I believe the main issue is that new topics are added more often than old ones are removed, leading to bloated posts. I'll not respond to a few of your comments to address this request. — keystone

Ok thanks. I realize that I myself write long posts.

Label the original string (-inf,+inf). — keystone

Define your notation, since you already told me this is NOT an open interval in the real numbers.

Did I not complain bitterly enough about this in my last post?

Cut it somewhere. Label the left partition (-inf,42). Label the right partition (42,+inf). Label the small gap between the strings 42. Now you have a new system: (-inf,42) U 42 U (42,+inf). But you seem to get hung up on those intervals number being continuous even though I'm saying that those intervals describe continua - abstract string in this case. — keystone

I don't know what your notation means and you are not going to tell me.

Moving forward, instead of writing "computer+mind", I'm just going to write "computer".

I believe that true mathematical rules exist independently of computers. These rules are necessary truths and finite in number. If one assumes they describe actually existing objects, such objects must exist beyond our comprehension, as no computer could contain them. — keystone

In another thread going on right now, it's been pointed out that there are uncountably many mathematical truths, and that most of them can't even be expressed, let alone proven. See

https://thephilosophyforum.com/discussion/15304/mathematical-truth-is-not-orderly-but-highly-chaotic

However, if we assume that mathematical objects must exist within a computer, then not all mathematical objects can actually exist and it becomes a matter of a computer choosing which objects to actualize. — keystone

I don't think anyone believes that, not even the constructivists. Maybe they do. It's pointless to argue constructivism with me, I know nothing about it.

Please allow me to use the SB-tree as something concrete to talk around. I acknowledge that any infinite complete tree will do. — keystone

I don't know why. It's not helpful to me at all.

We outline the rules for constructing the SB-tree and can mentally construct it to an arbitrary depth. Everything we ever actually construct is finite. Why insist on believing in the computationally impossible — the existence of the complete SB-tree? — keystone

I don't think it's productive for you to try to talk me out of my mathematical beliefs. I believe in the axiom of infinity and the higher transfinite cardinals.

You've said that the reals correspond to unending paths down the infinite complete binary tree, so indeed, there are potentially
paths that cannot be algorithmically defined. This doesn't mean the rules for constructing the tree are incomplete; it simply means there are paths computers can never traverse. Computers cannot exhaust these rules. — keystone

So what? My mathematical ontology is not confined to what's computational. Yours is. So you should study constructivism. It's pointless to try to discuss it with me, since I have already made good-faith efforts to get interested in constructivism, without getting interested.

Or here's how I see it. When I see the tree, I do not see paths and nodes. Instead I see a continua at each row, being cut by the numbers at each row. For example, I see the top two rows of the SB-tree as:

Row 1: 0 U (0,1) U 1 U (1,+inf)
Row 2: 0 U (0,1/2) U 1/2 U (1/2,1) U 1 U (1,2) U 2 U (2,+inf)
...

With this view, I would rephrase the conclusion as follows: computers cannot completely cut continua. Computers cannot exhaust cutting. Actually, I would go one step further and assume that computers are all that's available, so I would simply say that continua cannot be completely cut. But we know that already, you'll never cut a string to the point where it vanishes. — keystone

Whatever. I can't argue these points. Computers are woefully inadequate to express mathematical truth.

Sometimes I push back as a form of defense. Nevertheless I'll try and be more mindful of this. I'm very appreciative of our conversation. Thanks! — keystone

You're welcome. I just can't argue constructivism. There use to be some constructivists on this board. Long gone.

Given a particular countable language and meta-theory with a countable alphabet:

This is correct:

Given a countable set of symbols, there are exactly denumerably many finite sequences of symbols, thus exactly denumerably many sentences.

There are uncountably many subsets of the set of sentences. And any set of sentences can be a set of axioms. Therefore, there are uncountably many theories. But there are only countably many ways to state a theory, so there are theories that are not stable. — TonesInDeepFreeze

I considered that argument, that there might be uncountably many theories (interpretations) but I don't think it's correct. An interpretation must have finite length, yes? There are only countably many FINITE subsets of a countable set.

I'm pretty sure this is correct:

There are exactly denumerably many algorithms. And for every formal theory and set of axioms for that theory, there is an algorithm for whether a sentence is an axiom. So there are only countably many formal theories. — TonesInDeepFreeze

That's what I believe is true. Only countably many interpretations of each sentence. So the poster (sorry I didn't look back to find out who) who said natural language could be uncountable, is wrong. IMO anyway.

Given a language for a theory, trivially, there are uncountably many interpretations for the language, since any non-empty set can be the universe for an interpretation, and there are not just countably many sets. But there are only countably many ways to state an interpretation, so there are interpretations that are not statable. — TonesInDeepFreeze

I'd have to give that more thought.

Given any theory, there are uncountably many models of the theory, since there are uncountably many isomorphic models of the theory. But there are only countably many ways to state a model, so there are models that are not statable. — TonesInDeepFreeze

I don't know. Not knowledgeable about model theory.

Nicely phrase. Our new chum is propounding much more than is supported by the maths. Here and elsewhere. — Banno

Chumming the water, is our new chum. But actually I'd heard the same claim from Chaitin, and it wasn't till I read the paper referenced by the OP that I learned that this is a very trivial observation related to undefinable real numbers, and not a major insight. But perhaps Chaitin has taken it deeper.

Any readable proof of Cantor's Theorem will contain at most a finite number of characters. Yet it shows can be used to show* that there are numbers sets* with a cardinality greater than ℵ0. — Banno

Yes, a point made by the constructivists I believe. Math talks about the infinite but math itself consists of finite-length proofs.

And we are faced again with the difference between what is said and what is shown. — Banno

Moby Dick didn't really kill Ahab, even though we may have enjoyed the story. Moby and Ahab never existed. We can always use language to describe impossible things. "Fly me to the moon and let me play among the stars." What kind of pedant would complain about the illogic?

So will we count the number of grammatical strings a natural language can produce, and count that as limiting what can be - what word will we choose - rendered? That seems somehow insufficient. — Banno

Lost me here. Anyway I don't think I should get too involved in the question of natural language, even if I believe my countability argument applies. There are only countably many finite-length strings over a countable alphabet. I just don't see that there's any more to say; but again, I'd rather talk about Chaitin's idea than get tangled up in the complexities of natural language.

And here I might venture to use rendered as including both what can be said and what must instead be shown.
a
Somehow, despite consisting of a finite number of characters, both mathematics and English allow us to discuss transfinite issues. We understand more than is in the literal text; we understand from the ellipses that we are to carry on in the same way... And so on. — Banno

It's like a trip to the moon on gossamer wings. We have no trouble expressing the thought, even though it describes something that's not possible, even if we knew what gossamer wings are. Unless by gossamer wings we mean the hardware of the Apollo missions.

Symbolic language lets us express many fanciful ideas. Alice laughed. 'There's no use trying,' she said. 'One can't believe impossible things.'

I daresay you haven't had much practice,' said the Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.

Of course Lewis Carrol was a logician as well as a writer of children's stories, and perhaps he was making this very same point. That with language, we CAN believe, or at least express, impossible things.

But further, we have a way of taking the rules and turning them on their heads, as Davidson shows in "A nice derangement of epitaphs". Much of the development of maths happens by doing just that, breaking the conventions. — Banno

Yes. Math is all about believing impossible things! A point that I'm sure did not escape Carrol's notice.

Sometimes we follow the rules, sometimes we break them. No conclusion here, just a few notes. — Banno

Yes.

When were you in a grad math program? — jgill

'75 -'77.

Sounds like your professor just didn't like foundations.

That should read "countably infinite." We can think of endless permutations of language, but we could also spend and infinite amount of time saying the names of the reals between any two natural numbers. — Count Timothy von Icarus

Lost me. There are uncountably many reals between any two natural numbers.

Not sure what's being argued here. The cardinality of natural language is not relevant to the thread, it's a side topic. It is a fact that there are only countably many finite-length strings over an at most countably infinite alphabet. I think that addresses the issue, but perhaps there's some disagreement.

That seems to be true, so Benacerraf is right. — Ludwig V

Seems that way to me as well. @Michael is wishing that a limit point had an immediate predecessor, but it doesn't, and he's stuck there.

Doesn't it follow that both outcomes are consistent with the rules of the problem? — Ludwig V

Yes, Benacerraf and I conclude that.

If both outcomes are consistent with the rules of the problem, doesn't that imply that they are not self-consistent (contradict each other)? If so, Michael is right. — Ludwig V

Why? I drive down the road and come to a fork. One day I turn left. Then next day I drive down the same road and turn right.? What logical inconsistency do you see to there being multiple possible outcomes to a process that are inconsistent with each other, but each consistent with the rules of the game?

But if they contradict each other, doesn't ex falso quodlibet applies (logical explosion)? — Ludwig V

Not at all. What logical inconsistency is there if I turn left today and right at the same junction tomorrow?

Is geometry inconsistent because it allows both Euclidean and non-Euclidean geometry? They are inconsistent with each other; but they are each consistent in themselves.

The logical explosion implies your conclusion, that justifies your plate of spaghetti, doesn't it? So you are right.
End of discussion? Maybe. — Ludwig V

As far as I'm concerned this is a solved problem. And I do believe that if @Micheal would stop falsely imagining that there's an immediate predecessor to a limit point, he'd agree.

The rules must be consistent with each other where they apply. The problem is that the rules don't apply to the limit, because the limit is not generated by the function, that is, it is not defined by the function. — Ludwig V

Yes. Benacerraf explicitly says that. He says it's a mistake to assume that the terminal state bears any "logical relation," his words, to the preceding sequence of states.

The limit is defined, however, as part of the function, along with the starting-point and the divisor to be applied at each stage. In that sense, they are all arbitrary. But the idea that they could all be replaced by a plate of spaghetti is, I think, I mistake. — Ludwig V

Benecerraf explicitly says: "... Certainly, the lamp must be on or off at t1
(provided that it hasn't gone up in a metaphysical puff of smoke in the interval) ..."

In other words he is making the the point that for all we know, the lamp is not even constrained to be either on or off at the terminal state. And why should it be so constrained?

I am indeed gratified to see Benacerraf making this point.

Don't we need to say that these numbers are not defined by the function, but are assigned a role in the function when the function is defined, which is not quite the same as "arbitrary"? The range of arbitrary here, has to be limited to natural numbers; plates of spaghetti are neither numbers nor, from some points of view, natural. — Ludwig V

Why? It's an imaginary lamp to start with. I have previously analogized it to Cinderella's coach, which turns into a pumpkin at exactly midnight, but is a beautiful jewel-encrusted coach at and time prior to midnight.

What rule of the problem constrains the terminal state of the lamp? Thomson gives no such constraint. On the other hand @Michael may indeed have added his own such constraint, making the problem impossible. So Michael has not solved Thomson's lamp, he's solved Michael's lamp, by adding a condition that isn't present in the original problem.

C3 says it's not. — Michael

I'd find it helpful if you would write down a complete description of your version of the problem in one place, rather than pointing me to P1 here and C3 there. Just write down a complete description of the problem for my reference please.

Also, can you please note any divergences between Michael's lamp and Thomson's.

Your arbitrary stipulation that the lamp is on or off at midnight is inconsistent with P1-P4. — Michael

Mine, yes. But Benacerraf's also. I wish you'd carefully study his argument and respond to it. I mention this "appeal to authority" because Benacerraf is a big time mathematical philosopher, not just some anon rando such as my humble self.

The lamp can only ever be on iff the button is pushed when the lamp is off to turn it on. The lamp can only ever be off iff either it is never turned on or the button is pushed when the lamp is on to turn it off. Midnight is no exception. — Michael

Benacerraf addressed this point. Your assumption about what happens at the limit point is either (a) not part of Thomson's original formulation; or (b) logically inconsistent.

I address it all here. P1 is an implicit premise in Thomson's argument. He is asking "what happens to a lamp if we push its button an infinite number of times?", not "what happens to a lamp if we push its button an infinite number of times and then some arbitrary thing happens to it?". — Michael

P1 says, "Nothing happens to the lamp except what is caused to happen to it by pushing the button"

I do not know if that is something Thomson said or if you added that.

But in that case, then you are subject to Benacerraf's point. You haven't said ANYTHING about the terminal state. I don't know if a button is pushed or not at the terminal time. Who says it's not? You've just got your imagination running away with you about things that don't make sense.

If nothing happens to the lamp unless the button was pushed; then at the terminal time, if anything at all has happened to it -- turned on, turned off, turned into a plate of spaghetti, or as Benecerraf himself allows, just disappeared entirely -- then a button was pushed. If you say so.

There's no contradiction. You said if something happened, a button was pushed. Ok. Something happened and a button was pushed. What of it?

If your only way to make sense of a supertask is by introducing God or magic to fix the problem at the end then you haven't made sense of a supertask at all. — Michael

This is something you're adding, I don't know where you're getting it.

P1 is your own rule. If something happened to the lamp, a button was pushed. So at the terminal state if ANYTHING happened, a button was pushed. You're right. P1 is satisfied. And the lamp is on. Or the lamp is off. Or the lamp has turned into a plate of spaghetti.

You might as well try to resolve something like the grandfather paradox by using the same. Time travel into the past isn't physically possible but granting its possibility for the sake of argument doesn't then entail that anything goes, but that seems to be your approach to this issue. — Michael

I'd invite you to engage more deeply with Benacerraf's argument. I was indeed gratified that he even made my point about the spaghetti. "... provided that it hasn't gone up in a metaphysical puff of smoke in the interval ..." The terminal state is arbitrary.

