Ok, I'm reading up on "free group on two letters — alcontali

That's the heart of Banach-Tarski and it's purely syntactic, nothing to do with spheres. It's amazing. I hope you will look into this. I have a little writeup but I'm not sure this is the place for it. Depends on your level of interest. I notice that you're the thread starter here so perhaps it would be ok.

So, yes, serious levels of perseverance are for me often about not losing face and not letting key people down. — alcontali

Well see, that's a point I'm trying to make. It's not about face, it's just about learning what someone else happens to know. Your curiosity is wide so you shouldn't expect to master all this stuff in a few minutes of Wiki-ing. Especially topos theory and categorical logic. Grad students struggle with that stuff.

But Banach-Tarski, that's actually very approachable. The proof is long but the parts are simple. And the free group on two letters, the Wiki pages are good and it's a true veridical paradox, incredibly counterintuitive but requiring no questionable principles, not even the axiom of choice. No set-theoretic principles deeper than the union of five sets are used; and all the objects considered are finite strings and sets of finite strings.

I don't know what's the value of Coq or other proof assistants for someone who's trying to learn modern math. Better perhaps to just learn the math IMO.

I've found a machine-verifiable version of the Banach-Tarski proof for the Coq proof assistant. It looks endless. I first thought that this is because the Coq formalisms are excruciatingly lengthy, but the classical, annotated first-order-logic version is also 30 pages long. — alcontali

You hit a lot of really cool topics. I'll keep my answers brief. I hope we can focus on one or two things instead of branching out into too many diverse topics. You uptake new information rapidly but have a hard time staying focussed. Is that a good guess?

* Now re Banach-Tarski. First, a machine-formalizable proof would not be remotely understandable to a person who didn't already understand the standard proof. So that's not the way to go. Coq and machine-assisted proofs in general are an interesting subject, but let's hopefully not get sidetracked into that.

To that end, you should definitely read the @Mephist I linked earlier. He knows a lot more about Coq and constructive math than I do if he's around.

Secondly, it turns out that the proof of Banach-Tarski is very straightforward. The outline on Wiki is very good. And as I said, it is at heart a syntactic matter that could be programmed into a computer. Not the entire proof, but the core idea. If you are interested, the key buzzphrase is: "The free group on two letters has a paradoxical decomposition." That phrase leads to a web of interrelated Wiki pages that are very good. You'll halfway understand B-T from those.

But you know -- the fact is that Banach-Tarski is at heart a logical paradox. You lay out some very sensible axioms, and they lead to a non-sensible yet logically correct conclusion. Tarski and von Neumann and others were all interested in applying group theory to geometry. So this is a paradox in "geometric group theory." That's yet another buzzwordy way to say it. But Tarski was in the middle of all that.

It uses pyramidal vocabulary. For example:

Theorem 22 (Tarski’s Theorem)Let G act on X, and let E ⊆ X. Then there exists a finitely additive, G-invariant measure on X defined for all subsets of X and normalizing E if and only i fE is not G-paradoxical.

So, it requires figuring out what "G-invariant measure" and " normalizing" means. Otherwise, Tarski's theorem 22 will remain inaccessible. — alcontali

No no no!! It's not NEARLY so bad. I could fully explain the proof of the Banach-Tarski theorem to you in a perfectly understandable way. Just not here, because it's a little lengthy. There are several moving parts. But each part of the proof is quite straightforward.

You don't have to make your life difficult by wondering what those buzzwords mean. You could ask me, for example. But not in this thread. If someone starts a B-T thread that would be more appropriate. But the buzzwords are easy once they're explained.

The think about the Tarski theorem you quoted is that professional mathematicians can always summarize the proof in a sentence or two of jargon. The idea isn't to understand the jargon from the top down. You can understand the proof from the basic ideas on up. It's long but each part is surprisingly straightforward.

I can tell you want the buzzwords mean, but they won't shed light on the paradox until AFTER you understand the paradox. Tarski's using professional jargon.

It is interesting, though. If the proof were about formal language, I would probably keep drilling down in order to figure it out. — alcontali

I could lay that out for you in a page of exposition. Start a new thread, I feel bad enough about hijacking this one. The heart of the proof is about formal language. It's about all the finite strings you can make using two symbols and their inverses. It is the most syntactic thing you ever saw. It's a very natural construction that contains a surprising paradox. We then apply the paradox to Euclidean three-space.

