Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing. — Ludwig V

Oh I see. I prefer shorter posts so I don't get lost in the quoting!

That's a useful tactic. I shall use it in future. — Ludwig V

Yes, I just highlight the whole post and say Quote.

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

"Back then" wasn't that long ago, this scandal's just a few years old IIRC. I didn't follow the particulars. I'm sure it's just as common today. Or maybe not as 1950's, say. Thinks have changed since then. Still. The male-female thing, those are very deep energies being played with. You are not going to stamp it out with rules. You can change the form in which the scenarios are played out. What Eric Berne called the games. Games People Play, remember that?

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

There's another funny thing that goes on. Sometimes you make a definition, and it DOESN'T bring a mathematical object into existence. For example we write down the axioms for a group. But we have no idea if there are any groups, or if the axioms are perhaps vacuous. So the next step is to exhibit some groups, like the additive structure on the integers. This is a very common pattern in math: Make a definition, then show that there's something that satisfies the definition!

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

Right, potential/actual versus potential/completed. I've heard them both. Since they don't come up in math I kind of use them interchangeably. But they probably have more specific technical meanings or contexts I don't know about. But for me, I just regard the axiom of infinity as the sharp boundary between the two concepts. Induction on the one hand, versus a "completed loop."

That's a very helpful metaphor. — Ludwig V

Shoulders of road. Bounds. Glad that helped.

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

I quoted that entire exchange. I can't fathom your meaning. But you say you can't either.

Time is a concept in physics. You can see that, right? Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. 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.

If you can give an example of what you are thinking, that might be helpful.

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

Ok. You seem to be agreeing with me.

"A sequence does not approach its limit in time" makes no sense. — Ludwig V

And now you're not agreeing. 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. But 1/2 + 1/4 + 1/8 + ... IS 0; it does not "become" or "approach" zero. It's this language ambiguity that is the source of so much online confusion about the subject. See any .999... = 1 debate. You'll hear that .999... "approaches but does not reach" 1. But the sum is exactly 1 nonetheless, by virtue of the definitions. The math is designed to make it work out.

I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful. — Ludwig V

I'd like to know what the problem is, regardless. If for no other reason than to make sure you're understanding it correctly!

We have to accept that a sequence approaching its limit is not like a train approaching a station. — Ludwig V

Correct. It's a shame we use the word "approach," because many are confused by that.

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

Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world.

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

I think it's just a confusing use. When mathematicians use the word approach, in their minds they already have the full context of the theory of limits. So they are not confused. But non-mathematicians hear the word and associate it with is everyday context. No harm is done, till these misunderstanders show up to .999... = 1 threads online. Then we get problems.

Yes. I realize this is border country. Godel seems to live there too. — Ludwig V

With supertasks? I don't think so.

@Michael: I see now that a few of my previous comments, while not incorrect, were not helpful for understanding the problem. I'm still at provisional stages, but this is what I'm thinking now:

I would organize Thomson's argument differently from the way he organizes it. Near the end of his argument, he says "But the lamp must be either on or off." But he's actually invoking a premise. It is natural to regard the lamp as being either Off or On and not both, but in this highly hypothetical context, it would be good to say that as an explicit premise.

Then Thompson invokes infinitely divisible time. But not as a premise. I would include it as a premise. The advantage of doing that is that then the premise is explicitly a candidate for rejection to avoid the contradiction.

I simplified the language of your conclusions (we don't need all those tn, ti, tj and inequality symbols), and I don't think you need the conclusions to be biconditionals to derive that the lamp is neither Off nor On.

('r' for 'revised')

Premises:

rP1: At all times, the lamp is either Off or On and not both.

rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.*

*The pushing of the button and the change are together instantaneous, and the button can be pushed only once in any moment. This is not needed except to simplify the argument (especially to state rC1, rC2 and rP6).

rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.

rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.

rP5: The lamp is Off at 10:00.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.*

*Notice that T1 and T2 are in chronological order.

rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.*

*Notice that T1, T2 and T3 are in chronological order.

Premise:

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.*

* We could easily make that mathematically rigorous.

Conclusion:

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.


QUESTION: How do you state the arguments for rC1 and rC2 from the premises rP1-rP5?

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

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.

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.

It's a shame we use the word "approach," because many are confused by that. — fishfry

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.

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

In the case of platonism, yes. In the case of immanent realism, it would be true of some (physical) parts of the world. Now, a finer distinction: in the case of psychologism/conceptualism, true about our minds (so mathematics would reduce, prima facie, to psychology if psychologism is true).

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 know about neaty gritty details of platonistic matemathics — if mathematics is true of the physical world too or rather only applies to it —, if such shenanigans are even developed that deeply, but 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).

The incessant crank says, "Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers."

