If you’re reading this on the day of publication then it is Friday 11^th April 2025, the day before the start of the new season of Doctor Who, which we will call Day Zero. To be more precise, it’s Day Zero of Season Two (2025) of Doctor Who, or it’s Day Zero of Series 15 if you’re ignoring the break in numbering since the Disney co-funding partnership in 2023, or it’s Day Zero of Season 41 if you’re also ignoring the break in numbering since the show’s 21^st century revival in 2005 (respect!). This is an example of how counting systems used by Doctor Who fans can vary over time.

Such alternative counting systems can also vary by (geographical) space. Anyone in the UK who has purchased the latest entry in the stunning Doctor Who: The Collection Blu-ray boxsets will have on their shelf right now a copy of Season 7 (1970), the first season starring Jon Pertwee as the Third Doctor. But if you live in the United States of America and bought the exact same boxset (without importing it), then it’s not called Season 7; it’s now called ‘Jon Pertwee Complete Season One’. What about Season 25 (1988), the boxset released immediately prior? That’s now ‘Sylvester McCoy Complete Season Two’. And what about Season 15 (1977-78) before that? Well that’s ‘Tom Baker Complete Season Four’. It’s a whole new numbering system!

But wait, there’s more!

If you start looking into expanded universe releases and fan wikis, you can find yet more counting systems. Want to have a story with the Second Doctor and Jamie set after the end of The War Games (1969) but before Spearhead from Space (1970)? That’s Season 6B. What about a story with the Fifth Doctor and Nyssa, set after Time-Flight (1982) but before Arc of Infinity (1983)? Well that’s arguably Season 19.5. How about a story with the Eleventh Doctor and Big Finish-exclusive companion Valerie Lockwood, set in the gap between The Snowmen (2012) and The Bells of Saint John (2013)? That’s been dubbed Series 7V. And as for stories with The Fugitive Doctor, who chronologically existed before An Unearthly Child (1963), the very first Doctor Who story? Is that Season Zero? Can we have “negative” seasons of Doctor Who??

Regardless of what the Doctor Who Marketing and PR teams actually tell us, Doctor Who fans will always have their own counting systems. But for a show that has had such longevity, not to mention the sprawling canonicity baggage that entails, it was perhaps always inevitable that Doctor Who would need to start counting again.

Christopher H. Bidmead is a writer that taps into this notion through the use of the number zero in his serial Castrovalva (1982), the first TV adventure of the Peter Davison’s Fifth Doctor. Bidmead lacked clear guidance from the production team on how to characterise the new Doctor and so found himself in the position where he essentially had to write something out of nothing, an M.C. Escher-esque existential paradox (e.g. Drawing Hands, 1948) if there ever was one. And the number zero can be considered its own form of existential paradox.

I mean, think about it.

Zero is a number that represents nothing. But the symbol 0, which means the number zero, is something. It’s there, look, on the page – I can see it right there! The Romans thought there was no need for a number zero given that there’s no Roman numeral for zero; they just left a blank space. And quite right too, if there’s nothing to count then why show any evidence of counting? But then counting changed. Single-value numeral systems, like Roman numerals were out, and place-value systems, like our decimal system today, were in. The number zero is required.

One person credited with the formalisation of the number zero is the Indian mathematician and astronomer Brahmagupta (c. 598 – c. 668 CE). He called this number “shunya” or “sunya”, which is the Sanskrit word for emptiness or nothingness. The idea of one having a “nothing” or void inside of yourself is a philosophical concept taught within the Hindu and Buddhist religions, and Bidmead himself was interested in exploring Buddhist ideas within Doctor Who, as evidence by the inclusion of the I Ching in Warrior’s Gate (1981) and the initial commissioning of Kinda (1982) by playwright Christopher Bailey.

So firstly, the number zero is something that also represents nothing.

Looking at Castrovalva, you can quite clearly see that Bidmead really runs with the idea of “something out of nothing”. We are introduced to the Zero Room within the TARDIS, a room that’s entirely devoid of anything. It exists to help the Doctor stabilise in his new persona through meditation. Then there’s the appearance of the Big Bang aka Event One at the first episode’s cliffhanger. The Big Bang is a scientific theory that the entire universe came into existence through a massive intense explosion of energy. One moment there was nothing, and then there was something, created from nothing. (I mean this in a more visual sense rather than strictly physical sense, otherwise some physicists might get really angry at me.)

This idea even extends to the Fifth Doctor himself. Yes, he physically exists in the form of actor Peter Davison from the very start of the story, but in a more spiritual sense the character of the Fifth Doctor that we would come to recognise is not there yet. Something, and yet also nothing. He spends the entire story trying to find himself, and in a real-world production sense, actor Peter Davison is also trying to find himself within the role. It’s a bit like when the Eleventh Doctor was asked by a young Amelia Pond who he was and he replied “I don’t know yet. I’m still cooking.”

Secondly, the number zero also represents a new start point, a new beginning if you like.

This fits nicely with Castrovalva as it acts as a new starting point for viewers. It is the first story of the Fifth Doctor, the first story in Season 19, which first aired from January to March 1982, and the first full adventure together for the companions Nyssa, Tegan and Adric, having first met each other partway through the preceding serial Logopolis (1981). Moreover, every post-regeneration story, from The Power of the Daleks (1966) to The Woman Who Fell to Earth (2018), acts as its own form of start point within Doctor Who, because the lead actor has changed and this in turn can lead to changes in the writing style, themes and direction taken.

Some might argue that symbolically that the number one is a better numerical representation of a new start point, given its traditional status as the first counting number. That’s pretty fair, actually. But within the academic field of mathematics, we have a formal name for these counting numbers: the natural numbers. The number zero is just as valid as the number one as being a member of these so-called natural numbers. So you can start counting from zero if you like.

