"But mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions. For example, what is infinity minus infinity? Mathematically, you get self-contradictory answers."

It's hard to know what Craig really means here, because it is so confused. Mathematicians

*routinely*study "an actually infinite number of things", such as the natural numbers, the real numbers, and the complex numbers. No contradictions are involved.

But maybe Craig is talking about an actual infinity of things in

*nature*. Then he shouldn't be talking about mathematicians, but physicists. Even here, physicists

*do*discuss an actual physical infinity - without contradictions - such as Malament-Hogarth spacetime. Examples like Hilbert's hotel, that are often proffered as insoluble paradoxes, only show that infinity needs to be treated with care and may result in scenarios that seem counter-intuitive. But so does relativity.

Infinity minus infinity is not "self-contradictory", any more than 1/0 is "self-contradictory". Lane seems not to understand that not all functions are everywhere defined. The subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞). What's so hard to understand about that?

*Addendum*As I listen to more of the debate, it seems Craig retreats a bit from his claim about mathematics. He still seems to think that an actual infinite number of objects in the universe creates "contradictions", but he doesn't say explicitly what those contradictions are.

## 108 comments:

What I tell students is that if some aspect of mathematics appears to be paradoxical, like a Zen

koan, then it is because you haven't learned enough about what is being said,

or the statement is not correctly presented.

Have you read his 2009 presentation of the argument in Blackwell Companion to Natural Theology? He is of course much clearer on his position there than in popular debates.

No, I haven't read that.

I can't imagine it being much less confused. His issues seem to be pretty fundamental.

At his reasonable faith website, Q&A section, Craig has recently been engaging some weighty questions such (1) "does God know an infinite number of things" or (2) on 'infinite multiverses'. It's quite entertaining reading. here is one of the links:

http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=8618

Try as I might I can't but help concluding Craig is a confidence trickster who attempts to chat his quarry by confusing the audience. That said I don't understand why a guy like Krauss allowed Craig to define entities at will, on the fly.

Truti

I think Craig knows what he is talking about. By "actual infinity" he meant an infinity of things in the "actual" universe, i.e., the continent physical/mental universe. Now I believe that is a misuse of the word "actual" since I am a mathematical Platonist (about at least the natural numbers) but aside from that terminological dispute, I think he knows his stuff here.

He alluded to the "contradictions" a few times when he talked about someone possessing an infinite number of things and giving an infinity away. So I think for example he is referring to paradoxes such as that you cannot make sense out of the action "put two items in a jar and take one out" if there are an infinity of tasks possible. That is, there is no sense to describing the action like that because it depends upon which items you take out, precisely, that determines whether anything is left at the end and if so, what. (Could be infinite, could be nothing).

But that is not a problem with certain kinds of infinity existing, only some. Still, I don't think he can be criticized as nearly as strongly as Krauss, who said two nonsensical things "2+2=5 for large values of 2". Obviously false. And "the sum of the infinite number of numbers 1,2,3 is - 1/12, you may not like it, but it is". Again, obviously false. The analytic continuation of the power series defined for re(x)>1, sum 1/n**x, of course, known as the zeta function, is -1/12 at -1, and here the power series is 1+2+3.... But, uh, the zeta function is NOT EQUAL to that power series at -1, is it? That's just a stupid thing to say (even given the concept of Ramanujan "sum").

Krauss also flailed about in all sorts of other ways. E.g. "I don't like probability, but I'll use one for you: 90% of top scientists who looked at Craig's results don't believe in God". A nonsequitur and an appeal to authority, and also probably false since I think the stat he is citing does not establish these (the so-called "elite NAS scientists") have all studied Craig's work.

Another interesting piece of bluster was his "evidence has to be falsifiable". It is theories, not evidence, that in Popper's analysis of what constitutes science, that have to be falsifiable (BY evidence).

"Lane seems not to understand that not all functions are everywhere defined. The subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞). What's so hard to understand about that?"

No, he explicitly says in that debate that subtraction is not defined on infinite numbers although he is clearly talking about the usual sort of infinite numbers, ordinals and cardinals and not the "extended reals".

He says it is in order to avoid contradiction, though, which is not correct: it is simply because it does not make any sense. Normally subtraction is introduced as x-y = the z such that y+z = x. To use that definite description one must first establish there is only one such z).

Again, the "contradiction" (really just a paradox) would be something like: suppose you have an infinite number of things in a bag. You reach in and take one out to give it away. You keep going until you have given out an infinite number.

How many are left? There is no answer. Subtraction seems possible in the world, but it has no answer. (It is not possible unless you also assume supertasks are possible),

Actually you could probably put a probability measure on it. If you take them out at random, chances are there are an infinite number left over almost certainly. You need a very special technique to take them all out. Also you have to assume they are distinguishable ahead of time in some sense.

Anonymous:

Krauss's joke about 2+2 = 5 was not meant literally. I think it is silly to insist that it was.

As for 1+2+3+... = -1/12, this is the Ramanujan sum of the series. It may be taking a little bit of license to claim that this is

thesum of the series, but it is not entirely incorrrect. In physics one often manipulates diverging series in similar ways.The well-known "paradoxes" of the infinite are

notcontradictions, and Craig is wrong to imply that they are. I don't think he "knows his stuff".Craig is not trying to prove that the concept of infinity in mathematics is self-contradictory. Here's what Craig is thinking of.

Let N be the set of natural numbers. N obviously has an infinite number of members. Now, remove from N all the even numbers. We have removed an infinite number of members, and are left with all the odd numbers, of which there are still an infinite number. Simplistically, we might be tempted to conclude that: infinity - infinity = infinity.

But now take N and remove all the numbers greater than 3. The are only three numbers left, seemingly proving that infinity - infinity = 3. That's the "self-contradiction".

Craig's point is that, to avoid such contradictions, we must do what you said: "the subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞)". This is fine in mathematics - we make the rules. Craig has said elsewhere that he has no problem with the concept of infinity in mathematics.

The question now is: can such a restriction be placed on actual infinities in the real word? If we had an infinite number of real books in a real library, then nothing would stop us (with an infinite number of helpers) from taking all the odd books off the shelf, or all the books but the first three. If actual infinities existed in reality, we wouldn't be able to impose the restrictions on operations that mathematicians can to avoid contradiction.

Such is his argument. I can see a loophole. Craig want's to apply this to the past to see if it can be actually infinite. But there seems to be no way to remove time periods from the past in the same way as we can remove books from a library. This is especially true for Craig, who is an A theorist when it comes to time. The past then would be an example of a really existing actual infinite which avoided the contradictions in question.

If we had an infinite number of real books in a real library, then nothing would stop us (with an infinite number of helpers) from taking all the odd books off the shelf, or all the books but the first three.Yes, and so what? How does that result in any kind of contradiction or paradox?

This just seems like muddled thinking to me.

I have wondered about the point that Luke Barnes made. In A-time, time would be an actual infinite, would it not?

I knew this debate was coming up, but this is the first that I've seen that it happened. Thanks for the linky.

My impression of Craig is that he's a guy who's talented in the art of bullshit. He's not dumb; he knows what he's doing, and what he's doing is sounding smart to give the rubes an excuse that they don't need to examine their beliefs, because a smart guy already did that for them and he said it's OK.

His infinity argument is that the universe couldn't always have existed, because (somehow) that would be a contradiction of something or other. I've heard that claimed by others also, that the universe had to have a beginning in the finite past, because if it was infinitely old then "we could never get here." That's what they say anyway, and I don't understand their logic.

I think they're confusing the wording, thinking that saying the universe always existed means that the universe had a beginning but it was infinitely long ago. That's clearly nonsensical, so I think it's what they're thinking an infinitely old universe would mean.

I'm not a mathematician, but I can see the problem with that thought, and I can't see any logical reason that the universe couldn't be infinitely old. Seems like the same idea that the universe could extend infinitely into the future, but turned around the other way.

All goods points as per, Jeffrey.

However, sophistry and obfuscation are all part of William Lane Craig’s style. I seriously doubt whether he properly understands all the authorities he mines and all the nonsense he spouts.

Regardless of whether infinites can be actual or contradictory in themselves, Craig blatantly contradicts himself with this argument.

By bending over backwards to rule out the universe being eternal and uncaused he shoots himself in the foot by positing an infinite God to provide a definite beginning in space and time.

“Infinity is just something that exists in your mind,” Craig drawls.

Rather a lot like Yahweh then isn’t it, Bill?

So Mr. Craig doesn't like the notion of infinity - infinity. I guess then that he wouldn't like quantum electrodynamics, in the unlikely event that he knew anything about the subject matter. In computing the anomalous magnetic moment, mu, of the electron, one encounters a logarithmically divergent integral, I, from which is subtracted the electron mass, m. The electron mass m is redefined as the bare mass, m(0), which is then taken to be infinite, with the resulting expression I - m(0) set equal to -m. This, of course, is a mathematically preposterous exercise which is justified by the result that the computed value of mu agrees with the observed value to 10 significant digits! This procedure is known as mass renormalization.

Here is Krauss' post mortem at Pharyngula:

http://scienceblogs.com/pharyngula/2011/04/lawrence_krauss_vs_william_lan.php

William Lane Craig makes a very good point about infinity.

Infinity leads to paradoxes and downright contradictions.

I thought I had zero money in my pocket.

But then I realised that if zero existed, I could divide it by 2 and still have zero.

That is not a contradiction, but what happens if I divide zero by zero.

According to well-respected mathematicians, zero divided by zero is impossible. It cannot exist! It leads straight to absurdity.

So zero cannot be something which exists.

You can’t divide by it, as you can by any other number ,without producing absurdity.

Having proved that zero could not exist, I looked in my pocket, and guess what? There was some money there!

That is the power of logic.

"... zero divided by zero is impossible. It cannot exist!...

"So zero cannot be something which exists."

The latter doesn't follow (non sequitur) from the former. Simple flubs like these will cast doubt on the rest of what you say - watch it next time.

Jeffrey Shallit: "Yes, and so what? How does that result in any kind of contradiction or paradox?"

Didn't Luke Barnes answer that? There would only be three books left, seemingly proving that infinity - infinity = 3. That's the "self-contradiction". It is prevented in nature by the fact that there IS no actually existing infinite number of things.

Luke Barnes: "I can see a loophole. Craig want's to apply this to the past to see if it can be actually infinite. But there seems to be no way to remove time periods from the past in the same way as we can remove books from a library. This is especially true for Craig, who is an A theorist when it comes to time. The past then would be an example of a really existing actual infinite which avoided the contradictions in question."

Doesn't Craig argue that since Hilbert-like self-contradictions cannot exist in nature, that means that the past CANNOT be an example of an actual infinite which avoided the contradictions in questions. Wouldn't the alternative you are suggesting embrace not just paradox, but self-contradiction in nature?

There would only be three books left, seemingly proving that infinity - infinity = 3.No, that is not a contradiction. That just shows that "infinity" cannot be manipulated like an ordinary number. If this is all Craig has, he's got nothin'.

It is prevented in nature by the fact that there IS no actually existing infinite number of things.Pure assertion.

Dave S., that's Steven Carr's point. He's suggesting that WLC's argument is similarly wrong. (I agree, with the proviso that the oddities when you try to do arithmetic with infinity are worse than the oddities when you try to do arithmetic with zero, so much so that the mathematical structures we usually use are defined to permit a zero which can't be divided by, but not to permit infinities which can't be added and subtracted normally. But of course that doesn't in the least license an inference from "infinity can't be treated as a number" to "there are not infinitely many things in the universe".)

Gareth, you're suggesting Steve Carr, was using his example ironically to mirror Craig's argument and thus discredit it?

I had taken Steve Carr's post more at face value: as an indictment of logic itself, that it is faulty or at least untrustworthy (and thus somehow a proof of God?)

I wanted to say out that the logic in his example was too shoddy to make the point I think he was trying to make.

Jeffrey Shallit: "No, that is not a contradiction. That just shows that "infinity" cannot be manipulated like an ordinary number."

I'm not clear what you mean: are you saying that in the case of an actually existing infinite number of physical books, the operation described by Luke Barnes would not by physically possible? If that is not what you are saying, then what would the result of that operation be?

I'm not clear what you mean: are you saying that in the case of an actually existing infinite number of physical books, the operation described by Luke Barnes would not by physically possible?If you have infinitely many books, and you remove all but 3, of course you have three left. It is ridiculous to say that this is a "paradox" or a "contradiction".

