Aardvarks and Axioms: The Mathematics of Time in “Arthur”
ZACH BUSCH

I. Introduction

Arthur Logo (Source: Wikipedia)

In the 1996 Arthur episode “Buster Makes the Grade,” Arthur’s best friend Buster faces the daunting possibility of getting held back in school — if he doesn’t get his grades up, he risks failing third grade and having to repeat it.

Arthur 1x07b Buster Makes the Grade | Source: Ms Arthur TV/YouTube

This is a fear that children everywhere can identify with. Getting held back in school for any reason can be embarrassing or shameful. Moreover, children in grade school are at a point in their lives where every year brings enormous growth and change, making the possibility of getting left behind by one’s friends especially alienating.

In the end, of course, Buster perseveres. He works hard and builds better study habits with the help of his friends and ultimately passes the big exam. He will get to move on to the fourth grade next year, after all.

Except he won’t. He doesn’t know it yet, but Buster will repeat third grade many times. He will keep returning, year after year, to the same classroom, the same teacher, and the same level of education.

More than twenty seasons later, Buster Baxter is still in third grade. But don’t worry, he wasn’t left behind by his friends; they are all going to repeat third grade again, and again, and again, apparently without ever realizing it. And though years pass and friendships and relationships change, Arthur Read is still eight years old.

 … Or is he?

How is it possible that these characters are having formative experiences associated with getting older, while never getting older? More importantly, what is the nature of Arthur’s reality, and how do the processes of time, aging, and maturing work in his world? Finally, when — if ever — will Arthur turn 9? And when will he reach the fourth grade?

No matter which way you slice it, there is no way in which the events of Arthur  are contained within a single year.

II. Foundations

What does it mean to age?

We tend to think of aging as a process of change — usually physical, though sometimes mental, change that occurs as humans become older. We use terms like “aging well/poorly” or “aging gracefully” or even “ageless” to describe the effects of this process in relation to the passage of time. And when we use “aging” to describe food or drink, we are talking about a process that changes the characteristics of the product over time; a whiskey “ages” in the barrel, but it doesn’t “age” in the bottle.

Human aging is also largely connected to experiences. The difference between a twenty-year-old and a thirty-year-old isn’t just physical; it’s also a full 10 years of lived experience that one has but the other does not. In our society, age brings authority and clout. And when we use terms like “coming of age,” we are talking about events, experiences, lessons, rituals, etc. that define someone’s reaching a new level of maturity, independence, or adulthood; it is the experience, not the time itself, that makes that person come of age.

Perhaps the most universal definition of aging, though, is simply the question “How old are you?” For humans, someone’s numerical age (we will refer to this as nominal age) can put physical, mental, and emotional factors in context, along with lived experiences and accomplishments.

So when we talk about Arthur’s age, we need to bear multiple factors in mind: Arthur’s experiences, Arthur’s physical, mental, and emotional evolution, and the amount of time for which Arthur has existed.

By any reasonable metric, years and years are passing in the show — again, more than one year passed in the first season — yet no one is getting any older or moving up in terms of seniority.

The passage of time

The human conception of time is complex. Although we often see things like age, time, and maturity as intertwined, we can’t make that assumption when it comes to Arthur. We must therefore separate different concepts about time from one another.

  • Time will mean any metric for the causal sequence of events from first to last.
  • A year will mean the amount of time it takes for Arthur’s planet to complete a revolution of its sun or suns.
  • Aging will mean the process of changing as a result of passing through time.
  • Physical aging will mean the process of change that the body experiences as time passes.
  • Age will mean the amount of time that a person has existed continuously on a given timeline.
  • Nominal age will mean the number ascribed to the amount of time that a person has existed. Note that this is a nominal concept and does not necessarily align with how much time has passed since the person came into existence — ex. Arthur’s nominal age is 8, but that does not necessarily mean that Arthur has existed for 8 years.
  • Nominal aging will mean progressing through time such that others and/or society at large consider one’s nominal age to have increased.
  • Timeline will mean any unbroken forward-moving period in which time passes. We are not assuming that all events take place on one timeline.

Premises and conditions

  1. We will not be discussing Arthur and his friends as characters in a TV show, but rather as people and events being depicted by a TV show. In other words, the show Arthur will be viewed as an accurate, if incomplete, depiction of events occurring in the lives of the characters, and as textual evidence.
  2. The nature of Arthur as a TV show is not a valid justification of why things in Arthur happen the way they happen.
  3. Although the characters appear to represent different animals, we will consider them for all intents and purposes to be the same species with equivalent abilities and mental faculties.
  4. We cannot necessarily assume that the characters are mortal or immortal.

III. What We Know

Arthur  takes place over more than one year

No matter which way you slice it, there is no way in which the events of Arthur are contained within a single year. In fact, there are more episode segments than there are days in a year, and most of those segments take place over multiple days.

We also know that it’s not one continuous school year. Bizarrely, even within the first season alone, the school year starts and ends, after which the characters have summer break. Then, in the fall, they start a new year of third grade with Mr. Ratburn as their teacher. Between the changing of the seasons demonstrating the revolution of Arthur’s planet around the sun(s) and the evident cycle of the school year itself, it is not possible that this is just one year.

