A Mathemagician’s Apology – The Twelfth Doctor’s mathematical leanings

Image Credit: Gage Skidmore (CC BY-SA 2.0Flickr)

Image Description: Peter Capaldi at a SDCC 2017 Comic Con Panel

Evan Jones presents an unorthodox claim as to why he feels the Twelfth Doctor is really a mathematician hiding in plain sight.

The Doctor is a character who wears many hats – and no, I’m not referring to fezzes here. What I mean is that the Doctor takes on many roles within the show, ranging from scientist to explorer to revolutionary and even magician. This is quite understandable for a show that, at first, was pitched to help teach children about science and history, but has since covered a much broader range of topics from social, cultural and political spheres. 

If Doctor Who was to endure so many years and so many lifetimes, it would need to constantly adapt, to blend in with its surroundings – the titular character acting like some form of chameleon. How ironic, then, that the TARDIS refuses to change its form. However, I think it’s fair to say that the Doctor has rarely been ascribed the role of mathematician. But, quite brilliantly, this has precedent. 

The earliest example appears within the very first Dr Who Annual in a feature article entitled ‘The Equations of Doctor Who’. The article claims that the Doctor is “the greatest human mathematician” with the ability to visit “multi-dimensional spheres” otherwise inaccessible to humans. It’s a wild thing to read from a time before the show’s canon was firmly laid down, or even since it has been. 

An arguably more canonical introduction of the Doctor’s mathematical credentials can be seen in Episode One of The War Machines, in which he evaluates WOTAN’s performance using a maths problem. The Doctor asks WOTAN for the square root of 17422, with WOTAN subsequently printing out their answer for the Doctor to read: 131.933. If you check this on your own calculator, you’ll get an answer of 131.992424, which to three decimal places rounds to 131.992. WOTAN, rather amusingly, then gets the last decimal place wrong but the Doctor thinks it’s “near enough”, which is probably a fair assessment by the standards of 1960s computing technology.

Fast forward to almost a decade later in 1975 and the Doctor’s image was being used to teach children mathematics. The BBC Schools programme Mathshow (1975-76) aimed to teach secondary school children mathematics using comic sketches, some of which saw actor Tony Hughes dress up as the Fourth Doctor to teach various topics ranging from symmetry to probability and map coordinates. It was regularly repeated up to 1980 and led a generation of schoolchildren to think they had watched actual Doctor Who episodes at school.

But for my money, the most mathematical throughline in the show’s now almost 60-year history came in 2014 during the Twelfth Doctor’s debut set of adventures in Series Eight. Capaldi’s incarnation of the Doctor is initially presented as emotionally reserved, appearing to act more logically and more calculating than some of his predecessors, but by the time of his departure his flaws are plain to see. This piece, therefore, acts as a case for the Twelfth Doctor’s mathematical identity. The mathematician G. H. Hardy wrote an acclaimed essay in 1940 defending his life’s work, entitled A Mathematician’s Apology, so I have rather wittily dubbed this piece ‘A Mathemagician’s Apology’ in tribute to the original essay.

Deep Breath

The Doctor wakes up from his post-regenerative slumber and immediately begins searching the bedroom until he finds a stick of chalk; the radiator ending up with “abstruse calculations and  mathematics swarming all over the surface” according to the script’s stage directions. Chalk is one of the definitive tools used by mathematicians in their research and the communication of ideas, functioning “both as a metaphor and as a literal device in the construction and circulation of new concepts” according to Michael Barany and Donald MacKenzie in ‘Chalk: Materials and Concepts in Mathematics Research’.

The stick of chalk’s shape and integrity, as it traces along the surface of a blackboard, makes it ideally suited to the carefully unfolding nature of mathematical proofs and theorems; it’s instantly recognisable as one of the ubiquitous trademarks of a skilled communicator of mathematics. Chalk also features prominently in several popular culture depictions of mathematicians, such as Good Will Hunting (1997), A Beautiful Mind (2001) and Hidden Figures (2016).

A closer examination of the Doctor’s scribblings across the floorboards also reveals some genuine mathematics. There are hyperbolic functions, partial derivatives and algebraic expansions, all of which are neatly embedded among more Gallifreyan-esque circular symbols. The use of chalk and blackboards would subsequently become a prominent feature of the TARDIS console room during the Twelfth Doctor’s tenure.

Into The Dalek

The anxiety displayed by the Doctor over his mathematical identity ties in neatly with his overall arc during Series Eight, as he questions whether or not he is ‘a good man’. In Into The Dalek, he directly asks Clara this question and she, after hesitating, replies that she doesn’t know. The Doctor is frustrated by the lack of a clear binary response to his question: yes or no. Pure mathematicians like to work in discrete, binary terms. The Doctor continues to display anxiety throughout the series as he contemplates what kind of man he truly is, unable to find an answer that conforms to his binary constraints.