Whether it is physically possible or not I leave up to the physicists.

Jeffrey Shallit: "If you have infinitely many books, and you remove all but 3, of course you have three left. It is ridiculous to say that this is a "paradox" or a "contradiction".

So you have an infinite number of books, and you remove an infinite number of books (since the all the books greater than 3 are still an infinite number), you have three left, meaning infinity - infinity = 3.

If you have an infinite number of books, and remove all the odd numbered ones, you are left with an infinite number of even-numbered books, meaning infinity - infinity = ????

Perhaps you can help me out here, because these results seem paradoxical and/or self-contradictory to me.

Perhaps you can help me out here, because these results seem paradoxical and/or self-contradictory to me.No, there is no paradox or self-contradiction. The only thing you need to keep in mind is that when dealing with infinite quantities, the result can depend on the order of operations. This is why we do not assign a meaning to "infinity-infinity".

Mathematicians regard these things as utter trivialities.

Jeffrey Shallit: "The only thing you need to keep in mind is that when dealing with infinite quantities, the result can depend on the order of operations. This is why we do not assign a meaning to "infinity-infinity"."

Yes, I agree that under the rules of transfinite arithmetic you can just "not assign a meaning" to "infinity minus infinity." But in physical reality nothing stops you from physically removing all the odd-numbered books. Since that would produce a "meaningless" result, it would seem to indicate that in spacetime reality there ARE NO actual infinites upon which such operations could be performed.

Since that would produce a "meaningless" resultI think I see the root of your confusion - and Craig's.

You seem to think if the result depends on the order of operations, the outcome is "meaningless". But there is no reason to think this. Different infinite processes give different results; so what? This has

noimpact whatsoever on whether these processes can occur in nature.Again, mathematicians regard these distinctions as utterly trivial. I can only recommend to you (and to Craig) that you study a little set theory.

I noticed that in the "debate" (I put it into quotes because it seemed that Craig came to debate while Krauss came to have a discussion), Craig said something like "an actual infinite number of past events is logically impossible."

At that point I thought: wait, weren't there an infinite number of events just between 3:00 and 3:01 this afternoon?

Jeffrey Shallit: "Different infinite processes give different results; so what? This has no impact whatsoever on whether these processes can occur in nature."

Well, I am certainly no mathematician, I admit that. However, Craig, together with Jim Sinclair, have published peer-reviewed work on this in the Blackwell Companion to Natural Theology (http://tinyurl.com/3gaavgc) and the consensus is that their "Hilbert's Hotel" objection to an actual infinite has not been refuted (see discussion at this atheist website: http://commonsenseatheism.com/?p=2048). So I just disagree that "this has no impact whatsoever on whether these processes can occur in nature." It seems you would need to present an argument to that effect instead of just an assertion. But I appreciate the opportunity to interact on this issue. Thanks.

In excluding numbers divisible by 2 versus greater than 3 the issue has nothing to do with order of operations (exclusion of elements) but with which (infinite subset of) elements are excluded.

The problems of zero and of countable infinity are the same in the rationals, considered as ordered pairs plus a division operator. Reversing the pairs swaps a problem with zero for a problem with infinity.

Another way of looking at it is for every operation we choose that has a problem with zero there is another that has a problem with infinity, e.g. harmonic mean vs arithmetic mean. The problem with discussed operations on infinity or zero is that they are ill defined in the sense of having multiple solutions. Infinitely many (finite) numbers multiplied by zero give us zero, and infinitely many (finite) numbers subtracted from infinity give us infinity. Mathematics deals all the time with problems that have multiple solutions, in general any underspecified set of equations does.

As far as physics goes, lets do something analogous to counting integers that are or are not divisible by two or are or are not greater than three.

Suppose that at an instant of time there are an infinite number of objects (or 1 cubic metre cells in a cartesian grid centred on the sun) in the universe, and that there are only a finite number of humans and only one planet/star we call earth/sun. Then there are an infinite number of objects/cells that are not/do not include a human or that do not include our solar system.

We have a cartesian coordinate system and can number the cells uniquely by associating the cells with the defining grid point furthest from the origin, and can say where the sum of the coordinates is odd or even, and can quite meaningfully consider these cells separately.

So both kinds of subtraction of infinity from infinity make perfect sense in our real universe.

The result of 1 + 2 x 3 depends on the order of operations; it could be 7 or it could be 9. Does that mean "1 + 2 x 3" is meaningless? self-contradictory? paradoxical? physically impossible?

the consensus is that their "Hilbert's Hotel" objection to an actual infinite has not been refutedAs far as I can tell, they have no coherent objection at all.

But then, looking to theologians for an understanding of what is or is not possible in the physical world is like asking an architect what is possible in literature.

Jeffrey Shallit: "But then, looking to theologians for an understanding of what is or is not possible in the physical world is like asking an architect what is possible in literature."

Well, actually Sinclair is a physicist.

the consensus is that their "Hilbert's Hotel" objection to an actual infinite has not been refutedActually, when I go to the post you cited, I see that the mathematicians there agree with me that Craig's point is trivially silly (see, e.g., Matthew Johnston).

Jeffrey Shallit: "Actually, when I go to the post you cited, I see that the mathematicians there agree with me that Craig's point is trivially silly (see, e.g., Matthew Johnston)."

I was referring to the peer-reviewed publications that have addressed the issue, not to 3-paragraph blog comments. It seems unlikely to me that a "trivially silly" issue would be treated as a serious controversy in the peer-reviewed literature over a period of many years.

It seems unlikely to me that a "trivially silly" issue would be treated as a serious controversy in the peer-reviewed literature over a period of many years.Never underestimate the silliness of theologians.

Of course, there is a real controversy: whether there are infinite objects in the physical world.

However, I really doubt that anyone with a good mathematical education would find that Craig's misunderstanding of mathematics has any bearing on this question.

Dave S., many comments back: yes, I'm sure Steven Carr was parodying Craig's argument. He certainly was not arguing for the existence of God; he is an atheist.

Lane lost when he claimed that because Jesus rose from the dead it constituted evidence for God's existence.

If anonymous can't see why, there is no point explaining it.

Lane lost on several such points, but I suspect the extent of his bafflegab and supreme overconfidence tricks a lot of people into thinking otherwise.

Anonymous X

I have not seen the debate between Craig and Krauss, but relevant to the discussion here is something George Ellis wrote, in a short article in Nature, 20 Jan. 2011 'The untestable multiverse'. Ellis seems to be saying something relevant to this discussion.

Ellis writes ...

"

Greene, to his credit, devotes a chapter to the question of whether the multiverse idea is a scientific theory or not. He believes it is, and even supports the extravagant claim that infinities exist--infinite numbers of universes hosting countless galaxies. This leads to well-known paradoxes, such as the infinite repetition of everything because of the finiteness of possiblities. But again, there is no way to test it, because infinity is always beyond reach--and so will not plausibly exist in physical reality, as mathematician David Hilbert argued."infinite repetition of everything because of the finiteness of possiblitiesA common misconception. An infinite set need not contain "everything"; consider the well-known examples of uncountable sets of measure 0.

Hilbert did argue against the infinite in physics, but his arguments are not terribly convincing. They consist of trotting out examples like Hilbert's hotel and then saying, in effect, "See? That shouldn't be able to happen."

Judging from the blog entry, it seems the author isn’t aware of where Craig believes the infinity-minus-infinity contradiction is because the blogger never addresses it. The contradiction allegedly lies “in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result.”

If it isn’t clear how one can get an absurdity from that, note the following equation, where aleph-null represents the infinity of natural numbers.

aleph-null + 1 = aleph-null + 2

Both sides of the equation are of the same quantity; aleph-null. If we were to subtract equal quantities (the aleph-nulls) from both sides to get this:

1 = 2

we would get something of an absurdity if not a logical impossibility.

Wade:

You are confused. One cannot manipulate infinite quantities in exactly the same way one can manipulate finite quantities. But this does not constitute a "contradiction", "absurdity", or "logical possibility".

It seems neither Craig nor you have mastered basic college-level mathematics.

I agree that the fact that one "cannot manipulate infinite quantities in exactly the same way one can manipulate finite quantities" does not in itself constitute an absurdity, but I've never claimed otherwise. Rather, what I said was absurd was "1 = 2." Remember, the alleged contradiction comes from the

subtraction of infinities. Yes, mathematicians can impose rules forbidding the sort of subtraction I used to get 1 = 2, but that does nothing to remove the absurdities thatwouldresult if such subtraction of infinities were permitted.Perhaps it bears repeating that the alleged contradiction comes not from infinity being weird or being merely different from the finite in mathematical manipulation, but supposedly from the fact that one can subtract equal quantities from equal quantities and come up with non-identical results. The subtraction operation I did to get "1 = 2" should help illustrate why some would find this problematic.

supposedly from the fact that one can subtract equal quantities from equal quantities and come up with non-identical resultsI guess someone with little mathematical training might find this puzzling, but it is not a "contradiction" or "paradox", and only those with little understanding would regard it as one.

Exactly why is it that only those with

littlemathematical understanding find it paradoxical? It’s not as if only mathematical dunces accept that “1 = 2” is an absurd proposition; I’m fairly confident that most professional mathematicians would too. Given the equation I presented and how the result of subtracting the aleph-nulls (and therefore the same quantity) from each side of the equation led to 1 = 2, the claim that subtracting infinities in this way is neither contradictory nor paradoxical would seem to require justification. If you have such justification for your belief, may I ask what it is?Wade:

It's easy to come up with contradictions like "1=2"

ifyou casually evaluate functions where they are not defined.The function "-" is simply not defined for infinite quantities. What's so hard to understand about that?

Jeffery:

It is of course quite easy to understand how such subtraction would be undefined, but this does nothing to remove the apparent absurdity that

wouldresult if this subtractionwereallowed. Pointing out that this subtraction operation is undefined in transfinite arithmetic does nothing to mitigate the paradoxical nature of subtracting infinities in that way (if anything, it does the opposite). Presumably, if the actual infinite were real, there’d be no metaphysical constraint (and no mathematicians that couldn’t be locked up) barring someone from subtracting equal quantities from equal quantities to get non-identical results.To make matters worse for Craig (and his defenders like Wade), there *are* in fact perfectly respectable systems with infinite numbers in them, like the hyperreals of nonstandard analysis or Conway's surreals.

So the alleged paradox seems to go something like this: If there are actual infinities, *and* you insist on doing arithmetic with them, *and* you insist on not doing so using any of the well-established ways of doing arithmetic with infinite quantities, *then* you get nonsensical results. "Therefore" the universe can't be infinitely old.

The mind boggles.

Wade:

Your confusion continues to mount.

The mathematical process of "subtraction" is an

abstraction. It may or may not correspond to some physical action.When we deal with a finite number of apples, it just so happens that the

physical processof "taking away" corresponds to our mathematical notion of subtraction.But if we deal with an infinite number of physical objects, then this no longer need be the case. In the infinite case, the

orderin which the "taking away" is done controls the cardinality of the result.There is no paradox or contradiction; it is just that infinite quantities don't behave like finite ones.

I repeat, why is this so hard to understand?

Gareth McCaughan:

To make matters worse for Craig (and his defenders like Wade), there *are* in fact perfectly respectable systems with infinite numbers in them, like the hyperreals of nonstandard analysis or Conway's surreals.This is a bit of an

ignoratio elenchi; the existence of such a mathematical system is irrelevant to whether the actual infinite is metaphysically possible. Craig would happily concede the existence of transfinite mathematical systems, as would I. Indeed, transfinite arithmetic helps illustrate why subtracting infinities is so paradoxical regarding subtracting equal quantities from equal quantities and getting non-identical results. To recap what I wrote earlier, note the following equation, where aleph-null represents the infinity of natural numbers.aleph-null + 1 = aleph-null + 2

Both sides of the equation are of the same quantity; aleph-null. If we were to subtract equal quantities (the aleph-nulls) from both sides to get this:

1 = 2

we would get something of an absurdity if not a logical impossibility.

So the alleged paradox seems to go something like this: If there are actual infinities, *and* you insist on doing arithmetic with them, *and* you insist on not doing so using any of the well-established ways of doing arithmetic with infinite quantities, *then* you get nonsensical results. "Therefore" the universe can't be infinitely old.

