For years, advocates for web standards, performance, and accessibility have implored web developers to “use the platform”. I’ve often been one of those advocates.

The argument is simple: why build something yourself, in JavaScript, when the browser can do it for you? Whatever you build, it’s likely to have poorer performance and worse usability than something the browser could just give you out-of-the-box.

I think it’s worth taking the other side, though, if for no other reason than to understand where the “platform-skeptic” developers are coming from. If “use the platform” is so obvious, then why do so many people seem to need convincing?

The most obvious reason is historical: for the longest time, browsers were playing catch-up with the ecosystem on top of them. Libraries like jQuery filled crucial gaps while browsers implemented equivalent APIs – and even then, you might have to wait for laggards like IE6 to age out before you could actually use them. Today, most browsers are evergreen (Safari is debatable, although ~7 times per year ain’t bad), but up until the 2020s or so, web developers had to deal with a decidedly lumpy web. In that environment, rolling your own is a sensible choice.

Another reason is familiarity: when you’re used to looking for React components on npm, that’s what you tend to reach for, regardless of the problem at hand. If you search for “sticky positioning” on npm, there’s no package that says “just use CSS position: sticky, you dolt.”

And often, even with a robust standard, libraries on npm would fill a useful gap between framework ergonomics and the platform underneath it. I always found it intriguing that many React developers preferred to stick to JSX and React idioms – raw DOM APIs felt “icky” – but were perfectly happy to use lower-level libraries where raw DOM manipulations are common. For example, a virtual list library might happily use raw DOM APIs for pure performance, while exposing higher-level primitives that a novice React developer could better grasp. In a sense, the ecosystem of React components led to a natural division of labor where those with more expertise packaged up unfamiliar platform APIs in a more familiar form factor.

Some of this effect was also driven by documentation. Many npm packages have lovingly detailed READMEs or websites with examples, tutorials, and screenshots. Whereas until MDN became cemented as the go-to place for web documentation (with web.dev as Google’s more future-facing arm), documentation for the web platform was scattered across blogs, StackOverflow, and sites like CSS Tricks. And many of these sites would just tell you to use a well-known library like jQuery or GreenSock!

The Dragula site versus the Drag and Drop MDN page. Arguably the former is still more compelling.

If it were just about third-party libraries versus platform APIs, though, then I don’t think it could fully explain the antipathy toward “use the platform.” Developers who are lazy (or do I repeat myself?), and who just want a ready-made solution for whatever problem they’re facing, are unlikely to care whether that solution comes from npm, the browser, or copied off of someone’s random GitHub Gist. They want to solve their problem and move on. But there’s a different source of anti-“use the platform” that I want to explore.

For a certain type of developer, building things yourself is just more fun. And often the resulting code is easier to reason about, especially if you don’t have an encyclopedic knowledge of the web platform. And once you’ve built something, there can be a kind of IKEA effect where you want to maintain and tinker with your own homemade code.

As an example, let’s imagine you’re trying to build a modal dialog. You might visually understand how these are supposed to work: content appears on the screen, but the background is still visible although partially occluded, and maybe clicking outside the dialog dismisses it. So you might grab for position:absolute and z-index to position the dialog correctly – aha, but the background still scrolls, so you have to disable overflow on the body… And then if you understand something about accessibility, you realize you need to handle Esc to dismiss, and build a focus trap, and return focus to the element that launched the dialog, and…

For many developers, what I just described sounds like a nightmare (and a good way to build something that only half-works). But for many developers, this sounds like fun! Think of how much you learn as you start building this thing. And think about how you could start putting your own spin on it by adding animations, themes, an optional “close” button… Before you know it, you’ve built a library that’s ready to go on npm. That’s way more fun than just grabbing <dialog> and calling it a day – what a downer!

And for many of us, before APIs like <dialog> existed, this was how we learned the web platform! Many of the people who now advocate for “use the platform” were once themselves authors of polyfills, shims, and libraries. I know because I’m one myself! I spent years working on tooling for IndexedDB, WebSQL, and other browser storage APIs as part of my work on PouchDB, which eventually led to me feeling confident enough to sit in W3C standards meetings and even open issues and pull requests on the IndexedDB spec itself. Without the forcing function of a gap in the platform that needed to be filled, I don’t know if I would have found the interest or motivation to get to that level of expertise.

Of course, doing it yourself is not always an unalloyed good. Sometimes it just comes from pure ignorance. On the web platform in particular, I think one of the reasons there was such a proliferation of JavaScript solutions to problems that could be better solved by CSS, for example, is that many developers just didn’t take the time to deeply understand how CSS works.

And to be fair, CSS has historically been hard to understand! There’s a reason the site is called “CSS Tricks.” Things like the clear fix, floats, and the min-width: 0 trick are hardly intuitive. Rather than trying to understand CSS’s internal algorithm, it’s often much easier to just imagine the imperative logic you want and then express it in JavaScript. Plus, for years CSS didn’t have a straightforward way to express common patterns like line clamping, textarea resizing, scrollbar hiding, etc. So of course developers built it themselves using the tools they already understood.

I don’t even think this phenomenon of “avoiding the platform” is limited to the web. It can apply to any developer working on top of a platform they don’t fully understand. For example, at my work, we use ClickHouse for storing various kinds of analytics data. At one point, my coworker and I disagreed about how to store large JSON data in a column: he built a system for compressing it before storage, whereas I put the data in a separate key-value store and only inserted the key into ClickHouse. It turned out we were both wrong! ClickHouse automatically compresses data, and as a columnar data store you actually get better compression across rows if you just let ClickHouse handle it. And the separate key-value store was just a poor man’s version of what a columnar SELECT already does.

I only realized these things after actually taking the time to thoroughly read the ClickHouse docs and then write a benchmark to prove my hypothesis. In the end I was shocked that we had built something that was slower and clunkier than what the platform itself could give us out-of-the-box. The parallels with JavaScript and the web platform were hard to ignore.

I’m sure that if you’re a developer on iOS or Android, or someone building on top of a game engine, or really any kind of developer building on top of any platform layer, you probably have similar stories. There’s a reason that the stereotype of the grizzled senior engineer is someone who can take a junior’s baroque tangled mess of code and replace it with a single line. The more you learn, the more you’re able to wield your knowledge of how the entire system works end-to-end to create the smallest possible contribution to it (and thus reduce your maintenance burden in the long run).

I’ve been trying really hard not to talk about AI this entire post (because I’ve done it to death over the past year), but of course I can’t help but wonder how AI coding will impact this phenomenon. I actually have both an optimistic and a pessimistic take:

• Optimistic: because LLMs have an encyclopedic knowledge of whatever platform you’re working with, they can choose exactly the right platform API to deliver the experience the prompter asks for in vague English. And since this solution is likely faster and more correct than userland code, the agent will prefer it after rigorous testing and benchmarking. Furthermore, the “IKEA effect” goes away when developers are not actually writing the code themselves.
• Pessimistic: because LLMs seem to love duplicating code – for example, ignoring helper functions that already exist in favor of writing their own for the umpteenth time – the amount of custom, non-platform-idiomatic code will skyrocket. Developers won’t instruct their agents to test enough or to try enough alternatives, and will just commit the agent’s first draft. And because it’s always possible to add more epicycles, the agents will continue to iterate on over-engineered solutions that never should have existed in the first place.

In my own use of AI coding, I’ve seen both phenomena happen. I’d like to think that as models and coding harnesses get better we’ll start to veer more towards the optimistic outcome, but I can’t say for sure.

In any case, these are my longwinded and somewhat conflicting thoughts on “use the platform.” As a mantra I love it, because it succinctly captures a feeling I have when I’m looking at some overwrought pile of spaghetti code and thinking how much better and more elegant it would be if the author just understood the layers beneath them a bit better. At the same time, I’ve been that author, and I’ve felt the joy of building such beautiful, messy code (beautiful to me, anyway), so I think it’s worth understanding where such developers are coming from. For that reason, I’m sure we’ll be hearing “use the platform” for as long as there are platforms.

When Russian air strikes knocked out Ukrainian power plants across the country earlier this year, most of the Black Sea port city of Mykolaiv in southeastern Ukraine was unable to pump fresh water to the city’s housing blocs. But in an easterly suburb of Mykolaiv, drinking water flowed from kitchen taps, as it has since last year thanks to a small off-grid solar field tucked inconspicuously into its midst. Paul Hockenos reports.

The two sets of elevated arrays and battery, funded by the Danish government, run the underground pump on its premises.
‘There’s just enough sun [in January/this winter thus far] to make it work,’ says Olena Kondratiuk, project manager of Ecoclub, a non-profit that assists municipalities in obtaining renewables.

The city of Mykolaiv, only 60 kilometres from Russian-occupied territory, had been battered in 2022 when the Russian army launched its full invasion of Ukraine, pummelling the city with missiles, rockets and cluster bombs. The shelling laid waste to homes, hospitals, schools and factories. The bombardment also impaired the central water supply system, leaving most of the roughly half-a-million-strong town near the frontline without regularly flowing water.

