Wait I figured this out. Kamala is the white liberal's idea of an acceptable black person. Not unlike half-black Obama. That's the play. And the media are all calling her Black as if on cue. I'm thinking that actual African-American voters won't fall for it but we shall see.

Both are philosophical positions. All philosophical positions have some things in common with one another. However, I would think that any simulation would also require a plurality. The simulation, and what was being simulated... — creativesoul

Yes thank you, you are right. There's me and my simulator. So it's not solipsism.

Solipsism is a symptom of a gross misunderstanding of thought and belief hard at work. Solipsism is the name of a philosophical position/argument which outright denies the existence of anything other than the mind of the one forwarding the argument/position... — creativesoul

I have a question for you. As you know, the idea that the world may be a simulation running in a great computer "one level up" as they say. I personally don't believe any of this but many smart people do these days.

Do you consider simulation theory a form of solipsism? Or in this case would solipsism say I'm the ONLY program running; as opposed to the belief that I'm a program and you're a program and we share the same general environment or computational context?

Given that solipsism implies epistemological certainty of ones world, because there is nothing more to know than that what a solipsist knows about the world (implying that the world and reality are the same for a solipsist), then can a solipsist logically doubt? — Shawn

You can doubt you're a solipsist. "I can't prove that others exist, but what if they do anyway?" Seems like a perfectly reasonable thought. Gotta go it's feeding time at the vat!

ps to my previous post ... She doesn't bring in a state. California goes to the Democrat, period. Running up the score in a state you've already won does not help you, as Hillary found out. The more I think about this, the more I wonder if she's going to turn people off. As we saw in the primaries, she has superficial appeal to a lot of people but once they see her in action they don't like her. But what was Joe's other choice? Every other African-American female had major problems (Bass = commie, Demings = cop, etc) and Liz and Tammy were too pale for Maxine Waters. So it goes in the late stages of the empire.

Two lines of discussion:

1) Is it a good or bad pick politically; and

2) Is she or is she not a duplicitous, self-serving finger-in-the-wind politician who will say anything and do anything to advance her own career and who has not a single conviction or belief or principle she's not willing to abandon at a moment's notice when convenient.

(2) is easier because it's a matter of opinion so I can state mine and leave it at that. I prefer not to argue politics so I'll resist the temptation to clap back on any disagreements of opinion about the lady. I lived in the SF area and have watched her career since her Willie Brown days so my opinion is not casual but is the result of decades of following her career. I just mention a couple of items but there are dozens.

*When she was the Attorney General of California she busted large numbers of pot smokers; then when she declared her candidacy she gave an interview and bragged about being a stoner and gave her trademark cloying giggle which frankly makes me want to strangle her. It's so effing insincere and shows that she is emotionally disconnected from the serious consequences in people's lives.

* She also gave that sadistic and not-cute-though-she-thinks-it-is giggle when she was interviewed about throwing an African-American single mom into jail because her kid was truant. Expect to see the Trump campaign playing this clip.

https://twitter.com/WalkerBragman/status/1089831581030797312?ref_src=twsrc%5Etfw%7Ctwcamp%5Etweetembed%7Ctwterm%5E1089831581030797312%7Ctwgr%5E&ref_url=https%3A%2F%2Fwww.foxnews.com%2Fpolitics%2F2010-video-shows-kamala-harris-gleefully-talking-about-prosecuting-parents-including-a-homeless-mom-with-two-jobs-whose-kids-missed-classes

She was savagely criticized FROM THE LEFT for this policy. See for example

Also see https://www.latimes.com/politics/la-na-pol-kamala-harris-truancy-20190417-story.html

https://www.factcheck.org/2019/05/kamala-harris-spins-facts-on-truancy-law/

https://www.huffpost.com/entry/kamala-harris-truancy-arrests-2020-progressive-prosecutor_n_5c995789e4b0f7bfa1b57d2e.

* She sought to keep prisoners is jail longer than their sentences for the free labor. Apologies for the RW source but a lot of the Internet's been scrubbed in prep for her candidacy. Yeah that happens.

https://www.washingtonexaminer.com/politics/kamala-harris-office-sought-to-keep-inmates-locked-up-so-that-california-could-use-them-for-cheap-labor

Anyway I could pile on all day. She's devoid of compassion for the people she supposedly serves and has no principles or human decency whatever. I oppose her totally.

(1) That said, all in all I think it's a very good political pick, probably the best. One. Kamala's big with the Clinton and Obama people. She's part of the neoliberal centrist Dem establishment. The money will pour in from Wall Street and the big donors. The Biden/Harris ticket IS the centrist DNC. Now maybe a couple of months ago this wouldn't have been such a good idea. But today you have the Dems tacking back to the center amid the chaos in the cities. Rather than pick a bombthrower they're picking a prosecutor. The calculation is: The left will have to vote for us anyway because Trump. So the pick did NOT throw a bone to the left. They doubled down on the Biden wing. AOC and Bernie lost today no matter what kind of rhetoric they put in front of Biden to read. This is a centrist ticket. So the question is: Will the left show up for this pair of corporatists?