The mind boggles.

I don’t believe you’ve faithfully rendered Craig’s thinking. Rather, Craig’s reasoning is more likely to be something like this (where the type of possibility is metaphysical possibility):

(1) If an actual infinite is not possible, then an infinite past is not possible.

(2) If an actual infinite is possible, then subtracting infinities is possible (in the concrete world, one could lock up all the mathematicians and do subtractions on infinity regardless).

(3) Subtracting infinities is not possible (supposedly, it leads to contradictions or is at least too paradoxical).

(4) Therefore, an infinite past is not possible.

Incidentally, I am “defending” Craig in the sense that I don’t believe his point of view is being properly understood/appreciated; e.g. I’d like to swipe away red herrings, straw men, and misconceptions. I don’t actually find his “subtraction of infinities” argument quite convincing, though I think it does at least warrant

suspicionof an actual infinite’s metaphysical possibility. To say, for example, that there is nothing paradoxical about subtracting infinity in the way Craig describes appears misplaced (note the equation above).Jeffrey Shallit:

The mathematical process of "subtraction" is anabstraction. It may or may not correspond to some physical action.I agree, but part of the point is that subtraction of infinities would possibly correspond to some physical action if actual infinites were metaphysically possible (I’ll give an example shortly).

There is no paradox or contradiction; it is just that infinite quantities don't behave like finite ones.

I repeat, why is this so hard to understand?

Partly because you still have not justified your apparent assertion that there is nothing paradoxical going on here. Given the equation I presented and how the result of subtracting the aleph-nulls (and therefore the same quantity) from each side of the equation led to 1 = 2, the claim that subtracting infinities in this way is neither contradictory nor paradoxical would seem to require justification. I repeat, if you have such justification for your belief, may I ask what it is?

Or perhaps you concede that there is something paradoxical when this sort of subtraction is done in

mathematicsbut that there is nothing paradoxical when it is done in concrete reality. Yet since it would be possible for the equation and the subtraction operation to correspond to reality, it’s hard to see how that plausibly holds. If an actual infinite of, say, marbles were possible, one could physically instantiate the equation (e.g. two equally multitudinous groups of marbles with an equal sign between them) and the subtraction operation (taking the aleph-null pile-portion from each side). The paradox is essentially the same: after subtracting the same quantity from both sides of the equation, the equation is no longer true.Perhaps the best way to respond to this is to recognize the paradox for what it is and simply bite the bullet: claim it is a bizarre but acceptable consequence of an actual infinite existing in concrete reality. Reasonable people could then disagree whether the paradox is absurd or merely very strange.

the claim that subtracting infinities in this way is neither contradictory nor paradoxical would seem to require justification. I repeat, if you have such justification for your belief, may I ask what it is?Perhaps you should first try to elaborate what, exactly, you think the "contradiction" or "paradox"

is. So far I see a lot of confusion and misunderstanding, which I have tried to remedy, without avail.What level of mathematical training do you have?

The paradox is essentially the same: after subtracting the same quantity from both sides of the equation, the equation is no longer true.And why do you think it should be? We

knowthat infinite quantities cannot be manipulated like finite ones. And weknowthat varying the order in which "subtraction" is done with infinite quantities can affect the result. Why do you think this is surprising?For mathematicians this is an utter triviality.

Perhaps you should first try to elaborate what, exactly, you think the "contradiction" or "paradox"is.I stated what it was in my previous comment; you even quoted it in your own comment. I also illustrated it in the equation

aleph-null + 1 = aleph-null + 2

and when I subtracted the same quantity (the aleph-nulls) from both sides to get

1 = 2

The same quantity is subtracted from both sides of the equation and yet the equation becomes false.

What level of mathematical training do you have?I minored in mathematics when I went for my bachelor's in computer science, but it's important to keep in mind that we're

notdisagreeing over mathematical truths here. The issue has more to do with whether something is paradoxical, and that can be a fuzzy realm indeed. Sometimes one philosopher'smodus tollensis another’smodus ponens. To be somewhat tangential, how much training do you have in philosophy?“The paradox is essentially the same: after subtracting the same quantity from both sides of the equation, the equation is no longer true.”

And why do you think it should be? We know that infinite quantities cannot be manipulated like finite ones. And we know that varying the order in which "subtraction" is done with infinite quantities can affect the result. Why do you think this is surprising?

I don’t find it at all surprising; I consider it to be a more or less obvious implication. What is disputed is whether the paradoxes (e.g. that one can subtract the same quantity from both sides of an equation to make the equation false) are such that it is absurd and metaphysically impossible for the actual infinite to exist in the concrete world, or whether the implications are merely very strange (a similar thing could be said for Hilbert's Hotel; some believe the consequences to be absurd and some consider them to be merely odd). It's perhaps worth nothing that nobody is claiming that the actual infinite is

logicallyimpossible; rather the issue is whether it ismetaphysicallypossible.Is there a mathematical proof proving that “adding/subtracting the same real number from both sides of an equation preserves the truth of that equation”? Or is more of an axiom in mathematics, something intuitively perceived to be true?

Wade:

Where I come from, a "contradiction" means arriving at a statement like "0 = 1" by a

validseries of proof steps.You haven't produced a "contradiction" because the operation of "subtraction" on infinite quantities, the way you are doing it, is not logically valid; it does not correspond to a single well-defined answer. I've said this over and over, but it doesn't seem to stick.

a similar thing could be said for Hilbert's Hotel; some believe the consequences to be absurd and some consider them to be merely oddAnd some of us - those with mathematical training - regard the consequences as obvious and trivial.

Is there a mathematical proof proving that “adding/subtracting the same real number from both sides of an equation preserves the truth of that equation”?You seem to be asking, is the deduction

"a + b = c + b implies a = c" provable from, say, the Peano axioms. Certainly it is for systems like Z (the integers) and R (the reals). But it is not true and hence not provable for systems like Z U {+∞, -∞}, which is the system you seem to want to apply it to.

Your conceptual stumbling block seems to be that you want to apply your intuitions about equations that are true over Z or R to equations over some entirely different domain. You might as well spend your time getting all worked up over the fact that in the finite field of 2 elements we have 1 + 1 = 0.

At this point I think I've exhausted my ability to convince, and I find further conversation about it to be a waste of time.

Wade, *how* could "if you try to do arithmetic with infinite cardinals, subtraction doesn't work the way you expect" make it "metaphysically impossible" for (e.g.) past time to be infinite?

And, more specifically, how can it do so *without* making it "logically impossible"?

(And: does WLC also consider that these counterintuitive properties of infinite sets invalidate the idea that time might be infinitely divisible? Does it, in particular, show that Newtonian physics and Einsteinian relativity are absurd and could never have been correct?)

I think there's been some misunderstanding here; I agree with you more than you might think.

Where I come from, a "contradiction" means arriving at a statement like "0 = 1" by a valid series of proof steps.

You haven't produced a "contradiction" because the operation of "subtraction" on infinite quantities, the way you are doing it, is not logically valid; it does not correspond to a single well-defined answer. I've said this over and over, but it doesn't seem to stick.

Of course I agree that subtracting infinities does not correspond to a single, well-defined answer. And certainly subtracting infinity in the way I've done is an operation forbidden in transfinite arithmetic, but that does nothing to change what

wouldresult if one were to do the subtraction (namely, subtracting the same quantity from both sides of the equation would render the equation false); but neither of us really disputes this. Also, I amnotactually convinced that subtractions of infinities lead to a self-contradiction in the normal philosophical/mathematical sense, and while Craig feels justified in calling the subtraction of equal quantities from equal quantities yielding non-identical results a “self-contradiction” I personally believe the “self-contradiction” label is somewhat misleading. That is why I consider the results of subtracting infinities paradoxical but not logically impossible.a similar thing could be said for Hilbert's Hotel; some believe the consequences to be absurd and some consider them to be merely odd

And some of us - those with mathematical training - regard the consequences as obvious and trivial.

I have mathematical training, and I regard the consequences obvious as well. Nonetheless, that e.g. subtracting the same quantity from both sides of a physically instantiated equation renders the equation false is at least

odd--we do not for example have an infinite number of marbles at our disposal to actually do such a thing. Nor do we have anything like Hilbert's Hotel where a hotel jam-packed with guests can be made vacant merely by moving guests into different rooms. That the subtraction of infinities and Hilbert's Hotel allow such oddities is not really in dispute; the disputable point is whether such odd results warrant believing e.g. the Hotel to be metaphysically possible. Personally, while I believe instances like Hilbert's Hotel and the subtraction of infinities warrant suspicion of an actual infinite existing in concrete reality, like you I donotfind them convincing arguments against the possibility of an actual infinite.What eventually convinced me of the metaphysical impossibility of an actual infinite (somewhat against my will) is the Grim Reaper paradox. Unlike the subtraction of infinities, the Grim Reaper paradox really

doesyield a p&~p contradiction.[I think that a previous comment of mine here has disappeared into the void, and my latest attempt to comment produced only a vague error message from Blogger. Apologies, Jeffrey, if lots of redundant stuff from me ends up here.]

Wade:

First you say that any difficulty in subtracting infinite cardinals (i.e., one way of attempting to do arithmetic on infinite quantities) casts doubt on the metaphysical possibility of actual infinities in the physical universe.

Then you say that the existence of perfectly sensible number systems with infinite quantities in them (i.e., other ways of attempting to do arithmetic on infinite quantities) has no bearing on the metaphysical possibility of actual infinities in the physical universe.

I don't see how you can have it both ways.

Your reconstruction of Craig's reasoning is, I'm afraid, quite hopeless. (Craig's fault rather than yours, if the reconstruction is accurate.) In particular, it equivocates on the phrase "subtracting infinities", which sometimes means taking an infinite set and discarding an infinite subset -- which indeed ought to be possible if there are actual infinities in the physical universe, and which is mathematically absolutely unproblematic; -- and sometimes means performing subtractions on infinite cardinals and hoping that they will behave just like subtracting finite integers -- which I can see no reason to expect whether or not there are actual infinities in the physical universe.

First you say that any difficulty in subtracting infinite cardinals (i.e., one way of attempting to do arithmetic on infinite quantities) casts doubt on the metaphysical possibility of actual infinities in the physical universe.

Then you say that the existence of perfectly sensible number systems with infinite quantities in them (i.e., other ways of attempting to do arithmetic on infinite quantities) has no bearing on the metaphysical possibility of actual infinities in the physical universe.

I don't see how you can have it both ways.

Perhaps this will help. The dispute is not whether an actual infinite is

logicallyimpossible, but whether it ismetaphysicallypossible. Showing that an actual infinite is logically possible (as by pointing to the existence of mathematical systems that involve the infinite) is insufficient in showing whether the actual infinite is metaphysically possible. Anything metaphysically possible is logically possible (thus, if something is logically impossible then it is not metaphysically possible), but not all logically possible things are metaphysically possible (thus, showing that something is logically possible is insufficient in showing it is metaphysically possible). My point on subtracting infinite cardinals in transfinite arithmetic is the paradoxical result of subtracting the same quantity from both sides of the equation making the equation false. One could argue that this consequence warrants believing that subtractions of infinity cannot exist in the real concrete world, much as one could argue that the paradoxes of Hilbert’s Hotel warrant believing that such a Hotel could not exist in concrete reality.Your reconstruction of Craig's reasoning is, I'm afraid, quite hopeless. (Craig's fault rather than yours, if the reconstruction is accurate.)Craig might sometimes go into more detail in his writings than his oral debates. In the excellent book

God? A debate between a Christian and an atheist, Craig illustrates the kind of “self-contradictions” he has in mind. After giving examples of how one can subtract the same quantity from the same quantity and get different results when dealing with the actual infinite, on page 4 Craig writes, “It needs to be understood that in both cases we have subtracted identical quantities from identical quantities and come up with contradictory answers.” I think the alleged absurdity of what Craig describes here is better illustrated and made more apparent with transfinite arithmetic and the results of subtracting the same quantity from both sides of a true equation, but maybe that’s just me.Gareth McCaughan:

Wade, *how* could "if you try to do arithmetic with infinite cardinals, subtraction doesn't work the way you expect" make it "metaphysically impossible" for (e.g.) past time to be infinite?There’s a bit of a straw man here; it’s not just “doesn’t work the way you expect” but that the subtraction of infinities (arguably) results in absurdities just as Hilbert’s Hotel (arguably) results in absurdities.