There is no sophistry on my part. And no "replacing". I merely pointed out that proving that time is not continuous does not prove that time is not densely ordered (or infinitely divisible).

And the crank is so ignorant and mixed up about this very thread that he wildly infers that my remarks about the thought experiment vis-a-vis Michael's version of it entail that I have myself made certain claims about time beyond that not-continuous does not imply not-dense.

The garbage posting crank doesn't know what he's talking about, regarding continuity or denseness, or me. He is a bane upon reasoned inquiry.

/

The crank is on about .9... Whatever he's trying to say, in his usual thought salad way, we should at least recognize that the notation '.9...' is informal for the limit of a certain sequence.

Meanwhile he has his own utterly mixed up notions about what 'identical' and 'equal' mean. But he hasn't the least reasonability even to understand that his own having notions about what words should mean doesn't entail that everyone else is wrong for using the words both in their ordinary English senses and also in stipulated mathematical senses. He does not understand even the notion of stipulative definition, just as, in another instance, he does not understand even the difference between use and mention.

The crank falsely rails on and on against mathematics and modern logic, even though he has not read page one in a textbook in the subject. As he serves as a textbook example in crank sophistry.

Oh I see. I prefer shorter posts so I don't get lost in the quoting! — fishfry

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.

With supertasks? I don't think so. — fishfry

I didn't mean to imply that they were living together. That would be .... interestingly mnd-boggling.

See any .999... = 1 debate. — 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.

Peano arithmetic is potential and the axiom of infinity gives you a completed infinity. — fishfry

Since my earlier comment on this,

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'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.

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals. — fishfry

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.

Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. — 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.

Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world. — fishfry

Yes. But there are complications. How does math apply to the physical world?

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

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?

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

As I said before there are a number of ways to describe this. They're all a bit weird.

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

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.

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. 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.

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1. — TonesInDeepFreeze

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.

if mathematics is true of the physical world too or rather only applies to it — Lionino

I don't get the difference. If mathematics applies to the physical world, surely it is true of it?

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

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.

If mathematics applies to the physical world, surely it is true of it? — Ludwig V

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.

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

By the premises, there is no third state. Indeed, even if not a premise but a definition:

Df. 'On' means 'not Off'

there is no third state.

Or do you mean that it is not defined as on or off — Ludwig V

No, Thomson's argument is: The premises entail that at 12:00 the lamp is neither Off nor On, but the premises also include the stipulation that at all times the lamp is either Off or On, so the premises are inconsistent.

which leaves the possibility that it must be in one state or the other, we just don't know which. — Ludwig V

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.

I think it is just a result of thinking that you can write probability = 1 — Ludwig V

Who says anything about probability when merely mentioning that .9... = 1.

We prove that .9... = 1, from the definition of the notation '.9...'.

'.9...' stands for the limit of a certain sequence, and that limit is 1.

Anyone is free to regard '.9...' with a different definition and to get different results accordingly. But in context of the ordinary mathematical definition, we prove that .9... = 1.

I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. — Ludwig V

The adjective 'is potentially infinite' has no mathematical definition that I know of, including in alternative theories.

The adjective 'is infinite' is defined in mathematics.

The adjective 'is actually infinite' has no mathematical definition that I know of, including in alternative theories, unless it means simply 'is infinite'.

'is potentially infinite' is a notion about mathematics.

'is actually infinite', if not meaning simply 'is infinite', is a notion about mathematics.

Relax! I was talking about the traditional Aristotelian approach to infinity which was orthodox before Descartes but not since, so far as I know. Though I have since seen someone apparently still using the terms in Two Philosophers on a beach with Viking Dogs

Who says anything about probability when merely mentioning that .9... = 1. — TonesInDeepFreeze

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.

we prove that .9... = 1. — TonesInDeepFreeze

That's interesting. Can you refer me to a source?

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

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.

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

This issue was actually resolved a long time ago by Aristotle, in his discussions on the nature of "becoming". What he demonstrated is that between two opposing states (on and off in this case), there is a process of change, known as becoming. This process is the means by which the one property is replaced by the opposing property. If we posit a third state between the two states, as the process of change, then there will now be a process of change between the first and the third, and between the third and the second. We'd now have five distinct states, and the need to posit more states in between, to account for the process of change which occurs between each of the five. This produces an infinite regress.

So what Aristotle proposed is that becoming, as the activity which results in a changed state, is categorically different from, and incompatible with states of being. Further, he posited "matter" as the potential for change. "Potential" refers to that which neither is nor is not. As what may or may not be, "potential" violates the law of excluded middle. 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. This is the way that I understand Aristotle to have proposed that we deal with such activity, which appears to be unintelligible by violation of the law of excluded middle, neither having nor not having a specified property. The unintelligibility is due to a thing's matter or potential.

Can you refer me to a source? — Ludwig V

I won't refer you to a source.