She has some big weaknesses. In the primaries voters sensed her insincerity a mile away. Her campaign faded before the first primary contest.

She is great with prepared soundbites. "That little girl was me." The problem is she ALWAYS backs down the next day. After hitting Joe on busing it turned out that Harris's busing policy is basically the same as Joe's. And she did that same thing over and over: deliver a sharp line one day and walk her position back the next. Voters didn't like her in the primaries and she didn't get any more likable since then.

Her street cred is dubious. Black? Her mom is Indian, her dad's Jamaican, her grandfather owned a slave plantation. And this is an election season where a lot of African-American voters are tired of the Dems' condescension as exemplified by Joe "you ain't black if you don't vote for me" remark and that other foot-in-mouth about Latinos being more diverse than Blacks. Kamala does not mitigate those issues in the least in my opinion.

Expect to hear the names of Anthony Bologna and his sons Michael and Matthew in the coming weeks. They were gunned down in San Francisco by an undocumented immigrant with multiple prior violent felonies. The killer had been protected from deportation by SF's sanctuary city policy as enforced by then-mayor Gavin Newsom and you-know-who Kamala, who was the SF district attorney at the time. The family pleaded with Harris to seek the death penalty but she opposed that. [For the record I also oppose the death penalty. I'm just talking about what the GOP will be saying here, not taking sides].

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

But does any of that matter? Middle class and elite liberals love her. The left can go pound sand, this is the DNC we're talking about, they ALWAYS screw their base. And where are the AOC and Bernie crowd going to go?

All the other possible candidates had much worse baggage. Liz might have had a chance but once Maxine Waters said that Joe better pick someone Black, that was the end of that.

So like I say, I hate her as a human being but I am not surprised and all in all, it's a solid pick. She does have appeal to a large segment of the Democratic voters and the mainstream big donors. The rest can shut up and accept the corporatocracy. The DNC has decreed it. Again. Whatever happens, do not be fooled by any leftist rhetoric. This is the Wall Street and war ticket. A big winner in Dem circles lately, isn't that sad?

That's like saying - people love the thrill of speed, lets remove brakes from cars. The evaluation, assessment and guidance of governments/society should be more thoughtful - don't you think? — Saurabh Bondarde

You didn't include any of my quoted text and I'm afraid I don't know what you're referring to. The last thing I wrote in this thread was to agree that meat tastes good. I suppose your remark could refer to that. I myself prefer individual choice to State mandate, all things being equal. That puts me on the libertarian side of things. I might even go so far as to say that people like the thrill of speed, so let's have autobahns without speed limits. I'd certainly go that far. That's a better analogy than removing brakes.

I believe people should be allowed to jump out of airplanes if they make an informed choice to do so. That doesn't mean I'm opposed to parachutes. It does mean I'd oppose a government ban on recreational skydiving. I hope that analogy is clear.

Please stop the ad hominem attacks. — jgill

I apologize.

I only wish to tell you that with your background, every single thing that's been discussed on this forum that you think is beyond you, is actually trivially within your capabilities and knowledge.

I know you think I'm a dinosaur of a mathematician — jgill

I have never held such an opinion. When have I ever said such a thing? If you can link or quote anything I've ever written to that effect I'll stand corrected, but as I have never said any such thing you will not be able to. I have noticed that you very often refer to your status as a retired math professor; and that you often announce your ignorance of virtually everything outside of your specialty, and most of the standard undergrad math curriculum. You've said this probably a dozen times in the past couple of months, most recently in the post I just replied to. Perhaps you're projecting or perhaps you have feelings about this. I am certain I've never commented on these matters at all. You are the one who keeps bringing them up.

I don't understand why you post that you have no idea how to prove FTC from first principles, but then get angry when I ask if you are curious to learn how. I admit I don't get where you're coming from. I was, and still am, curious as to why you would post to the effect that you don't know something, without having the intention of either learning it or objecting to it. And that's what I asked.

From ZF the axiom of infinity gives us $\mathbb N$. We use an equivalence relation on $\mathbb N \times \mathbb N$ to form the integers; and another one on the integers to form the rationals. We build the reals as Dedekind cuts of rationals. One the reals are built we can make rigorous definitions of limits, continuity, derivatives, and Riemann integrals; and then any textbook proof of FTC will suffice.

Integers from naturals: http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_proofs_f2014/ccs_proofs_f2014_lecture5.pdf

Rationals from integers: https://en.wikipedia.org/wiki/Field_of_fractions

Reals from rationals: https://en.wikipedia.org/wiki/Dedekind_cut

I've never come across, nor contemplated tracing the proof of something like the fundamental theorem of calculus back to its set theoretic roots. — jgill

What do you mean to communicate here? Do you doubt it could be done? Do you wonder how it's done? Could you produce at least a high-level drilldown of the basic ideas if called upon to do so? Would you like to see what that would look like? Do you wish math would go back to the mid nineteenth century before the age of formalization? I'm wondering what is the purpose of your post. I know the formalisms, I'd be glad to outline them if that's what you're curious about.

