The Doctor told you to Google the Bootstrap Paradox, and now you’re here. OK. Let’s try and explain what on Earth the Doctor was talking about at the beginning of Before the Flood, all without using the words ‘wibbly-wobbly’.

In brief, the Doctor proposes a brain teaser. Imagine you have a time traveller who loves Beethoven and decides to travel back and meet his hero. However on arriving, he discovers Beethoven has not and will not write any of the music the time traveller loves so much. The time traveller, desperate, decides to copy out all of his favourite tunes for Beethoven. The plan is successful. Several centuries later, a certain time traveller is listening to his favourite composer, Beethoven, and decides to go meet the man himself… 

The question is: who really wrote Beethoven’s Fifth Symphony?

This is an ontological or ‘Bootstrap’ paradox, named for the story By His Bootstraps by Robert Heinlein, who wrote many classic sci-fi novels including Starship Troopers. In it, a time traveller finds a book that he later realises he himself wrote, by copying out his own copy line by line. But who wrote the original? The authorship makes as much sense as trying to lift yourself up by tugging on your own shoelaces.

It’s a trick that often pops up in sci-fi, usually as a throwaway joke. Take an earlier episode of Doctor Who, The Shakespeare Code, where the Doctor keeps referencing lines from plays that haven’t been written yet.

So to the Doctor’s question: who actually wrote Beethoven’s fifth? Let’s think about it. In this particular thought experiment a time traveller (we’ll call him ‘the Doctor’ for ease) is inspired by Beethoven’s Fifth and ends up ‘writing’ the tune. In effect, the Doctor inspired himself. The Doctor copied ‘Beethoven’ who copied the Doctor who copied ‘Beethoven’, on and on ad infinitum, without a beginning.

 

This is a regressive argument – in explaining one situation, we are forced to rely on a long sequence of events. Every stage in the argument requires a previous stage in order to explain itself, which in turn requires another predecessor. In trying to explain one thing – Beethoven’s Fifth – we have willed an infinite chain into existence. Since infinity has no end, we can’t know the beginning. In fact the only thing we are sure of is that Beethoven’s Fifth exists.

(Fun fact: Beethoven’s secretary described the opening bars as ‘fate knocking at the door’, so maybe it really is predestined.)

Bear in mind that not being able to puzzle out the beginning of a sequence doesn’t mean the results fade out of existence. Aristotle himself said you don’t have to prove ‘immediate knowledge’ – i.e. Peter Capaldi is playing Beethoven’s Fifth over the Doctor Who credits, so Beethoven’s Fifth must have been written in order to exist.

The classic example is to imagine you’re standing on top of a pillar so tall, the bottom is shrouded in clouds. The pillar may be infinite, you don’t know. All you know for sure is you’re not falling through the air.

Yet in terms of who ‘originally’ wrote Beethoven’s Fifth, we can’t give an answer – it’s Beethovens all the way down.

Now, the more observant among you will have noticed that the Bootstap Paradox doesn’t really describe a line, but its curvier cousin: a circle. The music caused itself to exist, which looks something like this. 

A ‘causal loop’, this arrangement seems more appealing than the infinite regress. When travelling along the outside of a circle, you don’t ask where it ‘begins’ or ‘ends’. A circle has no beginning and has no end, but your journey is a straight line from your perspective.

The tune can inspire itself without slipping into infinity, as a circle is a finite size. You can ride the merry-go-round as long as you like, but those ponies will never take you anywhere. Driving around a roundabout 100,000 times is not the same as travelling from John O’Groats to Lands End, and the Doctor and Beethoven can swap notes forever without issue. 

What’s more, this arrangement contains all of the information required for Beethoven’s Fifth to exist. Instead of an infinite chain of Doctors and Beethovens, we just need one leaving notes for himself. 

The issue is although it’s a tidier set-up, a circular argument is not hugely useful. A causal loop doesn’t ‘explain’ itself, it simply is. We have sidestepped the Doctor’s question – who wrote the music in the beginning – by rendering the concept of beginnings moot. We have given him a totally different sort of answer, which is the same as not answering. 

It’s like we’ve all started speaking like Matthew McConaughey in True Detective.

The Doctor: “Here pal, who wrote Beethoven’s Fifth Symphony?”

Matthew McConaughey: “Time is a flat circle.”

The Doctor: [Sigh] “...I miss Clara.”

The answer holds together, but it’s also infuriating.

(There are also issues with the nature of information and entropy that we’re not going to get into here. We wouldn’t want to get bogged down now, would we?)

If you’ve got this far, you’re probably the sort of person who’s now shouting about alternate timelines. (We’re going to ignore issues of whether alternate timelines exist in the Doctor Who universe, because trying to straighten out that canon would make our heads explode.) Here’s another Doctor explaining the concept.

This is more science fiction than philosophy, but essentially the theory goes that there must have been an original sequence of events in which Beethoven (or some other third party, but let’s not make this even more complicated) wrote the Fifth Symphony. 

However, this timeline was derailed by something that didn’t happen in the original sequence of events, and everything that happened afterwards is entirely new and unpredictable. With no-one else stepping up to write the Fifth, the time traveller is forced to do it himself.

From that point on we are in a new history – the reason we can’t go back and find Beethoven writing the music is because Beethoven didn’t write the music in this sequence of events. Nevertheless, the idea for the music did originally come from Beethoven. But what changed in the original timeline? 

How about the time traveller’s arrival itself?

Perhaps Beethoven was too busy arm wrestling the Doctor to hear an auspicious knock at his door, or perhaps meeting a fanboy put him off music altogether. Whatever the reason, the time traveller is forced to intervene. But what happens in the future of this new timeline, when a young time traveller decides to meet his hero Beethoven? And in the timeline that follows that one, when another Doctor comes hunting for an autograph? And the next one? And the next?

We have replaced a line stretching back into infinity with a branching tree, stretching forwards into infinity. The experience of the time traveller jumping track is a bit like…a bendy straw, we guess? It's twisted, but everything is moving in the same direction.

The big problem you’ll have noticed is that no matter how many timelines you’re dealing with, this theory begins with the assumption of an original author, and that this author was Beethoven. Which is what we're trying to prove!

The Greeks used to call this begging the question. We call it a massive headache.

So after all that, what have we learned?

1. The Fifth Symphony is a banging tune, no matter who wrote it.

2. Someone needs to smash the Doctor’s guitar.

3. The rules of time travel and logic really, really don’t get on. 

Or in other words, it's all… 

Well, we tried. 


Who do you think wrote Beethoven’s Fifth Symphony? Tell us below!