Warning, Long-assed post ahead. Please tell me if I'm on target with your concerns.

I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though. — Ludwig V

The jokes illustrate the principle. The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved.

It's called a performative speech act. Do you know about them? — Ludwig V

That tingled the circuit in my memory bank. Searle's doctoral advisor Austin talks about speech acts, and I believe Searle does too. That is everything I know about it. Not really clear what it's about.

Very roughly, the saying of certain words is the doing. The classic example is promising. A particularly important - and complicated - variety of speech act is a definition. Particularly interesting cases are the definition of rules. (Well, definitions are always regarded as rules, but there are cases that are a bit tricky.) — Ludwig V

Well I'm not sure I see what those examples are driving at. Speech where the speech is also an act. So, "It's raining out," is not a speech act, because I haven't done anything, I've only described an existing state of affairs. But telling you how the knight moves in chess (example of a rule] is a speech act, because I've brought the chess knight into existence by stating the rule. Something like that?

The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. — Ludwig V

It's very simple. First, by "+1" do you mean Peano successors? You used this notation several times in what follows and I am not sure I know exactly what you mean.

In Peano arithmetic (PA), we generate all the natural numbers with two rules:

* 0 is a number; and

* If n is a number, then Sn is a number, where S is the successor function.

We can use these two rules to define names like 1 = S0 and 2 = SS0 and so forth, and then use the successor function to define "+" so that we can prove 2 + 3 = 5 and so forth.

There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...

In particular, there is no container or set that holds all of them at once. The best we can do is say that there are always enough of them to do any problem that comes up in PA.

That gives you one logical system, PA, that has a certain amount of expressive power. We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth.

In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set.

Next step up is set theory, for example ZF, that includes the axiom of infinity. The axiom of infinity actually defines what we mean by a successor function for sets; and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X.

This gives you something PA doesn't: A "container that holds all of 0, 1, 2, 3, ... at once, in fact not just a container, but a set, an object that satisfies all the other axioms of ZF.

We can then show that the axiom of infinity lets us construct a model of PA within ZF; and we take that model to be the natural numbers.

The tl;dr is this:

PA gives you each of 0, 1, 2, 3, ...

ZF with the axiom of infinity gives you {0, 1, 2, 3, ...}; that is, all the marbles AND a bag to put them in.

Hope that wasn't too much information, but it's the way to think of "potential" versus "completed" infinities, which are philosophical terms that don't really find use in math.

Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). — Ludwig V

Mathematical sequences and supertasks are two entirely different, but strongly related, ideas.

There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. Supertask discussions talk about time, which is a physical concept; but then examples like Thomson's lamp posit circuits that can change state in arbitrarily short intervals of time, which is a decidedly NON-physical idea. It's a fairy tail (under currently known physics). That's where much of the confusion comes in.

So I hope that you can separate out these two concepts. Are you asking about mathematical sequences, such as 1/2, 1/4, 1/8, ... that have the limit 0? That is a completely understood subject in math.

Or are you imagining that someone "speaks these fractions out loud" in their corresponding amount of time, thereby "saying them all in finite time?" This is a totally nebulous, made-up conceptual fairy tail that is the cause of much confused thinking among philosophers.

But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. — Ludwig V

This is actually not much of an objection. It is far too weak. We cannot write out all the terms of any sufficiently large FINITE sequence, either. You can't write out the numerals from 1 to googolplex in y you lifetime at one number per second. It would take longer than the age of the universe.

So you are not making any substantive objection.

In PA the numbers are conceptually created one at a time, but they're really not, because there is no time. 0 is a number and S0 is a number and SS0 is a number, "all at once." You can call that completion if you like.

In ZF, it's more clear. There is a set that contains 0, 1, 2, 3, ... You can give the set a name and you can work with it.

But either way, your concept of completion involves time; and as I've noticed, that involves CONFUSING mathematical sequences, about which we have perfect logical clarity; with supertasks, about which we have much pretentious confusion.

In addition to the rule, there is a distinct action - applying the rule. That is where, I think, all the difficulties about infinity arise. — Ludwig V

No, that is something you are bringing to the table, but that I don't think is correct. There's no distinct action of applying the rules.

In the PA incantation: 0 is a number and Sn is a number if n is; that creates all the numbers. There is no time involved. Time is a factor that you are letting confuse you.

