Tuesday, May 06, 2008

Reply to William Lane Craig

Two readers of this blog have pointed out this post at William Lane Craig's blog. In the post, he responds to a question about my debate with Kirk Durston. Craig says I exhibit "ignorance on parade".

Well, there's a lot of ignorance to go around. My debate was with Durston, not with Craig. I was responding to Durston's claim (made at 05:37) that "Mathematics dictates that time itself would have had to have a beginning at some point in the past." In the debate that Durston took part in just a few days earlier at McMaster University, he claimed that Hilbert, in his 1925 paper, "On the Infinite" had proved mathematically that there could not be an infinite regress of causes." But this is not true. All Hilbert did in that paper was claim that then-current consensus about the physical universe was that no infinite quantities existed in it. That's a far cry from any kind of mathematical proof. William Lane Craig, like anyone else, can go read Hilbert's paper and verify that this is the case.

I pointed out that in fact, there is nothing mathematical that rules out an infinite regress of causes. For example, you could have an event at time -(n+1) causing an event at time -(n) for all positive integers n. Thus, an event at time -2 causes an event at time -1, an event at time -3 causes an event at time -2, etc. There is nothing logical or mathematical to rule this out. You can even have an infinite regress of causes if time has a beginning. If time begins at time 0, then you can have an event at time 1/(n+1) causing an event at time 1/n for all positive integers n. Thus, for example, an event at time 1/3 causes an event at time 1/2, an event at time 1/4 causes an event at time 1/3, etc. Again, nothing logical or mathematical rules this out.

Now you might say that once we bring our current state of physical knowledge into the picture, the first scenario is ruled out. But even modern physicists consider the possibility of infinite time-like curves that occur in the past of some other point; for example, in their study of Malament-Hogarth spacetime. Thus, I would contend that apologists like Durston and Craig have a really naive view of spacetime, one that is essentially based on the understanding of 100 years ago, not modern physics.

When I called Durston on this at the debate, his response was really comical. Here it is as I have transcribed it, beginning at 1:06:48:

"First, regarding Hilbert. He [Shallit] pulled a mathematical trick
there. Those of you who are used to summing infinite series
will know that if the x decreases exponentially, it comes to a
finite value. So let me explain how this really works.

Let's assume... now I don't know whether he's saying that.
Has he dodged the issue here, as to whether or not the past is
infinite or not? So let's assume the past is infinite. So
let's call this debate time 0, this hour here of the debate is
time 0. The next hour after this will be time 0+1, time 2, and
so forth. And in the past, we'll go to, the last hour before
this debate will be negative 1 hour, hour negative 2, and so forth.
Now if you want to assume, and this is to illustrate why there's
a problem of traversing an actual infinite series in
reality. Let's say that each step in the series is one hour
long. Now what he seems to be arguing, or what he's insisting
here, I'm not sure, is that Hilbert, that the past can be
infinite, that is, there's an actual hour infinitely separated
from this one here. So let's call that -infinity. We'll never
get there by getting in a time machine and going back, so let's
just take a quantum leap back into the past, we're now at minus
infinity. Now some of those of you who are familiar with
infinite set theory might be a little uncomfortable at this
point because if the past is, if you're saying it actually is
infinite, what you mean is that there actually is an hour back
there that is infinitely separated from this one. So let's
count our way down now. Infinity minus 1, infinity minus 2.
It's one hour each, not an exponentially decreasing amount of
time like that little equation he put up there, but just a
steady hour each time.

Or you could go with a multi-universe, this universe is a
product of another universe, and we're working our way down
from -infinity to the present. At what point in time will you
arrive at 0? You will never traverse an infinite series in
reality if you must stop at a discrete amount of time for a
constant amount of time in between. And that pretty much lays
to rest this notion that time itself can be infinite as far as
the past goes. It can be potentially infinite. you can do
lots of mathematical things, I can hold my hands together and
say there's an infinite number of mathematical points, no
problem, those are imaginary. But the moment you have a
discrete amount, that occupies a discrete amount of time, like
a minute, or a second, or an hour, and it is not decreasing
then suddenly you have a problem, if you want to actually
traverse that infinite series in reality."


After Durston's casual slur on my character at the beginning, this is completely incoherent. My equation was not "exponential", so that criticism is nonsensical. Secondly, it is perfectly possible to have an infinite past without having any point at infinite distance from the current time; for example, we could define times -1, -2, -3, etc., without having to define an actual point called "-infinity". (In exactly the same way, the negative integers are an infinite set that does not contain an integer called "-infinity".) This is not exactly a controversial point, but it is a misconception common to undergraduates. Mr. Durston, a graduate student, should know better.

Craig doesn't seem to understand what the debate was about. He says, "What's really peculiar is Shallit's "that was then, but this is now" move—as though views of mathematical existence are tied to the times!", thereby entirely missing the point. Hilbert's claim was about the 1925 understanding of physical existence, not mathematical existence. And anyway, views of mathematical existence do change through time. Consider, for example, the views of people like Brouwer.

Craig says, "On Shallit's view the universe still came into being a finite time ago and therefore requires an external cause." No, I didn't say that at all, and I don't hold that. In my second example, I said that "you can have an event at time 1/(n+1) causing an event at time 1/(n) for all positive integers n". This doesn't say anything about time 0, and it is logically possible to have an infinite chain of causes stretching back in this way, with nothing happening at time 0 at all - an uncaused beginning.

In general, Craig seems to have an extremely naive, almost childish view of infinity. Read Craig's reply to Sobel. On page 9 he says, "Imagine an actually infinite regress of past causes terminating in the present effect. In this case, the regress of causes terminating, say, yesterday, or, for that matter, at any day in the infinite past, has exactly the same number of causes as the regress terminating in the present. This seems absurd, since the entire regress contains all the same causes as any selected partial regress plus an arbitrarily large number of additional causes as well. Or again, if we number the causes, there will be as many odd-numbered causes as there are causes, which seems absurd, since there are an equally infinite number of even-numbered causes in the series in addition to the self-same odd-numbered causes."

It seems that what bothers Craig is perfectly understandable to any mathematician: namely, that the set of positive integers has the same cardinality as the set of integers greater than n (for any positive n), and the same cardinality as the set of even positive integers. All this was well understood 125 years ago, but it seems the Christian apologists haven't caught up.

Altogether, I would say these arguments by Durston and Craig are embarrassingly naive.

44 comments:

Anonymous said...

I believe you have a typo in your parenthetical remark:

"(In exactly the same way, the negative integers are an infinite set that does contain an integer called "-infinity".)"

Shouldn't that say "does not contain"?

Tom S.

Jeffrey Shallit said...

Thanks - I fixed that typo now.

Kirk said...

Jeffrey, thank you for this opportunity to clarify my response, and reply in more detail to your response. Your response is interesting, but I don't think it refutes the proposition that time had a beginning. Perhaps the best summary of my argument for the proposition that the past must be finite is the statement you quoted above,

You will never traverse an infinite series in reality if you must stop at a discrete amount of time for a constant amount of time in between.

Note that each event occupies a constant amount of time t, (say, a year); your equation does not have a constant t.

In your response, you have defined the time for each event such that each event does not occupy a constant amount of time. You use the equation,

t = -1/n ,

where t denotes time and n denotes an event number, where n = 1,2,3,…,
We can see that t, as portrayed in your equation, is not a constant; it gets smaller as one works ones way backward through past events. But the question was whether or not the past is infinite (i.e., an infinite number of years have actually transpired). Mathematically, there is no problem with letting n go to infinity, but I was talking about real events, where for each event, t is a constant (say, a year). Therefore, your response fails to show that the number of years that have actually transpired in the past are infinite.