‘Of course I’m happy to have running water again, I don’t care how,’ an older resident named Iryna told me. Until the solar park commenced operation, her taps discharged salty water, accessed from a nearby estuary, when anything dripped from them at all. She either purchased drinking water or filled plastic jugs with fresh water from mobile public tanks, and asked neighbours to carry them up to her fifth-floor apartment.

Other districts of Mykolaiv – the city is a country-wide pioneer in renewables – also receive fresh water on account of distributed solar technology. Five solar-powered desalination systems turn saline or polluted water into drinking water – when the sun shines. At night and on cloudy days, they are powered by the central grid – that is, if it’s up and running. The system, one of the largest of its kind in Europe, can supply 250,000 residents with 1.2 million litres of drinking water a day. The Mykolaiv children’s hospital and the maternity hospital will steady their energy supply with solar panels in the course of this year, as are a dozen others elsewhere in the country.

Since the war’s onset, Russia has relentlessly targeted Ukraine’s energy infrastructure – for the most part, overwhelmingly bulky, old-school thermal plants and the vast country-wide grid – in an effort to lame the war effort and beat down its people. Last winter was the gravest yet: attacks against coal, gas, cogeneration and hydropower plants, as well as transmission grids and their substations, left giant swaths of the country with irregular electricity and heat as temperatures have plummeted to minus 20 degrees centigrade. Many schools and other public services that were unable to reopen after Christmas are still closed today. Economists estimate that total damage to Ukraine’s energy sector now exceeds $56 billion.

US environmentalist Bill McKibben argues that Ukraine’s autonomous power sources are ‘comparatively invulnerable to attack’. In our new era of drone warfare, writes McKibben, centralized and complex energy facilities are now ‘sitting ducks for drone attack’. Smaller, ‘scattered’ generation sources with batteries and independent transmission grids, like the Mykolaiv fields, are not only harder for Russia to strafe, but they’re much more quickly repaired, too. This is in contrast to petrochemical refineries, which McKibben calls ‘one of the most complicated machines humanity has ever constructed’, and often cover hundreds of acres. These are ‘filled with highly complex equipment and highly flammable hydrocarbons; hit one corner with a drone and the flames and the damage are likely to spread quickly’.

‘A coal power station [is] a large single target that a single missile could take out,’ says Jeff Oatham of DTEK, Ukraine’s largest energy investor. ‘You would need around 40 missiles to do the equivalent amount of capacity damage at a wind farm.’ And, like wind parks, a solar field functions even when part of it is out of operation.

Ukraine is revamping its energy system at warp speed for the purpose of energy security, not climate protection. ‘Ukraine’s energy transition is not a slogan,’ says Levgeniia Kopytsia, a Ukrainian energy analyst at the think tank Institute for Climate Protection, Energy and Mobility. ‘It’s a security-driven transformation, unfolding under extreme constraints. It priorities decentralization, flexibility, and speed of recovery.’ In wartime Ukraine, renewables and local grids are coldly calculated security assets, Kopytsia says, that can boost the system’s resilience – and in Ukraine this translates into saving lives.

Ukraine’s shift away from fossil fuels began before the full-scale invasion: its negotiations to join the EU require it to adapt to the bloc’s climate criteria, and in 2021, Ukraine pledged to be coal free by 2035. For security reasons, there are currently no public figures available for energy production.

According to the International Energy Agency (IEA), this past summer, Ukraine made ‘strong strides’ in rebuilding and bolstering its system’s resilience. The renewables rollout was – and still is – led by rooftop solar, small-scale utility PV generation and local storage, as well as biomass combustion.

Before Russia seized three of them in 2022, there were 34 wind parks in Ukraine with around 700 turbines – and more in the works. Ukraine has seven gigawatts of wind power in the pipeline, ready to go this year, should conditions be fortuitous. This would more than triple the country’s current capacity, which in 2022 constituted over 20 per cent of its renewable mix.

The relentless waves of aerial attacks on infrastructure prompted an epiphany in Ukraine’s energy sector: a new, younger set of energy managers – that replaced the Soviet-style old guard – are fully committed to distributed energy – the faster the better, they say. They intend to double the country’s renewable energy consumption in just four years’ time.

While the war has sidelined the rollout of the likes of large-scale projects, such as wind turbines, households, businesses and public institutions have been installing solar at an unprecedented rate. Ukraine’s YASNO, a utility supplying electricity and gas to millions of Ukrainians, says its customers are snapping up the solar and storage packages that it offers. An astounding fact: on sunny days, Ukraine even boasts energy surpluses.
The German Ukrainian Energy Partnership, a platform for intergovernmental dialogue on energy matters, estimates that Ukraine’s potential to add solar capacity in the next five years is equivalent to about 80 medium-sized nuclear reactors. ‘The sector is emerging as one of the fastest-developing renewable markets in Eastern Europe,’ according to its website.

‘Individual consumers want to get off the grid any way that they can,’ explains the Ukrainian energy expert Andriy Martynyuk. ‘It’s largely a grassroots phenomenon and a bit chaotic now.’ He argues that the demand for renewables will shoot up further when the prodigious state subsidies for conventional energy eventually fall away.
This boom, of course, begs for storage options, and there too Ukraine is racing forward. In short order, the country’s national grid operator invested in half a gigawatt of storage capacity last year – an impressive score, according to experts who note that it is just under a quarter of Germany’s total storage supply. The battery projects that in Europe take two years to roll out, in Ukraine happen in six months, reports the Financial Times.

Renewables aren’t the only source of distributed supply. Ukraine envisions large-scale diesel generators, biomass facilities and smallish gas turbines, including combined heat and power plants, as integral to its scattered energy supply. ‘Distributed generation is those facilities that have up to 20 MW capacity,’ explains Roman Nitsovych of DiXi Group, a Ukrainian think tank. ‘Gas-fired units have several advantages: guaranteed output of available power, fast start, and flexibility.’

No matter how intent Ukraine is about diversifying its energy system, it’s going to require a centralized grid. The larger supply options, like the gas-turbines and wind parks, require a functional transmission network, including the substations that the Russians regularly target. The Ukrainians’ playbook is to put them underground – just as it now does its schools – making them impervious to Russian weaponry.

The views and opinions in this article do not necessarily reflect those of the Heinrich-Böll-Stiftung European Union | Global Dialogue.

We’ve passed a big milestone for our digital archive. Let’s talk about where our magazine collection is going next…

The Video Game History Foundation has added over 5000 total magazines to our digital archive, a major milestone in our project to build a first-in-class video game history research library.

When we launched our digital archive last year, we announced an initial collection of over 1500 out-of-print, text-searchable magazines, which are free for research access. In the time since we launched, we’ve more than tripled the size of that collection—now totaling nearly 600,000 pages, the equivalent of nearly 100 feet of shelf space!

Our magazine library now spans 46 years of video game history, from 1981–2026. Major new additions since launch include hundreds of trade magazines, a historically significant newsletter, and publications for niche gaming communities. Altogether, these account for about 60 percent of the Video Game History Foundation’s entire physical magazine collection that we’ve cataloged so far.

This milestone would not have been possible without support from community-run scanning projects. With our combined efforts, we’ve turned their magazine scans into a powerful research tool.

Around one-third of the magazines in our digital archive were digitized by the Video Game History Foundation, with the rest provided by external groups. We want to thank the groups and individuals who have helped:

• The community-run scanning groups Retromags, Out-of-Print Archive, Sega Retro, AtariAge, Atari Compendium, CGW Museum, The Odyssey² Homepage, TOSEC-PIX, eXoDOS, and Gaming Alexandria.
• Industry partners and alumni from Electronic Gaming Monthly, Game Developer, Game Informer, and MCV.
• …and a special shoutout to VGHF Patreon member Chris Chapman, who provided us with 157 scans and digital magazines, the most of any unaffiliated contributor.

It’s only been a year and a half since we launched our digital archive, but we’re already seeing its transformative effect on video game history research. In fact, the VGHF Digital Archive has already become the preferred reference source for out-of-print game magazines on Wikipedia, with over 500 citations and counting. We’re grateful that researchers have responded so quickly to our vision for a well-organized, freely accessible research platform.

We have also heard from the community about what additional features and series they’d like to see in our magazine library. We’ve taken that feedback to heart, and that’s helped us decide where we’re heading next.

Introducing Japanese magazines

You asked, and we’ve delivered: we are now supporting Japanese-language magazines in our digital archive.

Japanese magazines are a vital resource for studying video game history; in particular, they can be a rich source for interviews with developers—most of which have rarely if ever been reprinted—and first-hand accounts of Japanese player culture.

While there are a number of community projects to digitize Japanese magazines, they’ve often been met with unique cultural and legal challenges. We intend for our digital archive to build on the community’s work, much like we’ve done for English-language magazines.