Additionally, Buster taking a test to progress to the fourth grade is only coherent if the staff expect fourth grade to be right around the corner.

Age exists

The characters in Arthur are not ageless; each one is at a defined age and stage in life. And from what we see of Elwood City, people appear to be in every conceivable stage of life, from infancy to old age. As far as we can tell, these people age in much the same way we do. Children are smaller and less physically developed than their older counterparts. Older adults also show signs of physical and mental aging beyond younger adults.

We also know that age seems to come with greater experience, and this gives characters authority and clout, with adults in charge of society at large and older adults in positions of seniority. Mr. Haney, the principal, is older than Mr. Ratburn, and has greater authority. Mr. Ratburn, in turn, has authority over Arthur and his friends, who are children and students. Even Arthur, as an older brother, has a degree of seniority over D.W.

In our society, age brings authority and clout.

The past exists

The characters in Arthur experience frequent flashbacks to events that came to shape them, individually and as friends. This includes flashbacks to things that happened before third grade, before the age of 8. We know that things that happen to the characters at earlier stages of life affect them today; there is a causal connection between then and now.

We also know that history happened, and that Arthur’s world was shaped by historical figures, ancient civilizations, and even extinct species. There is a sense in which these things happened “before” the events of the show.

The characters expect  to age

When Buster faces having to repeat third grade, he is mortified at the prospect of being left behind. For all we know, third grade could repeat infinitely many times — but no one notices.

This is incredibly significant. It means that the characters are not aware of the conditions of their own existence. If they truly are moving through time without aging, then they do not know it. The children are in school for the same reasons that our children are: to acquire knowledge, build skills, and prepare for adulthood and an eventual career.

By any reasonable metric, years and years are passing in the show — again, more than one year passed in the first season — yet no one is getting any older or moving up in terms of seniority.

This is not Groundhog Day

When I first started investigating the Arthur timeline, I wondered if they might be caught in a sort of “time loop,” in which they repeat the same year of third grade, albeit with different results and occurrences happening each year.

However, this is easily disprovable; there is clear continuity between seasons. For example, D.W.’s Snowball continues to come up many seasons after its original appearance in Season 1. Elwood City is also not stagnant; new kids move in and others move out. Character development that occurs early on (ex. Binky’s evolution from a bully into a more loving, sensitive, and artistic character) is not undone.

ARTHUR: Binky’s Musical Daydream | Source: © Arthur Read/YouTube

Repetition is existentially scary to them

The characters’ fears frequently involve concepts like having to do the same thing for a long time or even forever. In the aforementioned episode, the characters envision Buster repeating the same grade into his 30s, which, of course, they consider a worst-case scenario (interestingly, he and Mr. Ratburn are both shown to age in this sequence). In another episode, Arthur falls asleep on a bus and has a nightmare in which he is stuck on the bus well into adulthood, never allowed to leave without sufficient fare.

Clearly, the idea of being stuck in space or time indefinitely is scary to them — which means that the repetition they experience within the show goes unnoticed or doesn’t “count.” If they truly understood their circumstances, they would undoubtedly experience a terrifying and traumatic existential crisis.

They start third grade in Season 1

Several key events in Season 1, such as meeting Mr. Ratburn, show that this is the beginning of their time in third grade. Regardless of whether or not third grade will continue infinitely far into the future, we can safely assume that third grade did not begin prior to Season 1.

[W]e are still no closer to understanding why Elwood City residents don’t seem to notice their strange purgatory, nor how their distorted experience of time will affect them as they age.

IV. What We Don’t Know

How does time flow? Does it flow?

We’ve already established that the characters in Arthur do, in some sense, have a past and a future in which they are older or younger. At some point, on some timeline, they have been younger and will be older. Yet the fact that they exist at different ages does not mean that they age.

This is a tricky concept to wrap your head around. When we talk about aging, we are describing a linear process — and we expect our nominal age to increase at a continuous rate. In other words, if you turn 8 years old, you know that you will turn 9 after one year. But while society considers you to be 8 until the day you turn 9, you are not actually the same age throughout; you are continuously getting older each day and eventually reach the milestone of 9.

In other words, we know that Arthur has existed in the state of being 7, and that he currently exists in the state of being 8, but we don’t know for sure that he becomes 8 from 7; it’s not clear if being 7 for long enough is sufficient or even necessary for being 8.

How long was Arthur in second grade?

Did Arthur also repeat second grade arbitrarily long without noticing? Or is third grade the first year of their lives where this bizarre time paradox is occurring?

Does the future exist?

We’ve established that the people of Elwood City expect to age, and the characters are consistently having nightmares and daydreams about what will happen when they are older. But can we say for sure that they will ever reach that age? Is the future merely delayed? There are occasional flash-forward sequences that appear to show how life will turn out for the characters, which suggest that there will come a point in the future where Arthur will be older. But the validity of this evidence is questionable.

