Something Rather Than Nothing II

Off-site someone linked me to this post by Richard Carrier, in which he comes to much the same conclusions as I did, but in much more depth, and also provides a more formal logical proof.

  • P1: In the beginning, there was absolutely nothing.
  • P2: If there was absolutely nothing, then (apart from logical necessity) nothing existed to prevent anything from happening or to make any one thing happening more likely than any other thing.
  • C1: Therefore, in the beginning, nothing existed to prevent anything from happening or to make any one thing happening more likely than any other thing.
  • P3: Of all the logically possible things that can happen when nothing exists to prevent them from happening, continuing to be nothing is one thing, one universe popping into existence is another thing, two universes popping into existence is yet another thing, and so on all the way to infinitely many universes popping into existence, and likewise for every cardinality of infinity, and every configuration of universes.
  • C2: Therefore*, continuing to be nothing was no more likely than one universe popping into existence, which was no more likely than two universes popping into existence, which was no more likely than infinitely many universes popping into existence, which was no more likely than any other particular number or cardinality of universes popping into existence.
  • P4: If each outcome (0 universes, 1 universe, 2 universes, etc. all the way toaleph-0 universes, aleph-1 universes, etc. [note that there is more than one infinity in this sequence]) is no more likely than the next, then the probability of any finite number of universes (including zero universes) or less having popped into existence is infinitely close to zero, and the probability of some infinite number of universes having popped into existence is infinitely close to one hundred percent.
  • C3: Therefore, the probability of some infinite number of universes having popped into existence is infinitely close to one hundred percent.
  • P5: If there are infinitely many universes, and our universe has a nonzero probability of existing (as by existing it proves it does, via cogito ergo sum), then the probability that our universe would exist is infinitely close to one hundred percent (because any nonzero probability approaches one hundred percent as the number of selections approaches infinity, via the law of large numbers).
  • C4: Therefore, if in the beginning there was absolutely nothing, then the probability that our universe would exist is infinitely close to one hundred percent.

Why is there something rather than nothing?

This is a question I get asked a lot as an atheist. And it’s a terrible question for two reasons. The first is that is presupposes that nothing is a possibility, that it is an option. Asking “why is there something rather than nothing?” only makes sense if nothing is a reasonable, possible alternative to something. But we have no examples of nothing to point to to demonstrate that it was ever an alternative, that it could have been possible for there to be nothing.

Even if we accept that nothing was ever a possibility, the second problem is that this question posits that nothing is not only possible, but also more likely than something. After all, if the questioner accepted that something was as or more likely than nothing, the question wouldn’t need to be asked (why is there x rather than y, when x is more likely or as likely as y, is always answered with “because x is as likely or more likely than y.”) So the question includes this other hidden hypothesis, that somehow nothing is more likely to have been, and the fact that there is something is somehow unexpected, unusual, or surprising.

But suppose that there were nothing. No space, no time, not even any sort of quantum vacuum a la Lawrence Krauss. In the absence of any stuff, in the absence of any laws of physics, of quantum mechanics, there are no restrictions on what can or cannot exist. In other words, if there is nothing, then there is also nothing to restrict the likelihood of or prevent the existence of something. The rest is mere probability: if there were nothing, then continuing to be nothing is one possibility, our universe or any other single universe beginning to exist is another, a multiverse with k universes is another, etc.

In other words, if there ever was nothing, it would be equally likely for there to continue to be nothing as it would be for our universe to exist. What is even more astonishing is that it is in fact more likely for a multitude of universe to exist that for nothing to continue being nothing or for one universe to exist (it is equally likely, for example, for there to be three universes as two universes as one as nothing, but more likely for there to be two or three universes than one or nothing.)

So, in fact, the answer to the question “why is there something rather than nothing?” is that, if there ever were nothing, it would be far more likely to cease being nothing and for many universes to exist than for nothing to continue being nothing.

.999… = 1

I’ve recently gotten involved in a number of online “debates” over the mathematical proof that .999… = 1. On the one hand, I don’t much see the point in debating this. People who refuse to believe that it is true typically seem to not understand basic mathematical concepts, and, on top of that, harbor a strangely deep mistrust of mathematicians. (Really, you don’t think any mathematician ever has thought of your very clever objection that it is just “really really close” to 1?)