But nevermind, this is not a forum for lengthy mathematical exposition. You can just google "Free group on 2 letters paradoxical" and there are a bunch of interrelated Wiki pages. That's probably better than my posting an exposition here.

Actually, Tarski does have theorems in the realm of formal languages, e.g. his undefinability theorem. I wonder how he ended up also working on abstract algebraic geometry and topology? — alcontali

These days logic is geometry. It's all come out via the mysterious categorical point of view. Buzzword: "topos theory."

But a lot of these old guys did a lot of different things.

Hilbert is also like that. He did his language-oriented Hilbert calculi but also his geometry-oriented Hilbert spaces. I wonder how people manage to not only handle both domains but even be original in both? — alcontali

Hilbert did much more than that. He's regarded as the number one mathematician of the first half of the 20th century. (Grothendieck wins the second half). He almost beat Einstein to general relativity. He's famous for a lot of deep work in a lot of areas. Very interesting guy, I'm sure you've seen the famous picture of him in his hat.

There could actually even be a deeper link between geometry and formal language, because the late Voevodsky saw homeomorphisms everywhere in formal language domains, while those are clearly a geometry concept. I still don't understand Voevodsky's univalent foundations. Everybody seems to be raving and hyping that stuff, but I still cannot wrap my head around it. — alcontali

That's something else I could put in context for you in fifteen minutes, but not this fifteen minutes. Make a thread so we don't go in all directions here.

There is of course nothing more fashionable than being able to come up with a finer point in the Voevodsky stuff. I cannot "show off" because I unfortunately don't get it! ;-) — alcontali

I hope you will allow me to a suggestion. It's not about showing off. It's about learning. You happen to be interested in a lot of things I know a little about (and some I know a lot about). I'm happy to yack about them. I'm pretty ignorant about most other things. Ask anyone.

There's a Zen saying. Beginner's mind.

Still, this twist is obviously not needed in upstream pure mathematics. So, it is a matter of correctly switching between both context-dependent views on the concept of witness. — alcontali

That's why they call it pure and applied math! And likewise the theory of computation. I hope you will agree that although Turing machines have an unbounded tape, no physical computer does. So I think you are making a rather commonplace observation. And also of course drawing the distinction between computability theory, the study of what is computable by a machine; and complexity theory, the study of how efficiently things may be computed. Again, true but not entirely earth-shaking. Yes?

I fully acknowledge, that in ZFC 2+2 is "the same" as 4. I am not denying this. I am saying that it is wrong, because it violates the law of identity, without any justification. — Metaphysician Undercover

I was skimming your reply looking for a point of reference, something I could understand. I came to this. I think it's a point of irreconcilable difference. I don't agree with your judgment, but I am incapable of rational response, because I cannot fathom the point of view.

This is my own personal limitation, I'm certain of that. My ignorance of philosophy is profound. I have no doubt that you have a point to make that, from your point of view, is a valid point.

I myself do not ever think I could agree with or even understand such a point of view. Within ZFC, at least, the statement 2 + 2 = 4 is a theorem that can be proven according to strict logical principles that are clearly expressed; and using assumptions that are clearly stated. Within this framework. 2 + 2 = 4 expresses an identity of sets. This is a technical fact that is beyond dispute. And set equality is defined directly in terms of the logical law of identity. I thought I explained it. The axiom of extensionality leverages the logical law of identity. So 2 + 2 = 4 is a perfect expression of the law of identity.

I acknowledge that you feel differently about this but I don't regard myself as being capable of ever understanding such a point of view. And I would not want to be able to understand such a point of view even if I could!

I'll read whatever you write on the topic in the hopes I might learn something, but I don't think it will be productive for me to engage on this.

↪fishfry My point is that the view has nothing to be said for it - until or unless we can explain in a rationally satisfying way how it is that an extended thing can be conscious, then positing that all extended things are conscious will do nothing whatever to help. — Bartricks

My quarks are really annoyed at this! They have a rich inner life, you know. At the very least, by virtue of being the constituents of the atoms that are the constituents of the organic molecules that are the constituents of the basic processes of life; and that make up the organs and the body and the nervous system and the brain, which somehow has managed to achieve self-awareness and consciousness.

Why should not the phenomenon of consciousness be experienced at every level of its existence?

After all, our network of neurons is a grid for transmitting electrochemical signals. And at the bottom of reality, at the level of the particle physicists, it's quarks and electrons. Tiny bits of electricity.

