The Maths of Doctor Who #4.2 – “There’s an awful lot of one, but there’s an infinity of the other.”

“Saving the day through a heartfelt sing song and the illogical powers of an emotional leaf felt like a distinct cop-out.”

– Mark Snow, IGN.1

“I caught the sound of a man airing the preposterous notion that the sum of all primes approaches infinity.”

– A complaint to BBC Radio 4 regarding an episode of More or Less.2

Infinity makes people cross. The very idea itself asks us to the imagine the most impossible of notions, that something can goes on forever. Fans of Doctor Who are likely to say that it is a show that will ‘go on forever’, but what they probably more accurately mean is that ‘Doctor Who will go on for the rest of my life and many, many years after’. If Doctor Who were to run for 100,000 years, then yes that might seem like a very, very long time indeed, but 100,000 years is just the teeniest tiniest drop in the ocean when compared to the all-encompassing enormity of infinity.

How then can we even begin to comprehend something like that using only our finite and comparatively tiny lived experiences? And when presented with rational arguments and logical conclusions on the consequences of such an idea, why do we intuitively decide to reject these answers as preposterous and absurd? Perhaps it is because it’s unlike anything else we know. Perhaps it has something to do we how we feel about infinity instead. This blog I hope will unravel the mysteries and shed some light on our understanding of infinity itself. But why have I actually brought up the idea of infinity? Because of The Rings of Akhaten (2013), of course.

The Rings of Akhaten and The Notion of Infinity

Judging by the Readers’ Poll conducted by Doctor Who Magazine in 20143, it would seem that The Rings of Akhaten is one of the most unpopular episodes among fandom at-large. This idea is examined in considerably more depth within William Shaw’s entry in The Black Archive range on The Rings of Akhaten, but Shaw has observed that “nearly all of the [contemporary reviews] were quite positive, or only mildly negative. The only strongly negative review from the time was in Doctor Who Magazine. Something about this episode seemed to hit differently with hardcore Doctor Who fans”4. One of the main points of critique is the episode’s climax, which not only involves the notion of infinity but also what Mark Snow of IGN has described as “the illogical powers of an emotional leaf”. So first off, let’s quickly recap the dialogue from that climatic moment:

CLARA: Well, I brought something for you. This. The most important leaf in human history. The most important leaf in human history. It’s full of stories, full of history. And full of a future that never got lived. Days that should have been that never were. Passed on to me. This leaf isn’t just the past, it’s a whole future that never happened. There are billions and millions of unlived days for every day we live. An infinity. All the days that never came. And these are all my mum’s.5

Upon Clara’s successful resolution to the problem, the Doctor then decides to come back to the fore and act like he knew this was the answer all the time:

DOCTOR: Well, come on then. Eat up. Are you full? I expect so, because there’s quite a difference, isn’t there, between what was and what should have been. There’s an awful lot of one, but there’s an infinity of the other. And infinity’s too much, even for your appetite.

This dialogue suggests a rather intriguing question: are we to accept that the Doctor’s past memories offered here are merely finite, whereas the lost future of Clara’s mum, symbolised by “the most important leaf in human history”6, represents an infinity of unlived days? What perhaps makes this even more unintuitive and radical a conclusion to the story here is that we generally perceive the Doctor as this immortal hero, who stars in a show that we like to think will go on forever. Yet here, in this particular moment, it actually pales in comparison to the seemingly finite, and tragically cut-short, lifespan of Clara’s mother. Allow me then an exciting digression into the ideas and consequences of infinity itself, in the hope that I will be able to answer this question more comprehensively. I do hope it’s not too much for your appetite.

Pictured: Clara, the Doctor, and the Most Important Leaf In Human History.

“Hey, do you mind if I tell you a story?” – The Early Days of Infinity

The earliest recorded mention of infinity is widely regarded to be from the philosopher Anaximander (c.610-546 BC) who used the word ‘apeiron’, which more literally means “unbounded” or “indefinite”, though many philosophers such as Heidegger and Derrida have debated the translation of this term. However, the Ancient Greeks were seemingly terrified by the notion, a fear which has since been termed as a “horror of the infinite”7. This is especially notable in the works of mathematician Euclid who, by using a clever recursive argument, first proved the existence of infinitely many prime numbers. Yet he deliberately avoids the word ‘infinite’ altogether. Instead, his proof in Book IX of The Elements translates into English as “Prime numbers are more than any assigned multitude of prime numbers”8. The Ancient Greeks may have been among the first to entertain the notion of infinity, but they certainly refused to take it seriously.

