The 2020 Black Archive Reviews – Part One (January to June)

This review block covers the set of Black Archives released in the first half of 2020. The stories covered by this set are Doctor Who and The Silurians, The Underwater Menace, Vengeance on Varos, The Rings of Akhaten, The Robots of Death and The Pandorica Opens/The Big Bang. At the end of the reviews, I pick out my personal favourite and explain why.

You can buy all of these from Obverse Books here.

The Black Archive #39: The Silurians by Robert Smith?

Key Themes: Technology, the 1970s energy crisis, the military, land rights, animal testing, science and ethics, and whether the Silurian plague could’ve killed us all.

Mathematicians typically review each other’s work. Whether it’s checking calculations or peer reviewing a new research paper, mathematics is very much a subject about teamwork and collaboration. Perhaps this goes against the prevailing stereotype that mathematicians are reclusive souls who solve hard problems on their lonesome, but the more common reality is that you need to work with others to ensure your arguments are communicated clearly and precisely; that we’re all singing from the same hymn sheet so to speak. This is the norm when it comes to mathematics, but it’s pretty rare when it comes to Doctor Who. And since Black Archive scribe Robert Smith? and myself are both mathematicians, this is one of those rare occasions.

Smith?’s specialism is mathematical biology, so it’s no surprise he’s opted to write about the second outing for Jon Pertwee’s incarnation. This entry by Smith? specialising in science and the spread of plague contains seven succinct chapters on seven separate themes over a seven-episode serial called Doctor Who and The Silurians, first broadcast in the year 1970, which makes it the most sibilant book in the series so far. Not content with last month’s controversial decision to omit Utopia (2007) from the analysis of the Series 3 finale, this Black Archive pretends that the initial three words of this serial’s title were never there, as the book predominantly refers to the story as “The Silurians”. I can however understand the latter decision a lot more given it was an in-house production error that led to this unusual title occurring.

The themes covered are diverse, ranging from the links to the 1970s energy crisis, morality in science, and the role of technology in the story. I strongly felt that the book developed in strength as each chapter went by, which gives it a nice crescendo in quality. The particular highlights were when Smith? enters his specialisms, providing unique and informed insights into questions on whether Doctor Who is a science show and whether the Silurian plague could have actually killed us all, a topic that has become surprisingly prescient with the current coronavirus outbreak happening right now. The book is also beautifully and thoroughly referenced, as is evident by the surprisingly lengthy bibliography on display.

However, by structuring the book around some rather broad questions, the analysis does sometimes lead to some rather general conclusions, such as those in the book’s initial two chapters saying that technology both can and can’t solve all our problems and that a Doctor Who story should be as long as it needs to be. The discussions had about these topics were certainly good reads, and I was particularly intrigued by Smith?’s passionate defence about the story’s exceptional length. But I really did feel these could have led to more interesting results. For example, I would have suggested that the book’s second chapter should really have been framed as “Does The Silurians really need to be seven episodes long?” instead. What I’m trying to say here is, I think there should have been another way.

In a fair number of my reviews I have neglected to mention the cover art and icon design by both Cody Schell and Blair Bidmead respectively. I’m actually quite the fan of this entry’s cover, which features the cave drawings of a Silurian with woodland creatures as it’s icon, brilliantly captured by Bidmead, as seen in the serial’s first episode. The choice by Schell to then overlap parts of the three creatures using white, brown and green outlines is inspired. It quite neatly represents the overlap of science and nature presented in the story here as well as the harmonious coexistence between the Silurians and the animals billions of years ago.

Concluding Thoughts

Smith?’s Black Archive entry breaks exciting new ground by looking at the themes of science and ethics in The Silurians, bringing unique and specialist insight on this particular serial. The discussions on morality, animal rights and pandemic plagues are well worth your time here and shows Smith?’s pedigree as a long-time critic of the show. I also do hope this encourages more scientific takes on the show in future entries. After all, “science leads” according to Kate Stewart, Head of Scientific Research at UNIT. She learnt that from her father, but did her father learn it from reflecting on this story’s events? Perhaps, but who knows?

 

The Black Archive #40: The Underwater Menace by James Cooray Smith

Key Themes: How does surviving material influence fan perception, should we even be allowed to take this story seriously, and is traditional fan wisdom bollocks?

There’s no doubt about it – The Underwater Menace (1967) is one of the strangest televised adventures in the entirety of Doctor Who’s 50+ year history. Its setting is mythical, taking place in the lost underwater city of Atlantis. Its science is ropey, with absurd ideas of draining the ocean into the Earth’s core. And its villain is utterly preposterous, a near-contemporary scientist, whom the Doctor is already aware of by reputation I might add, whose ideas of ‘supreme power’ will actually leave no-one in Atlantis alive to see the repercussions of his actions. But it does score highly on originality.

With all that in mind, why should a range like The Black Archive, which delivers thoughtful and serious critiques of any and all Doctor Who stories, dare to take it seriously? The idea that recurring contributor (and former editor of the range) James Cooray Smith would decide to hang his hat entirely around this premise initially seems a rather silly one, yet it actually achieves remarkable results. Much like his last contribution to the range, which examined The Ultimate Foe (1986), Cooray Smith delivers a sublime blend of analysis, document-based research and behind-the-scenes history, with its 110-or-so pages just flying by.

Given the frivolous nature of the story’s ideas and plotting, it is perhaps unsurprising that each of the book’s nine chapters are fairly short ones (about ten pages each) but what brings it all together is the use of a few key over-arching themes. Namely, these are how does missing material affect fan reception, taking full advantage of the rediscovery of Episode 2 back in 2011, and how collective fan wisdom can at times be sorely misplaced, which leads not only to some superlative myth-busting but also a few finger wags at the fandom-at-large. This even extends to the author himself who slaps himself on the wrist in a delightfully cheeky footnote.

The questions covered are an eclectic mix that honestly speak for themselves. Here are just some of the questions that this book presents well-informed and dutifully researched answers for your reading pleasure:

  • Why does fandom universally hate a story they’ve probably never seen?
  • What exactly happened at the BFI in 2011 when Episode 2 showed up?
  • Why was this story filmed despite being formerly abandoned?
  • Why do fans keep writing badly accented versions of the episode’s third cliffhanger?
  • Whatever happened to the Doctor’s hat in this story?
  • How does this story ultimately shape Troughton’s portrayal of the Doctor?

For me, there were two undisputed highlights during this read and both are towards the very end of the book. First, there’s a substantial appendix based on the Doctor’s note to Professor Zaroff signed “Dr. W”, which looks at whether the main character of the show is called ‘Doctor Who’ and whether the character is referred to as ‘Doctor Who’ both within and outside the fiction itself. The appendix doesn’t so much as cover but utterly annihilate the discourse surrounding these related questions, and with considerable aplomb too. Although, I must say the complete omission of the opening scene from World Enough of Time (2017) is a little baffling. I also sincerely hope his ‘Dame Shirley Bassey’ argument just catches on in general.

The second highlight was the book’s final chapter looking at the life, work and collaborators of its author, Geoffrey Orme. Little is known about the life of Orme as he was never interviewed about his work, not even by a single Doctor Who fan, and he died in 1978. The search into the archives detailed in this chapter in the hope to the reveal hidden depths about the story of The Underwater Menace is commendable and reveals subtle and astute observations. This chapter is the literary equivalent of an astounding new Toby Hadoke documentary, such as those which have looked into the previously shrouded lives of Peter R. Newman (Writer of The Sensorites (1964)) and Lennie Mayne (Director of four 70s Who serials). It is truly an excellent capstone to the book itself.

Concluding Thoughts

Despite perhaps being as utterly mad as Professor Zaroff’s plan, this book manages to be a resounding success. This entry takes full advantage of the story’s poor fan reception and partially missing status as an opportunity to re-examine the serial with fresh eyes. Covering a diverse range of topics surrounding its troubled production, obscure cult status, and its mysteriously disappearing hat, The Underwater Menace comes highly recommended to those who want to discover whether it has hidden depths. But if it should happen that fan historiography isn’t to your literary tastes then don’t worry, because there’s plenty more Fish People in the sea.