We understand how to apply the rule in finite situations. But not in infinite situations. — Ludwig V

We understand how to apply successors perfectly well in the infinite situation. In fact the rule that "If n is a number, then Sn is a number," is an instance of induction, or its close relative, recursion. These things are perfectly well understood.

Think of applying "countable" or "limit" to "+1". The concept has to be refined for that context, which, we could say, was not covered (envisaged) for the original concept. — Ludwig V

You are making this up out of some level of confusion involving time. Time is not a consideration or thing in mathematics. All mathematics happens "right here and now."

I am trying, I don't know if I'm getting through or not, but I am trying to get you to separate out your naive notion of timeliness in mathematics, with mathematics. Time matters in physics and in supertask discussions. It's important to distinguish these related but different concepts in your mind.

(By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?) — Ludwig V

Good question. A bound and a limit are two different things. A couple of examples:

* Consider the set {1/2, 1/4, 1/8, ...}.

-43, that is negative 43, is a lower bound of the set. 0 is the "greatest lower bound," a concept of great importance in calculus.

* But here's a more interesting example. Consider the sequence 1/2, 100, 1/4, 100, 1/8, 100 ...

It has two limit points, 0 and 100. But it has no limit, because the formal definition of a limit is not satisfied. To be a limit the sequence has to not only GET close to its limit, but also STAY close.

0 and 100 would be the greatest lower bound and the least upper bound, respectively.

Now I know this was too much info!! This is just technical jargon in the math biz, don't worry about it two much. But bounds and limits are different concepts. Limits are more strict.

Oh, yes, I get it. I think. — Ludwig V

Maybe that bit about the order topology was a little too much. My only point is that there is a mathematical context in which omega as the limits of the natural numbers is the same as calculus limits. That's all I need to say about that.

Forgive me for my obstinacy, but let me try to explain why I keep going on about it. I regard it as an adapted and extended use of the concept in a new context. (But there are other ways of describing this situation which may be more appropriate.) — Ludwig V

This didn't parse, I don't know what you are referring to. What is "it" and "this situation." Nevermind I'll work with the rest of the text.

My difficulties arise from another use of the "1" when we define the converging sequence between 0 and 1. It seems that there must be a connection between the two uses and that this may mean that the sense of "limit" here is different from the sense of ω in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context. — Ludwig V

This is a little convoluted and confused. What converging sequence between 0 and 1? Say we have the sequence 1/2, 1/4, 1/8, ... for definiteness.

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs $\frac{1}{2^n}$. I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter.

Now in order to formalize where the limit 0 fits into the scheme of things, we can say that the limit is the value of that function at the point $\omega$ in the EXTENDED natural numbers

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

Those are NOT the natural numbers. I've stuck a conceptual "point at infinity" at the end. I hope this is not confusing you. Tell me what your concerns are.

I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42. — Ludwig V

The "termination state" is 42. 42 is not the limit of the sequence 0, 1, 0, 1, ... The word limit has a very technical meaning. It's clear that the sequence does not "get near and stay near" 42.

That's why for purposes of analyzing supertasks I am DEFINING the phrase "termination state" of a sequence to be a value "stuck at the end," but that is NOT NECESSARILY A LIMIT.

I hope this is clear. The termination point is arbitrary, it can be 42 or a pumpkin. But in no case are those values limits in the calculus sense.

So we say that all limited infinite sequences converge on their limits. — Ludwig V

Hmmm. "Limited" is not a term of art in this context. Given a sequence, it either converges to a limit or it doesn't. A convergent sequence of course converges to its limit, but this is a tautology that follows from the definition of convergence to a limit. A convergent sequence converges to its limit, but that doesn't really any anything we didn't already know.

Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit. — Ludwig V

Glad it makes sense, but the limit is NOT repeat NOT part of the sequence.

It's part of what I'm calling the extended sequence, with the limit or termination point stuck at the end. But that is my terminology that I am making up just for these supertask problems.

Hope that's clear.

When I write my semicolon notation: 1/2, 1/4, 1/8, ...; 0

that is a fishfry-defined extended sequence. The sequence is 1/2, 1/4, 1/8, ..., and the limit is 0.

I use this notation to describe the termination state of a supertask: on, off, on, off, ...; pumpkin

The sequence is the on/off part; the pumpkin is the termination state.

Hope this is getting clearer.

I have completist tendencies. I try to resist them, but often fail. — Ludwig V

I don't even know what that means :-) What are completist tendencies?