Or are you asking why anyone would bother with such pedantry? It's those pesky nineteenth century guys again, noticing that their lack of rigor was causing trouble. For example before the great age of epsilons and deltas, even Cauchy famously wrote an erroneous proof by failing to distinguish between pointwise and uniform convergence.

tl;dr: Did you want to see an outline of a proof of FTC directly from ZF? Or are you wondering why anyone cares?

nonviolent protest appears to be more effective than violent protests. — Aleph Numbers

A researcher lost his job for reporting a study that said exactly that.

https://freebeacon.com/democrats/dem-data-firm-backed-by-former-google-ceo-dismissed-employee-for-linking-violent-protests-and-voter-turnout/

I don't think hamburgers and hotdogs (animal products in general) need a lobby. They taste so damn good. — Nils Loc

This.

There we go. That's what you meant. As they say in forumspeak "FIFY". — Outlander

I did not understand your point in the least. You altered my quoted text to make it appear that I wrote something that I did not write. That's bad form and bad ethics. How would you like me to start misquoting your quoted text and attributing my alterations to you? I can do that if you like.

I can't even figure out if you're agreeing or disagreeing with the point I made.

racism can be solved by posting a black-screen on instagram. — OnNothing

Racism can be solved by removing all depictions of ethnic minorities from pancake syrup, butter, and rice.

What I would suggest to you, though, is that, regardless as to how you feel about the cultural hegemony of the field, just ignore it and find something that you enjoy doing. What's the point of only really being able to deliver a somewhat vitrolic, but also somehow righteous, critique on Reddit? — thewonder

I'd say you tend to take things too seriously.

As to whether you should major in philosophy, that's up to you. You should be aware that the discipline of academic philosophy is NOT the place to express your own idiosyncratic ideas and/or fight the establishment. Rather, you'd first have to spend years sucking up. Decades maybe. You should consider that.

Study philosophy if it interests you. Not because you think it aligns with any personal mission you may have in life. You'll find that it doesn't.

My two cents. For the record, I'm not a philosopher by trade or by education. But I think my response is pretty much on target.

It's clear here that you're talking specifically about libertarianism as understood in the United States since the 1970s, — Pfhorrest

You're probably right. I never heard of libertarianism before I discovered a stack of Reason magazines (now reason.com) in the library one day and thought Wow these people get it! Free minds and free markets. I'm no theorist. I tend to agree with the perspective of reason.com, that's about as far as my theorizing goes.

To be honest, I haven't really put too much thought into questions like "Who will direct the traffic, and so, and so on?" Libertarianism slowly became vaguely synonymous with liberal social attitudes and laissez faire economics during the twentieth century, which is not necessarily what I'm suggesting by anti-authoritarianism. — thewonder

As I understand it there are two major strains of 20th century libertarianism. Cosmolibertarians, who are the liberal wing; and paelolibertarians, who are the conservative wing. Paleolibertarians want to be free from government interference so they can educate their kids in evangelical religious schools. Cosmolibertarians want to be free from government interference so they can smoke dope.

Cosmo = https://reason.com/

Paleo = https://www.lewrockwell.com/

Libertarianism sort of began as a left-wing philosophy, but I am specifically referring to anti-authoritarianism and do not mean for it to be synonymous Libertarianism. You could, however, be an anti-authoritarian Libertarian. — thewonder

I'm trying to figure that out. My position would be that a libertarian -- small l, to distinguish libertarians from the collection of oddball wackjobs that tend to make up the Libertarian political party in the US, at least -- sees the need for some authority, to keep order and run the schools and so forth. Libertarians just want as much of the authority as possible to be privatized. There's no reason in theory why police departments couldn't be privatized and quite a lot of other things people typically think of as communal. Highways and the like. But to have no authorities at all? That means that when it's time to build a road, we all have to get out there with picks and shovels?

I must not be understanding you. What does it mean to have no centralized authority at all? How can anything get done? We have to all get together in a big community meeting to decide how to build a road? Nothing could ever get done.

I tell people that I think that there should either be an informal set of a-systemic Liberal democratic governing assemblages and Anarchist communes or a "loosely affiliated set of freely associated societies", I use the term "society" not to refer to an "ordered community", but because "groups" doesn't really seem to refer to what I am attempting to describe. The former is really kind of a programmatic approach to the creation of what Murray Bookchin called the "Commune of communes". In short, and without political jargon, I do not agree that there must be some authorities as you say that there must. — thewonder

I'd like to see something like this too, in theory. Maybe the loose tribal societies when we lived in caves.

But what about human nature? What happens when the tribe over in the next valley comes under the spell of a charismatic leader who convinces them to come over and steal all our stuff and make us their slaves? We'd have to organize for self-defense. And by far the best way to do that is a command-and-control structure. You could not run a military operation as a commune. As George C. Scott said in Patton: "We defend democracy here. We don't practice it!"