So let us consider how old the universe is in terms of years, a unit of time that is constant. Let us call this event, event p. If an infinite number of years must elapse before we arrive at p, when will we arrive? The answer is never. Permit me to use a thought experiment that involves time machines, a concept most people can get their mind around. No matter how fast our time machine recedes into the past, it will never traverse an infinite number of years … ever. Now imagine that an observer is coming toward us from the infinite past in her own time machine that moves forward through time. When will the two time machines pass each other? Never. Yet those who argue that the past is an infinite number of years old, must defend the notion that an infinite number of actual years have unfolded, one year at a time, such that an imaginary traveler from the past has arrived at this event, just as we are about to climb into our time machine, here at event p. The fact that this event is occurring necessarily requires that the number of years that have elapsed up to this point are actually finite. Empirical evidence indicates that the past is finite, but the problems of actually traversing an infinite number of years also requires that the past is finite, consistent with empirical evidence.

Now let us consider cosmological models in general and Malament-Hogarth spacetime specifically. In theoretical physics, it is important to distinguish between mathematical models and reality. In a mathematical model, there is no problem with using concepts such as infinity and imaginary numbers (for the lay-person, think of an imaginary number as the square root of -1). In reality, an actual infinite cannot exist due to violations of the principle of non-contradiction, upon which mathematic and logic are founded. For example, we use imaginary numbers in everyday three phase electrical calculations, but imaginary numbers and imaginary infinites cannot, logically, exist in reality. No one, for example, would argue that I can have the square root of -1 apples in a basket. Hilbert's Hotel (begin reading at http://en.wikipedia.org/wiki/Hilbert%27s_Hotel and follow the references) is an example of the problem of logical contradictions with actual infinite sets. The bottom line is that mathematical cosmological models are free to use mathematical infinites, but they cannot exist in reality because of the violation of the principle of non-contradiction.

Now let us consider Malament-Hogarth (M-H) spacetime to see if that defeats the proposition that the universe (including spacetime) is of only a finite number of years old. I don't see how invoking M-H spacetime is goint to be of any help in arguing that the number of years that of actually elapsed in the past can be infinite (i.e., an infinite number of years old). It states that in the worldline, 'all events along λ are a finite interval in the past of p.' It states that proper time is infinite, but we must be very careful to distinguish here between a mathematical infinite, an actual infinite, and a potential infinite. If we are talking about reality the mathematical infinite is ruled out. An actual infinite is logically impossible if we want to avoid a violation of the principle of non-contradiction, so we are only left with a potential infinite, where time is finite and increasing towards infinity. Perhaps what you are thinking of here, if we wish to talk about real scenarios, is the possibility of a Kerr metric around a black hole, where relativistic effects essentially create two worldlines, the clocks in each of which run at different speeds. As a reference frame approaches the speed of light, a clock in that reference frame runs slower relative to a stationary reference frame. A clock in a stationary reference frame runs faster and faster relative to an observer in the relativistic reference frame as the relativistic reference frame approaches the speed of light. Once the relativistic reference frame achieves the speed of light, that observer observes an infinite number of years in the stationary reference frame pass by in a mathematical point of time.

There are a number of problems with this. First of all, in reality, one cannot achieve the speed of light. As one approaches the speed of light, ones mass becomes infinite (with associated infinite gravitational field) and ones length becomes zero. As a result of this, coupled with time dilation, the relativistic observer always gets closer and closer to the speed of light, but at a slower and slower rate with the result that the relativistic observer never actually achieves the speed of light and lever actually observes an infinite number of years pass by in a mathematical point of time in the stationary reference frame. One is then reduced, once again, to discussing mathematically infinite sets, not actual infinities. One other point; in both reference frames, the clocks never tick through an infinite number of years, the past is always finite, even if the potential future is infinite. In discussions of the age of our universe, we are concerned about our reference frame, not an imaginary reference frame traveling at the speed of light.

Overall, I don't see that you have shown that the age of the universe is infinite, or even that my proposition that our proper time must have a finite past, is false.

Best regards,
Kirk

Jeffrey Shallit said...

Thanks for replying, Kirk.

First of all, I'm not trying to show that the age of the universe is infinite. Why should I do this, when our current model says it is about 13 billion years old.

What I am trying to show is
(a) your claim about Hilbert is wrong;
(b) your claim that an infinite past is "mathematically" or "logically" dubious is incorrect.

Do you admit (a) yet? Have you read Hilbert's original paper?

As for (b), your claim that "You will never traverse an infinite series in reality if you must stop at a discrete amount of time for a constant amount of time in between" is irrelevant. You can have an infinite past without having to "traverse an infinite series", whatever that might mean.

You also get my equation wrong. I never said anything about -1/n. I talked about the sequences -n and 1/n, and you have somehow combined them. Do you admit that I had no exponentials in my example, as you falsely claimed?

You make another mistake when you say, "let us consider how old the universe is in terms of years, a unit of time that is constant." But in my example of an infinitely old universe, where events at time -(n+1) cause events at time -n, the age of the universe is infinite, which is not an integer constant.
So your reasoning from then on is bogus. There may not be an "event p" at an infinite distance in the past from now; yet the past could be infinite nevertheless.

You claim, " In reality, an actual infinite cannot exist due to violations of the principle of non-contradiction, upon which mathematic and logic are founded", but this is entirely bogus. You have given no contradiction arising from the existence of an actual infinity. Hilbert's hotel does not provide any logical contradictions; it merely provides situations that most people find counter-intuitive. But quantum mechanics and relativity are counter-intuitive, too.

In conclusion, you haven't adequately addressed my arguments at all. Nor have you admitted that your previous claims were wrong.

Unsympathetic reader said...

I wonder how old God is, then? Did He go through a maturing process?

That sounds a lot like Process Theology.

And can there be a "world without end" if infinite series are a practical impossibility?

Timmy said...

I am a mathematics major(senior)and I find this humorous. Obviously Kirk doesn't know his mathematics like a person with a graduate degree should. Hotel Infinity is actually a perfect example of how to think of infinity with regards to time. For his benefit, here is the short version:
There are an infinite number of rooms which are all filled. Suppose a bus containing 20 people show up, all needing separate rooms. The only way to solve it is to make everybody move over 20 rooms. But how can that be? Well, in mathematical infinity, there is always one more.
Also, who said that there has to be a singular point when time began? If you think of now as point 0 on a number line, everything after now is a positive real number, and everything before now is a negative real number, you can see that there is no problem. Every "now" is point zero. Which means there is an infinite negative side(past) and infinite positive side(future). It's all relative. In an infinite line such as time, there is no beginning and no end. There are only points relative to now(the origin on the number line). His problem is that he is thinking of infinity as a specific point n. But, there is always n-1 and n+1. There was no beginning. After taking Number Theory and Real, Complex, and Numerical Analysis, I have seen examples of infinities that would make Kirk shit bricks since he obviously didn't pay attention in school. I mean, technically there are many infinities, and some are bigger than others. For example, between any two numbers say, 0 and 1, there is an infinity. There is technically an infinite measurement between hours, minutes, or seconds. You could measure it as infinities, but time still passes. Sorry for rambling.

Bayesian Bouffant, FCD said...

Your response is interesting, but I don't think it refutes the proposition that time had a beginning.

But he most certainly refuted your argument that time must have had a beginning. Go back to school. learn something.

Bayesian Bouffant, FCD said...

Empirical evidence indicates that the past is finite

If you've got empirical evidence for that, you should produce it.

NAL said...

The question is: is time continuous? My theory is that time occurs in discrete steps. (This is my personal theory and I have exactly zero evidence to support it.) What is the measure of these discrete steps? I have no idea, maybe one half a Planck time. My theory also implies that length occurs in discrete units.

Erdos56 said...

It appears we have some absurd cross product of pidgeonhole principle with angels dancing on pinheads: pinheadhole principle?

paul01 said...

Just curious. I understand that in string theory there is more than one dimension of time. How would this fit into this discussion, if at all?

Kirk said...