There's electricity at the bottom of creation. And it's electricity that implements consciousness, or could we say at the very least, hosts it, as a computer hosts its software.

So yeah, you know, I was going to say that I don't have enough interest to defend my belief rationally. But it turns out I do have a bit of a plausible story. What do you think?

Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same? — Metaphysician Undercover

I walked through this in detail a few posts ago. In the Peano axioms they are both the number SSSS0. In ZF they are both the set {0, 1, 2, 3}. = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }.

See for example

https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals

This is absolutely false. The symbol 2 has a meaning, the symbol 4 has a meaning, and the symbol + has a meaning. Clearly 2+2 is not the same concept as 4. — Metaphysician Undercover

We must be talking past each other in some way. I cannot conceive of anyone claiming 2 + 2 and 4 are not the same thing. I can't respond because from where I sit you're talking nonsense. If you have some subtle philosophical point it eludes me. I just can't respond. Perhaps you have a reference to support your point of view.

Here's my earlier post where I described the Peano and ZF constructions.

https://thephilosophyforum.com/discussion/comment/323461

I acknowledge that there might well be some philosophical point of view that allows you to claim that 2 + 2 and 4 are not the same thing. I've never heard of it and I don't understand what you mean, but that could just be due to my own ignorance.

But you claim that 2 + 2 and 4 are not the same object in ZFC. And THAT is an area where I am not ignorant. You're just wrong. 2 + 2 and 4 represent the same set in ZFC.

But forget set theory. You claim that 2 + 2 is not 4? The last time I heard that it was in the novel 1984 when the protagonist Winston Smith is being tortured to obtain his submission to Big Brother. He's ordered to believe that 2 + 2 = 5; and in the end, he does.

You tell me how 2 + 2 is not 4. If it's not, what is it? And have we always been at war with eastasia?

The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity. — Metaphysician Undercover

Of course 2 + 2 is the same thing as 4. I cannot imagine the contrary nor what you might mean by that claim.

But more importantly, they are the same set in ZFC. So it's not an example of your claim that ZFC allows two distinct things to be regarded as the same.

But you hold that 2 + 2 and 4 are not the same? How so? Without quotes around them they are not strings of symbols, they are the abstract concept they represent. And they represent the same abstract concept, namely the number 4. You deny this? I do confess to bafflement.

ps -- I didn't read the S1 and S2 parts of the thread, if it might help I'll go back and review them.

I think that math infinities don't count as actual infinities, they just being potential infinities, and that actual infinities are impossible since they can't complete, much as the definition of 'Infinite' hints at, plus that 'infinite' is not really an amount or a number, also because it cannot be capped. — PoeticUniverse

I wonder if by actual you mean physical. In math the axiom of infinity gives actually infinite sets; that is, infinite sets all of whose elements can be corralled into a single set. Of course these are not physical sets.

length,
width,
depth,
4D—your world-line;
5th, all your probable futures;
6th, jump to any;
7th, all Big Bang starts to ends;
8th, all universes’ lines;
9th, jump to any;
10th, the IS of all possible realities — PoeticUniverse

What do you make of infinite-dimensional spaces? An example would be the set of all continuous functions from the real numbers to the real numbers. This set is an infinite-dimensional vector space.

This paradox arises out of the ill-defined nature of borders and nations. Are there other things we can apply it to? It might be helpful in rigorously defining concepts. — Paralogism

National borders are historically contingent. Surely this is a trivial point, not a profound one. The US was the US before and after the Louisiana purchase. It just had more land. Just as you are still you if you buy your neighbor's property and adjoin it to yours.

My understanding is that according to the I/D proponents our virtually infinite universe was created with the sole purpose of creating mankind, — Jacob-B

Isn't that the anthropic argument used by physicists? That the universe is fine-tuned for life? It comes up in multiverse theory. Point being that it's not only the theologists invoking theological arguments these days.

No, this is exactly the opposite to what I am arguing. When we adhere to the law of identity, then everything has an identity proper to itself, therefore its own meaning. This does not rob meaning from mathematics, it only establishes clear limits to the possibilities of mathematics, so that mathematicians will not believe themselves to have accomplished the impossible, like putting the infinite within a set. — Metaphysician Undercover

You made the statement that ZFC allows two different things to be equal. I said I know of no such example and you have not backed up your claim or put it in any context that I can understand. You must be thinking of something, I'm just curious to know what.