I happen to have seen some of your idealized society in action. In 2011 I attended many of the Occupy protests and was a regular visitor to the encampments in the San Francisco bay area. They had this crazy process of making decisions where everyone had to agree and it took hours to decide the most mundane things. "General Assembly" is the name for it. I'd always joke to myself that whatever the shortcomings of business, they know how to decide things in meeting. You get everyone in a room, everyone has their brief say, and if an obvious consensus doesn't develop, or there's a difference of opinion, the senior person in the org chart makes a decision and everyone else gets on board.

You simply can not function as a group by letting everyone have their say and argue till everyone agrees. There has to be a decision maker, they have to be willing and able to sometimes be arbitrary, and once the top authority figure decides, everyone else has to focus their energy on supporting that decision, even if they disagree with it. There is simply no other way for a group to function than some level of hierarchical power. Which we call authority.

What do you think? Why am I wrong?

So, I've recently come to the realization that anti-authoritarianism is the only political philosophy that any person should adopt. — thewonder

Are you distinguishing between libertarianism and anti-authoritarianism? Even libertarians like a police force to keep some semblance of order. They might argue for a privatized police force, that would be a libertarian stance. Even so they'd delegate the work to someone. I wasn't able to follow the distinction you're trying to make.

You can't be fully anti-authoritarian. I don't want the government constantly asking me how I feel about the soybean subsidies in the latest farm bill. I prefer to let my elected representatives decide these things so I don't have to be bothered. There must be SOME authorities, as open to abuse and corruption as authorities are. What would be the opposite? Popular vote on every single issue? That would be Twitter-ocracy and I think we've already seen that it's a disaster. Rule by the loudest and stupidest.

My Mind is not physical. — TheMadFool

I have read that scientists have measured human bodies at the moment before and after death, and can't detect the soul or spirit. I'm not sure this proves that there isn't one, or that it isn't somehow a product or at least byproduct of a physical process.

What's an analogy? A bunch of people get together and form a bowling team. A bowling team exists in the world. It's part of a league, it pays its entrance fee, it shows up together at tournaments. See the film The Big Lebowski for a fuller discussion of these matters.

Yet if we weigh the bowlers individually; and we weigh the team; we will find that no matter how finely we take our measurements, we will always find that the "team" has no weight other than what can be accounted for by the individual bowlers.

So there are abstractions that are real, and have existence and import in the world, but that are not physical. The the law is only the weight of the lawbooks. But the books are not the law. The law is a weightless, nonphysical abstraction.

Likewise perhaps the mind. Still, I agree that the mind is special in some way and your argument's not bad.

What if they prove that you can take a greater object from a smaller? — Gregory

Who's they and what would such a proof involve?

How is that much different than B\T? To me B\T says that is nothing discrete in geometry anymore — Gregory

I don't know what you mean by that. The best I can suggest is that you read through the Wiki pages that I've already linked on the Banach-Tarski paradox, the free group on two letters; and the idea of a paradoxical decomposition of a set.

The core of the proof is the paradoxical decomposition of the free group on two letters, which is explained in step 1 of the Wiki proof. As I've noted, this step does not require the axiom of choice and barely even uses infinite sets. It's just a surprising fact about finite-length strings made out of two arbitrary symbols and their symbolic inverses.

Another basic notion that might help is the idea of a nonmeasurable set. It turns out that there are sets of real numbers (and Euclidean space in general) such that it's not possible to assign them any sensible notion of size that analogizes the familiar length, area, and volume. The pieces involved in the B-T decomposition are such nonmeasurable sets. Our intuitions about volume fail on these types of sets, and that's why it's called a paradox.

I'm not sure how to respond to your concerns to repeat that the proof of B-T is a straightforward application of a number of very simple ideas. There's hardly any difficult math involved at all. But the key point really is that there are nonmeasurable sets that don't behave property with respect to intuition. To say anything more specific I'd have to go into the symbology but that's all on Wiki.

The B-T theorem doesn't break math or abolish discrete geometry, whatever you meant by that. It's just a counterintuitive fact about Euclidean space.

By the way have you seen the Vsauce video on B-T? He usually annoys me but his video on this topic is very good.

https://www.youtube.com/watch?v=s86-Z-CbaHA

ps -- It occurs to me that the word Euclidean could be part of the conceptual problem. What we think of as modern Euclidean space is very much different than Euclid's geometry. Euclid considered subsets of space like points, lines, and cones and so forth. But he never considered completely arbitrary sets of points that could not be assigned a sensible volume or size! Perhaps if we called $\mathbb R^3$ by the name set-theoretic space, we wouldn't expect it to behave as nicely as Euclid's space. We call $\mathbb R^3$ Euclidean space, but it's very different than what Euclid was thinking of.

I think Banach-Tarski disproves common mathematics the way Godel set out to do (but failed). It breaks math — Gregory

Oh gosh no. It's a very simple theorem. It just shows that the universe isn't literally identical to three-dimensional Euclidean space. And as I've said, the core paradox is in the free group on two letters. It's not a difficult argument.

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

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

The so-called "paradox" just shows that our mathematical intuition of physical space isn't actually true. Mathematically it's an easy theorem, accessible to anyone who puts in a little effort. The Wiki page has a very accessible outline of the proof.