Castrovalva is not only interested in the number zero, but also the concept of recursion.

At one point, Tegan mentions that her dad once said that ‘if’ was the most powerful word in the English language, which leads Nyssa to reply that recursion is a powerful mathematical concept. The concept of recursion was first formally defined by Richard Dedekind (1831 – 1916) in 1887. Within the history of mathematics, this rather neatly places it after the initial groundwork on computational mathematics done by Ada Lovelace (1815 – 1852) and Charles Babbage (1791 – 1871), who both feature in Spyfall: Part Two (2020), but before the pioneering work of mathematician and WWII code-breaker Alan Turing (1912 – 1954) who would go on to define computers in the modern sense (i.e. Turing machines). But Dedekind’s definition of ‘recursion’ is itself recursive. This is reflected in Part One of Castrovalva when Tegan says “if we had an index file, we could look it up in the index file, under index file.”

Dedekind’s definition of recursion refers back to a much older and more well-established idea within mathematics: induction. Induction is a highly powerful method of proof within mathematics. The most famous early example of induction is probably Euclid’s (c. 300 BC) proof that exist infinitely many prime numbers. The sequence of prime numbers is therefore a mathematical structure that goes on forever and ever, without end; much like the TV show Doctor Who. Recursion then is essentially induction but with a new lick of paint – “Same software, different casing.”

Proof by induction is a process that consist of two simple stages: proving the base case and proving the induction step.

Firstly we have the base case, which establishes a starting point for our proof. Algebraically this is typically noted as proving that the case n=1 exits. To provide a Doctor Who-themed analogy to this explanation, we need to show that the First Doctor, as portrayed by William Hartnell, exists. And despite the considerable efforts of the son of the writer Anthony Coburn to remove An Unearthly Child from existence within the Western world, I still have the DVD on my shelf. Therefore, we know that the First Doctor DOES exist.

Then we have the induction step, which establishes how we can if we know that one case exists then the next case along must also exist. Algebraically, this is typically noted as if the case n=k is true then the case n=k+1 must also be true. So if the case n=1 is true, then the case n=2 must be true, which means the case n=3 must also be true, and so on. To continue our Doctor Who-themed analogy to this explanation, we need to show that the regeneration process exists, which allows us to get from the First Doctor to the Second Doctor. And despite the considerable efforts of the BBC Archives to remove Episode 4 of The Tenth Planet (1966) from existence, I still have the first regeneration sequence on a DVD on my shelf, largely thanks to Blue Peter. Therefore, we know that the regeneration process DOES exist. Hence, we can transition from the First Doctor to the Second Doctor, to the Third Doctor, to the Fourth Doctor, to the Fifth Doctor, and so on…

Later on in Castrovalva, we can see this induction process illustrated by the Doctor’s inability to count from two to three whilst trapped within the walls of Castrovalva. Then a young girl helps him out, telling him that three comes after two, and so she helps sets the induction process on its way.

Lastly, the number zero also represents a point of transition.

Imagine a number line. Put the number zero at the centre of this line. As you move along the line from the centre to the right, you have all of the positive numbers – one, two, three, and so on. But equally as you move along the line from the centre to the left, you have all of the negative numbers – minus one, minus two, minus three, and so. Zero is neither positive or negative, it sits in its own little No Man’s Land between the two, acting as a gateway or portal between the two realms of number.

Bidmead has already explored this idea through the introduction of N-Space and E-Space in the season prior, Season 18 (1980-81). Warrior’s Gate is set in the void between E-Space and N-Space, acting as that very portal between the two – “Zero coordinates… Our N-space is positive and your E-space is negative.” The duology of Logopolis and Castrovalva, both penned by Bidmead, narratively act as this void space, transitioning between the end of the Fourth Doctor and the beginning of the Fifth Doctor.

Fan critics have on numerous occasions chosen to view Castrovalva as two ‘two-part’ acts rather than as one ‘whole’ four-part story – for example, this piece written by James Cooray Smith. There are a couple of key reasons for this. Firstly, within the narrative there appear to be two clear halves in terms of story location – the TARDIS predominantly features in Parts One and Two and the city of Castrovalva predominantly features in Parts Three and Four. Secondly, outside the narrative the episodes were originally broadcast twice weekly, meaning the real-time gap between Parts Two and Three was nearly a week whereas the real-time gap between Parts One and Two, and Parts Three and Four, was just a single day.

I choose to see things differently.

To me it makes far more sense to view Castrovalva not as two halves, but as one continuous journey, transitioning from the barely formed Fifth Doctor at the start to the more fully realised Fifth Doctor at the end. In Part One, we begin at the Pharos Project and journey into the TARDIS. In Part Two, we journey out of the TARDIS onto the nameless planet. In Part Three, we journey from the nameless planet into the walls of Castrovalva. Finally in Part Four, we journey out of Castrovalva and back to the TARDIS. The Doctor is constantly travelling throughout this story, constantly soul-searching. Castrovalva then is a journey of self-actualisation for the Fifth Doctor, a transition into his new identity. Out of nothing comes something. The number zero (0) brings us full circle.

Tomorrow on Doctor Who, a brand new series begins, with a brand new companion in the guise of Belinda Chandra. Someone will see this as a brand new starting point, someone else will see it as a continuation from the season prior, someone else will see it as the continuation of something that started several years, or even decades, ago. One person may even see it as something that is worth nothing. But if my reflections on Castrovalva and the number zero have taught me just one thing, it’s that maybe we fans should more often be willing to let go of seasons gone by and learn to love counting from the start once again.

Next Month: (in tune) I work at Bomb Disposal near the Pentagon!