Perhaps the most well-known use of the infinite in philosophical pop culture is Zeno’s Paradox about a hypothetical race between the hero Achilles and a tortoise. The tortoise is given a head start on Achilles and the race begins. By the time Achilles has reached the starting position of the tortoise, the tortoise has moved some distance ahead. Once Achilles has reached that position, the tortoise will have moved ahead some more, albeit a shorter distance. This process continues ad infinitum and so the argument here is that Achilles will never overtake the tortoise to win the race. Intuitively though, we know that in real life a man would easily overtake a tortoise in a running race and herein lies the paradox. What makes this paradox of infinity somewhat inadequate is that it does not explicitly recognise that an infinite sequence of events can still lead to a finite result. This is the entirely logical result of summing up sequences of numbers that converge towards a particular value and in mathematics we call that a ‘limit’.

Infinity, however, was still slow to catch on in the minds of mathematicians and doesn’t get its iconic ‘figure-of-eight’ symbol (∞) until 1665 when John Wallis first described an infinitesimal as the fraction 1/∞9. His idea caught on with the likes of Newton and Leibniz who would go onto independently discover calculus (formerly known as infinitesimal calculus) during the latter half of the 17th Century. By this point, it seems that infinity was here to stay.

“I’ve seen bigger.” – Are Some Infinities Bigger Than Others?

So far then, we’ve talked about the origins of infinity and things getting infinitely smaller, but what about when things get infinitely bigger? In fairness, mathematicians didn’t tackle that one head on until towards the end of the 19th Century. The principle figure behind this area of work was Georg Cantor, who introduced the radical new notion of cardinality, which essentially is a way to count the magnitudes of infinity. Rather than treat infinity as some flimsy piece of philosophical conjecture or just as an impossibly large number, Cantor decided that infinity should now be regarded as an entirely separate concept, complete with its own set of rules. Some of his contemporaries went as far as to describe this as ‘heretical’, which unfortunately led to him having a series of nervous breakdowns over his lifetime. David Foster Wallace has even identified this historical event as the origin of a stereotype he termed ‘The Mentally Ill Mathematician’ (with the most famous example of course being John Forbes Nash Jr., the pioneer of game theory, as the subject of the Oscar-winning film A Beautiful Mind (2001)) alongside others including the ‘Tortured Artist’ and ‘Mad Scientist’10.

To understand Cantor’s idea of cardinality, consider the set of all whole numbers, also commonly known as the natural numbers, and the set of all even numbers, which are all the numbers wholly divisible by two. If I asked you how big the set of all whole numbers is compared to the set of all even numbers, I would reasonably expect you to say that it is double the size. This seems intuitive because whilst both the sets of whole numbers and even numbers are infinite (as they both go on forever), the even numbers appear half as frequently throughout the whole numbers. Yet I can draw a one-to-one mapping of the whole numbers to the even numbers by pairing each whole number with the double of that number. This one-to-one mapping means they are in fact the exact same size and so have the same cardinality. Infinity then has now been repurposed to more precisely describe the size of any collection of objects and can be used to compare the relative sizes of infinity.

Pictured: A one-to-one mapping of the natural numbers N to the even numbers E. This illustrates the idea that both sets have the same cardinality and so are ‘equivalent’ in size.  They are both countably infinite. Image take from

Since mathematicians had run out of Greek letters to borrow, Cantor instead borrowed the Hebrew letter ‘aleph’ (ℵ) and so the cardinalities of infinite sets are written as ‘aleph numbers’ with ‘aleph null’ (ℵ0) being the smallest, and refers to the size of the natural (or counting) numbers. This cardinality of infinity, and any sets of identical magnitude are also known as ‘countable infinities’, with any higher cardinalities known as ‘uncountable infinities’. From this, you can identify the sets of integers and rational numbers as countable infinities, whilst the sets of all irrational numbers and real numbers are uncountable infinities. Whilst this might sound like a lot of work just to get a grasp on what infinity means, these ideas can help us with some more tangible problems ranging from the number of possible ways to slice a pizza to the number of possible chess games that can be played.