They're not totally disordered though. At any time you can state the position of each one relative to the others, and that's an order. When you say "they're totally disordered", that's just metaphoric, meaning that you haven't taken the time, or haven't the capacity, to determine the order which they are in. — Metaphysician Undercover

Ok. For things in the real world, they are already in some order, even if it's a complete state of disorder. Even a completely disordered collection of gas molecules in a container, at every instant each molecule is wherever it is. And that set of coordinates, locating every molecule in space, is the order.

I get that. But by the same token, there is no preferred order. Suppose for example that I got my schoolkids from the playground to line up single-file in order of height. And now YOU come along and say, "Ah, that is the inherent order, and all other orders are disorders of that."

But of course your observation was a complete accident. I could have lined them up alphabetically by last name.

So even among physical objects, if we allow that they are always in some order, even if it's disorderly; but nevertheless, there is no preferred or inherent order.

I believe you are saying there's an inherent order, have I got that right?

Those are just 'identified orders'. When the kids are running free, in what we might call a 'random order', what you called "totally disordered", there is still an order to them, it has just not been identified. — Metaphysician Undercover

Yes we see that the same way. Totally disordered gas molecules in a container, at every instant there is a list of all atoms and where they are in the box (more or less, quantum effects notwithstanding, but that's not the point I'm making). I'm agreeing with you that even the most disordered state is still an order. It's just the set of facts about where everything is.

So, for the principle "height", we could make a map and show at any specific time, the relations of the tallest, second tallest, etc., and that would be their order by height. And we could do the same for alphabetic order. So we number them in the same way that you would number them in a line, first second third etc., then show with the map, the positions of first second third etc., and that is their order. The supposed "random order", or "totally disordered" condition, is simply an order which has not been identified. — Metaphysician Undercover

Yes perfectly happy to regard whatever physical positions and attributes -- their state -- is regarded as the order they're in at that moment. I'm fine with that.

Yes, that's very apprehensive of you fishfry, and I commend you on this. Most TPF posters would persist in their opinion (in this case your claim of "totally disordered", which implies absolute lack of order), not willing to accept the possibility that perhaps they misspoke. — Metaphysician Undercover

You have indeed persuaded me that every collection of physical things has an order, even if they are apparently disordered. But sets, well you know ...

So that is the point, everything is in "SOME" order. — Metaphysician Undercover

Everything in the physical world.

Now, consider what it means to say "sets don't have inherent order". — Metaphysician Undercover

I have obviously spent much time considering that. As much time as I've spent explaining to you that sets don't have inherent order!

Would you agree that this sets them apart from real collections of things? — Metaphysician Undercover

Of course. Sets aren't real. They're a mathematical abstraction. I've never asserted otherwise.

A real collection of things, like the children, must have SOME order. And, this order which they do have, is very significant because it places limitations on their capacity to be ordered. — Metaphysician Undercover

Yes. I agree. But why does it matter? It doesn't apply to sets. You know why? Because that's what the axioms say. That's the ultimate source of truth. It's just axiomatics. I'm not sure why all this is important to you.

So when you said "first you have things, then you place them in order", we need to allow that the "things" being talked about, come to us in the first place, with an inherent order, — Metaphysician Undercover

No, I did not say that. I said that the things come to us WITHOUT an inherent order; and we place one on them.

And secondly, I was not talking about things in the world. I was talking about abstract mathematical structures. Objects of thought. Not of the world.

and this inherent order restricts their capacity to be ordered. — Metaphysician Undercover

Yeah it's harder to line the kids up by height depending on how they happen to be arranged on the playground. But sets aren't kids on the playground. Sets are a conceptual gadget for thinking, what would it be like to have a collection of things that don't have any order? What could we still say about them?

It's a philosophical game like that. And it's useful. It lets us separate out the facts about sets that depend on their order, from the facts that don't.

This is just the essence of mathematical abstraction.

For example, let's say that the things being talked about are numbers. We might say that 1 is first, 2 is second, 3 is third, etc., and this is their "inherent order". — Metaphysician Undercover

Well now that you mention it, no. 1, 2, 3, ... is NOT the inherent order of the set $\mathbb N$, believe it or not. On the other hand it sort of is, in a sneaky way. Von Neumann defined the symbols 1, 2, 3, ... in such a way that $n \in m$ if it happens to be the case that we want n < m to be true.

But we still have to (1) define what 1, 2, 3, ... are as set; and then (2) define that what we mean by n < m, is that $n \in m$ as sets.

So even though it's kind of rigged for 1, 2, 3, ... to be the natural order -- in fact that's what they call it, the natural order -- it is still explicitly defined. And without that definition, the natural numbers have no inherent order.

I know this is hard for normal humans to accept, since it's pretty obvious that 1 < 2 < 3 and so on. But mathematicians insist on being picky about how numbers and other things are defined. In the set-theoretic view of modern math, the numbers 1, 2, 3, ... are defined as particular sets, with no inherent order; and then we impose their order by leveraging the $\in$ operator.

Now I'm going to meet you halfway on something. I admitted that the set definitions of 1, 2, 3,... are already set up to leverage the $\in$. But you could say -- and I am going to agree with you -- that when John von Neumann invented the modern set-theoretic definition of the natural numbers; he already had a pre-intuition of the inherent order 1 < 2 < 3 ... and that's why he defined things to work out that way.

So even though formally we've removed the scaffolding and there's no inherent order in the finished mathematical theory; the inspiration for designing the theory that way was in fact the inherent order of the natural numbers, no matter what set theory says.

I am going to agree with you about that. Mathematics is mysteriously influenced by something "real" about even the most abstract things, like numbers. Formalism is defeated in the end. It's NOT just about the symbols. Math is expressing something real about the world.

Is any of this along the lines of your thinking?

This is the way we find these "things", how they come to us, 1 is synonymous with first, 2 is synonymous with second, etc., and that is their inherent order. — Metaphysician Undercover

I just talked myself into (almost) agreeing with what I take to be your point of view:

* Even though the numbers 1, 2, 3, ... have an inherent order; when we do set theory we pretend they don't, purely for the sake of the formalism. But the formalism is missing something important. The set-theoretic formalism denies that the numbers have an inherent order. But the counting numbers DO have an inherent order that is obvious to every school child. Therefore the set-theoretic natural numbers do not capture the full metaphysical properties of the natural numbers.

Have I got any of that right?

The proposition of set theory, that there is no inherent order to a set, removes this inherent order, so we can no longer say that one means first, etc.. — Metaphysician Undercover

Only to quickly put it back. And as I just acknowledge, the method of defining numbers as sets, and then being able to "define" the order 1 < 2 < 3 ... was obviously set up to facilitate just that. Showing that von Neumann had a priori knowledge of the order he was formalizing.

But that's not surprising, really. Even formalists don't think everyone's just making everything up. Math is "about" something "out there" in the world. Right? Am I making any sense to you?

Now there cannot be any first, second, or anything like that, inherent within the meaning of the numbers themselves. This effectively removes meaning from the symbols, as you've been saying. — Metaphysician Undercover

From the formal perspective of set theory. But not to deny that numbers don't have inherent order. Our formal model of numbers has no inherent order. But that's a virtue. It lets us study those aspects of sets that don't depend on order. I think I said that earlier. It's a process of abstraction, not lying for ill intent or metaphysical error.

Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers. — Metaphysician Undercover

Yes if you'll accept that it's really all that's going on. Like when you go to the dermatologist to remove a mole, he doesn't send you in for a bunch of tests on your pancreas. He deals with one particular level of your entire being. He's not denying you have all these other organs. He's just focussing on one thing at a time.

It's perfectly ok to think of sets as having an inherent order and all their other possible orders, but we're just not concerned about them today. We only want to look at the property of membership.

Yes, I think I see this. I would say it's a type of formalism, the attempt to totally remove meaning from the symbols. — Metaphysician Undercover

Yes, so that we can reason precisely about the objects the symbols represent.

The problem though, is that such attempts are impossible, and some meaning still remains, as hidden, and the fact that it is hidden allows it to be deceptive and misleading. — Metaphysician Undercover

Maybe you mean that von Neuman secretly knew that 1 < 2 when he formalized the natural numbers in such a way to make 1 < 2 come out true.

But of course he did! So maybe you are getting at the intuition that guides mathematicians to do things the way they do.

In other words math is discovered inductively; and only presented deductively.

So, by the abstraction process you refer to, we remove all meaning from the symbols, to have "no inherent order". Now, what differentiates "2" from "3"? — Metaphysician Undercover

What differentiates any set from any other set? All together: The axiom of extensionality!

In von Neumann's clever encoding, we make the following symbolic definitions:

$0 = \emptyset$

$1 = \{0\}$

$2 = \{0, 1\}$

$3 = \{0, 1, 2\}$

and so forth. One virtue of this encoding is that the cardinality of each number is what it "should" be. The set representing 3 has cardinality 3. Which of course isn't within the theory, it's outside the theory. We secretly already know was 3 is even before we define it.

Is that one of the thing's you're getting at?

Anyway, back to the question. How do we know that 2 and 3 are not the same set?

Well $2 \in 3$, but $2 \notin 2$.

Therefore by extensionality, $2 \neq 3$, because they don't have exactly the same elements.

Perhaps you can begin to see the virtues of working a the set level separately from its order properties. We can see the mechanics of how to use the axiom of extensionality. No order properties are needed to determine that 2 and 3 are different sets. It's just a matter of ignoring hypotheses that you don't need for a particular argument.

Nobody is saying that a given set doesn't have an order, as well as a lot of other stuff. A topology, some algebraic operations, a manifold structure perhaps. But we can learn a lot just from restricting our attention to the membership relation and seeing what we can learn just about that.

They are different symbols, with different applicable rules. If what is symbolized by these two, can have "no inherent order", then the rules for what we can do with them cannot have anything to do with order. This allows absolute freedom as to how they may be ordered. — Metaphysician Undercover

Yes, math is often concerned with the most general kind of structure or conceptual framework.

What most people think of the order of the counting numbers: 1, 2, 3, ..., mathematicians call the "standard order" or "usual order," in contrast with many other interesting orders we could define.

While normal people only think about the usual order, mathematicians are people who think about all the different ways a set like the natural numbers can be ordered. In fact there's a beautiful order to the ways that the natural numbers can be well-ordered. Those are the ordinal numbers.

In other words the collection of all the ways a set can be ordered ... can itself have a natural order that we can study.

So that's the kind of thought process mathematicians enjoy, when they go from order to orders. From an order like 1 < 2 < 3, to the concept of order itself in its most abstract form.

However, we can ask, can the two numbers,2 and 3, be equal?
— Metaphysician Undercover

I believe I already demonstrated from first principles, from the ZF axioms, that 2 and 3 are not the same set; which, in the von Neumann encoding, proves that they are not the same number.

I don't think so. — Metaphysician Undercover

I just did. 3 contains an element, namely 2, that is not an element of 2. Therefore they're not the same set by the axiom of extensionality.

Therefore we can conclude that there actually is a rule concerning their order, and there actually is not absolute freedom as to how they can be ordered. — Metaphysician Undercover

I hope you have taken the foregoing to heart. We define 2 and 3 as particular sets, and by design they are different sets under extensionality. They do not have the same elements therefore they are not the same set. Under the identification of sets as representatives of numbers, they are not the same number.

The two symbols cannot have the same place in an order. — Metaphysician Undercover

I hope I've already convinced you that 2 and 3 are not the same number, and that we can demonstrate that using nothing more than the axiom of extensionality. In other words we do not need to use any order properties to prove that 2 and 3 are not the same number.

Therefore, there actually is "SOME" inherent order to the set, a rule concerning an order which is impossible. — Metaphysician Undercover

No. I proved $2 \neq 3$ using only their set membership properties and without any need to invoke their order properties, inherent or assigned.

And this is why I say that these attempts at formalism, to completely remove meaning which inheres within what is symbolized by the symbol itself, are misleading and deceptive. — Metaphysician Undercover

Then you have been proven wrong. I don't need to mention or consider or use any of the order properties of 2 and 3 to determine that they're different numbers.

We simply assume that the formalism has been successful, and inherent meaning has been removed (we take what is claimed for granted without justification), and we continue under this assumption, with complete disregard for the possibility of problems which might pop up later, due to the incompleteness of the abstraction process which is assumed to be complete. — Metaphysician Undercover

Entirely without rational basis. This para is a wild generalization of your complaint about 2 and 3, but I already showed how we can distinguish 2 and 3 using only their membership properties and not their order properties. If that was the basis for this paragraph, you have no basis. But even then, the para makes wild unsupported accusations.

You will need a much better example -- well an example, period -- of the formalism failing. It certainly held up to your first test. I used the axiom of extensionality to prove that 2 is not 3. And a good thing, too!

The set theoretic abstractions have held up for a century, from Zermelo's 1922 axiomitization to today.

Then when a problem does pop up, we are inclined to analyze the application as what is causing the problem, and the last thing we would do is look back for faults in the fundamental assumptions, as cause of the problem. — Metaphysician Undercover

You are thrashing away at a strawman you've created out of your imagination, and under the mistaken belief that we can't tell 2 from 3 without their order properties. But we can.

As described above, you need to look for what is inherent within the meaning of the symbol. Formalism attempts the perfect, "ideal" abstraction, as you say, which is to give the imagination complete freedom to make the symbol mean absolutely anything. However, there is always vestiges of meaning which remain, such as the one I showed, it is impossible that 2=3. The vestiges of meaning usually manifest as impossibilities. Any impossibility limits possibility, which denies the "ideal abstraction", by limiting freedom. — Metaphysician Undercover

You know you have stopped being clear and coherent in the last few paras. All based on a mistaken belief. This last para does not parse. Not for me anyway. Formalism is a tool, it's not the goal.