I'll refer you to this:


Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).

Let f(k) = SUM(j = 1 to k) 9/(10^j).

Show that lim(k = 1 to inf) f(k) = 1.

That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.

First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).

Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).

Inductive hypothesis: 1 - f(k) = 1/(10^k).

Show that 1 - f(k+1) = 1/(10^(k+1)).

1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).

By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).

Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).

So by induction, for all k, 1 - f(k) = 1/(10^k).

Let e > 0. Then there exists n such that, 1/(10^n) < e.

For all k > n, 1/(10^k) < 1/(10^n).

So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n) < e.


(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. But arguments that subtract infinite rows are handwaving since subtraction with infinite rows is not defined.)

Relax! — Ludwig V

I was quite relaxed when I provided the information.

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

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. You can go back to the argument to witness it step by step. Best is to read Thomson's paper that is not long and not abstruse, free to download.

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 agree with everything you wrote in this post.

@Michael

Two presentations that are equivalent.

I would like to know how C2 and C3 are derived in Michael's version. That is rC1 and rC2 in my version.

But we get them anyway from my premise rP6 (the antecedent of Michael's C6). I'll show that presentation too (PRESENTATION2). And I think it is closer to Thomson's argument.


MICHAEL'S PRESENTATION

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]


TONESINDEEPFREEZE'S PRESENTATION:

Premises:

rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

Premise:

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.

Conclusion:

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.


Again, I don't know how we derive Michael's C2 and C3 (my rC1 and rC2). But we don't need them anyway:


TONESINDEEPFREEZE'S PRESENTATION 2:

Premises:

rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
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.

Conclusions:

rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
rC1: The lamp is neither Off nor On at 12:00. Contradicts rP1.


So, we don't have to be concerned whether rP1-rP5 entail rC1-rC3. Rather, we see easily that rP1-rP6 entail rC1-rC3. It's a clean and correct inference that way.

So, unless we do have a proof of Michael's C2 and C3 from his P1-P4, he has his argument out of order: we need my rP6 in the premises. And that seems to be flow of Thomson's argument too.

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

I've bowed out of the supertask discussion, having not typed anything new in weeks. It would be inappropriate for me to comment on anything @Michael said, since he'd then be obliged to reply and we'd be right back in it again. @TonesInDeepFreeze merely made the point that 1/2, 1/4, etc only requires the rational numbers. Perfectly sensible observation.

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'm fully supertasked out.

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

Let's keep the Infinity theory in that thread. Well I'm not a moderator here so nevermind, do what you like. I prefer not to engage in these thread-hijacking points here. Every discipline has its terms of art, which confuse non-practioners. When a doctor tells you your liver is "unremarkable," that's great news and not an insult.

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

Oh I see. That's what I like about discussion forums. You can pick up a topic weeks or even months later.

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

Oh my. We must have a conversation about probability sometime. You're wrong about that. 1 is a perfectly sensible probability. But worse, probability 1 events may be false. 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.

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

1 is a perfectly sensible probability. Your friend is misinformed. As Mark Twain said, if you don't read the newspapers, you're uninformed. If you do read the newspapers, you're misinformed.

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

It's mathematically unhelpful to think of a infinite sequence as the "completion of an infinite number of tasks." It leads to confusion. It's not how mathematicians think about sequences.

You notice that maths outside time is metaphorical, right? — Ludwig V

No, it's literally true. Of course math as a human activity is historically contingent. But math itself speaks to truths that are outside of time.

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

Ok, but IMO it's deeper than a semantic point.

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

Right. We don't even have good words to talk about things outside of time. Timeless present is a pretty good phrase.

Yes. But there are complications. How does math apply to the physical world? — Ludwig V

As in Wigner's famous paper on the "unreasonable effectiveness" of math in the physical sciences. If math doesn't actually refer to anything, why's it so useful?

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

I have no idea. I have myself argued from time to time that 5 was not a prime number before there were intelligent beings to observe that fact. I don't actually believe that, but I've argued it.

As I said before there are a number of ways to describe this. They're all a bit weird. — Ludwig V

This was in reference to 0, 1, 2, ... existing "all at once" in PA. What ways are there to describe this? Is it a timeless present? That's a great locution.

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

Yes. "Approach" is a term of art in mathematics. It has a specific technical meaning that is unambiguous. It is not the same as the everyday meaning.

An old word given a new definition. Perhaps. — Ludwig V

Term of art. A lovely legal phrase. Lawyers commonly have to deal with the jargon of whatever discipline a a particular dispute is about.

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

There's a class math majors take called Real Analysis, where they teach you all this; and after which you are forever clear in your mind about things that were formerly vague and fuzzy. Sadly nobody but math majors takes this class, leading to so much confusion.

