On analogies, parallels and thought experiments

Nilima Nigam, Professor at Simon Fraser University

(What follows are long-winded  personal reflections around AI and mathematics. I’ll conflate ‘AI’ with LLMs for the most part for the rest of this.)

Cursed as we are to live in ‘interesting times’ which are changing rapidly thanks to AI, we reach for historical parallels and analogies. As our colleagues in the humanities may caution, both analogies and parallels are rarely exact. While relying on them to shine light on dilemmas, it is also fruitful to construct and ponder stark thought experiments which lay the quandaries bare.

Analogies

Centaurs: An oft-repeated analogy of late (for instance on LinkedIn), this one induces chuckles. The analogy is intended to evoke the following: equipped with access to frontier models, we are merging our cognitive selves with tools of superhuman strength.  Just as the progeny of Ixion and Nephele, this projected fusion of man and tireless AI suggests we’ll transcend our boring mortal limitations. The giggle comes because the centaurs memorably are also associated with wild parties, and an abortive attempt to kidnap a queen and all the women of Lapith. How far, then, should we take this current-day centaur analogy?

Food: This analogy thanks to Terence Tao is much more apt. Just as humans transitioned from an era of food scarcity to food abundance, leading to a change in how we structure societal priorities, so must we imagine a transition from proof scarcity to abundance. Our goals shift from searching (painfully, slowly) for proofs, to preparing them for digestion. (I paraphrase considerably.) The analogy makes a lot of sense to me in some regards, but it contains within it a moral imperative: only a moral monster could object to food abundance. [Indeed, my children have accused me as such when I tried to curtail their access to a surfeit of Hallowe’en candy.]

And yet: mathematical proofs, and indeed mathematics itself (central, universal, true and beautiful though it is) are not primal human needs. The moral imperative of abundance is different.

Tulips: A modification of the food analogy hopefully makes my point clearer: we used to live in an era of tulip scarcity. Tulips are beautiful, enrich our lives, and some (distressingly small) number even have magical properties beneficial for other human pursuits.  Many people used to spend time looking for tulips. Now we could enter an era of tulip abundance.

This modified analogy allows us to ask: what are the pros and cons of moving to this new era of more tulips? Should we go from too-few to too-many within the span of one Spring term, or phase it in more gradually? Does it matter that a generation of tulip-growing apprentices will find themselves fairly quickly redundant? How do we train them?

Parallels

The industrial revolution: This historical parallel is frequently brought up. Who could object to new technologies that lead eventually to an improvement in living standards for billions? But I believe this parallel to be both incomplete and instructive. Technological advances that impact societies do, in fact, impact societies. So how we regulate these changes, how we frame policies to manage the transition, and how deliberate societies are, is actually consequential. The Industrial Revolution (actually there are two) indeed eventually brought about a rise in GDP, and benefits across people. Another historical parallel is to the Luddites – surely only a fringe person would turn their back on a technology.

However, the transitions brought about were far from painless. A pretty good question to ask is: could one design policies to usher in the industrial revolution, that would have prevented (say) the growth of slave plantations for cheap cotton, exploitation of children, and extreme pollution?¹

If we believe that we’re at the cusp of a new industrial revolution, surely now is the time to really wonder about possible adverse impacts, and how to minimize them. And as mathematicians we can’t just decide this is someone else’s problem. Leaving this to the wisdom of the markets may not be wise. Mathematics is an incredibly powerful ‘use-case’ for AI companies. However excited we are about our new tools, I’m fairly confident the marketing we’re providing these companies is invaluable. ‘Look, our model solved math! Now for $199.99, you can have the same model to run your spreadsheets, too.’²

Calculators and computers: A mathematics-specific parallel I’ve seen draw is to calculators, computers and symbolic computation tools. Surely we cannot object to the introduction of these tools? But the parallel is flawed: these are tools which automate tasks a human identified and determined were of no value to do by humans. AI-automation of large elements of mathematical work – proving theorems, finding counterexamples, proof verification and eventually human-digestible exposition as well as judgement – this is categorically different.

Fossil fuels and fertilizers: Human-made fertilizers have increased crop yields, and ensured billions of humans are able to eat. Their production is associated with high rates of greenhouse gas emissions. Much of the fertilizer industry relies on burning fossil fuels. It is irresponsible to ignore the benefits or the environmental costs of human-made fertilizers. Governments and societies have known about both for several decades, yet policy and regulatory frameworks have focused on abundance. The mitigation of the risks of burning fossil fuels has not been a priority. We know the risks associated with climate change, but corporations and politics ensure we’re not able to respond adequately or in time. Technological progress routinely outstrips policy responses, and market-mediated technology even more so. The parallels with AI development are eerily close, in my view.

Thought experiments

I like thought experiments that our Philosophy colleagues may study. They are deliberately outrageous, and seek to isolate a moral question. The trolley case, or the transplant surgeon with 5 dying patients scenario famously test our intuitions. Here’s my attempt at a thought experiment for mathematicians.

The Bionic Graduate Student: you are a mathematician with a graduate student named Tulip, whom you’re obliged to employ and supervise for the next N years. You cannot get rid of them, but they are measurably slower than the last AI-for-math model FrodoBaggins42 (you already use it).  By the time you train them on prompting FrodoBaggins42, hundreds upon thousands of conjectures remain unproven. Your lab partner has perfected a simple and safe lobotomy procedure: you replace 50% of Tulip’s brain by BilboBaggins42, a bio-compatible version of FrodoBaggins42. You both agree mathematics as a whole will be enriched by the results achievable with this procedure. Immediately your student would be able to contribute to the formulating and proving of all these outstanding conjectures. You would not need to have many conversations gently nudging them towards their own discoveries.  Tulip, noting their career prospects are severely impacted if they don’t agree, provides consent. After all, it’s only 50% of their brain.

