Obama caged fewer kids. There's a reason for that. When you separate a kid from his ALLEGED parent long enough to determine whether it's an actual parent or a trafficker, you then have to keep the kid safe from violent sexual predators, so you put them in an caged enclosure. The cages are to PROTECT the kids. But the optics are terrible.

It's much better politically to just turn the kids over to the alleged parents. In the end you cage fewer kids but turn more kids over to sex traffickers. That's the route Obama chose.

But why are we talking about who put more kids in cages? Many liberals actually disbelieve that Obama put ANY kids in cages. It was a big shock a few of months ago when Jeh Johnson, Obama's director of Homeland security, admitted that Obama built the cages. Most liberals simply had no idea. Myself, I despaired at the deadly combination of arrogance and ignorance embodied by those liberals.

You dare call me unsympathetic to the plight of Mexican and central American refugees? I am far more sympathetic to them, having followed this ongoing tragedy for decades, than the many ignorant liberals who think history started when Trump got elected. They don't take any responsibility for the anti-immigrant rhetoric of Bill and Hill and Obama. They don't know about the foreign aid conditioned on perpetuating the drug war, leading to the human misery that underlies the upwelling of desperate migrants from central America and Mexico. They don't have any idea how the border has been militarized over the last 25 years. All they know is Orange Man Bad. That's not compassion. It's preening ignorance.

Like which kind of policies? Ones that lead to greater wealth disparity? — frank

Didn't really want to go into the past thirty years of border policy. You lost me on the wealth disparity question. What I'm talking about is that the Dems talk compassion, support sanctuary cities and drivers licenses for undocumented immigrants. But they don't want to be called soft on immigration so they vote to militarize the border. They don't get out in front of these efforts, but they vote for the harsh measures the GOPs propose. Likewise the GOP, they talk tough on immigration but love the cheap labor. So you get decades of hypocritical policy. We let people die of thirst in the desert but if they make it we give them a job. It's a cruel, sick system. And it's bipartisan.

Another aspect is monetary aid. A lot of US aid to Mexico is conditioned on Mexico fighting a bloody war on drugs. Americans don't want to admit that we're the users and without a demand there wouldn't be a supply. Instead we pretend Mexico is "pushing" drugs on us. Hillary and DiFi in particular are two prominent Dem pols sometimes called "liberals" when they're anything but.

Like I say I don't want to go research chapter and verse on every bad law that militarized the border, added surveillance, and so forth. You can Google around and find Hillary and Obama and Bill Clinton speaking out strongly against illegal immigration. And suddenly the Dems want to blame it all on Trump. It bothers me.

↪fishfry Did you feel like Clinton's comment about deplorabless was directed at you? — frank

Oh no! Thank you for asking. I agree with Hillary that HALF of Trump's supporters are a basket of deplorables. She's entirely right. I'm in the half that aren't. If the Dems would ever stop to ask who these people are, they'd understand the country better. I'm sick of the hypocrisy and corruption, sick of the endless wars (Hillary's vote for the Iraq war was the moment I finally left the Dem plantation), sick of the neocon/neoliberal consensus that's wrecking the country.

Trump ain't no great savior in this regard but he was the alternative to what Hillary represented, which is business as usual. The candidate I actually liked in 2016 was Jim Webb. He didn't last long in the Dem primaries. This time around I like Tulsi. You see what kind of disaffected Democrat I am. I like the candidates who make sense but who have no chance in the Dem primaries.

Can you point out public records that help to substantiate your version? — praxis

Won't waste my time. Go look it up. Jeh Johnson, Obama's immigration guy, admitted a few months ago to reporters that Obama built the cages. This is exactly the kind of denial I mean.

But you're picking at the margins, the fine points of who caged kids and why. On the larger point of Dem complicity in the ongoing 30-year program of militarizing the border and creating the current humanitarian disaster, you're conspicuously silent.

I just explained this. When the symbol "4" is used twice in "4+4=8", it must signify a different thing in each of the two instances, or else 4+4 would not equal 8. — Metaphysician Undercover

I can not relate this sentence to anything that I know nor to anything that makes any rational sense. I hope you'll forgive me, I cannot continue this conversation. You are factually wrong about this and you are not making me understand your reasoning. In other words if I thought you were wrong but I could say, "Ok I see where he's coming from," that would be fine. But I can't even do that.

The '4' in s 4 + 4 = 8 signifies two different things else 4 + 4 would not equal 8?

I find that absurd beyond the point of my being able or willing to respond. Of course the '4' signifies the exact same thing every time it is used, namely the number 4. If you are not willing to stipulate that then we have no common basis for conversation.

I get that you are sincere in your beliefs. From my viewpoint you give me nothing rational to respond to.

But perhaps you could give me a reference that supports your view. Earlier I noted that Wikipedia supports my view that logical identity and mathematical equality are the same thing. I'm willing to grant that Wikipedia is often wrong. But at least I have one reference. Give me something to work with, else I can't respond.

ps -- I should add this so you understand why you are wrong. It's a basic principle of math that the same symbol means exactly the same thing each time it's used in an argument or equation. For example when we say that for all even natural numbers n, 2 divides n, then even though n ranges over all possible even numbers, in each particular instance n means the same thing each of the two times it's used.

Likewise when we say 4 + 4 = 8, it's basic to all rational enterprise that the symbol '4' refers to the exact same thing each time it's used. Without that, there could be no rational communication at all. Natural language is symbolic. If I say that today it's raining and today it's Thursday, and you claim I can't assume that "today" refers to the same day each time I use it, then we'd all still be in caves. You couldn't say "pass the salt" without someone saying, "What do you mean pass, what do you mean salt, what do you mean "the"? You are denying the foundation of all symbolic systems from natural language to computer programming to math.

What exactly do you mean that '4' refers to two different things in 4 + 4 = 8? The burden is on you to justify denying the entirety of scientific and indeed rational discourse.