 

The Black Archive #41: Vengeance on Varos by Jonathan Dennis

Key Themes: Orwellian literary influences, capitalist realism, depictions of violence in 1980s television and “video nasties”.

There’s a certain footnote in this book that I think quite neatly encapsulates the experience of reading of this Black Archive. On page 95, there’s a lone sentence which clarifies that 1982’s The Running Man takes place in the fictional totalitarian United States of America in the year 2019 and not the actual totalitarian United States of America in the year 2019. Dennis is here to talk to you about 1985’s Vengeance on Varos, and he ain’t pulling no punches.

The contents page of this entry looks exactly like what you’d expect if you were to identify the central ideas of Philip Martin’s first Who serial: Orwellian literature and dystopias, the portrayal of rampant capitalism, sadistic violence and “video nasties”. A light introduction on the garish nature of the Sixth Doctor’s coat gets things underway and neatly sets the style and tone for the analysis that follows. I can’t quite figure out how but this book manages to just feel so eighties, which is odd considering I don’t actually remember them – I was born in the decade that came after!

The first main chapter strikes while the iron is hot, swiftly drawing allusions between the political system found on Varos and the political mess we in the UK and those in the USA currently find themselves in. The second covers neoliberalism and Doctor Who’s unfolding relationship with capitalism as both theme and setting, drawing natural comparisons to both Oxygen (2017) and Kerblam! (2018). In my review of the Black Archive on the latter story I suggested that those interested in the story’s politics would be disappointed but now it seems there’s a book for those people as well.

The next chapter was my personal favourite as it helped contextualise the social and political zeitgeist around television violence in the 1980s and the relatively short-lived anxieties surrounding “video nasties”, which are excessively violent films that eluded classification from the BBFC due to a loophole and these caught the attention of a certain Mary Whitehouse. The chapter’s title “They also affects dogs” initially left me mystified until I subsequently learnt within the prose that this was a quote from a Tory backbencher who on national TV claimed there was research that shows “video nasties” not only caused children to become more violent, but dogs as well. This certainly made me feel that we live in a more enlightened age, albeit for a few precious seconds. A short discussion on how television is nothing without somebody to observe it brings the book to a sublime conclusion.

Concluding Thoughts

Other entries in the range do all the hard work of researching their themes and topics in depth, before leaving the reader to come to their own conclusions. You won’t get that with this one. Dennis is pretty blunt when it comes to his perspective on politics, economics and the media but he puts in such clear crystal prose that you are left in no doubt why he thinks that way and he’ll leave you wanting more. It’s a thoroughly recommended read and it confirms in my eyes that the somewhat underappreciated Sixth Doctor is getting some of the strongest critique in the range to-date.

 

The Black Archive #42: The Rings of Akhaten by William Shaw

Key Themes: The Doctor as New Atheist, feminist and post-colonial theory, the episode’s critical reception with fans, and the story’s engagement with ‘anniversary anxiety’.

I’ve personally always been a fan of The Rings of Akhaten; it’s far from my favourite Doctor Who story but I wouldn’t hesitate in telling you that it’s a pretty good one, with many charming aspects, and probably the strongest episode in an otherwise maligned series. To see it land in the bottom ten of DWM’s story poll in 2014 was somewhat baffling, and so I did hope that this episode would be critically reappraised one day soon. Not only then am I delighted that William Shaw has stood up to bat for The Rings of Akhaten as part of the Black Archive’s first foray into Series 7, but I’m also terribly surprised that someone else who rather likes the story has such radically different reasons for doing so. It seems that Doctor Who is once again a broader church than I had previously conceived.

Shaw frames the episode as a critical reflection of the show during its fiftieth anniversary year, highlighting numerous aspects that have previously been underappreciated, looking back at its flaws and shortcomings, whilst also looking ahead towards its future of as-yet unrealised potential. I could have easily predicted that such a book would examine the episode’s religious and patriarchal overtones as well as the poor critical reception among vocal fans; I just didn’t expect New Atheism.

Chapter 1 provides a fresh, exciting and radical viewpoint on the episode never-before-seen, arguing the episode is a subtle critique of the New Atheist movement. Those who remain unconvinced by the inspired introductory section linking Dawkins to post-2005 Who will have to be very patient. An early subsection entitled ‘Doctor Who and the New Atheist Movement’ runs for around seven pages without a single actual mention of Doctor Who itself. But I feel my patience was rewarded, Shaw’s introduction of several key texts manages to bridge the gaps between the episode and his reading, shedding light of the show’s broader relationship with the movement. It is by far the book’s most substantial chapter that could have been hastened by getting to its Doctor Who analysis more snappily.

The second chapter examines the episode through feminist and post-colonial theory. It kicks off with a fantastically astute observation of the post-titles opening scene of Clara waiting for the Doctor with her book ‘101 Places To See’. It’s a much stronger engagement with the episode than the first but a short section focussing on the episode’s ‘Long Song’ knocked its stride, telling the reader what the music is doing and when, with little how and why. Some reference to the music’s emotion and how this is achieved would benefit the musical analysis. This is a minor nitpick though.

Chapter 3 then seeks to rationalise why the episode did not have a good critical reception among fans, notably highlighting that most public reviews were actually quite positive. It comprehensively looks at multiple lines of inquiry including the show’s format, on-screen representation, thematic shortcomings and even the divorced popularity of the episode’s ‘Long Song’ speech on YouTube (don’t forget to subscribe to the Official Doctor Who YouTube channel), providing an ample set of reasons for the story’s arguably muddied execution.

Not content with one radical concept about The Rings of Akhaten, Shaw delivers us another one in his final chapter on ‘Anniversary Anxiety’. It’s easily my favourite of the book, lucidly realising that the subtext of Clara as a proto-Doctor was always there from the get-go and makes her subsequent development in Series 8 and 9 all the more expected. It also grapples with tedious yet recurring internet arguments of Clara as a Mary Sue and Manic Pixie Dream Girl. Whilst the episode may not be perfectly executed, Shaw does establish how it pointed to the show’s narrative trajectory of Series 7 and beyond, again linking it to the story’s denouement of unrealised potential.

A set of three appendices makes the book all the more substantial with an examination of whether Akhaten is a sun or a planet, an interview with director Farren Blackburn, and a previously unseen production document for the episode; all providing key talking points in the book’s wider analysis. I find it difficult to imagine what else could be added to this book. A real-life autumnal leaf? Perhaps not, for I fear it may radically alter the entire projection of my life.

Concluding Thoughts

It is rare to find an entry in the Black Archive range that is simultaneously this comprehensive, holistic and unconventional in its take on a story. Shaw reframes The Rings of Akhaten as a story with radical and reformatory politics that fell short in the execution of delivering its message. Perhaps you won’t be as taken by its viewpoints as I was, but you’ve got to admire its sheer ambition and endeavour alone. The continuous introduction of bold, new ideas is what keeps discourse about the show fresh and exciting. Shaw’s conclusion points to an episode about listening and learning from others. I certainly learnt a few things reading it myself and I hope to follow through on its message in the years to come.

 

The Black Archive #43: The Robots of Death by Fiona Moore

Key Themes: Development from script to screen, Modernist and Expressionist influences, the character of D84, themes of class and power, diverse casting and the serial’s legacy.

With the recent release of Season 14 on Blu-ray, a lot of fans will have been rewatching The Robots of Death (1977). So, with the release of this Black Archive by writer and academic Fiona Moore (which would have been out in the same month were it not for a short delay), it seems like excellent timing to re-examine one of the most beloved Classic Who serials within the fandom-at-large.

First airing in early 1977, The Robots of Death was broadcast during one of the most popular eras in the show’s history. Furthermore, it was also one of the first Doctor Who stories to be made available for purchase on videotape, and then it was the very first Doctor Who story to be made available on DVD. The Robots of Death then is a serial that has enjoyed an exceptionally long shelf-life, especially for a forty-year-old piece of cheaply-made television. It has also probably been examined by fandom a lot more than other serials as well, so the question here really is whether there’s anything more that can be said about The Robots of Death. The answer, rather delightfully, turns out to be yes.