It doesn't "disprove common mathematics." It disproves the common intuition that physical space is the same as 3D Euclidean space. It isn't. The paradox depends on the set-theoretical viewpoint that we can form arbitrary subsets of such a space. There's no physics theory to match or allow that.

Nice defense of the AOC ! [/url]

LOL. When I read that sentence in my mentions I thought I must have said something good about Alexandria Ocasio-Cortez in one of the political threads!

— jgill

Nice defense of the AOC ! [/url]

:cool: I think that those who spend time studying set theory and foundations have a much deeper appreciation for those subjects than do those of us who are briefly exposed to it and move on to different topics. It may have been a shift towards foundations the math department at the U of Chicago initiated back in the late 1950s that caused a rift with the physics department and resulted in physics students being required to take all their math courses in the school of physics. But I don't know how long that lasted; it may have been a temporary policy. — jgill

As a math undergrad I called up the physics dept and asked them if there was way to study physics in an accelerated manner since I already knew a lot of the math. They told me absolutely no way, I had to start with freshman physics and take only physics classes. They were quite snippy about it! There was definitely some friction between the math and physics departments.

And without the AOC or something similar we wouldn't have non-measurable sets. :sad: — jgill

The question of why we choose one axiom over another gives insight into mathematical philosophy. As I've mentioned, AC regularizes infinite sets. It makes them well-behaved and gets rid of pathologies. So we adopt AC based on principles like convenience and naturality. That is to say, highly pragmatic and human grounds, not logical ones. Emphasizing again that math is in large part a human activity. Raising the question of which parts aren't, if any!

Why do I say that AC is natural? Because a choice set is a legislature. Let me explain. Say you have a country that's divided up into states. We say that two people are in the same "equivalence class" if they live in the same state. This is an equivalence relation that partitions the country into pairwise disjoint equivalence classes.

Can we choose a representative from each state? Sure, that's the US Senate. The US Senate is two applications of the axiom of choice. And it's perfectly natural. Each state is nonempty, so we can pick a representative from each state and send the representatives to Washington.

The House of Representatives is a choice set on the nation's Congressional districts. We say two people are equivalent if they live in the same CD, then we choose a representative from each district.

Now suppose there's a country somewhere hat has infinitely many states. For that matter suppose there are uncountably many nonempty states. Why can't we simply pick a legislature? It would be wildly unnatural for you to tell me that we can't pick a representative of each state. Of course we can, whether there are finitely many or infinitely many states.

A choice set is just a legislature. And you can always pick a legislature. Adopting AC is natural; adopting its negation would be unnatural.

Eg for every 5 trials they reduce your sentence by one year or something like that — Gitonga

For example they might let a hardened young killer out of prison if he offers to undergo experimental treatments to cure him of his violent tendencies.

If we have alien stuff — RogueAI

This was a very strange announcement from a government agency. I for one don't think anyone's discovered space aliens or even vehicles "we couldn't make ourselves." There's something off about this story. Myself I'd withhold judgment.

What is an effective way to curb corruption in 3rd world countries — Gitonga

And what shall we do about the corruption in a certain first-world country I could name?

for petty corruption (mainly) set up a dedicated phone line to report corrupt officials, cops, etc. — Olivier5

What could possibly go wrong? East Germany here we come.

That is bull shit. — Metaphysician Undercover

Tell it to Wiki, which yesterday you claimed was the ultimate authority on these matters and which, once you realized it agrees with me, you no longer have any interest in.

so be it. — Metaphysician Undercover

So be it.

The B-T paradox, as has been said several times in this forum, depends upon the Axiom of Choice. Not one of Euclid's. Sometimes the Axiom of Choice is involved in what are called "pathological" examples in mathematics. — jgill

Just perusing recent threads so no worries if this topic's no longer of interest. But you should be aware that the negation of the axiom of choice is an even more fruitful source of pathologies: a vector space with no basis; a surjection with no right inverse; a commutative ring with unity with no maximal ideal; a Dedekind-finite set that is nevertheless infinite; infinite sets that are not bijectively equivalent to any Aleph; an infinite set whose cardinality changes if you remove a single element; an infinite set not bijectively equivalent to any of its proper subsets; and many others. The axiom of choice serves to make infinite sets behave as they should and is therefore the more natural choice than its negation. In fact without the axiom of choice and with a single inaccessible cardinal (which amounts to nothing more than assuming ZF is consistent) the real numbers can be expressed as a countable union of countable sets, making it impossible to get modern probability theory off the ground.

Ever see the infinite hat problem? That's my favorite bizarre consequence of the axiom of choice.

The Banach-Tarski theorem uses the axiom of choice in a very natural way. You have an equivalence relation and you choose a representative from each nonempty equivalence class. It would be much more counterintutive if you couldn't do that than if you could.

The actual heart of the paradox is much simpler and depends on nothing more than the fact that the free group on two letters is paradoxical. The core idea is in fact syntactic and has little to do with 3-space and nothing to do with the axiom of choice. I wanted to write this up for this forum a while back but the person I was talking to lost interest.

