Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing. — Ludwig V
That's a useful tactic. I shall use it in future. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
That's a very helpful metaphor. — 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?
— fishfry
Yes.
I cannot fathom what you might mean.
— fishfry
Nor can I. That's the problem. — Ludwig V
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.) — Ludwig V
"A sequence does not approach its limit in time" makes no sense. — Ludwig V
I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful. — Ludwig V
We have to accept that a sequence approaching its limit is not like a train approaching a station. — Ludwig V
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. — Ludwig V
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. — Ludwig V
Yes. I realize this is border country. Godel seems to live there too. — Ludwig V
And then again, we have no evidence that the mathematical real numbers are even a decent model for time. The real numbers are continuous, but nobody knows if time is. — fishfry
It's a shame we use the word "approach," because many are confused by that. — fishfry
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? — Ludwig V
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? — Ludwig V
I don't say that selecting and organizing the quotations is easy. It fits better with the fact that I tend to get slabs of time when I can pursue these discussions but in between, I'm not available at all. So the quick back and to is more difficult for me.Oh I see. I prefer shorter posts so I don't get lost in the quoting! — fishfry
I didn't mean to imply that they were living together. That would be .... interestingly mnd-boggling.With supertasks? I don't think so. — fishfry
Don't get me started. What particularly annoys me is that so many people seem absolutely certain that they are right about that. I think it is just a result of thinking that you can write probability = 1, when 1 means that p cannot be assigned a probability, since it is true. A friend once conceded to me that it was a degenerate sense of probability, which is like saying that cheese is a degenerate form of milk.See any .999... = 1 debate. — fishfry
Since my earlier comment on this,Peano arithmetic is potential and the axiom of infinity gives you a completed infinity. — fishfry
I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. So "potential" and "completed" can be fitted together after all.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. — Ludwig V
I think I shall stick to my view that defining an infinite sequence or getting a beer from the fridge is the completion of an infinite number of tasks. I don't think it gives any real basis for thinking that supertasks are possible.The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals. — fishfry
You notice that maths outside time is metaphorical, right? I prefer to say that time does not apply to maths, meaning that the grammatical tenses (past, present and future) do not apply to the statements of mathematics. I like "always already" for this. There is a use of language that corresponds to this - the "timeless present". "One plus one is two" makes sense, but "One plus one was two" and "One plus one will be two" don't.Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. — fishfry
Yes. But there are complications. How does math apply to the physical world?Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world. — fishfry
We have a choice between insisting that Non-Euclidean geometries are not created but discovered and insisting that they are not discovered but created - though they exist, presumably, forever. But if we create them, what happens if and when we forget them?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. — fishfry
As I said before there are a number of ways to describe this. They're all a bit weird.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
It sounds as if you are saying that "approach" is a simply two different senses of the same word, like "bank" as in rivers and "bank" as in financial institutions. An old word given a new definition. Perhaps.The word "approach" is colloquial. It is not intended to evoke images of panthers stalking their prey, or arriving at your destination in a car. Not at all. It's just the word we use for the limiting process. — fishfry
That's a very neat definition. I'll remember that. But you can see, surely, how difficult it is to shake off the picture of a machine that sucks in raw materials and spits out finished products. But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying.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). — fishfry
I don't really understand this. If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. Or do you mean that it is not defined as on or off, which leaves the possibility that it must be in one state or the other, we just don't know which.rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1. — TonesInDeepFreeze
I don't get the difference. If mathematics applies to the physical world, surely it is true of it?if mathematics is true of the physical world too or rather only applies to it — Lionino
Yes. Different geometries apply in different contexts. That's only a problem if you think that just one of them must be absolutely true, which appears to be false.Euclidean geometry applies to a car going from the theater to the restaurant (the surface of the city is flat), non-Euclidean to an airplane going around the Earth (spherical geometry) or things interacting in space-time (hyperbolic geometry). — Lionino
If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. — Ludwig V
Or do you mean that it is not defined as on or off — Ludwig V
which leaves the possibility that it must be in one state or the other, we just don't know which. — Ludwig V
I think it is just a result of thinking that you can write probability = 1 — Ludwig V
I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. — Ludwig V
Yes, I didn't think of the possible application of that idea to this discussion. I've only ever encountered it in the context of probability.Who says anything about probability when merely mentioning that .9... = 1. — TonesInDeepFreeze
That's interesting. Can you refer me to a source?we prove that .9... = 1. — TonesInDeepFreeze
I'm sorry. It's probably not worth pursuing, but I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it.No, it's not a matter of knowledge. Rather, at 12:00 the lamp is neither Off nor On, which contradicts that at all times the lamp is either Off or On. — TonesInDeepFreeze
If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. — Ludwig V
Can you refer me to a source? — Ludwig V
Relax! — Ludwig V