Moore’s monograph opens with a chapter contextualizing the conception and development of the serial, suggesting that a perfect storm of ingredients and individuals involved helped the story achieve its renowned status. It also takes the opportunity to bust some long-touted myths about the serial such as being an ‘Agatha Christie-style’ murder mystery and a story about ‘robot rebellion’. It then swiftly moves onto an analysis of the rehearsal and camera scripts to see how Taren Capel’s backstory became obscured between drafts as well as what happened to ‘Jan’, the crew member that never made it to screen!

The middle three chapters were what held my attention the most with an examination of the serial’s influences from Modernism and Expressionism, a character analysis of D84 and a discussion of the themes of class and power in the works of writer Chris Boucher. I particularly enjoyed how Moore delves much deeper into the collected works of Isaac Asimov and Frank Herbert in order to uncover more than just the usual surface-level links typically mentioned in reference material. It is precisely this kind of discourse that keeps me returning to the Black Archive range. All three of these chapters delighted me with new pieces of context and points of reference that allowed me to appreciate the story a bit more, in spite of me not being a huge fan of the serial.

The final two chapters have a more compilatory feel to them with a chapter on diverse casting followed by a concluding chapter on the serial’s legacy in the TV show and expanded media. The former has little to say specifically about the serial in question and so quickly spills over into the show’s broader casting history as well as the portrayal of Leela more generally, a topic that has been covered more comprehensively in The Black Archive #27: The Face of Evil by Thomas L. Rodebaugh. Meanwhile, the latter is an exercise in gathering all the bits of various continuity in novels, such as Corpse Marker (1999), TV stories, like Voyage of the Damned (2007), and of course the Big Finish audio dramas, such as Robophobia (2011).

Concluding Thoughts

The primary challenge with this Black Archive entry was to find new things to say about a serial that is much-loved, well-documented and oft-discussed in the Doctor Who fandom, and to that end it has succeeded admirably. It’s clear that Moore has a deep appreciation for the serial being discussed and this shines through in the writing of this monograph. This comes recommended to those who want to learn about the serial’s literary influences, the blurred line between man and machine, and how Boucher develops his ideas of class and power within his other works. You can throw your money for this at Obverse Books right away, but please do not throw hands at them.

 

The Black Archive #44: The Pandorica Opens/The Big Bang by Philip Bates

Key Themes: The Epic and the Intimate, Pandora’s Box and other fairytales, anomalies, the trouble with time travel, the story’s relationship with time, and the beginning and the end of the universe.

Steven Moffat’s first series finale remains one of the most popular episodes he ever wrote for Doctor Who. On its tenth anniversary, and just after a fairly recent re-release on Steelbook Blu-Ray (with the most glorious artwork by Sophie Cowdry), the Black Archive has given us its take on The Pandorica Opens/The Big Bang (2010) looking at how it deals with the grand and small scales, fairytales, time travel and, quite naturally, the Big Bang itself. Penned by first-time scribe Philip Bates, this book is a passionate and emphatic celebration of what Bates describes as his personal favourite Doctor Who story.

Bates opens with a sketch of the universe, asking us to consider its various perspectives. Ranging from expansive far-off galaxies to the movement of quantum particles, from the giants of history to the ordinary unknown faces of society, from the epic and the extraordinary, to the small and the intimate, wherever we look we are part of the universe and we are all stories in the end. Perhaps Bates can apply for the role of lecturer at St Luke’s University in Bristol, now the Twelfth Doctor has left a vacancy?

The first chapter looks at the storytelling devices in Moffat’s box of tricks to help convey the scale and complexity of the narrative here. It may shock you to read this but it never actually occurred to me that the fez in The Big Bang and The Name of the Doctor (2013) serve precisely the same narrative function and disappear as soon as their work is complete! It was great to read how Bates broke it all down, illuminating the connections and themes with Series 5 as a whole as well as Moffat’s other scripts. Chapter 2 looks at, perhaps unsurprisingly, at the legend of Pandora’s Box as well as the broader fairytale motifs on display in this story too.

It’s from chapter 3 onwards where things start to get a bit knotty as the remaining chapters look at anomalies, the rules of time travel, the various representations of time and how the universe is thought to have started and later how it might end. This, in my mind, was always going to be the trickiest part of analysing a story that plays fast and loose with the typical rules of the show without much of a rational or scientific basis to go off. Consequently, Bates goes for a defence arguing why the episode is entertaining, emotionally satisfying and earns the right to break some of the standard rules.

Whilst the overall book is certainly an easy read for Doctor Who fans, I would like to have seen more points of comparison with other time travel stories, like the Back to the Future films, to strengthen the analysis on the rules of time travel rather than solely relying on Doctor Who for reference points. Furthermore, the book’s latter chapters provide some sound insight into current scientific understanding on matters ranging from black holes, neutron stars and even 10-dimensional string theory, and I felt this was a remarkable improvement on the range’s previous entry covering The Impossible Planet/The Satan Pit (2006). Yet I was personally frustrated that the two lines of inquiry, those being the narrative and the scientific, didn’t seem to intersect. Why put the two together if they don’t seem to connect?

The back of the book has a brilliantly thought-out appendix, providing us with not one but six different reasons for why the TARDIS exploded in the finale, trying to reconcile the loose threads and thematic connections across the entire Matt Smith era. Will a subsequent novelisation confirm one of these theories or provide an entirely different one? Who knows!

Concluding Thoughts

The Black Archives come in many shapes and forms; some allow readers to re-contextualise the serial during the original time of broadcast, whilst others provide subject-based lenses to examine a particular story. However, Bates’ entry on The Pandorica Opens/The Big Bang is a celebration of one of Who’s most popular stories, helping us to better understand the ideas that made it resonate with fans in the first place, and an ideal jumping on point if you haven’t started the range already. I’m not convinced it will persuade others who don’t see it as one of the greats but otherwise I have no hesitation in recommending, should this one take your fancy. Now where did I put my fez…?

 

My Top Pick – #42: The Rings of Akhaten by William Shaw

If I had to, I could bat for any of these Black Archives for being the best one of the set – they’re all worth a purchase if you fancy them and they are all brilliantly different. But I’m going for #42: The Rings of Akhaten for its sheer level of ambition and originality. It even inspired me to write my own piece on the episode, helping to bridge the gap between my enjoyment of mathematics and the story itself. Clearly, it struck a chord with me, and I hope because of that I’ve managed to help someone else learn something about the episode, in much the same way that I did from Will’s book. I do hope there are many more places to see.

You can buy all of these from Obverse Books here.

The Maths of Doctor Who #5 – “It’s like it’s some kind of game, and only you know the rules.”

The Seventh Doctor likes to play games. Not little ones mind, but really big ones. He likes to challenge opponents to games of strategy, like chess, but mix it in with the high-stakes winnings of gambling games, like poker. He’s not afraid to use real people as the game pieces, including his closest friends and allies, and the outcome of his games will ultimately determine the fate of entire worlds and cause the toppling of empires. Like he once said, quoting the former British Prime Minister Benjamin Disraeli, “Every great decision creates ripples…”1.

Arguably there is no story that makes this on-screen characterisation of the Seventh Doctor clearer than 1989’s The Curse of Fenric, a story which sees him do battle once again with an ancient and terrible evil known as Fenric. The Doctor challenges Fenric to a chess problem and Ace, along with us the audience, learns that the story’s unfolding events are all part of a real-life chess game being played between them. A game within a game, if you will, one an abstract representation contained within the other.

This story then employs the ideas of an area in mathematics known as ‘game theory’, and the serial itself explicitly invokes these ideas with the Doctor’s reference to the Prisoners’ Dilemma, perhaps the most well-known problem within game theory. We can even see, as we are told, a logic diagram for the Prisoners’ Dilemma on one of the blackboards in Dr. Judson’s offices. Whilst these ideas are present in the background of the story, they are never expanded upon or explained fully within the serial, which is unsurprising given how much is already going on – they were certainly pressed for time as it was when it came to the broadcast edit!

However, I feel that these ideas of game theory and the Prisoners’ Dilemma have stronger thematic relevance to the story than has been realised among fans, and that these ideas are remarkably suited to a story set during the height of the Second World War. So then, without further ado… Guys, it’s time for some game theory.

Game Theory and the Mathematics of War

“Real mathematics has no effect on war. No-one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.”