Hello all. It's been a jammed day and the next two days are full as well. However, I will aim to post a response to Jeffrey's (a) and (b) before midnight Friday. In the meantime, having read through the new comments, I would strongly encourage all interested in this topic to get a copy of Hilbert's paper 'On the Infinite', in Philosophy of Mathematics (1964) and read it before my next post. Pay special attention in the paper to concepts I mentioned in my first post, such as 'potential infinite' and 'actual infinite', and the difference between the two. Note also Hilbert's definition of the principle of non-contradiction that I mentioned, and the role it plays in his conclusion. That will make our discussion a lot more fruitful.

Also, Timmy, you may want to expand on your statement that some infinities "are bigger than others" for the benefit of those readers who may not have your extensive knowledge of infinite set theory.

Sorry for the delay till Friday in posting a response. I will respond specifically to Jeffrey's (a) and (b), as well as use Hilbert's Grand Hotel to illustrate a point in my initial post and clear up some confusion I see in one or two responses. This is a fascinating topic with important implications, so check Saturday for my response. Again, my apologies for the delay.

Cheers,
Kirk

Timmy said...

On different infinities:
Sorry, I should've explained my statement about "some infinities are bigger than others." The set of Natural numbers(our counting numbers: 0, 1, 2, 3, 4...) is an infinite set(Georg Cantor called this aleph-null). Now, if you look at the real numbers, this gets interesting. Look between 0 and 1, for example. There is an infinite number of real numbers between those two. Actually, you can take any two real numbers and there is an infinity of numbers between them. So, the real numbers are an infinity that is bigger than the natural numbers.

On my personal hypothesis about time and the universe, I try to make things simple. My hypothesis is this: time and space are infinite. Time had no beginning and will have no end. Time is just a concept that we invented to measure duration of events. I also think that the "Big Bang" was "local" in that what sense would it make for all of the matter in the infinite universe to be in a singularity? If you go back to before the Big Bang, what else was out there? I think that it was simply localized and that there are or have been Big Bangs further than our technology can measure. And I think that it happens over and over and a body gains mass and gravity and collects all the matter in its "local" area and blows up itself. I think our Big Bang that produced our measurable local area was just one of many. Anyway, I'm going off topic so I will stop rambling.

improbable said...

Oh dear... just stumbled across this. Infinity is hard to get used to even if you're not trying to make it support a religious position.

Kirk is unfortunately not alone. John Ellis (of Hawking-Ellis fame, and recent Templeton prize) has some strange views about a spatially infinite universe being inherently contradictory, which I do not understand. http://www.mth.uct.ac.za/~ellis/

Although perhaps I'm overestimating the importance of religion here: a famous inflationary cosmologist I know bases much of his work on the idea that time not being infinite (to the past) is somehow a problem.


Timmy:Actually, you can take any two real numbers and there is an infinity of numbers between them. So, the real numbers are an infinity that is bigger than the natural numbers.

It it true that there are more real than natural numbers, but it's not quite this simple to show. In fact there are as many rational numbers (fractions) as naturals, even though they too are dense (an infinite number between any two).

paul01: String theory has many spatial dimensions, but only one time. (Sometimes we construct a space with one time by embedding it in a space with two, AdS in R^2,6, but no reality is attached to the space with two timelike directions.)

improbable said...

Umm, make that George Ellis, not John. Sorry.

Kirk said...

I've rapped this out in William's Coffee Bar under some time constraints, so haven't had time to proof-read. Hoping for the best.

I would like to respond to three items in Jeffrey's response to my original post. The three items are:
a) his challenge to my claim that David Hilbert has shown that the past must be finite
b) his challenge to my claim that an infinite past is mathematically or logically dubious
c) his challenge to my claim that in reality, an actual infinite cannot exist due to violations of the principle of non-contradiction, upon which mathematic and logic are founded.

First, my apologies for translating your (Jeffrey's) "event at time 1/(n+1) causing an event at time 1/(n) for all positive integers n" the way I did, and sticking a negative sign out front. I was focused on the fact that, defined that way, the interval of time was not a constant and I got sloppy in translating your sentence into an equation. Very bad form on my part, and it's clear that you have no exponents in your example. Now to the three tasks at hand as outlined above. In answering (a), (b) will also be answered. I will deal with those two first, and then proceed to (c).

The problem of contradictions and inconsistencies In his paper On the Infinite,, Hilbert's objective is to resolve 'the problem of the infinite'. He spends his first two pages with preliminary remarks, in which he introduces the concern of contradiction and consistency, and the 'inanities and absurdities which have had their source in the infinite.' He concludes his opening remarks by stating, 'The foregoing remarks are intended only to establish the fact that the definitive clarification of the nature of the infinite ….. is needed for the dignity of the human intellect itself. He observes that, 'as it seemed very plausible to identify the infinite with the "very large", there soon arose inconsistencies which were known in part to the ancient sophists., the so-called paradoxes of the infinitesimal calculus.' Nine pages into his paper, in the course of which his concern with contradictions, inconsistencies and paradoxes that arise out of the problem of the infinite, he summarizes it all by stating, 'the present state of affairs where we run up against the paradoxes is intolerable. He then states two 'desires and attitudes, the second of which ends with 'contradictions and paradoxes arise only through our own carelessness.' His solution is that 'these goals can be attained only after we have fully elucidated the nature of the infinite. All this to show that the principle of non-contradiction, which Hilbert holds dear, was very much a motivation for his paper. If this is not enough, when it comes time for him to develop a theory of mathematical proof, one of the principle axioms he chooses is II(i), the 'law of contradiction'. He then goes on to state the 'essential condition' of his work, an 'absolutely necessary one', as proof of consistency, i.e., it 'does not cause contradictions'. Having laid out a theory of mathematical proof, which includes what he regards as the essential axiom, the law of contradiction, he concludes that such a mathematical theory of proof solves the problem of the infinite. Specifically, he closes with

Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought - a remarkable harmony between being and thought.

I conclude, therefore, that Jeffrey's claim that my claim that, "In reality, an actual infinite cannot exist due to violations of the principle of non-contradiction, upon which mathematic and logic are founded" is 'bogus', does not stand, …. especially in light of Hilbert's assertion that 'proof of consistency' is an 'essential condition' of his work, and the role his axiom of the law of contradiction in achieving this.

The principle of non-contradiction or 'the law of contradiction' is an axiom for both Hilbert's theory of mathematical proof (which, among other things, resolves the problem of the infinite) and an axiom of logic and reasoning. Indeed, science assumes it as axiomatic as well in the scientific method regarding the roll of falsifiable predictions. Predictions can only be falsified if the law of non-contradiction is assumed. I see it being assumed in this discussion when various people claim that so-and-so is 'wrong'.

A finite past: With regard to how all this applies to the past being finite, note Hilbert's transfinite axiom III (i), the inference from a universal to a particular. If Hilbert's result that the infinite is nowhere to be found in reality, then that result is a universal proposition about reality. The statement, 'the past consists of an actual infinite number of years' is a particular. From axiom III (i), then, it follows that if Hilbert's universal conclusion is true, then it also is true that an infinite past is also 'solely an idea' that is 'nowhere to be found in reality.'

Using Hilbert's clarification of the two infinities, the potential and the actual, I think we can agree that if this discussion is occurring at time p then the future is a potential infinite of years. The past is not a potential infinite; it must either be an actual infinite set of years, or a finite set of years. But Hilbert has shown that, given his theory of proof, one of the axioms of which is the law of contradiction, an actual infinite is impossible in reality. Therefore the past is a finite set of years.

Another way of writing my argument is as follows:

1. If the past contains an infinite number of years, then it is an actual infinite
2. An actual infinite in reality is impossible
therefore, the past cannot, in reality contain an infinite number of years.

