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Lamrot Hakol (Despite Everything)

Musings and kvetchings and Torah thoughts from an unconventional Orthodox Jew.

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"I blog, therefore I am". Clearly not true, or I wouldn't exist except every now and then.

Friday, May 12, 2006

Omnipotence, Trolling, and Dividing by Zero

When I was in high school, someone brought in a proof that 1 = 0. Some of you have seen this before:

  1. Set x = 1
  2. multiply both sides by x: x2 = x
  3. Subtract one from both sides: x2-1 = x-1
  4. Factor x2-1: (x+1)(x-1) = x-1
  5. Divide both sides by (x-1): x+1 = 1
  6. Subtract 1 from both sides: x = 0

We sat around trying to figure out what was wrong with it, since everything worked out great. In the end, it turned out that since we'd started by defining x as 1, that made (x-1) equal to 0. And we divided by it. You can't divide by zero, because it's meaningless.

So the other day, a 19 year old kid on some forum raised the age-old question "If God is omnipotent, can He make a rock that's so heavy that He can't lift it?" God save me from 19 year olds generally, but this one was apparently quite pleased with himself.

It made me think about that proof for 1 = 0. Because, you see, there's nothing wrong, generally speaking, with dividing by (x-1). It's only because we started by defining x as equal to 1 that it causes a problem. And the same exact thing is true of his question.

When we speak, or write, words have to have meanings. Otherwise, we're not really speaking at all. We're just bibbling our lips. So in that question, the word "God" has to be defined. And it can either be defined as something with limits or something without limits. That should be clear. So consider:

If the word "God" means something without limits, then the last four words of the question "He can't lift it" have no meaning. And the question containing that sentence fragment also has no meaning. If we didn't start by defining God as having no limits, this wouldn't be the case, just like it's fine to divide by (x-1), provided you haven't already set x = 1. But we did, so the sentence becomes gibberish.

That means that for the sentence to make any sense, we have to start from the assumption that the word "God" means something that does have limits. But the question starts out by asking what the case is if God is omnipotent. So it's asking a question about something that it has already ruled out.

And in fact, it turned out that this 19 year old had come to the forum, and posted his question in a new topic, entitled "Can God be omnipotent", despite the fact that he himself doesn't believe in God. What he did by creating that thread was precisely what the question itself does, at its best.


Trolling is about trying to stir things up. Trying to tweak people by pretending that you're interested in a particular subject, when all you're really interested in is eliciting a reaction. Why? I don't know. Maybe because it's fun to watch. Throw a piece of raw meat into a group of sharks and watch the feeding frenzy. It's the same dynamic. And some of us are as clueless as those sharks when a troll stops by. Sometimes we bite because we think the question is being asked honestly. I'm going to try and bear in mind one of the Rules of the Internet:

Do Not Feed The Troll.

Wish me luck.


Blogger Milhouse said...

CS Lewis wrote: "Meaningless combinations of words do not suddenly acquire meaning simply because we prefix to them the two other words 'God can'... Nonsense remains nonsense even when we talk it about God."

OTOH, Chabad theology insists that God is above the sort of human logic that cannot tolerate paradoxes, that He is nimna hanimna`ot, and keshem sheyesh lo koach bevilti gvul, kach yesh lo koach bigvul. According to this theology, God can make a stone He cannot lift, and He can lift it anyway, and our incapability of understanding how this could be is a flaw in us, not in the situation.

Mekom ha'aron einah min hamidah – the aron was 2.5 amot long, the kodesh hakodashim was 20 amot wide, and yet the distance from each wall to the aron was 10 amot. And the chanukah oil was only enough to last one night, it burned at its natural rate, no new oil was created, and yet it lasted eight nights. According to Chabad theology, these were two instances where the very reality of the universe, including the laws of logic, gave way before an overwhelming kedusha.

2:33 AM  
Blogger Lisa said...

If Chabad theology says that, then I disagree with Chabad theology. I don't accept that A can be not-A, period. It's not like a law of nature, which could be different in different domains (gravity having a different force, for instance), but rather a concept which is inherently true.

I don't think you can draw comparisons to miracles like the aron. Space can curve. There's not enough information to know exactly what was going on there. And with regards to the Hanukkah oil, you're assuming that it burned at its natural rate. I don't know of any source that says so, nor any way such a claim could have been made.

You talk about "laws of logic" as though they're comparable to "laws of science". They aren't.

1:22 PM  
Blogger Joshua W. Burton said...

You talk about "laws of logic" as though they're comparable to "laws of science". They aren't.

And therein, the thorn. Science, handmaiden to experiment, can get by with rough-and-ready near-logic, as long as it's sufficiently straight and true to support kitchen math (well, Riemannian geometry, Lie algebras and so on, as well as finite arithmetic -- but nothing foundational).