2 + 2 and 4 represent the exact same mathematical set. '2+ 2" and '4' are distinct strings of symbols. I don't know any mathematicians confused about this. And, as you agree, the discovery that these two strings of symbols represent the same set, is a nontrivial accomplishment of humanity and is meaningful.

I really don't understand your remark that ZFC allows distinct things to be regarded as the same. Unless you mean colloquially, as in the integer 1 and the real number 1 being identified via a natural injection.

Here is your quote.

ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover

I categorically deny that claim. Please put it in context for me. As stated it's flat out false as far as I know. Of course one thing may have multiple representations; and it may have taken years, decades, or centuries to discover that fact.

The explanations for the Banach-Tarski paradox in the following video are very intuitive: — alcontali

What's interesting about Banach-Tarski is that it's a purely syntactic paradox. The free group on two letters has a paradoxical decomposition, and this is purely a matter of meaningless syntax. One then lifts the paradox to Euclidean 3-space since the isometry group of 3-space (that is, the group of rigid transformations) contains a copy of the free group on two letters by virtue of the existence of a pair of independent, non-commuting rotations.

In any event, the subject was the use of the word witness, and my reference to that book was incidental.

↪fishfry How does assuming everything is conscious help explain how a lump of meat can be? It's no explanation at all.

Plus, the 'problem' is not explaining how meat can be conscious. The problem is that we have rational intuitions that represent all material things to be lacking in mental properties. And it's not a problem, unless you've started out assuming that we're material things. — Bartricks

Like I say, I'm not in a position to defend panpsychism intellectually, since I haven't studied the literature. It's just a personal belief, and one not strongly held.

The first time I saw it was in Stan Wagon's book, The Banach-Tarski paradox. I found it confusing and had to devote brain cycles to figuring out what it meant, let alone what the mathematical argument was. I don't like it, it's pretentious and confusing. You could say "2 witnesses the falsity of the claim that all primes are odd"; but why make a statement harder to read? It's a pretty common locution but I'm not a fan.

I will say that it's a stretch to go from the colloquial meaning of witness to Boolean satisfiability. I don't think there's necessarily a functor from, "While Googling X I ran across Y" to "X and Y are deeply related." If anything, it's the opposite. Google knowledge is all surface, no depth. It gives the illusion of understanding.

It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy" — alcontali

@Mephist and I had a monumental pages-long conversation about constructive math a while ago. You might find it interesting. All in all I learned far more about constructivism than I ever did before, and even read some technical papers on the subject. But in the end I never gained any affinity for the subject. I still regard constructive math as unnecessarily tying your hands behind your back. But of course neo-intuitionism is making a comeback via computer science and homotopy type theory. I call it Brouwer's revenge.

https://thephilosophyforum.com/discussion/5791/musings-on-infinity

I
Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-) — alcontali

LOL. Of course I was only damning with faint praise, complementing your Wiki research but not your understanding. In that particular matter, at least. I'm perfectly well aware of the history of intuitionism and the Hilbert-Brouwer debate. FWIW I honestly did not make the connection from what you wrote, to the subject of intuitionism and constructive math. Could just be me.

The categorical vocabulary itself, however, seems to be spreading like wildfire. — alcontali

I'm glad I could turn you on to this paper.

I saw a little category theory back in grad school many moons go, then left math. When the Internet appeared in the 90's I was amazed to discover that category theory was being used in loop quantum gravity in theoretical physics. (This was of course from John Baez's This Week in Mathematical Physics on Usenet and later on his blog). Now category theory is in economics, biology (also via Baez) and of course computer science. Functional languages are the big thing now and they have monads, so Youtube is full of CS lectures on category theory. I find it all quite amazing to have seen this mind virus grow over the decades. Then again it takes a long time. Category theory was invented/discovered/whatever in the 1940's and was confined to math till probably the 90's. Another reason that "useless abstract math" should be valued. You never know what's going to eventually be important.

This by the way is my annoyance with the "indispensability argument" for mathematical existence. Category theory was not indispensable for computer science in 1940 but today it is. Therefore it has to be retconned as retro-indispensable; and then by analogy, everything is. Because in math, not everything is indispensable; but everything is potentially indispensable.

By the law of identity, two distinct sets cannot be the same. If they actually are the same, then they are necessarily one, the same set. It's contradictory to say that two things are the same. If it is the same, it is only one. — Metaphysician Undercover

Yes I do understand take this philosophical objection. If I say 2 + 2 = 4 then if they are the same object they're the same. I'm saying nothing! If I stand up and say, "I am me!" I am saying nothing. I've only affirmed the law of identity.