How we’re doing it

Today, as our pilot project, we’re launching research access to Neo Geo Freak, a magazine that covered games for Neo Geo home and arcade platforms from 1995–2000.

Neo Geo Freak felt like a good practical starting place for us. Last year, a generous community member donated the entire run of Neo Geo Freak to our library, and the magazine had already been digitized by the user Japanese Magazine Scans, both of which reduced the amount of work on our end to start exploring what it would take to add Japanese materials to our digital archive.

A snapshot of Neo Geo fan culture in Japan, from Neo Geo Freak, Volume 5, Issue 11, November 1999 (p.132).

More importantly, though, Neo Geo Freak is also a good example of the unique material that can found in Japanese publications. Neo Geo platforms performed especially well in Japan relative to the rest of the world, making this an excellent candidate for the sort of sustained coverage you wouldn’t find in the English-language press.

However, adding Japanese magazines to our library was not as simple as uploading a bunch of PDFs. We maintain a high standard for how we present materials in our archive, and for a content-dense magazine written in a language that our staff isn’t fluent in, meeting those standards was easier said than done.

First, we worked with our community of peers to determine the best methods for Japanese script recognition. Professional archivists in Japan pointed us towards their preferred libraries for Japanese document transcription, which we integrated into the text mapping toolset that we use for English-language magazines.

We believe the result is the most accurate text-searchable Japanese magazines currently in any digital archive. We verified the results with game history researchers who use Japanese materials to ensure that the output is reliable and usable at scale.

An example of Japanese-language OCR in action, from Neo Geo Freak, Volume 2, Issue 2.

To make Neo Geo Freak more discoverable for an English-speaking audience, we took the additional step to index the major articles from complete run of the magazine in English. Rather than attempting to translate 10,000 pages of Japanese text, we assembled a group of Japanese-fluent volunteers to create a list of major articles and interviews for each issue.

We think this is a practical and efficient method for making these magazines’ contents more accessible, while avoiding the pitfalls associated with automated bulk translation. Users are encouraged to translate magazine text using their own tools of choice as needed—even just using their phone camera.

Full support for Japanese magazines also required some improvements to our digital archive platform. New features we’ve added for this pilot project include filtering items by language, displaying right-to-left page layouts, and supporting Japanese text without interword spacing in search results.

We also went back and re-processed Japanese materials that were already in our digital archive, such as our collection of FromSoftware promotional material, to match our new standards.

An example of Japanese-language search results, searching for “Twinkle Star Sprites” in katakana.

What’s next

This is—hopefully!—the beginning of a commitment from us for more international materials in our library. The Video Game History Foundation does not have the most extensive collection of Japanese-language magazines, but all this new structure will make it easier for us to work with the materials we do own.

We also are open to working with other organizations and collectors who have more extensive libraries of Japanese material, but would need support to turn them into a research resource.

This project was made possible thanks to the efforts and input of our community volunteers, who had the subject and language expertise we needed to even consider undertaking this pilot. Thanks to Brian Clark, Rob Curl, Misty De Méo, Dave Fishel, Akito Inoue, Stef “pidgezero_one” Kischak, Lillian McIntyre, Stephen Meyerink, and Adria Miller for their help.

How are we doing?

We want to hear from you about how we’re doing so far. Adding Japanese publications to our research archive is an exciting new step for us, and we want to be sure we’re treating these materials with the respect and fidelity they deserve.

If you work with Japanese-language materials in your research, please let us know what you think. Positive feedback is welcome! We’re eyeing another batch of (gulp) 70,000 pages that we’re planning to add in the coming years, so we want to make sure we’re on the right track before going further.

(And if you are proficient in Japanese, we’d love your help indexing the next magazines we’re planning to add! Please reach out to our library team.)

Thank you to everyone who uses our digital archive and has encouraged us to continue expanding our collections. We’re building our library to support to the study of video game history, and we’re excited every day by how you use these resources.

A Parent’s Guide to Understanding the Pattern, Supporting Connection, and Navigating Friendship

What Rejection Sensitivity Is (and Is Not)

Rejection sensitivity is a heightened tendency to perceive, anticipate, and react intensely to rejection or criticism, including situations that are neutral or ambiguous. It exists on a spectrum. Most people experience some version of it. For gifted and twice-exceptional children, that baseline sensitivity is typically amplified by the same wiring that makes them intellectually intense: a nervous system built to detect and process signals quickly and deeply.

Rejection Sensitive Dysphoria, or RSD, is a more specific and more severe term. Psychiatrist William Dodson coined it to describe the extreme, almost physically painful emotional reaction some people, particularly those with ADHD, have to perceived rejection or a sense of falling short of their own or others’ standards. Dodson found that the large majority of adolescents and adults with ADHD he surveyed recognized this experience in themselves, and a third named it the most impairing symptom they had.

RSD is not a formal diagnosis, and researchers have not fully settled where rejection sensitivity ends and clinically significant RSD begins. That does not make the experience less real for a child living through it. It means parents should treat RSD as a useful, well-supported description of a pattern, not as a diagnostic label to apply on their own.

What This Is and Is Not

What this is: nervous system intensity combined with a child’s identity becoming fused to performance and belonging.

What this is not: manipulation, entitlement, weak character, or a result of bad parenting.

Four Overlapping Populations, One Pattern

Rejection sensitivity shows up across several groups that commonly overlap in twice-exceptional children. Each has its own research base and its own account of why the sensitivity develops. A child rarely fits neatly into just one.

ADHD: Emotional Dysregulation and Reward Sensitivity

This is the most researched population. Dodson’s clinical work links RSD to ADHD’s core but often overlooked feature of emotional dysregulation. ADHD brains tend to be more reactive to both immediate reward and immediate threat, and social approval and social criticism land in exactly those categories. Working memory limits compound this: a child in the moment often cannot hold both “I got corrected” and “my teacher likes me overall” at the same time. The correction temporarily erases the bigger picture.

Autism: A Different Mechanism, a Similar Intensity

Qualitative research with autistic adults describes rejection sensitivity as profoundly overwhelming and exhausting, often accompanied by physical tension, pain, and reliving past rejections. One notable difference from the ADHD pattern is that some researchers suggest autistic individuals may have difficulty consciously processing social evaluative cues in the moment, meaning a child may not always recognize that what they are experiencing is a reaction to rejection, even while the physiological load of it is very real. This matters for parents because a strategy that depends on a child noticing and naming the feeling as it happens may need to be adapted.

High Sensitivity: A Trait, Not a Disorder

Psychologist Elaine Aron’s research on sensory processing sensitivity describes a trait found in roughly 15 to 20 percent of the population. Highly sensitive people show increased emotional reactivity and a stronger response to criticism specifically. Aron’s own framework for supporting highly sensitive people includes reframing how they interpret perceived failures, rejections, and judgment from others, which is directly useful language for coaching both children and the parents raising them.

Giftedness: Overexcitability and Emotional Intensity

Kazimierz Dabrowski’s theory of overexcitabilities, further developed by researcher Linda Silverman, describes gifted individuals as having heightened, inborn sensitivity to stimuli across several domains, including emotional overexcitability marked by intense connectedness with others, deeply felt experience, and strong, well-differentiated feelings about the self. It remains the most widely used framework for describing emotional intensity in gifted children, and many clinicians and parents find it accurate to their child’s lived experience. The research support, though, is weaker than its popularity suggests: meta-analyses show only small, inconsistent differences between gifted and non-gifted samples, and the difference largely disappears when giftedness is measured by cognitive testing rather than by prior gifted identification, which raises the possibility that the pattern reflects labeling and environment as much as innate neurology. Some researchers argue the construct overlaps with the established personality trait of openness to experience and may not need to be gifted-specific at all. The practical takeaway for parents is unchanged: many gifted children report this kind of intensity, and the strategies in this guide still apply, even though the claim that it is a proven, distinct neurological feature of giftedness remains contested.

Twice-exceptional children, gifted alongside ADHD, autism, or another difference, commonly sit at the intersection of two or more of these frameworks at once. That layering is usually why their reactions feel more intense and less predictable than a single-cause explanation can account for.

Why It Happens: The Mechanics Underneath the Reaction

Micro-Shame Activation

Gifted children tend to fuse performance with identity very early. Feedback on an assignment is rarely just about the assignment. It becomes a referendum on whether the child is still who they believe themselves to be.

Social Magnification

A peer doesn’t sit next to them, doesn’t laugh at their joke, or gets picked for something they wanted. What follows is rarely proportional: “no one likes me,” a wave of rage, or a full shutdown. ADHD brains, in particular, are wired to track social status and belonging cues intensely, and differences in reward and threat circuitry seem to make these signals land louder than they would for other children.

Anger as a Defense

In boys especially, though not only boys, rejection sensitivity can present as irritability, dismissiveness, or an argumentative stance: “I didn’t even want that anyway.” This is protective armor. Underneath it is usually plain embarrassment.