You would hate the rational numbers then. They are not complete. For example the sequence 1, 1.4, 1.41, 1.412, ... where each term is the next truncation of sqrt(2), does not have a completion in the rationals.

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals.

Well I wrote a lot, let me know if any of this was helpful and let me know what's still troubling you.

tl;dr to this entire post:

Mathematical sequences are clear and rigorous. We have a fully worked out theory of them.

Supertasks are nebulous and vague. Reason: There is no time in math. Time is a concept of physics. And Supertask problems always involve physical impossibilities, like flipping a lamp in arbitrarily small intervals of time. That's the source of all the confusion. Supertasks are fairy tales, like Cinderella's coach; and you can no more apply logic to a supertask problem than you can to the coach turning into a pumpkin.

This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him. — TonesInDeepFreeze

I traced back to your mention of the axiom of infinity, and I still fail to see the relevance of the remark in context. I apologize for lashing out regardless. "wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement.

"wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement. — fishfry

wut?

wut? axiom of infinity. what's wrong with you tonight? — fishfry

My response was to 'what's wrong with you tonight?', not so much to 'wut?'.

Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.

There was nothing wrong with what I posted that night. You just snapped-at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written.

Hey, I get your whole "Aw shucks, I'm just a scorpion who's gonna do what a scorpion's gonna do. I don't mean nothin' by it" routine. But it doesn't mean jack to me as far as feeling any less right in answering right back.

My response was to 'what's wrong with you tonight?', not so much to 'wut?'.

Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.

There was nothing wrong with what I posted that night. You just lashed out at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written. — TonesInDeepFreeze

Ah. The what is wrong with you and not the wut. I can see that now that you mention it.

I am terribly sorry to have offended you once again.

You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.

Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque.

@Michael

I've not gone back to review all that's been said in this thread, and I need to catch up to your replies, but starting again from the beginning of your argument.

I surmise that the reason you put your argument in numbered steps is so that it can be seen to be airtight.

Is your argument intended to be Thomson's argument?

You have mentioned different conclusions you draw:

(1) The conditions (the premises) of the lamp are inconsistent.

(2) Supertasks are impossible. (But can we infer from the impossibility of Thomson's lamp that all supertasks are impossible?)

(3) Time is not continuous. (I've suggested that what you actually seem to dispute is that time is not densely ordered (infinitely divisible), which is a stronger claim.)

(4) Benacerraf is wrong.

Here's Thomson's statement of the problem:

"There are certain reading-lamps that have a button in the
base. If the lamp is off and you press the button the lamp goes
on, and if the lamp is on and you press the button the lamp goes
off. So if the lamp was originally off, and you pressed the
button an odd number of times, the lamp is on, and if you
pressed the button an even number of times the lamp is off.
Suppose now that the lamp is off, and I succeed in pressing the
button an infinite number of times, perhaps making one jab
in one minute, another jab in the next half-minute, and so on,
according to Russell's recipe. After I have completed the whole
infinite sequence of jabs, i.e. at the end of the two minutes, is
the lamp on or off? It seems impossible to answer this question.
It cannot be on, because I did not ever turn it on without at
once turning it off. It cannot be off, because I did in the first
place turn it on, and thereafter I never turned it off without at
once turning it on. But the lamp must be either on or off. This
is a contradiction."

Here's your presentation:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00

From these we can then deduce:

C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on

From these we can then deduce:

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction] — Michael

I want to get back to looking at this more closely, but in the meantime, do you consider your presentation equivalent with Thomson's statement of the problem?

Zeno's paradox concerns analysis of an actual physical event. Thomson's lamp concerns analysis of a hypothetical state-of-affairs. One difference is that with Zeno's paradox, we read a conclusion that a certain fact is impossible, which is impossible. With Thomson's lamp, according to Thomson, there is a derivation of a contradiction; but that comes from non-factuals. Are there other crucial differences between Zeno and Thomson?

When possibility is part of the analysis, the analysis can get complicated. We should be careful that our inferences regarding the modalitiy are proper.

You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.

Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque. — TonesInDeepFreeze

Are you just committed to picking fights with me? I've apologized several times tonight, for sins real and imagined. And some cosines too. Enough bro'.

The issue here is that we really know very little about the nature of the passing of time. — Metaphysician Undercover

I won't argue with that. For some reason, I've never been able to get my philosophical head around that topic. Just like Augustine, all that time (!) ago.