In normal discussions, I say little about these things, because I find that most people have thought little about the problem of the infinite. Instead, I give the time machine analogy in the hope of helping them see the problem. In this venue, however, since those interested have had an opportunity to read Hilbert's paper, I would say the problem in people's thinking is that they treat the past as a potential infinite. Standing at this point of history, they look to the future and see no reason why time should end, and they do the same for the past. The problem is that there is an arrow of time that moves in the forward direction, not the reverse. Past history is actual, the future is not. Here's another attempt to show the problem of traversing an infinite past. Imagine that history is a line and this discussion is occurring at time p. The potential future extends before us as a geometrical ray, but in reality, there is nothing there yet; it has not been actualized yet. Those who believe the past is infinite cannot say the same about the past. It has been actualized. That is, in their mind, there is an actual infinite number of years that have elapsed in order to arrive at this point. In other words, the past is like a geometrical ray that would require one to count down from infinity one year at a time until arriving at this point. But just as you can never begin at the point and count up to infinity, so you cannot count down from infinity to this point, one year at a time.

Hilbert's Hotel and the law of contradiction: Finally, let me respond to Jeffrey's contention that "Hilbert's hotel does not provide any logical contradictions; it merely provides situations that most people find counter-intuitive. But quantum mechanics and relativity are counter-intuitive, too." I will now show that this is not the case.

As I have suggested above, and as Hilbert also holds, the principle of non-contradiction is an axiom not only for mathematics, but for logic, for human reasoning in general, and for science. Hilbert makes it evident from the sheer number of times he uses words like 'contradiction', 'paradox' and 'consistency' that the problem of the infinite was primarily that it violated the principle of non-contradiction. Jeffrey does allow that Hilbert's Hotel 'provides situations that most people find counter-intuitive'. I agree, but we need to be more precise than that. What, exactly, is counter-intuitive? Hilbert makes it clear that it is the inconsistencies, contradictions, and paradoxes, all of which he seems to use interchangeably as violations of the axiom invoking the law of contradiction (or the law of non-contradiction, as it is also known). So, to be more precise, what is counter-intuitive about some situations that arise out of Hilbert's Hotel is that it produces situations that violate that axiom that is fundamental to both mathematics and logic. Let me illustrate.

Assume the owner of Hilbert's Hotel decides that he only wants people in rooms that have a number ending in zero (i.e. 10, 20, 30, …). Everyone else must leave the hotel which, for those of you who are not familiar with the hotel, has an actual infinite number of rooms. The concierge begins to go down the hall, beginning with room number 1, and notify the occupants whether they can stay or not, depending upon the criterion that the room number must end with a zero. The doorman, who is a bit of an accountant, decides he wants to keep track of the number of people who have been notified (denoted by n) and, out of that number, how many have left the hotel (denoted by g), how many have stayed, (denoted by s), and the difference between the two values (denoted by d). He realizes that as long as he knows the value of n, he can calculated the other values as follows:

g = 9n/10 (1)

s = n/10 (2)

d = 8n/10 (3)

As the concierge notifies more and more people, all there values increase, but not at the same rate, with the result that the difference between the number of people who have left and the people who have stayed gets larger and larger. As n approaches infinity, the difference between the number of people who have left and the number who have stayed, approaches infinity. Finally, the concierge realizes that he will never notify everybody if he does it one room at a time, so he gets on the PA system and notifies the entire hotel. At that point, n becomes an actual infinite, with g, s and d becoming actual infinites as well. According to the three equations, the doorman knows that infinitely more people have left the hotel (d) than have stayed, so g should be infinitely larger than s. But the problem is that it is a property of an actual infinite that the cardinality of an actual infinite set (denoted by the famous aleph-null) is the same as its proper subsets. We are left with something that is counter-intuitive, specifically, a violation of the law of non-contradiction. It is true that, from Eqn. (3), infinitely more people have left the hotel than have stayed. Yet because g, s, and d, are all actual infinites, they have the same cardinality (aleph-null). For example, as n approaches infinity, d approaches infinity (i.e., d gets larger and larger). If n is infinite, so is d, but if the cardinality of g and s must be Aleph-null, then g and s are equal and so d must be zero when n is infinite, but since d is also an actual infinite, its cardinality must be aleph-null as well; a violation of the law of contradiction. This is similar to Bertrand Russell's famous 'Tristram Shandy paradox'. An actual infinite is not the same size as a potential infinite, as Hilbert points out, but all actual infinites have the same cardinality, contrary to what I think some have said here.

In conclusion, Hilbert's Hotel does prove a counter-intuitive, specifically, a violation of Hilbert's axiom II (i), the 'law of contradiction'. This was and is, I contend, the primary 'problem of the infinite', as Hilbert put it. All this to argue that past history cannot be composed of an actual infinite set of years.

Kirk

IvanM said...

Altogether, I would say these arguments by Durston and Craig are embarrassingly naive.

Indeed.

Kirk said: “...check Saturday for my response.”

That reminds me of something...

IvanM said...

Kirk, you're digging yourself into a bigger hole, but I do want to thank you for prompting me to read Hilbert's paper On the infinite.

Your conclusions are based on quote mining ("Our principal result is that the infinite is nowhere to be found in reality") combined with garbled reasoning (I'm puzzled by the fascination with the axiom of contradiction). From reading the paper, it is clear that this "result" is merely an informal conclusion based on some plausible ideas from quantum mechanics (small scale) and cosmology (large scale). It is not a logical or mathematical result, nor could it be.

Hilbert is concerned with promoting the establishment of formal proof systems involving finitary reasoning. He is not deducing anything about the physical universe (or about mathematics either-- it's just an essay); although if the universe were indeed finite in space and time, it would perhaps be, as he says, "a remarkable harmony between being and thought."

Bayesian Bouffant, FCD said...

I would like to respond to three items in Jeffrey's response to my original post.

This is tangential, but what is with this annoying modern habit of assuming one is on a first-name basis with another who is not a close friend?

Bayesian Bouffant, FCD said...

So let's call that -infinity. We'll never
get there by getting in a time machine and going back, so let's
just take a quantum leap back into the past, we're now at minus
infinity. Now some of those of you who are familiar with
infinite set theory might be a little uncomfortable at this
point because if the past is, if you're saying it actually is
infinite, what you mean is that there actually is an hour back
there that is infinitely separated from this one. So let's
count our way down now. Infinity minus 1, infinity minus 2.


Hey I have an idea: instead of counting down, let's count up! Infinity plus 1, infinity plus 2...

What does it mean to say that you would go back to the beginning of something that (, as you have already agreed,) does not have a beginning?

Timmy said...

Improbable:
While it is true that there are as many rationals as natural numbers, the really wacky thing is that there are MORE irrational numbers than rationals. I'm surprised some religious mathematician hasn't tried to make that into a proof for a deity's existence.

To those that think time had a specific beginning, I must ask: What do you think was around before time began? When did that time begin? You get yourself into an infinite regression when you take an infinite concept and try to make it finite and infinite at the same time. If you consider the beginning of time to be when the universe came into existence, then you are talking about the amount of time that has elapsed since then. But, then I must sincerely ask: What do you think is out there once you reach the edge of our "universe"? If you could fly to the edge of the last bit of visible matter, what is left if you think the universe is finite? A wall? Something to stop you from going any further? But then someone will say that when they speak of the universe that they are talking about all matter, not space itself. What do you think space is then? That implies that there would be a finite "outer space", which can't be the case. Again what is going to stop you from going any further once you reach the last bit of matter? A wall?
Unless I see evidence otherwise, I stand by my hypothesis that it is illogical for there to be a finite amount of matter, a finite "outer space", and a finite time-line.

Bayesian Bouffant, FCD said...

Using a time machine to disprove a finite past

(This is at least as rigorous as Durston's use of a time machine to disprove an infinite past.)

Imagine you have a time machine. Imagine the universe began N years ago (14 billion, or any other number you are fond of). Set the dial on your time machine to N+1 years in the past. You might protest that nothing exists in that time. But this is clearly false, since the time machine exists then. Bingo! Creation date disproven. This works for any value of N, therefore the past must be infinite.

secondclass said...