But at the same time, I think that there are some objections that stem from simply genuinely misunderstanding how decimal notation works (and, let’s admit, the tricky ideas of infinity and forever). And that’s worth at least some effort to explain.

The proof that is most often shared goes like this:

(1) Let x = .999…..

(2) Then 10x = 9.999….

(3) Then 10x – x = 9.999… – .999…

(4) Then 9x = 9

(5) Then x = 9/9 = 1

(6) Since x = .999… and x = 1, .999… = 1

Line (1) rarely seems to cause any problems. All we’re doing is letting an arbitrary variable, x, be .999… . The second line occasionally causes problems, with the objection that we’re multiplying by 10 on the left, but adding 9 on the right. This is incorrect: in decimal notation, multiplying by ten moves the decimal point over one place (you can try this out on any calculator you like: do .99*10 and you will get 9.9. If you disbelieve the calculator, consider that 1 = 1.000.  Then 1.000*10 = 10.00 and 10.00*10 = 100.0 and 100.0*10 = 1000. You can see how multiplying by ten simply shifts the decimal place one over.)

The objection usually follows that then the .999…. after the 9 is one less that the original .999…. But this ignores that the  9’s after the decimal go on forever. Removing one doesn’t stop them going on forever. After all, suppose it did. Then if we divide by ten and get that 9 back on the other side of the decimal, we’d have a finite list of 9’s plus another 9. But that’s still finite, when our original list restored should go on forever again. So despite multiplying by ten, we still have an infinite, unending list of 9’s after the decimal.

The real objections usually turn up with (3) and (4). Here we subtract x, or .999… (which, you’ll recall, is x) from both sides. The idea that 10x-x = 9x is usually uncontested, but the fact that 9.999… – .999… = 9 sometimes causes problems. But if you think about what decimal notation really means, it shouldn’t. All decimal notation is is a series of sums. 9.999…. is really just a shorthand for an infinite sum: 9 + 9/10 + 9/100 + 9/1000 + …… Now, .999… is also shorthand for a series of sums: 0 + 9/10 + 9/100 + 9/1000 + ….

Notice that .999…. expanded is exactly the same as 9.999….. expanded, just without the original 9. Suppose we subtracted these from each other, instead of the decimal shorthand. Then we’d have:

9 + 9/10 + 9/100 + … – 9/10 – 9/100 – ….

For each term after the 9 in 9.999…. we have its negative, that is, we subtract it away and it cancels out. We are left, then, with just 9.

This leads us, finally, to the statement in (4) that 9x = 9. I hope it is uncontroversial that it follows from (4) that x = 1.

Since two things equal to the same thing are equal to each other, as x = 1 and x=.999…, 1 = .999…. I’ve seen objections here that we just “assume” that x = .999… and that that creates a circularity. I honestly don’t understand how anyone could think this well enough to refute it, except to say that, since we originally chose x arbitrarily to equal .999…, and x has no value itself, we’ve proven here that *anything* equal to .999…. is also equal to 1. So start with any number or representation which is equal to .999…. and it will, necessarily, be equal to 1.

UPDATE: 8/17/2012

Commenter Thaumas Themelios posted another excellent eposition,read the whole comment below:

One variation I’ve used is to ask the person “What is 1/3 in decimal notation?” They say 0.333…. Next, “If 1/3 is one third, i.e. 1 thing divided into 3 parts, and taking only 1 of those parts as being ‘a third’, then what do you get if you put the three thirds back together again?” Obviously, 1. “So, 1/3 multiplied by 3 is equal to 1, correct?” Yes.

“Okay, now take 1/3 written as a decimal and multiply it by 3, so 3 x 0.3333…. = 0.9999…., correct?” Yep. “So, in the first case, you took 1/3, and multiplied it by 3 and got 1. And in the second, you took the same 1/3 and multiplied by 3 and got 0.9999… If you accept the first is true, that 1/3 in decimal is *equal* to 0.3333…, then you must also logically accept that the second is true, that 1 is equal to 0.9999….”