You only need the axiom of choice to lift the paradoxical decomposition of the free group on two letters to the group of rotations of the unit sphere. Wikipedia has an excellent overview of the entire proof that is quite accessible. It's a long proof made up of very simple steps that can be followed one by one.

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

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Also nonstandard analysis and in particular the construction of the hyperreals is about an hour's worth of work for a competent undergrad in math. Given the basic notions of set theory like set inclusion and partial orders and so forth, one defines filters, ultrafilters, and nonprincipal ultrafilters. You then mod out the reals by a nonprincipal ultrafilter in a manner analogous to how you construct the reals from equivalence classes of Cauchy sequences of rationals. The entire construction is surprisingly simply. The only hard part is the model theory that tells you exactly when you can transfer results back and forth from the reals to the hyperreals. Terence Tao has written some beautiful expository articles on nonstandard analysis and ultrafilters that are extremely enlightening, which I can link if anyone's interested.

The tl;dr here: (1) The axiom of choice is far more natural than its negation; (2) The Banach-Tarski theorem can be understood by a high school student who's willing to put in some work. There are no advanced concepts involved. The Wiki proof is very good. There are a lot of steps but each is relatively easy. (3) The construction of the hyperreals can be understood easily by a math undergrad in about an hour. Or a few days if you want to go back and nail down every detail and read the Tao articles and achieve some grokitude. There's much less to the subject than the folklore around it would indicate.

Sorry, I don't normally use Wikipedia — Metaphysician Undercover

But you're not above trying to quote-mine an off-topic page to make a point.

and I only looked at the page on "equals sign". — Metaphysician Undercover

But the equal sign is not the topic of discussion. The concept of mathematical equality is. No wonder you name-checked Robert Recorde. You looked up the wrong thing.

Nevertheless, what is at issue is whether a so-called "mathematical object" is an object identifiable according to the law of identity. It is not, because two equal, but different things, such as the addition operation of 2+2, and the number 4 are said to be the same object. Therefore, despite what the Wikipedia quote indicates, and many mathematicians might claim, these two different things, the operation represented by "2+2", and the number represented by "4", cannot be "the same" if we adhere to the law of identity, which denies that two distinct things are the same object... The mathematical axioms which state that these two distinct things are the same thing are nothing more than deception. I know you'll continue in your denial, but so be it. — Metaphysician Undercover

Completely ignoring the point I just made. Anyway I just wanted to note that you looked at the wrong Wiki article and when pointed at the right one, which 100% supports my point of view, you change the subject.

2 + 2 and 4 point to or refer to or represent the exact same object. It's not possible to do math without that understanding.

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page. — Metaphysician Undercover

Prompted by our conversation I went and looked at what Wikipedia says. It 100% absolutely agrees with what I said. You're being quite disingenuous to claim otherwise.

From Equality (mathematics):

"In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. [my emphasis] The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct. [my emphasis]"

That's exactly what I'm telling you. Baffled that you claim that Wiki says otherwise when plainly it says exactly what I'm saying

I think they operate quite rightly, provisionally treating the theory as if it is the thing is part of how it works I think. If the discussion we'd have is "what properties of a model can be treated as standing in for a property or behaviour of the thing", that'd be quite different from "are all models merely epistemic" - the first would actually be about the uncertainty principle, the second is a much broader realism vs anti-realism of scientific content debate. If you and I have to go through the latter to get to the former, that's fine with me, both are interesting. — fdrake

I'm really interested in diving into this but haven't had time to think about it much or read your other post to me on the subject. Instead of deferring my response to your posts indefinitely in the false hope of eventually saying something clever, for now I'll just retreat into what I think my position is.

The Planck scale is the scale at which our current physical theories can not be applied. We don't know what's going on at distances and times smaller than the Planck length and time, respectively. That's why I say it's epistemic.

I am not sure I follow the argument from Fourier series to saying that "therefore the Planck scale is ontic."

I don't think I'm conflating the problem of models in general with the idea of the Planck scale. Except that you can't apply a model outside of its domain of applicability, which is the same as with the Planck scale. Good question. With models in general, they don't so explicitly tell us what we can't know. If we have a model, people can disagree about whether it applies to a given situation. But with the Planck scale, everyone agrees on where the theory applies and where it doesn't. I'm afraid I don't have any better thoughts on this topic at the moment. But it is important to me because I have a strong belief that the Planck limits tell us what we can know, not what is. I wish I understood your argument better.

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page. — Metaphysician Undercover

I am hardly responsible for anything on Wikipedia, let alone your own interpretations of same. I didn't spend any time looking at that Wiki page. I know the math. Critiquing Wiki on this issue is not of interest to me at the moment. Some Wiki pages are better than others. Especially when it comes to the lead sentence or paragraph of a technical subject, many simplifications are made and many subtleties ignored. They're trying to give naive readers a sense of an idea. Summary and overview. Not to be taken literally and used as some kind of trump card in an online discussion. Jeez man you are reaching.