Yet ironically, we have a world full of people standing up and saying "I am me!" and this is of great psychological and sociological and political importance! Going off topic a little but noting the irony.

But yes I've seen this argument before. Math is meaningless because in the end it's all tautologies and saying that a thing is equal to itself.

I don't think that can really be true though. Math IS useful and meaningful because it takes human effort to determine whether two different representations of a thing are actually the same thing. Don't you agree? 2 + 2 = 4 is formally a tautology. But historically, it was a really big deal for humanity. Agree or no?

I seem to recall the old philosophical standby of the morning star and the evening star, which appear to be two different things but (upon astronomical research that took millennia) turn out to be the same thing, namely the planet Venus and not a star at all.

If you reduce everything to the law of identity, you are saying those millennia of observation and theory and hard work by humans means nothing. I don't accept that.

ZFC was initiated by Cantor and Dedekind in the 1870s — alcontali

I agree with you re installed base or established mindshare. There are substantial developments in new foundations these days, category theory and homotopy type theory being the two leading candidates. In the end, foundations don't matter to the vast majority of working mathematicians. As an example if ZFC were found inconsistent tomorrow morning, it wouldn't affect group theorists or topologists or anyone else. They'd keep doing their work while the set theorists patched the problems. You'd be surprised how little attention working mathematicians pay to foundations.

ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover

I'm afraid I don't follow this at all. I know of no such instance.

The axiom of extensionality depends on the law of identity, which is a principle of logic and not of set theory. A thing is equal to itself. Then we define two sets to be equal if they have "the same" elements, meaning that we can pair off their respective elements using the law of identity.

It's true that in math we often identify sets as being the same type of "something" in a given context. For example the integers mod 4, the set {0, 1, 2, 3} with addition mod 4, are a very different set from the integer powers of the complex number i, {i, -1, -i, 1}. Yet the integers mod 4 (with the operation of addition mod 4) are isomorphic, as groups, to the powers of i under the operation of complex number multiplication. A group theorist will say these are "the same group" while being perfectly well aware that they're not the same set.

Another example is that if we define the natural numbers via the Peano axioms then use them to define the rationals and reals, then the natural number 1 and the real number 1 are entirely distinct sets. We regard them as the same via the "natural inclusion" of the naturals into the reals. If pressed on the details, any working mathematicians would explain this just as I have and there is never any confusion.

These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

I noticed that the example of 2 + 2 = 4 was given. 2 + 2 and 4 are exactly the same set. In the Peano axioms, 0 is a number and if n is a number, Sn, the successor of n, is a number. This gives an endless sequence 0, S0, SS0, SSS0, SSSS0, ... which we can use to name all the numbers. However this notation soon gets cumbersome so we adopt the definitions: S0 = 1, S2 = 2, S2 = 3, and so forth.

Now we can define addition inductively: n + 1 = Sn; and n + Sm = S(n + m). With this definition in hand, 2 + 2 = 4 is an easy theorem. If we then lift Peano to set theory via the axiom of infinity, 2 + 2 and 4 are easily seen to be the same set.

It's perfectly true that 2 + 2 and 4 are distinct strings of symbols. If we are studying strings, they're distinct. But in number theory they're the same number; and in set theory they're the same set.

I don't think time is a dimension. Dimensions are infinitely divisible. But nothing can be infinitely divisible. Therefore, no dimensions exist. Time does exist. Thus, time is not a dimension. — Bartricks

Well there's an answer to that. We distinguish between time, a particular aspect of the universe, and time as modeled in physics. It's traditional to model time as the mathematical real numbers. But that is not a claim that time itself is like that. After all when one embarks on the mathematical study of the real numbers, one soon finds that they are very unreal. The mathematical real numbers are an abstract technical construction that has no necessary relation to the thing in the universe that we call time. A lot of physicists don't understand that. They're taught: "This is time, and here are these equations," and they spend their life in that environment and often never step back and realize that they are only working with the map, not the territory. After all the world didn't change when we went from Aristotelian to Newtonian to Einsteinian gravity. Only the model changed.

I don't understand your reply. — Bartricks

I am not a philosopher. I am not competent to defend anything I wrote in this thread. Consider it just an idle drive-by post, not to be taken as authoritative in any way.