Then the point which marks the limit, midnight or whatever never comes — Metaphysician Undercover

I was going to reply that slowing down isn't stopping. I didn't realize that the slowing down was a convergent series. Perhaps slowing down can be stopping.

I agree with this, but I'd describe it as how we apply mathematics to space and time. — Metaphysician Undercover

Well, we could if we wanted to do. But why would we want to? Apart from the fun of the paradox. Mind you, I have a peculiar view of paradoxes. I think of them as quirks in the system, which are perfectly real and which we have to navigate round, rather than resolve. Think of the paradoxes of self-reference. Never permanently settled. New variants cropping up.

I don't ask for apologies. But it's okay if you want to give them. But you embed into your apologies yet more items that I feel deserve response. Your apologies themselves are snarky; "sins imagined" e.g. I don't even object to snark, except it's your way of ostensibly apologizing while still turning it back on me.

If I misconstrue someone's math or philosophy points, especially to mischaracterize them, then if the person calls me on it or I discover it myself, before posting back to that person again, I should post my recognition of my mistake. That's my ethos. Yours might be different. But I will stick with my prerogative to reply when I like.

And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. In various thread, you have posted a lot of inaccuracies and misconceptions about math, and now lately about me. I respond to that.

And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. — TonesInDeepFreeze

Ok no more snark.

Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it.

I want to get back to looking at this more closely, but in the meantime, do you consider your presentation equivalent with Thomson's statement of the problem? — TonesInDeepFreeze

Yes.

When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. — fishfry

No I'm not. I accept that one of the premises of the thought experiment is physically impossible. That doesn't then mean that we cannot have another premise such as "there are no spontaneous, uncaused events".

You seem to think that because we allow for one physical impossibility then anything goes. That is not how thought experiments work.

It is physically impossible for me to push a button 10¹⁰⁰¹⁰⁰ times within one minute, but given the premises of the thought experiment it deductively follows that the lamp will be off after doing so. Your claim that the lamp can turn into a plate of spaghetti is incorrect.

I was going to reply that slowing down isn't stopping. I didn't realize that the slowing down was a convergent series. Perhaps slowing down can be stopping. — Ludwig V

In many cases of common language usage, "slowing down" is stopping, but that implies the end, not yet achieved. The point is that "stopping" is distinct from "stopped'. And if the slowing down never reaches the point of being stopped, then the term "stopping" is not justified. The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved.

In the modern physical world of relativity, "stopped" is arbitrarily assigned according to an inertial reference frame. This implies a sort of equilibrium, or stability within that specific reference frame, but it's highly unlikely that it is a true case of "stopped", more likely very slow movement, misrepresented as "stopped". We like to round things off.

Well, we could if we wanted to do. But why would we want to? Apart from the fun of the paradox. Mind you, I have a peculiar view of paradoxes. I think of them as quirks in the system, which are perfectly real and which we have to navigate round, rather than resolve. Think of the paradoxes of self-reference. Never permanently settled. New variants cropping up. — Ludwig V

Why would we want to? Because we are philosophers seeking knowledge. Understanding is the primary objective. I look at such paradoxes as indications of a lack of understanding. The principles applied do not adequately map to the reality which they are being applied, this is a failure of our knowledge. Then we need to subject all the principles to skeptical doubt, to determine the various problems. We could just live with quirks in the system, but that's unphilosophical. Knowledge evolves, and that evolution is caused by people attempting to work out the quirks in the system.

I missed these posts.

Perhaps I misunderstood. What then? — fishfry

The objects that constitute both Euclidean and non-Euclidean (the unending many of them) spaces are abstract and both exist. Those objects may be applied in our scientific theories because a description of these objects can also describe some phenomenons in the real world. The problem is how do we get knowledge of these objects, if they are not physical? That is Benecerraf's problem.

If both of these are true, then we need to be very careful about what we mean by "the world". There is an application that takes "the world" to exist in space and time. — Ludwig V

This is not one of those cases. The world here is meant by everything that is not created by the mind (realism X anti-realism), not just what is located in space-time (physicalism).

Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it. — TonesInDeepFreeze

I am so appreciative that you straightened me out on this extensionality thing that I can't argue with you about anything. I accept all your criticisms. You say I've done these things and I don't deny them. I make no defense nor explanation.

I do have a sarcasm gene and that rarely works online. You'd think I'd learn.

