Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way.Sorry about that. I typically select the entire post and hit Quote, and it seems to lose a lot of the attribution. — fishfry
Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake.The limit is not part of the sequence. so that doesn't run the sequence backward. I am not sure what point you are making about the sequence. The dots merely indicate that the sequence progresses indefinitely. — fishfry
Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes."If you know what you're doing you're not learning anything." Think I read that somewhere. — fishfry
If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.)I'm out of my depth on that. Don't understand what's meant by realism or anti-realism. Simply don't believe that 2 + 2 = 4 has a truth value before some intelligent entity shows up to pass judgment. — fishfry
I'm glad it made sense.Ok. Don't think I disagreed with anything you said. — fishfry
I followed your link and found this quotation from Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics. I've put the passages of interest in bold and italicized the passage quoted from Thompson for clarity.I see no contradiction in Thompson's lamp, only a failure to define the terminal state.
— fishfry
See here. — Michael
Thomson's first argument, concerning the lamp, is short, imaginative, and compelling. It appears to demonstrate that "completing a super-task" is a self-contradictory concept. Let me reproduce it here:
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. ... 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 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.
contradictsIt 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.
That seems to be true.But the lamp must be either on or off.
That also seems to be true. The three sentences in bold in the first passage are not individually self-contradictory, but the conjunction of the three (the concept of a supertask) could be described as self-contradictory. Nor are Benacerraf's A or B self-contradictory. They could be both true, if a third state that is neither on nor off were possible. Perhaps Benacerraf was assuming that there isn't.Rarely are we presented with an argument so neat and convincing. This one has only one flaw. It is invalid. Let us see why. Consider the following two descriptions:
A. Aladdin starts at t0 and performs the super-task in question just as Thomson does. Let t1 be the first instant after he has completed the whole infinite sequence of jabs – the instant about which Thomson asks "Is the lamp on or off?" – and let the lamp be on at t1.
B. Bernard starts at t0 and performs the super-task in question (on another lamp) just as Aladdin does, and let Bernard's lamp be off at t1.
I submit that neither description is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).
That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light. — Ludwig V
There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why.Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent. — Metaphysician Undercover
I agree that this isn't really about anything empirical, but it sort of seems to be.Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear". — Metaphysician Undercover
Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent — Metaphysician Undercover
There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why. — Ludwig V
I agree that this isn't really about anything empirical, but it sort of seems to be. — Ludwig V
In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. — jgill
It's a bit more complicated than that. Bulbs like fluorescent ones flicker, but the light really is constant. It's like what is called "motion illusion" or the φ phenomenon. Film and television both rely on it. In a sense, the motion is an illusion, but in another sense, it isn't. The illusion of constant light, paradoxically, is real.In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. — jgill
Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems.The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent. — Metaphysician Undercover
I don't have a problem with ideal principles. They are very useful. We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical.This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. — Metaphysician Undercover
I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. But even that depends on the level of description you are applying, or, if you prefer, the level of analysis you are using.In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility. — Metaphysician Undercover
You may be right. I'm afraid that I'm like Augustine. I don't know what time is, though I do know, of course, what time it is right now and what time I woke up.Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing. — Metaphysician Undercover
Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems. — Ludwig V
We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical. — Ludwig V
I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. — Ludwig V
The colour of something isn't divisible at all. — Ludwig V
.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way.There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible. — Metaphysician Undercover
That is the best representation of colour that physics can manage. But most people do not know about wave-lengths or Fourier transforms. So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light.It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform. — Metaphysician Undercover
So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring?No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. — Metaphysician Undercover
Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way. — Ludwig V
Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake. — Ludwig V
"If you know what you're doing you're not learning anything." Think I read that somewhere.
— fishfry
Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes. — Ludwig V
If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. — Ludwig V
Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.) — Ludwig V
Ok. Don't think I disagreed with anything you said.
— fishfry
I'm glad it made sense. — Ludwig V
P1. If we can recite the natural numbers at successively halved intervals of time then we can recite every natural number in finite time
P2. We cannot recite every natural number in finite time
C1. Therefore, we cannot recite the natural numbers at successively halved intervals of time — Michael
I see no contradiction in Thompson's lamp, only a failure to define the terminal state.
PS Since I started writing this, the link to the post that I copied this quotation from seems to have become non-functional. Very odd. — Ludwig V
Can you clarify which sense you mean? — fishfry
When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number. — fishfry
I still can't find it. I copied the quoted passage into my message, but not the commentary. Which is a pity.Was this from you to me? That post of Michael disappeared for me as well. — fishfry
There's another strictly philosophical issue. I know that metaphysics overlaps with logic. I'm still trying to work out whether it is identical with logic.c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time — Michael
suggests to me that it is a question of logic.Supertasks cannot be performed in any possible world. — Michael
↪Michael
OK. You and fishfry both believe that the supertask is impossible. But you believe that is because it is contradictory and fishfry believes that it is because the last step is not defined. Am I right about that? — Ludwig V
.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way. — Ludwig V
So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light. — Ludwig V
So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring? — Ludwig V
An afterthought. Do I understand rightly that your analysis of wholes and parts applies to physical objects, and not to mathematical ones? — Ludwig V
Can you clarify which sense you mean?