So to answer your question, the order which is inherent is not one of the orders you can give the set, it is a preexisting limitation to the orders which you can give. When we receive the items, what you express as "first we have the items", there is always something within the nature of the items themselves (what you call "SOME order"), as received, which restricts your freedom to order them in anyway whatsoever. — Metaphysician Undercover

That works for numbers. What about kids? What is the inherent order, the playground or the single file?

There are many different ways that "same" is used. You and I might both have "the same book". The word "set" used here is "the same word" as someone using "set" somewhere else. So it's like any other word of convenience, it derives a different meaning in a different sort of context. In common parlance, mathematicians might say "they are the same set", but I think that what it really means is that they have the same members. So that's really a qualified "same". — Metaphysician Undercover

No, you are consistently wrong about this. If A and B are sets and I can prove that A = B, then A and B are the same set. They are in fact the identical set, of which there is only one instance in the entire universe. They are NOT "two copies" or two distinct entities that we are calling the same by changing the meaning of the word "same."

"mathematicians might say "they are the same set", but I think that what it really means is that they have the same members."

DUH that is what it MEANS to be the same set. That is the ONLY thing it means to be the same set. Sets don't have any existence other that what the axioms say. There is nothing else to know about sets.

I can't believe you wrote that. Yes that is what it MEANS for two sets to be the same. That they have the same members. That's ALL it means and EVERYTHING it means.

You simply can't accept that and I don't know why. The knight in chess moves the way it does. Not for any reason other than those are the rules. Likewise two sets are the same when they have the same members. Period, end of story. That's it.

There is ONE SET {1, 2, 3}. That's the only one. There is exactly one instance of every set. If two supposed sets are equal they are the same sets. Like the morning star and the evening star, they're the same object. I don't know how many times I've explained this over the years and I can't understand why it eludes you or troubles you.

Actually I got annoyed with Tones rapidly, when we first met, but now he just amuses me. — Metaphysician Undercover

If you say so ...

This was my first thought. Natural languages would seem to need to be computable, which would entail countably finite. — Count Timothy von Icarus

I wonder about that. Think of the slang the kids of every generation come up with. No algorithm could predict that. Humans not just using, but constantly recreating their own language. New words and ideas and phrases are constantly coming into existence, sometimes gaining traction in the general culture, sometimes fading away. I think that the way humans evolve their own languages in so many ways, is something that humans can do that perhaps algorithms can't. I'll just put that out there.

This reminds me a little of Chomsky's generative grammars. He revolutionized linguistics by saying there are structural elements common to all human languages. Not programs, per se, but structures. Just my impressions, that's all I know about it.

I was trying to convey that the representation itself is not important; what matters is the behavior. If in my mind x+x=2, then x behaves like 1. Similarly, if y+y=2, then y also behaves like 1. In this scenario, 1 has multiple representations (x and y) in my mind, but that isn't an issue because they both behave the same. — keystone

Yes but who determines that? You said that numbers come into existence when some computer represents them. If one computer represents "XLII" and other represents "42", does your creation engine instantiate one or two numbers? How does God or whoever decides what numbers are instantiated, immediately know whether those two expressions always behave the same? If you think about it it's a hard problem.

But I must highlight that to conclude that x=1, I don’t work through an infinite checklist, considering all possible arithmetic equations involving 1. No, I'm mindful of the consistent and finite set of rules associated with the construction and arithmetic of the SB-tree (or equivalent tree), so all I need to do is declare that x will behave like the node occupied by 1. At that moment, I bring 1 into existence and it is representation in my mind is the character x. — keystone

"... the consistent and finite set of rules ..."

Oh, you use an axiomatic BOTTOM UP system after all!!!!!

We each are the god of the mathematical systems that inhabit our own minds. If we want to compare my (-inf,1) U 1 U (1,+inf) with your (-inf,42) U 42 U (42,+inf) we need to agree to the SB-tree and compare the nodes where my 1 and your 42 lie. If they correspond to the same node, then our systems are equivalent. While nobody explicitly does this, it's the unspoken agreement we make when comparing systems. I don't see why we would need a third party to arbitrate the comparison. — keystone

If you have a set of rules (a bottom up concept) that let you know when two representations denote the same number, then why do you need the computer? Why not just accept that the rules themselves bring all possible numbers into existence already?

Since no human artifact can be infinite, is it fair to say that you believe in the axioms but not in the infinite objects they describe? — keystone

I believe in the mathematical existence of the abstract objects they describe. I make no claim that they are physically real. Just like when I read Moby Dick I accept the existence of Ahab and the whale, without making any ontological commitments outside of the context of the novel.

If so, this directly supports my thesis—forget about the existence of infinite sets and instead focus on the (Turing) algorithms designed to generate these infinite sets. — keystone

There simply aren't enough algorithms to generate all the sets. There are countably many algos and uncountably many subsets of the natural numbers.

I'm receptive to a constructivist approach to the axiom of infinity. If were talking about computable infinite sets in the same way that Turing talked about computable real numbers I have no problems, provided we do not assert the existence of infinite objects. — keystone

Ok. You should study constructivism. I can be of no assistance, I don't know much about it.

What harm is there in relying on Newtonian mechanics when it performs admirably for slow-moving objects like ourselves? — keystone

Completely wrong analogy. Infinitary math always works. It just doesn't apply to physics (as far as we know). And if it DOES someday turn out to be important for physics, it will be a good thing if the pure mathematicians have been studying it all along.

Just as Riemann developed non-Euclidean geometry 70 years before Einstein needed it for general relativity.

Or just like Diophantus studied number theory some 2200 years before public key cryptography became the foundation of online commerce.

Similarly, what harm is there in embracing General Relativity when the singularities it predicts are distant from our everyday experience? There's a certain beauty in their simplicity; as a mechanical engineer, I rely on Newtonian mechanics daily and will continue to do so regardless of advances in physics. — keystone

Then you are agreeing with my point. I read something interesting once about sailing. Sailboat technology did not get really good until sail was no longer a useful means of practical transportation. Today's high-tech sailboats were developed by purely recreational sailors, after sailboats became obsolete in commerce.

Yet, as physicists began pulling the loose threads of classical physics, a more fundamentally robust and aesthetically compelling framework emerged: Quantum Mechanics. There is something to be said for pulling the loose threads. — keystone

I disagree with the analogy. Constructivism is not a deeper form of conventional math. But I can see the argument being made. Just not by me.

It would be much easier if you would just roll with the intuitions for a little while so we wouldn't get stuck on the first step. Let's sweep through the whole idea informally and if it has any merit then we can sweep through again and formally define things. Think of a line as a piece of string. Think of a cut as what you do with scissors to partition the string. You're making this more complicated than it needs to be. — keystone

I'm fine with cutting strings. You have never explained to me how this serves as a new foundation for math.

No, I'm not. Yes, I'm referring to (Turing) algorithms that produce rational approximations with arbitrary precision, but the algorithm itself is exact. The algorithm perfectly encapsulates the essence of the real. That's why I'm emphasizing the algorithm itself, not its output. — keystone

That's fine, you're making a constructivist argument. I'm the wrong person to have that discussion with, since I have no feel for constructivism, despite making a run at the subject several times.

I agree to both sentences! (1) That's what I'm trying to do and (2) I'm just trying to throw a 'potential' in front of the 'infinitary math'. — keystone

I'm not stopping you. You have a ways to go in terms of developing a comprehensive, logical theory that I can understand. It could just be me.

Cantor has already received considerable acclaim, making it difficult to envision greater recognition for him. What I meant to convey is that Cantor unearthed something monumental, yet his interpretation was poop (actual infinities). I believe that in the future, it will be recognized that his true discovery lay in articulating the potential within continua and mathematics as a whole. — keystone

That's a stretch. Please don't be a Cantor crank. I'd be disillusioned. If you are one, it's better to keep it to yourself.

I'm using open interval notation to describe the bundle (line) the lies between its endpoints. This bundle cannot be described as an infinite set of individual points, because as I mentioned before, we can only talk about individual points when the line has been cut. For this reason, I'm reluctant to say that I've been proposing open sets. — keystone

So after all this time your interval notation does not not stand for its conventional meaning?

How nice of you to let me know.

Can you see that in terms of communication, you could have either made that clear up front, or defined a different notation, like <a,b>. Can you see why I've been utterly confused for weeks?

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything, much like classical physics. They both suffer the same flaw...and I believe are both addressed with the same resolution: a top-down view. — keystone

I would say that the noncomputable reals show us the limitations of algorithms. And, being a skeptic of simulation theory and the trendy thesis that human minds are Turing machines, this is an important plank in my platform. Algorithms are vastly insufficient to express what's important, either about math or reality. This is perhaps why the constructivists annoy me.

Yeah, you're going to lose some things with constructive mathematics, be it LEM or the axiom of choice. But by and large I'm proposing a much more beautiful structure. Just as classical physics was a natural stepping stone to QM, actual-infinitary-math is a natural stepping stone to potential-infinitary-math. — keystone

I have no doubt you have a beautiful structure in mind. Your challenge is to express it. Maybe I'm a poor sounding board.

I can reframe my examples in Python if that's your preference. The main drawback however is that my posts would get longer. — keystone

Jeez people's code fragments definitely make my eyes glaze. No code please.

It's frustrating to think that I'm running out of ways to communicate my ideas so I'm starting to think that the conversation might end prematurely with the least desirable conclusion (that you don't know whether my ideas are right or wrong). — keystone

No risk of that. I already know your ideas are wrong. Hey you set yourself up for that one :-)

But perhaps it's too soon to talk about the end. I'm getting value out of every post you and I write so I'd be grateful if we keep going and just take it one day at a time. — keystone

That's fine with me. I can't promise to meet your expectations about what you wish I'd write. I've responded sensibly and I think relevantly to each point you've made. I don't have to agree with your overall point of view, or even understand it.

But of course you could start by defining your notation.

Those parenthesis (a,b) don't stand for standard open sets. Can you see that you redefined a notation that's so universal that I had no choice but to be sent into a state of confusion about your meaning? Can you see why I've been confused for weeks?

It's just that at some point we'll need to talk beef and I'll need to figure out an alternative way to communicate the bullet post. — keystone

I make no commitment to meet your conversational expectations. What you see is what you get. You have expectations about my state of mind that are unlikely to be met. I'd ask you to accept that rather than continually expressing disappointment with my posts.

According to the Buddhists, unfulfilled expectations are the source of suffering.

Like I say, if you want to communicate, don't redefine extremely well-known standard notations without announcing it clearly.

And also, if I could make this request ... can you write shorter posts? Short and to the point.

Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. Just by way of fairness, Pullum and Scholz argue against assuming that natural languages are even infinite. — Banno

I'll check out those links. But if they deny natural languages are even infinite, then they surely aren't uncountable.

I do think natural language is infinite, in the sense that there are infinitely many legal sentences. The sun rises in the morning, I know the sun rises in the morning, I know that I know the sun rises in the morning, etc. There is a countable infinity of those.

Langendoen and Postal do not agree that "a natural language consists of a collection of finite-length strings". — Banno

I'm confused by that. If you allow infinite length strings then there are uncountably many of them, though most aren't grammatical. Are they making an argument about grammatical constraints?

Does mathematics also "consists of a collection of finite-length strings made from an at most countably infinite alphabet"? — Banno

Formally, yes. Every mathematical statement or proof has finite length. There are only countably many mathematical statements. That's the argument in the paper. There are uncountably many truths, but only countably many of them can be expressed.

Also, doesn't English (or any other natural language) encompass mathematics? It's not that clear how, and perhaps even that, maths is distinct from natural language. — Banno

As formal systems, probably not much difference. But natural language is much messier than math, I'm not even sure if a computer could determine whether a string is legal in natural language. Not once we include slang and the language of pop culture and the young.

All of which might show that the issues here are complex, requiring care and clarity. There's enough here for dozens of threads. — Banno

Well I argued in my earlier post that there's less to that paper than meets the eye. The inexpressible truths in the paper are trivial and unimportant. I wonder if there are nontrivial truths that can't be expressed, and what that would even mean.

But it is still the case that the lamp cannot arbitrarily be on (whether at midnight or any other time). It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ω. — Michael

I've been thinking about your button. I was going to try to address your argument on your own terms, but first I thought I'd go back to Benecerraf's paper to review his argument, which is the one that makes sense to me. He directly addresses some of your concerns.

A note on notation. You like 12:30-1:00 and I prefer 1/2, 3/4, 7/8, ... We can convert between my notation and yours as 12:30 + that fraction of 30 minutes. So my t = 1/2 is your 12:45. Hope that's acceptable. Benecerraf uses my notation, or rather he says that $t_0$ and $t_1$ are the initial and terminal times of the lamp. He doesn't go into any more detail, but this corresponds to my idea of 1/2, 3/4, 7/8, ...; 1 as the times at which we observe the lamp.

I find it helpful to analogize with the mathematical sequence 1/2, 3/4, 7/8, ..., which converges to 1.

Now to Benecerraf's argument.

He posits that two individuals, Aladdin and Bernard, each "perform the supertask." At time $t_1$, the terminal state, Aladdin finds the lamp on, and Bernard finds it off.

Benacerraf argues that neither outcome is inconsistent with the rules of the problem, for the reason that the rules are defined at each of the times t = 1/2, 3/4, 7/8, etc., but NOT at t = 1.