Perfectionism and Fragility

Gifted children are used to being right, being fast, and being praised for their intelligence. When ADHD or another difference introduces sloppy mistakes, executive function gaps, or emotional impulsivity, the mismatch between self-image and lived experience creates distress: “I am supposed to be exceptional, so why did I mess up?” Because the identity is more invested, the shame lands harder.

Asynchronous Development

A child’s cognitive age and emotional regulation age can be years apart; for instance, an eleven-year-old mind managing feelings with the regulation capacity of a six-year-old. That gap is where a great deal of the combustion happens.

Identity Investment

When adults consistently praise a child’s intelligence rather than their process or effort, mistakes stop feeling like normal setbacks and start feeling existential. Parents who praise effort over intelligence are ahead of the curve here, but a child’s friends and classmates often have not had that same experience.

How It Shows Up Around Authority Figures

One of the clearest everyday expressions of this pattern is a child’s dread of being called into the principal’s office, a note home from a teacher, or any adult wanting to talk privately, even when nothing is wrong. A highly sensitive or gifted nervous system typically cannot distinguish a neutral request for a conversation from an actual reprimand, so it responds to both with the same alarm.

The underlying fear in these moments is usually not “I will get in trouble.” It is closer to “I will be misunderstood, reduced to one moment, or unfairly categorized.” That distinction changes how an adult should respond. Separating the message from the delivery, staying calm and low-affect, and explicitly saying, “This isn’t trouble; I want to understand what happened,” does more to defuse the reaction than reassurance about consequences ever will.

How It Shows Up in Friendship

The One-Friend Pattern

Gifted children tend to reach for the kind of deep, trust-based friendship researchers call the “sure shelter” years before same-age peers are looking for anything beyond a play partner. Some gifted girls as young as six or seven already hold conceptions of friendship that typical children do not develop until eleven or twelve. When a gifted or twice-exceptional child finally finds someone who reciprocates that intensity, the relief can tip into over-attachment. A school counselor working with gifted students has described this pattern plainly: these kids can become “thirsty” for friendship. Read that way, clinginess is less a character flaw and more a response to scarcity.

Misreading Ordinary Friction as Rejection

Every friendship includes small everyday frictions; researchers call these “slights.” How a child handles slights is one of the strongest predictors of whether a friendship lasts. A highly sensitive or gifted child’s heightened threat detection, the same mechanism behind the rest of this pattern, means an offhand comment or a friend’s momentary preference for someone else often gets processed as a much bigger signal than it was ever meant to be.

Becoming Bossy or Controlling

These children frequently have more advanced, adult-like ideas about fairness and group organization than their peers. When they try to direct play toward what seems logical or fair to them, and peers resist, they can get labeled bossy, snobbish, or rigid. A related but distinct thread is perfectionism: some children need control over how a game unfolds because unpredictability itself, not just losing, is what feels distressing.

What Actually Helps

Strategies below draw on both clinical guidance for 2e children and the friendship research above. None of these require diagnosing a specific condition to be useful.

Normalize the brain, not the behavior

Language such as “I think your brain has a fast alarm system when it feels judged” reduces shame without excusing the behavior that follows the alarm.

Separate performance from self

Explicitly teaching “your work isn’t you, it’s something you made” does not come naturally to most gifted children and needs to be said directly and often.

Pre-correct before feedback

A short framing sentence before any critique, such as “I’m going to give you some ideas to strengthen this; that doesn’t mean it’s bad,” protects the child’s nervous system before it activates rather than trying to calm it afterward.

Spot the spike

A simple model many 2e kids respond well to: trigger, story, body reaction, choice. Most gifted children enjoy this kind of meta-cognitive tracking and can approach it almost like a scientist studying their own reactions. For autistic children, remember that conscious awareness of the rejection response may be less reliable, so this tool may need to lean more on noticing body signals than on naming the social trigger itself.

Build tolerance through micro-repair

Rather than shielding a child from all correction, practice small feedback, immediate repair, and visible appreciation right afterward. This is slow, deliberate work to rewire what the child expects to happen after a mistake.

Broaden the friend pool

Because the one-friend pattern often comes from scarcity of like-minded peers, structured groups built around shared interests tend to work better than simply increasing exposure to same-age children. The goal isn’t to discourage the close friendship, but to make sure it isn’t the only source of connection.

Coach the slight-versus-rejection distinction directly

Naming this distinction out loud, and helping a child check their read of a situation before reacting, gives them a tool that plain reassurance does not.

Model repair after your own conflicts

Children absorb far more from watching a parent apologize, disagree, and reconnect with another adult than from being told that friendship conflict is survivable.

When to Look Beyond Parenting Adjustments

Some of what is described in this guide is temperament and environment, both of which shift with awareness and the strategies above. Some of it is closer to clinical anxiety or depression and needs support beyond what a parent or educator can offer alone. Consider bringing in an outside professional if a child shows significant avoidance of school or social situations, physical symptoms tied to social stress, school refusal, or any signs of hopelessness or self-harm. Not every case resolves through parenting changes alone, and it helps to say so plainly.

Books to Read Together, by Age

Picture books and early-elementary titles work well because the story does the teaching. Tween and teen titles work differently: they tend to sit in ambiguity rather than resolve into a tidy lesson, which fits the developmental stage better but usually means a parent needs to carry more of the discussion afterward.

Ages 3 to 7

• Gossie & Gertie, by Olivier Dunrea, a gentle look at one friend being bossier and both learning to share control
• A Bad Case of Stripes, by David Shannon, the fear of losing friends by not conforming
• Something Else, by Kathryn Cave, wanting a friend exactly like yourself
• Elephant & Piggie: Happy Pig Day, by Mo Willems, jealousy when a close friend makes a new friend

Ages 6 to 10

• Teamwork Isn’t My Thing and I Don’t Like to Share, by Julia Cook, directly addresses bossy and controlling behavior with peers
• Enemy Pie, by Derek Munson, misreading a peer as a rival
• Pink Tiara Cookies for Three, by Maria Dismondy, jealousy and exclusion when a two-person friendship becomes three

Ages 9 to 13

• Real Friends, by Shannon Hale, a graphic memoir showing the ambiguity and shifting loyalty of real friendships without a tidy moral
• The Science of Friendship, by Tanita S. Davis, built around real research on friendship and stress, pairs naturally with this guide’s framing
• Harriet the Spy, by Louise Fitzhugh, a classic portrait of an intense, socially awkward, gifted-type child

Ages 11 and Up

• Skills for Rejection Sensitive Dysphoria: A Workbook for Tweens and Teens, a graphic novel workbook that directly teaches kids to recognize their own rejection alarm and use regulation skills, closer to a teaching tool than a mirror story

Sources and Further Reading

• William Dodson, M.D., “Rejection Sensitive Dysphoria,” ADDitude Magazine and CHADD’s Attention magazine
• Elaine Aron, Ph.D., The Highly Sensitive Person and hsperson.com
• Linda Silverman, “Dabrowski’s Overexcitabilities: A Layman’s Explanation,” and the Davidson Institute’s gifted education resources
• Dana Winkler and Alan Voight, meta-analysis on giftedness and overexcitability, published in Gifted Child Quarterly, 2016
• Paula Olszewski-Kubilius and colleagues, meta-analysis on overexcitabilities and giftedness, published in Gifted Child Quarterly, 2026
• Salvatore Mendaglio, “Overexcitabilities and Giftedness Research: A Call for a Paradigm Shift,” Journal for the Education of the Gifted, 2012
• Julie MacEvoy, Ph.D., Boston College, research on friendship and response to social slights
• Miraca Gross, research on friendship stage development in highly gifted children
• van Asselt, Roke, Begeer, and Scheeren, “Feeling Constantly Kicked Down,” a 2025 qualitative study of rejection sensitivity in autistic adults
• Davidson Institute for Talent Development, gifted-blog archive on peer relationships and overexcitability

Blindsight is a hard science fiction novel by Canadian writer Peter Watts, published by Tor Books in 2006. It won the Seiun Award for the best novel in Japanese translation (where it is published by Tokyo Sogensha) and was nominated for the Hugo Award for Best Novel, the John W. Campbell Memorial Award for Best Science Fiction Novel, and the Locus Award for Best Science Fiction Novel. The story follows a crew of astronauts sent to investigate a trans-Neptunian comet dubbed "Burns-Caulfield" that has been found to be transmitting an unidentified radio signal, followed by their subsequent first contact. The novel explores themes of identity, consciousness, free will, artificial intelligence, neurology, and game theory as well as evolution and biology.

Blindsight is available online under a Creative Commons Attribution-NonCommercial-ShareAlike license. Its sequel (or "sidequel"), Echopraxia, came out in 2014.

Plot

In the year 2082, tens of thousands of coordinated comet-like objects of an unknown origin, dubbed "Fireflies", burn up in the Earth's atmosphere in a precise grid, while momentarily broadcasting across an immense portion of the electromagnetic spectrum, catching humanity off guard and alerting it to an undeniable extraterrestrial presence. It is suspected that the entire planet has been surveyed in one effective sweep. Despite the magnitude of this "Firefall", human politics soon return to normal.

