Politics, Technology, and Language

If thought corrupts language, language can also corrupt thought — George Orwell

God, the universe, and nothing else

Posted by metaphorical on 17 February 2007

The juxtaposition earlier this week of atheism and Andzrej Mostowski (my Berkeley math logic professor) proved surprisingly combustible, igniting some new thoughts last night. I don’t know what Mostowski’s feelings about religion were; I assume, but have no way of knowing, that he would have been some sort of nearly-agnostic Deist, along the lines of what I understand Einstein to have believed.

As Mostowski explained Intuitionism, it too was agnostic, at least about some mathematical entities and ideas. Among them are infinities; it didn’t disbelieve in them so much as withheld belief. And from infinity, it’s just a short hop, at least when you’re drifting off to sleep, to God.

The Wikipedia entry on Constructivism (Intuitionism is a form of it) is helpful here:

In fact, L.E.J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as abstracted from finite experience, and then applied to the infinite without justification.

[The law of the excluded middle says that for every proposition p, either p or not-p (in logical notation, p v ~p) is true. It seems even more obvious than Euclid's parallel-line postuate, so it's unsurprising that everything changes when you don't accept it. We'll come back to the topic of infinity in a minute.]

For instance, Goldbach’s conjecture is the assertion that every even number (greater than 2) is the sum of two prime numbers. It is possible to test for any particular even number whether or not it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.

But there is no known proof that all of them are so, nor any known proof that not all of them are so. Thus to Brouwer, one cannot say “either Goldbach’s conjecture is true, or it is not.” And while the conjecture may one day be solved, the argument applies to similar unsolved problems; to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution.

Atheism is sometimes seen as a macho version of agnosticism, but this is a burly agnosticism that anyone could be proud of. Imagine withholding not just belief, but a truth value, to a proposition simply because we cannot construct a proof!

To jump ahead for a moment, I call myself an atheist, but this industrial-strength agnosticism is probably more to my liking. I’ve said in the past that I find the proposition “God exists” meaningless, in the literal sense that I can’t assign meanings to the constituent words. It’s probably more accurate, though, to say that I don’t assent to the proposition that “God exists or God does not exist” (p v ~p), because I cannot imagine how to construct a proof of either side of the disjunction.

Mostowski spent a lot of time talking about infinity and what the Intuitionists think about it. He said pretty much what the quote above says, that mathematicians sometimes say things about finite sets and then say the same things about infinite sets as if the difference didn’t matter. One example is the different “sizes” of different infinities—Cantor “proved” that while the set of odd numbers is just as large as the set of odd and even numbers, even though it is a proper subset of it. He also “proved” that the infinity of the real numbers is fundamentally larger than the infinity of whole numbers. I put the word “proved” in quotes here because these are not necessarily proofs that all mathematicians would be happy with.

As I understood it from Mostowski, the Intuitionists weren’t very comfortable with talking about infinities at all, and they could be quite circumspect about it. “For every number x, there is an x+1″ is a sentence an Intuitionist is very comfortable with. “There are an infinite number of numbers,” is not, and they would recast the one into the other. “Add 1″ is a clear (and finite!) method of construction.

My point here isn’t to examine those proofs or Intuitionism itself, but to draw from it the basic lesson that, to paraphrase the description of Brouwer, we cannot take ideas abstracted from finite experience and then apply them to the infinite without justification.

I think the same can be said of “the universe,” a concept much related to that of infinity. We sling the word around as if it were a finite concept, like “the White House” or “the Earth.” Physicists in particular have put ideas out into the world that make it easy to talk this way. “The universe is 14 billion years old,” “The universe is largely composed of dark matter,” “The universe is expanding.”

Physicists have a technical sense in which they are using the term “universe” (at least I hope they do), and these sentences can make sense, they can be true or false, evidence can be marshaled in favor or against, in that technical sense. But the sentences bleed out into ordinary speech, and once the enter the atmosphere of everyday life, the meaning of the term “universe” loses whatever spark of precision it had within the Leyden jar of the physical sciences.

And so we talk of the universe as if it were one thing in the world among others, instead of as the totality of all things. Having done so, it becomes easy to ask, Who created it? If, instead, we were as circumspect as the Intuitionists, and talked instead about the totality of all things, the question becomes harder to ask: Who created the totality of all things—including everything and everyone that creates anything?