He writes: [Benecerraf's italics as indicated, but I didn't mark up all of them]

I submit that neither description [Aladdin or Bernard's - ed] is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

According to Thomson, Aladdin 's lamp cannot be on at $t_1$, because Aladdin turned it off after each time he turned it on. But this is true only of instants before $t_1$! From this it follows only that there is no time between $t_0$ and $t_1$ at which the lamp was on and which was not followed by a time also before $t_1$ at which it was off. Nothing whatever has been said about the lamp at $t_1$ or later. And similarly with Bernard's lamp. The only reasons Thomson gives for supposing that his lamp will not be off at $t_1$ are ones which hold only for times before $t_1$. The explanation is quite simply that Thomson's instructions do not cover the state of the lamp at $t_1$, although they do tell us what will be its state at every instant between $t_0$ and $t_1$ (including $t_0$). Certainly, the lamp must be on or off at $t_1$ (provided that it hasn't gone up in a metaphysical puff of smoke in the interval), but nothingf we are told implies which it is to be. The arguments to the effect that it can't be either just have no bearing on the case. To suppose that they do is to suppose that a description of the physical state of the lamp at $t_1$ (with respect to the property of being on or off) is a logical consequence of a description of its state (with respect to the samne property) at times prior to $t_1$. I don't know whether this is true or not, and in section II I shall briefly investigate some matters that bear on this issue. But, true or not, the argument is invalid without the addition of a premise to that effect.
— Benecerraf

I could not have said that any better. Though I'd have inserted some paragraphs. I left it as is since that's what he wrote.

I note in passing that he's anticipating my plate of spaghetti. When he says, "... the lamp must be on or off at $t_1$ (provided that it hasn't gone up in a metaphysical puff of smoke in the interval) ..." he is making the point that the terminal state of the lamp is entirely arbitrary. There are no constraints on the terminal state in Thomson's description of the problem, not even the necessity of being either off or on.

He also denies that the terminal state of the lamp is a logical function of the prior states. He is directly addressing your belief that the terminal state must somehow arise from what's gone before. He says that without additional assumptions, the argument is invalid.

Can you respond to Benecerraf's argument? Clearly he is responding to Thomson's version, and perhaps you have additional assumptions.

My sense of your view is that you intuitively wish that the limit 1 had an immediate predecessor among 1/2, 3/4, 7/8, ... But of course it doesn't. Limits are like that. They don't have immediate predecessors. In fact if you plotted the points 1/2, 3/4, 7/8, ..., and 1 on the number line, you can see that any step left from 1, no matter how tiny, necessarily jumps over all but finitely many of the terms of the sequence.

It's true that if the terminal state is on, then it was off at some time in the past. And if it's off, then it was on at some time in the past. Benecerraf points that out as if anticipating your thought process.

I don't hate on individual math professors. They are just pawns in the game.

This is the part I don't get. The administrators and bean counters and HR reps of the world, I can understand your frustration with institutional stupidity on such a grand scale.

But math professors aren't any of that. They're dreamers who sit in their office and push around symbols to prove theorems about things that nobody but other mathematicians understand.

They are totally harmless.

They serve the other academic departments by teaching calculus classes to the engineers and such. Everything else, the math major undergrad and math grad school, is all about training professional mathematicians. All of this has got nothing at all to do with the administrative stupidity you decry.

Why have you got it in for the math professors?
— Tarskian

One (or rather two) of the things I don't like, is the combo of academic credentialism combined with the student debt scam. Like all usury, it is a tool to enslave people. The banks conjure fiat money out of thin air and them want it back along with interest from teenagers who were lied to and most of whom will never have the ability to pay back. The ruling mafia even guarantees to the bankstering mafia that they will pay if the student does not. First of all, though, they will exhaust all options afforded by the use of violent threats of lawfare. — Tarskian

I bet you're not a fan of fractional reserve banking either.

If an electron is 'composed' of position, momentum, spin, charge and mass; aren't these properties more fundamental than the electron? — Treatid

They're properties of the electron. Position, momentum, etc. are how we talk about the electron. A property isn't a constituent part or component of an electron or anything else.

What is more fundamental than electrons are quantum fields. I am very far from being knowledgeable in physics, but I do watch YouTube videos. That counts for some kind of knowledge these days.

A particle: be it an electron, a quark, a gluon, whatever; is nothing more than an excitation in a quantum field. Imagine space filled with the fluctuations of the quantum field. If there's a region where the fluctuations are highly concentrated, that's where we see a particle. And out of all that, all the rest of the stuff that the particles make.

So we're all just fluctuating vibrations in the quantum field.

This is my understanding of fundamental physics these days. Particles are excitations in the quantum field.

From Wiki:

QFT treats particles as excited states (also called quantum levels) of their underlying quantum fields, which are more fundamental than the particles. — Wiki

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

I would never have wanted to be staff, though. When we talk about "bottom line", the only one that mattered to me was my own "bottom line". I was not interested in selflessly "sacrifice" myself for someone else's bottom line. I cannot identify with the profit of the company. I can only identify with my own profit. I understand that C-level execs somewhat care, since they receive payments for when the stock goes up, but the other salaried office drones? Seriously, why would anybody else care? — Tarskian

Come the revolution it will all be made right.

I worked in corporations and share some of your opinions about them. But why hate on the math professors? Math is so cool, why can't you let them do their thing?

Most mathematical truth is unprovable and therefore unpredictable, if only because most of its truth is ineffable ("inexpressible"). — Tarskian

Going back to your top post:

I read the paper. It's true, but there's less there than meets the eye.

They're only talking about undefinable sets. Sets of natural numbers that can't be characterized by a predicate. There are uncountably many sets, and only countably many predicates (a predicate being a finite string over an at most countable alphabet). So "almost all," all but countably many sets of natural numbers, can't be described.

So if $S$ is one such indescribable set (whose elements are essentially random: there's no way to describe them like "the prime numbers," or "the even numbers," or whatever), we have a bunch of true facts like $s \in S$ for each element $s$ that happens to be in $S$. And so forth.

So there are uncountably many facts about the powerset of the natural numbers that can not be expressed, but that are clearly true.

This is perfectly correct as far as it goes. But how far does it go?

There is nothing particularly interesting about a random set. It has no characteristic property that lets us determine whether a particular number is or isn't in the set. We have to look and see if it's in there. There's no rhyme or reason to the members of a random set.

These are all mathematical truths, but they're not very interesting mathematical truths.

What mathematicians do is find the interesting mathematical truths. The ones that form an overarching structural narrative of math. Perhaps the author of the paper, or Chaitin, are saying that this narrative is an illusion; that the mathematical truths we discover are a tiny, almost irrelevant subset of all the mathematical truth that's out there.

I think the opposite view could be taken. That the work of mathematicians in developing interesting, axiomatic mathematical truth, has value. It's what humans bring to the table.

How about this metaphor. Mathematicians are sculptors. Out of the uncountably infinite and random universe of mathematical truth, mathematicians carve away the irrelevant and uninteresting truths, leaving only the beautiful sculpture that is modern mathematics, those truths that we can express and prove. They're special just for being that.

But a sentence is not the same as a string. — Treatid

Sorry, I did no follow the intent of the rest of your post.

The question I replied to was from @Banno, who asked: "Why should we suppose that natural languages are only countably infinite?"

I gave a proof that the set of finite-length strings over a countably infinite alphabet is countably infinite.

That is, there are at most countably many finite-length expressions, strings, words, sentences, books, in any natural language.

I was confused by your quoted point here, since a sentence is a particular kind of well-formed finite-length strings. So the sentences are a proper subset of the strings. If the strings are countable so are the sentences.

Can you clarify your post? I may have missed something.

The interpretation of a sentence depends on the context/axioms. The same string in two axiomatic systems is two distinct sentences. — Treatid

I'm talking syntax, not semantics. There are only countably many finite-length strings.

I do see your point. Even if there are countably many strings, each string could be given a different interpretation, so that there could be lots more meanings.

That's a point about models, or interpretations, or semantics, and I'm not sure it's the appropriate context for the question. On the other hand you think it is, so at least let me try to respond in that context as well.

If there are at most countably many axiom systems or interpretations, then there are still only countably many sentences.

The only way you could make your idea work would be to have uncountably many interpretations. Do you have that many interpretations?

Also I haven't hear "sentence" used in this way. I thought the idea about sentences was two expressions that say the same thing, for example in different languages.

I haven't heard a sentence as you are defining it, as a syntactic string plus an interpretation. Is this something people do?

In any event, as long as you only have countably many interpretations per string, you'll still only have countably many sentences in your definition, assuming I'm understanding your post.

However, the assertion that natural languages are countably infinite no longer holds given there are an uncountable infinite number of contexts for any given sentence. — Treatid

Ok I see you anticipated my point. Can you suggest a context in which I can conceptualize an uncountably infinite number of contexts for interpreting a language?

And again, I haven't seen "sentence" used this way. There seems to be a blurring of syntax and semantics.

I'll sign off if I feel I'm done. Don't like to ghost a conversation. Your post was way off the mark, which made it very easy to keep the reply short. — noAxioms

Hope you'll explain where sims with bodies live. The phrase doesn't even make sense.

I don't have much more to say on all this. If I'm incapable of understanding where the sims live, so be it. To have any idea what I'm talking about I should read the rest of Bostrom, but I may not get to that.

No factory anywhere. No bodies in the GS world. The bodies are in this world. I, like most people, Bostrom included, presume I have a body. — noAxioms

But our world is imaginary. An artifact of a computation. There is no "sim world" that is a physical world that's created anywhere. Your idea is incoherent. And like I say, I don't have to talk you into that. I can live with agreeing to disagree, pending my reading of the rest of Bostrom's paper, which is way down the to-do list.

You're thinking of an android. — noAxioms

Yes, that's the only way a sim could have a body.

A simulated anything is the product of a computer simulation. — noAxioms

Correct. It has no physical instantiation or existence anywhere. When the execution of my Euclid program finds the GCD of two integers there is no matter created anywhere.

A storm simulator has one simulated storm. The storm is probably not created, but is rather already there, part of the initial state. The purpose of simulating it is to see where it goes, and how strong it gets, and which areas need to evacuate. — noAxioms

It's not a physical storm. I prefer to agree to disagree on this point rather to debate it. Here is where we stand:

* I think you are expressing an idea that is incoherent;

* You think I'm far off the mark and failing to understand something very basic about simulations.

This is not going to get better. I remember at the beginning I asked you if Ms. Pac-Man has an inner life, and you said yes. I believe you are still in this (a) delusion, or (b) funny way of using words that makes it true.

Then we're pretty stuck. Most people can at least get that much out of Bostrom's abstract. If you can't, but rather insist on this weird replicant track, I don't know how to unmire you. — noAxioms

Mired I am, then. I think you must be reading something into Bostrom that isn't there. What on earth can it mean to simulate a physical body ... somewhere? I don't know if the error is yours or Bostrom's. Regardless, what you are saying is incoherent. In my mired opinion, of course.

We definitely agree on where we're stuck. I could live with a graceful quiescence of the convo soon.

You don't think you have a body then? — noAxioms

Not if I'm an artifact of a mind-instantiating algorithm. I'm Descartes, but where even his mind is not his own. A truly horrifying reality.

You think perhaps you were created in a factory instead of being born of your mother? I said that nobody (but you) suggests this, but you persist. — noAxioms

No, you are saying that. But the factory isn't physical either, it's an executing program. Where is the body? How are bodies created? Or is this Ms Pac-Man's inner life again?

Do I have a body like Ms Pac-Man? Is that what you mean? I'm in a 3D display of some sort?

Never mind I don't want to know. I'll stipulate to being mired. I wish I could dispatch a clone to take yet another look at Bostrom's paper, but I probably won't get to it myself, and I'm all out of clones. I'll go with Sabine when she says the simulation hypothesis is pseudoscience. I'm content to leave it at that.

cc @TonesInDeepFreeze since your name got referenced a lot here.

Apology accepted. — Metaphysician Undercover

Thanks.

As I indicate in my latest post in the supertask thread, Tones has a knack for taking highly specialized definitions designed for a particular axiomatic system, and applying them completely out of context. Be aware of that. — Metaphysician Undercover

I often don't follow the purpose of his symbology.

You guys should stay out of that thread, you're not discussing supertasks and I can see why @Michael moved you.

But nobody claims mathematical equality is identity.
— fishfry

Tones does, obviously. — Metaphysician Undercover

Yes I didn't realize that. I don't see how it can be, but I'm not aware of how the logicians and set theorists resolve this.

I dropped out of abstract mathematics somewhere around trigonometry, for that very reason. I got hung up in my need to understand everything clearly, and could not get past what was supposed to be simple axioms. I had a similar but slightly different problem in physics. We learned how a wave was a disturbance in a substance, and got to play in wave tanks, using all different sorts of vibrations, to make various waves and interference patterns. Then we moved along to learn about light as a wave without a substance. Wait, what was the point about teaching us how waves are a feature of a substance? — Metaphysician Undercover

Yeah that business about waves without a medium is pretty murky. I've seen videos where they say, "It's a probability wave!" as if that explains anything. Probability waves are purely abstract mathematical gadgets, they aren't physical. Leaving unexplained the question of what electromagnetic waves are waving.

It's a well known problem that physics is no longer about the physical world, but rather about esoteric models that seem to work, without telling us much about the physical world. The "shut up and calculate" school of quantum physics.

But Tones is a bit different. Tones forges ahead with misunderstanding of fundamental axioms. Tones insists that the axiom of extensionality tells us when two sets are identical. He refers to something he calls "identity theory", which I haven't yet been able to decipher. — Metaphysician Undercover

Aha! I just asked him about this very point.