I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it. — Ludwig V
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. — TonesInDeepFreeze
I think that this is what the so-called "paradox" of supertasks is all about. What is revealed is that at least one or the other, space or time, or both, must not be continuous. I think that's what Michael has been arguing since the beginning. Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers. — Metaphysician Undercover
The real issue is that if one of these, space or time, is not continuous, then it cannot be modeled as one thing. There must be something else, a duality, which provides for the separations, or boundaries. But I don't think anyone has shown evidence of such a duality, so we have no real principles to base a non-continuous ordering system on. — Metaphysician Undercover
I'd say this is similar to Tones' use of "identity" in set theory. We take a word, such as "approach", which clearly does not mean achieving the stated goal, and through practise we allow vagueness (to use Peirce's word), then the meaning becomes twisted, and the use of the word in practise gets reflected back onto the theory. So we have the theory stating one thing, and practise stating something different, then the meaning of the words in the theory get twisted to match the practise. Practise says .999... is equal to 1, so "approach" in the theory then takes on the meaning of "equal". Practise says that two equal sets are identical, so "equal" in the theory takes on the meaning of "identical". These are examples of how theory gets corrupted through practise when the words are not well defined. — Metaphysician Undercover
I don't say that selecting and organizing the quotations is easy. It fits better with the fact that I tend to get slabs of time when I can pursue these discussions but in between, I'm not available at all. So the quick back and to is more difficult for me. — Ludwig V
Don't get me started. What particularly annoys me is that so many people seem absolutely certain that they are right about that. I think it is just a result of thinking that you can write probability = 1, when 1 means that p cannot be assigned a probability, since it is true. — Ludwig V
A friend once conceded to me that it was a degenerate sense of probability, which is like saying that cheese is a degenerate form of milk. — Ludwig V
I think I shall stick to my view that defining an infinite sequence or getting a beer from the fridge is the completion of an infinite number of tasks. I don't think it gives any real basis for thinking that supertasks are possible. — Ludwig V
You notice that maths outside time is metaphorical, right? — Ludwig V
I prefer to say that time does not apply to maths, meaning that the grammatical tenses (past, present and future) do not apply to the statements of mathematics. — Ludwig V
I like "always already" for this. There is a use of language that corresponds to this - the "timeless present". "One plus one is two" makes sense, but "One plus one was two" and "One plus one will be two" don't. — Ludwig V
Yes. But there are complications. How does math apply to the physical world? — Ludwig V
We have a choice between insisting that Non-Euclidean geometries are not created but discovered and insisting that they are not discovered but created - though they exist, presumably, forever. But if we create them, what happens if and when we forget them? — Ludwig V
As I said before there are a number of ways to describe this. They're all a bit weird. — Ludwig V
It sounds as if you are saying that "approach" is a simply two different senses of the same word, like "bank" as in rivers and "bank" as in financial institutions. — Ludwig V
An old word given a new definition. Perhaps. — Ludwig V
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).
— fishfry
That's a very neat definition. I'll remember that. — Ludwig V
But you can see, surely, how difficult it is to shake off the picture of a machine that sucks in raw materials and spits out finished products. — Ludwig V
But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying. — Ludwig V
It seems to me that Benacerraf is skipping that condition. — TonesInDeepFreeze
And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf. — TonesInDeepFreeze
If Benacerraf is not skipping the condition, then where does he recognize it? — TonesInDeepFreeze
Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible — TonesInDeepFreeze
I'm not deeply versed in Aristotle, but my impression is that he did indeed resolve the issue, as it was understood in his time (and what more than that could he possibly resolve?). In doing so, he invented or discovered or recognized the concept of categories, which was a titanic moment in philosophy. It's a pity that there seem to be so many people around who are completely unaware of it.This issue was actually resolved a long time ago by Aristotle, — Metaphysician Undercover
I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."The unintelligibility is due to a thing's matter or potential. — Metaphysician Undercover
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. 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.So in the example, when the lamp is neither on nor off, rather than think that there must be a third state which violates the excluded middle law, we can say that it is neither on nor off, being understood as potential. — Metaphysician Undercover
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.As what may or may not be, "potential" violates the law of excluded middle. — Metaphysician Undercover
P13 Some infinitist claim, however, that at t_{b}, after performing Thomson’s supertask, the lamp could be in any unknown state, even in an exotic one. But a lamp that can be in an unknown state is not a Thomson’s lamp: the only possible states of a Thomson’s lamp are on and off. No other alternative is possible without arbitrarily violating the formal legitimate definition of Thomson’s lamp. And we presume no formal theory is authorized to violate arbitrarily a formal definition, nor, obviously to change, in the same arbitrary terms, the nature of the world (Principle of invariance). It goes without saying that if that were the case any thing could be expected from that theory, because the case could be applied to any other argument.
P16 At this point some infinitists claim the lamp could be at S_{b} by reasons unknown. But, once again, that claim violates the definition of the lamp: the state of a Thomson’s lamp changes exclusively by pressing down its button, by clicking its button. So a lamp that changes its state by reasons unknown is not, by definition, a Thomson’s lamp (Principles of Invariance and of Autonomy).
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