No I'm not. I accept that one of the premises of the thought experiment is physically impossible. That doesn't then mean that we cannot have another premise such as "there are no spontaneous, uncaused events".

You seem to think that because we allow for one physical impossibility then anything goes. That is not how thought experiments work.

It is physically impossible for me to push a button 10100100 times within one minute, but given the premises of the thought experiment it deductively follows that the lamp will be off after doing so. Your claim that the lamp can turn into a plate of spaghetti is incorrect. — Michael

I respectfully leave this conversation. We've said it all. i've enjoyed our chat.

The objects that constitute both Euclidean and non-Euclidean (the unending many of them) spaces are abstract and both exist. Those objects may be applied in our scientific theories because a description of these objects can also describe some phenomenons in the real world. The problem is how do we get knowledge of these objects, if they are not physical? That is Benecerraf's problem. — Lionino

Yes I see what you meant. Thanks.

I can't argue with you about anything — fishfry

Then you can't argue with me that you can argue with me.

Warning, Long-assed post ahead. Please tell me if I'm on target with your concerns. — fishfry

It's very helpful, so that's fine. I get my revenge in this post.

The system is not helping me here, because it invites me to link to specific comments, but I'll do my best to make clear what I'm responding to.

The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved. — fishfry

:grin:
Perhaps that's why philosophers keep tripping up on them. It is well known that they don't notice what's on the floor - too busy worrying about all the infinite staircases and the fall of man.

You would hate the rational numbers then. They are not complete. For example the sequence 1, 1.4, 1.41, 1.412, ... where each term is the next truncation of sqrt(2), does not have a completion in the rationals. — fishfry

That was not a very well thought out remark. I would certainly have hated them in the long-ago days when the Pythagoreans kept the facts secret so that they could sort it out before everyone's faith in mathematics was blown apart. But now that mathematicians have slapped a label on these numbers and proved that they cannot be completed, I'm perfectly happy with them.

That tingled the circuit in my memory bank. Searle's doctoral advisor Austin talks about speech acts, and I believe Searle does too. That is everything I know about it. Not really clear what it's about. — fishfry

Yes. Austin invented them, Grice took them up, Searle was the most prominent exponent for a long time, although he has now moved on to other things now. It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic.

Well I'm not sure I see what those examples are driving at. Speech where the speech is also an act. So, "It's raining out," is not a speech act, because I haven't done anything, I've only described an existing state of affairs. But telling you how the knight moves in chess (example of a rule] is a speech act, because I've brought the chess knight into existence by stating the rule. Something like that? — fishfry

Something like that. The initial point was to establish that there are perfectly meaningful uses of language that are not propositions (i.e. capable of being true or false), in the context of Logical Positivism. I doubt that you would welcome a lot of detail, but that idea (especially the case of the knight in chess) will be at the bottom of some of the later stuff.

Hope that wasn't too much information, but it's the way to think of "potential" versus "completed" infinities, which are philosophical terms that don't really find use in math. — fishfry

It was very helpful to me. I have doubts about the terminology "potential" vs "completed", but the idea is fine. I particularly liked "don't really find a use in math".

Now I know this was too much info!! This is just technical jargon in the math biz, don't worry about it two much. But bounds and limits are different concepts. Limits are more strict. — fishfry

Too much or not. It helped me. Someone else started talking about bounds and I couldn't understand it at all. I may not understand perfectly, but I think I understand enough.

Glad it makes sense, but the limit is NOT repeat NOT part of the sequence. — fishfry

I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.

Now in order to formalize where the limit 0 fits into the scheme of things, we can say that the limit is the value of that function at the point ω in the EXTENDED natural numbers — fishfry

... because "1/2, 1/4, 1/8, .." gets near and stays near 0. Yes?

The "termination state" is 42. 42 is not the limit of the sequence 0, 1, 0, 1, ... The word limit has a very technical meaning. It's clear that the sequence does not "get near and stay near" 42. — fishfry

I understand that distinction.

There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. — fishfry

I am trying, I don't know if I'm getting through or not, but I am trying to get you to separate out your naive notion of timeliness in mathematics, with mathematics. Time matters in physics and in supertask discussions. It's important to distinguish these related but different concepts in your mind. — fishfry

Many of my notions are naive or mistaken. But this separation is my default position. I'm not making an objection, but am trying to point out what may be a puzzle, which you may be able to resolve. On the other hand, this may not be a mathematical problem at all.