Kirk, a few responses:

Now imagine that an observer is coming toward us from the infinite past in her own time machine that moves forward through time. When will the two time machines pass each other? Never.

Why never? You seem to be assuming your conclusion, namely that the other observer cannot have already traversed infinite time.

For example, we use imaginary numbers in everyday three phase electrical calculations, but imaginary numbers and imaginary infinites cannot, logically, exist in reality. No one, for example, would argue that I can have the square root of -1 apples in a basket.

It's true that number of apples in a basket must be a real number, but it's also true that the impedance of an inductor must be an imaginary number. The idea that imaginary numbers can't exist in reality, with the implication that real numbers can, doesn't make sense. Neither real nor imaginary numbers exist as physical objects, but both are useful for modeling physical phenomena.

If n is infinite, so is d, but if the cardinality of g and s must be Aleph-null, then g and s are equal and so d must be zero when n is infinite, but since d is also an actual infinite, its cardinality must be aleph-null as well; a violation of the law of contradiction.

There is no reason that d must be zero. Infinity minus infinity is an indeterminate form, so there is no contradiction.

Anonymous said...

Kirk,

Thank you for responding. I thought your debate with Shallit was masterfully done. Not only did you beat him in a verbal debate, but you finished him off quite well with your literary skills here in his blog!

Jeffrey,

I hope you would at least be humble enough to admit you deliberately misquoted Dr. Craig. You claimed he said Hilbert said an infinite regress of causes was mathematically impossible. Dr. Craig never said that. Instead of admitting your bold-faced lie you make the laughable excuse that he misunderstood you. As we can see he's not the one with that problem.

IvanM said...

Timmy: Space can be finite without having a boundary wall, in exactly the same way as the two-dimensional surface of a sphere has finite area but no boundary. There are many possibilities for the topology of the universe, whether finite or infinite in volume.

Jeffrey Shallit said...

Anonymous:

If you're going to comment on the blog, please exhibit some politeness.

" You claimed he said Hilbert said an infinite regress of causes was mathematically impossible."

No, I said Durston said that. And I said Durston was misled by Craig, which is correct.

Even assuming I quoted Craig incorrectly, why do you think this would be a "lie" as opposed to a simple mistake?

Anonymous said...

Jeff,

If you're going to comment on the blog, please exhibit some politeness.

May I ask that you do the same, too? You've made ad hominem remarks that Craig and Kirk were acting childish with respect to their understanding of mathematics. That is not polite, sir. You're obviously exhibiting some hypocrisy.

"You claimed he said Hilbert said an infinite regress of causes was mathematically impossible."

No, I said Durston said that. And I said Durston was misled by Craig, which is correct.


So, let me get this straight, you NOW admit that Craig never said it, but that Durston said it. But then Durston was misled by Dr. Craig who, uh, never said that an infinite regress of causes was mathematically impossible, anyway! So, how in the world can Dr. Craig mislead Durston if Dr. Craig never conceded it? LOL! You really don't know what you're talking about, do you? No, Jeffrey, that is not correct.

Even assuming I quoted Craig incorrectly, why do you think this would be a "lie" as opposed to a simple mistake?

So, you made a mistake, then? Well, you just admitted that Dr. Craig never said an infinite regress of causes was mathematically impossible via Hilbert. Can you at least apologize for making that mistake then? And can you apologize to Durston for saying he was misled by Dr. Craig? Since you admitted Dr. Craig never made said claim about Hilbert.

Jeffrey Shallit said...

Anonymous:

You say, "You've made ad hominem remarks that Craig and Kirk were acting childish with respect to their understanding of mathematics."

You are incorrect. I said (and it is easy to verify) that "In general, Craig seems to have an extremely naive, almost childish view of infinity", and I stand by that. Having a "childish view" is not even remotely like "acting childish". And you don't even know what "ad hominem" means, either, since my comment was not "ad hominem".

Sorry, Mr. Anonymous, you're out of here.

Timmy said...

IvanM:
Thanks for that link! That was interesting. But, one reason I am not a theoretical physicist is that there are things that I can't process for myself. The idea that the universe has a shape of any kind and a finite area and volume is something that I can't process. My problem lies here: if the universe is finite, and somehow I get close to the edge, what is there to stop me from going further and what is beyond that boundary? I hope my question makes sense :-P

Maybe there is some explanation for this where I have to think differently, like when I found out about the Riemann Spiral(I think that's what it is called). For those that don't know what I mean, I am talking about this "spiral" that Riemann came up with that spirals through all 3 dimensions and needs a 4th dimension to spiral into to exist. When I first read that it I couldn't think anymore the rest of the day, like it drained my brain-battery. Maybe I would have to think like that. I guess that's what I'm getting at.

It's questions like these: infinite time and space; that make me wish some super-evolved smart and wise aliens(I think statistically they do exist) would come along and give us a hand with these things ;-)

Sorry, Jeffrey, for getting off topic. I do tend to do that.

Erdos56 said...

Going in a slightly different direction, Quentin Smith works a bit on this topic and has reprints here.

The arguments move quickly into the arcana and hermeneutics that professional philosophers love so well, but it's nice wrapping one's head around Putnam or Kripke from time to time.

Jim Lippard said...

Craig (and apparently Durston) are presuming what J. M. E. McTaggart called A-series time, where "now" is an objective moment that traverses time, passing through past events into the future. Most physicists advocate B-series time, where all events, past, present, and future, are part of space-time. (Perhaps it's not fair to say "presuming," as Craig does argue for it.)

Richard Sorabji's 1983 book, _Time, Creation, and the Continuum_ has a critique of Craig's kalam cosmological argument which focuses on his arguments about the alleged impossibility of an infinite past.

Graham Oppy has also had several exchanges with Craig regarding the argument, and in "Time, Successive Addition, and Kalam Cosmological Arguments," he specifically takes issue with the idea that past events must be a temporal series produced by successive addition.

Bayesian Bouffant, FCD said...

Bertrand Russell, from The Scientific Outlook (1931) as included in Bertrand Russell on God and Religion (ed. Al Seckel, (1986):

The purely intellectual argument on this point may be put in a nutshell: is the Creator amenable to the laws of physics or is He not? If He is not, He cannot be inferred from physical phenomena, since no physical causal law can lead to Him; if He is, we shall have to apply the second law of thermodynamics to Him and suppose that He also had to be created at some remote period. But in that case He has lost His raison d'etre. It is curious that not only the physicists, but even the theologians, seem to find something new in the arguments from modern physics. Physicists, perhaps, can scarcely be expected to know the history of theology, but the theologians ought to be aware that the modern arguments have all had their counterparts at earlier times. Eddington's argument about free will and the brain is, as we saw, closely parallel to Descartes's. Jeans's argument is a compound of Plato and Berkeley, and has no more warrant in physics than it had at the time of either of these philosophers. The argument that the world must have had a beginning in time is set forth with great clearness by Kant, who, however, supplements it by an equally powerful argument to prove that the world had no beginning in time. Our age is rendered conceited by the multitude of new discoveries and interventions, but in the realm of philosophy it is much less in advance of the past that it imagines itself to be.

Kirk said...

Sorry for taking so long to respond. I find it difficult to squeeze in the time to participate in internet discussions, but this is an interesting topic, and the readership here is not necessary in favor of what I'm arguing for, which I like. I'll briefly address several points raised here, and especially the question of whether an infinite regression of a causal chain is mathematically possible.

On Bayesian Bouffant's objection to the use of first names
I see blog discussions like this as akin to going for coffee after class to hash out some interesting issue, where the use of first names contributes to an informal and collegial atmosphere. In no way do I mean disrespect to anyone in this forum and especially to Jeffrey. Dr. Shallit is a professor at one of Canada's top-ranked universities and automatically commands respect for that accomplishment. Of course, I'm happy to call anyone whatever they wish. I'll continue to use first names unless otherwise requested. One additional thing; I especially respect anyone who uses their real name in these discussions, even it it's only their first name.