As I said earlier, Cantor was criticised by some mathematicians at the time but some came staunchly to his defence. One of those was David Hilbert who described Cantor’s work as “the finest product of mathematical genius” and defiantly exclaimed that “no-one shall expel us from the Paradise that Cantor has created”11. No sensationalism detected whatsoever. Hilbert is expressing Cantor’s work on infinity here in terms of a state of afterlife, a place of eternal happiness, yet it may be worth noting that Hilbert himself was agnostic (he was raised as a Protestant though). In this moment, infinity is not so much what it actually is, mathematically speaking, but rather what you actually believe in.

“There are billions and millions of unlived days for every day we live.” – How Does The Rings of Akhaten Handle Infinity During Its Climax?

We can clearly divide the climax into two key events: the Doctor’s speech that fails to resolve the situation and Clara’s speech that manages to succeed instead. The Doctor offers to the Sun God12 his memories but this fails to satisfy its appetite. His passionate speech conjures up these incredible, awe-inspiring and seemingly impossible imagery such as watching “universes freeze and creations burn” and “universes where the laws of physics were devised by the mind of a mad man”. The Doctor’s strategy here then appears to be to overload the Sun God with these extraordinary tales. But this is a massive oversight on his part. Whether a story is short or long, probable or impossible, factually accurate or entirely fiction, it doesn’t matter: it is still a story. What will end the Sun God’s existence is not the nature of these stories, but the number of them.

But surely the Doctor has an infinity of stories to tell? Indeed, if we are to take all of the licensed expanded universe stories (and, just for good measure, all the unlicensed fan fiction as well) then we can see gaps between adventures that can contain an uncountable infinity of adventures, even in places where logically there shouldn’t be more adventures (otherwise known as The Law of Big Finish). One could suggest a multitude of reasons to get around this, ranging from the television show not considering these canonical to perhaps the Doctor having a finite capacity to his memory, but I think the most reasonable answer is also the simplest one: the Doctor is selecting a finite set of memories to offer. His adventures may take an infinity of forms but his chosen memories are a finite number. In fact, throughout the entire story, he is unwilling to sacrifice anything of his, whether it be his sonic screwdriver, Amy’s glasses, or his entire past, and so he continues to fail at understanding the situation at hand13.

Clara, however, doesn’t just offer her past memories but “a whole future that never happened”, all the uncountable possibilities of the days she could have shared with her mother, “passed on to [her]”. I would argue here that it’s entirely intuitive that she figures this out given that she has already made one sacrifice earlier on in the story, her mother’s ring – in order to gain access to the space moped. Unlike the Doctor, Clara is willing to offer everything, willing to demonstrate the unbounded sentimental value she holds of her most treasured possessions, and this is why she succeeds. It would be impossible to map all the days that could have happened to the days Clara expected her mum to live out with her; an uncountable infinity of days. Just like Hilbert proclaiming that Cantor had created a ‘Paradise’ from his work on infinity, and just like Clara’s mother’s ring, a never-ending circle representing a union that lasts forever, it’s actually the sentiment of infinity that truly counts here. And infinity is too much, even for the Sun God’s appetite.

This emphasis on sentimentalism over mathematical rational is not just present here in The Rings of Akhaten but in Neil Cross’s other work too. In an episode of Luther, Alice Morgan tells John Luther about the size of the observable universe:

MORGAN: Did you know that the observable universe just got bigger? […] Last time I saw you, we assumed there were about 200 billion galaxies. The revised estimate puts it at two trillion, so what we believed to be absolutely everything was basically just a round error. Closer to zero than the true number.14

Yes, Alice. That’s how scale factors work.

The intended effect here is imply that our place in the universe is so much smaller than we had previously thought, but anyone with some sense of mathematical intuition would realise that on such a large scale, even though the absolute difference of nearly two trillion seems a gargantuan number, it is actually relatively tiny. Two trillion is still nowhere near everything, not a scratch on infinity. The emphasis here yet again is not on understanding what it actually means, but on what it makes you feel.

Whilst it may be disappointing to see that the Eleventh Doctor fails to understand what is taking place during The Rings of Akhaten, he does learn his lesson eventually. In The Time of the Doctor (2013), he sacrifices the remainder of what he believes is his thirteenth and final life to defend the town of Christmas on the planet Trenzalore. And just like Clara’s sacrifice of “the most important leaf in human history”, it manages to change their future. Perhaps then that is why we hear a reprise of The Long Song just before the Eleventh Doctor regenerates.