With that in mind, recall the argument I put forth earlier that I believe more or less faithfully represents Craig’s reasoning:

(1) If an actual infinite is not possible, then an infinite past is not possible.

(2) If an actual infinite is possible, then subtracting infinities is possible (in the concrete world, one could lock up all the mathematicians and do subtractions on infinity regardless).

(3) Subtracting infinities is not possible (supposedly, it leads to contradictions or is at least too paradoxical).

(4) Therefore, an infinite past is not possible.

You could attack (2) by saying some instantiations of an actual infinite are possible (such as an infinite past) and some are not (such as an actual infinite number of marbles all existing at the same time), but I don’t think this strategy would work. It seems that if an actual infinite sequence of past events is possible, then so is an infinite number of e.g. marbles. How? Just have each event in some infinite set of past events create a marble; this seems metaphysically possible if an infinite sequence of past events is.

Geez, if philosophers come up with things like the Grim Reaper "paradox" and are not immediately shot down by the physical reality, then philosophy is even in worse shape than I thought.

1. Instantaneous action doesn't seem possible, according to our current understanding.

2. Planck time puts a bound on how discretely we can set off events, according to our current understanding.

The grim reaper paradox only shows the metaphysical impossibility of actual infinities if the *only* thing preventing it from happening is that impossibility.

In our universe there are reasons why you couldn't do it that have (prima facie) nothing to do with t.m.i.o.a.i.; for instance, you'd need infinitely many Reapers arbitrarily close to Fred (so that lightspeed delays don't stop them doing their job) and then what actually kills him is the black hole they produce :-). More generally, in our universe it appears that either time is quantized or everything that happens happens smoothly, according to a bunch of differential equations, and neither of those situations permits infinitely many grim reapers acting on the same target at a set of times that has an accumulation point.

Perhaps you could argue: without t.m.i.o.a.i. there is nothing that would prevent some hypothetical universe (with different physical laws) allowing Grim Reapers. But that hypothetical universe would have to be very different from ours -- in particular, it would need to allow the sort of discontinuous change that our universe doesn't -- and I don't see why we should trust our intuitions that (e.g.) if the reapers collectively kill Fred then one of them individually must kill Fred, when dealing with a universe so different from ours. So I don't see a route from "if X, then in a universe very unlike ours, something very counterintuitive could happen" to "X is metaphysically impossible".

(Even in our universe there are phenomena that seem strange in the same sort of "having an effect without doing anything" way. Consider, e.g., the Elitzur-Vaidman bomb-tester.)

Wade said:

It seems that if an actual infinite sequence of past events is possible, then so is an infinite number of e.g. marbles. How? Just have each event in some infinite set of past events create a marble; this seems metaphysically possible if an infinite sequence of past events is.Wouldn't a universe with finite mass and energy not allow an infinite number of marbles but still allow an infinite sequence of past events?

Jeffrey: You can rescue the Grim Reaper argument from your first objection by giving reaper n the interval from 1/(2n+2) to 1/(2n+1) to work with. Then each has a nonzero amount of time, and each reaper's operation is separated from the others' by a nonzero amount of time. (I do think the argument is hopeless, so this is just a quibble.)

Wade: You didn't merely say that showing something's logically possible doesn't prove it's metaphysically possible. You said: "the existence of such a mathematical system is

irrelevantto whether the actual infinite is metaphysically possible". (Emphasis mine.)Aside from that, I think you're just repeating yourself now, and in any case arguments of the form "X is

metaphysically impossible, which I know becauseit seems likeif X were true then we could make Y true too, andit seems likeY doesn't make any sense" are obviously hopeless unless you have an absurdly inflated opinion of your intuition. So I'm done with this. Thanks for the discussion.Anonymous:

Wouldn't a universe with finite mass and energy not allow an infinite number of marbles but still allow an infinite sequence of past events?Perhaps, but recall this is about

metaphysicalpossibility, not physical possibility. The conservation of mass-energy is a contingent physical law and its existence could have been otherwise (hence the necessity for empirical observation for us to know about it) and even so some speculate there’s a way to get around it by creating some positive matter at the expense of negative matter. That’s highly speculative, but it is strictly within the physical law. Regardless, if an infinite sequence of past events is metaphysically possible, so is an infinite number of marbles; just have each event in some infinite set of past events create a marble (perhaps at the expense of creating negative matter).Gareth McCaughan, what is “t.m.i.o.a.i”? You mentioned it at one point but didn’t define it. Anyway, regarding the Grim Reaper paradox,

More generally, in our universe it appears that either time is quantized or everything that happens happens smoothly, according to a bunch of differential equations, and neither of those situations permits infinitely many grim reapers acting on the same target at a set of times that has an accumulation point.Here you seem to be saying that time being continuous prohibits the infinite Grim Reaper team to check on Fred (and kill him if he’s alive) at the times they do in the paradox, but why on earth would

thatbe true? You’ve given no explanation and no argument here. Recall also that if Fred is already dead, the Grim Reapers do nothing. A Grim Reaper kills Fred only if Fred is alive, and at most only one Grim Reaper can kill him in this scenarioYou said, “I don't see why we should trust our intuitions that (e.g.) if the reapers collectively kill Fred then one of them individually must kill Fred” but Pruss specifically addresses the mereological summation (i.e. they kill Fred collectively) objection, noting that the Grim Reapers “check if Fred is already dead before they do anything, so in the present setting, none of them actually

doanything—and if they don't do anything, how can they kill Fred?” I suspect the question is rhetorical; it seems reasonably clear that the Grim Reapers cannot kill Fred if none of them do anything.Wade: You didn't merely say that showing something's logically possible doesn't prove it's metaphysically possible. You said: "the existence of such a mathematical system is irrelevant to whether the actual infinite is metaphysically possible".Fair enough, but (1) don’t forget the spirit of my objection: you said the existence of such a mathematical system made things worse when clearly its existence is irrelevant (not only is the actual infinite’s logical possibility not disputed, but it’s logically possibility doesn’t entail it being metaphysically possible); (2) what I said was true anyway, e.g. even if we humans had yet to construct such a mathematical system, this by itself wouldn’t make an actual infinite metaphysically impossible.

Aside from that, I think you're just repeating yourself now, and in any case arguments of the form "X ismetaphysically impossible, which I know becauseit seems likeif X were true then we could make Y true too, andit seems like Ydoesn't make any sense" are obviously hopeless unless you have an absurdly inflated opinion of your intuition.It’s easy to find a counterexample. Let X = “There is a married bachelor” and Y = “There is a married man who is unmarried.” For me to conclude that X entails Y and that Y doesn’t make sense hardly necessitates my having “an absurdly inflated opinion” of my own intuition. Nor would I need to an absurdly inflated opinion of my intuition to conclude that the Grim Reaper paradox—which results in a self-contradiction—is metaphysically impossible.

Jeffrey Shallit, regarding the Grim Reaper paradox:

1. Instantaneous action doesn't seem possible, according to our current understanding.

2. Planck time puts a bound on how discretely we can set off events, according to our current understanding.

(1) with the advent of quantum entanglement, it’s at best unclear whether instantaneous action is physically impossible. (2) Planck time by itself is merely a unit of measurement, and it isn’t clear whether time is composed of discrete units of Planck time any more than space is composed of discrete units of Planck length. For example, cosmologists speak of what the universe was like within the first Planck unit of time, but recognize that we don’t yet have the physics knowledge to describe it (we’d need some suitable marriage of relativity and quantum mechanics).

That said, (1) and (2) are irrelevant anyway. As Gareth McCaughan pointed out, the Grim Reaper paradox does not require instantaneous action, and since the paradox is more about metaphysical possibility than physical possibility, (2) misses the point. It’s surely

metaphysicallypossible for a universe to have continuous time, just as surely as we’d need empirical investigation to determine whether our own universe is composed of discrete units of time. Yet if that’s true, it seems that if an infinite sequence of past events (or for that matter, an actual infinite in general) is metaphysically possible, then so is an infinite number of Grim Reapers as well as the Grim Reaper paradox.Who cares whether something is "metaphysically possible" (whatever that is supposed to mean)? We were discussing whether the universe we inhabit could have infinitely many objects, not whether some universe you imagine does. In our universe, the Grim Reaper paradox is countered by our current understanding of the quantization of time.

[Damn, Blogger ate my comment again, and again and again and again. Apologies if two versions of this appear. Or more. From past experience I think they won't. I've tried authenticating differently; apologies if that makes extra work for Jeffrey somehow.]

Wade, I said I was done but it would be rude for me not to answer your questions so here goes.

"t.m.i.o.a.i." is short for "the metaphysical impossibility of actual infinities". I was just bored of typing that phrase over and over again. I thought the meaning was obvious; my apologies.

I think it's obvious (kinda: see below) that physical laws of the "classical" differential-equation sort don't permit an infinite convergence of grim reapers as required for the "paradox". I haven't attempted to give a proof because that would be a lot of work, and I'm not the one making extravagant metaphysical claims here. Feel free to refute me by coming up with an actual set of physical laws of that kind and solutions to them that behave in the required manner. (I'd find that surprising but not astonishing; "obvious" above means not "obviously provable, if I took a little time" but "the most obvious guess, which seems very much more likely right than wrong"; but mathematics is surprising sometimes, and of course I might be wrong.)

And, while I'm here, super-brief comments on the rest.

Pruss didn't refute the none-individually-but-all-collectively argument; he just restated his intuition that it couldn't be so (which one should not trust when considering a universe different from ours in a way that allows an infinite convergence of grim reapers).

I think it's reasonable to trust your intuition about simple *logical* impossibilities more than your intuition about *metaphysical* impossibilities; married bachelors are in the former category, actual infinities and infinite swarms of grim reapers in the latter. So your alleged counterexample isn't.

Jeffrey, Wade's argument goes like this: (1) I can imagine a universe in which an infinite converging swarm of grim reapers happens; (2) that's obviously impossible; (3) therefore something about that scenario is impossible; (4) the only possible candidate for what's impossible is the existence of "actual infinities"; (5) therefore actual infinities are metaphysically impossible.

I think #1 is only right if "imagine" is taken in a rather weak sense; #2 is highly debatable; #3 is a reasonable inference from #1 and #2; #4 is absurdly wrong; #5 is a reasonable inference from #3 and #4. All, of course, assuming that a coherent meaning can be given to "metaphysically impossible", which I agree is doubtful.

Unsound as the argument is, what invalidates it isn't the fact that the scenario it describes can't happen in our universe.

Gareth McCaughan, regarding the Grim Reaper paradox:

I think it's obvious (kinda: see below) that physical laws of the "classical" differential-equation sort don't permit an infinite convergence of grim reapers as required for the "paradox". I haven't attempted to give a proof because that would be a lot of work, and I'm not the one making extravagant metaphysical claims here.To claim that continuous time makes it metaphysically impossible for the Grim Reapers to check on Fred at the times they do in the paradox would be a pretty extravagant metaphysical claim. Similarly, the claim that the Grim Reapers collectively kill Fred when none of them do anything is a

veryextravagant metaphysical claim.Feel free to refute me by coming up with an actual set of physical laws of that kind and solutions to them that behave in the required manner.We can conceive of a kind of universe where Grim Reapers are supernatural beings or the naturalistic equivalent (e.g. because they are composed of exotic matter X, the normal laws don’t apply to them in that they can check whether Fred is alive in any positive nonzero time interval, can kill Fred in any positive nonzero time interval, and so forth). It’s very difficult to see how continuous time would prevent such entities from existing. If you have a good argument for such a claim, please let me know.

Pruss didn't refute the none-individually-but-all-collectively argument; he just restated his intuition that it couldn't be so (which one should not trust when considering a universe different from ours in a way that allows an infinite convergence of grim reapers).He did more than that; he posed a question that you still have not answered. Pruss said that the Grim Reapers “check if Fred is already dead before they do anything, so in the present setting, none of them actually

doanything—and if they don't do anything, how can they kill Fred?” If none of the Grim Reapers do anything, it’s very implausible that they collectively kill Fred.I think it's reasonable to trust your intuition about simple *logical* impossibilities more than your intuition about *metaphysical* impossibilities; married bachelors are in the former category, actual infinities and infinite swarms of grim reapers in the latter.Married bachelors actually fit the category of both. My intuition suggests that self-contradictory propositions are metaphysically impossible. You could say my intuition is wrong here, but that wouldn’t be very plausible.