Time is not a consideration or thing in mathematics. All mathematics happens "right here and now." — fishfry

There are other ways of putting the point. What about "Mathematics is always already true"? Or mathematics is outside time? Or time is inapplicable to mathematics?

But your example of making a rule in chess. Note that as soon as the rules are made, we can starting defining possibilities in chess, or calculating the number of possible games and so forth. It's as if a whole structure springs into being as we utter the words. So a timeless structure is created by our action, which takes place in time. Isn't that at least somewhat like a definition in mathematics? And the definition is an action that takes place in space and time.

More difficult are various commonplace ways of talking about mathematics.

In PA the numbers are conceptually created one at a time, but they're really not, because there is no time. 0 is a number and S0 is a number and SS0 is a number, "all at once." You can call that completion if you like. — fishfry

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals. — fishfry

It's clear that the sequence does not "get near and stay near" 42. — fishfry

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs 1/(2 to the power of n). I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter. — fishfry

If n is a number, then Sn is a number, where S is the successor function. — fishfry

At first sight, these seem to presuppose time (and even, perhaps space) Perhaps they are all metaphors and there are different ways of expressing them that are not metaphorical. Is that the case? I recognize that I may be talking nonsense.

Right, except for the kinds of realism that make it about the physical world, but that is one type among many. — Lionino

This is not one of those cases. The world here is meant by everything that is not created by the mind (realism X anti-realism), not just what is located in space-time (physicalism). — Lionino

So when you use the appropriate sense of the "world", and say that realism is true of the world, you are saying that realism is true of some parts of the world - the abstract parts?
I strongly approve of defining the context in which one is using "real" or "realism", but using it of the world, defined as everything that exists independently of the mind, you are simply re-asserting the basic thesis that both geometries are true independently of the mind. Since they are both true in the abstract world, but not simultaneously in the physical world, would it not be helpful to add that explanation?

The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved. — Metaphysician Undercover

Yes, I agree with that. I was suggesting that a slowing down according to a convergent series might count as stopped, since it would never reach the limit or "0".

We like to round things off. — Metaphysician Undercover

If you are right about relativity, I wouldn't disagree.

I was suggesting that a slowing down according to a convergent series might count as stopped, since it would never reach the limit or "0". — Ludwig V

Some time ago I mentioned time dilation in relativity theory in this regard.

Then you can't argue with me that you can argue with me. — TonesInDeepFreeze

Correct. Which is why I acknowledged your complaints and said nothing else. If I did, you'd complain that I was minimizing my apology by contextualizing it, either with snark or denial.

So I didn't even apologize. I acknowledge your complaints and I stand mute. I have nothing to say at all.

It's very helpful, so that's fine. I get my revenge in this post. — Ludwig V

Glad to know. Revenge? What do you mean? By writing a long post? Well I write long posts but prefer when others write shorter ones. I haven't solved this dilemma yet.

The system is not helping me here, because it invites me to link to specific comments, but I'll do my best to make clear what I'm responding to. — Ludwig V

Not sure what you mean. I generally quote the whole post then stick in quote tags around the specific chunks of text I want to respond do.

Perhaps that's why philosophers keep tripping up on them. It is well known that they don't notice what's on the floor - too busy worrying about all the infinite staircases and the fall of man. — Ludwig V

I don't know many philosopher jokes.

That was not a very well thought out remark. I would certainly have hated them in the long-ago days when the Pythagoreans kept the facts secret so that they could sort it out before everyone's faith in mathematics was blown apart. But now that mathematicians have slapped a label on these numbers and proved that they cannot be completed, I'm perfectly happy with them. — Ludwig V

Sorry maybe I was off track about the rationals.

Yes. Austin invented them, Grice took them up, Searle was the most prominent exponent for a long time, although he has now moved on to other things now. — Ludwig V

Yes he got in trouble for harassing his female doctoral students.

It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic. — Ludwig V

Ok. Why did you bring it up relative to math? Oh I remember. "Let x = 3" brings a variable x into existence, with the value 3. So statements in math are speech acts, in the sense that they bring other mathematical objects into existence. I can see that.

Something like that. The initial point was to establish that there are perfectly meaningful uses of language that are not propositions (i.e. capable of being true or false), in the context of Logical Positivism. I doubt that you would welcome a lot of detail, but that idea (especially the case of the knight in chess) will be at the bottom of some of the later stuff. — Ludwig V

Ok. Not entirely sure where you're going.