I loathe modern society and everything it entails. — madworld

I haven't read this thread, just jumping in with something I know about this, which is that Freud had this same concern. Wild and crazy passionate erotic animalistic humans trying to get through life sitting at desks and wearing suits and being polite to one another. Some of this arrangement seems to be fraying at the seams lately if you read the papers.

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

What if mathematics IS metaphysics? — Eremit

Could not be, ever since Riemann and others discovered the logical consistency of non-Euclidean geometry in the 1840's. Prior to then it was believed that mathematics was physics and that the world conformed to the Euclidean paradigm. Riemann discovered that other geometries were possible; and then Einstein showed that not only were they possible, but that they were actually true.

Mathematics can not say what the truth of the universe is. Math can only say which mathematical ideas are logically consistent. And there are logically consistent but mutually inconsistent theories out there.

Even in contemporary physics there are many interesting new theories out there that all use math but can't all be true. Math is just a tool for modeling ideas. It can't tell you what's true.

The equal sign means that the two sides are equal, just as "equal" indicates. Here's what Wikipedia says:
"The equals sign or equality sign, =, is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover

Yes that's very cool. I do happen to know that Robert Recorde invented the equal sign. I keep hoping someone will ask me someday.

Clearly you are wrong to say that it's everyone's rule, that the equal sign means that the right and left sides refer to the same object. This rule is an expression of your idiosyncrasy. — Metaphysician Undercover

You're wrong mathematically and we are not getting anywhere. I tried to beg off the conv a while back but seem to be having difficulty executing on my intention. We're not making progress. I have nothing new to say.

I already went through your converse error, but I'll explain it to you again, as you don't seem to get it for some reason. I believe the formal fallacy is called affirming the consequent. If two symbols refer to the same thing, then there is necessarily equality between what the symbols refer to. But this does not mean that two equal things are the same thing. Do you understand this so far? — Metaphysician Undercover

I understand your belief that two things that are mathematically equal are not the same thing. You're wrong. If X = Y then X and Y necessarily refer to the same abstract object. There is no question about it.

Many things are equal, like two human beings, two dogs, or two cats, in the sense that the two distinct things can be given the same value. A human being might be equal to a dog if the evaluation criteria is being an animal. — Metaphysician Undercover

Yes, and this is called the fallacy of equivocation. Using the same word, equality, with two distinct meanings within the same argument. In human affairs, equality has a different meaning. When Thomas Jefferson wrote that "All men are created equal," he of course did NOT mean that they were mathematically equal, as 2 + 2 and 4 are equal; but rather equal under God and nature as human beings.

Mathematically, no two human beings are equal. They could be equivalent modulo various properties. If they both live in California or if they are of the same race or work in the same profession or so forth, we could call them equivalent. Or if we want to assert the desirability of the state of affairs in which they each have the same chances and possibilities in life, that would be another form of human equality.

But it's not mathematical equality. I'd like to say it's beneath you to stoop to such a low rhetorical trick. But I guess it's not beneath you after all. Frankly it's beneath ME to have to explain this in words, it should be obvious that mathematical equality and Jeffersonian equality are not the same thing.

And do you see that the equal sign means that the right and left side are equal, as the Wikipedia articles says? How can you conclude, without the fallacy of affirming the consequent, that two equal things are necessarily the same thing? — Metaphysician Undercover

Because that's what mathematical equality is. That's how mathematicians define equality. Ultimately you have the same set on both sides of an equation. Once again you erroneously take your ignorance of mathematics as profundity in philosophy.

If two expressions do NOT refer to the same abstract object, then they are NOT equal.

That is a mathematical fact. Though of course it is not a fact in the Jeffersonian sense of equality. Two distinct people could be equal under the law. Equivocation. Same word different meaning depending on context.

Right, now if you were in Green Bay, and followed the directions of how to get from Milwaukee to Sheboygan, you would not get there from Green Bay. Likewise, if you were at 6, and followed the directions of how to get to 4 from 2, i.e. "+2", you would not get to 4 from 6, following those directions. — Metaphysician Undercover

WHAT? That is completely nonresponsive to the point, which is that Sheboygan is still Sheboygan regardless of how you got there.

Nevermind. If you respond I'm going to try not to. Just for my own sanity. Interacting with you is fun in a tongue-in-my-sore-tooth kind of way, but I need a little break please.

Is it your rule that the left side of an equation refers to one object, and the right side of an equation refers to the same object? — Metaphysician Undercover

Yes. My rule and everyone else's. That's what the equal sign means.

Ah. Harkens back to the last words of the state as they executed the heliocentrist. Before they were disproved and later overthrown of course. How reminiscent. — Outlander

For sake of conversation, I don't think they actually executed any heliocentrists. If you're thinking of Giordano Bruno, Wikipedia points out that ...

"Starting in 1593, Bruno was tried for heresy by the Roman Inquisition on charges of denial of several core Catholic doctrines, including eternal damnation, the Trinity, the divinity of Christ, the virginity of Mary, and transubstantiation. Bruno's pantheism was not taken lightly by the church, as was his teaching of the transmigration of the soul/reincarnation. The Inquisition found him guilty, and he was burned at the stake in Rome's Campo de' Fiori in 1600."