Soon afterwards, a comet-surveying satellite stumbles across a radio transmission originating from a comet, subsequently named 'Burns-Caulfield'. This tight-beam broadcast is directed to an unknown location and in fact does not intersect the Earth at any point. As this is the first opportunity to learn more about the extraterrestrials, three waves of ships are sent out: the first being lightweight probes shot out for an as-soon-as-possible flyby of the comet, then a wave of heavier but better-equipped probes, and finally a crewed ship, the Theseus.

Theseus is propelled by an antimatter reactor and captained by an artificial intelligence. It carries a crew of five cutting-edge transhuman hyper-specialists, of whom one is a genetically reincarnated vampire who acts as the nominal mission commander. While the crew is in hibernation en route, the just-arrived second wave of probes commence a compounded radar scan of the subsurface of Burns-Caulfield, but this immediately causes the object to self-destruct. Theseus is re-routed mid-flight to the new-found destination of the signal: a previously undetected sub-brown dwarf deep in the Oort cloud, dubbed 'Big Ben'.

The crew wakes from hibernation while the Theseus closes on Big Ben. They discover a giant, concealed object in the vicinity, and assume it to be a vessel of some kind. As soon as the crew uncloaks the vessel, it immediately hails them over radio and, in a range of languages varying from English to Chinese, identifies itself as 'Rorschach'. They determine that Rorschach must have learned human languages by eavesdropping on comm-chatter since its arrival, sometime after the Broadcast Age began. Over the course of a few days many questions and answers are exchanged by both parties. Eventually Susan James, the linguist, determines that 'Rorschach' does not really understand what either party is actually saying.

Theseus probes Rorschach and finds it to have hollow sections, some with atmosphere, all filled with levels of radiation that render remote operation of machinery virtually impossible and would kill a human in a matter of hours. Despite this and over Rorschach's objections the whole crew except the mission commander enters and explores in a series of short forays, using the ship's advanced medical facilities to recover from the damage the radiation inflicts on their bodies. They discover the presence of highly evasive, fast-moving nine-legged organisms dubbed 'Scramblers'. They kill one and capture two for study. The 'Scramblers' appear to have orders of magnitude more brainpower than human beings but use most of it simply to operate their fantastically complex musculature and sensory organs; they are more akin to something like white blood cells in a human body. They are dependent on the radiation and EM fields of Rorschach for basic biological functions and seem to completely lack consciousness.

The crew explore questions of identity, the nature, utility and interdependence of intelligence and consciousness. They theorize that humanity could be an unusual offshoot of evolution, wasting bodily and economic resources on the self-aware ego which has little value in terms of Darwinian fitness. Open warfare breaks out between the humans and the Scramblers and Theseus eventually decides to sacrifice itself and its crew using its antimatter payload to eliminate Rorschach. One crew member, the protagonist and narrator Siri Keeton, is shot off inside an escape vessel in a decades-long fall back to Earth to relay the crucial information amassed back to humanity.

Characters

Crew of the Theseus

• Siri Keeton is the narrator and protagonist. Debilitating brain surgery for medical purposes has cut him off from his own emotional life and made him a talented "synthesist", adept at reading others' intentions impartially with the aid of cybernetics. He is assigned to Theseus to interpret the actions of the specialized crew and report these activities to Mission Control on Earth.

• Major Amanda Bates is a combat specialist, controlling an army of robotic "grunts".

• Isaac Szpindel is the ship's primary biologist and physician. He is in love with Michelle, one of the Gang's personalities.

• Jukka Sarasti is a vampire and the crew's nominal (and frightening) leader. As a predator from the Pleistocene, he is alleged to be far smarter than baseline humans.

• The Gang are four distinct personalities in the mind of one woman, the ship's linguist. They are tasked with communicating with the aliens, if possible. A single personality "surfaces" to take control of their body at any given time. The active personality reveals itself through a change in tone and posture. These personalities express offence when referred to as "alters". The personalities are:

  • Susan James, whom the others refer to as "Mom". She is the "original" personality.
  • Michelle is a shy, quiet, synaesthetic woman who is romantically involved with Szpindel.
  • Sascha is harsher and more overtly hostile towards Siri.
  • Cruncher, a male personality, rarely surfaces and serves as an advanced data-processing facility for James.

• Robert Cunningham, Szpindel's backup, is a secondary biologist/physician. He possesses a set of enhancements that allow him to process data with additional senses, and somewhat inhabits the machinery connected to him.

• The Captain is the ship's artificial intelligence. Throughout the story, the Captain remains inscrutable and mysterious, generally communicating directly only with Sarasti.

People on Earth

• Robert Paglino, Siri's childhood best friend and a practical example of Siri's muted emotions: Siri cannot actually feel "friendship" following his brain surgery, but intellectually knows how he is expected to behave as a friend and continues to play the part.
• Chelsea, Siri's ex-girlfriend. A professional tweaker of human personalities.
• Helen Keeton, Siri's mother, whose consciousness has been connected, brain in a vat style, to a virtual utopia called "Heaven". As a parent, she traumatized Siri with emotional demands and intrusiveness into his private life.
• Jim Moore is Siri's father, a colonel involved with planetary defense.

Aliens

• Rorschach, an alien vessel or organism in low orbit around the sub-brown dwarf Big Ben. While it has a superhuman intelligence, it gradually becomes apparent that Rorschach completely lacks true consciousness or self-awareness.
• Scramblers, 9-legged anaerobic aliens that inhabit Rorschach and appear to be part of it in some sense. Like Rorschach, they are more intelligent than humans but not conscious or self-aware.

Major themes

Consciousness

The exploration of consciousness is the central thematic element of Blindsight. The title of the novel refers to the condition blindsight, in which vision is non-functional in the conscious brain but remains useful to non-conscious action. Other conditions, such as Cotard delusion and Anton–Babinski syndrome, are used to illustrate differences from the usual assumptions about conscious experience. The novel raises questions about the essential character of consciousness. Is the interior experience of consciousness necessary, or is externally observed behavior the sole determining characteristic of conscious experience? Is an interior emotional experience necessary for empathy, or is empathic behavior sufficient to possess empathy? Relevant to these questions is a plot element near the climax of the story, in which the vampire captain is revealed to have been controlled by the ship's artificial intelligence for the entirety of the novel.

Philosopher John Searle's Chinese room thought experiment is used as a metaphor to illustrate the tension between the notions of consciousness as an interior experience of understanding, as contrasted with consciousness as the emergent result of merely functional non-introspective components. Blindsight contributes to this debate by implying that some aspects of consciousness are empirically detectable. Specifically, the novel supposes that consciousness is necessary for both aesthetic appreciation and effective communication. However, the possibility is raised that consciousness is, for humanity, an evolutionary dead end. That is, consciousness may have been naturally selected as a solution for the challenges of a specific place in space and time, but will become a limitation as conditions change or competing intelligences are encountered.

The alien creatures encountered by the crew of the Theseus themselves lack consciousness. The necessity of consciousness for effective communication is illustrated by a passage from the novel in which the linguist realizes that the alien creatures cannot be, in fact, conscious because of their lack of semantic understanding:

"Tell me more about your cousins," Rorschach sent.
"Our cousins lie about the family tree," Sascha replied, "with nieces and nephews and Neanderthals. We do not like annoying cousins."
"We'd like to know about this tree."
Sascha muted the channel and gave us a look that said Could it be any more obvious? "It couldn't have parsed that. There were three linguistic ambiguities in there. It just ignored them."
"Well, it asked for clarification," Bates pointed out.
"It asked a follow-up question. Different thing entirely."

The notion that these aliens could lack consciousness and possess intelligence is linked to the idea that some humans could also have diminished consciousness and remain outwardly functional. This idea is similar to the concept of philosophical zombie, as it is understood in philosophy of mind. Blindsight supposes that sociopaths might be a manifestation of this same phenomenon, and the demands of corporate environments might be environmental factors causing some part of humanity to evolve toward becoming philosophical zombies.

Bicameral mind

See Bicameral mentality

Transhumanism

Blindsight also explores the implications of a transhuman future. Within the novel, humans no longer engage in sex with other humans for pleasure, instead choosing to use virtual reality to find idealized partners, and many choose to withdraw from reality entirely by living in constructed virtual worlds, referred to as "Heaven". Vampires are predators from humanity's distant past, resurrected through recovered DNA, and live among the humans of the late 21st century. These vampires operate with diminished sentience presented as comparable to high-functional autism with comparable dysfunction in affect and speech, but have the advantage of multiple simultaneous thoughts occurring in parallel within their minds. Enhanced pattern-matching skills comparable to some forms of autism combine with this "hyperthreading" to make them invaluable in developing unusual and often very effective approaches to solving complex problems.