How often do you wonder what a cup of tea thinks like? — Bartricks

I have a distinct recollection of being around six years old, outside playing with a ball. I had a strong sense of wondering what it was like to be the ball. The impression is with me decades later. I do have a personal gut sense of panpsychism. I find it very plausible. As I say I don't know the literature and can not converse substantively on the topic. My belief isn't very strong nor has my interest ever even risen to reading the Wiki page I linked. It's just an idle belief. I have many others.

That's the old mind-body problem. If there's a nonphysical realm of existence where my mind dwells, what exactly is it? And what else lives there? The Flying Spaghetti Monster? The baby Jesus? The answer to the Continuum hypothesis? On the other hand if my mind is just an emergent property of my atoms acting according to physical law, then I'm just a machine, a clockwork orange. My life has all the meaning of a bunch of billiard balls bouncing off each other.

You could start reading the philosophical literature on this and never finish.

To the extent that I've thought about it at all, I like panpsychism. That says that everything is conscious. The rocks, the atoms, everything. That makes more sense than the other theories. You don't have to figure out how a clump of brain tissue becomes self-aware. It is because everything is.

https://en.wikipedia.org/wiki/Mind%E2%80%93body_problem

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

Yes, now what about the way that mathematics links space and time, when will that be overturned? — Metaphysician Undercover

I'm afraid my knowledge of physics is limited to reading Scientific American articles back in the day, and watching Youtube videos lately. My understanding is that general relativity would not be overturned, but rather incorporated by a more refined model just as relativity incorporates rather than overturns Newtonian gravity. By incorporates I mean, "Approximates as a special case."

There's a lot we don't know. The century-long inability to integrate electromagnetism with gravity is a definite clue that there's a deeper theory out there. So then we get into the string theory wars. Way more than I know about. It's actually very interesting. Physics is stuck. The Large Hadron Collider didn't find supersymmetry. That means there's no new physics to be found at any energy that taxpayers are likely to pay for. Always money for wars but not much for knowledge.

I wouldn't state it that way. If we mean first order Peano arithmetic (PA), then there are not in PA definitions of 'set', 'class', and 'proper class'. Meanwhile, in set theory, the domain of the standard model of PA is a set.
19 hours ago — GrandMinnow

I suppose so. But even in PA there are infinitely many numbers. There's just no completed set of them. So they are not formally a proper class, but we can use this idea as an analogy to what a proper class is. It's a collection that's too big to be a set. If I stated this as just a useful mental visualization or metaphor and not as a formal fact, we'd be in agreement I think.

The link is synthetic, we link space and time with mathematics. How space and time are really related we haven't the foggiest idea. That's because we do not know what neither of these is, nor can we even describe what space or time is. — Metaphysician Undercover

I certainly agree with you that reality is one thing, and our historically contingent scientific models are another. I don't believe that "the universe follows the laws of physics," as I've heard some people say. Rather, the laws of physics are our current mathematical model this week, to be overturned tomorrow or in a century. I think we're in agreement on this.

This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects. — GrandMinnow

I agree that mathematical infinity is "true" in the abstract realm of math. But that's like saying that the way the knight moves is "true" in chess. But it has no physical meaning in the world we live in. It's only true within a formal game played for entertainment. That's what I was trying to say.

What would this link be? I'd like to know if you're willing to teach. — TheMadFool

I don't know enough physics to serve as an explainer. I couldn't even find a decent explanation online. For ex

"The logical consequence of taking these postulates together is the inseparable joining together of the four dimensions, hitherto assumed as independent, of space and time. "

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

In relativity, there aren't four independent axes like there are in Euclidean 4-space. Rather, the time dimension is mathematically dependent on the spatial dimensions and vice versa. If I run across a halfway decent explanation online I'll post it.

The point isn't that we can talk about when something happens as well as where. The point is that space and time aren't independent. They're not like Euclidean 4D space. Space and time are mathematically linked.

I'm confused by the distinction actual vs potential infinity? — TheMadFool

There's a straightforward and unambiguous mathematical distinction.

The inductive axiom of the Peano axioms say that whenever n is a number, n + 1 is a number. So we have 0, and 1, and 2, and 3, ... [The fact that 0 is a number is another axiom so we can get the induction started]. However we never have a "completed" set of them. In any given application we have as many numbers as we need; but we never have all of them assembled together into a single set.

The axiom of infinity says that there is a set containing all of them.

So with the Peano axioms we may write: 0, 1, 2, 3, ...

