{"id":681,"date":"2026-08-07T04:38:59","date_gmt":"2026-08-07T04:38:59","guid":{"rendered":"https:\/\/cityflowjournal.com\/?p=681"},"modified":"2026-08-07T04:38:59","modified_gmt":"2026-08-07T04:38:59","slug":"bachs-lesson-for-the-ai-age","status":"publish","type":"post","link":"https:\/\/cityflowjournal.com\/?p=681","title":{"rendered":"Bach’s lesson for the AI age"},"content":{"rendered":"
\n

I\u2019m not much of a mathematician which may seem a surprising admission for a statistician. However, the question of whether \u201cStatistics\u201d (the discipline, with a capital S) is a field within, or an applied science that overlaps with, \u201cMathematics\u201d (capital M) was, certainly in the Eighties, a lively debate.<\/span><\/p>\n

Read more The twilight of the ladyboss<\/a><\/p>\n

Note too that \u201clively debate\u201d 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 \u201cdoing\u201d or \u201cusing\u201d math.<\/span><\/p>\n

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. <\/span>I believe that the <\/span>in<\/span><\/i>ductive (statistical inference) and <\/span>de<\/span><\/i>ductive (mathematical) logics are all we need to explain \u2014 in the sense of \u201cmake valid claims about\u201d \u2014 the universe. Mastery of these fields are among our species\u2019 proudest achievements.<\/span><\/p>\n

All of which makes me <\/span>worried<\/span><\/i>, 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 <\/span>proofs<\/span> for some of the trickier unproven theorems. Which leads to the question: if a machine can prove a theorem, what\u2019s the point of trying to do it oneself?<\/span><\/p>\n

Kirwin Hampshire, who is in the process of working on his own PhD at the University of Victoria (in the department of Mathematics <\/span>and <\/span><\/i>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 \u201c<\/span>a profound spiritual crisis<\/span>\u201d and, in his worst moments of paranoia, wonders if it be the \u201cexpress goal of these [AI] companies to make me kill myself\u201d.<\/span><\/p>\n

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

That reminded me of music, and I might suggest this to Kirwin as part-antidote to his fears (though he won\u2019t 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.\u00a0<\/span><\/p>\n<\/p>\n

Nothing I fumble out at the keyboard (I\u2019m a constant piano student) will ever touch the divine musicality of Bach\u2019s genius or the technical mastery of my favorite virtuosi. But <\/span>it really doesn\u2019t matter<\/span><\/i>. Because few things give me more pleasure than <\/span>trying<\/span><\/i>. Few things feel more satisfactory than to master a complex passage <\/span>even though I know no-one will ever hear it other than myself \u2014 <\/span><\/i>and that even I would prefer to listen to Glenn Gould doing it properly.<\/span><\/p>\n

In my analogy, the existence of digital recordings of music are the AI machines in Kirwin\u2019s nightmare: they obviate the need for the task which gives my life meaning (\u201cI am a piano student\u201d) and pleasure (\u201cI need to hear this music\u201d). If a machine can \u201cdo\u201d 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.<\/span><\/p>\n

I suspect some similar \u201cbehavioral\u201d 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\u2019t rob humans of their need to do the algebra for themselves, nor diminish the beauty of any particular proof.<\/span><\/p>\n

To which Kirwin might reply: so what? You\u2019re talking about a hobby. I\u2019m talking about the pitface of human endeavor, about <\/span>what it means to be a mathematician<\/span><\/i>. As he says \u201cMathematicians are paid to prove theorems\u201d and, if you don\u2019t need them for that, do you need them at all?<\/span><\/p>\n

I cannot comprehend \u2014 this might be a failure of imagination \u2014 how the answer to that question could ever be \u201cNo, we do not need them\u201d, not while there are humans in the world and a human society about which to be concerned. Math is special because it\u2019s universal; no other discipline \u2014 save, perhaps, music \u2014 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 \u201cproving\u201d and I can\u2019t imagine how that could be achieved were those humans not themselves mathematicians.<\/span><\/p>\n

Read more How corruption could bring down Zelensky<\/a><\/p>\n

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: \u201cUnfortunately we have not been able to prove this Lemma [a proposition].\u201d In other words, I couldn\u2019t demonstrate whether Chapter 4\u2019s \u201cgreat idea\u201d had legs.<\/span><\/p>\n

Having read Kirwin\u2019s essay, I wondered if an AI could pick up where I\u2019d left off in 1993. I wish I hadn\u2019t. Here is what Grok told me:<\/span><\/p>\n

You correctly identified the obstacle. You tried to adapt Eggermont\u2019s 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 <\/span><\/i>not<\/i> hold in general, and the extra diagonal projector, which \u201cswitches on\u201d only the current subset, prevents the global residual from decreasing monotonically. Consequently the Lyapunov argument collapses exactly where you left it.<\/span><\/i><\/p>\n

There was a lot more. TL;DR: you couldn\u2019t prove it because it was obviously unprovable, and you should have known this. It was quite clear that Grok already \u201cknows\u201d 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\u2019s point.<\/span><\/p>\n

But <\/span>in any case<\/span> 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\u2019s early successes: trained models can make better guesses at true images than my Pooterish attempts in Chapter 4.<\/span><\/p>\n

But even without machine learning, I doubt many people\u2019s 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\u2019ll rise above that. (Almost.)<\/span><\/p>\n

As with Bach, so with PhDs: the <\/span>process <\/span><\/i>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.<\/span><\/p>\n

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.<\/p>\n

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.<\/span><\/p>\n

Most PhD students have spent nearly 20 years of school and undergraduate life being told: \u201cThis is what you need to understand in order to pass an exam,\u201d and most of us are <\/span>extremely good <\/span><\/i>at that way of learning. The PhD environment in mathematics and statistics \u2014 no classmates, no solutions to tutorials, possibly no solution at all to the problem you\u2019re investigating (see Chapter 4 for details!), no one to tell you if the roll of the dice you\u2019ve just made for the next three years will have any outcome beyond failure \u2014 is <\/span>very<\/span><\/i> different. It would almost be bizarre for a young person in such a situation <\/span>not <\/span><\/i>to develop angst, an existential dread.<\/span><\/p>\n

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.<\/span><\/p>\n

Read more New York\u2019s war on its Jews<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

I\u2019m not much of a mathematician which may seem a surprising admission for a statistician. However, the question of whether \u201cStatistics\u201d (the discipline, with a capital S) is a field within, or an applied science that overlaps with, \u201cMathematics\u201d (capital M) was, certainly in the Eighties, a lively debate. Read more The twilight of the […]<\/p>\n","protected":false},"author":1,"featured_media":680,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[103],"tags":[],"class_list":["post-681","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ai"],"yoast_head":"\nBach's lesson for the AI age - \u0421ity Flow Journal<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cityflowjournal.com\/?p=681\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bach's lesson for the AI age - \u0421ity Flow Journal\" \/>\n<meta property=\"og:description\" content=\"I\u2019m not much of a mathematician which may seem a surprising admission for a statistician. 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