– G. H. Hardy, 1940.2

Mathematics had a considerable effect on bringing about the end of the Second World War in 1945. Not only had number theory been used by the cryptographers working at Bletchley Park to crack the Enigma Code and potentially shorten the war by around two years, but also the mathematics of relativity assisted with the development and subsequent testing of the first atomic bomb. Whilst Hardy, a highly regarded mathematician of his time, provides an emphatic defence about the pursuit of mathematical studies for its own sake in his landmark essay A Mathematician’s Apology, his aforementioned quote is perhaps one of the finest examples of Things That Have Aged Poorly. At times, his thoughts even stray into blatant misanthropy (“most people can do nothing at all well”3) and I would consider such an attitude against the narrative ethos of Fenric as well as Doctor Who more generally (e.g. “We’re all capable of the most incredible change”4).

However, Hardy was known to detest the militaristic applications of mathematics and so naturally did not play a considerable role in the efforts of the Second World War, but had he known about the highly secretive work of his contemporaries then he may have sooner revised his earlier statement. One such contemporary was John Von Neumann, a Hungarian-born mathematician from a wealthy Jewish family who emigrated to America comfortably before the outbreak of the Second World War. Writer Alex Bellos describes Neumann as “the mathematician who shaped the modern world”5. Whilst not a cryptographer like Alan Turing, he played a central role in the development of the modern computer, designing the fundamental internal architecture of the electronic device you are currently using to read this blog, as well as working on the Manhattan Project which developed the first nuclear bomb. He was also the central figure behind the field of game theory.

Game theory is “an area of mathematics concerned with modelling how participants behave in situations of conflict and cooperation”6. Neumann coined the term ‘game theory’ himself in 1944 when he co-wrote the book The Theory of Games and Economic Behaviour. However, his ideas weren’t simply used for recreational purposes but to predict the behaviour of competitive market forces in economic scenarios as well as develop military strategies for US intelligence during the Cold War. As Simon Singh notes, generals were now “treating battles as complex games of chess”7. This is precisely what the Doctor is up to in The Curse of Fenric when he arrives at the secret military base near Maiden’s Point.

But more than that, the story presents us with a dramatic representation of game theory in motion, set at the point in history when it first came into formal existence. Because in the year 1943, as the Doctor is masterminding a plan to prevent Fenric and the Ancient One detonating a set of devastating chemical bombs that will poison and pollute the entire world, Von Neumann is taking up his post on the Manhattan Project, pursuing the development of a weapon that will have similar consequences.

Perhaps it’s unlikely that writer Ian Briggs knew this detail within the history of mathematics, but nevertheless the inclusion of game theory in a story set at this exact point in history is extremely pertinent. As Una McCormack observes in her Black Archive, “The wartime setting of The Curse of Fenric is very far from being window dressing, and the moment in the war is crucial.”8 Neumann’s choice to apply his knowledge of mathematics to military warfare, in what can be read as an attempt to re-lay the global chessboard, creates the very future that we inhabit today. Just like in The Curse of Fenric, the history of the past continues to unfold within our present moment.

Zero-Sum Games and The Prisoners’ Dilemma

JUDSON: You’re familiar with the Prisoner’s Dilemma, then?

DOCTOR: Based on a false premise, don’t you think? Like all zero-sum games. But a neat algorithm nevertheless, Doctor Judson.9

This quote gives us a nice insight to the Seventh Doctor’s moral philosophy here, as he states that all zero games are based on a ‘false premise’. Game theorists will assign a value, sometimes referred to as ‘utility’, to every possible outcome for each player in a game. A zero-sum game is one where if you add up all the possible values, the sum of all the utility, you get zero. This means that if one player gains some points then another player must lose an equal number of points; the sum total of points remaining constant. If you were to apply this idea to all real-world contexts, it would suggest that there must always be winners and losers in each game. The concept of a mutually beneficial outcome for all players doesn’t exist! There is significant research10 to suggest that people tend to have a cognitive bias towards zero-sum games. They believe, intuitively or otherwise, that this is how the world works.

Consequently, this suggests that the Doctor believes life more accurately reflects a non-zero-sum game, meaning that there exists at least one outcome where all the players can gain utility, that it is indeed possible for to achieve mutually beneficial outcomes. This remark then foreshadows the story’s conclusion where the British and Russian soldiers, Bates and Vershinn, join forces to fight the common enemy. This is a rejection by them of the ideology of zero-sum games as they embrace the possibility for the first time that both sides can win. Moreover, this is a rejection of Thatcher’s own political philosophy by the narrative, as is pretty much every other story produced under the tenure of script editor Andrew Cartmel. It also managed to pre-empt Geoffrey Howe in his resignation speech in 1990 (“The European enterprise is not and should not be seen like that – as some kind of zero-sum game”).

What about the Prisoners’ Dilemma then? How does that fit in with all this? Below I have presented the problem as formalised by Albert W. Tucker in 1950:

“Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

  • If A and B each betray the other, each of them serves two years in prison.
  • If A betrays B but B remains silent, A will be set free and B will serve three years in prison.
  • If A remains silent but B betrays A, A will serve three years in prison and B will be set free.
  • If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).”11

We can more easily refine the description of this problem with a pay-off matrix, a grid which shows all the values in an easy-to-read layout, like so:

Criminal B remains silent Criminal B betrays
Criminal A remains silent [1, 1] [3, 0]
Criminal A betrays [0, 3] [2, 2]

For each set of outcomes, the first number represents the jail term of criminal A and the second number represents the jail term of criminal B. So if A betrays and B remains silent, then A spends 0 years in prison whilst B spends 3 years in prison, just as its stated in the second bullet point above. It is also not a zero-sum game, allowing the two prisoners to decide whether they want to cooperate or compete with each other.

What outcome might we expect if we let the two criminals play the game? Well, one way that a game theorist might predict this is to investigate whether there is a dominant strategy here. A dominant strategy is an action that a criminal can take that will always provide the better outcome, regardless of what the other criminal chooses to do. We can see that such a strategy is indeed present here.

If Criminal B expects Criminal A to remain silent, then they should choose to betray because they will spend zero years in prison instead of one. But if Criminal B expects Criminal A to betray them, then they should still choose to betray them because they will spend two years in prison instead of three. Whatever Criminal A chooses, it would seem the rational choice for Criminal B is to betray.

Another approach is to use the minimax algorithm, meaning here that each criminal wants to minimise their maximum sentence. A quick look at the pay-off matrix shows that the maximum sentence possible for each criminal is three years and this can only occur if they remain silent. So, in order to avoid the worst possible outcome for themselves individually, they will each choose to betray the other and so consequently end up with two years in jail each. Again, this reveals the dominant strategy of the game presented here.

This individualistic and supposedly rational mindset to decision making reveals the inherent tragedy of the Prisoners’ Dilemma, because whilst they have individually avoided the worst outcome for themselves (three years in prison) they have ended up in the worst-case scenario as a collective (four years combined in prison). If the prisoners had decided to cooperate instead of compete, by both remaining silent, then they would have collectively spent only two years in prison, which would have been the best-case scenario for the prison gang.

You can change the actions, the points and the context of the scenario, but if your pay-off matrix reveals this same basic conclusion as described here then it is yet another example of the Prisoners’ Dilemma. The tragedy then is that by choosing to avoid the worst-case scenario, the players of the game fail to achieve the best-case scenario.

Chess Problems and Mind Games

“But the ‘great game’ of chess is primarily psychological, a conflict between one trained intelligence and another”

 – G. H. Hardy.12

This fundamental idea behind the Prisoners’ Dilemma appears in a number of ways throughout the story. Perhaps the most obvious of these is the chess puzzle presented by the Doctor as a challenge for Fenric to solve. The solution is revealed to be an unintuitive yet rather straightforward move involving opposing pawns uniting in order to reach checkmate, but logically this seems rather bizarre. As Sandifer duly observes, “the fact that the chess puzzle and its solution are completely non-sensical, that a mate-in-one puzzle that stumps an ancient god for ages is ridiculous”13.