This episode also sees the introduction of Danny Pink, a former soldier who now teaches mathematics at Coal Hill School, the same school as then-companion Clara Oswald. For fans of the show who have followed it from its very beginnings, this echoes the show’s history in two key ways. Firstly, it calls back to the show’s educational roots with the first companions, Ian Chesterton and Barbara Wright, being school teachers from Coal Hill School. Secondly, and perhaps more pertinently, it calls back to Mawdryn Undead (1983), though this is more obvious in hindsight after viewing the series finale. Mawdryn Undead saw the return of Brigadier Lethbridge-Stewart, who just like Danny Pink was also a former soldier now teaching mathematics, albeit at Brendon Public School. This episode marks the beginning of Danny acting as a mathematical rival for the Twelfth Doctor, which will develop throughout the series.

Robot of Sherwood and Listen

The Doctor’s love of chalk reoccurs in the opening scene of Listen, in which he argues for the existence of a creature with perfect hiding skills. Whilst presenting his case, he uses overtly mathematical terms such as ‘proposition’ and ‘conjecture’. Arguably, if the Doctor was presenting a scientific theory, he would use terms like ‘hypothesis’ and ‘experiment’ instead. But his choice of language here signals that the Doctor is presenting a mathematical proof, a rigorous argument consisting of a series of logical statements that aims to convince his audience that the statement must always be true. And yet, his proof is deeply flawed.

Firstly, the Doctor’s argument is self-evidently not rigorous, with no clear definition of what defines a ‘creature’, or why there are just three mutually exclusive groups of these so-called ‘perfect’ creatures: ‘hunters’, ‘defenders’ and ‘hiders’. Ideally, the Doctor would present his proof for peer-review under the watchful eyes of other mathematicians, ensuring his argument is suitably robust, but instead he chooses not to share it beyond the walls of the TARDIS console room. 

Secondly, and more damningly, his argument fails to convince the one audience member present to witness it: himself! A mathematician who has complete faith in their argument would be satisfied that their work is done, but the Doctor remains unconvinced. In the words of esteemed mathematician and astrophysicist Sir Arthur Eddington, “Proof is an idol before which the mathematician tortures himself.”  The fact that the Doctor never conclusively finds the creature he was looking for is perhaps evidence enough for the veracity of his ‘proof’. 

This isn’t even the first time the Twelfth Doctor’s attempts at logical thinking have failed him during the series. Whilst Listen sees the Doctor arguing for the existence of something that he isn’t able to find, Robot of Sherwood sees him attempt the exact opposite; he tries to assert that there is “no such thing as Robin Hood” and then is immediately faced with evidence to the contrary. Robot of Sherwood and Listen therefore act as mirrors to each other, both of them reflecting back the flaws in the Doctor’s supposedly rational beliefs for the viewer to plainly see.

Time Heist

The successful heist of the Bank of Karabraxos carried out by the Doctor and his crew is actually down to some classic mathematical thinking. Co-writer Stephen Thompson, who studied mathematics at university and is himself a former maths teacher, has cited the ‘Crossing-the-River Problem’ as the basis for the episode’s plot structure. 

The problem is a logic puzzle that involves a man transporting a wolf, a goat and a cabbage across a river. The man can only carry one item at a time in his boat, but he cannot leave the wolf and goat alone (or else the wolf will eat the goat), nor can he leave the goat and the cabbage alone (or else the goat will eat the cabbage). What then is the smallest number of boat journeys required by the man to transport all three across the river? This problem is left as an exercise for the reader. 

Thompson himself explicitly linked the two in an interview whilst describing how he managed to work out the Doctor’s plan:

“That is precisely the mathematical problem we were faced with in creating Time Heist… This is how the Doctor actually cracks it: what he does is very quickly in the TARDIS he takes the plan for the heist and cuts it into four bits. And then very quickly in the TARDIS he beams over to the bank and leaves those four bits of the plan and then beams back before the [Teller] can read his mind. Then he wipes his own memory and sends himself back to the bank, and bit by bit he gets the plan. But of course he never knows all of it, so the [Teller] can never read his mind.” 

The Teller cannot identify the intent of the bank robbers because the robbers themselves do not know what they intend to do; each part of the heist is enacted without them knowing where it’s all going. Without all the required information, the Teller cannot therefore solve the problem they are tasked with: defending the bank. This is the same predicament that the Twelfth Doctor faces when trying to resolve his moral and mathematical identities: how can he possibly reach an answer when you don’t have all the pieces to get there?