If axioms are rules, then they mean something. They dictate how the "formal game" is to be played. If the rules are misunderstood, as is the case with Tones, then the rules will not be properly applied. — Metaphysician Undercover

I can't comment on Tones's opinions in that area. I haven't been reading this thread.

Tones is a monster, not of my own creation. — Metaphysician Undercover

I can't tell if he knows a lot of logic but doesn't always explain himself, or is just typing stuff in. I had a very hard time with his last post to me explaining how identity was set equality.

It is self-contradicting, what you say. " First you have things, then you place them in order." — Metaphysician Undercover

No it's perfectly sensible. You have a class of screaming school kids, eight year olds say, on the playground. They're totally disordered. The only organizing principle is that you have a set of kids.

Then you tell them to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order.

It's an everyday commonplace fact that we can have a set of things in various orders.

Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.

But sets don't have inherent order.

If you have things, there is necessarily an order to those things which you have. To say "I have some things and there is no order to these things which I have, is contradictory, because to exist as "some things" is to have an order. Here we get to the bottom of things, the difference between having things, and imaginary things. — Metaphysician Undercover

Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers.

But mathematical sets by themselves have no inherent order till we give them one. It's just part of the abstraction process.

You are taking Tones' misrepresentation. I fully respect, and have repeatedly told Tones, that sets have no inherent order, exactly as you explained to me, years ago. — Metaphysician Undercover

I'm moved that I had an effect. It was not in vain. I'm happy.

What I argue is that things, have an order to their constituent elements, and this is an essential aspect of a thing's identity. — Metaphysician Undercover

Yes, I am starting to come around to your point of view. But tell me this. Since, given a set, there are many different ways to order it, how do you know which one is inherently part of it?

So I've been trying to explain to Tones, that the "identity" of a set (as derived from the axiom of extensionality) is not consistent with the identity of a thing (as stated in the law of identity). — Metaphysician Undercover

I would go so far as to say that identity isn't set equality, because identity applies also to things that are not sets.

But if you ask me whether I think that two sets that are equal are identical, I'd have to say yes. Because if they're equal, they're the same set. Not because of metaphysics, but because of set theory. Set theory only talks about sets, and doesn't even say what they are. Nobody knows what sets are. They're fictional entities. They obey the axioms and that's all we can know about them.

But Tones is in denial, and incessantly insists that set theory is based in the law of identity. — Metaphysician Undercover

I can see that you've developed a bit of a, what is the word, obsession? attitude? annoyance? with him.

Agreed. But sometimes the person will not see the flaw in their argument unless explicitly identified by someone else. — keystone

I am trying to understand you.

By "digest" I didn't mean to suggest that you would accept it. But there's value in being able to entertain a thought without accepting it. — keystone

I'm trying to understand your argument or thesis or idea.

If a human thinks of a duck and somehow in their computations the duck behaves exactly like the mathematical object 42, then (within that person's thoughts) the duck represents an instance of the number 42. As the old saying goes, "If it swims like a 42, and quacks like a 42, then it probably is a 42." I think we both agree that absent of an intentional being giving mathematical meaning to the duck (or to electrical activity within a computer), no mathematics is going on. — keystone

You said that numbers get instantiated when they appear in a computation. I asked you whether one number or several numbers get instantiated when various representations exist. Who determines that they act the same? Where is that process, that brings a number into existence?

But if at a later time the human's thoughts of the duck do not correspond to the number 42, then the duck is no longer an instance of 42. The number 42 is contingent on thought. It's existence is temporary. — keystone

Is God watching all this and keeping track of everyone's version of each number? This seems like a cumbersome idea.

If we frame our views within this context, the difference is that you believe in an infinite consciousness whose thoughts eternally encompass all numbers. On the other hand, I believe there is no such preferred consciousness; rather, there are only finite consciousnesses whose thoughts can hold only a finite number of numbers at any given time. — keystone

I believe no such thing, what are you talking about? I believe in the axioms of ZF and not much else. They are purely a human artifact.

Computers perform calculus, and everything they do is finite. So, you're essentially arguing that there's a disconnect between the theory and the practice. Remember, in the case of calculus, the practice came first, and mathematicians later developed an actual-infinity-based theory to justify the practice. Might it be possible that a potential-infinity-based theory could provide a better justification for the practice? This one-minute video by Joscha Bach, titled "Before Constructive Mathematics, People Were Cheating," eloquently captures my view: https://www.youtube.com/watch?v=jreGFfCxXr4 — keystone

Yes, that's how constructivists think. Thanks for telling me it was a short vid, you got me to watch it. Math doesn't need a justification. It doesn't have to make constructivists happy.

While I haven't done much research on logic, I have a reasonably strong grasp of basic classical calculus. I understand that continuity is essential for classical calculus—my view starts with continua. I also understand that limits are essential for classical calculus—my view achieves the same ends by using arbitrariness. If you don't want to entertain my ideas simply because clever people weren't able to make calculus work within a finitist framework, that's fine as well. But let's be clear—it's not that you can't digest my ideas; it's that you won't entertain them. — keystone

You are making a grandiose claim that's likely to be false. But there are plenty of productive finitists, and the constructivists are on the march these days due to computer proof systems. But there is something to be said for infinitary math. Why shouldn't we enjoy having such a lovely theory of the infinite? What is the harm?

I understand how my claims appear. I'd like to support my position but it's quite hard if you don't look at my figures or words. You ask for the beef but the only comments you respond to are the bun. — keystone

LOL. It's hard to develop a theory of the reals without the axiom of infinity. The figures and words haven't helped much so far. You say, Start with a line. Make a cut. I don't know what these things are. You're just approximating the reals. I don't see anything to grab on to.

I believe my view is naïve in the same sense that Naïve Set Theory is naïve (minus the contradictions). — keystone

Your idea isn't naive. It's grandiose. And let's talk about something else please.

Joscha Bach seems quite confident that classical mathematics is filled with contradictions. — keystone

You say that like it's a bad thing!!

Paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating[clarification needed] way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, which reject the principle of explosion. — Wiki

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

But some guy thinks standard math has contradictions. He could be right. And if it did, the contradictions would be repaired. People wouldn't stop doing infinitary math. If a contradiction were found in ZF, it wouldn't affect group theory or differential geometry .

You’ve probably heard the story of Penzias and Wilson, who struggled with persistent background noise on their radio receiver, initially attributing it to pigeon droppings. It turned out to be the cosmic microwave background radiation from the Big Bang, earning them a Nobel Prize. I believe Cantor has interpreted his incredible discoveries as mere pigeon droppings. — keystone

I don't follow the analogy you're making. Cantor has underestimated or overestimated his discoveries?

I don't think I mentioned open sets. — keystone

You have been making use of open intervals all along, haven't you?

Funny you mention this. I skimmed through it a few days ago and then watched his YouTube lecture by that name yesterday. Now, I'm in the middle of his lecture on LEM. I'm really excited about watching his lectures. — keystone

Ok. Well if I turned you on to constructivism I'm happy and I've done some good. But I can't go down that road with you too far, because I have tried to understand constructivism a few times and it just doesn't speak to me. I like infinitary math and I think that if you reject the noncomputable, you are missing a lot.

You know, that's a particular bit of philosophy I can assert. The noncomputable reals are telling us something. Infinitary math is telling us something. The history of math is expansive, never contractive. Nobody says, "Those complex numbers, they were a step too far." But they say that about infinitary math.

Is your preferred format essay?? How did you become a mathematician and not an english major? — keystone

Well I didn't become a mathematician! I got to grad school and my eyes glazed. Well there were a lot of things going on. I ended up in programming. In math when you're stuck, you're stuck. In programming you can always code something, get something running, solve a bug, do something useful. I don't have an eye glaze factor when I'm coding, but I do when I do math. Interesting, I never thought of it that way but it's true.

But seriously, how am I supposed to communicate my ideas to you? — keystone

Are you getting frustrated? I'm sorry, I thought I was helping the best I can. You're doing fine, I've understood a lot. It's a tall order to reformulate analysis without the axiom of infinity. Even constructivists have infinite sets, not not noncomputable ones. It's like if you told me you were going to do brain surgery. You might be able to learn to do it, but you are not there yet as far as I can see.

This might not be the best chat forum etiquette, but would you be open to a Google Hangout? ...Please feel no need to even respond to that idea... — keystone

No thanks, this forum's all I can handle and barely at that. Plus I hate Google. They went from Don't be evil to being evil.

I don't know why you are acting as if I'm not attending to what you say. I sense a difference of perspective that I'm not privy too. Everything seems fine at my end.

Apologies for slow reply fishfry, but another topic has consumed much of my attention and I didn't even see your notify in my mention list. — noAxioms

Ok thanks. I was wondering if perhaps my last post was so far off the mark that you gave up on me (possible); or so brilliant that I thoroughly refuted your argument (unlikely); or you just got bored (also possible. I'm simulated out myself).

The last thing I remember is that you said the sims have actual bodies, made in the sim factory operated by the simulators. If I understood you correctly, that has massive implications and I find it hard to believe this is what Bostrom had in mind.

I meant to say that 'we are 'simulated (biological beings)'. — noAxioms

What on earth is a simulated biological being? Like an android with a soul? Like Data on Start Trek, but with a biological body? A manufactured human. What else can you mean? By simulated to you mean manufactured?

Or are you falling back on saying the simulation exists only in the execution of the computer?

Your interpretation of those words was 'we are (simulated biological) beings', which is perhaps what Data is. Data is an imitation human in the same world as its creator. — noAxioms

Yes. Just not biological, but that's an implementation detail. More like the replicants in Blade Runner.

The sim hypothesis is that we're biological beings in a different (simulated) world. — noAxioms

I do not know what that means. I gave a couple of examples. In Westworld (the tv series) the Hosts, as the lifelike bots are called, are geo-fenced within the park. The theme of the show is that they escape.

But I gather you don't mean that. You mean something else, but I can't fathom what that is.

I've said this over and over, included in the very statement you quoted above your response there. — noAxioms

I'm sure the fault is in my own understanding, but I have no idea what you're talking about. How would we simulate a physical bot that is not in the same world as us? Explain this point to me because you have lost me completely.

No, it's not Blade runner. No robots/replicants. — noAxioms

No robots, no replicants. Ok I misunderstood you.

But when WHAT? There is a factory that rolls sims off the assembly line, imbues them with self-awareness and will (illusory or no) ... but these sims don't live in the world of their makers? Where do they live?

You seem quite determined to paint a very different picture from the one Bostrom posits. — noAxioms

Not at all. I'm just trying to understand your interpretation of it, which frankly is crashing up on the rocky shoals of a point that you are being terminally vague about. The sims have bodies but the bodies are not in the world of the simulators. Where are they?

Your running with this idea for most of the post seems more designed to disengage than to communicate. — noAxioms

Not at all. You said the sims have bodies. That's a massive assumption that leads to all kinds of problems for anyone who claims that. I pointed those out.

(me) I say your mind is just your own subjective experiences and thoughts.
This works. — noAxioms

Yay. You agree. We can talk about minds without discarding physicalism.

In my world, I do both. I am not in the GS world, so I don't do either there. — noAxioms

Where are you?

I find 'process' not to fall under the term 'object'. It's not an assertion of ontology, just how I use the language. — noAxioms

You have a very funny way of putting things. You think you have explained to me that the sims have bodies that live in their own world. I can't make any sense of this.

Yes, Yanofsky's paper also mentions Chaitin's work: — Tarskian

Thanks, I'll check out that paper.

Yanofsky's paper mentions an even larger class of random mathematical truth: unprovable because ineffable ("inexpressable"). There is no way to prove truths that cannot even be expressed in language. — Tarskian

True yet inexpressible in language. Great concept.

Surprised nobody has reactivated this thread. Julian Assange pleaded guilty to one count of violating the US Espionage act, in return for a sentence of 62 months with credit for time served, the 62 months he just spent in 23 hour a day solitary confinement in Britain's notorious Belmarsh prison. He's a free man.

There's good news and bad news.

For Julian, I am thrilled. I've been hoping for his freedom for years. Few of us could have stood up to his five year incarceration, preceded by seven years holed up in the Ecuadorian embassy in London. Julian Assange is tonight a free man, and I am glad for that.

The bad news is that the US has established the precedent that journalism is espionage. That's a step down a slippery slope that few of us are going to like. Every journalist in the world got the message today. Reveal the US government's crimes and they destroy your life. A lot of other governments in the "free world" too. Forget that pesky First Amendment and the notion of a free press.

Assange did exactly what the New York Times did in the Pentagon Papers case. He was treated a lot differently, and the world of mainstream journalism deserted him and hung him out to dry. When you're declared an enemy of the state, few will stand up for principle at the cost of being seen defending you.

Today even the Times, the voice of the establishment, agreed with the risk of this deal to press freedom.

Assange’s Plea Deal Sets a Chilling Precedent, but It Could Have Been Worse

The deal brings an ambiguous end to a legal saga that has jeopardized the ability of journalists to report on military, intelligence or diplomatic information that officials deem secret

https://www.nytimes.com/2024/06/25/us/politics/assange-plea-deal-press-freedom.html

From the Times:

The agreement means that for the first time in American history, gathering and publishing information the government considers secret has been successfully treated as a crime. This new precedent will send a threatening message to national security journalists, who may be chilled in how aggressively they do their jobs because they will see a greater risk of prosecution.