Should you give Tulip a lobotomy?³

My concerns

Having entertained myself with analogies and thought experiments, maybe I should come clean about where I stand.

I’m profoundly concerned about how AI-for-mathematics is being rolled out – frontier models are highly concentrated amongst commercial actors with vested interests, we don’t really understand how they work, we don’t know what future to train our students for, and we are developing new norms much slower than we’re ripping up the form of our discipline. I don’t think mathematicians are in the driver’s seat, and yes, we should be.

I don’t think we’re listening enough to the worried voices amongst our younger colleagues and students.⁴ We’re presenting to them excitement (and it is an exciting time!), certainty (the discipline is irrevocably changed!), a compressed timeline (we must change now!) and opaqueness (we cannot predict very much). We envision somehow the distant contours of mathematics (where we’ll pick over already-proven theorems, examine them in the light and decide they are beautiful to us) where we become like art critics rather than artists. Or maybe we’ll be mathematical equivalents of Picasso and Braque, our job to invent new frameworks like Cubism. But we cannot tell our students today what kinds of jobs they’ll be able to get within the next 2-5 years. And while we’ve had several years of experiencing the thrill of doing mathematics as our predecessors, we’re telling our students this isn’t for them anymore. Many of them in the system have done everything we have asked of them until now, only for us to, well, decide it wasn’t what we want.⁵ I don’t think this is fair to them.

Some predictions

I predict

• With high confidence (70-80%):  AI tools will become standard for literature search, editing and fill-in-technical-detail work within the next 2-5 years. I think this will not impact jobs much, will be readily adopted in our community, and will be minimally disruptive. De-skilling or no-skilling impacts will not be apparent in this time-frame.
• With moderate confidence (40-50%):  Within 2-5 years AI will routinely produce complete, correct proofs of moderately hard open problems in already well-formalized sub-fields. This goes beyond locating counterexamples. The amount of prompting required will require mathematical sophistication at the level of a current 2nd year graduate student. Efficacy will not be uniform across fields. This will negatively impact entry-level jobs in those sub-fields.
• Separately, and with moderate confidence (40-50%): In an already-tight funding environment (due to assorted other reasons), universities at large will seek to cut back on new tenure-track hiring for research positions in mathematics. AI will be stated as a causal factor in university communications around hiring.
• With very low confidence (20-30%): by 2032, grant panels, hiring committees and prize committees will have developed new and clear criteria for how and what to assess. These changes will most significantly impact young researchers.

In closing

Feedback I received on this draft from colleagues has been invaluable.  I also used Claude to spell-check this. I’ve signed the Leiden Declaration. Since I don’t really know how best to train my students with AI tools for the very uncertain future facing them – we’re operating on the patient while learning basic anatomy – I’ve decided to eschew the use of LLMS for any other purpose in my mathematical work. This is a niche and odd position, perhaps, and tenure helps. I’m into the evening of my own journey, and the tools would unequivocally help. But I have yet to figure out how to transmit mathematical knowledge, painfully-gained experience (judgement?) and efficacious use of AI tools, without really understanding how the latter impacts the development of the former. I’ll revisit this position with a ‘first-do-no-harm’ lens.

Research mathematicians on this planet number perhaps 150,000.⁶ Our motivations for our research work are manifold. Society at large sees our utility primarily as teachers, and (much more rarely) because some beautiful part of mathematics finds a great deal of excitement-generating relevance or captures our collective awe at what the human mind can do. This does not deter us, because we find frustration, exhilaration and fulfilment in the research we do, even as its practice enriches us as teachers. Is it obvious, then, that we must race to upend how and what we do, at a pace set by anyone but ourselves?

1. Note the way the question is phrased puts a thumb on the scale of the argument.
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2. Another interesting feature of the present-day technological miracle is the non-zero probability being assigned to catastrophic impacts, whether through malicious humans or rogue AI is up for debate. Maybe it’s not crazy to ask if we absolutely it must know if the Riemann Hypothesis is true this year instead of a decade from now, if we could in the meanwhile ensure a slower pace of AI development allowed for better AI alignment.
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3. This, clearly, is about the value we place on mathematics compared to human mathematicians. Alert readers will note the allusion to the ship of Theseus.
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4. I’ve learned a lot, for instance, from Tasmin Chu, Marcel Goh and Kirwin Hampshire who write both beautifully and courageously.
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5. The story which comes to mind frequently is of Eklavya, from the Mahabharata. Eklavya wants to learn archery from Dronacharya, the teacher of kings. Drona refuses because Eklavya is of lower status. The kid nonetheless is determined, and makes himself a mud statue of Drona while he teaches himself archery. Versions differ, but in a poignant one I remember from my childhood, Eklavya occasionally hides behind the trees and watches Drona’s instructions to the princes. He’s found out. Drona demands, as ‘gurudakshina’, Eklavya’s right thumb, effectively ending his career as an archer. This isn’t an analogy, or a parallel. It’s just a story about teachers who don’t do the right thing for the humans in their pastoral care.
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6. This is a Fermi estimate. ↩︎

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Received 11 August 2026.