Quote mining to misrepresent vs. quoting to support a summary
IvanM, I think I did the latter. Hilbert sets out to write his paper to address the problem of the infinite. On page two he writes, "It is, therefore, the problem of the infinite in the sense just indicated which we need to resolve once and for all." In his summary at the end of the paper, he believes he has resolved the problem and writes,
Our principle result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought – a remarkable harmony between being and thought.
What happened in between? First he lays out the problem, violations of the law of contradiction, also referred to as paradoxes and inconsistencies. Second, he lays out a series of axioms that need to be applied to this problem. Third, he emphasizes the need for consistency as the "one condition" and an "absolutely necessary one" which is ensured by axiom II (i), the law of contradiction. He does present a theory of proof, as IvanM mentions, but IvanM overlooks the primary reason and context within which Hilbert presents that theory, the problem of the infinite. Speaking of the sketchy use of quotes, I would also question how IvanM uses the quote at the end of his response. See my longer quote above for the context of that quote.

On the nature of time in this universe
Both Timmy and Bayesian Bouffant have stated a view about space-time that is corrected by IvanM. I want to add something and use Bayesian Bouffant's N+1 disproof of the notion that time had a beginning (where N=the age of the universe). Einstein's General Theory of Relativity requires that space-time is inseparable. When astronomers and cosmologists talk about the beginning of the universe, they are talking about the beginning of space-time itself. In other words, there is no time = N+1. There is no 'before'. The controversy is the nature of the beginning.

On the beginning of time
During my years doing an undergraduate degree in physics at the U of Manitoba, I learned that most physicists do not think in terms of an A-theory of time or a B-theory of time, indeed, most would have never heard of those terms. To physicists, time is a physical property of the universe, inseperable from space, and seems to flow only in one direction on the macroscopic level (see the arrow of time http://en.wikipedia.org/wiki/Arrow_of_time ). Some hold (I call this the Newtonian view of time, which may or may not be correct) that physical time is not quantized and that the universe began with a singularity at time t=0, where t denotes a point. The singularity, itself, is a violation of axiom II (i), which is why many physicists have a problem with it. Others believe that at the microscopic scale, time may be a quanta within which there is no arrow of time. In this view t=0 is not a point but a quanta. In other views, such as the pea-instanton theory, time might begin as an extended surface on a small sphere, to use the pea analogy. In all cases, time has a beginning, 'before' which there is no space-time. Only a Newtonian view of time insists that t=0 must represent a point. t=0 can represent the first quanta, the extended surface of space-time on the pea instanton, or the beginning of the arrow of time whatever the nature of that beginning.

On the use of 'indeterminate' when it comes to infinite set theory
Secondclass wrote, "Infinity minus infinity is an indeterminate form, so there is no contradiction." Well, that might be true so long as we did not worry about why the word 'indeterminate' is used in these cases. The word 'indeterminate' is used precisely because violations of Hilbert's axiom II (i) arise, as illustrated in my previous post about Hilbert's Hotel if we take the application of actual infinites to reality seriously. I want to remind the reader that we are talking about the cardinality of aleph-null when it comes to actual infinites.

Problem with Jeffrey's examples:
In his opening post in this discussion, Jeffrey suggests that 'we could define times -1, -2, -3, etc. without having to define an actual point called "-infinity".' In his second example, Jeffrey suggests that we could have an event at time 1/(n+1) causing an event at time 1/(n). There is a problem with this in that neither example can apply to the causal chains of the history of the universe. If the history of the universe had no arrow of time, or the arrow of time could run in reverse, then Jeffrey would be right. In both his examples, time relies on a potential infinite. In the first example, one works backward through time and, of course, never reaches infinity. This is a classic example of Hilbert's concept of a potential infinite (page 189). In Jeffrey's second example, as n approaches infinity, t approaches 0. But since n never reaches infinity, the curve never reaches zero on the t-axis.

I was talking about the history of space-time, and whether it had a beginning. In the actual world, the arrow of time does not run backward, which entails that the sequence of causal events does not run backward either. Causal chains unfold only in the forward direction (which does allow for simultaneity in macroscopic quantum events). Thus, Jeffrey's first example fails to show that the past history of space-time could be an actual infinite. The second example is also a potential infinite, but this time it is the variable n. Since in actual space-time, the arrow of time flows only in the positive (forward, or future) direction, this forces the variable n to descend from infinity. For a mathematical model to apply to actual space-time, every critical component of the model must have some correspondence with something in the real world. What does n represent in the actual world? If n represents an actual infinite in reality, then it violates Hilbert's axiom II (i), and n cannot be a potential infinite given the arrow of time for this universe. Thus Jeffrey's second example, although valid for potential infinites, fails to show that the past history of space-time could be infinite.

On the mathematical impossibility of an infinite regression of causes in actual history
I will invoke the mathematical concept of Hilbert's actual infinite and Hilbert's axiom II (i) in his mathematical theory of proof.
1. Because the arrow of time in actual space-time moves only in the forward direction for macroscopic events, an infinite regression of causes requires that some real component of past history is an actual infinite.
2. An actual infinite cannot occur in reality (due to violations of axiom II (i))
therefore, an infinite regression of causes in actual space-time are mathematically impossible from the mathematical concept of an actual infinite and violations of axiom II (i) if an actual infinite is applied to reality.

Jeffrey Shallit said...

Kirk:

The distinction between "actual" and "potential" infinity does not exist in mathematics. There is a well-established theory of infinite quantities, and "potential infinite" is not one of them. Infinite quantities do exist in mathematics, and are manipulated all the time; therefore, to claim that Hilbert had a mathematical proof that there is no infinity in nature is simply silly.

Hilbert was writing 80 years ago, before many discoveries of modern physics. Modern physicists do consider infinite quantities, such as the example of Malament-Hogarth spacetime I cited.

In my example of a past time existing at -n for all positive integers n, this is not a "potential" infinity, but an actual infinity, in the sense that mathematicians use: I can make a one-one correspondence between the positive integers (an infinite set) and time coordinates.

I completely dispute your claim that "But Hilbert has shown that, given his theory of proof, one of the axioms of which is the law of contradiction, an actual infinite is impossible in reality." Please support this by providing a citation to the exact page number where this is "shown".

You say, "it is a property of an actual infinite that the cardinality of an actual infinite set (denoted by the famous aleph-null) is the same as its proper subsets." Of course, this is nonsense. The integers are a subset of the reals, but they don't have the same cardinality.

You say, " Finally, the concierge realizes that he will never notify everybody if he does it one room at a time, so he gets on the PA system and notifies the entire hotel. " But this is supposed to be an actual physical experiment, not just a thought experiment. How can we, in physical space, "notify the entire hotel" in a finite time? We can't, because the speed of light is finite. So your thought experiment fails immediately, right there.

I also dispute your claim "If n is infinite, so is d, but if the cardinality of g and s must be Aleph-null, then g and s are equal and so d must be zero when n is infinite, but since d is also an actual infinite, its cardinality must be aleph-null as well; a violation of the law of contradiction." There is no contradiction here, only your misunderstanding about how to manipulate infinities.

Anonymous said...

"Infinite quantities do exist in mathematics, and are manipulated all the time"

Caveat: the existence of an infinite set is an axiom of ZFC, but is not part of the Peano axioms.

secondclass said...

Kirk: The word 'indeterminate' is used precisely because violations of Hilbert's axiom II (i) arise...

No, the word "indeterminate" is used because it's indeterminate. You resolved the indeterminacy by algebraically subtracting s(n) from g(n), showing that g(n)-s(n) goes to infinity, which is correct. This contradicts your notion that removing a countably infinite number of members from a countably infinite set must result in a null set. But that notion is incorrect, so there is no contradiction.