The Caretaker and Kill the Moon

When the Doctor meets Danny Pink for the first time in The Caretaker, he repeatedly refers to him as “P.E.”, as if unable to grasp the notion that he really is a mathematics teacher. Whilst this can be read as the Doctor expressing his apparent disdain for soldiers and the military, a theme that also runs throughout the duration of Series 8, it can also be seen as the Doctor projecting his anxiety over his mathematical identity onto Danny Pink.

Ultimately though, the Doctor’s uncaring front backfires on him after the events of Kill the Moon, with Clara leaving the TARDIS and him on unfavourable terms. Clara turns to Danny for help managing her feelings and he in return provides a sympathetic ear and some advice based on personal experience. Danny then is more able to manage his plural identities of soldier, partner and mathematics teacher, whilst the Doctor continues to struggle with what kind of man he really is.

Mummy on the Orient Express

This episode, like Time Heist, is another example of the Doctor successfully using mathematical thinking to save the day. The Doctor manages to use his skills in logical deduction and problem-solving to solve the cryptic riddle of the Foretold. 

Firstly, he casts doubt on the initial assumption they’ve made that all the victims are chosen at random. He proceeds to use a data-informed approach to tackling the problem, asking for all available information on the passengers so that he can deduce who the next victim will be. He’s also the first person to critically question the specificity of the sixty-six second time limit, which in turn allows him to draw the conclusion that it must be an artificial process for the transfer of energy. These are precisely the kind of thinking that a mathematician would use to solve real-world problems.  Sometimes the only equations you have are bad ones, but you still have to solve them.


The Doctor’s strategy for establishing contact with the Boneless in Flatline is perhaps the best illustration of the strengths and weaknesses of the Twelfth Doctor’s mathematical thinking throughout Series Eight. On the one hand, it shows the Doctor’s ingenuity for thinking outside the box, a metaphor that becomes physically realised within the episode when the Doctor uses his hand, Addams-family-style, to move the shrunken TARDIS out of the path of an oncoming train. He is also successful in translating between second- and third-dimensional spaces by inventing the ‘Toodis’, a device that looks like a clever gizmo with a calculator slapdashed onto it. 

On the other hand, the Doctor’s plan ultimately fails to work. He was unable to translate the language of the Boneless so that he could understand what they wanted. Once he manages to restore the TARDIS to its usual dimensions with the help of Clara and Rigsy, the Doctor draws his own conclusions about the events of the episode: 

“I tried to talk. I want you to remember that. I tried to reach out, I tried to understand you, but I think that you understand us perfectly. And I think you just don’t care. And I don’t know whether you are here to invade, infiltrate or just replace us. I don’t suppose it really matters now. You are monsters. That is the role you seem determined to play. So it seems I must play mine. The man that stops the monsters.”

He cannot prove outright that the Boneless intended to cause harm, despite the many people who have died as a result of their actions. But in order to bring closure, the Doctor chooses to provide his own answers instead, by asserting that they understand us well enough, that they are not ‘aliens’ but ‘monsters’, and that he is the one that must stop them. The Doctor, satisfied with the identities he has ascribed, sends the Boneless back to their own dimension. He even gives them a name in the process, one that plainly acknowledges their absence of a logical structure or form. Perhaps that’s the key reason why the Boneless disturb the Twelfth Doctor’s mathematical tendencies.

In the Forest of the Night

The Doctor’s confrontational attitude towards Danny persists in this episode, where he refers to him as “the PE teacher” whilst talking to Maebh. There is no resolution in sight for the conflict between these two mathematical rivals.

However, we also witness a flashback in this episode to one of Danny’s maths lessons. In the lesson, Danny tries to teach algebra to a class of pupils, one that is surprisingly small for a secondary comprehensive school in London, even for a ‘Gifted and Talented’ group. Danny’s imprecise use of language causes one of the pupils, Ruby, to develop a common misconception about the notion of ‘finding X’. Instead of recognising that X is a symbolic representation for an unknown number whose value needs to be found, she instead identifies the physical location of the X, at the top of the board.

Whilst this scene is arguably here to further cement the episode’s main theme of differing perspectives between adults and children, it’s nevertheless a rookie mistake for a fully qualified maths teacher like Danny to make. He only realises the possible misunderstanding he has given to the children after the event has occurred, despite it being a well-known misconception. This, in turn, displays anxiety from Danny Pink about his own mathematical ability, reflecting the Doctor’s own display from earlier in the series. It’s also a good justification for why Danny appears to be rather wounded by the Doctor’s repeated dismissal of his identity as a maths teacher.