Jeffrey, Wade's argument goes like this…Again, you’ve made a bit of a straw man. The argument I’m referring to is the same one Pruss presented, which went something like this:

(1) If an infinite sequence of past events were possible, then Hilbert’s Hotel (a hotel with infinitely many rooms and guests, and some nice infinity paradoxes) would be possible.

(2) If Hilbert’s Hotel were possible, then the Grim Reaper paradox could happen.

(3) The Grim Reaper paradox cannot happen.

(4) Therefore, an infinite sequence of past events is not possible.

Like the argument or not, it is deductively valid, and if sound has the bonus of entailing that Hilbert’s Hotel is not possible. While Pruss didn’t specify the type of possibility he’s using, I suspect it’s metaphysical possibility. I was unconvinced that the actual infinite is metaphysically impossible until I saw the Pruss’s argument above. Like you and Jeffrey Shallit, I did not find Craig’s infinity subtraction argument convincing.

Jeffrey Shallit,

Who cares whether something is "metaphysically possible" (whatever that is supposed to mean)?A proposition is metaphysically possible if and only if it describes the way the world is or could have been like. For example, a universe with different physical laws is metaphysically possible (hence the need for empirical investigation in science) whereas married bachelors are metaphysically impossible. If the infinite past is metaphysically impossible, then our universe cannot be infinitely old.

William Lane Craig agrees that actual infinites are logically possible, but also believes that they are metaphysically impossible.

But again, I think this kind of reasoning is a waste of time. Either the universe allows infinitely many objects, or it doesn't. But we cannot decide this by reasoning about

supposedly possibleuniverses when (a) we have no evidence that such universes could exist and (b) we recklessly insist on applying our intuition about finiteness to infinite cases.For example, in your new account of Pruss, you insist that the grim reapers "check" to see whether you are dead before killing you. In our universe, provided the grim reapers are at finite distance from you, this check cannot be done faster than the speed of light, and as Gareth already observed, we can't pack infinitely many grim reapers in a finite spatial interval. So Pruss's "paradox" disappears because of relativity.

So far I am completely unimpressed. I must say, it confirms my already poor impression of theist philosophy.

But again, I think this kind of reasoning is a waste of time. Either the universe allows infinitely many objects, or it doesn't. But we cannot decide this by reasoning aboutsupposedly possibleuniverses when (a) we have no evidence that such universes could exist and (b) we recklessly insist on applying our intuition about finiteness to infinite cases.None of this really attacks or even addresses a premise of the argument. With regard to the premises, point (a) is irrelevant; none of the premises even remotely require that such universes exist, much less require evidence for them. Point (b) doesn’t seem to apply at all. Surely it’s not a reckless practice of intuition to conclude that self-contradictory situations (e.g. Fred is both a live and not alive) are metaphysically impossible, whether dealing with infinite cases or not. If point (b) is to have any useful relevance, it will need to be more fine-tuned.

I mention that you didn’t attack the premises because the conclusion of the Grim Reaper argument follows logically and necessarily from the premises (I can prove this via symbolic logic if you wish, thanks be to analytic philosophy!). A deductive argument can fail to be sound in only one of two ways: it is invalid (basically, that the conclusion doesn’t follow from the premises) or at least one of the premises is false. Yet with the Grim Reaper argument the conclusion does follow, and the premises are justifiably true. Do you agree? If not, what reason do you have to reject the premises? I am genuinely interested in hearing a mathematician’s rebuttal to the veracity of the premises because unlike Craig’s subtraction of infinities argument, I don’t see any good objections here.

For example, in your new account of Pruss, you insist that the grim reapers "check" to see whether you are dead before killing you. In our universe, provided the grim reapers are at finite distance from you, this check cannot be done faster than the speed of light, and as Gareth already observed, we can't pack infinitely many grim reapers in a finite spatial interval.First, my account of Pruss isn’t new. It’s the same as it was before (Gareth presented a straw man earlier). Second, your objections raise points that are irrelevant to the veracity of the premises. The fact that it would be physically impossible in our universe for (physical) Grim Reapers to exist does nothing to show that it would be

metaphysicallyimpossible. For example, we can conceive of a universe where Grim Reapers are composed of exotic material X whereby the Grim Reapers can come together in the same space without creating a black hole any more than 10^900 photons coming together in the same space would. At the very least, we don’t have any known metaphysical constraints (e.g. it being self-contradictory, like a married bachelor is) that imply such a thing is metaphysically impossible (perhaps this is analogous to a hypothetical physical event being consistent with known physical laws). The first two premises appear true, as does the third premise (the Grim Reaper paradox cannot happen if it yields the self-contradiction it appears to produce). If you have a good and relevant objection to a specific premise of the argument I would very much like to hear it.Wade: Now you seem even more detached from reality than before!

Pruss's argument fails because of his claim (2): our infinitely many hotel rooms have to occupy unbounded space; therefore it will be impossible to build infinitely many GR's that successfully "check" (in an arbitrarily small time scale) to see if Fred is dead before attempting to kill him, since such a check cannot (in our current understanding of the universe) be done faster than the speed of light.

If the fact that our universe doesn't allow the scenario you describe doesn't sway you that the argument has nothing to do with the possible infinitude of objects in our universe, then obviously nothing will.

Also, Gareth put forth another objection that I think is quite relevant:

"I don't see why we should trust our intuitions that (e.g.) if the reapers collectively kill Fred then one of them individually must kill Fred, when dealing with a universe so different from ours."

I agree with Gareth, in the following sense: if the game you want to play (and it seems to be) allows conjuring up arbitrary universes with arbitrary physical rules, then you can get paradoxical results much more easily, e.g., with the Banach-Tarski paradox. Why not just say, "See, if we had infinitely many rooms we could perform the Banach-Tarski construction and create matter out of nothing?" That's much cleaner than the grim reaper "paradox" you put forth.

Pruss's argument fails because of his claim (2): our infinitely many hotel rooms have to occupy unbounded space; therefore it will be impossible to build infinitely many GR's that successfully "check" (in an arbitrarily small time scale) to see if Fred is dead before attempting to kill him, since such a check cannot (in our current understanding of the universe) be done faster than the speed of light.Again, you’ve raised a point that’s irrelevant to the veracity of the premises. For the nonce, let’s ignore the possibility of wormholes. As I said earlier, the fact that it would be physically impossible in our universe for (physical) Grim Reapers to exist does nothing to show that it would be

metaphysicallyimpossible. Neither the speed of light (299,792,458 meters per second) nor the speed of light being the speed limit is metaphysically necessary. Assuming Pruss’s argument refers to metaphysical possibility, premise (2) says that if Hilbert’s Hotel weremetaphysicallypossible, then the Grim Reaper paradox would bemetaphysicallypossible. Pointing out that the Grim Reaper paradox isn’tphysicallypossible (due to e.g. limitations imposed by the speed of light) doesn’t quite hit the target.If the fact that our universe doesn't allow the scenario you describe doesn't sway you that the argument has nothing to do with the possible infinitude of objects in our universe, then obviously nothing will.First, it should be immediately clear that the “fact that our universe doesn't allow the scenario” I describe doesn’t relevantly attack any premise of the argument. Don’t you agree? Second, is it possible for me to be swayed that the argument is unsound? All that needs to be done for that is to show that a premise is false! Considering that a false premise is the only possible way for the argument to be unsound (given its deductively validity), I don’t think this is too much to task for.

The conclusion and the corollary regarding Hilbert’s Hotel clearly has

somethingto do with an infinitude of objects in our universe. The conclusion says that it’s (metaphysically) impossible for the universe to have a backwards infinite sequence of events, and a corollary is that Hilbert’s Hotel is (metaphysically) impossible. Still, I agree that the argument in its current form doesn’t show that an actual infinitude in general is impossible; for that you’d need to add a premise like “If an actual infinite is metaphysically possible, then Hilbert’s Hotel is metaphysically possible.”I agree with Gareth, in the following sense: if the game you want to play (and it seems to be) allows conjuring up arbitrary universes with arbitrary physical rules, then you can get paradoxical results much more easily, e.g., with the Banach-Tarski paradox….That's much cleaner than the grim reaper "paradox" you put forth.First, it doesn’t seem like you’re agreeing with Gareth’s point in any meaningful sense there (being able to create easier paradoxes doesn’t seem to have anything to do with what Gareth said). Second, while the Banach-Tarski paradox is an interesting one, it’s more complicated and it doesn’t yield a self-contradiction like the Grim Reaper paradox does. Self-contradictory situations (e.g. a married bachelor) are the gold standard for metaphysical impossibilities.

1. Pruss's argument is either existential or universal. His blog post is so poorly worded I can't tell which he means. I took it to refer to our universe, but you insist this is not the case.

2. If it is existential, I take it to apply to some universe of his own construction. Then who cares what it proves? It has no impact at all on our universe, since then one of his premises is false, as I have shown

3. If it is universal, then it must apply to all universes, including ours. Then

my objection explains why it is not a paradox in our universe.

In all three cases it proves

nothingof interest or relevance to our discussion. If you disagree, explain which of 1-3 you disagree with.I guess you didn't understand my point about Banach-Tarski, either. The point is, if you are basing your argument on intuition about what it means to be alive, and assuming your intuition will hold in every possible universe, then you might as well base your argument on the impossibility of making larger spheres from smaller ones. I think this is a "paradox" in

exactlythe same way you are insisting grim reaper is.Jeffrey Shallit, regarding the Grim Reaper argument, do you believe any of the premises are false? If so, which one and why? So far you haven’t given a relevant attack against any premise of the argument.

1. Pruss's argument is either existential or universal. His blog post is so poorly worded I can't tell which he means. I took it to refer to our universe, but you insist this is not the case.

2. If it is existential, I take it to apply to some universe of his own construction. Then who cares what it proves? It has no impact at all on our universe, since then one of his premises is false, as I have shown

I don’t see how you’ve shown that any of his premises are false. Where did you do this? If you’re referring to the objection of your previous comment, what about my rebuttal? Remember, the fact that the GR paradox would be physically impossible in our universe does nothing to show that it would be

metaphysicallyimpossible.I’m not entirely sure what you mean by “universal” or “existential” in this context (i.e. in a way that would involve it being a relevant attack against a premise of the argument); could you precisely define how you’re using these terms and how either 1, 2, or 3 would make a premise false or the conclusion irrelevant? For example, suppose it is true that the antecedent and the consequent of the second premise refer to metaphysically possible worlds that are not our own. To say that this makes the conclusion useless doesn’t seem to follow since the conclusion is

stillthat there cannot be an infinite sequence of past events (i.e. that such a sequence of past events does not exist in any metaphysically possible world). Or maybe you think this conclusion doesn’t follow from the premises (from your “It has no impact at all on our universe” remark), but if you believe this I’ll happily prove the argument’s validity via symbolic logic.I guess you didn't understand my point about Banach-Tarski, either. The point is, if you are basing your argument on intuition about what it means to be alive, and assuming your intuition will hold in every possible universe, then you might as well base your argument on the impossibility of making larger spheres from smaller ones.(I thought the paradox had to do with making two spheres from the same size from a sphere that was the same size as the other two, but that is neither here nor there.)

You may argue that a married person who is not married is metaphysically impossible. But if you’re basing your argument on intuition on what it means to be married, and assuming your intuition will hold in every possible universe, then you might as well base your argument on the impossibility of making larger spheres from smaller ones.

Obviously that doesn’t quite work, because the law of noncontradiction applies regardless of how we define “alive” or “married.” Whatever “alive” is, the Grim Reaper makes Fred “not alive” and the self-contradictory state of being both alive and not alive at the same time and in the same context is as metaphysically impossible as a married bachelor.

And really, the Grim Reaper and Fred’s alive-ness is bit arbitrary. We can create a self-contradictory result with a jar of marbles and an infinite number of marble thieves (no marble thief

ntakes the marbles, because there’s always a marble thiefn + 1before him that would have stolen the marbles…).Wade:

The distinction between universal and existential is a basic one in logic and mathematics.

Suppose your space of possibilities is U. A universal claim is a proposition P that is true for every element of U. An existential claim says that P is true for at least one element of U. Which do you understand Pruss to mean?

I don’t see how you’ve shown that any of his premises are false. Where did you do this?We're really getting nowhere.

