A Response to Scalia

During the arguments for the Prop 8 case, Scalia said the following:

JUSTICE SCALIA: Mr. Cooper, let me — let me give you one — one concrete thing. I don’t know why you don’t mention some concrete things. If you redefine marriage to include same-sex couples, you must — you must permit adoption by same-sex couples, and there’s – there’s considerable disagreement among — among sociologists as to what the consequences of raising a child in a — in a single-sex family, whether that is harmful to the child or not. Some States do not — do not permit adoption by same-sex couples for that reason.

It’s obvious to anyone familiar with the scientific literature that Scalia is either ignorant or lying: to pick just one of many, the American Academy of Pediatrics recently released a statement:

A great deal of scientific research documents there is no cause-and-effect relationship between parents’ sexual orientation and children’s well-being, according to the AAP policy. In fact, many studies attest to the normal development of children of same-gender couples when the child is wanted, the parents have a commitment to shared parenting, and the parents have strong social and economic support. Critical factors that affect the normal development and mental health of children are parental stress, economic and social stability, community resources, discrimination, and children’s exposure to toxic stressors at home or in their communities — not the sexual orientation of their parents.

According to the policy statement, the AAP “supports pediatricians advocating for public policies that help all children and their parents, regardless of sexual orientation, build and maintain strong, stable, and healthy families that are able to meet the needs of their children.”

But what if Scalia were correct? What if there actually was sufficient debate among experts and a sufficiently contradictory evidence so that we could not draw any scientific conclusions regarding the effect of gay parents on children? Then, in the absence of any evidence to sway the probabilities in any single direction, we’d be left with three equally likely possibilities (on average): negative effect, no effect, or positive effect. In which case it is more likely that there is either no effect or a positive effect than that there is a negative effect.

So even if Scalia were correct, even if we ignore all the scientific evidence (which would weight the probabilities further against “negative effect”), still we find that it is less likely for gay parenting to have a negative effect, on average, than not. Of course, he doesn’t acknowledge this, bundling two cases (positive effect or no effect) into one, and with masterful sleight of hand implying that the probabilities are equal, thus implicitly attempting to grant a stronger probability to the case of negative effect. Unfortunately for him, math just doesn’t work that way.

Proof by Contradiction: a Fairy Tale

Once upon a time there was an evil king. He had all he could desire: a huge castle, servants, riches, and more power than he could shake a stick at. But as time wore on, even these could not comfort him, nor could they solve his one problem (and this wasn’t his usual problem, which was that he was so powerful even he had to obey himself.) No, this problem ran much deeper. Even with everything he had, there was one thing he couldn’t have: friends. He would try to become friends with people — but he could never have a true friend, because eventually friends disagree, and he was an evil king, and you don’t disagree with evil kings. So he became lonely. And lonelier. And as he became lonelier and lonelier, he became bitterer and bitterer and bittererer.

Finally, he became so enraged at everyone who had friends, that he began to devise an evil plan. He would hold an enormous feast, and after the feast, all the guests would be lined up. “And then,” he said aloud, in his evil planning voice, “I will begin to draw numbers from a hat, and with each number drawn, one guest shall die! Moreover, as I have no friends, each guest will die when the number of friends they have at the feast is called!” His plan completed, the King began to put it in motion. Until he hit a fatal snag.

He spent weeks and weeks trying to devise a guest list, but as you’ll recall, he was so powerful he had to obey his own commands, and he had said that with each number called, one guest would die. But try as he might, he could not figure out a guest list (even with his evil party planner) that would invite guests where each guest had a unique number of friends at the feast. Finally, the king sent for his evil royal mathematician, the evilest mathematician there was. “O King,” said the mathematician, “how my I serve your Royal Evilness.”

“Help me with my guest-list, and I shall reward you beyond your wildest imagination,” the king responded, detailing his dastardly plan.

Cowering in sudden fear, the mathematician spoke, her voice trembling. “Your Majesty, it — it is not possible.”

“Explain yourself,” the king roared.

“Well, Your Evilness,” she spoke again, “suppose you invite k guests. Since you aren’t counting guests being friends with themselves, for each guest there are k-1 other guests to be friends with. So the most friends any guest can have is k-1. The least number of friends any guest can have, clearly, is 0. Which means that for any guest there are k possible numbers of friends: 0, 1, 2 ,3, …, or k-1. But now, you see, you have k guests and k numbers-of-friends you need to pair. If you want each guest to have a unique number of friends, you must therefore pair some guest with each of the k numbers-of-friends. Thus one guest, Xanthia, will have 0 friends, and also another guest, Bartholomew, will have k-1 friends. But since Bartholomew has k-1 friends and there are k people counting him, he must be friends with Xanthia. But then Xanthia and he are friends, which makes no sense, since Xanthia has no friends and thus Bartholomew and Xanthia aren’t friends. This is clearly impossible, and so each guest cannot have a unique number of friends, and so therefore there must always, no matter who you invite, be two guests with the same number of friends.”

And with that, the king suddenly ceased to exist. For, you see, his existence had become the battleground of two great Necessities: the Necessity of his power said the feast must exist, and the Necessity of the Laws of Mathematics said the feast must not exist. Thus, his existence was a contradiction, and therefore, he did not exist.

And so, children, the kingdom was saved from the evilest, powerfulest, loneliest King in the world.

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.

.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….”