I am a bit surprised at this take. The Times has not been outspoken in Assange's defense as far as I know, but I admit I haven't followed their coverage over the years.

Well I'm happy for Julian Assange tonight, and I'm sad they put another big dent in the First amendment and the public's right to know about government malfeasance.

The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic. — Tarskian

I didn't read the rest of this interesting thread yet so I'm just responding to the top post.

I believe Chaitin made a similar point. He has a proof of Gödel's incompleteness theorems from algorithmic complexity theory. I believe he says that mathematical truth is essentially random. Things are true just because they are, not because of any deeper reason.

This sounds related to what you're saying.

Why should we suppose that natural languages are only countably infinite? — Banno

The set of finite-length strings over an at most countably infinite alphabet is countable. There are countably many strings of length 1, countably many of length 2, dot dot dot, therefore countably many finite strings.

If you allow infinite strings, of course, you can have uncountably many strings. That's the difference between positive integers, which have finitely many digits; and real numbers, which have infinitely many. That's why the positive integers are countable and the real numbers uncountable. It's the infinitely long strings that make the difference. But natural language doesn't allow infinitely long strings. Every word or sentence is finite, so there can only be countably many of them.

a. Form the set of all subsets of S0, P(S0). — The Vastness of Natural Language

I didn't completely follow what you're doing, but in taking the powerset of a countably infinite set, you are creating an uncountable one. There aren't uncountably many words or phrases or strings possible in a natural language, if you agree that a natural language consists of a collection of finite-length strings made from at an most countably infinite alphabet. I think this might be a flaw in your argument, where you're introducing an uncountable set.

There are three ways we could approach for set theory: — TonesInDeepFreeze

Ok I finally made a run at this, and I am having a little trouble understanding your meaning.

I believe you are trying to convince me that logical identity is the same thing as set equality as given by extensionality.

I'm a little unclear on what you mean by logical idenity. Do you mean the law of identity, everything is equal to itself? or identical to itself? Or did you mean something else?

I could use some specifics to help anchor my understanding.

(1) Take '=' from identity theory — TonesInDeepFreeze

I already got in trouble here! I looked up "identity theory." Both Wiki and IEP say it's a theory of mind. So that's not what you're talking about. Wiki has a disambiguation page that led me to Pure identity theories, a linkable paragraph within an article called List of first-order theories.

So if you could just define "identity theory" for me, and tell me what "=" means in that theory, I'll understand you better.

, with the axioms of identity theory, and add the axiom of extensionality. In that case, '=' is still undefined but we happen to have an additional axiom about it. — TonesInDeepFreeze

Even if I knew what you mean by identity theory, I still did not understand this. Still undefined but additional axiom. Sorry I don't follow. I'm probably missing your point I'm sure.

I have a question for you.

In ZF, I define $R = \{x \notin x\}$ and $S = \{x \notin x\}$, two definitions of the Russell set.

I ask: What is the truth value, if any, of $R = S$?

How should I think of this? In ZF? In set theories with classes?

The axiom of extensionality is not a definition there. And, with the usual semantics, '=' stands for the identity relation. It seems to me that this is the most common approach. — TonesInDeepFreeze

Ok, I hope I will understand this when you tell me what the identity relation is. But if extensionality is not an axiom, what is it? Axioms and definitions are the same thing. You can take them as "assumed true," or you can take them as definitional classifiers, separating the universe into things that satisfy the definition and things that don't.

For example, take the axioms for group theory. As axioms, they are assumed true for every group. But we actually use the axioms as a definition. If a mathematical structure obeys the axioms it's a group; and if not, not. So we can use axioms as a definitional boundary between everything we're interested in, and everything we're not. Axioms and definitions are the same thing viewed from different perspectives.

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z) — TonesInDeepFreeze

I don't really understand this. What are you trying to say in the axiom? That if two sets satisfy extensionality (the definition) then any set one of them is an element of, the other is also an element of? Am I getting that right? I think that already follows from the definition. In fact I convinced myself I could prove it, but did not work out the details. So I could be wrong about this.

But what is the intent?

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols. — TonesInDeepFreeze

I didn't get all this, what's the intent of the axiom, what does it all mean?

As far as 1 + 1 = 2, I've explained to @Metaphysician Undercover that these are two expressions that refer to the same set. By extensionality there is only one set, and two different representations of it. Other posters have mentioned that the intentional meanings of 1 + 1 and 2 are different, and the extensional meanings are the same. Mathematicians use the extentional meaning of a symbol. 1 + 1 = 2 "point" to the same abstract number in abstract number land, wherever that may be.

Mathematicians don't think of numbers as uninterpreted symbols. But perhaps if pressed to explain where numbers live, what kind of existence they have, it's safer to revert to formalism.

Suppose X and Y are objects in the universe, but they are not sets?
— fishfry



In set theory, contrary to a popular notion, we can define 'set':

x is a class <-> (x =0 or Ez z e x)

x is a set <-> (x is a class & Ez x e z)

x is a proper class <-> (x is a class & x is not a set)

x is an urelement <-> x is not a class — TonesInDeepFreeze

I don't know much about set theory with classes. I'm just a humble ZF guy. I have no choice! (set theory joke).

Then in ordinary set theory we have these theorems:

Ax x is a class

Ax x is set

Ax x is not a proper class

Ax x is not an urelement

If our meta-theory for doing models has only sets, then all members of universes are sets.

If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model'). And no proper class is a member of a set.

If our meta-theory for doing models has urelements, and '=' stands for the identity relation, then the axiom of extensionality is false in any model that has two or more urelements in the universe or has the empty set and one or more urelements in the universe. — TonesInDeepFreeze

Didn't get to read much of this closely. Perhaps we can revisit later.



I did note one thing I disagreed with. You wrote:

"If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model')"

Perhaps we're using different terminology. When they do independence proofs, models are sets. So for example to prove that ZF is consistent, we are required to produce a set that satisfies the axioms. It's no good to just provide a proper class, since the class of all sets satisfies the axioms but isn't sufficient to prove the consistency of ZF. (As usual I'm still confused about how ZF can see this, since it doesn't have proper classes).

Secondly, I know universes that are not sets. For example:

* The von Neumann universe and Gödel's constructible universe, both of which are proper classes (however you regard them) and are commonly called universes.

So perhaps I am not sure what is your definition of a universe.

Even informally, in ZF the universe is "all the sets there are." The axioms quantify over all the sets. And the universe of sets is not a set.

Summary of all this

These are some of my thoughts on reading your post. I'm sure I missed a lot. I'm still interested in understanding why you think that the logical identity (whatever that is, I'm still a little unclear) is the same thing as set equality under extensionality. I just can't see how that could even be. Once we get outside of the world of sets, we need a new definition. Because extensionality only applies to sets. And if we have a new definition, by definition we do not have the SAME definition. We have a DIFFERENT definition, which is to be consulted whenever we are wondering about the equality of two things that are not both sets.

It's like if I'm a computer multiplying integers, I use one algorithm. If I'm multiplying floats, I use another. So if we have two sets, we use extensionality to tell if A = B. If at least one of them is not a set, we have to use some OTHER way of telling. Which is your logical identity. It's a different thing, not the same thing.

Since I've convinced myself from first principles that logical identity and set equality are different things, I have a bit of a hard time following your arguments, since they must be wrong, or we must not be talking about the same thing.

Perhaps you can walk me through this slowly and clearly.

This is the assumption we allow for to examine the possibility of supertasks.

But it is still the case that it cannot arbitrarily be on. It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ω. — Michael

Jeez I didn't even finish editing before you snapped back. Have a nice evening. You are trolling me now. Doing a good job of it.

The problem is that you seem to fail to acknowledge how lamps work. — Michael

Lamps that switch state in arbitrarily small intervals of time? I missed that day in lamp school.

The lamp is as much a fairy tale as Cinderella's coach, which is why I use that example. It's a magic lamp. You are the one invoking magic, and then acting like it's a real lamp to try to argue a pointless point. That's another source of your confusion.

"How lamps work?" Lamps don't work that way! Maybe YOU missed that day in lamp school.

He doesn't push the button at midnight. He only pushes it at 11:00, 11:30, 11:45, ...

Also, pushing the button will only turn the lamp on if it was off and will only turn the lamp off if it was on. — Michael

I appreciate your enthusiasm, but surely you can see that nothing new has been said between us for a long time. We could just reread each other's old posts. I hope you'll forgive me for declining to engage. If you had better insight into infinite sequences and their limits, you wouldn't be troubled by what happens at midnight. I say that because the code you posted exactly reflects your misunderstanding of the nature of infinite sequences. Think of Cinderella. Why does the coach turn into a pumpkin? It does so because that's the premise of the story. The lamp is in some state at midnight because you say it is. And it doesn't matter which.

The first sentence is true and is the proof that "supertasks are senseless" (as Thomson says). — Michael

@TonesInDeepFreeze asked for Benacerraf's argument, of which I provided my own version without any change in meaning. I also linked Benacerraf's paper per Tones's request. That was my motivation for posting.

As mentioned several times, the implicit premises are that the lamp continues to exist (as a lamp) at 12:00 and that nothing other than pushing the button can turn the lamp on or off. — Michael

This is a red herring of no relevance. I've responded to the button pushing argument as many times as you've mentioned it. If "nothing other than pushing the button turns the lamp on or off," then at midnight, the button pusher pushes the button and turns the lamp on or off, per your premise. The terminal state is On or Off. In either case "the button was pushed" since you insist on it. No matter the final state, the button was pushed, and your premise is satisfied. Neither On nor Off is to be preferred since the sequence 1, 0, 1, 0, ... has no limit. Thompson himself makes this observation in his paper.

I did enjoy our earlier conversation, but no additional typing on my part could be of any use to you. If it could be, it would have already been.

I regard Thompson's lamp as a solved problem. I'll leave it at that, and refer you to my existing posts on the subject.

Sour Grapes. — Vera Mont

I think so. What's funny is that I'm the one who should have sour grapes. I'm a math grad school dropout. The fault was not in my stars, but in myself.

What exactly would I envy? Dealings with the HR department of a university? I have never had to go through any HR department. I find the practice insulting. — Tarskian

Can you separate out your thoughtful critique of how post-secondary education should be done (and I agree that the current system often fails the creative and the gifted) on the one hand, with your emotional resentment of ... HR departments? Having spent my life working for corporations, I share your detestation of HR. But you seem to take it personally. You hardly ever interact with them, you fill out some forms when you get hired and then have an exit interview when you leave. Hopefully you stay out of their clutches the rest of the time. Maybe you just find administration insulting. Bureaucracy, regimentation, rules and regulations, the trappings of organizational order.

Or the very trappings of modern life. Is that you're main driver?

You are unhappy with students being taught the state of the art in their field. AND you are unhappy, viscerally so, with the administrative structure that supports organizations. Both for the same reason? You have a lot of feelings invested in all this.

Can you see that resentment of grad school pedagogy and resentment of HR department are two entirely different resentments? Frankly you come across as resentful.

It says everything about your station in life. — Tarskian

Not really. The president of the United States draws a paycheck. He has taxes taken out, someone cuts him a check or does a direct deposit. Maybe it's the payroll department, maybe it's HR. How does the president have a low station in life because he works in an organization that follows the law with regard to employment and taxes and compensation?

I bet when the new president takes over he has to fill out a ton of mindless HR forms. Of course he has people to do it fo him. But you can't run an organization without an HR function.

According to Google, "... the HR (Human Resources) department is a group who is responsible for managing the employee life cycle (i.e., recruiting, hiring, onboarding, training, and firing employees) and administering employee benefits." In every organization someone's got to keep track of the legal employment paperwork, payroll, benefits administration.

Why on earth this upsetting to you?

Stephen Wolfram writes on this subject:
... (Wolfram quote omitted)

So, in order to know everything there is to know about mathematics, you need to read 3 million papers. Did I read them? Did I ever said that I read them? Did I even read 0.1% of them? — Tarskian

I don't think I see the relevance of any of this. There's plenty to learn, more than anyone could master in a lifetime. Much of it is obsolete, but that's why you go to school. To have experts guide you along a productive path.

Knowledge is a gigantic database of (claim,justification) two-tuples that is for 99.999% stale and irrelevant. The only meaningful way of finding out what is relevant, is to work your way back from solutions that solve problems all the way into the math that directly or indirectly facilitates the solution. — Tarskian

You need to acquire the basic skills first. That generally takes a lot of schooling. You seem to think we should educate everyone as if they were geniuses. But that wouldn't work very well, you'd leave almost everyone behind. You get a class of eight year olds. "Ok see if you can all develop quantum physics." Or even 18 year olds, if you only want to apply your "sink or swim" approach in college.

Is that what you're saying?

So, is knowledge a good thing? Possibly, but it is first and foremost, utterly useless. — Tarskian

Newton mastered the work of the ancients. And people tell myths that Einstein failed math and whatnot, but he was excellent at math in school and was thoroughly steeped in the physics of Newton and Maxwell.

The idea of feeding students with some arbitrary excerpt from such knowledge database, assuming that it will ever be useful to them, is misguided and nonsensical. — Tarskian

If it's arbitrary, yes. But if the knowledge is curated by leading-edge practitioners of the craft, that's exactly how you come to understand the modern state of science. That's what universities are for.

I hardly think you'd take a math undergrad and toss them a copy of Grothendieck's Éléments de géométrie algébrique, in the original French of course, and tell them to figure it all out or go screw themselves. I don't think you are making a rational proposal.

Grothendieck himself was one of your savants. From Wiki, quoting someone else talking about him: He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics.

But we can not all be Grothendieck. He was the greatest mathematician of the second half of the twentieth century.

