Adios, Infinity

MIT physics professor John Brockman advances a surprising, perhaps even heretical, thesis: The idea of infinity should be retired from physics. Why? Because it causes too many problems. Here's an excerpt from his lede:

The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT—and, indeed, all of modern physics. But it’s an untested assumption, which begs the question: Is it actually true?

There are in fact two separate assumptions: “infinitely big” and “infinitely small.” By infinitely big, I mean that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum—the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously. The two assumptions are closely related, because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely.

The theory of inflation has been spectacularly successful and is a leading contender for a Nobel Prize. It explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, uniform universe, with tiny density fluctuations that eventually grew into today’s galaxies and cosmic large-scale structure—all in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments. But by predicting that space isn’t just big but truly infinite, inflation has also brought about the so-called measure problem, which I view as the greatest crisis facing modern physics.

Physics is all about predicting the future from the past, but inflation seems to sabotage this. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!

This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞.

Brockman goes on to discuss this problem and to cite mathematicians who dismissed the idea that an actual infinite could exist. Philosophers, or at least some of them, have long disdained the idea of infinities because of the paradoxes to which they lead. For example, Hilbert's Hotel, the brainchild of mathematician David Hilbert who imagined a hotel with an infinite number of rooms, leads to the paradox that even if every room was occupied there'd still be an infinite number of rooms available. You could have an infinite number of rooms occupied and an infinite number unoccupied in the same hotel:

One consequence of removing infinity from the physicist's tool box is that it also removes an objection to one of the most famous families of arguments for the existence of a creator of the universe. The arguments are collectively called the Cosmological argument and one version, put simply, goes something like this:

1. Whatever begins to exist has a cause.
2. Nothing causes itself to exist.
3. The universe is not infinite and therefore began to exist.
4. Therefore, the universe has a cause outside of itself.

With the addition of a few more premises it can be shown that the cause is very much like the God of traditional theism, but that's a discussion for another time. The point here is that an objection to this argument is the claim that premise 3. is a false assumption. It's been more difficult to sustain this objection, of course, since the 1960s when the Big Bang cosmogeny was pretty much confirmed, but there have nevertheless been attempts to deflect the force of the above argument by insisting that there have been an infinite number of big bangs in the history of the universe which has been oscillating between phases of expansion and collapse forever.

Infinity is also employed against the argument for God from the fine-tuning of the universe. The argument here is that even though it seems astronomically improbable that a universe like ours, suitable for human life, should exist solely by random chance, nevertheless, given that there are an infinite number of different universes (at least hypothetically) every possible universe must exist. Thus, ours exists.

In other words, infinity has been an escape hatch for those who wish to avoid the conclusion that the universe is a product of a transcendent creator. By invoking it they've been able to justify, at least to themselves, their refusal to accept the conclusion of arguments such as the above.

It's interesting, then, to read a prominent physicist calling for the retirement of the concept.