Craig makes the same kind of mistake, imputing properties of finite numbers to transfinite numbers (along with other fallacies such as question-begging and appeals to intuition/incredulity).

Kirk said...

Jeffrey:
By now, it's probably just you and I reading this exchange. However, I appreciate this opportunity to clarify what I am arguing for and not arguing for.

Summary of what I have been arguing:
1. The past history of the universe is finite. Empirical evidence supports this, the impossibility of having traversed an infinite set of actual years, one year at a time, supports this, and the impossibility of an actual infinite in reality supports this. Our discussion has centered on the latter, specifically can past history be composed of an actual countable infinite number of years.
2. I do not take a platonic view of numbers and mathematics. I make a very clear distinction between mathematical equations/models and reality. A mathematical equation fails to be relevant to reality if each variable in the equation does not correspond to a physical property or attribute in our actual physical universe. Thus, neither of your examples are relevant to the real world. Your first example requires the arrow of time to move in reverse. Your second example has a variable n that corresponds to no component in the physical universe.
3. If something is mathematically impossible then, necessarily, it is impossible in the real world, but that is not a bi-conditional (if and only if) statement. Something may be impossible in our particular universe, but still mathematically possible or possible in another universe. An example of this would be the arrow of time. In mathematics, it can assume the reverse direction, not so in our physical universe, although there may be other universes within which the arrow of time runs in reverse.
4. I am not (and I want to be crystal clear about this), not, arguing that we cannot or do not use the concept of an actual infinite (e.g., the set of natural numbers) as an idea within mathematics. I am arguing that, because of the paradoxes, inconsistencies, and contradictions that arise when trying to apply actual infinites to reality, it is irrational to believe they can exist in reality (i.e., a violation of axiom II (i)).
5. I am assuming we are not Platonists, where the set of all natural numbers have actual ontological existence. I see axiom II (i) as the defeating axiom for Platonism.

A need for caution:
In response to my statement that it is a property of an actual infinite set that the cardinality of an actual infinite is the same as its proper subsets, you wrote, "Of course, this is nonsense. The integers are a subset of the reals, but they don't have the same cardinality."

Keep in mind here that we are discussing Aleph-null and, therefore, countably infinite sets. The set of real numbers is not a countable infinite set with cardinality Aleph-null. Care must be taken in distinguishing between a countable and uncountable infinite set. Within the context of the discussion of whether the past history of the universe is composed of an infinite number of years or not, we are talking about a countable infinite. See the discussion on Aleph-null. It states,

Aleph-null (\aleph_0) is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. A set has cardinality \aleph_0 if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers, the set of all integers, the set of all rational numbers, the set of algebraic numbers, and the set of all finite subsets of any countably infinite set.

As one examines this definition of Aleph-null, and reflects on the requirement that the cardinality of the set of all natural numbers is the same as the cardinality of the set of all prime numbers, one can see why contradictions can arise, as I illustrated using Hilbert's Hotel. This is where Secondclass's objection needs to be addressed.

Secondclass wrote: " No, the word "indeterminate" is used because it's indeterminate. You resolved the indeterminacy by algebraically subtracting s(n) from g(n), showing that g(n)-s(n) goes to infinity, which is correct. This contradicts your notion that removing a countably infinite number of members from a countably infinite set must result in a null set. But that notion is incorrect, so there is no contradiction."

First, it is not my notion that 'removing a countably infinite number of members from a countably infinite set must result in a null set." I showed that the difference in cardinality between g(n) and s(n) increases as n approaches infinity according to d(n)=8n/10. However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite. This is a paradox/contradiction/inconsistency.

Jeffrey wrote: "The distinction between "actual" and "potential" infinity does not exist in mathematics. There is a well-established theory of infinite quantities, and "potential infinite" is not one of them. Infinite quantities do exist in mathematics, and are manipulated all the time; therefore, to claim that Hilbert had a mathematical proof that there is no infinity in nature is simply silly"

Eric Schechter writes in the section titled 'History and Controversy', "Nearly all research-level mathematicians today (I would guess 99.99% of them) take for granted both 'potential' and 'completed' infinity, and most probably do not even know the distinction indicated by those two terms." My own response is to consider y=5/x. There is a very big difference between treating x as a potential infinite 'as x approaches infinity' and as an actual (completed/definite) infinite 'when x is infinite'. We may use the idea of the actual infinite in mathematics all the time, but as I pointed out in (2) and (3), we must be very careful to make a distinction between a useful idea or concept in mathematics and the physical universe. It really does not matter what modern physics has discovered recently (and it certainly has not discoved an example of an actual infinite). Discoveries and advances in modern physics do not trump mathematical axioms. I.e., if axiom II (i) is accepted as true in mathematics (which it must if it is to practice formal proofs), then it must be accepted in the real space-time continuum in which we find ourselves (see (3) of my initial summary). Physicists certainly use the idea of infinity, but they get into trouble when they attempt to apply them to reality, whether it be the infinitesimally small (e.g., a singularity) or the infinitely large. I did a quick survey of the web and I would have to say that you are very much in the minority in arguing that actual infinites can exist in reality. I think you need to provide an example in the physical universe (and not a mathematical model …. I already dealt with the Malament-Hogarth spacetime and showed how it breaks down at the speed of light). A mathematical model is not to be confused with reality. A chronic problem often encountered in physics with mathematical models is their incompleteness and/or approximation. Some variables transpose beautifully and others either have no counterpart in reality, or they violate axiom II (i), the principle of non-contradiction (as in the singularity).

Jeffrey wrote: " I completely dispute your claim that "But Hilbert has shown that, given his theory of proof, one of the axioms of which is the law of contradiction, an actual infinite is impossible in reality." Please support this by providing a citation to the exact page number where this is "shown".

That depends upon whether we are talking about an informal proof or formal proof. Hilbert does not offer a formal proof but I think he does offer an informal proof ("…high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience.", Wiki, ref above). I will outline it here, including page numbers as requested:

Principle axiom: II (i), p. 198 (see also last paragraph on p. 199 for verification that this is the principle axiom)
Problem: Treating the actual infinite as anything more than a useful idea or concept for mathematics leads to violations of axiom II (i), ('… the literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite.' p. 184 …. there are also several other references to the paradoxes, contradictions, and inconsistencies that arise out of treating the infinite as real, or something more than just a useful mathematical concept or idea.)
Conclusion: The infinite "neither exists in nature nor provides a legitimate basis for rational though …", p. 201 (and rational thought includes mathematical discussions of the infinite).

I would think that Hilbert's informal proof is sufficient, given that axiom II (i) coupled with the paradoxical 'problem of the infinite' he mentions, makes it evident that the actual infinite cannot exist in reality. I would also not use the term 'exists in mathematics' when discussing the actual infinite, unless one very carefully defines what one means by 'exist', for mathematical 'existence' is certainly different from ontological, or 'real' existence.

If one wants to construct a formal proof, then I would suggest using the last paragraph on p. 199 as the starting point, with the objective of showing that 1≠1 if one assumes an actual infinite to be real.

Jeffrey wrote re. Hilbert's Hotel thought experiment: "But this is supposed to be an actual physical experiment, not just a thought experiment. How can we, in physical space, "notify the entire hotel" in a finite time? We can't, because the speed of light is finite. So your thought experiment fails immediately, right there."

No, it's not supposed to be a physical experiment. It is a thought experiment to see if an actual infinite can exist in reality. If the thought experiment produces a violation of axiom II (i), then it will fail in reality as well (see my (3) at the outset of this post). Of course, there isn't enough mass in the universe to build Hilbert's Hotel, humans will never produce an infinite number of offspring to fill Hilbert's Hotel, the speed of light limits things (as you point out), etc. (By the way, we could use a PA system that depends on quantum simultaneity to get around the limitation imposed by the speed of light.)