That is the reason why the education system fails. Its knowledge-acquisition strategy simply does not make sense. — Tarskian

I certainly agree that the system fails geniuses. But 99.999% of people are not geniuses. The system of training scientists and mathematicians is flawed, so you should work out your own theory and publish it. But you seem like you are not coming from, "How can I improve the system," so much as, "I have some kind of irrational and substantially unfounded resentment of the system," and that just leaps out of your own words. You don't like the professors, and you don't like the HR department. What about the cafeteria workers, are you ok with them?

The only way to pick the right things to learn, is by going in exactly the opposite direction. You start by trying to solve a practical problem, for which there exists someone willing to pay for the solution, and only then you learn knowledge as required for producing the solution. — Tarskian

All the 18 year olds are apprenticed out to people who will pay them even though they're completely ignorant? You can't be serious. What are you talking about?

Maybe you think all the geniuses should be issued family farms, like Newton; or be given jobs as patent clerks, like Einstein. Or live with their mother in Russia, like Perelman. But these are the one percent of the one percent of the one percent. Literally one in a million. What about everyone else?

I am semi-retired now. If I was ever going to work again, I'd rather swear fealty as a serf to the lord of the manor than to deal with an HR department. — Tarskian

So maybe you're against large organizations.

You're like Marlon Brando in the 1953 film, The Wild One. Girl asks Brando, "What are you rebelling against?" And he answers: "What have you got?"

Okay then, maybe just a very dull axe. — Vera Mont

LOL.

Trying to make my ideas clearer so that your eyes might not glaze over has indeed helped me collect my thoughts. I've also benefitted in other of your recommendations, such as construtivism which I really appreciate. So thanks for the glass half full. But there will come a point where no further progress can be made if I can't produce post that you are able to digest. — keystone

I don't see why. One of the best ways you can respond to someone who brings a problem to you is to just ask them to explain it all to you in detail. By the time they're halfway done they usually solve it themselves. It's their explaining that does the trick, not my understanding. Smart bosses do that.

Also, the thing is, and I thought of mentioning this to you the other day, I don't actually care about anyone's ideas about what the real numbers are or how math should work. There's nothing at stake for me here. I enjoy trying to relate your ideas with things I know in math, but there's never going to come a point where I "digest" this. All this is a very half-baked brew, to mix metaphors.

I do plan to do a deeper investigation into Constructivism and certainly Brouwer will be a part of that. Thanks. — keystone

Welcome. If you think everything's generated by algorithms, constructivism's your math philosophy.

I, on the other hand, am particularly drawn to intuitionism because I find it to be the least 'out there'. In this perspective, what exists are not infinite, eternal abstract objects in some inaccessible realm, but rather the finite set of objects currently being 'thought' by active computers. In my view, if the number 42 is not presently within the thoughts of any computer, then 42 does not currently exist. — keystone

Computers don't have thoughts. In fact, there aren't any numbers in computers. There are electrical circuits. It's the humans that interpret certain bit patterns in certain circuits as numbers. The Chinese room does not understand Chinese.

Let me ask you a question. Suppose there is a computer in the world that contains, at this moment, a bit pattern corresponding to the character string "XLII". Then suppose there's another computer somewhere else in the world, and it contains the bit pattern 101010. And in yet a third computer, we find a bit pattern corresponding to the character string "42".

Do these three computers each instantiate the existence of the same number 42? And how would you know?

I am drawing your attention to the problem of representation. There is never a number inside the computer. There are only bit patterns. And depending on the encoding scheme, the same bit pattern may mean different things; and different bit patterns may mean the same thing.

In line with my intuitionistic view, I'm not constructing any infinite set, rather constructing computable reals one cut at a time. More importantly, I can stop at any point and still have a working system. There's no need to complete the impossible task of constructing all the real numbers...after all, computers do math without ever having the complete set of real numbers in memory. — keystone

You can do engineering like that, but you can't do math. In finitism (rejecting the axiom of infinity) we can do a fair amount of number theory, but not analysis. You can't do calculus, you can't do physics. You can do finite approximations, but the underlying theory is infinitary.

I disagree with this decision. I believe it is possible to perform analysis without relying on the axiom of infinity. — keystone

You should research that claim rather than just proclaim it. This is one of the reasons I am never going to "digest" your ideas. Many clever people have given these matters considerable thought. You should do a literature search on this idea to clarify your thinking.

Can you see that grandiose claims made without sufficient background come down to untrained feelings and intuitions? Not that there's anything wrong with that. But it supports my belief that there is nothing to digest.

While I don't have formal rules or detailed structures yet, I possess concepts that would be found in an introductory calculus textbook, or perhaps an introductory engineering calculus textbook. Admittedly, this is a significant claim that requires substantial support...it's just that your eyes glaze over... — keystone

There's a mathematician and and engineer joke in there somewhere. And eye glazing is something else. I think you have a bad idea, not in the sense that it's absolutely wrong; but in the sense that you have a very naive understanding of what's involved, so that it seems grandiose.

But that is NOT what makes my eyes glaze. Certain diagrams and definitely long lists make my eyes glaze. Eye glazing is entirely independent of reasonableness. Your claim about calculus is unreasonable. That doesn't make my eyes glaze. The diagrams and lists make my eyes glaze. Hope that's clear. Someone could show me a diagram or list that was 100% correct and brilliant, and my eyes would still glaze. Eye glazing is no measure of the quality of an idea, it's just how I process or have difficulty processing it. But grandiose claims that you personally have figured out how to do analysis in the absence of the axiom of infinity, that's a bad idea. My eyes are perfectly clear about that.

Hope I made this distinction. Some of the best ideas make my eyes glaze.

Cantor's proofs are quite fascinating. Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. These individuals are in the minority, as most mathematicians do not share this view. I'm intrigued by the idea of a mathematics that does not rely on actual infinity, as I believe this approach is more aligned with true mathematics. It promises to be free of contradictions and brings with it the potential for beauty and advancement. — keystone

Why? Infinitary set theory is perfectly clear of contradictions. Well, as far as we know. We can't prove its consistency without assuming more powerful systems.

Are you a Cantor crank by any chance? A long time ago (when I was younger and I suppose less harmless looking than I am these days) a traffic cop pulled me over. As he was writing my ticket, he asked me if I'd ever been arrested. "I just want to know who I'm dealing with," he said. That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with.

Why do you talk of everything, such as 'all the open sets'? I can't imagine a computer holding this infinite set in memory. I'd rather talk about what I know is possible, such as a computer which holds a few open intervals with rational endpoints. As for infinitary operations, my long post with many bullets (let's call it the bullet post) addresses my view. — keystone

The open sets were your idea. And the standard topology on the reals is generated by the open intervals with rational endpoints. There are only countably many of those.

Nobody's saying you can't approximate things with computers. You're imagining some kind of mathematical metaphysics that isn't really there. But the constructivists have this all worked out. Did I post this Andrej Bauer article, Five Stages Of Accepting Constructive Mathematics? Give it a read, tell me if any of it makes sense to you. I read it a while back, don't remember much. I never get very far trying to understand constructivists.

https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf

Yes, the bullet post. — keystone

Ok good, thanks.

I'd like to read Benacerraf's paper that disputes that there can't be a state at 12:00 — TonesInDeepFreeze

I wandered over and happened to notice this remark. I can say something about it. I was pointed at Benacerraf's paper a few weeks ago, and since I had developed the same solution, I felt vindicated and at the very least sane. I know Benacerraf from his great paper, What Numbers Could Not Be, and has a lot of credibility with me.

So I feel qualified to explain his point.

Here is Benacerraf's argument in my math-y conception. It's the same argument.

The lamp problem is best modeled as a function defined on the ordinal $\omega + 1$, with output in the set $\{0, 1\}$. (I don't recall if Benacerraf explicitly uses this mathematical approach but his argument is the same ).

$\omega + 1$ can be visualized as the sequence

0, 1, 2, 3, ...; $\omega$

where 0, 1, 2, 3, ... ranges through all the natural numbers; and there is a "point at infinity" after them all, traditionally called $\omega$ in this context. It can of course be formalized as an ordinal but that's not even important in this context.

If it makes anyone happy in the context of this problem, we can model it as

1/2, 3/4, 7/8, ...; 1

where the values are times getting ever closer to 1, with 1 being the mathematical limit, as well as the time on a clock. Or 1/2, 1/4, 1/8, ...; 0 where the times are seconds before midnight, and 0 is exactly midnight.

I analogized that last idea to Cinderella's coach, which is a beautiful coach at 1/2, 1/4, 1/8, ... seconds before midnight; but becomes a pumpkin at midnight. It's not usually thought of as such, but Cinderella's coach is a supertask puzzle. If Cinderella checks her watch at each time 1/2, 1/4, etc., it's always a coach. How does it become a pumpkin?

It's the same mathematical insight that applies to most of these supertask puzzles. Sequences don't ever "reach" their limit, but every sequence can be assigned an arbitrary terminal state. Sometimes the terminal state is the limit of the sequence, and sometimes it's not. You can always assign a terminal state because it's just a mathematical function. It's a function defined not on $\omega$, as infinite sequences usually are; but on $\omega + 1$, to include a terminal state.

We can use this structure to model the lamp. For each natural number 0, 1, 2, 3, ... the output alternates on/off, which we can denote as 1, 0, 1, 0, ...

So the lamp problem defines the function for each of the inputs 0, 1, 2, 3, ... but does not specify the value at $\omega$. Therefore we are free to define it any way we like.

Both outcomes are entirely consistent with the premises of the problem.

Additionally, unlike in some supertask puzzles, in this case there is no natural or preferred solution. That's because in neither case can the terminal value be the limit of the sequence.

Contrast this, for example, with the sequence 1/2, 3/4, 7/8, ...; 1. That notation says that there's an infinite sequence; and after the entire sequence, we have a value, 1.

Now just as before, the assignment of the value of the terminal state is arbitrary. But in this particular case, the value 1 is natural, in the sense that 1 is the mathematical limit of the sequence. This idea of adjoining the limit of a convergent sequence to the end of the sequence is a convenient and natural formalism.

With the lamp, there is no possible way to assign a terminating value that makes any particular sense. Instead, absolutely any answer will do. On, Off, or as I facetiously said earlier, a plate of spaghetti; to emphasize the arbitrariness of the choice.

I've visualized the lamp problem that way for a long time. And when I found out that Benacerraf gave the same argument (minus the spaghetti), I was happy.

That's Benecerraf's argument in my words. Here's his paper.

Tasks, Super-Tasks, and the Modern Eleatics

@Metaphysician Undercover, @TonesInDeepFreeze, Ah this is the Infinity thread. I was totally confused. I thought this was the supertask thread. Sorry for the confusion. Meta you did not hijack that thread. I can't believe they are still going on about that stupid lamp. I can't believe I get lost on this forum.

Meta, once I understood that Tones was arguing that set equality is the law of identity, I realized why you're arguing this point. I entirely agree with you. I apologize to you for jumping to multiple wrong conclusions.

I noticed that you posted then deleted a response to me, so perhaps you at first objected to my post then realized that by the end, I was in agreement with you. I'll think of it that way.

I haven't yet worked through Tones's reply to me outlining his argument, so I should reserve judgment. But at this moment it seems to me that set equality is a defined symbol in a particular axiomatic system. As such has no referent at all, any more than the chess bishop refers to Bishop Berkeley. It doesn't refer to anything concrete, nor anything abstract. It simply stands for a certain predicate in ZF. It can't possibly "know" about logic or metaphysics. It can't refer to "sets" since nobody knows what a set is. A set is whatever satisfies the axioms. And set equality is a relation between sets, which have no existence outside the axioms; and have no meaning even within the axioms.

Meta I agree with you on this point and had no idea that I was jumping into an ongoing conversation in a thread I didn't realize I was in. My bad all around. Tones, I look forward to working through your post. I'm sure many clever people must have considered this very problem of set equality and identity and there's a lot I don't know.

I didn't get the business about the Eleatics. Did they have this conversation back in the day?

I've moved your discussion on set ordering and the meaning of equality to this discussion — Michael

Explains a lot ...

Yeah, mostly. There is other stuff in the post, but credence is accurate. Not like the view of one particular unknown probability but how you consider occurrences (where you know the chance) when there are multiple times they could occur or multiple things that could happen. — Igitur

Yes, it's a nice concept. As I understand it it's basically the same math as probability theory. But we are interpreting it differently. Instead of probability residing in the coins or dice, credence works in ourselves. It's a murky question to try to say what probability actually is, and why a coin or a dice should care. But credence is belief, and belief is not murky at all. I just ask myself what my degree of belief is, and that's what it is.

Credence makes life simpler, since we then don't have to think about what probability really means.

I did read your post and wasn't sure exactly what you were getting at, so my apologies if I missed the main point.

The concert pianist actually intends to solve a problem. So does the athlete. — Tarskian

Mathematicians solve problems. They solve problems in math. Like Wiles solving Fermat's last theorem, a problem pure mathematicians had been struggling to solve for 357 years.

For some reason your posts seem to dismiss or denigrate this type of work. I am perhaps not understanding your thesis, but I do feel that some of the things you say about math aren't actually true.

What problem does the math graduate intend to solve except for teaching math? — Tarskian

Wiles had to slog through all the proofs before him. He was already an established professional mathematician when he started his seven year quest to solve FLT. Imagine how many boring and tedious proofs he's gone through in his life.

This statement, that math grad students are there to become math teachers, is so wrong that it's the reason I say you haven't much understanding of math. That is not credentialism. That's just reading your posts, here and on the other forum. You have an ax to grind with math, and you are not articulating it very well, particularly as many of your premises are false.