The Borromean Rings of Akhaten – A Conclusion

I have one last piece of mathematics to bring up in this discussion: Borromean rings. The Borromean rings consist of three interlocking circles forming what is known in knot theory as a Brunnian link. What this simply means is that if one of the circles were removed, then all remaining circles would become unlinked. If you inspect the picture closely you may realise that this is a geometrically impossible shape; no-one could physically construct such an object using real rings. It does become possible once you make the rings elliptical but then these aren’t proper rings, are they?

Pictured: The Borromean Rings. Image taken from

The name itself comes from the coat of arms of the Borromeo family, an Italian aristocratic family from around the 17th Century. They certainly weren’t the first to use the symbol as it dates back to the Vikings of Scandinavia, who called it ‘Odin’s triangle’ or a ‘walknot’. Besides Viking runes, it has also been found in early Buddhist art and Roman mosaics. It frequently appears in religious scripture as a symbol of strength in unity and to represent sacred trinities, groups of three-into-one, such as the Holy Trinity of the Father, the Son and the Holy Ghost, for example. These Borromean rings then are not just a symbol of unity, but also of faith and belief. In the final scene of The Rings of Akhaten, the Doctor returns to Clara her mother’s ring:

DOCTOR: They wanted you to have it.

CLARA: Who did?

DOCTOR: Everyone. All the people you saved.

Whilst I think it’s a great shame that we don’t get to see the people of Akhaten do this in person, it nevertheless shows that they greatly value what she has done for them. The return of her mother’s ring then is a symbol of their belief in Clara. But Clara was not alone. She was also supported by Merry, the Queen of Years, and by Ellie Ravenwood, her own mother, symbolised here by “the most important leaf in human history”. In his recent Black Archive, William Shaw argues that it is these three characters who can provide an alternative positive, feminist version of the future in response to the Doctor’s patriarchal assumptions that are present in The Rings of Akhaten14. Here then I propose a new trinity, one that symbolises unity in sisterhood all across the “Seven worlds orbiting the same sun” and beyond. I shall call it the Trinity of Akhaten, and it consists of Clara, Merry, and the Most Important Leaf in Human History. Long may they continue to be with us, now and forever more.



Whilst all internet references have been highlighted throughout, my primary sources of inspiration and reference points were the following three books:

  • Alex’s Adventures in Numberland by Alex Bellos.
  • Things to Make and Do In The Fourth Dimension by Matt Parker.
  • The Black Archive #42: The Rings of Akhaten by William Shaw.



1 Snow, Mark, ‘Doctor Who: “The Rings of Akhaten” Review, IGN, 6 April 2013,

2 Parker, Matt, Things To Make And Do In The Fourth Dimension, p403.

3 Griffths, Peter, ‘The Results in Full!’, DWM #474, cover date July 2014.

4 Maleski, Sam, ‘INTERVIEW – William Shaw, the Leaf and “Rings of Akhaten”’, Downtime, 25 April 2020,

5 All transcripts are taken from and edited for clarity at the author’s discretion.

6 All quotes are taken from The Rings of Akhaten unless otherwise stated.

7 Hutten, Earnest H., The Origins of Science: An Inquiry into the Foundations of Western Thought, George Allen & Unwin Ltd, p. 135. Why not have a listen to this BBC Radio 4 programme to learn more about the Ancient Greeks and the ‘Horror of the Infinite’?

8 Heath, Sir Thomas Little; Heiberg, Johan Ludvig, The Thirteen Books of Euclid’s Elements, The University Press, p. 412 (Book IX, Proposition 20).

9 Bellos, Alex, Alex’s Adventures in Numberland, p400.

10 Bellos, Alex, Alex’s Adventures in Numberland, p400.

11 Parker, Matt, Things To Make And Do In The Fourth Dimension, p404.

12 Or is it a Planet God? See Appendix 1 of William Shaw’s Black Archive on The Rings of Akhaten for more discussion on whether Akhaten is a planet or a sun.

13 For more on how the Doctor misunderstands the events of the episode, see Chapter 1 of William Shaw’s Black Archive on The Rings of Akhaten.

14 Luther, Series 5, Episode 2 (2019). I have never actually seen an episode of Luther but this quote is referred to on pg92-3 in the Black Archive on The Rings of Akhaten.

15 For more on how these three offer a feminised vision of the future, see Chapter 2 of William Shaw’s Black Archive on The Rings of Akhaten.

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