Reception

Carl Hayes, in his review for Booklist, wrote, "Watts packs in enough tantalizing ideas for a score of novels while spinning new twists on every cutting-edge motif from virtual reality to extraterrestrial biology." Kirkus Reviews said about the book, "Watts carries several complications too many, but presents nonetheless a searching, disconcerting, challenging, sometimes piercing inquisition." Jackie Cassida in her review for Library Journal wrote, "Watts continues to challenge readers with his imaginative plots and superb storytelling." Publishers Weekly wrote, "Watts puts a terrifying and original spin on the familiar alien contact story."

Elizabeth Bear, an award-winning author in the science fiction field, declared the following:

It's my opinion that Peter Watts's Blindsight is the best hard science fiction novel of the first decade of this millennium – and I say that as someone who remains unconvinced of all the ramifications of its central argument. Watts is one of the crown princes of science fiction's most difficult subgenre: his work is rigorous, unsentimental, and full of the sort of brilliant little moments of synthesis that make a nerd's brain light up like a pinball machine. But he's also a poet – a damned fine writer on a sentence level...

Adaptations

In October 2020 a non-commercial Blindsight short film was released. Watts describes it as, "snatches of Blindsight recalled by Siri Keeton during one of his waking interludes in the aftermath of that novel. Spectacular highlights arranged in reverse order, Memento-like". In April 2025, Neill Blomkamp was set to adapt the novel into a feature film.

Headlines and History

The headlines keep coming: such and such an AI system has solved such and such a math problem. And more and more I’m hearing people saying: maybe we don’t need people doing math research anymore; maybe we should just delegate it all to more and more powerful AIs.

I must admit that I’m getting a bit impatient with some of what’s being said. Because it seems to me too often to involve fundamental misunderstandings of what math is really about—and also, frankly, what AI is about.

Perhaps I have a unique piece of personal history that informs this. After all, back in 1988, when we first introduced Mathematica, there was also some of the same kind of talk about math being taken over, and made pointless. Of course that’s not how it worked out at all. Instead, Mathematica (now Wolfram Language) just raised the level of math that can be done—and over the years led to all sorts of important new math.

If working out symbolic integrals was what one thinks doing math is really about, then, yes, Mathematica has essentially replaced it. But while that kind of problem solving is what’s needed in many applications of math, it’s not the core of what math itself, in its pure form, is about.

The enterprise of pure mathematics is an old one, crucially entwined with the history of civilization. From the time of Plato and Euclid pure mathematics was the defining example of a place where abstract, rational thought could build an ever larger structure. And over the centuries, pure mathematics has come to be the single largest intellectual edifice that our civilization has built.

It’s not been without its pathologies and limitations. And even among those building the edifice one runs into misunderstanding about what’s important, how things should be done, etc. Is mathematics fundamentally about producing proofs, by whatever means necessary? Is mathematics always ultimately justified by its applications? Is there some inevitable “book of right answers” that it is the goal of mathematics to discover?

Again, I suppose, I have some personal history in all of this. Because my efforts in basic science have led me to ask questions about the foundations of many things, including mathematics. So I’ve studied questions like what the space of all possible mathematicses is, what the limiting structure of the network of all theorems might be, and, notably, what the role of humans is in defining the thing we call mathematics.

The Value of Modern AI

We’ll talk later about the general character and value of pure mathematics. But before that, let’s address the issue of the moment: the role of modern AI.

And the first thing to say is that it’s unquestionably useful, sometimes very useful. For me, its greatest use in mathematical pursuits has been its ability in effect to thematically mine the knowledgebase of human mathematics. Starting back in the 1970s, being able to do keyword searches of the scientific literature was a crucial enabler of quite a bit of the research that I did. And now, with modern AI, one can do so much more. Because somewhere inside those LLMs—in a way that we don’t yet scientifically understand—there’s what amounts to a representation of raw ideas gleaned from all those millions of papers and books about mathematics.

And, at its best, it’s not just about retrieving things. It’s also about making connections. Of being able to see that this result here can be put together with that result there to come up with a surprising and useful conclusion. Humans routinely do that too. But they tend to have only read hundreds of papers; AIs have effectively read millions, and it’s cheap for them in effect to try out lots and lots of possible combinations.

So can one expect to just launch an AI off and have it come back with great math? As we’ll discuss, great math is—more than anything else—defined by the questions it asks. Yes, the AI can successfully automate things that humans would normally have had to do themselves before. But—as we’ll discuss later—at the core of pure mathematics is the human imagination that guides what questions to ask.

It’s worth understanding the difference between what modern AI does, and what pure computation does. Modern AI is, first and foremost, a way of leveraging the existing corpus of human knowledge. Computation is—at its most powerful—an open-ended way to generate things that are fundamentally and irreducibly new. Start from some rule or axiom, and just repeatedly run the computation of applying it, and my all-time favorite phenomenon of computational irreducibility guarantees that you’ll go on getting fresh, new results that can’t be reached except by doing all those computational steps.

(For those who aren’t already familiar with it, computational irreducibility is an idea I introduced in the 1980s to capture the notion that many computational processes—even when defined by simple rules—allow no general shortcut: the only way to determine their outcome is to explicitly run each step. It’s turned out to be a phenomenon that’s quite ubiquitous in the computational universe of possible programs—and to be connected to a long sequence of foundational issues in many areas of science, as well as in philosophy, etc.)

So, yes, having nothing to do with AI, computation can generate an infinite sequence of new theorems, representing an infinite sequence of new facts, etc. And among those theorems there’ll be all sorts of “originality” and, in effect, “surprise”. But there’s a catch. Those theorems are in a sense just “results plucked from the computational universe”. And as such they certainly fall under the purview of the new—and I think very important—field of ruliology on which I have spent so much effort. But are they math?

What Is Math Anyway?

Well, of course that depends on what math really is—which is exactly what we need to understand. In its early history, math was thought of as a way of making precise and formal statements about the world, say about arithmetic or about geometry. But by the later part of the nineteenth century higher levels of abstraction had been reached, no longer tethered to features of the world as we experience it. And there swept across mathematics an increasingly formalistic view: that ultimately math really is just the collection of theorems that can be “mechanically” (i.e. computationally) derived from certain axioms, say the axioms of set theory.

Gödel’s theorem put a small dent in that picture. But even today, if pressed, many mathematicians will try to define mathematics as being the formal study of the consequences of certain chosen axioms. But while they may say that, it’s not a good description of what they actually do. The vast majority of actual pure math research does not operate at the level of axioms and mechanical derivations. Instead, it works at a much higher level, building and studying abstract structures and their interrelationships.

It’s not obvious that this should be possible. It could be that the only way to get right answers in math would be to operate at the lowest level, say directly in terms of axioms. But it’s an essential—if typically unspoken—feature of pure mathematics that in practice one can reason at the level of, say, the Pythagorean theorem, without constantly having to dive down and talk, say, about the axiomatic definition of real numbers. I’ve recently argued that this happens for much the same reasons that in physics we can successfully do fluid mechanics, without always having to dive down and trace the collisions of individual molecules.

Ultimately the fluid is made of all those molecules. But the point is that observers like us typically sample it only at the much more human level of overall fluid motions, etc. And so it is also with mathematics. Human mathematicians typically sample the vast metamathematical web of underlying axiomatic derivations only in overall, collective ways, in terms of “human-level” structures and concepts.

Ultimately the story actually seems to be very much the same in mathematics and in physics. At the lowest level, everything is full of computational irreducibility, so that one can figure things out only by mechanically taking every computational step. But within such computational irreducibility there are always pockets of computational reducibility where it’s possible to “jump ahead” in a “higher-level” way. And it’s within these pockets of reducibility that our laws of physics—and our human-level mathematics—reside.

We can see physics—or indeed natural science in general—as an effort to take the complexities of the natural world and find aspects that can be described by narratives that fit in finite human minds. Mathematics can be seen in very much the same way: as an effort to take the complexities of the metamathematical world and find aspects that can be described by narratives—now mathematical ones—that fit in finite human minds.

In other words, the problem of advancing pure mathematics is at some level not so much about pushing back some raw metamathematical frontier as about finding human ways to represent what’s there.

The history of human mathematics has been characterized by the building of ever taller towers of concepts—that capture ever greater abstraction. And one might suppose that this process would somehow in the end inevitably “reveal all mathematics”. But the story is more complicated. One can imagine that as an ultimate limit one could start from all possible axiom systems and generate all possible theorems. What one gets this way is a unique object that I call the ruliad—that corresponds to the entangled limit of all possible computational processes. But the issue is that finite minds—like ours—can only perceive a tiny part of the ruliad. And that means that we can have no “absolute mathematics”, only mathematics based on how we sample the ruliad.

I’ve argued elsewhere that some aspects of this sampling follow inevitably from general features of the way we are as observers of the ruliad. And this leads to certain “general laws of mathematics”, such as the very fact that higher-level mathematics is possible—at least for observers like us. But other aspects of the sampling we can see as historical accidents: choices that the mathematical community made to explore one direction rather than another. And once again it’s important to recognize that it’s inevitable that one has to make such choices.