It was very helpful to me. I have doubts about the terminology "potential" vs "completed", but the idea is fine. I particularly liked "don't really find a use in math". — Ludwig V

Ok I was only trying to be philosophical. Aristotle (I think) made the distinction. It doesn't come up in math, nobody ever uses the terminology. But the way I understand it is that Peano arithmetic is potential and the axiom of infinity gives you a completed infinity.

Too much or not. It helped me. Someone else started talking about bounds and I couldn't understand it at all. I may not understand perfectly, but I think I understand enough. — Ludwig V

Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails.

I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.
— Ludwig V

I understand that distinction. — Ludwig V

That was about limits versus "termination state." I should emphasize that limits are perfectly standard mathematical terminology. But "termination state" is my own locution for purposes of talking about supertasks. The termination state is like a limit in the sense that we can conceptually "stick it at the end" of an infinite sequence; it just doesn't have to satisfy the definition of the limit of a sequence. Like 1/2,/ 3/4, 7/8, ...; 42

The semicolon notation is my own too. I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing.

Many of my notions are naive or mistaken. But this separation is my default position. I'm not making an objection, but am trying to point out what may be a puzzle, which you may be able to resolve. On the other hand, this may not be a mathematical problem at all. — Ludwig V

I am not aware of what problem or puzzle you are expressing.

There are other ways of putting the point. What about "Mathematics is always already true"? Or mathematics is outside time? Or time is inapplicable to mathematics? — Ludwig V

The subject matter of mathematics does not speak about time. That's different than saying "math is outside of time," although it's kind of related. Physics talks about time, and physicists use math to model time, but that is a very different thing.

It's the difference between a loop in math versus programming.

In math when we say that 1/2 + 1/4 + 1/8 + ... = 1, we mean "right now," though even that is a reference to timeliness. The equality "just is."

But in a programming language when we write a loop that keeps adding each term to a running total, that notation stands for a physical process that takes place in a computing device and requires time and energy to execute, and produces heat. A programming loop is a notation for a physical process.

But your example of making a rule in chess. Note that as soon as the rules are made, we can starting defining possibilities in chess, or calculating the number of possible games and so forth. It's as if a whole structure springs into being as we utter the words. So a timeless structure is created by our action, which takes place in time. Isn't that at least somewhat like a definition in mathematics? And the definition is an action that takes place in space and time. — Ludwig V

Ok. So when I write down the rules of set theory, I instantiate or create all the complex world of sets as studied by set theorists. And you speculate that this might be an event that takes place in time.

There is another point of view. The structures of the sets were there. Mathematicians discovered the structures. So the discovery of set theory is historically contingent and takes place in time, around 1874 or so with Cantor's first paper on set theory. But the sets themselves, the structures of set theory, are eternal!

In other words this is the old "invented or discovered" question of mathematical philosophy.

Now chess, I think we can agree, was invented and not discovered. But math is somehow different. Math is somehow wired into the logic centers of our minds, and perhaps the universe.

More difficult are various commonplace ways of talking about mathematics. — Ludwig V

Are you referring to what I just talked about?

At first sight, these seem to presuppose time (and even, perhaps space) — Ludwig V

I cannot fathom what you might mean. A sequence does not approach its limit in time. The limit of 1/2, 1/4, 1/8, ...is 0 right now and for all eternity. The fact is inherent in the axioms of set theory, along with the usual constructions and definitions of the real numbers and calculus. In that sense the fact "came into existence" when Newton thought about it, or maybe when Cauchy formalized it, and so forth.

But the history of our understanding of the fact is not the same as the fact itself. The earth went around the sun even before Copernicus had that clever idea. Likewise every convergent sequence always converged to its limit, independently of our discovery of those limits, and our understanding of what a limit is.

Is this your point of contention or concern? That you think that time is hiding in there somewhere? I profoundly disagree. You greatly misunderstand mathematics; or you have an interesting and original philosophy of mathematics; if you believe there's time hiding inside mathematics.

Perhaps they are all metaphors and there are different ways of expressing them that are not metaphorical. Is that the case? I recognize that I may be talking nonsense. — Ludwig V

If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.

Have I got that right?

It was merely a math quip.

Some time ago I mentioned time dilation in relativity theory in this regard. — jgill

I either skimmed past it or forgot it. Sorry. Not having been trained for it, I wouldn't want to comment on it. But it is that left field plausibility that I always appreciate.

