♠ Friday, June 11, 2004
On the Nature of Time and Infinity
Darn, I can't find a photo online of the Pythonites in the "Australian Philosophers" sketch to replace the Joker for this article. It might be more appropriate. (No poker here.)
Gil and I have exchanged a number of drunken, uhh, exchanges, on the nature of time and infinity. The particular problem is time in the past. (Gil says this is all an antimony of Immanuel Kant's, but really, how good a philosopher could Kant be, if he never wrote a Poker book?)
The problem is this. Assume a steady-state universe, that is, one that has always existed and will always exist. (This seems to be contrary to the way the universe actually works, but facts shouldn't get in the way of a thought experiment.) Does the fact that we find ourselves at a particular moment in time, mean that the time that has gone before cannot be infinite?
We can make this question apply to the universe as we know it if we take the blinders off of one particular facet of modern cosmology. Cosmologists begin Time at the moment of the Big Bang. They are generally silent as to the nature of the universe before the Big Bang, reasoning that since there is no way to know what went before, and what went before could have had no effect on the universe as we know it, everything before the Big Bang is irrelevant, so it may as well have not happened, and let's decide that in fact it didn't. But, there's no way to know that it didn't, so we can, if we want to apply this argument to the universe as we know it, we can include in the universe whatever happened before the Big Bang—perhaps a previous universe dying in a Big Crunch? In any case, we must assume that Time existed before the Big Bang. (This is actually a separate argument: one I'm sure Gil and I will discuss, sometime. But it's a digression, here, to allow you to apply this argument to the real world if you wish to.)
As I understand Gil's argument, he says that since we got to Now, past time can't have been infinite, because it would have been impossible to travel an infinite amount of time to reach Now.
I think the first problem with this is conceptual. I think Gil imagines a First Day that was "infinity" days ago. But that's not the way infinity works. Infinity is not A Really Big Number; there's no such thing as "infinity days ago." In fact, by postulating past time as infinite, one is saying that there was no First Day. Think of a day really, really far in the past, and the number of days before that, is also infinite. It might seem that the semantic difference between "there are an infinite number of days in the past" and "there are infinity days in the past" is small, but it is not. The first sentence treats each day as a single unit, while the second seems to treat them as an aggregate. There are infinity days in the past? What happened the day before that? It's telling that one could substitute "five" in place of "infinity" in the second sentence without changing the structure at all, while the first sentence would have to be dramatically restructured to do so. All of this being said, though, I'll grant that if Gil's conceptualization was really the way infinity worked, his argument would be correct; it would be impossible to reach Now from that First Day.
(Now you see why I put that stuff in about the Big Bang: The way the universe really seems to work, there was a First Day. We currently think it was between fifteen and twenty billion years ago. For all of this discussion to make sense, we have to postulate a steady-state universe (contrary to apparent fact), or consider the question of what happened before the Big Bang (contrary to modern cosmology).)
So, we have an infinite number of days before Now. Does this mean that Gil is right? I don't think so. Pick a day in the past; it doesn't matter how near or far. The distance (in time) between that day, and now, is finite. For example, the distance between 25 days ago, and Now, is 25 days. It is possible to have reached Now from 25 days ago, because that distance is finite, and we know that it took 25 days to do it. Now, pick a different day in the past, let's say 500,000 days ago. The distance between 500,000 days ago, and Now, is 500,000 days, a finite number. It is possible to have reached Now from 500,000 days ago, because that distance is finite, and we know that it took 500,000 days to do it. It's easy to see that any day we pick in the past will be a finite distance from Now, and that we can have reached Now from any day in the past. Conversely, it is impossible to pick a day which is not a finite distance from Now.
I think Gil's conceptual problem might be giving him difficulty here. He may be trying to pick a day that is "infinity days ago," where the distance between then and Now is not finite. He's argued several times to my formulations of the preceding paragraph, that there must thereby be a day in the past that one cannot reach. I didn't get that, but it would make sense under Gil's conceptualization of infinity. The answer: No, there is no day in the past that one cannot reach. There was no day that was "infinity days ago." The phrase, and the concept it represents, are both nonsense.
