I’m writing a book. I’ve got the page numbers done.
Steven Wright, comedian
It is impossible to be a mathematician without being a poet in soul.
Sofia Kovalevskaya, mathematician
Sum Thing to Talk About
Numbers possess a certain inevitability. Especially round this neck of the woods. It is for example a truth universally acknowledged that a number in possession of a certain value must be in want of a Tiffinian to do something with it. Yet, for the majority of human history, arithmetic was as useful as fitting wheels to a tomato, to borrow the phrase. Indeed, the whole number business was a conundrum that intrigued even those evolution wunderkinds, Darwin and Wallace: why, they puzzled, should all humans possess an innate grasp of arithmetic when, for most of our existence it wasn’t needed?
It’s an interesting question. Wallace and Darwin didn’t know for a fact that all humans possessed arithmetical ability, they assumed it, but it does seem a fairly safe assumption. It’s just part of our nature to understand that there are infinitely many natural numbers, and that when you add them, it works this way, rather than some other way, and so on. That does seem to be part of universal human nature. Certainly among the universal humans I have met. What puzzled W & D particularly about this was how this capacity came to be there. It could not possibly have been the result of ‘natural selection’ since it’s only been used in a tiny recent period of human history.
Whether they fell out over it, history does not relate but Wallace proposed a theory that significantly diverged from Darwin’s views. He suggested that arithmetic ability must have evolved through mechanisms beyond natural selection; speculating that human cognitive faculties might have developed in tandem with a spiritual or intellectual evolution that was not strictly tied to survival. Wallace was, at this stage in his career, increasingly convinced that some aspects of human intelligence, especially higher reasoning, artistic ability, and moral sensibility, could not be fully explained by Darwinian processes alone. He proposed that an external guiding force may have played a role, an idea he referred to as “intelligent evolution.” This view placed him in stark opposition to Darwin, who saw it as an unwarranted retreat into mysticism.
Darwin, ever the pragmatist (you’d have to be with a beard like that), as well as a staunch advocate of natural selection, countered Wallace’s assertions by emphasising the role of survival in shaping human cognition. He posited that the capacity for basic arithmetic would have provided significant advantages in early human societies, facilitating trade, resource management, social organisation and A* Grades in Further Maths (not really). Darwin believed that, while arithmetic skills may not have been essential for survival in the way that hunting or foraging were, they nonetheless conferred practical benefits that could enhance an individual’s social standing and reproductive success. Thus, he framed arithmetic as an evolutionary adaptation, shaped by the pressures of survival and social interaction rather than an abstract or spiritual evolution. Yet neither Wallace nor Darwin ever really arrived at a satisfactory conclusion. The origins of our numerical abilities remained shrouded in mystery.
Until now!
It turns out that one possible answer (dear reader, you may need to sit down for this), is that our arithmetical knowledge could be an offshoot of language. Now, bear in mind, we’re not talking about numbers themselves; that’s an entirely different barrel of onions. If there’s time, we could get to them, but for now we’re just dealing with our knowledge of arithmetic and where that comes from. You see, it turns out, if you take the most elementary principles that yield linguistic structures, and you reduce them to their absolute minimum (a lexicon, which contains one element) you get the successor function (the idea that each number follows the last), as well as something like addition. So, you get the rudiments of arithmetic. And thus it’s possible that the reason we have knowledge of arithmetic is because we have language. The very structures underpinning human speech, such as rules, recursion, sequences etc. may have paved the way for mathematical thought long before tally marks were ever inscribed in clay. This connection extends even further into the realm of music. Studies reveal structural similarities between language and music, particularly concerning syntax and pattern recognition. Given that mathematics also relies on pattern-based thinking, one might ponder whether numerical ability is a byproduct not solely of language but also of humanity’s innate propensity for structured sound. In columns past, you will recall, we have indeed discussed how singing preceded speaking. If both music and language shape our cognition, could it be that arithmetic is, at its core, a harmonic interplay between sound, symbol, and structure?
This theory gains traction if we think about the origins of counting. Most early number systems are base 10, a quirk easily explained by the fact that most of us have ten fingers. The act of counting likely began with our hands, and contemporary neuropsychological studies reveal a profound connection between numerical cognition and finger recognition. Patients with damage to the left parietal lobe; a region critical for both arithmetic and finger awareness, often encounter difficulties with basic calculations. Our mathematical instincts, it appears, are literally at our fingertips. A little Director joke there..
The Babylonians mastered positional digit value, and by around 500 BC, the concept of zero emerged; a remarkable abstraction for a species whose ancestors initiated counting with knuckles and thumbs. The cognitive leap from tally marks to equations was substantial, and all the more interesting to think of it as fuelled by an existing human skill: the capacity to structure thought into ordered, generative patterns. In other words: language.