For some reason, the term God has never been used in quite the same ways. The following exchange is a standard piece of discourse:

“Who created the universe?”
“God.”
“Who created God?”
“God isn’t something that’s created.”

Somewhere along the line, “the universe” lost its connotation as something that can’t be created, if it ever had it in the first place. It would be nice to regain it (or gain it, if it never had it).

It would be a progression that’s been tried once before. The first statement in Wittgenstein’s Tractatus is,

The world is all that is the case.

and the last one is

Whereof one cannot speak, thereof one must be silent.

Wittgenstein eschewed talk of “the universe,” preferring to use the word “totality.” He would say that the universe isn’t something that we should say exists. But then, he would also say that we shouldn’t say that the White House and the Earth exist, though he wouldn’t say they don’t exist either. “The world is the totality of facts, not things,” Wittgenstein said. That’s an ontological parsimony that might give even an Intuitionist pause.

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8 Responses to “God, the universe, and nothing else”

  1. seneca said

    A very interesting post, my friend!

    Thank you for not jamming God, or the lack of God, down my throat!

    “from infinity, it’s just a short hop…..to God” is, to me, the Leap of Faith.

    But that Leap does not describe God! Is God good? Is God just? If there is a God, I don’t think there is any evidence that God is either good or just.

    What does science tell us?

    The earth was once flat. Then the sun circled us. Now we circle the sun.

    Newton added gravity, then Einstein determined that gravity does not exist.

    We are an expanding universe. But we might be steady-state, or expanding to an eventual contraction.

    One respected physicist thinks time does not exist, that we are a pulse, a beat, in a universe forever expanding and contacting.

    Regarding physical laws, I’m not sure we’ve discovered any. They only work within limited specifications. Don’t we come up with a law, get new data, then revise the law? That implies to me that we never had the law right in the first place.

    Someone wrote that the laws of the universe must be written in eleven dimensions, and we are capable of writing in only four.
    I believe that, if we ever find a way to write in eleven dimensions, we will find that it takes 121 dimensions.

    We are my dog. He sees the Sun come up and sees the Sun set. He is incapable of understanding the evidence right in front of him, that the Earth circles the Sun. Like my dog, we see, but do not see.

    Unfortunately, we will die not knowing what it is all about.

    In the “Apology”, Socrates tells us (he is to be executed via ingesting hemlock that evening) that he will shortly be with the famous and honorable in History or he will be in an infinitely long and restful sleep. Our views have not changed much in 2,500 years.

    My friend, for all our inquiries into the nature of the Universe, I truly believe that it is beyong the human brain, and our inquiries will always be to no avail.

    However, they’re fun anyway.

  2. ClaireDePlume said

    Are our inquiries to no avail, I wonder? If we wish to investigate universal theories of a universe, we might be prompted to look within.

    Like a tiny universe, our bodies act as a mortal coil encompassing myriad biological cells which arrive into existence. All the while others implode . If we seek super-novas, creationism, birth of life and the universe, infinite possibilities exist with the finite walls of every cell we call our own bodies.

    I favour the following words. Somehow they strike a resonance within the confines of my brain and my mind, and once again there is cerebral excitement, rife with birth and explosions of thought within me and reaching ever outward:

    “My goal is simple. It is complete understanding of the universe, why it
    as it is and why it exists as all.
    – Stephen Hawking

  3. Blue Athena said

    I think this concern is a result of a mathematician’s preference for early Wittgenstein. Assuming that everyday language follows the same types of logic as mathematics or possesses “deep meaning” leads to the same kinds of errors you saw in mathematician-linguist Chomsky, which oddly enough led him to reliance on “god” for his explanations of language.

    Once you abandon the idea than language is a mirror of reality, or that it is comprised of tightly defined logical concepts, so much falls under the “silence” requirement that you would be unable to speak if you still followed Wittgenstein’s early recommendations. The next step is to accept that you can speak, but have to abandon some of your old ideas of what makes an argument.

  4. It’s that damn Axiom of Choice that spoils everything.

  5. ClaireDePlume said

    yes andrew. well put. if not for the law of free and willful entropy, we would have a singular reality and a cohesive, coherent universe.

  6. Ernst Zermelo has a lot to answer for then.

  7. It *is* the Axiom of Choice’s fault, cool that you would know that. Mostowski spent quite a bit of time picking through its nuances, Zorn’s Lemma, Hausdorff’s Principle, and all the rest.

  8. I guess I retained something while getting those two math degrees.

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