With the axiom of infinity we may write: {0, 2, 3, ...}.

The brackets mean that there is a single completed object, the set of all natural numbers. That's Cantor's great leap. To work out the mathematical consequences of completed infinity.

I'm sure from a philosophical point of view there may be some quibbles. But this is how I think of it. the axiom of mathematical induction gives you potential infinity. The axiom of infinity gives you completed infinity.

Note that even with potential infinity, there are still infinitely many numbers. It's just that we can't corral them all into the barn. In fact in Peano arithmetic, the collection of all the natural numbers is a proper class. This is a good way to visualize what we mean when we say that a given collection is "too big" to be a set.

Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool

Yes perfect sense. The axiom of infinity is a humongously ambitious claim for which there's currently no evidence in the real world. It's a bold statement. On the other hand without it, we can't get a decent theory of the real numbers off the ground. So the ultimate reason to adopt the axiom of infinity is pragmatic. It gives a much more powerful theory. Whether it's "true" in any meaningful sense is, frankly, doubtful.

So, you may think that advanced set theory is too "hard" to read, but sorry, it is a walk in the park compared to what we do, — alcontali

You're smart in one area and a delusional bullshit artist when you talk about things you clearly know nothing about. You got busted. Give it a rest. You're a buzzword jockey utterly lacking in self-awareness. Who do you think you're fooling?

I will read up on the details when he does finally finish his work. — alcontali

LOL. One (you, me, anyone) would need a Ph.D. in set theory and several years of specialized postdoc work, and probably more than that, just to read what he's done so far. You keep making this laughable claim that you'll deign to read his work when he's done. You're embarrassing yourself.

Ok. happy now? — alcontali

You bluffed and got called. Have a nice evening.

I keep getting irate remarks from fishfry in another thread because I refuse to read up on the nitty-gritty details of the cutting-edge research on the Continuum Hypothesis (in math). He seems to insinuate that my point of view -- I have to draw the line somwhere, don't I? -- is pure evil. — alcontali

None of that is remotely true. You started talking about set theory here ... https://thephilosophyforum.com/discussion/comment/317954 . Note that YOU were the one who first brought up set theory in that thread. I corrected some of your errors and you started making wild extrapolations of things you didn't understand. I called you out on your additional errors. A perusal of that thread will confirm my account.

I don't care if you study set theory or not. But if you pretend to understand more than you do and make elementary errors, I'll surely correct you. And point out that you like to throw out buzzwords without understanding their meaning. In that particular thread you came off like a bs artist and you got called on it.

Ok so in formalism, we have string manipulation rules that have no inherent meaning. But someone can use them by interpreting them; that is, by assigning the symbols meaning.

which can involve meaning as part of those rules, — Zuhair

What does that mean? How is meaning "part of those rules?" For example we can do basic arithmetic from the Peano axioms without assigning any meaning to the symbols, then we can balance our checkbooks or do number theory using those symbols. But it's still formalism. There's no meaning inherent to the symbols.

So what does it mean for meaning to be part of the rules?

The following statement is false
The previous statement is true

Is that logically possible ? :meh:
It can be possible if we drop law of the excluded middle. — Wittgenstein

This doesn't appear to be what the OP is talking about. Can you explain?

Can you please give a concrete example?

So mathematics is about studying rule following games. — Zuhair

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

https://plato.stanford.edu/entries/formalism-mathematics/

I've already explicitly stated in my head post that I don't agree with formalism, — Zuhair

Hmmm. What does "mathematics is about rule following games" mean then?

I thought it was a rule that if infinite sets are countable, they're equal. No exceptions. — RogueAI

The even natural numbers and the odd natural numbers are both countable but they're not equal as sets. They're cardinally equivalent, but that's not the same thing as set equality. In set theory two sets are equal if and only if they have the same elements. That's the axiom of extensionality. https://en.wikipedia.org/wiki/Axiom_of_extensionality

In any event simulation theory is essentially a theological argument. If I say God makes every blade of grass, made me the way I am (so THAT explains it!), is omnipresent, omniscient, and omnipotent, you'd call me a religious true believer or a religious nut, depending on your manners and theological beliefs. If I said the great computer in the sky did it, you'd invite me to give a TED talk. There's no difference in the argument. Whether I'm a program running in the great computer or a thought in the mind of God, what is the difference? Both claims are strictly outside of science. They're not subject to observation or experiment. What kind of experiment could distinguish the two, and distinguish either of them from naturalism, the idea that the stuff out there is real and we live in a physical universe that is explainable by natural processes?