However, thematically it ‘rhymes’ with the narrative at-hand. The Doctor’s chess puzzle is a mirror of the real-life game happening right now at the secret military base, and is used by him to showcase the flaw in Fenric’s strategic outlook; he cannot fathom the possibility that the pawns might not kill each other at the first possible chance, the clear dominant strategy, or to actively choose to work against the premise of the game itself. The pawns then, represented by Bates and Vershinn, choose to work together in order to achieve the best outcome for themselves rather than as individuals. Cooperation over competition.

Then there’s the Ancient One. For most of the narrative, he14 is used as a game piece by Fenric, who belittles and barks orders at him, in order for him to reach his desired outcome of the chemical pollution of the entire world. But I mentioned earlier that we witness a game within a game and this allows the Doctor to redefine the game being played. He persuades the Ancient One to stop being a pawn in Fenric’s game, essentially exiting the chessboard, and instead becomes a player in the game, substituting into the Doctor’s place. This entraps Fenric once again in a game where he cannot foresee the winning move, and now he must face the consequences of mistreating his own game piece. And since the Ancient One by this point already believes that mutual cooperation between them is no longer possible, they are left only with the option to betray each other: mutually assured destruction. This is the flipside to Bates and Vershinn. The Curse of Fenric’s resolution presents us with both ‘winners’ and ‘losers’ of the Prisoners’ Dilemma. Of course, this reading assumes that we actually witnessed the end of Fenric, but the expanded universe may have other ideas.15

Margaret Thatcher once famously said, “There is no alternative.” But unfortunately for her, there is. So what is it? The alternative is that we witnessed just one of many iterations in the ongoing battle between the Doctor and Fenric. Much like how in Heaven Sent (2015) we initially see one iteration of the Doctor running about the castle, in fact. What then does such a game look like? Let’s dare to imagine that we can even comprehend such a thing.

Consider then that the Seventh Doctor and Fenric are playing the most elaborate and extraordinary game. One with an impossibly large number of options for each of them to choose from, and perhaps not limited to a mere two-dimensional display of outcomes but many, many more. And the potential pay-offs are not just points on a scoreboard but the lives of countless individuals, people like you or me, and the continued existence of our world. The whole of reality as we know then is at stake here. A ‘rather neat’ algorithm, as the Doctor put it, that started so very long ago and will continue from now until the end of time. Making decision after decision. Iteration after iteration. Game after game.

The end of history? Far from it.

“We play the contest again, Time Lord.”

 

Bibliography

  • Alex Through the Looking Glass by Alex Bellos
  • A Mathematician’s Apology by G. H. Hardy
  • Fermat’s Last Theorem by Simon Singh
  • The Black Archive #23: The Curse of Fenric by Una McCormack
  • The Simpsons and Their Mathematical Secrets by Simon Singh

All internet references have been highlighted throughout.

Footnotes

1 Remembrance of the Daleks (1988)

2 Hardy, G.H., A Mathematician’s Apology, p44

3 Hardy, G.H., A Mathematician’s Apology, p7

4 The Woman Who Fell To Earth (2018)

5 Bellos, Alex, Alex Through The Looking Glass, p261

6 Singh, Simon, The Simpsons and Their Mathematical Secrets, p99

7 Singh, Simon, Fermat’s Last Theorem, p167

8 McCormack, Una, The Black Archive #23: The Curse of Fenric, p41

9 The Curse of Fenric: Special Edition (2019)

10 For example, see “Belief in a Zero-Sum Game as a Social Axiom: A 37-Nation Study” and “Your gain is my loss”: An examination of zero-sum thinking with love in multi-partner romantic relationships and with grades in the university classroom.

11 I’ve quoted this as presented on the Wikipedia page on the Prisoner’s Dilemma. Accessed 3rd August 2020.

12 Hardy, G.H., A Mathematician’s Apology, p30

13 Sandifer, Elizabeth, Take Hitler and Put him in the Cupboard Over There (The Curse of Fenric)

14 The television story identifies the Ancient One as male with he/him pronouns but the novelisation tells us that the Ancient One is female and uses she/her pronouns. I do not agree with TARDIS Wiki insisting on referring to the Ancient One as “it”.

15 See Gods and Monsters by Alan Barnes and Mike Maddox.

 

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.

ClaraLeafAkhaten
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.

Picture2
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 https://en.wikipedia.org/wiki/Cardinality.

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?

Picture1
Pictured: The Borromean Rings. Image taken from https://www.ancient-symbols.com/symbols-directory/borromean-rings.html

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.

 

References

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.

 

Footnotes

1 Snow, Mark, ‘Doctor Who: “The Rings of Akhaten” Review, IGN, 6 April 2013, https://www.ign.com/articles/2013/04/07/doctor-who-the-rings-of-akhaten-review.

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, https://downtime2017.wordpress.com/2020/04/25/interview-william-shaw-the-leaf-and-rings-of-akhaten/.

5 All transcripts are taken from http://www.chakoteya.net/DoctorWho/33-8.htm 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.

Review: The Black Archive #40 – The Underwater Menace by James Cooray Smith

Key Facts:

  • Story No. 32. Written by Geoffrey Orme. Directed by Julia Smith.
  • Episodes 1 and 4 are missing. They are survived by episodes 2 and 3.
  • Key Themes: How does surviving material influence fan perception? Should we even take this story seriously? Is fan wisdom bollocks?
Screen Shot 2020-04-11 at 15.30.21
The Underwater Menace: I’d planned to include more fish puns in this review but later decided to scale back on them.

The Review

There’s no doubt about it – The Underwater Menace (1967) is one of the strangest televised adventures in the entirety of Doctor Who’s 50+ year history. Its setting is mythical, taking place in the lost underwater city of Atlantis. Its science is ropey, with absurd ideas of draining the ocean into the Earth’s core. And its villain is utterly preposterous, a near-contemporary scientist, whom the Doctor is already aware of by reputation I might add, whose ideas of ‘supreme power’ will actually leave no-one in Atlantis alive to see the repercussions of his actions. But it certainly does score highly on originality.

With all that in mind, why should a range like The Black Archive, which delivers thoughtful and serious critiques of any and all Doctor Who stories, dare to even take it seriously? The idea that recurring contributor (and former editor of the range) James Cooray Smith would decide to hang his hat entirely around this premise initially seems a rather silly one, yet it actually achieves remarkable results. Much like his last contribution to the range, which examined The Ultimate Foe (1986), Cooray Smith delivers a sublime blend of analysis, document-based research and behind-the-scenes history, with its 110-or-so pages just flying by.

Given the frivolous nature of the story’s ideas and plotting, it is perhaps unsurprising that each of the book’s nine chapters are fairly short ones (about ten pages each) but what brings it all together is the use of a few key over-arching themes, namely how does missing material affect fan reception, taking full advantage of the rediscovery of Episode 2 back in 2011, as well as how collective fan wisdom can at times be sorely misplaced, which leads not only to some superlative myth-busting but also a few finger wags at the fandom-at-large. This even extends to the author himself who slaps himself on the wrist in a delightfully cheeky footnote.

The questions covered are an eclectic mix that honestly speak for themselves. Just take a look below at some of the questions this book presents well-informed and dutifully researched answers for you:

  • Why does fandom universally hate a story they’ve probably never seen?
  • What exactly happened at the BFI in 2011 when Episode 2 showed up?
  • Why was this story filmed despite being formerly abandoned?
  • Why do fans keep writing badly accented versions of the episode’s third cliffhanger?
  • Whatever happened to the Doctor’s hat in this story?
  • How does this story ultimately shape Troughton’s portrayal of the Doctor?

For me, there were two undisputed highlights during this read and both are towards the very end of the book. First, there’s a substantial appendix based on the Doctor’s note to Professor Zaroff signed “Dr. W”, which looks at whether the main character of the show is called ‘Doctor Who’ and whether the character is referred to as ‘Doctor Who’ both within and outside the fiction itself. The appendix doesn’t so much as cover but utterly annihilate the discourse surrounding these related questions, and with considerable aplomb too. Although, I must say the complete omission of the opening scene from World Enough of Time (2017) is a little baffling. I also sincerely hope his “Dame Shirley Bassey’ argument catches on in general.

