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. Read the rest of this entry »