Note too that “lively debate” means something different to statisticians from normal people. You would be unlikely to split your sides laughing at conversations about whether building a model to use clinical data to predict outcomes (a core statistical activity), counts as “doing” or “using” math.

Whatever: my undergraduate degree required a lot of math, and my PhD thesis made use of lots of algebra and calculus. I call myself what I am: a statistician. But I do think of myself as belonging to the wider mathematical brotherhood. I believe that the inductive (statistical inference) and deductive (mathematical) logics are all we need to explain — in the sense of “make valid claims about” — the universe. Mastery of these fields are among our species’ proudest achievements.

All of which makes me worried, I think is the word, to read of the impact artificial intelligence and machine learning is having on (at least some) academic mathematicians. Recently, AI has started constructing proofs for some of the trickier unproven theorems. Which leads to the question: if a machine can prove a theorem, what’s the point of trying to do it oneself?

Kirwin Hampshire, who is in the process of working on his own PhD at the University of Victoria (in the department of Mathematics and Statistics, ha!), has just published a Substack essay in which he confesses that the ability of AI to solve problems is causing nothing less than severe existential angst. He feels “a profound spiritual crisis” and, in his worst moments of paranoia, wonders if it be the “express goal of these [AI] companies to make me kill myself”.

His essay is beautiful. To do him the disservice of summarizing it in a paragraph, he argues that the process of learning and proving is as important as the outcome, and that a machine-generated mathematics is no mathematics at all.

That reminded me of music, and I might suggest this to Kirwin as part-antidote to his fears (though he won’t like much of what I say). I am very fond of the Bach two-part and three-part inventions and have multiple recordings of these by some of the ablest pianists ever recorded: Glenn Gould and Andras Schiff are favorites. 

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AI and the false consciousness trap

By Yanis Varoufakis

Nothing I fumble out at the keyboard (I’m a constant piano student) will ever touch the divine musicality of Bach’s genius or the technical mastery of my favorite virtuosi. But it really doesn’t matter. Because few things give me more pleasure than trying. Few things feel more satisfactory than to master a complex passage even though I know no-one will ever hear it other than myself — and that even I would prefer to listen to Glenn Gould doing it properly.

In my analogy, the existence of digital recordings of music are the AI machines in Kirwin’s nightmare: they obviate the need for the task which gives my life meaning (“I am a piano student”) and pleasure (“I need to hear this music”). If a machine can “do” Bach for you, why try and do it yourself? Yet the pleasure persists, as does the choice to practice. I cannot prove this, but I believe it: a robot will one day produce music that sounds Bach-like. I do not believe it will improve upon him.

I suspect some similar “behavioral” consequence might happen to much of mathematics. A machine will be able to do algebra at least as well as a human. But that won’t rob humans of their need to do the algebra for themselves, nor diminish the beauty of any particular proof.

To which Kirwin might reply: so what? You’re talking about a hobby. I’m talking about the pitface of human endeavor, about what it means to be a mathematician. As he says “Mathematicians are paid to prove theorems” and, if you don’t need them for that, do you need them at all?

I cannot comprehend — this might be a failure of imagination — how the answer to that question could ever be “No, we do not need them”, not while there are humans in the world and a human society about which to be concerned. Math is special because it’s universal; no other discipline — save, perhaps, music — comes close. It is true that a machine could prove theorems in such a way as to be understandable and useful only to other machines, but if we reach that point then not only mathematicians should worry about superfluousness. In the meantime, humans will need to understand what the AI is “proving” and I can’t imagine how that could be achieved were those humans not themselves mathematicians.

Back to my own early practice at math. The point of my thesis was to take a noisy, distorted picture of something important (like a remotely-sensed heart scan) and reconstruct an approximation to the real object. In Chapter 4, I ended a few pages of algebra with this melancholic sentence: “Unfortunately we have not been able to prove this Lemma [a proposition].” In other words, I couldn’t demonstrate whether Chapter 4’s “great idea” had legs.

Having read Kirwin’s essay, I wondered if an AI could pick up where I’d left off in 1993. I wish I hadn’t. Here is what Grok told me:

You correctly identified the obstacle. You tried to adapt Eggermont’s argument (his Lemma 6.1 and the subsequent KL + residual Lyapunov function). The natural ordered-subset extension leads to the candidate inequality (your Lemma). That inequality does not hold in general, and the extra diagonal projector, which “switches on” only the current subset, prevents the global residual from decreasing monotonically. Consequently the Lyapunov argument collapses exactly where you left it.

There was a lot more. TL;DR: you couldn’t prove it because it was obviously unprovable, and you should have known this. It was quite clear that Grok already “knows” more about image reconstruction algorithms and their associated mathematics than I could ever hope to master. I think I preferred feeling sad, to feeling both sad and inadequate. Which is, of course, a version of Kirwin’s point.