The second highlight was the book’s final chapter looking at the life, work and collaborators of its author, Geoffrey Orme. Little is known about the life of Orme as he was never interviewed about his work, not even by a single Doctor Who fan, and he died in 1978. The search into the archives detailed in this chapter in the hope to the reveal hidden depths about the story of The Underwater Menace is commendable and reveals subtle and astute observations. This chapter is the literary equivalent of an astounding new Toby Hadoke documentary, such as those which have looked into the previously shrouded lives of Peter R. Newman (Writer of 60s serial The Sensorites (1964)) and Lennie Mayne (Director of four 70s Who serials). It is truly an excellent capstone to the book itself.

under1
The Fish People: Far from being background characters, they actually rose up and seized the means of food production.

Concluding Thoughts

Despite perhaps being as utterly mad as Professor Zaroff’s plan, this book manages to be a resounding success. This entry takes full advantage of the story’s poor fan reception and partially missing status as an opportunity to re-examine the serial with fresh eyes. Covering a diverse range of topics surrounding its troubled production, obscure cult status, and its mysteriously disappearing hat, The Underwater Menace by James Cooray Smith comes highly recommended to those who want to discover whether it has hidden depths. But if it should happen that fan historiography isn’t to your literary tastes then don’t worry, because there’s plenty more Fish People in the sea.

The Maths of Doctor Who #3 – “We would have to consult our top scientists”

Doctor Who has always been recognised as science-fiction show and its earliest serials used time travel as a narrative device in order to tell stories set on either far-distant planets or in the long-distant past that not only aimed to entertain families between Grandstand and Juke Box Jury but also teach children about both science and history. But by 1966, this approach to the show’s production shifted significantly, abandoning history in favour of science. This coincided with the introduction of writer Kit Pedler and script editor Gerry Davis. For this blog entry, I want to have a look at how Pedler and Davis’ serials brought a surge in STEM representation in the show’s format, looking at how mathematics and, more broadly, science, is used in their storytelling. But first, a bit of background.

Christopher Magnus Howard “Kit” Pedler was born on 11 June 1927 and initially worked as a British medical scientist at the University of London, where he was head of the electron microscope department. His first contribution to British television was, perhaps unsurprisingly, Tomorrow’s World and would later go onto co-create and co-write Doomwatch (1970-72). However, he is arguably best known for his work on Doctor Who, for which he has three credited scripts (The Tenth Planet (1966), The Moonbase and The Tomb of the Cybermen (both 1967), provided initial ideas for three further stories (The War Machines (1966), The Wheel in Space (1968) and The Invasion (1969)) and generally acted as the show’s unofficial scientific advisor under Innes Lloyd’s tenure as producer, who wanted to inject more hard science into the show’s format.

Kit’s frequent collaborator was television writer Gerry Davis, who was Doctor Who’s script editor for over a year (running from episode 4 of The Massacre (1966) to episode 3 of The Evil of the Daleks (1967)) and so was part of the production crew that oversaw the transition from William Hartnell to Patrick Troughton. He too was a co-writer for The Tenth Planet and The Tomb of the Cybermen but also contributed The Highlanders (1966-67), which introduced long-serving companion Jamie McCrimmon, as well as Revenge of the Cybermen (1975) during Tom Baker’s first season, though this was heavily re-written by the then-script editor Robert Holmes. The original version, now entitled Return of the Cybermen, will be released as a Big Finish audio drama in November 2021. Together then, it seems we have a duo whose primary skills complement each other: Pedler having the cutting-edge scientific ideas that he wants to fashion into stories but lacking in television experience, whilst Davis has such experience writing TV soaps and drama but can use such scientific ideas to make socially and culturally relevant stories for BBC broadcast. But how did that translate into Doctor Who itself?

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Pictured: Doctor Who writers Kit Pedler (left) and Gerry Davis (right). There seem to few photos of them together, with this being the most common by far.

The War Machines and STEM Representation in Late-1960s Who

Kit Pedler’s first story idea for Doctor Who to get made became The War Machines, written by Ian Stuart Black, and at one stage had the working title of “The Computers”1. Not only is it one of the few complete serials from Season 3, it is also the only entirely complete serial to feature companions Ben and Polly, which also happens to be their debut story. The story is set in contemporary time, which is highly irregular for the show at this point, and the plot mainly revolves around a highly advanced supercomputer called WOTAN (which stands for Will Operating Thought ANalogue) who turns out to be surprisingly malevolent.

Episode 1 sees the Doctor meet WOTAN’s creator, Professor Brett, before later attending a meeting of the Royal Scientific Club, immediately presents scientists as members of the upper echelons of British society, hanging around with the likes of aristocrats such as Sir Charles Summer and security figureheads such as Major Green. It also establishes a link between science and the military that would be become a lot more prominent during the first few seasons of Jon Pertwee’s tenure2. You only have to look as far as Summer’s coining of the term C-Day for Computer Day, which naturally invokes thoughts of the D-Day landings during the Second World War, to further cement the link.

Another interesting and perhaps quite alarming statement from Sir Charles Summer is that WOTAN “is merely a brain which thinks logically without any political or private ends. It is pure thought. It makes calculations, it supplies only the truth.” The complete disassociation between science and logic from politics and ethics here is later demonstrated to be spectacularly misjudged when WOTAN starts hypnotising people to construct the eponymous War Machines and attempt to take control over London, unless one considers total conquest of the world to be an unquestionable truth about how life should be. As Commander Millington remarks about computers in 1989’s The Curse of Fenric, “Whose thoughts will they think?” With plans to link WOTAN to computers around the world, the story presents science not only as a subject that will shape the future of our society, but also one that will be highly influential in the unfolding geopolitical landscape, with Parliament, the Kremlin and the White House all name-checked in the background of Summer’s press conference.

WOTAN’s presence in the story not only brings science into the show but also mathematics. Later on in Episode One, we have the very first maths problem to feature in Doctor Who when the Doctor asks:

DOCTOR: Er, what is the square root of 17422?

(The machine whirrs, then prints a number on a piece of paper.)

BRETT: Correct?

DOCTOR: One moment please. 131 point 993. Yes, that’s near enough.3

I hope I’m not the first person to have actually checked this but if you type that question into your calculator you should get an answer of 131.992424, which to three decimal places would round to 131.992, not 131.993. With this in mind then, the Doctor’s comment of “near enough” lends an alternative reading of the scene. Rather than being in awe of WOTAN’s computational speed and accuracy, the Doctor is actually aware of the machine’s slight calculation error beyond the second decimal place and that he now realises Summer’s complete faith in its calculations are misplaced. It would also imply that the Doctor has superior mental maths skills to the villain.

The plot’s resolution involves the Doctor using his own scientific knowledge to reprogram a captured War Machine and then gives it new orders so that he can use it against WOTAN; the War Machine firing repeatedly at it until WOTAN is destroyed. Far from bringing world peace as was intended, it seems the technology of WOTAN is just yet another new tool that can be used in warfare and is also capable of destroying itself. Only when science and technology are used, not in isolation as intended by Summer, but in conjunction with the Doctor’s ethics can they be used to prevent the invasion of London and so subsequently bring peace.

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Pictured: Sir Charles Summer (left) and Dr. Who (right) arguing whether or not the show was political back in 1966. The apolitical War Machine is required.

STEM Representation After The War Machines

Far from being an outlier in Doctor Who’s cavalcade of serials during the 1960s, The War Machines presents a decisive shift in the characters and themes used in the show’s storytelling. Judging by the progression of serials under Innes Lloyd as producer, he seems to have declared that historical ones are now out and scientific ones are firmly in. Scientists would now feature as key characters in the majority of stories. Starting from Hartnell’s swansong, The Tenth Planet, we would get the introduction of the Cybermen, who would become Doctor Who’s second villain after the Daleks, as well as the trope of teams of scientists being in space stations or other remote locations, extending all the way until the end of the 1960s.

Communities of scientists working together and facing some form of mechanical menace feature in every story where Kit Pedler is credited, clearly showing it is a staple of his storytelling within the show. Pedler would also revisit the ideas of international communications (and magnetic forces) from The War Machines in his final story contribution to the show, The Invasion, which features the remarkable rise of International Electromatics (or International Electromatix if you’re reading the novelisation) and it even features a Head of Security figure called Packer, again linking science with national security.

