I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though. — Ludwig V
It's called a performative speech act. Do you know about them? — Ludwig V
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
The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. — Ludwig V
Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). — Ludwig V
But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. — Ludwig V
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
We understand how to apply the rule in finite situations. But not in infinite situations. — Ludwig V
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
(By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?) — Ludwig V
Oh, yes, I get it. I think. — Ludwig V
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
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
I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42. — Ludwig V
So we say that all limited infinite sequences converge on their limits. — Ludwig V
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
I have completist tendencies. I try to resist them, but often fail. — Ludwig V
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
"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? 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 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
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
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
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.The issue here is that we really know very little about the nature of the passing of time. — 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.Then the point which marks the limit, midnight or whatever never comes — 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 agree with this, but I'd describe it as how we apply mathematics to space and time. — Metaphysician Undercover
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
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
When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. — fishfry
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
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
Perhaps I misunderstood. What then? — fishfry
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
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
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
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
I can't argue with you about anything — fishfry
It's very helpful, so that's fine. I get my revenge in this post.Warning, Long-assed post ahead. Please tell me if I'm on target with your concerns. — fishfry
:grin: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
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.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
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.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
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.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
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".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
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.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
I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.Glad it makes sense, but the limit is NOT repeat NOT part of the sequence. — fishfry
... because "1/2, 1/4, 1/8, .." gets near and stays near 0. Yes?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
I understand that distinction.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
There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. — 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.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
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?Time is not a consideration or thing in mathematics. All mathematics happens "right here and now." — fishfry
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
Right, except for the kinds of realism that make it about the physical world, but that is one type among many. — 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?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
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".The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved. — Metaphysician Undercover
If you are right about relativity, I wouldn't disagree.We like to round things off. — Metaphysician Undercover
Then you can't argue with me that you can argue with me. — TonesInDeepFreeze
It's very helpful, so that's fine. I get my revenge in this post. — Ludwig V
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
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
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
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
It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic. — Ludwig V
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
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
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
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
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
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
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
More difficult are various commonplace ways of talking about mathematics. — Ludwig V
At first sight, these seem to presuppose time (and even, perhaps space) — Ludwig V
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
Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing.Revenge? What do you mean? By writing a long post? — fishfry
That's a useful tactic. I shall use it in future.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
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.Yes he got in trouble for harassing his female doctoral students. — 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. 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
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 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
That's a very helpful metaphor.Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails. — fishfry
Yes.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
Nor can I. That's the problem.I cannot fathom what you might mean. — fishfry
That's the starting-point.The subject matter of mathematics does not speak about 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.A sequence does not approach its limit in time. — fishfry
Yes. I realize this is border country. Godel seems to live there too.I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing. — fishfry
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
use the successor function to define "+" — fishfry
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
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
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
The axiom of infinity actually defines what we mean by a successor function for sets — fishfry
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
lets us construct a model of PA within ZF; and we take that model to be the natural numbers. — fishfry
PA gives you each of 0, 1, 2, 3, ... — fishfry
the axiom of infinity gives you {0, 1, 2, 3, ...} — fishfry
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