Revenge? What do you mean? By writing a long post? — fishfry

Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing.

Not sure what you mean. I generally quote the whole post then stick in quote tags around the specific chunks of text I want to respond do. — fishfry

That's a useful tactic. I shall use it in future.

Yes he got in trouble for harassing his female doctoral students. — fishfry

He did indeed. It was very common back in the day. It was disapproved of by many, but not treated as unacceptable. I don't think anyone can really understand how horrible it is unless they've actually experienced it.

Ok. Why did you bring it up relative to math? Oh I remember. "Let x = 3" brings a variable x into existence, with the value 3. So statements in math are speech acts, in the sense that they bring other mathematical objects into existence. I can see that. — fishfry

Exactly. There's a lot of refinement needed. But that's the basic idea. What those objects are is defined entirely by their use in mathematics.

Ok I was only trying to be philosophical. Aristotle (I think) made the distinction. It doesn't come up in math, nobody ever uses the terminology. But the way I understand it is that Peano arithmetic is potential and the axiom of infinity gives you a completed infinity. — fishfry

I was just being pedantic. It was a thing in the era before Descartes &c. But I understood that the distinction was "potential" and "actual". Nonetheless, the idea of a "completed" infinity catches something important.

Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails. — fishfry

That's a very helpful metaphor.

If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.
Have I got that right? — fishfry

Yes.

I cannot fathom what you might mean. — fishfry

Nor can I. That's the problem.

The subject matter of mathematics does not speak about time. — fishfry

That's the starting-point.

A sequence does not approach its limit in time. — fishfry

Why is this a problem? The traditional view is that mathematics, as timeless, cannot change. Our knowledge of it can, but not the subject matter. (Strictly that rules out creating any mathematical objects as well, but let's skate over that.) "A sequence does not approach its limit in time" makes no sense.

I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful.

We have to accept that a sequence approaching its limit is not like a train approaching a station. The train is approaching in space and time. But you can't ask what time the sequence left its origin and when it will arrive at its limit.
You can call the sequence approaching its limit a metaphor or an extended use. The train approaching the station is the "core" or "paradigm" or "literal" use. The sequence approaching its limit is a different context, which, on the case of it, makes no sense. So we call this use is extended or metaphorical.
We can explain the metaphor by drawing a graph or writing down some numbers and pointing out that the different between n and the limit is less than the difference between n+1 and the limit is less and that the difference between n and n-1 is greater.
And so on.

I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing. — fishfry

Yes. I realize this is border country. Godel seems to live there too.

@fishfry: Probably some of these points you already know ; I'm mentioning them just to fill out the picture.

In Peano arithmetic (PA), we generate all the natural numbers with two rules:

* 0 is a number; and

* If n is a number, then Sn is a number, where S is the successor function. — fishfry

If PA here is first order, then PA does not have a predicate 'is a number' nor those axioms.

use the successor function to define "+" — fishfry

Just to be clear, that occurs in set theory, not in PA. In PA, '+' is not defined. It is primitive.

There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...In particular, there is no container or set that holds all of them at once. — fishfry

Of course, that's correct regarding PA.

We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth. — fishfry

Right.

In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set. — fishfry

Right.

The axiom of infinity actually defines what we mean by a successor function for sets — fishfry

The axiom of infinity does not define anything, including the successor operation.

The successor operation only requires pairing and union:

Df. the successor of x = xu{x}.

That is logically prior to the axiom of infinity. Then the axiom of infinity only says that there is a set that has 0 and is closed under successor.

Then we prove that there is a unique set that is a subset of all sets that have 0 and are closed under successor.

Then we define w = the set that is a subset of all sets that have 0 and are closed under successor.

and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X. — fishfry

Not "and". All it says is what you said after the "and": "there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X".

lets us construct a model of PA within ZF; and we take that model to be the natural numbers. — fishfry

Rather than "the model" I would say "the standard model". There are other models too. And models not isomorphic with the standard model.

PA gives you each of 0, 1, 2, 3, ... — fishfry

the axiom of infinity gives you {0, 1, 2, 3, ...} — fishfry

Both are right, and well said. In both PA and Z without infinity (even in Z with the axiom of infinity replaced by the negation of the axiom of infinity), we can define each number natural number, and in Z we can prove the existence of the set of all and only the natural numbers.