Notably, some languages, such as Pirahã, spoken by a particular Amazonian tribe, exhibit limited numerical systems, encompassing only “one,” “two,” and “many.” This raises the intriguing possibility that while arithmetic may emerge from language, its significance is not universally emphasised. Could it be that certain human cultures, at specific historical junctures, resembled the AI model more closely i.e. handling numbers as symbols devoid of deeper conceptual understanding?
I don’t know, but it does give us an excuse to return to our old friend, AI.
AI excels in arithmetic; unlike humans, it does not grapple with long division or forget to carry the one. But does it truly comprehend numbers? Or is it merely executing syntactical operations, mechanically adhering to rules devoid of conceptual insight? Gödel’s incompleteness theorems suggest that there are always truths in mathematics that lie beyond formal computation. Perhaps another distinction between AI and human mathematical intuition.
This week, I rectified the omission that whilst I had been aware of the great Alan Turing and even popped along to see The Imitation Game, I had never actually read his famous (and very well written) paper: Computing Machinery and Intelligence. A. M. Turing (1950) Mind 49: 433-460.) In this groundbreaking work on artificial intelligence, Turing writes, I propose to consider the question, “Can machines think?” Now, I don’t want to spoil the read for you but at some turns in the paper he suggests that inquiring whether machines can think is as futile as questioning whether submarines can swim. Indeed, Turing starts off by pointing out that the question whether machines can think is too meaningless to deserve discussion. For someone like linguistic philosopher and all-round clever clogs Chomsky, it’s what might be called a terminological question, like whether submarines can swim, or airplanes can fly. In a way they do, but it depends what you mean by ‘swim’ and ‘fly’. I like to think that Ned Washington and Oliver George Wallace had this question in their minds when they penned the wonderful When I See An Elephant Fly. Singalong with the Director:
I seen a peanut stand, heard a rubber band
I seen a needle that winked its eye
But I be done seen ’bout ev’rything
When I see an elephant fly
…
I seen a front porch swing, heard a diamond ring
I seen a polka-dot railroad tie
But I be done seen ’bout ev’rything
When I see an elephant fly
No? But anyway, a similar inquiry could be posed regarding arithmetic: does AI perform mathematics, or does it merely execute a series of computational manoeuvres that looks like they’re ‘doing’ arithmetic.
Humans, dear reader, possess an intuitive grasp of numbers that transcends mere calculation. From childhood, we recognise that numbers continue indefinitely. The concept of infinity feels natural, perhaps even inevitable to us. Yet, for AI, infinity merely represents another dataset, another function to process. It cannot, and does not, fathom the boundless nature of numbers in the same way a human does. While it can compute an infinite series, it does not take a moment in its AI garden on a summer’s day to marvel at the endless expanse of numbers.
The deeper question then, is not whether AI can perform arithmetic, but whether it can ever count in the manner we do. For us, numbers are not mere symbols but integral components of a complex web of thought, one perhaps spun from the very structures that enable us to communicate, reason, and imagine.
With our restless pattern-seeking minds, we are compelled to count, calculate, and impose order upon the world. If arithmetic is indeed an offshoot of language, then the limitations of AI may lie not in its computational capabilities but in its inability to engage in the profoundly human act of simply stating, “one, two, three.” Consider if you will, by way of analogy, the Greenland shark, mysteriously gliding its sharky way through Arctic waters. It inhabits a realm characterised by slow instincts and deep time, utterly untroubled by the recursive structures of human language. Its lifespan spans hundreds of years, yet it harbours no need to count them. Hypothetically, if a Greenland shark were to develop arithmetic, it might take 400 years merely to tally its own age, and even then, one suspects it would not be in a hurry.
And for those of us who work in schools? Well, whilst we more and more frequently traverse the intersection of language, numbers, and AI in education, we must underscore the importance of critical thinking and inquiry that enables students to comprehend these concepts on a profound level. By encouraging exploration of how numerical data can enrich their narratives, we foster an appreciation for the intricate and nuanced stories that numbers can convey; stories as rich and complex as the language we employ to articulate them.
You never know, we might cultivate a generation that not only excels in arithmetic but also grasps the profound interconnections between numbers and the human experience. After all, in a world increasingly shaped by data, mastering the language of numbers is not merely a skill; it now represents an essential facet of what it means to be human.
Until next time, Happy Reading/Doing ‘rithmetic!
Coincidentally, the Director’s mailbag this week contained a joke sent in by a Mr B Lavery, which I print here for you in lieu of this week’s Director’s tip. Thank you, Sir and if you could drop me another email explaining it, I would be most grateful.
A philosopher, a physicist, a mathematician and a computer scientist were travelling through Scotland when they saw a black sheep through the window of the train.
“Aha,” says the philosopher, “I see that Scottish sheep are black.”
“Hmm,” says the physicist, “You mean that some Scottish sheep are black.”
“No,” says the mathematician, “All we know is that there is at least one sheep in Scotland, and that at least one side of that one sheep is black!”
“Oh, no!” shouts the computer scientist, “A special case!”