Neil deGrasse Tyson somewhat misrepresented the Bruno case on his tv show. Bruno's fatal heresy was religious and not necessarily scientific.

Nor was Galileo executed. He was confined to house arrest. Even this story is more nuanced than commonly known. The Pope was not a science denier but was in fact an enlightened, scientific man; and Galileo was the Pope's good buddy. Galileo went out of his way to cause his old friend trouble and that's a large part of what happened to him.

And nor, as long as I'm here, does the earth revolve around the sun in any absolute way. If you stood outside the galaxy and plotted the paths of the earth and sun, they'd spiral around each other as the spirals moved through the universe. Only by drawing a coordinate system with the sun in a fixed position can we write down the "orbit" of the earth. And if we wanted to, we could assume a fixed earth and write down the orbit of the sun. A fable like "The earth revolves around the sun" involves simplifying assumptions that are not metaphysically true; unless by truth you mean historically contingent scientific convenience and consensus.

I'm thinking of taking up geocentrism. It might be the last sane stance left in a crazy world.

No I'm not making such an identity. But I'm saying that if you assert that "2+2" is the same object as "4" you are claiming such an identity. — Metaphysician Undercover

This is the same error @jgill pointed out that I made the other day. I do not assert that 2 + 2 and 4 are identical. They're obviously different symbol strings. They do refer to the same abstract object, as does 3.999..., 6 - 2, and the smallest positive integer $n$ such that there are two non-isomorphic groups of order $n$. All those descriptions refer to the same abstract object; same as in identical.

The "means" or "process" of getting to the abstract object is irrelevant. In fact given the abstract object represented by 4, all the other representations are already inherent in it. You deny that. That is the crux of our disagreement I believe.

The process of adding two with two will make four, but it is not four. I am saying that the process which makes four is not the same as the number four itself. A cause is not the same thing as its effect. You are insisting that the two are the same. — Metaphysician Undercover

The abstract number represented by '4', which I represent as 4 (but do not confuse the referent with the symbol), is not an effect. It stands alone, and in its very existence incorporates all of its representations and the processes that they bring to mind. But the processes are not the thing. There are multiple processes that may lead to the same abstract thing, as this example shows. But I don't think of this as a process leading to 4. I think of 4, as the primary object, that already incorporates all of the processes that could lead to it. 2 + 2, 3.9999..., etc.

I really can't believe that you do not see the difference between what "2+2" represents and what "4" represents. — Metaphysician Undercover

That's because I'm not hallucinating "causes" or "processes" where there are none. The causes and processes are secondary to the essential existence of the abstract number 4. You have your ontology backwards.

The former indicates two quantities of two, with an operation of addition also indicated. The latter indicates one quantity of four. To interpret "2+2" as representing the number four is very clearly a misinterpretation. — Metaphysician Undercover

You're stubborn, I'll give you that.

The proof is that we can represent the relationship between two distinct 2s in many different ways, such as "I have 2 dogs, and 2 cats", "2,2", "2+2", "2X2", etc. In all such instances of representing two distinct quantities of two, they must be interpreted as two distinct quantities, to avoid misinterpretation. If you do not follow this simple rule of interpretation you completely disregard the application problem of adding apples and oranges. If two groups of two are automatically four there is no way to avoid the category mistake described as adding apples and oranges. In other words, you do not allow any provision for the reason why they were represented as distinct in the first place. Therefore, in insisting that "2+2" represents the same thing as "4", you are denying any valid reason for representing the two 2s as distinct in the first place, and nullifying that representation, of two distinct 2s, as an invalid representation. — Metaphysician Undercover

Oh, proof. Why would I waste my time looking at, or even acknowledging that you've provided a proof? You don't offer me the same courtesy.

But again you're just repeating your confusion. The number 4 incorporates within it 2 + 2 cats or 3.999... or whatever. They all point to the same thing. They're not "ways of getting to" the thing. I can't imagine why you have such a strange idea.

Sheboygan, Wisconsin is the same identical city whether you get there from Milwaukee or Green Bay. 4 is the same number whether you "get there from" 2 + 2 or 3.999... Why you think these two cases are different I can't imagine. Unless you think that $\text{Sheboygan}_{\text{Milwaukee}} \neq \text{Sheboygan}_{\text{Green Bay}}$. Is that what you think?

The fact that a different process can be utilized to make an object indicates that the process is not the same thing as the object. — Metaphysician Undercover

Are you identifying an object such as the number 4 with the process that "creates" it? I think that's a pretty big stretch. I can't imagine Aristotle wouldn't have been that wrong, so I'm guessing your interpretation is.

In other words you seem to be associating 4 with the "process that created it," so that a 4 stamped whole on God's forge is different than one made by jamming 2 and 2 together.

This notion makes no sense to me. The number 4 is the number 4, and it's inherent in its nature that it can be represented many different ways. Are you claiming Aristotle said that a thing is actually "thing plus method of making it?" And what makes him right and everyone else wrong?

You know, no one I've told the story too ever thought about it like that. — fdrake

I have that happen to me a lot.