You may say that it is possible to divided indefinitely, but that does not mean that infinite divisibility is possible. Take pi for example. You can get a computer to produce the decimal extension for pi, "indefinitely", but you never succeed in reaching an infinite extension. Divisibility is the very same principle. Some mathematical principles allow one to divide indefinitely, but you never reach infinite division. That is because infinite division, therefore infinite divisibility, is logically impossible. — Metaphysician Undercover
People have, I think, confused saying (1) it is conceivable that each of an infinity of tasks be possible (practically possible) of performance, with saying (2) that is conceivable that all of an infinite number of tasks should have been performed. They have supposed that (1) entails (2). And my reason for thinking that people have supposed this is as follows. To suppose that (1) entails (2) is of course to suppose that anyone who denies thinking (2) is committed to denying (1). Now to deny (1) is to be committed to holding, what is quite absurd, (3) that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed. But no-one would hold (3) to be true unless he had confused logical possibility with physical possibility. And we do find that those who wish to assert (2) are constantly accusing their opponents of just this confusion. They seem to think that all they have to do to render (2) plausible is to clear away any confusions that prevent people from accepting (1).
var isLampOn = false function pushButton() { isLampOn = !isLampOn } var i = 120 while (true) { wait i *= 0.5 pushButton() } echo isLampOn
Infinity minus one equals infinity
Would the above qualify as a paradox, or just be silly in "the" non abstract and possible realm but fit into the abstract and possible realm? Or the reverse? — kazan
wait(i *= 0.5)
let isLampOn = false
console.log(isLampOn)
wait(i *= 0.5)
division by 2 ... results in even number of total divisions. — SpaceDweller
Under assumption that those 2 minutes must pass the lamp will therefore be off with console.log(isLampOn) — SpaceDweller
var isLampOn = false function pushButton() { isLampOn = !isLampOn } while (true) { pushButton() } echo isLampOn
I don't know what you mean by this. — Michael
The wait(i *= 0.5) simply means that pushButton is called at successively halved intervals of time, i.e. it is called for the first time after 60 seconds, for the second time after a further 30 seconds, for the third time after a further 15 seconds, and so on. — Michael
Given what while (true) { ... } means, it is logically impossible for console.log(isLampOn) to ever run. — Michael
I have no problem acknowledging that 2+2=4. I have a problem with people who claim that "2+2" symbolizes the same thing that "4" does. — Metaphysician Undercover
And so, I refused to accept your claim to have proven that "2+2" signifies the very same thing as "4" does. — Metaphysician Undercover
Simply put, if the right side of an equation does not signify something distinct from the left side, mathematics would be completely useless. — Metaphysician Undercover
You can say that I have a problem with formalism, — Metaphysician Undercover
because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. — Metaphysician Undercover
That is, to remove all content from a logical application, to have a logical system which is purely formal. If such a thing was possible we'd have a logical system which is absolutely useless, applicable to nothing whatsoever. — My vetaphysician Undercover
Attempts at formalism end up disguising content as form, — Metaphysician Undercover
producing a smoke and mirrors system of sophistry, — Metaphysician Undercover
which is riddled with errors, due to the inherent unintelligibility of the content, which then permeates through the entire system, undetected because its existence is denied. — Metaphysician Undercover
You wanted something new so see the above. — Michael
What is wrong with that? — Ludwig V
2 + 2 and 4 symbolize the same set. You are the one strawmanning the claim that somebody says they're the same thing. — fishfry
If math is a flagrant fiction, why's it so darn useful? — fishfry
My point has always been that "same" in this context is not consistent with "same" in the context of the law of identity. So, to say " 2+ 2 and 4 symbolize the same set" is to use "same in a way which is in violation of the law of identity. — Metaphysician Undercover
Whether we are talking about "same thing", "same set", "same number", or "same kick in the ass", is irrelevant. The point is that this specific use of "same" is very clearly in violation of the law of identity. — Metaphysician Undercover
If the law of identity indicates that only a thing can be said to be "the same", and you do not believe that a set is a thing, and you want to say that a set is the same, then I suggest that you do not agree with the way that "same' is used by the law of identity. Is this the case? Do you believe that mathematicians have a better definition of "same"? — Metaphysician Undercover
If math is a flagrant fiction, why's it so darn useful?