As for mathematics, Episode One of The Moonbase is the first Doctor Who episode to feature mathematicians as named supporting characters when Hobson says “Nils, our mad Dane, is an astronomer and mathematician as is Charlie here.” The show would also go on to get its first mathematical companion in the form of Zoe Heriot, who introduces herself in The Wheel in Space by saying “I’m an astrophysicist. Pure mathematics major.” However, the juxtaposition of these two sentences is quite striking to a mathematician like myself. The two areas mentioned here could not be further apart. Astrophysics is a subdomain of physics that involves the study of planetary bodies and would involve substantial applied mathematics such as the mechanics of celestial bodies, whereas pure mathematics is generally used to describe the study of mathematics devoid of any context or application, including areas such as geometry, analysis and number theory. Perhaps GCHQ aren’t hiring anyone in 2079?

Whilst the use of scientific ideas in the stories by both Pedler and Davis can be at times wishy-washy and at worst just plain inaccurate, their consistent and topically relevant inclusion in the show’s format was arguably a good thing. It provided the show one of its most iconic villains in the form of the Cybermen as well as several memorable stories that viewers and fans have enjoyed over the years. All but one of Kit Pedler’s six contributions are available to buy on DVD either because they are fully intact or have been completed with animation, and I suspect The Wheel in Space is not far off being animated itself – though that’s just a personal hunch. However, I do have one bone to pick with Pedler and Davis, and it really is a rather petty one, but there’s a certain scene in one of their stories that I find just absolutely atrocious. I doubt most people will find it as annoying as I do, but there’s no harm in trying to explain why. So, let’s talk about Episode One of The Tomb of the Cybermen.

The Tomb of the Cybermen and Appropriating Mathematics as Technobabble

Thought to be yet another long-lost Troughton classic, the serial The Tomb of the Cybermen (1967) was recovered from a TV station in Hong Kong in 1991, and then quickly released by BBC Enterprises on VHS in 1992. With a gap of almost twenty-five years between its initial broadcast and initial commercial release, Tomb has now been available for fans to view longer than it had been lost4. It has received wide acclaim from the fandom, with some proclaiming it as “quite simply, the best [Cyberman] story”5 and “one hundred minutes of sheer magic”6. It was even the first Troughton-era serial to be released on DVD back in 2002, later getting its own Special Edition in 2012. Simply put, it is a highly-regarded serial among fans, coming in at number 23 in a 2014 DWM Poll7, that perhaps, I might dare to suggest, are being a bit too generous towards it. But I digress…

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Pictured: Troughton (centre) may not know what he’s talking about but he certainly knows how to look smart with just a notepad and pen.

For those who need a quick reminder, the opening of Tomb sees the Doctor and co. arriving on the planet Telos at the same time as an archeological expedition. They have discovered an ice tomb which they believe contains the last remains of the Cybermen and, somewhat concerningly, the Doctor keeps drip-feeding them the answers to all the puzzles and traps set for anyone who tries to enter. One such person he assists is Eric Klieg, who delivers this quite remarkable line of dialogue:

KLIEG: But take this mathematical sequence, for example. I’m really no nearer to its solution. I’ve tried every possible combination. You’d hardly call that easy!

At this point in the story, it has already been established that Klieg has helped financed the expedition (so he’s probably very well-off) and we later learn he is a member of the Brotherhood of Logicians, though we never learn what this actually means beyond having sympathies towards the Cybermen. However, the aforementioned line of dialogue suggest quite positively that Klieg is no expert in mathematical logic. In fact, he seems to have a flimsy grasp of the basics of maths itself.

A sequence can be defined as a set of numbers that follow each other in a logical pattern: all we need is a starting point (or first term) and a pattern (or term-to-term rule). Arithmetic sequences involve adding the same number each time and we can use this to generate our times tables (For example, the three times table is 3, 6, 9, 12, 15, 18, 21, 24 …). Geometric sequences involve multiplying by the same number instead, and this can generate the powers of two for example (1, 2, 4, 8, 16, 32, 64, 128…). Other sequences are more playful, such as the Fibonacci numbers, where you get the next number by adding the previous two (1, 1, 2, 3, 5, 8, 13, 21, 34…) or one of my personal favourites, “say-what-you-see” sequences, where the next number is a numerical description of the previous number (1, 11, 21, 1211, 111221, 312211…)8.

We can see that sequences have starting points and rules, but they don’t have solutions, unless perhaps you’re trying to get the next number? But then if you don’t know the next number, how do you know you have a sequence? Furthermore, what are these combinations that Klieg is using to try and solve it? Combinatorics is the area of mathematics that looks at possible outcomes or combinations of events, such as shuffling a pack of cards or rolling a set of dice. It’s certainly not the sort of maths I would use to solve an unknown sequence. I can only begin to understand what he means by his bold claim of trying every possible option by thinking he must be highly incompetent. And to be fair, that’s probably what he’s supposed to be. The arrogant fool who overestimates his intellectual abilities, and requires a much smarter character to help him, who somehow thinks he can become the new leader of the Cybermen. So maybe the writers had intended this all along then… except then the Doctor opens his mouth:

DOCTOR: You see, if you take any progressive series it can be converted into binary notation. If you take the sum of the integrants, and express the result as a power series, then the indices show the basic binary blocks. Only I wouldn’t do it if I were  you. Oh no, I really wouldn’t do it!

If you listen to the DVD commentary of this scene, you will hear Frazer Hines talking about how terrible he was at maths and I’m not surprised as this is what is must sound like to those who don’t understand. Never mind the remarkable logical leap that expects you to convert your numerical sequence into binary numbers (unless we are to believe that Cybermen think entirely like simple computers?), what firmly put this into the realm of nonsense is the word ‘integrants’ – there are no such things in mathematics. You can have integrals, integrands and integration, but not ‘integrants’. However, integrant is a word in the OED relating to something that is integral. This then I would conjecture is a classic case of Patrick Troughton learning an approximation of his lines, rather than what was on the script. Or maybe he did just misremember? Perhaps it was even a typo? We can’t know for sure, but it does seem to fit a wider observation about Troughton’s overall performance.

Even if we substitute it with a near-sounding replacement like integrals or integrands, it doesn’t help elicit any understanding. Why would you consider taking the terms of a sequence and turn them into a sum of integrals or integrands? I should probably clarify these terms first. An integral is an equation that invokes the process of integration, in much the same way that a sum invokes the process of addition, so it’s basically a fancy sum. Integrands are the functions that you are wanting to integrate, like how in a sum you have numbers you want to add. As for integration, well Klieg starts blabbing on about it some more straight after the Doctor’s not-so-subtle hints:

KLIEG: Look! Sum between limits of one and nine one integral into power series. Yes! Yes! Then you differentiate…

At this point, Klieg seems to be your stereotypical mad scientists having some eureka moment, but my final curious observation here is that he has an integral and then… differentiates? This makes little sense since. Differentiation is the process used to find rates of change of mathematical functions, usually the gradients of curves, whilst integration allows you to find the area under the curve. The Fundamental Theorem of Calculus shows us that differentiation and integration are the inverse processes of each other, meaning if you were to integrate a function and then differentiate it you will get back to where you started. If Klieg manages to solve anything here, I haven’t got the faintest idea how.

Whilst one might consider commending the use of mathematics within a popular children’s TV show, for those who understand the language being used may be frustrated by the lack of any coherent logic to it. And as for those who don’t, like Frazer Hines’ comment stated earlier for example, it brings back school day memories of mathematical anxiety, where people remember have frustrated and confused feelings about not understanding what is happening in the lessons. I would therefore be inclined to draw the conclusion that such representation does more to hinder than to help the subject’s image. I have already highlighted some instances where Pedler and Davis’ representation is much more, shall we say, integrated into the stories they tell, but this scene falls below the mark in my opinion.

But this is just the start for the show’s relationship with mathematics, and more broadly science. Doctor Who will go on to have far more sophisticated representations of mathematics in stories like Castrovalva (1982) by Christopher H. Bidmead, which employs recursion and Escher’s art, Flatline (2014) by Jamie Mathieson, which sees creatures transcending between the second and third dimensions, and Extremis (2017) by Steven Moffat, which employs the not-so random nature of random number generators to help deliver a key plot revelation in that particular story. But I think it’s somewhat fair to say that it all got started back in 1966 when Pedler and Davis wanted to bring their interests and ideas into the stories of Doctor Who. Just so long as you don’t start peer-reviewing their work.

