The Problem with Infinity

About a year ago, I started listening to podcasts. (I know, I’m late to the game. Moving on…) While there are so, so many to choose from, I’ve found myself very much gravitated towards BBC Sounds’ podcasts, especially the ones that involve history or science. I suspect that might be in part to my love of Time Team and a show I first learned about in the early 1990s, Connections.

Connections, in particular, was a watershed moment for me. I’d watched all kinds of science shows in the years previous, but for the first time, I was seeing science and technology presented in a whole new way. James Burke himself labelled it as “an alternative view”, and some of the connections are tenuous, but the goal was never to be a definitive explanation, it was to make you think about how we begat technology, and how it changed the world.

These shows – and the podcasts I listen to – are about communication: explaining topics to the general populous that are usually well beyond regular comprehension. We don’t learn about archeology in school beyond digging up old bones, yes there’s so much more about the process and how dating can be achieved without having to go to big, complex machines. And while we learn science in school, it never covers the impact of science, or how science has changed our society.

That they also tend to be entertaining, even funny, just increases one’s ability to listen.

I went through 13 Minutes to the Moon too quickly. I ran through the entirety of You’re Dead To Me far more quickly than I had wished. I switched to Evil Genius, but soon found myself missing the deeper knowledge.

Then I found The Infinite Monkey Cage. It’s a (comedy) science and reason show, presented by Professor Brian Cox and comedian Robin Ince. They play the roles of “scientist” and “fool”, respectively, though Cox is often the butt of jokes and Ince is far wiser (and well read) than might be expected. (I used “comedy” in brackets as while the show is funny, you often need some element of knowledge in order to find the humour.) They tackle nearly every topic with a panel of experts, usually bringing in one more well-known person, which has included actor Brian Blessed, astrophysicist Neil deGrasse Tyson, and even James Burke, himself.

To say that I love this podcast truly belies my passion for it. I feel it has changed my view of science communication as much – if not more – than Connections some 30 years ago. If you have any interest in the sciences at all, I can’t recommend this podcast highly enough.

That said, I have have a particular quibble. The name.

The name, obviously, is a reference to the “infinite monkey” theorem: an infinite number of monkeys with an infinite number of typewriters will eventually write the works of Shakespeare (lampooned by The Simpsons, of course.)

I’m not the only one with concerns about the name, the first two series regularly had feedback from listeners (as edited versions play on BBC Radio 4) who struggled with what the name meant, in a more scientific view: what is an “infinite” monkey?

Which got me to thinking: the name is wrong, as it describes a scenario that cannot exist.

The issue is “infinity” itself. (I’m using the generic “infinity”, by the way. Apparently there’s an infinite set of infinities of various sizes, and yes, some infinities are bigger than others, and that hurt my head as much as it’s hurting yours.) An infinite amount of anything is painful enough to comprehend, but then we have to consider the physical impact of an infinite amount of something. Excuse me whilst I channel Randall Munroe

The name “infinite monkey cage” suggests one of three possible scenarios:

  1. Something that contains a monkey of an infinite size
  2. Something that contains a infinite number of monkeys (of indeterminate, but otherwise irrelevant size)
  3. At least one (possibly more, though also irrelevant) monkey that resides within a cage of infinite size

Let’s deal with the last one first: an infinite cage. We could apply our modern, human view of a cage: an orthogonal box of bars with a nice lock on it. From an engineering perspective, it would be quite the challenge to build it, and it might be possible to gather together all the dark matter in the universe (because there’s nowhere near enough normal matter), we could potentially build a cage. We’d run out of stuff before we got anywhere near “infinite”, even if we made the edges and bars a mere atom thick, and the structural capacity to retain even a single monkey would be a challenge.

In short, the cage is inconsequential.

That leaves us with the other two, which share a commonality: mass. The monkey(s) has(have) mass. The cage has mass. Even if we look at only one item – the monkey or the cage – the adjective “infinite” must apply. And we don’t even have to get to an infinite size before mass becomes a problem: too much of it, and things will go catastrophically wrong.

Barring a cosmic change in physics and Earth-based biology (for sake of argument, we’ll assume the monkey is Terran), there’s a point at which the mass of monkey meat – whether singular or infinite – creates a gravitational force sufficient to overwhelm the skeletal structure – singular or infinite – and the monkey(s) basically implode.

As awful as that sounds, we’re dealing with “infinite”, so this will get significantly worse. No, we’re not just talking about a “mole of moles”-sized moon, we’re going to have to look at a mass of monkey(s) that rivals our own sun. Godzilla would be quaking in his Shibuya boots at that thought. Of course, that many monkeys would be dead, by this point, their skeletons having collapsed and organs having totally failed – that’s what happens when you have 2,000,000,000,000,000,000,000,000,000 of them (approximate estimate of how many Bonnet macaques it would take to fill the sun’s volume). But even a singular monkey of that size would also be dead. After all, there’s no air in space for the poor thing to breath. Plus, -273C tends to hamper one’s ability to stay warm.

But I digress, because it still gets worse.

Once we have a few solar masses of monkey(s), gravity really starts to have fun, and we’ll reach the Schwarzschild radius, at which a black hole is inevitable. The formula to figure that out is Rs = 2MG/c^2 – the Mass of the object multiplied to the Gravitational constant, over the speed of light, squared.

At the sun’s mass, the Schwarzschild radius would have to be less than 24 metres before we’d get a black hole. That doesn’t seem like nearly enough mass for that to happen, so that should give us a modicum of relief (as if a sun-size ball of monkey wouldn’t be terrifying in and of itself). But if we keep scaling up the size of this mass of monkeys, eventually we’ll hit a point where the centre of the ball will be so compressed, so hot, that atoms will start fusing in weird, new ways. And just like the sun, the force of that fusion will be retained under the intense gravitational pull of all that primate matter.

Somewhere, and I couldn’t say where, but I figure a monkey mass merely the size of our solar system would tip the scales somewhere, causing the threshold to be reached, and the universe would be witnessed to the first bonobo hole. One must wonder what sound it would make, were it possible to hear the scream.

In short, we’ll never get to infinity, with either the monkey(s) or the cage. So, really, the show should be called “The Simian Singularity”.