In a sense the ruliad is too big for it to be otherwise: our finite minds can’t span all directions, so we have to choose just some. And then we have to summarize what we find in terms of some limited set of concepts—out of which we can then build our mathematical narratives.

It’s very much like with human language. Out of all possibilities we choose certain concepts to represent, say by words. And these concepts are how we “encapsulate thoughts” to be able to communicate them in finite ways, and be able to fit them in our minds.

At a raw computational level it’s straightforward to do what amounts to mathematical ruliology and start axiomatically generating huge numbers of new theorems that have never been seen before. But with overwhelming probability this will lead us only to “alien mathematics” that doesn’t usefully connect with mathematics as we know it—or, for that matter, with anything that we can view as a higher-level concept at all.

Generating Math with AI

But, OK, let’s say we’re using modern AI. Can’t it operate directly with higher-level mathematical concepts? Certainly LLMs manage to deal with human language. And indeed, just as LLMs successfully learn the general patterns for putting together words in human language, so also they can learn the general patterns for putting together constructs in mathematics.

But generating useful mathematics is a much more exacting activity than generating language. A written story can’t really be “wrong”; a written piece of mathematics certainly can be. Of course, it helps a lot when the LLM can call on Wolfram Language—as humans do—to reliably compute things. (And, yes, everyone should spend the few seconds it takes to connect their AIs to our Wolfram MCP system!) But a serious piece of math research will typically involve fitting together many parts, doing many steps in an argument, etc. And the issue is that the fundamentally statistical nature of how an LLM works inevitably makes it progressively less likely that, as things get more complicated, they will work out correctly. And it can become something of a game to try to find cases where the LLM happens to do the right thing.

But how can one even tell? It’s a frustrating feature of modern times that someone like me gets sent many AI-generated documents every day that have the “statistical texture” of math papers, but that one at least expects have a very low probability of being meaningfully correct (and that were, one assumes, mostly made by people who didn’t particularly understand math themselves, but just told their AIs to “make math” for them). So, yes, the LLM can in effect work in terms of human-level mathematical concepts, but with the same kinds of uncertainties and imprecisions that affect human thinking.

The Power and Challenge of Formalization

But what about turning those human-level constructs into something precise and formal—something at a more axiomatic level? And, yes, I’ve been involved in thinking about such things for several decades. There’s been quite a bit of excitement of late in the idea of autoformalization: take “human-level math”, and automatically turn it into something formal and axiomatic, that can for example be verified by a proof assistant system. At first it sounds like a good idea. But there’s a problem. Yes, the proof assistant might verify a proof. But was it a proof of what you thought it was a proof of? The weak link is being sure that the formalization of your human-level math is correctly capturing what you were trying to express.

I’ve had the experience quite a few times now: I try to autoformalize something, and an AI will tell me “I did it; look, the proof checks out!” But, actually, in some sense it cheated: instead of formalizing what I intended, it found a (sometimes very squirrely) way to interpret what I asked so that it could successfully prove it. It’s often very hard to tell, though, that this is what happened—not least because the formalized versions of things (say as expressed in popular proof assistant systems) tend to be very low level, very verbose and very hard for us humans to understand.

So what can one do? Well, I actually think there’s something very good one can do, that’s centrally based on the tower of ideas and technology that we’ve been developing for so long in the Wolfram Language. Our goal with the Wolfram Language has always been to create a language—or in effect a notation—that can represent in precise computational terms things we humans think about. It’s not like a programming language where one’s just representing raw constructs and operations in a computer; instead it’s based on representing actual things in the world, and actual concepts we think about. And also, it’s not just a way of telling a computer what to do; a bit like a vast generalization of mathematical notation, it’s a language that one can read, and think in.

In the past couple of years a powerful general workflow has emerged: tell an AI what you want, then have the AI write Wolfram Language to give a precise representation of what it thinks you mean. If it’s properly done, this representation should be something you as a human can understand—then check and modify as desired.

A New High-Level Language for Pure Mathematics

Over the past four decades we’ve steadily grown the Wolfram Language to cover a great many domains, including most of what one can think of as today’s “applicable mathematics”. And indeed what the Wolfram Language does has proved extremely useful in pure math research. But in the past—notwithstanding our efforts a bit more than a decade ago—we’ve never quite been able to justify building broad capabilities specifically for pure math research. But thanks in part to the general advance of the Wolfram Language platform, and in part to the changing economics of development associated with AI, we are in the middle of a large effort to extend the design of Wolfram Language to encompass the constructs of pure math research, from sheaves to Lie groups to Clifford algebras.

Our goal is to use our methods and long experience in computational language design to make pure math broadly computational—in a way that both humans and AIs can use. And, yes, we can then at long last expect to have papers where underneath every mathematical statement is a precise and unambiguous computational version, that one can systematically execute and process computationally.

But also we’ll have a real target for autoformalization—in effect a nexus of human mathematics, AI and computation. So let’s say we have a statement comprehensibly formalized in our pure-math-extended Wolfram Language. Often what we’ll in practice want to do with it is to compute a result from it. But sometimes—following the traditions of pure math—we’ll just want to abstractly prove it.

Automated Theorem Proving

So can we do that automatically? Well, there’s automated theorem proving, many types of which are, for example, built into the Wolfram Language. Sometimes automated theorem proving is in effect implicit (like, “Solve showed this polynomial has no real roots”). Sometimes (like with FindEquationalProof) it can be more explicit, with a precise—and easily verifiable—symbolic representation of all steps in a proof.

One might imagine this would be something broadly powerful for pure math. But at least for the past many decades it hasn’t been. A large part of the reason is that it’s been hard to break typical research-level pure math down to a sufficiently granular axiomatic level to make it amenable to such methods. But another part has been that—thanks to the phenomenon of computational irreducibility, and undecidability—proofs can be unreachably long.

And indeed I think it’s fair to say that in the whole history of automated theorem proving there’s actually only one example of something that can even plausibly be considered a real mathematical result that wasn’t already believed to be true, and that was found for the first time with automated theorem proving. It’s something I did in 2000—to find the minimal axiom system for Boolean algebra. But there was another problem here: the proof that was found is very long, very low level, and very alien. And in the past 26 years it’s never been possible—with AI or otherwise—to find any human-level version of it. In effect, so far as we can tell, it’s a kind of prong in the metamathematical universe, unconnected to existing human-level mathematical concepts.

Of course, we can just use the result of the proof—as a kind of “black-box piece of mathematics”. But insofar as our objective in mathematics is to provide human-level mathematical narrative this represents a gap in the narrative.

But let’s say instead you take a proof that was already known in the mathematical literature, and you somehow formalize it—whether by AI or by direct human effort. If it’s a complicated proof, then this is certainly an achievement. And if you’re sure that you actually formalized the right thing, then it’s a good check that the proof really is correct. But one shouldn’t expect to get new understanding, or to surface new reusable components. And then there’s the question of actual mathematical content. Yes, a formal result has been established—but for example what underlying axiom system did it use (and, no, it’s probably in the end not something as familiar as set theory)?

(It’s worth commenting that formalized proofs can themsleves become objects of study. For example, with many proofs of a given theorem, one can start asking about the geometry and topology of proof space. And with proofs of many theorems one can start doing “empirical metamathematics” to understand their fundamental relations.)

Problem Solving in Math

As we’ll discuss, much of the most important progress in pure mathematics comes from the invention of new concepts and structures (often encapsulated in new definitions). But there’s still a vast amount to do operating with existing concepts and existing structures. Sometimes the goal is to “compute an answer”. Sometimes it’s to “find a proof”. Sometimes it’s to “find an example or counterexample”. And sometimes it’s to “figure out what’s true”. In all of these types of cases there’s plenty of powerful algorithmic computation that can be used (e.g. in Wolfram Language) to get results, often very directly and quickly.

AIs can certainly use such computation (say through Wolfram MCP). But how can modern AI directly contribute? One can imagine just straightforwardly asking it “chatbot style” to generate results. And often, yes, it will come up with a result. But how can one tell if it’s correct? Sometimes the only way is in effect to formalize the whole process of getting it, as we discussed above. But sometimes we can just take the result and do a computation on it to check if it’s correct. For example, if the AI claims to solve an integral, we can just differentiate its solution and see if we get back something equivalent to the integrand. Or if the AI comes up with a counterexample to something, then we can just do a computation to check whether it is in fact a counterexample.

We don’t know much about what’s “going on inside” when an LLM comes up with a result. But it’s probably not a bad model to think of it as a process of fitting together digested pieces from the millions of mathematical papers on which it’s been trained. In the end, though, what it does may actually amount to something that’s algorithmically quite simple. The AI has the advantage of great breadth. And in my experience it’s rather common to find that it’s come up with something that would be quite simple—or even obvious—if one just happened to look in the right place. But having found it, it can then be implemented as a piece of pure computation (e.g. in Wolfram Language), that doesn’t need any AI around.