Concerning my academic background in a branch of applied math, if it were still relevant after decades, it would mean that I wouldn't have done anything meaningful in the meanwhile. — Tarskian

I have no doubt that you have many meaningful accomplishments. I can only go by what you write. So far you seem to be unhappy about something, but the things you're unhappy about are strawmen. You say grad students are training to be teachers. Nothing could be further from the truth. Grad students are training to do research math.

The math department needs to teach calculus as a service course for the engineers and physicists and economists and the like. They teach the undergrad math major curriculum as the introduction to pure math, as professional mathematicians see it. It's the start of training to do research in higher math, and grad school is that on steroids.

The math teaching curriculum is actually separate. The Ph.D. track is about research. If you're a good teacher and a lousy researcher, you're out. If you're a terrible teacher and a good researcher, you have a job for life.

Surely you must have a sense of this.

If a degree matters after your first job, it simply means that your first job did not matter. — Tarskian

Not relevant to my point. I'm judging your thesis by your posts. I don't care about your resume. How could I? On the Interned we only have our words and our ideas, not our credentials.

My stints in pure math came much later. Sometimes because I was looking under the hood of the software I was using. Sometimes just out of interest. — Tarskian

Perhaps you have learned a lot but still don't know everything there is to know, and perhaps you have made some wrong assumptions.

I don't understand the level of your annoyance or pique or whatever with math. Of course I agree with you that the grade school, middle school, and high school math curriculum has been an unmitigated disaster for decades. But you seem unhappy with grad-level professional training in math research. That part I don't understand.

For example, I did my first foray in abstract algebra by looking under the hood of elliptic-curve cryptography. In fact, you understand abstract algebra much better if you have first been exposed to subjects like ECDSA and Shnorr signatures. The other way around is not true. — Tarskian

That's a very interesting background. Reminds me of how functional programmers are into category theory. There are many paths to abstract math.

I will certainly agree with you that when an earnest undergrad first takes groups, rings, and fields, it's like being thrown into the deep end of the pool. It takes a while to understand why they're doing all that. I agree that perhaps they should motivate it better.

You have a credentialist view on knowledge. — Tarskian

No not at all. I am judging the posts you've written on this forum and the other one. I don't understand your antipathy to the math establishment, or the purpose of graduate school, or whatever.

That is typical for teaching associates at university. — Tarskian

I don't know if that is or isn't. What does it have to do with me?

They think that credentialism matters. — Tarskian

They're just trying to survive, like grad students everywhere.

Well, they have to, because their hourly rate clearly does not matter. — Tarskian

Yes, grad students are cheap labor.

The academia are full of postdocs and other idiots who think they know but who in reality have nothing to show for. — Tarskian

This is certainly the case. But better a lame math postdoc than a plagiarizing dean. I mean, if we are talking rot in the academy, surely the math postdocs are among the least affected. You have to get a Ph.D. in math by doing a piece of original research. Then you have to find a job in a very overcrowded and competitive job market, and now you have to start your career on the publish or perish treadmill.

These are not soft sociology degrees in grievance studies, not to get political. These are early career researchers in pure math. Why do they upset you?

I will concede that, just like ninety-nine percent of every profession, most of them in the end are mediocre. Still, they do their research, they write books, they go to faculty tea.

Why are you angry at these low-paid hard working young research mathematicians?

Furthermore, the relevant math is elsewhere. — Tarskian

Interesting point of view. The academic mathematicians are putzes and the real work's being done in AI and quantum cryptography at Google.

Well maybe. The tech companies certainly employ mathematicians, as do the insurance companies. But for pure math, the academy is the only place they're going to pay you actual money to do the kind of work most pure math researchers do.

They really do not understand, not even to save themselves from drowning, which areas in pure math power technology. — Tarskian

So what? They're not doing applications. If they are they are, and if they aren't, they aren't. A lot of pure math doesn't find application for decades or even centuries. Number theory was supremely useless for two thousand years till public key cryptography came about in the 1980s.

Why does this bother you?

That is why they are stuck in areas that are irrelevant. — Tarskian

Irrelevant to you, you mean. Clearly not irrelevant to their university math departments.

So what's your beef? Why is this a particular concern of yours? Is this an objection to pure math, or universities, or grad school, or what exactly?

Vitalik Buterin — Tarskian

Oh he's a brilliant guy, no question. But we can't all be Vitalik. Why does it bother you that some mathematician is sitting in a cramped office working on his research into pure math?

A bit like Bill Gates (Microsoft) or Steve Jobs (Apple), Vitalik had to stop wasting his time and drop out of his university undergraduate in order to do something more important: — Tarskian

Yes that's true. Newton had his miracle year when the plague shut down Cambridge university in London, and he spent a year and a half on his family farm. Einstein's miracle year was when he was working as a patent clerk after being unable to get an academic position. Perelman was only able to resolve the Poincaré conjecture by refusing tenure-track appointments so he could focus on his work and not be bothered having to publish anything till he was ready.

And of course Wiles himself had a full time day job as a tenured math professor at Princeton, while he worked secretly in his attic to solve FLT.

So yes, for some people. dropping out is the better path, or just not being at the university. For most of the rest of us who aren't geniuses, there's always grad school.

I take your point. But so what? What is the nub of your irritation about this?

That was said to Metaphysician Undercover. — TonesInDeepFreeze

Yes I understood that. I haven't been in this thread in a couple of weeks and when I checked it out, @Metaphysician Undercover had evidently introduced his favorite theme, that mathematical equality is not metaphysical identity, which seemed a little afar from the original topic.

Actually, I am the one who took up his misconception that sets have an inherent order. — TonesInDeepFreeze

Yes I did actually understand that! I was just startled that @Meta was still going on about order being an inseparable and inherent aspect of a set, when I had already had such a detailed conversation with him on this subject several years ago. I did actually realize you were quoting him -- I was just surprised to see him still hung up on that topic.

I don't consider that "hijacking", since his posts in this thread about tasks need to be taken in context of his basic confusions about mathematics, as mathematics has been discussed here. — TonesInDeepFreeze

@Meta's basic confusions in math are too big to to be placed within context. He's completely unwilling to meet any mathematical idea on its own terms.

What? In ordinary mathematics, '=' does stand for identity. It stands for the identity relation on the universe. — TonesInDeepFreeze

Ah ... you said that? Well I understand why my friend Meta is unhappy with you then. I am not sure if I agree with your statement.

After all if = is the identity relation on the universe, why does ZF need to redefine it then? Is the = of ZF the same as the identity relation on the universe?

I do not think so. Because the ZF version is a definition. It's a defined symbol that wasn't defined before.

Answer me this, maybe I'll learn something. Suppose X and Y are objects in the universe, but they are not sets?

Well in ZF they don't exist. And even if they did, how would you define X = Y?

Oh ok, that's great to hear. Yeah, sorry for the large number of line items... — keystone

Your line items are helpful to you, and that is the ultimate goal. Technically it doesn't matter whether I ever understand your ideas or not, as long as I am useful as a sounding board. So if you will take the glass half full approach to my not relating to your charts and graphs and lists, then you can feel free to keep posting them and my eyeballs will feel free to be glazed.

You're right, I'm likely a constructivist/intuitionist. I say 'likely' because there's a lot of material to go through, and I need more time to fully understand it all. However, my views align with the key principles of constructivism. My main frustration with the material I've found so far is that it doesn't seem to address what I'm talking about... — keystone

Check out this guy.

https://en.wikipedia.org/wiki/L._E._J._Brouwer

I must say that in my modest studies of those subjects, constructivism seems more reasonable. I'm perfectly ok with working out the consequences of restricting the real numbers to the computable reals.

But intuitionism, with its active intelligence creating sequences as they go ... that's just a little out there for my taste.

Given this, it's pretty clear that I'm not constructing the familiar reals. — keystone

You're not constructing the familiar reals? First time I'm hearing this. Maybe you're constructing the computable reals. Is that what you're doing?

I think it is more correct to say that I have an alternate view of continua for which reals only play a supporting role. If mathematics were reformulated to be entirely absent of actual infinities would that be significant? — keystone

I'm pretty sure, but have no specific info about this, that people already decided you can't do analysis, that is calculus and the theory of the reals, without the axiom of infinity. But I could be wrong. I think if you could do analysis without the axiom of infinity that would be impressive.

I'm working towards a foundation free of actual infinities. — keystone

But infinities are one of the most fun and interesting part of math! I always liked infinities. I think I just don't understand the psychology of someone who doesn't like the axiom of infinity.

Tell me, what makes you interested in trying to do math without infinite sets?

Okay, I was too rash to take the bundles away. I think they're a useful way for us to find common ground. One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. — keystone

I would interpret that as your intuition that the open intervals with rational endpoints are a basis for the usual topology on the reals. All the open sets are unions (perhaps infinite) of open intervals with rational endpoints. But then again ... do you allow infinite unions and intersections of sets? Do you want to get rid of infinitary operations as well as infinite sets?

I comment on this in the long post which you haven't responded to. — keystone

Did I miss a post? Or do you mean the long list of definitions and principles that glazed my eyes a bit?

By the way my eyes glaze over frequently at many things. It's nothing personal.


Yeah, the other post of mine was more beefy.[/quote]

Sorry I'm still confused. Did you mean the big list?

The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle. He proposed this principle as a means of refuting the arguments of sophists such as those from of Elea, (of which Zeno was one), who could use logic to produce absurd conclusions. — Metaphysician Undercover

I can't believe this thread is still going. I see you've hijacked it to your hobby horse.

For my edification, can you explain the above paragraph?

My understanding of your point is this, and do correct me if I'm wrong.

My concept of your thesis: The law of identity says that a thing is equal to itself. Mathematical equality is not metaphysical identity. Therefore math is wrong. Or something.

But nobody claims mathematical equality is identity. It may be spoken of that way in casual conversation, and by mathematicians who have not given the matter any thought and mean nothing by it.

When pressed, a mathematician would readily admit that mathematical equality is nothing more than a formal symbol defined within ZF set theory in the logical system of first order predicate logic. It's not actually the same as mathematical identity. It's not the same as anything.

If we like, we can visualize that equality is identity. Why not? Everything in our mathematical world is a set; if there are things that are identical but not equal as sets, they're entirely out our consciousness. So they often think of it that way. But they don't mean it as any kind of metaphysical thesis. It's just a manner of speaking, like jargon in any field.

Any logically or philosophically-oriented mathematician will immediately concede the point; and the rest would have no opinion at all. Most working mathematicians don't spend any times thinking about whether mathematical equality is logical identity. The question doesn't enter their minds.

So that's what I understand about your thesis, and I don't know anything about the Eleatics. What is your point with all this?

Discussion with you about this is pointless because you make statements like the one above, where you acknowledge the difference between the mathematical concept of "identity" and the ontological concept of "identity", but you claim that the only relevant concept of "identity" is the mathematical one. — Metaphysician Undercover

I'm just jumping into this convo between you and @TonesInDeepFreeze, and I can't speak for why he wrote what he did.

But I can tell you that I would agree with what he said, in the context of mathematics.

That is, if I'm a mathematician, all of the objects I deal with in my life are sets of one kind or another, and we know what equality for sets, we defined it via the axiom of extension.

You would be fun in set theory class. You're entirely hung up on the very first axiom. "Class, Axiom 1 is the axiom of extensionality. It tells us when two sets are equal." You, three years later: "But that's not metaphysical identity! You mathematicians are bad people. And you don't understand anything!" And your professor goes, Meta, We still have a countable infinitely of axioms to get through! Can we please move on?

But axioms don'g mean anything. They're just rules in a formal game, like chess. As I say, if you asked a mathematician if mathematical equality is metaphysical identity, a few of them would have an educated opinion about the matter and they'd immediately agree with you. The rest, the vast majority, wouldn't understand the question and would be annoyed that you interrupted them.

Of course, relevance depends on one's goals, and truth is clearly not one of yours. — Metaphysician Undercover

Such a civil one. It would be impolite even if you weren't also completely wrong.

In practise the math always refers to something. — Metaphysician Undercover

I found this in an old post of yours. It it exactly your misunderstanding.

Nobody claims that math refers. That's your straw man.

Now math can be useful. So we often play the game of (1) Assume math refers to this particular situation; and (2) Use math to improve your control of the situation, whatever it is.

Just because we constantly apply math as if it refers to the world; and just because it so often turns out to be really useful; does not mean that math itself refers to the world. That's the brilliant essence of math. Math refers to nothing; but is locally useful in almost everything.

You just don't get that. You're fighting a straw man of your own creation.

Clearly "identity" by the law of identity includes the order of a thing's elements, as it includes all aspect of the thing, even the unknown aspects. So the ordering of the thing's elements is therefore included in the thing's identity, unlike the supposed (fake) "identity" stated by the axiom of extensionality. — Metaphysician Undercover

I remember fondly when I spent weeks trying to explain order theory to you, back when I thought you were trying to understand anything. You are still at this. If two sets have the same elements and the same order, they are equal as ordered sets. It's just about layers of abstraction, separating out concepts. First you have things, then you place them in order.

Somehow this offends you. Why?

I have been thinking a lot recently about the idea of "perceived probability" — Igitur

Could you perhaps be thinking of the concept known as credence?

"Credence or degree of belief is a statistical term that expresses how much a person believes that a proposition is true. As an example, a reasonable person will believe with close to 50% credence that a fair coin will land on heads the next time it is flipped (minus the probability that the coin lands on its edge)."