Dark Water/Death in Heaven

The two-part finale of Series Eight aims to draw a conclusion to the Doctor’s mathematical anxiety and his crisis of faith to make the best decisions for those around him, i.e. to be a good man. It is reasonable to assume that this personal exploration was inspired by Clara’s interest in the Roman emperor and philosopher Marcus Aurelius, as observed in both Deep Breath and The Day of the Doctor: “Waste no more time arguing what a good man should be. Be one.” G. H. Hardy himself made a flippant remark about the worthiness of arguing over trivial matters in his essay A Mathematician’s Apology, asserting that “criticism… is work for second-rate minds”.

In The Caretaker, we learn that Danny accepts his identity as a soldier, whilst the Doctor refutes the identity of ‘an officer’ that Danny attaches to him: “I’m a soldier, guilty as charged. You see him? He’s an officer”. Just like the Doctor’s inability to recognise Danny as anything other than a ‘P.E. teacher’, this also has the effect of distancing the Doctor and Danny from each other as they conflict over their own identities, mathematical or otherwise.

As already discussed, Danny Pink echoes the return appearance of the Brigadier as a maths teacher in Mawdryn Undead. The tacit parallel between soldiering and mathematics is the idea of following instructions, whether they be the orders of a commanding officer or the steps of a well-defined algorithm. Both of these disciplines are far more concerned with ‘how’ questions than ‘why’ questions. This is the idea being employed by head writer Steven Moffat, who is himself a self-confessed fan of Mawdryn Undead

The Brigadier appears in two guises within Death in Heaven: his portrait on prominent display within the presidential aeroplane transporting the Doctor and UNIT, and his cyber-converted corpse in the graveyard at the episode’s climax. It is only through the reintroduction of the Brigadier, a character who shares all three identities of soldier, officer and mathematician, that the Doctor and Danny’s conflicts over their identities manage to reach a resolution.

The Doctor finally accepts that he’ll never find the logical answers he was searching for, to determine whether or not he is a good man. Instead, he provides his own philosophical case for his identity. Occam’s Razor tells us that the simplest solution is always the best one, and the Doctor applies this rule by saying:

“I am an idiot, with a box and a screwdriver. Just passing through, helping out, learning”.

Whilst Hardy’s conclusion to A Mathematician’s Apology is considerably more elaborate, I feel it elicits much the same sentiment:

“The case for my life, then… is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.”

Hardy’s reference to “some kind of memorial” even has echoes of the finale’s themes of life after death and remembrance. And isn’t it funny how both of these men reached their conclusions on personal identities with the graveyard looming over them in the background – metaphorically for Hardy, and literally for the Doctor? 

Beyond Series Eight

The mathematical undercurrent of Series Eight continues to appear in various forms over the remainder of the Twelfth Doctor’s tenure. In The Magician’s Apprentice, he proclaims to the locals of Essex that he has given them “some top-notch maths tuition in a fun but relevant way”. His mathematical abilities are also put to the test in Heaven Sent by having to make some very quick calculations before jumping out of a window. The episode’s director, Rachel Talalay, actually majored in mathematics at Yale, and used kinematic equations to calculate the duration of the Doctor’s fall during the episode’s production. She even showed her workings out on Twitter.

The Doctor then becomes a lecturer at St Luke’s University before the events of Series Ten, teaching about not just mathematics but practically everything, as seen during The Pilot and Oxygen. The openings of both of these episodes prominently feature the use of chalk by the Doctor to help communicate his lectures to the students. His subsequent use of a transparent whiteboard and marker pen to convey scientific exposition in World Enough and Time appears all the more striking then, as if to suggest that something is not right here, adding to the viewer’s sense of unease before his impending regeneration.

He also shows off his mathematics expertise when he is faced with the Shadow Test in Extremis. The Doctor tells Bill that “Computers aren’t good with random numbers. If you ask a computer simulated person to generate a random string of numbers, it won’t truly be random.” Realising that the numbers being selected are in fact pseudo-random allows him to figure out that he exists inside a computer simulation created by the Monks, which in turn enables him to inform the ‘real’ Doctor of their plans to conquer the Earth.

Lastly, we see the Twelfth Doctor die at the hands of the Cybermen whilst he is on board a hijacked spaceship, the exact fate which befell his former companion Adric, himself a self-proclaimed mathematical genius. The Twelfth Doctor not only lived the life of a mathematician but managed to die the death of a mathematician as well. Even at the very end, he was thinking about stars…

Tides 48 is, at time of publication, available to buy through this link

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s