But in any case none of the work I produced in 1993 is even vaguely relevant today, because the field has moved on. Ironically, image reconstruction was one of machine learning’s early successes: trained models can make better guesses at true images than my Pooterish attempts in Chapter 4.

But even without machine learning, I doubt many people’s theses from 30-odd years ago bear re-reading. I could make a joke here about some more recent contributions to the sociological literature, but I’ll rise above that. (Almost.)

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Must data centres be so ugly?

By Tom Ough

As with Bach, so with PhDs: the process is what matters. Learning how to read literature; how to build a hypothesis and investigate it, mathematically and empirically: this is why we train people to doctoral level in Mathematics and Statistics.

And the more machines produce technology or any other solution to any other problem, the more, I should think, we will need men and women like Kirwin.

I did, though, have long dark nights of the soul during those PhD; years, in fact, when I felt overwhelmed with futility. I remember walking towards the Math building in University Gardens and then just sort of veering off at the last minute, because of the sick feeling in my stomach. Not because of a machine, but because of my inadequacy.

Most PhD students have spent nearly 20 years of school and undergraduate life being told: “This is what you need to understand in order to pass an exam,” and most of us are extremely good at that way of learning. The PhD environment in mathematics and statistics — no classmates, no solutions to tutorials, possibly no solution at all to the problem you’re investigating (see Chapter 4 for details!), no one to tell you if the roll of the dice you’ve just made for the next three years will have any outcome beyond failure — is very different. It would almost be bizarre for a young person in such a situation not to develop angst, an existential dread.

And yet the sun rises, and yes, the work has purpose. Kirwin, your abilities will be needed in the age of AI more than ever before. Maybe just switch off your computer and close your notebook for a while, and think of something else. It was in those long dark nights, I recall, that I first fell in love with Bach.

What did the people coding away in the glass towers know that I did not? What did they actually do, these data scientists and developers — and why did it pay so much better than what I did? Initially, I agreed with the argument, sometimes aired in The Guardian and other organs of correct thinking, that if our tech overlords were going to have such an outsize impact on society, they had a duty to study literature, history and philosophy. But then I realized that I knew nothing of their area of expertise, and was hardly in a position to criticize.

And so I began to read books about technology and the history of computing. Much to my surprise, they were very interesting. I learned about Charles Babbage and his Difference Engine; Alan Turing and his imaginary machines; and ENIAC, the first computer, which had to be literally “debugged” because the heat and light emitted by its 17,468 vacuum tubes attracted moths. I learned, too, about the 1947 invention of the transistor and “Moore’s Law”, which held that the processing power of a microchip would double every two years, even as costs remained relatively stable, a prediction that proved accurate for decades.

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Frank Zappa: America's last rebel

By Daniel Kalder

The biggest surprise, however, was discovering that many of the core concepts underpinning the information age were the work of one man: Claude Shannon. Every time you send a text, make a phone call, search the internet or stream music or video, you rely on ideas he was the first to develop. In terms of impact, he appeared to me to be on a par with Einstein (at least). Yet while Einstein’s name is synonymous with genius, Shannon’s is unknown to the broad public. His first biography wasn’t published until 2017, almost two decades after his death and 80 years after he published the first of two major works that would revolutionize how we do, well, almost everything.

The childhoods of people who change the world always seem mundane in contrast to their later achievements, and in this, Shannon was typical. Born in 1906, he grew up in Gaylord, Michigan, a small town surrounded by farmland. His father was a furniture maker and probate judge, his mother a teacher. As a child, he enjoyed tinkering with radios, and went on to study electrical engineering and mathematics at the University of Michigan. In 1936, he took a position as a research assistant at MIT, where he learned to program a “Differential Analyzer”, a mechanical contraption the size of a room that solved equations with shafts, gears, wheels and discs. The next year, he worked as an intern for the US telephone monopoly AT&T, which had been established by Alexander Graham Bell 60 years earlier.

Shannon now started to theorize about a different kind of thinking machine. The relay circuits in the telephone systems at AT&T were comprised of switches that could be switched on, allowing electrical current to flow, and off, stopping the current. Shannon then made a conceptual leap, applying a system of logical reasoning, developed by the 19th-century English mathematician George Boole, to circuitry. Boole’s binary system was based on true or false questions, and Shannon proposed that if an open circuit was used to represent one and a closed circuit to represent zero, then circuits could be used to perform logical tasks such as decision making and processing rules. In 1937, he articulated this idea in his master’s thesis, “A Symbolic Analysis of Relay and Switching Circuits”. Though no one realized it at first, he had laid the theoretical groundwork for digital computing. Today, it is recognized as the most influential master’s thesis of all time. Shannon was 21 years old.