Footnotes

  1. Source: http://www.shannonsullivan.com/drwho/serials/bb.html
  2. For more on this, I would recommend Robert Smith?’s Black Archive on The Silurians (1970) which investigates further the link between science and the military.
  3. All quotes are taken from the transcripts provided on http://www.chakoteya.net/ with a few minor spelling and grammar edits by myself.
  4. Source: https://www.herocollector.com/Article/doctor-who-tomb-recovered
  5. Martin Day in Cloister Bell 10/11, dated March 1985
  6. Jeff Stone in TSV 29, dated July 1992.
  7. http://doctoroo.blogspot.com/2014/06/from-doctor-who-magazines-first-50.html
  8. If you haven’t quite understood this then here’s a longer explanation. The first number is 1, which can be described as one one, so the next number is 11. This can be described as two ones, so the next number is 21. This can then be described as one two and one one so the next number is 1211… and so on. Wikipedia calls them Look-and-say sequences but they are the exact same thing!

Ten Women in STEM I’d like to see in the next series of Doctor Who

If you’re reading this around the time of publication, then you’ll surely be aware that the latest series of Doctor Who has just finished, and now the long wait for a new special or series begins again. If you also happen to be reading this on the day of publication itself, then today, March 8th, is International Women’s Day, a day for celebrating the social, economic, cultural and political achievements of women around the world.

A personal highlight for me in Series 12 was the appearance of Ada Lovelace in Spyfall: Part Two, marking not only a rare positive portrayal of a mathematical figure within the show (certainly in comparison to the likes of Adric, the Sylvest twins and the Logopolitans), but also the first female historical STEM figure (that’s Science, Technology, Engineering and Mathematics) to feature in the TV show. Interestingly, Lovelace has already made an appearance within a Big Finish audio play, The Enchantress of Numbers (which I wholeheartedly recommend to you), alongside Tom Baker’s Fourth Doctor.

Alongside the other appearances of Nikola Tesla and Thomas Edison, it’s been a good year for STEM figures in the show and I hope Chibnall will continue this trend during his time as showrunner and executive producer. Should he happen to stumble upon this blog post, then perhaps he can use this as a starting point for finding other female STEM figures to include in a future episode of Doctor Who. Here are ten possible suggestions:

Marie Curie (1867-1934)

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Born as Maria Sklodawska, Curie was the first woman to win a Nobel prize and the only woman to have won two Nobel prizes to date (which were Physics in 1903 and Chemistry in 1911). A Polish chemist and physicist, she conducted pioneering work into the subject of radiation and radioactivity alongside her husband Pierre Curie, who was a French physicist. Curie also named two radioactive elements that she discovered: radium and polonium (which is named after her home country of Poland). She is undoubtably one of the most well-known female scientists in history. Perhaps we could have an episode where the Doctor is facing some sort of radioactive extra-terrestrial and she needs some expertise to help save the Earth?

Grace Hopper (1906-1992)

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Why stop at Ada Lovelace when you can have another outstanding computer programmer? Born Grace Brewster Murray in 1906, Hopper (nicknamed “Amazing Grace”) invented the first compiler for a computer programming language as well as popularising the idea of machine-independent computer languages (“They told me computers could only do arithmetic.”). She has a PhD in mathematics from Yale, served as an Admiral in the US Navy and helped to popularise the term ‘debugging’ after removing a moth that had got stuck in a Mark II computer running at Harvard. Maybe she could help the Doctor out if she ever has a rematch with Daniel Barton, the CEO of VOR, a company more powerful than most nations, in the Series 12 opener Spyfall.

Rachel Carson (1907-1967)

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Carson was an American biologist, conservationist and author of a trilogy books based around marine life. Her incredible work helped raised awareness of serious environmental problems, such as those caused by artificial pesticides, and inspired the global conservation movement. This eventually led to the creation of the Environmental Protection Agency (EPA) in America and President Jimmy Carter posthumously awarded her the Presidential Medal of Freedom. An appearance from her would not only continue post-2005 Who’s love of visiting historical authors, but also follow up the environmental themes seen in episodes like Orphan 55 and Praxeus.

Rosalind Franklin (1920-1958)

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A prominent British Jewish chemist, Franklin is best known for her work on X-ray crystallography that has helped us understand and identify the structures of deoxyribonucleic acid (or DNA). Tragically, her contributions to science were only truly recognised after her premature death at age 37, caused by ovarian cancer. Her work led to the discovery of the double helix structure we all associate with DNA now but   she was sadly not recognised for this contribution by the Nobel Committee. An episode featuring her would certainly help draw more attention to her scientific contributions.

Maryam Mirzakhani (1977-2017)

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Mirzakhani was the first Iranian and the first woman to be awarded the Fields Medal, the most prestigious prize within the field of mathematics (it is awarded once every four years and only to people under 40). Her specialist region of research was on the symmetry of curved surfaces, an area of maths that blends dynamics with geometry. Tragically, she died aged 40 from breast cancer but her legacy has been profound: her birthday May 12th is now recognised as International Women In Mathematics Day, her international fame has significantly progressed the societal representation of Iranian mothers in her home country, and students at the University of Oxford set up a society in her name for women and non-binary students who study mathematics there. An ideal episode would not only recognise her excellent mathematical ability but also the social and cultural progress that she has inspired.

Caroline Herschel (1750 – 1848)

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Herschel is most notable for being the first woman ever to discover a comet. A German astronomer, she worked alongside her brother William and took meticulous notes and records of her observations. She was also the first woman to be salaried as a scientist and the first woman in England to hold a position in government. Doctor Who has rarely seen older female scientists represented (such as Professor Rumford in The Stones of Blood) but Herschel would most definitely be a great choice to include in the TV show.

Florence Nightingale (1820-1910)

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Arguably one of the more well-known figures on this list, Nightingale is perhaps best regarded as a social reformer and founder of modern nursing, becoming an icon within Victorian culture known as “The Lady of the Lamp”. But what is perhaps less celebrated is her aptitude for mathematics. She was a trailblazer in the art of data visualisation (indeed, she popularised the use of the pie chart) and used graphical representations of data to convince others of her observations. Her meticulous and comprehensive study of sanitation of hospitals during the Crimean War enabled her to effectively lobby the British Government for improved sanitation through the Public Health Acts of 1874-75. It is perhaps one of the earliest instances of evidence-based policy in the history of British Government. She truly was a badass statistician.

Rosalyn Sussman Yalow (1921-2011)

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Yalow was the second woman to be awarded a Nobel Prize and the first in Medicine, for which she developed the technique of radioimmunoassay, which allows scientists to trace substances in the bloodstream. Despite its commercial potential, Yalow refused to patent the method. This kind of scientific innovation could certainly work in a story about alien infection, and would most definitely bring her work to the attention of a much wider audience. 

Jocelyn Bell Burnell (1943-)

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The first person to discover (the first four) pulsars, highly magnetized rotating neutron stars that emit regular pulses, Bell Burnell was denied the honour of a Nobel Prize for Physics years later on the grounds that she was a postgraduate student (She even went on record to say she believed such a recognition would demean the prize itself). Her discovery is regarded as “one of the most significant scientific achievements of the 20th Century” and she is the only person on this list who is still alive. She has since held distinguished positions such as president of the Royal Astronomical Society and the Institute of Physics.

Marie-Sophie Germain (1776-1831)

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Germain was a French mathematician, physicist and philosopher who persevered through considerable opposition (including her parents, male mathematical peers and just the patriarchal nature of society in general) to become one of the most influential polymaths of recent centuries. She would smuggle candles to her bedroom to allow herself to study through the night, and submitted academic work under the nom de plume Monsieur Le Blanc so that her male mathematical peers would take her seriously, yet despite recognition from distinguished figures like Lagrange and Gauss she was sadly unable to make a true career out of mathematics. She was a true pioneer in the fields of elasticity theory and number theory, and the French Academy of Sciences now has an annual mathematics prize named in her honour. Her final works, which were published posthumously, were on philosophy and she passionately argued that there were no differences between the humanities and the sciences. Put simply, she was brilliant.