The phenomenon of our own consciousness, and still more so the ineffable Mind of God, accept no such pragmatism; either they're solid all the way down, or where are we? And down at the bottom, you always find a Gödel sentence, worrying at the very foundation of A=A for any given formal system.

Here is something nontechnical I wrote about this a long time ago, not an answer but (I think) a fair formulation of the question. That thread as a whole has a few highlights, and Douglas Gresham the (apostate Jew and) adopted son of C. S. Lewis winds up on the receiving end of some well-deserved abuse. B'te'avon.

5:18 PM  
Blogger Lisa said...

Uh. Dammit, Josh. You're one of the only people I know who consistently makes me feel stupid.

I don't have time to read that again (I just read it twice) and try and figure out what you mean. I'm going to have to read the whole thread, too, I imagine. For context.

Maybe I'll just print it out and you can explain it to me this Shabbat. Unless you have a Cliff's Notes version, I mean.

Also, we're going to have to discuss your understanding of the term "nontechnical".

6:19 PM  
Blogger Joshua W. Burton said...

Maybe I'll just print it out and you can explain it to me this Shabbat.

Hee, that was sort of my ulterior motive. (Query: how many adjectives, besides "ulterior," only work with one noun? "Pendulous," I suppose. "Abject," "furtive," "sordid," "ebbing." Uh, where was I?)

The one-foot version is that self-biting sentences sometimes happen "by accident," even after you work really hard to define your terms. So, you might get rid of "This sentence is false" by demanding that sentences talk about things while only meta-sentences talk about sentences. (Sing along with me: "Anything you can do, I can do meta- !") But then, comes

"Yields a falsehood when preceded by its quotation" yields a falsehood when preceded by its quotation.

which talks, not about itself exactly, but about the sentence you get when you take the quoted phrase and modify it in a certain way...which just happens to be the sentence itself. And not only does it talk about itself, it lies!

This trick, which sounds stupid and trivial, turns out to be frighteningly robust. A mathematician named Kurt Gödel, around 1930, used it to utterly destroy Whitehead & Russell's Principia Mathematica, the last best hope to put mathematical logic on a final, firm A=A sort of footing. So far, the only way anyone has found to avoid Gödel's jujitsu move (which exploits an inferential system's own strength against it) is to stick with logical systems that are weaker than number theory. But human reason (and, presumably, God's Truth) isn't weaker than number theory, so it is ineradicably riddled with "rock he himself can't lift," "this sentence is false" type paradoxes.

In words, you can say that terms need consistent definitions, and that if we just speak plainly we don't hit any paradoxes. But once you really go and write it out in rigid symbols, Gödel's recipe can construct the "accidental" holes in your particular precise axiomatic system, so your only defense is not to have one. Ugh.

Lots of good stuff to say about all this, and since I don't have any answers it's a good open-ended chat for a lazy Saturday afternoon. Or, not.

7:03 PM  
Blogger Joshua W. Burton said...

Shorter Joshua:

Gödel's sentence A: "Sentence A has no proof in axiom system X."

Gödel's trick: for any axiom system X, come up with a bulletproof way to list all the grammatical sentences and all the proofs, so that sentence A can refer to sentence A "by accident" (by its number on the list, say) even if your system doesn't let sentences talk about sentences directly.

Now, let X be "God's Truth." A, or not-A?

7:21 PM  
Blogger Lisa said...

I need to think more about this.

On the subject of adjectives, however, I can think of abject poverty and abject misery. A sordid tale and a sordid past. A furtive glance and... well, furtive behavior, maybe, but that's pushing it a bit. And both tides and strength can ebb. But pendulous and ulterior do seem to only have a single noun each.

9:16 PM  
Blogger Milhouse said...

I dunno, Webster thinks you can have pendulous vines, or pendulous jowls. Googling "pendulous vines" produces 5 hits; one is the Webster dictionary itself, two are copies of the same instance from Herman Melville's Omoo, one is from Emanuel Swedenborg's The Delights of Wisdom Pertaining to Conjugial Love, and one is, of all things, from a report on a trip to St Lucia, published in the Greater Framingham Running Club News!

Googling "pendulous jowls" is more successful, producing 29 hits, from a wide variety of sources. And "pendulous orbs" produces 21 hits, including some that are not anatomical, or at least not referring to that part of the anatomy.

Meanwhile, "ulterior shore" finds approximately 6 distinct instances, including two separate ones by Belloc. "ulterior -motive -motives" gets what Google claims to be 2,470,000 hits, though there's probably something wrong with that.

5:46 PM  

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