In our universe, his claim (2) is not true based on our current understanding. So even if it is true forsomeuniverse, that says nothing at all about ours.I ask again, which of my (1) - (3) do you dispute?

the self-contradictory state of being both alive and not alive at the same timeI take it you've never read about SchrÃ¶dinger's cat.

I thought the paradox had to do with making two spheres from the same size from a sphere that was the same size as the other twoThat's one statement of it, but not the only one, as you could have easily verified yourself by consulting Wikipedia.

(Addressing only questions, accusations of dishonesty, and flagrant errors. I am done with substantive philosophical debate with Wade here, as already mentioned. -- Oh, and Blogger's been screwing me around even more, which is why this is so long delayed. I don't know what the underlying problem is.)

Wade, it isn't possible to answer the question "how do they kill Fred?" since we've got no information about how even a single Reaper would kill him. (It can't be by any process that's physically possible in our universe.) You're just postulating Reapers as an unanalysed feature of how the world works, and then "when you have infinitely many Reapers arranged like so, their victim dies at time t0" is a provable consequence; asking "how" in the absence of further information about what the Reapers are doing is like asking "how" one massive body attracts another at a distance when all you've got is the laws of Newtonian mechanics.

I have not said that "continuous time makes it metaphysically impossible [etc.]"; please do not misrepresent me.

Duh, of course if something is logically impossible then it's metaphysically impossible. The point is that if something is logically impossible then you don't need any specifically-metaphysical intuition to know it's metaphysically impossible, whereas if it isn't then you do.

Steps 1 and 2 in the argument you present are (or at least) require arguments themselves. Neither you nor Pruss have presented a deductively valid argument for either of them. I think it's a bit off to say "This is a deductively valid argument" when what you're really doing is taking an invalid argument and collapsing the invalid steps into premises.

The distinction between universal and existential is a basic one in logic and mathematics.Yes, I understand the general definition of the terms, my question however was exactly how they applied here. For example, you said,

Suppose your space of possibilities is U. A universal claim is a proposition P that is true for every element of U. An existential claim says that P is true for at least one element of U. Which do you understand Pruss to mean?But you didn’t specify

whichclaim you were referring to. Are you referring to the antecedent of the first premise? The first premise in general? The conclusion? The conclusion, as I indicated earlier, is clearly making a universal claim, i.e. that an infinite sequence of past events does not exist in any possible world.In any case, among the first things that came to mind when you spoke of the existential and universal is the “possible world” semantics of modal logic (I’m guessing you might be talking about this when you described U as a “space of possibilities”), which I kind of alluded to this when I referred to “metaphysically possible worlds” in my previous comment. If we use this semantics in our modal logic we get a mixture of universal (true/false in all possible worlds) and existential (true/false in at least one possible world) claims in the argument. So when you claim, “Pruss's argument is either existential or universal” and ask me which one it is, you can see why I asked for clarification as to how you’re using the terms, because the argument uses a mixture of both if you were referring to possible world semantics.

Recall that a proposition is metaphysically possible if and only if it describes the way the world is or could have been like. A “possible world” is a complete description of a metaphysically possible world. In possible world semantics, “metaphysically possible” ends up being “true in at least one possible world” and “impossible” ends up being “false in all possible worlds.” With that in mind, and assuming Pruss’s argument was using metaphysical possibility, let’s reframe the argument as follows:

(1) If in infinite sequence of past events were metaphysically possible, then Hilbert’s Hotel would be metaphysically possible.

(2) If Hilbert’s Hotel were metaphysically possible, then the Grim Reaper paradox would be metaphysically possible.

(3) The Grim Reaper paradox is metaphysically impossible.

(4) Therefore, an infinite sequence of past events is metaphysically impossible.

Using possible world semantics, we have a mixture of existential and universal claims, with the existential claims being the antecedents and consequents of premises (1) and (2), and the conclusion itself being a universal claim (“an infinite sequence of past events exists” is false in all possible worlds).

With the above clarification, let’s revisit 2 and 3:

2. If it is existential, I take it to apply to some universe of his own construction. Then who cares what it proves? It has no impact at all on our universe, since then one of his premises is false, as I have shown

3. If it is universal, then it must apply to all universes, including ours. Then my objection explains why it is not a paradox in our universe.

It’s a paradox in our universe all right but only in the sense that the scenario so described yields a contradiction. We agree that it isn’t a paradox in our universe in the sense that it could actually

occurin our universe, but the argument doesn’t require otherwise. Nor does the argument require that the Grim Reaper paradox could occur in all possible worlds. Rather, the second premise just says that if Hilbert’s Hotel were metaphysically possible (i.e. it exists in some possible world), then the Grim Reaper paradox would be metaphysically possible (i.e. it would exist in at least one possible world).You could say that this premise has no impact in our own universe, but it

doeshave an impact when coupled with premises (1) and (3). If all three premises are true, then the conclusion follows inescapably.I take it you've never read about SchrÃ¶dinger's cat.Yes I have, but the interpretation that the cat is both alive and not alive at the same time and in the same context is just bad philosophy of science (for that matter, interpreting superpositions in general as violations of the law of noncontradiction is bad philosophy of science).

That's one statement of it, but not the only one, as you could have easily verified yourself by consulting Wikipedia.Fair enough; I read only about the other version (though not in Wikipedia).

We're still not getting anywhere.

I repeat my question: which of (1) - (3) do you take issue with?

It’s a paradox in our universe all right but only in the sense that the scenario so described yields a contradiction.No, it yields no contradiction, because Pruss's (2) is not satisfied in our universe.

We've made so little progress here that I think it is fruitless to continue.

Please let me know when Pruss's paper appears in a journal somewhere, because I am going to have fun writing a rebuttal.

I'm going to restate Pruss as I understand him. If you disagree with something let me know:

1. Fix a possible universe U. If U allows an infinite sequence of events, Hilbert's Hotel would be possible, in U.

2. If U permits the construction of Hilbert's hotel, then U permits the possibility of constructing all those grim reapers.

3. But the grim reapers allow someone to be killed by all without being killed by any individual grim reaper. Pruss and you find this absurd.

4. Therefore, there cannot be a backwards infinite sequence of events in U.

Once we make it clear what universe we are talking about, suddenly the supposed "contradiction" disappears, and claim 2 is just wrong. It could well be that U permits the construction of a Hilbert hotel, yet does not permit the construction of the grim reapers with the properties they are supposed to have. Indeed, U = our universe is a possible counterexample.

Claim 3 is also suspect, as previously pointed out.

Gareth McCaughan, regarding the Grim Reaper argument,

Wade, it isn't possible to answer the question "how do they kill Fred?" since we've got no information about how even a single Reaper would kill him.Sure we do; a Grim Reaper swings his scythe and it’s all over for Fred. But if none of the Grim Reapers do anything, how do they collectively kill Fred? It doesn’t seem very plausible that they would.

Perhaps modifying the scenario in the following way will make the problem more visible: suppose Fred has 100 hit points and is dead only when his hit point level is less than or equal to 0 hit points. Each Grim Reaper has a “scythe attack” that can take off 100 hit points from Fred, thereby killing him if he isn’t already dead. Cumulative Grim Reaper attacks are additive, e.g. three Grim Reapers each doing a scythe attack collectively take off 300 hit points. At 11 a.m. + 1/

nminutes, Grim Reapernchecks on Fred to see if he is alive. If Fred is alive, the Grim Reaper does his scythe attack, taking 100 hit points away from Fred and thereby killing him. If Fred is already dead, the Grim Reaper does nothing and thereby takes away exactly 0 hit points away from Fred. Yet it seems for any Grim Reapern, that Grim Reaper doesn’t do anything (i.e. takes away 0 hit points) because there’s another Grim Reapern + 1before him. And if none of the Grim Reapers do anything (they each swipe away 0 hit points), they collectively take away 0 hit points from Fred, thereby not killing him. The idea that the Grim Reapers collectively kill Fred just doesn’t work.I have not said that "continuous time makes it metaphysically impossible [etc.]"; please do not misrepresent me.I’m sorry if I misunderstood you, but then it’s hard to see how your objection could relevantly attack any premise of the argument.

Steps 1 and 2 in the argument you present are (or at least) require arguments themselves. Neither you nor Pruss have presented a deductively valid argument for either of them. I think it's a bit off to say "This is a deductively valid argument" when what you're really doing is taking an invalid argument and collapsing the invalid steps into premises.The argument as Pruss presented it is deductively valid, and the fact that the premises themselves require arguments (and I agree they do) doesn’t change this. Nonetheless, Pruss presented arguments for the premises, and I have yet to see a good rebuttal for them. If you know of any please let me know.

I repeat my question: which of (1) - (3) do you take issue with?If you’ll recall I took issue with “Pruss's argument is either existential or universal” which was (1). The reason is that some parts of the argument are existential and some parts of it are universal (assuming you’re referring to the possible world semantics of modal logic). (1) as it was didn’t seem to make sense (hence my repeated questions of clarification); it was as if you weren’t understanding the argument correctly. On that note, let’s visit this:

I'm going to restate Pruss as I understand him. If you disagree with something let me know:

1. Fix a possible universe U. If U allows an infinite sequence of events, Hilbert's Hotel would be possible, in U.

2. If U permits the construction of Hilbert's hotel, then U permits the possibility of constructing all those grim reapers.

3. But the grim reapers allow someone to be killed by all without being killed by any individual grim reaper. Pruss and you find this absurd.

4. Therefore, there cannot be a backwards infinite sequence of events in U.

This misunderstands Pruss’s argument. There is no fixed possible universe U running through all four lines of the argument, and I think this misunderstanding was what was impeding communication between us. I tried to clear up what Pruss’s argument was in a previous comment but I’ll try again, bringing out the possible world semantics more explicitly this time. Assuming Pruss was using metaphysical possibility in his argument, we get the following:

(1) If an infinite sequence of past events were to exist in at least one possible world, then Hilbert’s Hotel would exist in at least one possible world.

(2) If Hilbert’s Hotel were to exist in at least one possible world, then the Grim Reaper paradox would exist in at least one possible world.

(3) It is not the case that the Grim Reaper paradox exists in at least one possible world (the Grim Reaper paradox is metaphysically impossible).

(4) Therefore, it is not the case that an infinite sequence of past events exists in at least one possible world.

I preferred dispensing with the more explicit use of possible world semantics since I feared they might foster misunderstanding (e.g. perhaps the reasoning behind Pruss’s justification for the premises would be less graspable), but in this case perhaps it’ll help you correctly understand the argument. Again, there is no fixed universe running through all four lines of the argument, or even in the first two lines.

I am still unsure what your claim 4 is intended to mean. Is it:

4. It is not the case that P holds, where P is the proposition "there exists a possible world W having an infinite sequence of past events".

Because if it is, I can create a possible world where this happens: namely a world W like our world, existing infinitely far in the past, where at every time -t in the past a hotel of size 1/2 comes into existence at position t. Further, in such a world, time and space are quantized and relativity applies. In such a world the grim reaper "paradox" is not possible, because the hypotheses you demand cannot be put into place, as we've discussed. So W seems a counterexample to 4.

That is a non-sequitur. P does not imply any such capability of your creating arbitrary possible worlds like the one you propose, and indeed I can define

P and not quantized (and/or not relativized)

This world satisfies P but precludes your construction.

I can further construct such a possible world P that is not assumed to have any minimum quantum of time by merely relaxing the quantum assumption in our known world, e.g. by defining a new unit of time the silisecond such that time in siliseconds forward or back from the time of this comment are defined by an integral of t^-2 seconds

such that the first silisecond forward or backward is of duration 1/1 second, the second silisecond is of duration 1/4 second, the third is of duration 1/8 second.

Our whole infinite universe forward and backward now fits into ±2 seconds, where the infinite number of events of interest in either direction are defined to occur as each integral number of seconds passes.

Another infinite universe takes places in achileseconds, each successive achillesecond being the time it takes for Achilles to cover the lead of the tortoise after the previous achillesecond.

At the time Achilles overtakes the tortoise there is a past of an infinite number of such achillesecond events in Xeno's paradox.

David:

There is some confusion here. If the conclusion is "it is not the case that P holds", then this conclusion can be defeated by producing a single counterexample. This is what I have done. Talking about worlds other than the one I have suggested is not relevant.