— fishfry
In case you have never noticed, fiction is extremely useful. I suggest you begin with a look at the obvious, deception. Deception demonstrates that fiction is very useful in convincing others, to help us get what we want from them. And, so is mathematics. — Metaphysician Undercover
BUT!!!!!! You have not defined the terminal state. So why do you think there should be a sensible answer for what it is?
I don't see how your expressing the problem in pseudocode adds anything. We all have agreed to it long ago, even before you wrote it down. That's the premise of the problem. But the question is about the terminal state, which is not defined. — fishfry
After all, there is no number that can serve as the limit of the sequence 0, 1, 0, 1, ... — fishfry
We need not use the word same if it bothers you. — fishfry
Then why are you disagreeing with me? — fishfry
And if you say it's a useful deception, that's fine. — fishfry
May I ask, is chess similarly a useful deception? Language? You didn't respond to my point earlier that language is also a formal symbology that attempts to capture, however imperfectly. some aspect of abstract thought. — fishfry
We need not use the word same if it bothers you.
— fishfry
Great, I prefer the word "equal". It's better suited for that purpose. — Metaphysician Undercover
"Equal" generally allows that the two things which are said to be equal are not necessarily the same. — Metaphysician Undercover
"Same" is defined by the law of identity as indicating one thing only. — Metaphysician Undercover
That is the commonly expressed difference between "same" and "equal". — Metaphysician Undercover
"Equal" indicates a similarity of two things by both sharing an identifiable property, while "same" means that you are referring to one thing only. — Metaphysician Undercover
Generally I disagree with your wording, as indicated above. — Metaphysician Undercover
The axiom of extensionality indicates what is required for two sets to be equal, yet you state this as "the same". That I take as a mistaken use of words. — Metaphysician Undercover
I didn't say that though. I simply gave an example of how fiction is useful, one that was obvious. — Metaphysician Undercover
Many times fiction is used in ways not intended to deceive, like the use of counterfactuals in logic, for example. So, the issue is complex, because mathematics, like fiction in general has many uses. — Metaphysician Undercover
These are broad generalizations which I can't relate to because I do not accept them as valid
generalizations, so I do not reply. — Metaphysician Undercover
For example, you say that language "attempts to...". But language doesn't attempt anything, individual people attempt to do things with the use of language. And, there is such an extremely broad range of things which people attempt to do with language, that it doesn't make sense to make the generalization that what people attempt to do with language is to capture "some aspect of abstract thought". — Metaphysician Undercover
The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times. — Michael
We're discussing the consequence of having pushed a button an infinite number of times, not the limit of some infinite sequence of numbers. These are two different things. — Michael
As Thomson says, "the impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be associated arithmetical sequence is convergent or divergent." — Michael
You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. — fishfry
You're confusing yourself on this point.
That's funny, because according to SEP, he used the asymptotic density of the sequence 0, 1, 0, 1, ... to argue that the final state must be 1/2. He made that exact argument using that exact sequence. So you are mistaken, because Thompson has used the exact same reasoning I did. — fishfry
What is the sum of the infinite divergent sequence +1, -1, +1, ...? Now mathematicians do say that this sequence has a sum; they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on.
In set theory, two sets that are equal are the same set. — fishfry
I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.
What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use e and i, because why not?
The logic of the lamp just has nothing to do with numbers at all. — Michael
Are you arguing that Thompson's sequence is finishable hypothetically, but without possessing a definite end value? — sime
In which case your argument would be closer to constructive mathematics based on intuitionistic logic, rather than to intuitionism. — sime
By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises — sime
It seems pretty clear that "same" does not have the same (!) meaning in the context of set theory that it has in the context of the law of identity. How could it? The definition that applies in the context of the law of identity is inapplicable to the context of set theory, and vice versa. So why don't you conclude that the use in the context of the law of identity violates the use in the context of set theory? It seems to be an arbitrary choice.Conclusion: set theory is in violation of the law of identity. I've explained to you why this is the case. Do you agree with me? — Metaphysician Undercover
The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined.The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times. — Michael
Surely, the contradiction is the result of the lack of any definition of the terminal state. If the terminal state could be a plate of spaghetti, why couldn't be a lamp that is neither on nor off?You have not defined the terminal state. So why do you think there should be a sensible answer for what it is? — fishfry
The plate of spaghetti is a great dramatic way of making the point that there is no definition. But the series is defined on the basis that its limit is 1. You can't derive 1/2 from a plate of spaghetti.You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. It's not a real light bulb, it's not driven by a real circuit. — fishfry
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