One of the more intriguing arguments for the existence of life of other worlds is the argument from infinity. Although it is not known whether our universe is finite or infinite, it is certainly very large (from a human perspective, anyway) and for all intents and purposes can be treated as infinite.

So the argument goes: we have no evidence for life anywhere in the universe except on Earth, but since our presence on Earth demonstrates that it is possible for life to develop in our universe, and given that our universe is for all practical purposes infinite, then it follows that life almost certainly exists on other worlds.

In an infinite universe, any event that occurs once will in fact occur an infinite number of times. Even if life only arises on one in every googolplex worlds, there are still an infinite number of such worlds. To put it another way, any given phenomenon will either be impossible, in which case no examples will exist, or possible, in which case there will be an infinite number of examples even if we cannot observe them. In an infinite universe there are really only two possible outcomes: zero and infinity.

To some of our more provocative thinkers, an infinite universe means there will be an infinite number of Earths just like ours. And an infinite number of Earths just a little bit different to ours. And an infinite number of planets that are wildly different to ours but still harbour life of some form or other. Some have even calculated the number of possible quantum states in the observable universe and then used that to estimate how far the nearest exact copy of Earth is to us (it was a very long way away).

But the problem with this line of argument is that infinities are not necessarily uniform. Science philosopher Jesús Mosterín says:

The suggestion that an infinity of objects characterized by certain numbers or properties implies the existence among them of objects with any combination of those numbers or characteristics [...] is mistaken. An infinity does not imply at all that any arrangement is present or repeated.

Example: a singular prime

A prime number, for those who do not know, is a number divisible only by 1 and itself. The first ten prime number are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. The largest known prime is 2^43,112,609 – 1; if written out, this number would be nearly thirteen million digits long, so mathematicians often call the number “M45.”

In the universe of prime numbers, we can only see as far as M45. But we know that there are an infinite number of primes. Even though the identity of M45 was hidden to us until modern computing technology was available, the proof of infinite primes is one of the oldest proofs in history, having been formulated by Euclid in around 300 BC.

But even though there are an infinite number of primes, there are not an infinite number of primes with the same properties. The most compelling example is the number 2, which is the only even prime number. For obvious reasons, every even number is divisible by two and is therefore not a prime number…except for the number 2 itself.

Even though there are an infinite number of even integers, only one of them is a prime. And even though there are an infinite number of primes, only one of them is an even number.

It is possible then that the Earth is the singular example of a life-bearing world in an infinite universe. For my own opinion, I very much doubt we are alone. The universe may or may not be infinite, but it certainly is big. Very, very big. Even if life only thrives as rarely as one out of every trillion stars, there will still be around a 10 million such stars in the known universe. This would mean, however, that living planets would occur in every hundredth galaxy — and for all practical purposes we might as well be alone.

We don’t really know how to crunch the numbers — to be more accurate, we’ve had a good approach to crunching the numbers ever since the Drake Equation, but we don’t know what numbers to feed into the equation’s maw. It is easy to justify numbers that generate the answer one wants. If it is emotionally satisfying to think that life is common in the universe, you choose numbers in the manner of Isaac Asimov, who calculated that there are 530,000 technological civilisations in our galaxy alone. If it is emotionally satisfying to believe that we are the Universe’s chosen race, then you can choose numbers to suit and share Barrow and Tipler’s conclusion in The Anthropic Cosmological Principle that not only is humanity the only intelligent life form in the cosmos, but that our very existence is what made the universe come into being 14 billion years ago.

Appealing to infinities is simply a more sophisticated way of plugging presumptuous numbers into equations. While it is possible to manipulate infinities mathematically using cardinal numbers, when we talk about the number of life-bearing worlds we are using real numbers. And in the realm of real numbers, multiplying by infinity is the same as dividing by zero.

Tags: drake equation, euclid, even primes, infinity, jesus mosterin, life, M45, prime numbers, unique infinities, universe