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?

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.

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…

Screen Shot 2020-04-05 at 17.34.52
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.


  1. Source:
  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 with a few minor spelling and grammar edits by myself.
  4. Source:
  5. Martin Day in Cloister Bell 10/11, dated March 1985
  6. Jeff Stone in TSV 29, dated July 1992.
  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!

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