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

Yes, that's a "function machine," a visualization when we teach functions to high schoolers. And of course mathematical functions are routinely applied to real world processes. A vending machine is a function of two variables: put in money and push a particular button, and the appropriate product comes out.

So the picture, or visualization of a function as a process or a machine is perfectly valid. The mathematical abstraction that strips away the process or machine interpretation is for the purpose of clarifying our ideas.

But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying. — Ludwig V

Yes. The elements of a sequence have a timeless relation to the index set 1, 2, 3, ...

@Michael @fishfry

I haven't yet read all of Benacerraf's paper, but at least where he disscusses Aladdin and Bernard, it seems to me that he's not addressing Thomson's problem but only offering a different problem that does have an easy solution.

With Thomson's problem we have:

If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.

and

If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

It seems to me that Benacerraf is skipping that condition. And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf.

@Michael

Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible (or that it is not possible that time is infinitely divisible - and the modality there may make this more complicated). If I understand correctly, Thomson does't announce such a view about time, though, of course, what Thomson may believe doesn't determine our own conclusions.

It seems to me that Benacerraf is skipping that condition. — TonesInDeepFreeze

@Michael's point, about which he and I disagree.

And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf. — TonesInDeepFreeze

Don't believe so. But by expressing disagreement I invite rebuttal. I am supertasked out, really. Hope I have the strength to not get sucked in again.

If Benacerraf is not skipping the condition, then where does he recognize it? [EDIT: Actually he does address it, but, as far as I can tell, he gets it wrong when he addresses it.]

What essential difference is there between Aladdin/Bernard and Cinderella?

Benacerraf:

"A. Aladdin starts at to and performs the super-task in question just as
Thomson does. Let t1 be the first inistant 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 to and performs the super-task in question (on an-
other 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"

But that contradicts:

If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.

and

If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.

Benacerraf is saying the lamp gets switched in a way that is not possible given Thomson's conditions.

When we describe the events, we have to look closely and exactly at whether they may occur given Thomson's premises. We can't dissolve Thomson's argument merely by ignoring the premises of the argument.

/

Benacerraf:

"According to Thonmson, Aladdin's lamp cannot be on at t,
because Aladdin turned it off after each time he turned it on.
But this is true only of instants before tl!"

There he does recognize the premises, but, it seems to me, he mistakes them. The premises don't cover just what happens before 12:00. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.

"Nothing whatever has been said about the lamp at t1
or later."

That seems to me to be incorrect. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.

Benacerraf:

"The explanation
is quite simply that Thomson's instructions do not cover the state
of the lamp at t1, although they do tell us what will be its state at
every instant between to and t1"

The instructions don't need to specify what happens at 12:00. The instructions specify what happens at all moments and also what happens before 12:00, but what happens at 12:00 still must conform to the instructions that apply to all moments.

The issue is not that the instructions don't specify what happens at 12:00. The issue is that the instructions entail that at 12:00 the lamp is Off and at 12:00 the lamp is On. Thus the instructions are contradictory.

If Benacerraf is not skipping the condition, then where does he recognize it? — TonesInDeepFreeze

Please accept my regrets for not engaging. I have little interest in supertask puzzles in general, and this thread has long since exhausted any points I could possibly make.

Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible — TonesInDeepFreeze

The simple reasoning is that if time is infinitely divisible then pushing a button an infinite number of times within two minutes is theoretically possible. Pushing a button an infinite number of times within two minutes entails a contradiction and so isn't theoretically possible. Therefore, time is not infinitely divisible.

Although I think perhaps this variation of Zeno's paradox might be better at questioning the infinite divisibility of spacetime.

This issue was actually resolved a long time ago by Aristotle, — Metaphysician Undercover

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.

The unintelligibility is due to a thing's matter or potential. — Metaphysician Undercover

I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."

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. 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.

As what may or may not be, "potential" violates the law of excluded middle. — 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.

Thomson’s lamp revisited makes much the same points I have made:

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.

i.e. the lamp can't turn into a pumpkin.

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).

i.e. the lamp is on if and only if the button is pushed (when the lamp is off) to turn it on (and not then pushed to turn it off).

Infinite divisibility doesn't entail a contradiction. Rather, infinitely divisibility along with the other premises entails a contradiction. Moreover, you are adding another premise (call it 'DT'): if infinite divisibility, then tasks can be performed at each of the infinitely many times. Therefore, we are entitled to question any of the premises, including the new one DT, not just infinite divisibility.

I'm not rejecting anything.

I'm saying:

(1) What is the proof of C2 and C3 from the premises? (Though we don't need it, if we adopt my rP6.)

(2) Instead of rejecting infinite divisibility, we may reject other premises instead.

That comment was directed at fishfry who claims that the lamp can turn into a pumpkin or spontaneously and without cause be on at 12:00.