Gil didn't like it when I argued from the analogy of a number line, because Time (he says) is not so abstract as a number line. But, I think the analogy holds up well. Now would correspond to the Origin (Zero) on the number line, and all of the negative numbers would correspond to an equivalent number of days, years, minutes, or whatever in the past, and the positive numbers would correspond to equivalent numbers in the future. It is easy to see that there would be a one-to-one correspondence between distance in (pick your unit) from Now, and a corresponding point on the number line. And, on the number line, there is no point marked "infinity" or "negative infinity," noplace that is "infinity points away" from the Origin. Every point on the line is a finite distance from the Origin, and every point on the line is "reachable" from the Origin. Does it not make sense to infer that the same is true of the phenomenon (Time) that we're using the number line to model?
Gil had another problem with the nature of infinity. He pointed out that each day in the past is a finite length. If one adds up a whole bunch of finite things, says Gil, then one cannot arrive at an infinite; past time cannot be infinite. That's true, say I, if one is adding a finite number of finite things. But if one adds an infinite number of finite things, then one not only can arrive at an infinite, but one must arrive at an infinite. Gil has an objection to my response; apparently in Gil-land "an infinite number of finite things" is nonsense. It is easy to demonstrate an example, though; the set of whole numbers is infinite, and it is arrived at by continually (infinitely) adding 1, which is a finite number. An infinite number of finite things. (As a weird aside, if one adds an infinite number of things infinitely small (infinitesimals), then one gets ... One. 10∞ × 10−∞ = 1, by the same math which holds that 1010 × 10−10 = 1.)
I'm stuck for how to end this. If this were a courtroom, I'd say "the prosecution rests," Gil would get to mount his defense, and the judge would then rule in my favor because I'm clearly right and judges always want to make the right decision. (There's an inside joke there.) In any case, the prosecution rests.
[Edited: I'm including the first 16 (I think) comments from Haloscan for posterity's sake; I don't know what will happen to the responses as this message falls deeper and deeper into the archives. I'm also cleaning up Gil's punctuation, which is horrid.]
Man, Gil really takes a beating in today's blog. He always does. How do you guys remain friends? ;-)
Dan | 06.11.04 - 9:08 am | #
Dan actually refutes his own case. Infinity is not just a big number but is, well, infinite. This implies that you can't add up any number of finite measure and reach infinity. Therefore if a day is finite and we to the present, that track must be finite. To say that one can have an infinite number of finite things makes no sense, because of the same threshold problem—if we go a billion days into the past, we still have a finite number, and no matter how many days we add, we still have a finite number. The fact that the past is, well, past, means that we didn't yet keep adding a day forever or we wouldn't be here.
By the way we remain friends because Dan having a lot of wrong ideas is part of his charm.
Gil | 06.11.04 - 9:29 am | #
Can you really DEFINE infinity by comparing it to time? or distance? or measures of any sort?
MuddyRoad | Email | 06.11.04 - 1:35 pm | #
Muddyroad's question is valid, but I think easily dispensable, using my example of correspondence to the number line. Say you have a gumball, and then add another gumball, and add another, out forever. Assuming you don't run out of gumballs, how many do you end up with? Answer: You don't. The same is true if you're measuring inches or days or pounds. In a way, it's easier to talk about inches or days or pounds, because we don't have to postulate a never-ending supply of gumballs. :-)
Here's another thing: In decimal (everyday) mathematics, 1 ÷ 3 = 0.333333333 · · · without end. How many 3's are there? When do you write the last 3? The answer is that you don't; you have to write an infinite number of 3's. Which is why in everyday life we write either · · · or a ¯ macron over the part that repeats (the last 3 that we write).
And then to Gil's reply: His reply and the voice conversation we had this morning make even more clear that Gil misapprehends infinity.
He comes close to understanding when he keeps adding days in the past. Infinity means that you don't get to stop adding days. And every time you add a day, or a billion days, or a googol days, you still are a finite distance in the past, and yes, there are still an infinite number of days before that. One never reaches "infinity days ago." I think that's part of Gil's problem; he believes that at some point you get to stop, and say "Okay, we have infinity now." That, simply, is not the way infinity works.
LordGeznikor | Email | Homepage | 06.11.04 - 2:49 pm | #
Indeed, I went to school an infinite number of days in the past. ,,,,,.!?-- (put them wherever you like).
The key point is that for the past to be past it must be finite or else the present isn't possible.