I believe the best way to compare two infinite sets of same cardinality is by density measure, — Wittgenstein

Natural density is useful in some contexts, but it's not countably additive hence is not, strictly speaking, a measure or a probability measure. That's its drawback. It does have its uses, but to claim it's the best way to compare countably infinite sets is asking it to do too much. There's no uniform probability measure on a countably infinite set, and that's a fact we all have to live with.

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

As I said, Woodin's strategy to prove CH from PD -- while staying clear of ZFC in any form or shape -- sounds really interesting, but he has also clearly said that he still hasn't managed to do it. I am just repeating Woodin's own words. — alcontali

You're repeating them out of context without understanding. I'm always up for a chat about set theory but this ain't it.

Without looking it up, what is a projective set, and what is projective determinacy? What are some arguments as to why it might be true? False? What other well-known axioms does it conflict with? This stuff is basic basic in advanced set theory.

If Woodin said that he's "not finished," he means he's not finished with his decades-long research project. He's got plenty of published proofs, he's one of the top set theorists in the world. Your comment that there's nothing on Wikipedia because he doesn't have a finished proof is, I apologize for my directness, laughable. I don't know what else to say. You're deeply misconstruing this entire topic. You don't know what you don't know.

you can see that the only workable proof strategy is exactly about avoiding set-theoretic knowledge. — alcontali

That's your strategy. Avoiding knowledge. If I came here and conversed with demonstrable ignorance about your own area of technical competence, you'd recognize it. If you mentioned finite fields and I said, "Oh yeah like a wheat field bounded by corn fields?" you'd laugh at me, you wouldn't placate me. First, of course, you'd try to explain what a finite field is in abstract algebra. And if I said, "Ignorance of abstract algebra is the only workable proof strategy," then you'd laugh.

I just somehow hope that the nitty-gritty distinctions between "extremely large", "super huge", and "incredibly out-sized" cardinals won't be needed. — alcontali

Why not? May I ask if these are technical terms with which you're familar? You mis-stated CH in such a way as to give me the impression you haven't studied much set theory beyond the basics. If you're a specialist in advanced set theory who made an elementary error then you have my apologies. It's just that I don't understand why you are trying to discuss or comment on things that are far outside of your (and my) competence. If I've got this all wrong just tell me.

I
Furthermore, I was more interested in the remark by/on Woodin, "why the Axiom of Projective Determinacy, PD, should be accepted". I do not think it matters "why". What matters is that If PD is provably independent from ZFC, and that Woodin also manages to prove CH from PD. Then, Woodin will have finished the job. The real problem is what he writes at "the argument is still incomplete ...". That is why I said that I am waiting for him to complete the argument. Unfortunately, we cannot do particularly much with just half the argument ... — alcontali

Do you know what PD is? Do you have an opinion on its truth or falsity, and knowledge of how it relates to other candidate axioms floating about? Do you know a lot about infinite games and projective sets? I saw a lecture on AD once, the axiom of determinacy. That says that EVERY set of reals is determined. PD says that every projective set is determined, which is weaker.

I
That is why I said that I am waiting for him to complete the argument. Unfortunately, we cannot do particularly much with just half the argument ... — alcontali

My point is that neither of us is remotely qualified to discuss Woodin's work at all, unless you have set-theoretic knowledge far in excess of what you have demonstrated so far. Nor is his argument incomplete. Google for "extender models" and you'll probably find some papers.

I'm honestly having trouble here. By your comments I've placed you at a certain level in set theory. And you seem perfectly confident of your ability to understand what Woodin is talking about. There's a massive disconnect. He's one of the world's leading set theorists. This is bleeding edge theoretical work. It's not absent from Wiki because it's "not complete." It's absent because there are as yet no humans who can understand his work and translate it for the public; even that segment of the public with a technical degree or two. There are one or two popular articles on the subject that are no help at all. You wouldn't expect to jump into the latest bleeding edge research in biology or physics.

I had a professor once, Fields medalist, very famous guy. In grad-level real analysis he forgot the wording of Zorn's lemma and asked for help, Which I, bright-eyed young genius that I was at the time, quoted for him perfectly. That was the high point of my math career. Point being that you misconstrued CH either because: (a) you haven't studied much set theory; or (b) you are so advanced you can no longer be bothered with the basics. There's a disconnect and it's bothering me.