Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino
I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.I'll refer you to this: — TonesInDeepFreeze
Perhaps it would serve our purposes. I could probably get the point even if it isn't completely rigorous.I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. — TonesInDeepFreeze
The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze
The contradiction is created here - specifically in the last two words, which make it impossible to know whether it has been pushed an even or odd number of times since 11:00.rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.* — TonesInDeepFreeze
Which one do you think should be rejected?The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze
If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe.For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals. — fishfry
.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? (I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it.1 is a perfectly sensible probability. — fishfry
Rather, infinitely divisibility along with the other premises entails a contradiction. — TonesInDeepFreeze
var isLampOn = false function pushButton() { isLampOn = !isLampOn } var i = 120 while (true) { wait i *= 0.5 pushButton() } echo isLampOn
var i = 120 while (true) { wait i *= 0.5 }
while (true) { }
Here's how I look at it. I think that everyone will agree that a formula is not about anything specific and, in itself is neither true nor false. x + y = z doesn't make any assertions, until you substitute values for the variables. So 2 +1 = 4 is false, but 2 + 3 = 5 is true. So there's a temptation to think it must be true of something. Hence realism. But 2 + 3 = 5 is itself like a formula in that once we specify what is being counted, it does make an assertion about the world - 2 apples + 2 apples = 4 apples. It is true of the world. Of course, 2 drops of water plus 2 drops of water doesn't make 4 drops of water, (until we learn to measure the volume of water). The domain of applicability and truth is limited.Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino
If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe. — Ludwig V
.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. — Ludwig V
So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? — Ludwig V
(I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it. — Ludwig V
I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential." — Ludwig V
I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. — Ludwig V
A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc. — Ludwig V
I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time. — Ludwig V
The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected.
— TonesInDeepFreeze
Which one do you think should be rejected? — Ludwig V
Say we accept that Thomson's lamp entails a contradiction; the lamp can neither be on nor off at 12:00.
I take this as proof that having pushed a button an infinite number of times is metaphysically impossible. — Michael
You seem to take this as proof that having pushed a button an infinite number of times is metaphysically impossible only if the premises are true. — Michael
let's say that our button is broken; pushing it never turns the lamp on. In such a scenario we can unproblematically say that the lamp is off at 12:00. But this does not then entail that it is possible to have pushed the button an infinite number of times. — Michael
we can imagine a lamp with two buttons; one that turns it on and off and one that does nothing. Whenever it's possible to push one it's also possible to push the other, and so if it's possible to have pushed the broken button an infinite number of times then it's possible to have pushed the working button an infinite number of times. Given that the latter is false, the former is also false. — Michael
Having pushed a button an infinite number of times is an inherent contradiction — Michael
Having the button turn a lamp on and off, and the lamp therefore being neither on nor off at the end, is only a way to demonstrate the contradiction; it isn't the reason for the contradiction. — Michael
is also why Benacerraf's response to the problem misses the mark. — Michael
It's not a contradiction in and of itself. — TonesInDeepFreeze
Having the operation be to push a button, and having this button turn a lamp on and off, is simply a way to make this inherent contradiction even clearer. — Michael
what is the reasoning behind the claim that if some wizard steps in at 12:00 to magically turn the lamp on — Michael
I didn't say they end. — TonesInDeepFreeze
Again, the contradiction comes from the conjunction of the premises. — TonesInDeepFreeze
(3) We don't have a satisfactory definition of 'metaphysical possibility' here. — TonesInDeepFreeze
One of the contradictions does; the state of the lamp at 12:00. — Michael
The other contradiction is the inherent contradiction of an endless sequence of operations coming to an end. — Michael
Just incase you missed my edit to my previous post:
Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00. — Michael
Finding some way to resolve the former does not retroactively resolve the latter. — Michael
The definition of a super-task is as you say. But your listed premises don't say anything about completion or ending. — TonesInDeepFreeze
The contradiction is: The lamp is either On or Off T 12:00 and the lamp is neither On nor Off at 12:00.
But that contradiction comes from a set of premises, each of which is not logically true, and dropping any one of the premises blocks deriving the contradiction. It would help if you would at least tell me that you understand that. — TonesInDeepFreeze
It is begging the question merely to declare it is a contradiction that denumerably many tasks can be executed in finite time. — TonesInDeepFreeze
It merely says that metaphysical possibility may be logically possibility and that there's another notion that the article describes ostensively. So is it just the same as logical possibility, and if not what is a proper definition that is not merely ostensive? — TonesInDeepFreeze
"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?". — Michael
I am no longer pushing the button at any time after 12:00. My infinite button pushes has allegedly ended. — Michael
I use the phrase "metaphysical impossibility" rather than "logical impossibility" simply because it's the weaker claim. — Michael
But completion is not in your premises. — TonesInDeepFreeze
P5 is an inherent contradiction — Michael
It doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same. — TonesInDeepFreeze
it makes no difference if it's an antecedent in a conclusion or as a premise. — Michael
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