Further, it makes no sense to say that there can be an infinte number of finite things. Plus, Dan's argument about repeating decimals shows that there is not an exact correspondence of numbers to the "real world."
Gil | 06.11.04 - 3:25 pm | #
Actually, the repeating decimals is an artifact of it being, well, decimal, or base 10. In base 3, where the only digits are 0, 1, and 2, the same problem is 1 ÷ 10 = 0.1—which is no problem, right?
Your other suggestion about the past comes close to being a real argument. But every point in the past is a finite distance in the past. What's the problem?
LordGeznikor | Email | Homepage | 06.11.04 - 3:37 pm | #
As I have stated before because the past is, in fact, past—we must be able to get to the present from each and every day in the past.
Gil | 06.12.04 - 9:22 am | #
Exactly ... each and every day in the past is a finite distance from now, which is what you seem to mean. Even in an infinite past, there is no day from which Now cannot be reached.
LordGeznikor | Email | Homepage | 06.12.04 - 4:02 pm | #
So your condition is met by an infinite past, is all I mean by that. Your belief that it could be otherwise only shows us your continued incomprehension of the nature of Infinity.
LordGeznikor | Email | Homepage | 06.12.04 - 4:08 pm | #
If each and every day in the past is finite then the past is finite
Gil | 06.12.04 - 8:18 pm | #
being an applied math major (talk about a waste of infinite time), i can say lord g is right. your analogy to the number line is dead on. you're also right with your "an infinite number of finite values". consider this: what is the sum of every positive natural number? each number is finite (1 or 123 or whatever), but since we have infinite postive natural numbers, they total to infinity.
NemoD | Email | 06.13.04 - 5:20 am | #
In regard to NemoD clearly the number line is infinite, but the total number line has never been, nor never can be, counted (there is no number we can't add 1 to, which is what we mean by infinite). However, every day in the past has occurred, that is, it has been counted, therefore it is finite. It might make sense to say that the future can be infinite but the past, qua past, is not.
As a student of philosophy, let me point out that this is actually Plato's problem of the one and the many. The one here is the whole, and in one sense, the whole is comprised of parts, the many, but the one is not simply all the parts.
Gil | 06.13.04 - 10:30 am | #
Uhhh, no. I won't get into the Plato thing, because that is at best a tenuous analogy, and at least your telling of it here is a confusing jumble. We should stay on topic here.
Every past day is a finite distance from Now. It is possible to have travelled from any day, every day, each and every day (whichever phraseology you wish), in the past to Now. This is true even if the past is infinite.
What I've been looking for from you all along here is a logical proof of your assertion, that it is impossible for past time to be infinite. You haven't done that. You seem to have tried, but you haven't connected the dots. My assertion is that no such logical proof exists, without twisting infinity in a way that it doesn't go.
Here's my proof.
- It is possible for any finite distance to be contained in the past. (I am not proving this statement because we both agree that it is true. For example, it is possible for "10 days ago" to be in the past. By this I mean it is possible to have got to Now, from 10 days ago.)
- The addition of two finite distances results in a finite distance. (Again, I could prove this, but I think we agree.)
- 1 is a finite number, and 1 day is a finite distance. (If I have to prove this, we're in more trouble than I thought.)
- By (2) and (3), for any finite distance x, distance x + 1 is also finite.
- (This isn't really part of the proof, it just explains the implication of (4). At the beginning, the only finite number we assume is 1. So x=1, and therefore 2 is finite. So x=2, and therefore 3 is finite. And so forth, to include all of the natural numbers as definitionally finite, or at least as representing finite distances in time.)
- By (1) and (4), for any value of x, it is possible for "x days ago" to be contained in the past.
- The set of all possible values of x is infinite:
- The set of all possible values of x corresponds to the set of all integers greater than or equal to 1.
- The set of all integers greater than or equal to 1, is an infinite set.
- Therefore, by (6), from any and all days in the past, it is possible to have got to Now, and by (7), the past contains an infinite number of days.
I expect that you will respond by saying "no, it's not, because it's past," or some such other thing that demonstrates your incomprehension of infinity. I submit that to prove your point, you must either (a) find the flaw in my logic, or (b) prove with equal logical rigor that the opposite assertion is also true, from premises we agree upon.
LordGeznikor Email | Homepage | 06.14.04 - 3:10 am | #