Yes I have, but the interpretation that the cat is both alive and not alive at the same time and in the same context is just bad philosophy of science (for that matter, interpreting superpositions in general as violations of the law of noncontradiction is bad philosophy of science).I'm not so sure about that. Maybe quantum mechanics says something deeper about the structure of our universe. Maybe some propositions about the physical world have no absolute truth value, but only a probabilistic distribution of truth.

But I guess this is just a side distraction from our current discussion.

I like these metaphysical "proofs" because the rules for deciding what you are allowed to do, and what constitutes a contradiction, are so vague, it seems you can prove almost anything.

Let's "prove" that one of the requirements of the grim reaper paradox - that you have a killing machine that can both test if you are dead and kill you if you aren't, within an

arbitrarily short time period- is metaphysically impossible.Suppose in some possible world all those grim reapers exist. They function in arbitrarily small time intervals, and act simultaneously, so in some possible world we can also turn lights on and off within an arbitrarily small time period. In my possible world lights are either on or off and they change state instantaneously.

I will just use a single light switch and a light. When the light switch is flipped, the light goes on (if it was off) or off (if it was on). Initially it is off.

Now at time t = 1/2, 3/4, 7/8, ... I flip the switch. Question: is the light on or off at time t = 1? It has to be one or the other because of my imposed requirements.

Well, if it is on at t = 1, this can only be - since it was initially off - from the fact that it was turned on at some point. Which point? If you say because it was turned on at 1 - 2^{-n}, I counter by saying, no, it was turned off at the later time 1 - 2^{-(n+1)}.

Now it is turned on at t = 1/2, so by the same argument, if it is off at time t = 1, this can only be because it was turned off at some point after t = 1/2. Which point? Same argument applies.

Therefore it cannot be either on or off, a contradiction. So arbitrarily small time periods are metaphysically impossible.

David:

That is a non-sequitur.If you’re saying the Grim Reaper argument is a non sequitur, say the word and I will happily prove otherwise via symbolic logic.

“Yes I have, but the interpretation that the cat is both alive and not alive at the same time and in the same context is just bad philosophy of science (for that matter, interpreting superpositions in general as violations of the law of noncontradiction is bad philosophy of science).”

I'm not so sure about that.

Violations of the law of noncontradiction are (by definition) logically impossible. Unsurprisingly then, I have yet to see a good argument for this bizarre metaphysical interpretation of quantum mechanics. Of course, one could say that asking whether an electron is spin-up or spin-down when the electron is in a certain superposition is a category error, like asking whether the number 2 tastes like chicken or roast beef. But this conception of a superposition, odd as it might be compared to the macro-world, would be very different from the lunacy of accepting the logically impossible as possible. You can’t get much more irrational than believing in logically impossible propositions.

Incidentally, the adherent of the Copenhagen interpretation need not accept that the cat is in a superposition of alive/dead; it seems that the most sensible Copenhagen adherent would conclude that the wave function collapses before then (say, by the time the radioactive particle detector determines whether to release the poison gas).

I am still unsure what your claim 4 is intended to mean.Recall that I used possible worlds semantics for modal logic (if you’re unfamiliar with modal logic or symbolic logic in general, I recommend reading up on it; a mathematician such as yourself might enjoy it). Since we’re using possible world semantics, line (4) means “It is not possible that an infinite sequence of past events exists,” or, since the type of possibility we’re using is metaphysical possibility, line (4) means “It is not metaphysically possible that an infinite sequence of past events exists.”

Is it:

4. It is not the case that P holds, where P is the proposition "there exists a possible world W having an infinite sequence of past events".

Because if it is, I can create a possible world where this happens: namely a world W like our world, existing infinitely far in the past, where at every time -t in the past a hotel of size 1/2 comes into existence at position t.

Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself. Your intuitions may tell you that the conclusion of the Grim Reaper argument (line 4) is wrong, but don’t you agree that the argument is both deductively valid and that the premises are justifiably true? If you do, then you can reject the conclusion only on pain of irrationality. If you do not believe the premises are true, which one do you believe is false and why? If you have any good objections to the premises of the argument I would be genuinely interested in hearing them, because so far I haven’t heard any.

To recap the argument (again assuming Pruss was using metaphysical possibility):

(1) If in infinite sequence of past events were metaphysically possible, then Hilbert’s Hotel would be metaphysically possible.

(2) If Hilbert’s Hotel were metaphysically possible, then the Grim Reaper paradox would be metaphysically possible.

(3) The Grim Reaper paradox is not metaphysically possible.

(4) Therefore, an infinite sequence of past events is not metaphysically possible.

Pruss’s justification for lines (1) and (2) are as follows:

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result. (Recall that the “could be” presumably refers to

metaphysicalpossibility; not physical possibility)Argument for (2): If Hilbert's Hotel were possible, each room in it could be (again, think metaphysical possibility) a factory in which a Grim Reaper is produced. Moreover, it is surely possible that the staff in room

nshould set the Grim Reaper to go off at 11 am + 1/nminutes. And that would result in the Grim Reaper paradox.Now at time t = 1/2, 3/4, 7/8, ... I flip the switch. Question: is the light on or off at time t = 1? It has to be one or the other because of my imposed requirements.This is actually a famous thought experiment called Thomson’s lamp, but I don’t think it yields a self-contradiction.

Meanwhile, back to the Grim Reaper argument, do you concede that the premises are more plausible than their denials? If not, which one do you think isn’t and why?

With reference to the lamp, what you linked to was about its *logical* impossibility, but you're the one claiming that logical impossibility is not the same as metaphysical impossibility. Anyway, I found the argument there to be without any merit.

So what flaw do

yousee in my argument? You have not provided any.Of course, I see my argument as having the same epistemological worth as the Grim Reaper argument: none. You can hypothesize all you like about "possible worlds", but the conclusions you ultimately draw have to be tied to what is possible in

thisworld.As for the Grim Reaper, I've said over and over what I think is wrong with it, so I think it is fruitless to continue with that.

Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself.The world I have constructed is completely consistent with everything in Pruss's argument - and it invalidates the Grim Reaper paradox by denying the possibility of the Grim Reapers' existence. I find that completely convincing.

Even Pruss seems to recognize this, since he says "The argument has an additional premise, namely that time is not necessarily discrete."

As for the Grim Reaper, I've said over and over what I think is wrong with it, so I think it is fruitless to continue with that.Is there any objection you’ve made that I haven’t refuted? Remember, the Grim Reaper argument can fail to be sound only in one of two ways: it is deductively invalid or a premise is false. Since the argument is deductively valid, the only way it can fail to be sound is if a premise is false. Yet your objections to a premise of the argument tended to rely on a misunderstanding. Assuming you correctly understand the argument now, notice that it’s apparently not until very recently that you really understood the argument (you seemed to think that Pruss was referring to a fixed universe through all lines of the argument, but this was mistaken). With such misunderstanding it’s perhaps no surprise then that you have yet to provide any good, relevant objection to any premise of the argument. If you think I’m mistaken here, please cite one specific example.

Remember, the Grim Reaper argument can fail to be sound only in one of two ways: it is deductively invalid or a premise is false. If you believe the argument to be unsound, do you believe it to be invalid? Do you concede that the argument’s premises are more plausible than their denials?

“Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself.”

The world I have constructed is completely consistent with everything in Pruss's argument - and it invalidates the Grim Reaper paradox by denying the possibility of the Grim Reapers' existence.

I find your claim that your construction of the world is consistent with everything in Pruss’s argument very surprising, because it is logically impossible for “the world I have constructed is completely consistent with everything in Pruss's argument” to be true. Why? Because it is logically impossible for the premises of Pruss’s argument to be true and the conclusion (that a world in which an actual infinite sequence of past events exists is

notmetaphysically possible) to be false. If you don’t believe me, I’ll happily prove so via symbolic logic.With reference to the lamp, what you linked to was about its *logical* impossibility, but you're the one claiming that logical impossibility is not the same as metaphysical impossibility.That is true, but your argument seemed to suggest that the proposition in question is metaphysically impossible as a result of yielding a self-contradictory situation (which of course is a logical impossible state of affairs).

So what flaw doyousee in my argument?The same one that the Stanford Encyclopedia of Philosophy entry sees: the end result isn’t a genuine self-contradiction. If you want me to be more analytical about it, please frame your argument with clearly delineated premises and conclusion, and I’ll be happy to tell you if I believe the argument is deductively invalid or if the argument has a false premise.

Wade says that Pruss presented arguments for the most dubious premises in his argument, but he really didn't. Here are Pruss's "arguments".

For "If backward-infinite series of events possible, then Hilbert Hotel possible": "If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed." *This is just a restatement of the premise*. "If p is possible, then p-and-q is possible". It's ridiculous.

For "if Hilbert hotel possible, then paradoxical grim reapers possible": "If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes." Again, the alleged argument is basically just a restatement of the premise. Worse: consider a universe with the same rules as ours that's spatially and temporally infinite. (Our actual universe may or may not be such a universe.) In such a universe, (a) the Hilbert hotel scenario is possible but (b) the GR "paradox" is not. So Pruss's intuition for what the Hilbert Hotel staff could "surely" do is demonstrably broken.

I have no idea what the point is of asking me to say what's wrong with Pruss's arguments: Pruss doesn't offer any arguments. He just states his premises, states them again, and apparently thinks he's given good support for them.

"a Grim Reaper swings his scythe and it’s all over for Fred": well, if *that* is what looks to you like a description of *how* a Reaper does its job then (1) your infinite-converging-swarm-of-Reapers scenario is "metaphysically impossible" for reasons that have nothing to do with a general impossibility of infinities -- namely, "scythe" denotes a particular sort of object that can only exist in a universe very like this one, and in this sort of universe Reapers are physically impossible -- and (2) an equally good answer to "how do the Reapers collectively kill Fred?" is "They all stand their with their scythes ready to kill him, and as a result he dies". Which is, yes, a pathetic excuse for an explanation: just like yours, which I cannot believe you intended seriously.

The point of the thing about continuous time, differential equations, etc., is: The GR "paradox" depends on intuitions like "if none of them kills Fred then all-of-them-collectively do not kill Fred", and no matter how you bluster the fact is that this is just a statement about your intuitions for what various configurations of things can do. Those intuitions might be of some use when it comes to the actual world in which they were formed (though, as I've pointed out a couple of times and been ignored, the actual world is host to actual phenomena that run counter to exactly those intuitions) but should obviously not be trusted when it comes to universes where (e.g.) time and causality are very different from in ours. And it sure seems like the nature of time and causality would have to be very different in a universe in which the possibility of completed infinities sufficed to enable the Reaper "paradox": and *that* is the point of what I was saying. It wasn't meant to attack any specific premise of any deductive argument because no one's given any non-laughable deductive argument for the Reapers: the only such arguments you've so much as waved in the general direction of are based on very dubious intuitions. (If you are content to call an argument a valid deductive argument simply because you've packaged the invalid steps into premises, fair enough; there's some value to doing that. But the value in doing it isn't that the argument isn't stronger that way.)

Wade:

Honestly, I don't see any value in simply repeating my arguments. I've already said over and over again what I think is wrong with the argument. I even provided a counterexample universe for which the argument fails, but you simply dismissed it for no good reason I can see.

And you also provided no reason to think my argument with the light is incorrect.

So at this point I don't have much more to say.