— fishfry
Metaphysical impossibility. Supertasks cannot be performed in any possible world. P3 is a tautology, P2 follows from P3, and so C1 is necessarily true. — Michael
Here are three distinct propositions:
a) 1/2 + 1/4 + 1/8 + ... = 1
b) there is a bijection between this geometric series and the natural numbers
c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time
(a) and (b) are true and (c) is false. — Michael
Your argument rests on the assumption that (c) follows from (a) and (b), but it doesn't. — Michael
(c) is proven false by P3, as well as arguments like Thomson's lamp. — Michael
You can continually assert that (a) and (b) are true, and I will continually agree, but until you can present actual evidence or reasoning to support (c), I will always reject it as per the above. — Michael
I expect we'll survive.Lost in the ether, forever. — fishfry
Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail.I have not said that. I have said that I have no strong opinion about supertasks and am entirely comfortable arguing either side. — fishfry
I would be very grateful if you could help me clarify this. When you say:-One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series. — Ludwig V
That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence?When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number. — fishfry
Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real.Ok. Possible worlds. I actually took a class where we talked about that, but I have a hard time understanding the concept. There are people who think possible worlds are real. I'm not one of them. And the whole metaphor is lost on me. — fishfry
Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.And even then I reject the claim on its own merits. I could argue (not that I do, but that I could -- hope that's clear) that if time is modeled by the real numbers (agreed, that is a dubious assumption) then I perform a supertask every time I get up to go to the kitchen for a snack. I named my refrigerator Zeno. — fishfry
It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery.You could probably help me out by clearly defining metaphysically impossible. — fishfry
I simply do not understand why you jump to saying that means it's metaphysically impossible. — fishfry
You could probably help me out by clearly defining metaphysically impossible. — fishfry
Quite so. And we know that it is an approximation because we know what more and less accurate or precise measurement is. The exact measure, in the physical world, is the limit that empirical measurements can approach and never reach. That's mathematics and logic.Again, I do not follow. Metres can be divided. We have centimetres and millimetres. But when we measure, at some point an approximation is made, a rounding off. — Metaphysician Undercover
I'll take that. I wouldn't put it the same way, but it's near enough. I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. But that's not my problem.I don't think that this is relevant. I believe the analysis applies to all objects. But there is a problem with supposed "mathematical objects", and this is that we assume them to be infinitely divisible. And this assumption creates incoherency. This incoherency renders the supposed objects as not true objects. — Metaphysician Undercover
Your beer will never be finished. — Ludwig V
I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. — Ludwig V
At your local metaphysical beer shop, of course. I'm sure Google knows its address and will give you directions. (Shops never stock both metaphysical and mathematical beers at the same time. They fight, you know - very messy!)Where do i get one of these metaphysical beers? — Metaphysician Undercover
Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem?I clearly explained though, it isn't "infinite" which is incoherent, it is "infinite divisibility" which is. "Infinite divisibility" is a specific application of the term "infinite" which is incoherent. It is incoherent because the concept of "infinite" is incompatible with, inconsistent with, or contradicts, what is implied by the concept "divisible". Therefore the two together as "infinite divisibility" is self-contradicting. — Metaphysician Undercover
I would be happy to accept that there are two concepts of infinity here. I think that may be because their concept has its roots in mathematics, whereas the metaphysical concept has roots elsewhere..Mathematicians have made "infinite" into a new term, which really has very little resemblance to its metaphysical roots. — Metaphysician Undercover
So we just have a case of Domains of Magisterial Authority, and no need to fight about it.This leaves mathematics, and mathematicians in general, as fundamentally incapable of dealing with the metaphysical problems involved with the concept "infinite". — Metaphysician Undercover
I think it would be better to stick with the strong claim. At least it is more comprehensible.But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim. — Michael
Yes Kripke does claim that. But he waters down the meaning of "necessarily". For him, it no longer means "in all possible worlds", but "in all possible worlds in which certain conditions hold". But contingent means, or used to mean, "true or false depending on certain conditions". So, on this account "necessarily" means what "contingent" used to mean. Talk about having your cake and eating it!Metaphysical impossibilities are things which are necessarily false; e.g. see Kripke's Naming and Necessity in which he argues that "water is H2O" is necessarily true even though not a priori (i.e. logically necessary). — Michael
Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem? — Ludwig V
So we just have a case of Domains of Magisterial Authority, and no need to fight about it. — Ludwig V
Our only remaining issue is whether the problem of Achilles and the tortoise and Thompson's lamp is a mathematical problem or a metaphysical problem. — Ludwig V
Or maybe it's just a question of understanding two solutions to the same problem. They clearly won't be incompatible. — Ludwig V
Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail. — Ludwig V
One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series. — Ludwig V
I would be very grateful if you could help me clarify this. When you say:-
When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number.
— fishfry
That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence? — Ludwig V
Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real. — Ludwig V
Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.
There are infinite ways in which I can mark out the plank, and they are all true at the same time and the physical object that is the plank is unaffected by any of them. True, the marks will be physical objects, so there will be limits to what I can do. But the system allows me infinite possibilities, including a convergent series. None of these makes the slightest difference to the plank. So when you visit Zeno for a beer, the fact that there are infinitely many analyses of your journey does not make the slightest difference. It's all in your head. — Ludwig V
(Here's a thought. When you drink your beer, you have to drink 1/2 of it and then 1/4 of it and then... Your beer will never be finished. :smile: But then, a similar argument would show that you can't even start drinking it. :sad: ) — Ludwig V
You could probably help me out by clearly defining metaphysically impossible.
— fishfry
It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery. — Ludwig V
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