Conclusion:
Jeffrey, I am not sure whether you are just playing the devil's advocate here (and I do not use that term pejoratively, as I apply that term to myself sometimes in discussions) or you truly belief that:

a) actual infinites can exist in reality (i.e., map to some real component of the physical universe),
b) an infinite regression of causes can occur where the arrow of time runs only forward.

I see very little support for your position in my brief survey of the web, although my sampling may well be insufficient and truth is not decided by numbers. I think for (a), you will need to supply a physical example where each variable in your mathematical model maps to a real component of the universe. For (b), I don't think you actually hold this position, but are playing the devil's advocate; it's indefensible in reality.

Again, I appreciate your hospitality in permitting me to clarify and defend my position here. It is an interesting topic, though it may be unrealistic to think we are going to resolve it here.

Timmy said...

I must admit that you all have gone above my head with the physics and math mix here, lol!

Anyway, regarding the comments about g(n) - s(n) = null. Unless I am mistaken, you can add to aleph-null and still have aleph-null. Does it not also work with subtraction? Theoretically can't you subtract an infinity from aleph-null and still have aleph-null?

Also, as I've stated before, it is my hypothesis that you can think of every "now" as a zero on a number line. There is infinite time behind us and infinite time ahead. My reasoning is this: if time had a finite starting point, then what was going on before that? I, personally, can't think of an answer to that without committing some kind of logical fallacy, assuming I haven't already done that myself. Assuming that if we could not get to this point if time was infinite is the same fallacy, IMHO, as the old greek argument that motion is impossible from A to B because you would have to go half that distance and before that, half that distance, and before that half that distance, ad infinitum. It is obvious that motion is possible since you simply just move along and your frame of measuring has no bearing on your motion. I propose that is the same way with infinite time. It is obvious that time passes. It passes whether we measure it or not. So, going in reverse, with A being negative-aleph-null, why can't you go back infinitely? I can't see a reason myself.

IvanM said...

By now, it's probably just you and I reading this exchange.

Naw, I like to drop by once in a while to have a nice choke on my beverage. To practice my spit-takes, y'know.

Hilbert does not offer a formal proof but I think he does offer an informal proof ("…high-level sketches that would allow an expert to reconstruct a formal proof...") ... If one wants to construct a formal proof, then I would suggest...

Bzzzzt. Wrong. Thanks for playing. I'm sure it's fun to pretend you can argue mathematics with the big boys, but this is ridiculous.

(If I seem too scornful, consider the magnitude of your offenses against sound mathematical reasoning. And since Jeffrey is being the good cop, I figure someone has to be the bad cop.)

On a more constructive note, if you really want to learn about this stuff you might check out this course. It's really fascinating material, and the first step in understanding something is to admit what you don't know. Wikipedia and historical essays can be good resources, but often require some nuances of understanding to appreciate properly.

secondclass said...

Kirk: First, it is not my notion that 'removing a countably infinite number of members from a countably infinite set must result in a null set."

Sorry, I was trying to guess at the basis for your claim that "d must be zero when n is infinite". Whatever hidden assumption that claim is based on, it's false.

However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite. This is a paradox/contradiction/inconsistency.

Your scenario involves a particular mapping from S to G -- there are nine elements in G for every element in S. But you can't determine whether two sets have the same cardinality by looking at a single mapping. Other mappings from S to G are bijective, so S and G have the same cardinality. That fact is not contradicted by anything in your scenario, not even by the fact that d=infinity for your 9-to-1 mapping. Your claim that "the difference in their cardinality is infinite when n is infinite" is simply wrong.

On a more fundamental note, I find it strange that you think that actual (physical) infinites can be formally disproven. Whatever axioms you base your proof on, you'll have to show that those axioms model reality accurately, and that's an empirical issue. Good luck.

On a final note, if you think that your position (i.e. that actual infinites can be mathematically disproven) is a majority position, I'd be interested in references to modern mathematicians who agree with you.

Bayesian Bouffant, FCD said...

The past history of the universe is finite. Empirical evidence supports this, the impossibility of having traversed an infinite set of actual years, one year at a time, supports this

It's impossible at one year at a time? Well then, traverse it two years at a time! Or five. But seriously, it is only "impossible" when you imagine that the traversal has to be done in a finite time, by a creature with a finite lifespan. I.e. you do not have a mathematical proof, but only an argument from incredulity.

Bayesian Bouffant, FCD said...

Both Timmy and Bayesian Bouffant have stated a view about space-time that is corrected by IvanM.

I have no idea what you are writing about. The way to disprove something reductio ad absurdum is to start with a legal proposition within a system. Then you work within the rules of a system until you wind up at a position contrary to your original position. When you say you go back to the beginning of a universe with no beginning, do you really think that is working within the rules of the system? Your time traversal "proof"fails.

You then proceed to slide from your time traversal "proof" to comments about the empirical nature of space-time and a presumption that the Big Bang was the beginning of the universe, which is a separate empirical claim. These cannot be part of your time traversal thought experiment. Also, there are already cosmologists considering what may have preceded the Big Bang.

Jeffrey Shallit said...

Kirk:

No offense, but you don't know what you're talking about. You demonstrate this when you say, "I am arguing that, because of the paradoxes, inconsistencies, and contradictions that arise when trying to apply actual infinites to reality, it is irrational to believe they can exist in reality (i.e., a violation of axiom II (i))." But there are no paradoxes (in the mathematical sense) involving the infinities we are talking about. There may be paradoxes involving your intuition about how infinite quantities should behave, but this just indicates your intuition is wrong. People's intuitions have been wrong when confronted with relativity and quantum mechanics, but I don't see you claiming these are impossible. The fact is that mainstream physicists deal with the possibility of infinite quantities existing, as in the example of Malament-Hogarth spacetime. So you can't offhandedly dismiss this as "paradoxical".

You're still wrong about the claim that the cardinality of an infinite set is the same as the cardinality of its subsets. I already gave you one example, which you simply dismissed for no reason, so here is another. Consider Z and {0}. {0} is a proper subset of Z, but it does not have the same cardinality as Z.

You write, "However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite. This is a paradox/contradiction/inconsistency." No, there is no paradox - you are simply giving an example of something well-known to every mathematician: a statement that is true for every finite prefix of an infinite set need not hold for the infinite set as a whole. Consider, for example, the well-known theorem that the union of two regular languages is regular. This is also true for every finite union, but not true for infinite unions. This is not a "paradox/contradiction/inconsistency", but merely a well-understood phenomenon of infinite sets. I think you need to take a refresher course in set theory.

I claimed that mathematicians don't talk about "potential infinity", and in response you give me a web page by a person who says in the first line, "I am not a leading researcher on infinite sets". Good job! I'll ask again: go look at any book on set theory, and show me where "potential infinity" is defined.

You say, "My own response is to consider y=5/x. There is a very big difference between treating x as a potential infinite 'as x approaches infinity' and as an actual (completed/definite) infinite 'when x is infinite'." Kirk, this depends entirely on what your domain is. Are you working in the reals, or the extended reals?

No offense, but your understanding of this topic is woefully deficient.

Francisco Antonio said...

Newton da Costa and I & coworkers have proved a theorem that asserts:

Consider the set of spacetimes. Then the subset of spacetimes that are exotic and have no `cosmic time' property is topologically and measure-theoretically generic.

Spacetime: a 4-manifold, real, smooth, endowed with a smooth 1-foliation (the ``arrow of time'')

Exotic: pick up any good recent book on the topology of 4-manifolds and learn from it.

Cosmic time property: the property that allows one to say things like, ``the universe began 14 billion years ago.'' Time, on the contrary, can well be a `local' property. Worse: at very very large scales, time may not even exist, and result from a broken symmetry phenomenon at our scale.

(I won't even delve on the infinities of set theory...)

F. A. Doria