V. Theories for the Passage of Time

1. Aging does not exist

As previously discussed, just because Arthur is currently 8 does not mean that he will ever be 9; he may simply not be aging at all. And even if he will eventually be 9, that doesn’t mean he will become 9; he may simply shift to a different age state.

Yet the fact that they exist at different ages does not mean that they age.

1.1 One possibility is that Arthur is perpetually 8 — at least in terms of nominal age. In this version of events, Arthur has always been 8 and will always be 8, despite thinking otherwise. His recollection of times when he was younger may be part of a separate timeline, or he might recall events that never technically “happened.” We can cautiously rule this out given evidence that they begin third grade in season 1.

1.2 Another way this could play out is that Arthur changes between ages without aging. He might have been 6, then 7, then 8, but he did not age to gain each year; rather, he simply jumped from one constant age of 7 to a constant age of 8. Plotted on a graph with time passed on the x-axis and age on the y-axis, this would look like a series of unconnected horizontal lines; instead of his age increasing from 7 to 8, he would simply jump from 7 to 8 at an infinite slope.

2. Arthur will be 8 for infinitely long

2.1 First, Arthur could be each age for infinite time. We can assume that Arthur hasn’t been 8 for infinitely long, but he may only turn 9 infinitely far in the future. Furthermore, he may have been 7 for infinitely long before reaching the age of 8. 

This sounds deeply counterintuitive, but such is the nature of infinity. The type of infinity we are dealing with is called “countable infinity,” and it is, in fact, a number — the number of seconds in forever. It is simultaneously how many odd numbers there are, how many even numbers there are, and the number of odds and evens combined. Incredibly enough, it’s also how many Rational numbers there are, which includes all possible fractions (it’s not how many Real numbers there are, though; that number is bigger). And countable infinity times countable infinity is still countable infinity. Therefore, Arthur could spend an infinite number of seconds in third grade, and an infinite number of seconds in fourth grade, and still have infinite time left over to approach infinite age.

2.2 Another variation is that Arthur aged normally until 8, but will remain 8 for infinite time. This would be extremely strange, but it would be even stranger if time in Arthur were intuitive to us mortals. Under this scenario, he may become 9 after infinite time, or not.

2.3 A third, possibly even more counterintuitive possibility is that Arthur is aging, but extremely slowly — in fact, slower and slower as time goes on. In this scenario, Arthur’s age converges to 9 as x approaches infinity; eventually, his age will become virtually indistinguishable from 9, yet he will only reach the nominal age of 9 after infinite time. Happy birthday.

3. Arthur’s age increases logarithmically

3.1 If we want to avoid the possibility of a literally infinite third grade, this may be the most plausible remaining theory. In this scenario, Arthur ages along a logarithmic curve from 0. The passage of time begins fairly normally from Arthur’s birth, yet the more time passes, the time it takes to reach the next age will increase. This means that Arthur and friends will eventually reach age 9, and even 10 and beyond to adulthood — but fourth grade will be significantly longer than third grade, and so on.

Arthur’s Birthday | Source: Arthur Read/YouTube

VI. Conclusion

Despite the best efforts of human logic, it may be impossible to conclusively prove the mysteries of Arthur. And even if we could know for a fact which of these models describes Arthur’s true reality, we are still no closer to understanding why Elwood City residents don’t seem to notice their strange purgatory, nor how their distorted experience of time will affect them as they age. Yet as is the case in many areas of conceptual physics and mathematics, just because a theory cannot be physically proven doesn’t make it any less important.

The axioms we use in transfinite mathematics may be human constructs, but they produce real-world results. Likewise, proposed models like String Theory, M-Theory, and Quantum-Loop Gravity may be impossible to prove, but they are our best insights so far into the fabric of the cosmos. And throughout mathematical history, we have seen abstract ideas develop that seemed to have no practical applications, only to later emerge as keys to understanding fields such as physics, engineering, or epidemiology. The same may be true for Arthur

More importantly, what is the nature of Arthur’s reality, and how do the processes of time, aging, and maturing work in his world?

Strange, counterintuitive, and abstract ideas are critical to our understanding of reality and the advancement of human knowledge. It may seem strange to you, that we should seek an answer to Arthur’s dark mysteries when we know we might not find one — but it shouldn’t. And although we may never have all the pieces to assemble a comprehensive Theory of Arthur (ToA), we must press on. Because our purpose here on Earth is not to know all the answers; only God can know everything. And we are human, not gods. And that’s what humans do: we seek. We seek answers, we seek knowledge, we seek truth. We reach out our hands in the dark, feeling for a light switch that might not even be there, yearning for illumination. We seek the light because we are human, and we are human because we seek the light.

And Arthur? Well, he’s an aardvark.


1 comment

  1. I’m flattered that you used my special scene with the fractal. It may be the first infinite fractal in a cartoon. I think you are first one to notice. You would not believe the endless hours of discussions and arguments we had about our continuity of Arthur’s life timeline over the years.
    I really appreciated reading your article. It was very interesting and amusing
    All the best.
    Greg Bailey
    Director of ‘Arthur’

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