Sorry, maybe one final comment is worth saying. The Stanford article claims "what can be deduced logically from this way of acting will apply only to instants in the t-series". This is clearly false,

ifwe also specify our model of time explicitly: namely, if the lamp is in state s at time t, and no action is taken between time t and time t' > t, then the lamp is still in state s at time t'. With this proviso, it is clear that "what can be deduced" includes other times than those in the t-series alone. So if that's the counter-argument, it is not convincing.For "If backward-infinite series of events possible, then Hilbert Hotel possible": "If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed." *This is just a restatement of the premise*No it’s not, the premise is “If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible” and the argument goes into further detail as to

whythat would be true (it describes how the hotel could be created).For "if Hilbert hotel possible, then paradoxical grim reapers possible": "If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes." Again, the alleged argument is basically just a restatement of the premise.Again, the argument allows us to go into greater detail; it explains

whyHilbert’s Hotel being possible would entail the Grim Reaper paradox being possible, and I found such an explanation illuminative. Of course, both arguments for the premises are very short, and we could quibble over how “basically” it is like the premises. At the end of the day though, if we want to say the arguments for the premises are unsuccessful one ought to have a good counter-argument (it seems the arguments work unless some counter-argument is available).I haven’t seen any good counterargument against the premises or the (albeit brief) justification for the premises. The closest you offered was this:

Worse: consider a universe with the same rules as ours that's spatially and temporally infinite. (Our actual universe may or may not be such a universe.) In such a universe, (a) the Hilbert hotel scenario is possible but (b) the GR "paradox" is not. So Pruss's intuition for what the Hilbert Hotel staff could "surely" do is demonstrably broken.Suppose it’s true that in such a universe, (a) is possible but (b) is not. This doesn’t relevantly attack the premise or the justification given for it. The argument’s second premise doesn’t say that “For any world W, If Hilbert’s Hotel were nomically possible in W, then the Grim Reaper paradox would be nomically possible in world W.” Rather, the claim is more modest, something like “If Hilbert’s Hotel were metaphysically possible (if were to exist in at least one possible world), then the Grim Reaper paradox would be metaphysically possible (it would exist in at least one possible world).” So the fact that in some possible worlds the antecedent is nomically possible and the consequent isn’t doesn’t relevantly affect the veracity of this premise.

The second premise also appears more plausible than its denial; if Hilbert’s Hotel is metaphysically permissible, it doesn’t appear that any of our known metaphysical constraints would prohibit e.g. each room having a Grim Reaper factory. If you think otherwise and have a good argument for thinking that this is would be metaphysically impossible if Hilbert’s Hotel were metaphysically possible, I’d very much like to hear it.

"a Grim Reaper swings his scythe and it’s all over for Fred": well, if *that* is what looks to you like a description of *how* a Reaper does its job then (1) your infinite-converging-swarm-of-Reapers scenario is "metaphysically impossible" for reasons that have nothing to do with a general impossibility of infinities -- namely, "scythe" denotes a particular sort of object that can only exist in a universe very like this one, and in this sort of universe Reapers are physically impossible -- and (2) an equally good answer to "how do the Reapers collectively kill Fred?" is "They all stand their with their scythes ready to kill him, and as a result he dies".For (1), it isn’t clear why scythes are incompossible with infinitely many entities that can kill Fred at any positive nonzero time interval. In any case, if need be we can use “scythe” in a somewhat more metaphorical way to refer to a weapon that is shaped like a scythe but when it touches Fred it emits a lethal energy field capable of killing Fred in any positive nonzero time interval. Point (2) doesn’t seem to work; if they none of them actually

usetheir scythes to kill Fred, how do they collectively kill Fred? And let’s not forget my “hit point” variant of the paradox to illustrate the problem more clearly. In that modified scenario, all cumulative Grim Reaper events are the sum of their individual hit points, e.g. three Grim Reapers each doing a scythe attack collectively take off 300 hit points, and three Grim Reapers doing nothing collectively take away 0 hit points. If none of the Grim Reapers do anything, they collectively take away 0 hit points away from Fred, and therefore they do not collectively kill Fred.Jeffrey Shallit, for the Grim Reaper argument I’ll finally use some symbolic logic that might help understand the situation better. But first…

Honestly, I don't see any value in simply repeating my arguments. I've already said over and over again what I think is wrong with the argument.Again, have you given any rebuttal that I haven’t already refuted? I apologize if I missed a rebuttal, but I don’t believe I have. As I said, your rebuttals tended to rely on misunderstandings of the argument, and it wasn’t until relatively recently that you correctly understood the argument (assuming you correctly understand it now). If you have a rebuttal I have missed, by all means please point it out to me.

I even provided a counterexample universe for which the argument fails, but you simply dismissed it for no good reason I can see.My reason for dismissing it was that your counterexample is logically impossible. Don’t you consider that to be a good reason? Perhaps it’s best if I finally use some symbolic logic, though I’ll have to approximate the symbols with text.

Let []-> represent the counterfactual/subjunctive conditional, e.g. A []->B translates into “If A were true, then B would be true.” Let <> represent the modal operator for possibility, e.g. <>A is “A is possibly true.” The tilde will represent negation, e.g. ~A translates into “not-A” or “A is false.” Let

Grepresent “the Grim Reaper paradox occurs,”Irepresent “an infinite past sequence of events exists” andHstand for “Hilbert’s Hotel exists.” Thus, since we’re using metaphysical possibility, <>H stands for “Hilbert’s Hotel is metaphysically possible” (or to use possible world semantics, “Hilbert’s Hotel exists in at least one possible world”). The proof can go as follows:(1) <>I []-> <>H

(2) <>H []-> <>G

(3) ~<>G

(4) ~<>H, 2, 3,

modus tollens(5) ~<>I, 1, 4,

modus tollensSo as you can hopefully see now, your counterexample of a possible world in which

Iis true and yet is consistent with everything in Pruss's argument (e.g. the first three premises of the argument) is logically impossible. The conclusion that no such possible world exists follows logically and inescapably from the first three premises.Because the argument is provably valid, the only way for the argument to be unsound is for a premise to be false. There’s simply no way to get around this. Incidentally, do you concede that the premises are more plausible than their denials?

I’ll comment on your lamp-argument next.

And you also provided no reason to think my argument with the light is incorrect.I linked to the SEP, but if you want me to come up with something tailor-made to your argument, please delineate your argument with clearly defined premises and conclusion (as I did for the Grim Reaper argument).

The Stanford article claims "what can be deduced logically from this way of acting will apply only to instants in the t-series". This is clearly false, if we also specify our model of time explicitly: namely, if the lamp is in state s at time t, and no action is taken between time t and time t' > t, then the lamp is still in state s at time t'. With this proviso, it is clear that "what can be deduced" includes other times than those in the t-series alone.It’s hard to see how that assertion applies to Thomson’s lamp or saves your argument, because there is no last “let’s flip the switch” time t where there is a time t’ > t for the lamp to be still in state s at time t’ and for there not to be another time t’’ > t’ that is a member of the

t-series and thus in which another flip-the-switch action is done at t’’, and if that’s the case “we only act on it at instants in thet-series, and so what can be deduced logically from this way of acting will apply only to instants in thet-series” as the SEP article says. Now it may be possible that I’m not correctly understanding your argument’s train of thought here, but if so I again request that you put your argument in the form of clearly listed premises and conclusion.A popular style of philosophy is known as analytic philosophy, which emphasizes clarity, rigor and precision in its practice. For example, an analytic philosopher might clearly list out the premises of an argument (as I have) and use symbolic logic to prove the argument’s validity (as I have) or attack a specific premise when his or her opponent clearly lists out the premises of an argument (as I have on other occasions). I recommend we adopt this style not only because of your mathematical background, but because it makes communication and dialogue a bit more fruitful.

Wade:

I'm sorry, but I'm not interested at all in continuing. I've already pointed out that even Pruss admits his arguments needs the infinite divisibility of time to work, and I already presented an argument which I consider rigorous to show, using your own methods, that this is not sensible.

While I’m sorry to hear that you’re not interested in continuing, I thank you for your input and would like to introduce a stronger version of your argument (a supertask that relies on arbitrarily small time intervals and

doesyield a self-contradiction). But first…I've already pointed out that even Pruss admits his arguments needs the infinite divisibility of time to work, and I already presented an argument which I consider rigorous to show, using your own methods, that this is not sensible.Well, not

quitemy own methods. Compared to the Grim Reaper argument I put forward, you were substantially less analytical and less rigorous than I ended up being (which I find a bit ironic since you’re the one who earned a Ph.D. in mathematics). Recall that I constructed a deductively valid argument with clearly listed premises and I later proved the argument’s validity via symbolic logic. Not only did you not use symbolic logic, you didn’t even clearly delineate your argument’s premises even after I asked you to. That said, I’ll try to introduce a strengthened form of your argument anyway.It is true that Pruss’s argument requires infinite divisibility of time in the sense that it requires that arbitrarily small time intervals between events be metaphysically possible (more specifically, premise 2 requires this). One could claim that “arbitrarily small time intervals between events cannot exist in any possible world,” but I find such a metaphysical constraint very strange and implausible, akin to saying that an infinite number of marbles can exist but subtracting from an infinite marble set is metaphysically impossible (with no apparent motivation other than to avoid the paradoxes of infinity subtraction while still believing in the actual infinite).

So why on earth think that arbitrarily small time intervals between events are metaphysically impossible? Your approach, if I understand it correctly, goes something like this:

(1) If arbitrarily small time intervals between events are metaphysically possible, then supertask X is metaphysically possible.

(2) Supertask X is not metaphysically possible.

(3) Therefore, arbitrarily small time intervals between events are not metaphysically possible.

The key problem with the premise (2) is that the supertask you’ve chosen (Thomson’s lamp) doesn’t yield the self-contradiction that you’ve claimed. Not to worry though, because there are supertasks that rely on arbitrarily small time intervals between events that

doyield self-contradictions. A quick example is the Grim Reaper paradox. If it makes you feel any more comfortable, we can modify the paradox to be a “Duo” version in which Grim Reaper 1 checks on Fred at 11 a.m. + 1/(2n) minutes and Grim Reaper 2 checks on Fred at 11 a.m. + 1/(2n+1) minutes. Still, let’s assume arguendo that the Grim Reaper Duo supertask and your own lamp supertask both yield self-contradictions. The problem is that these self-contradictions ariseonly when there is an infinite sequence of past events. No paradox is generated when only a finite quantity of such events occur (e.g. the Grim Reaper Duo checks on Fred only finitely many times). So claiming that the argument against an infinite number of past events fails because arbitrarily small time intervals between events are metaphysically impossible—and then justifying that claim by pointing out self-contradictions resulting from cases where there are an infinite number of past events—isn’t going to sound terribly convincing.So the suggestion is that we can take Grim Reaper types of arguments to show that there is a minimum unit of time. That's a really controversial conclusion.

Moreover, it is a conclusion that helps the Kalaam arguer. For when we combine the claim that there is a minimum unit of time with the arguments that the past has only finite length (whether based on the Big Bang or on inflationary considerations), we get the claim that there were only finitely many past moments, and hence there cannot be a pastwards infinite regress. That certainly would help Craig.

I'm not a very smart guy, I'm not even a professional philosopher, but I wonder if this MIGHT solve the Grim Reaper paradox, sort of;-):

The Grim Reaper paradox shows that there must be an indivisible unit of time. Okay, so, lets say that God wants to find out what the smallest possible unit of time is. He visits a World, W1, and finds that the Planck time, P1, is equal to 10^-43 seconds. But it is logically possible that P is one order smaller and generate no logical paradox at al. So he goes to W2 and finds P2 is 10^-44 seconds. But of course, God can keep jumping to possible worlds, W, and find P, one order smaller than the previous where no logical paradox is generated. In fact there is no W where P could not be one order smaller and not generate a logical paradox. Therefore, time is infinitely divisible. But that can't be right, right?

I think, with great trepidation, that any way the skeptic of infinite temporal divisibility answers, the skeptic of the Grim Reaper paradox can answer the same, which would mean the the argument against temporal infinity from the Grim Reaper paradox is false. Here are some examples:

Objection 1: It's just a brute fact that there is a W where P is indivisible.

Well, it could also be a brute fact that one of the GR kills Fred.

Objection 2: God cannot create a W where the indivisible is divided as he cannot do the logically impossible.

Okay, then one of the GR must kill Fred as it is logically impossible that there as a W where none of the GR kills Fred.

Objection 3: It is logically necessary that there is a W where P is indivisible, even though it appears as though there could be an infinity of W where P is one order smaller.

Same again. Its logically necessary that a GR kills Fred even though it looks like no GR kills Fred.

Objection 4: The paradox is, in some way, incorrectly presented and isn't analogous.

Again, same could be said of the GR paradox: in some way it isn't analogous to a backwards temporally infinite sequence. How? For example, a backwards temporally infinite sequence has no beginning, yet in the GR paradox, Fred is alive at 11:00pm, which means, somehow, the GR at that instance, didn't kill Fred instantaneously, which isn't analogous to a universe with a beginningless past.

Anyway, I'm not confident enough to think I am certainly right. But this does seem a reasonable conclusion, at least to me.

From what I can see the GR paradox is dependent upon the assumption that the infinite amount of universes are causually connected. If the assumption isn't met then the paradox doesn't occur.

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