In 1941, Shannon took a job at Bell Labs, the research wing of AT&T. During the war, he applied his mathematical skills to anti-aircraft targeting and was part of the team that developed the “X-system” which was used to encrypt communications between Churchill and Roosevelt. This work was significant in itself, but it was after the war that he continued the work of establishing the information age. In “A Mathematical Theory of Communication” Shannon set out to solve the problem of “reproducing at one point either exactly or approximately a message selected at another point”. The problem was that analog signals, such as the electrical waves used in telecommunications systems, were susceptible to noise build up: static, errors, interference and so on. But boosting the signal also boosted the noise, which had the effect of rendering the original message unintelligible when it arrived at the other end — for instance in a long-distance phone call.

Shannon’s solution was to shift the focus from waves and current and to rethink the problem: what, exactly, was being conveyed in a message? At first, he referred in a letter to the “transmission of intelligence” before settling on the word “information”. He coined the term bit (from “binary digit”) to describe the smallest unit of measure. Shannon showed that all information — images, text, sound — could be broken up into bits, transmitted as zeros and ones, and then reconstructed clearly at the other end, no matter the medium, whether phone lines or radio waves.

So ubiquitous now is the idea of measuring information digitally that it is difficult to believe it is so recent. Shannon published his magnum opus in The Bell System Technical Journal, in 1948, and his colleagues were quick to see its significance; in the early Sixties AT&T moved the US telephone network to a fully digital system. Meanwhile, researchers at Bell Labs had invented the transistor in 1947, which would make the binary logic circuits Shannon had theorized about a decade earlier a practical reality. The age of the computer now began in earnest. And as computer and communications technology advanced, so more and more information could be created, processed, and manipulated: all at ever faster speeds. Shannon had paved the way for everything from the moon landings to the guided missiles hitting their targets with unprecedented accuracy in Operation Desert Storm; to the rise of the internet and social media; to the vast fortunes of Bezos, Gates, Zuckerberg and Musk.

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Why I’m boycotting AI

By Sam Kahn

When Shannon died of Alzheimer’s in 2001, it seemed that his theory had already transformed the world — even DNA could be measured in bits. Yet, in many ways, the information age still resembled its analog predecessor. Digital information was still stored on physical artifacts such as CDs, DVDs, floppy disks and hard drives, just as analog had paper, tape and vinyl. A shift was taking place, however, as information became increasingly disembodied. Print increasingly gave way to websites; music and film moved online and could be downloaded as torrents of bits. Then, as technology advanced, you didn’t even need to download any of it anymore. Information was everywhere and nowhere, afloat in the ether, accessible through phones, computers, smart TVs and smart fridges — even sunglasses.

We are, of course, still grappling with the consequences of this shift. Few doubt that putting everybody in touch with everybody else instantaneously has driven us all a bit mad. Other changes are more subtle. In “A Mathematical Theory of Communication”, Shannon had declared that meaning was irrelevant to the “engineering problem”. This was not intended as a value judgment, but as all forms of media are reduced to zeroes and ones and made instantly available, it is hard not to think that a certain flattening has occurred. In the age of digital streaming, everything from the Lord of the Rings to Looney Tunes cartoons to Bach’s Goldberg Variations and the songs of Taylor Swift have been equally reduced to “content” — always on, yet also easy to switch off. But more than that, digital information is easily adapted and modified so that the “content” itself has become unstable, increasing its sense of weightlessness. Anyone can modify a digital file, and with the advent of AI, also fundamentally shaped by Shannon’s theories, we can remix, reshape and transform simply by describing what we want to a model.

And so the revolution set in motion by Shannon continues, though we may already have passed through its most radical stage. This occurred during the pandemic when, locked in our homes, we attempted to become information ourselves, like characters in a William Gibson novel. Gazing into our webcams, our voices and images were captured and broken down into bits, transmitted and then reconstructed as digital simulacra on screens elsewhere. As society attempted to render itself virtual, everything was digitized, from church services, to school classes, to wine tastings and even funerals. So excited was Mark Zuckerberg by the prospect that in 2021 he proclaimed we were going to live in the Metaverse, and that “immersive digital worlds” would “become the primary way that we live our lives and spend our time”. But this was also the moment that we started to learn the limits of life-as-information. Zoom fatigue kicked in and the Metaverse quickly became a punchline — its Reality Labs division has reportedly lost over $60 billion since 2020. It turned out that we did have bodies after all.

Ironically, the Father of the Information Age himself may have something to teach us here. For all that Shannon dealt in abstractions, he was deeply rooted in the physical world. He loved jazz and played the clarinet; he taught himself to juggle and ride a unicycle, two skills that require coordination and rhythm. He liked to build machines: at Bell Labs he created a robot mouse named Theseus that could find its own way through a maze. Later, he lived with his wife and three children in a Victorian mansion with a view of Mystic Lake in Winchester, Massachusetts, and built a chairlift to carry them there. His workshop was full of projects that reflected a playful imagination, including a flame-throwing trumpet. A modest man uninterested in fame, he found joy and fulfillment in solving interesting problems and making things. In 1936, T.S. Eliot asked: “Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?” Perhaps Shannon knew all along.