When humans do math and solve problems an important part of the process tends to be the use of intuition. But what actually is intuition? At some level it seems to be a kind of “I know how this will go” procedural pattern matching. And—speaking of intuition—my intuition is that at least in its basic form this isn’t a capability that’s fundamentally out of reach for LLMs.

But one of the challenges is that even to solve a problem stated in terms of existing mathematical concepts it’s common to have to “break out of the system” and invent new concepts—a classic example being the need for complex numbers in finding even real solutions to cubic equations. At some level this is yet another manifestation of computational irreducibility, and the way it limits any given pocket of computational reducibility. But its effect is that it forces what might seem like “pure problem solving” to actually often involve the invention of new concepts and structures.

Open-Ended Math

So what about what one can think of as open-ended math? In ruliology it’s very common to just go out into the computational universe and “look for interesting things”. In math that’s traditionally not something that’s common. Of course, one can certainly in principle imagine, say, systematically enumerating possible theorems based on some set of axioms, and then asking: which of these are “interesting”? Operationally one might ask “Which of these theorems would one mention if one was writing a book or paper?”

Years ago I looked at the very simple case of possible theorems of Boolean algebra and found empirically that there was a criterion for which of them are typically given names in textbooks of logic: it’s basically ones that can’t be proved from theorems earlier in a lexicographically ordered list of possible theorems—or, in some sense, the simplest theorems that “give new information”.

In more complicated cases one might imagine that an AI trained on millions of examples of theorems that people chose to publish in the literature of mathematics might be able to channel human criteria for interestingness. But in my experience this doesn’t really work. And I think the reason is that in all but the simplest cases interestingness is not something that can be determined theorem by theorem: rather, it requires one to map out the whole historical development of a particular area of mathematics. And, yes, then one can ask what aspects of the arc of that development are somehow inevitable, and what are just historical accidents.

I’ve asked a somewhat similar question recently for biological evolution. And in that case I’ve discovered that there do seem to be some (fundamentally computational) general principles. For example, adaptive evolution according to a computationally limited fitness function quite generally seems to lead to the presence of modular structures (analogous to organelles, organs, etc.) Those structures don’t “come with an explanation of what they are”. But they do in effect represent “raw pockets of computational reducibility”.

In the development of pure mathematics one thing that’s clearly critical is the identification of new structures and concepts. Historically such structures and concepts have always been introduced by humans, and have in effect “come with explanations”. But could one for example imagine new structures and concepts that are automatically found, say by an AI? And could one then build new mathematics with these?

New Mathematical Concepts

Those two questions turn out to be somewhat separate. At least at a simple level, it’s fairly straightforward (say by looking at autoencoders) to identify in neural nets particular patterns of activation that one can think of as summarizing pockets of computational reducibility, or in effect “representing new concepts”. But insofar as mathematics is about defining mathematical narratives for us, this isn’t immediately helpful.

Yes, the AI can in effect identify zillions of new concepts. But our human minds can only deal with limited numbers of them. It’s very much like with natural language. We can imagine an AI looking at patterns of human discourse and figuring out zillions of new words that could be introduced to summarize them. But we humans seem to only be able to know about a limited number of distinct words (typically a few tens of thousands).

So what about in mathematics? An AI could in principle come up with a zillion definitions of new mathematical concepts. But something built from them will inevitably seem about as alien as something built “ruliologically” from raw axiomatic-level mathematics. The building blocks might be more “human level”. But they’re not familiar to us humans, and so we can’t readily think in terms of them.

There are of course new words introduced into human languages, and new concepts introduced into mathematics. But it’s a gradual process, that in practice tends to involve a certain “societal consensus”. (And indeed in developing the design of the Wolfram Language over the past four decades, I’ve quite explicitly limited the rate at which new concepts are introduced, to make sure to let people keep up.)

The Goals of Math

But, OK, so there are lots of possible new concepts and new structures—all corresponding to pockets of computational reducibility—that are “out there in the metamathematical universe”. But in developing mathematics suitable for us humans, we can only ever pursue a tiny fraction of them.

In other words, it’s inevitable that the development of mathematics has to make choices about where to go. It can’t meaningfully “pursue all possibilities”. It has to have definite goals.

But where do goals for pure mathematics come from? In technology development, goals tend to be set quite directly by what people find useful. In natural science, by what one observes and has tools to interpret in the natural world. But in pure mathematics, there don’t at first seem to be such obvious “external” ways to define goals—making the math that’s been done seem more determined by the internal choices of the mathematical community and the leaders within it.

But actually if one looks at the historical arc of human mathematics there are at least definite trends to be seen. The journey of mathematics began with basic formalizations of things like numbers and space that somehow reflect everyday human perception of the world. And although a century or so ago mathematics seemed intent on trying to “generalize to all possibilities”, most of the strongest trajectories of mathematical development over the past century have actually stayed much closer to what humans can intuitively understand and visualize.

There is still plenty of freedom in where to go in math. But to be meaningful to us humans we somehow have to have a certain familiarity with whatever choices are made. Yes, an AI could, say, pick a direction at random, perhaps informed by the observed statistics of existing human choices. But there’s then no reason for humans to care about it; it’s not something that relates to our actual experience of mathematics.

At a more practical level, some part of the doing of math is about figuring out how to achieve goals that have been set—and this is something we can imagine AI doing. But what’s ultimately more important is the setting of the goals in the first place. And almost by definition, these must come from “outside the system”. Or, in particular, from us.

It’s a notable observation that the most significant reported successes for AI in math so far tend to come from some of the most skilled human mathematicians. And in some sense we should not be surprised—because it’s the setting of goals (or, in effect, knowing the questions to ask) that is the most important and unique part. Indeed, it’s often the case that once one really knows the question to ask, one’s already done the lion’s share of the work to answer it.

But just where do goals for pure mathematics in practice come from? There tend to be two basic sources. One is what amounts to problem solving; the other to what amounts to creative building. Problem solving is in a sense more cut and dried, measurable, and amenable to AI. Take a problem that’s been defined in the past, and worked on for a while. Then the goal is to solve it. And if one succeeds, there is a clear way to say one has done something (“an Erdős problem was solved!”, etc.), even if it sometimes seems more like a sporting achievement, with at best muddy ultimate intellectual content.

The Aesthetics of Math

But then there’s what one might think of as “creative building”: coming up with new structures and new concepts for mathematics. At some ultimate metamathematical level, these structures and concepts must already “be there”, as pockets of computational reducibility. So then one can think of the goal as being to identify particular ones, and bring them into the “lexicon” with which one talks about mathematics.

But of all the possibilities, which ones does one want? One could imagine various criteria. But mostly they come down to wanting to create abstractions that best unify, summarize and simplify disparate things. And put this way, one might think this would be an objective that could be automated, say by AI. But there’s a crucial wrinkle. One might imagine one could define something as being “simpler” if it’s shorter to state. But how long something is to state depends on the language it’s stated in. So again there’s nothing immediately absolute; it’s something that inevitably depends on a whole history of interconnected choices.

Often people describe the choices as aesthetic ones. What makes a structure more elegant, more perfect, etc.? Ultimately these are things grounded in human preference and human experience, and extended by what one can describe as human initiative. But does one actually need a human to make the choices? Or could one instead just have some kind of generative AI trained on existing human choices? The issue is that to usefully create new building blocks for mathematics one needs more than just to invent new concepts and structures; they also have to be knitted into the fabric of mathematical culture.

It’s like introducing a new word in a human language. For the word to be useful, its meaning has to be widely enough known for it to routinely stand for a particular concept. And spreading this meaning is inevitably at some level a social process. And so it is for new concepts and structures in mathematics. Insofar as mathematics is about developing mathematical narratives for human minds, concepts and structures have to be shared if they’re going to usefully be used as building blocks. It’s a bit like the issue in a typical automated proof. If one can say that a particular step is from “X’s theorem”, that provides a kind of conceptual anchor. But if every step is in some sense bespoke, one ends up with something that in effect seems alien.

So where does this leave AI in math? It’s similar to AI in many other places. Properly define a goal and AI can be very helpful. Without a goal, it doesn’t know where to go. Exploring possibilities in a ruliological way—and in effect doing experimental mathematics—can bring one lots of surprising new results. But in a sense they’re “born alien”, not immediately connected to what we normally think of as mathematics.

Why Do Pure Math Anyway?

But, OK, so AI can help in extending the enterprise that we can call pure mathematics. But it’s still inevitably going to take human effort and human initiative. So why do it?

A common argument is that it’ll eventually end up being useful for something “practical”. Often the notion is that pure math somehow manages to lay down certain “beacons” that science and technology will eventually reach. But I think the true picture is rather the reverse. The math that we create provides ways of thinking. And it’s when we have these ways of thinking that we end up creating science and technology that makes use of them. It’s not that there’s a convergence of what’s been done in math with what’s been discovered in science. It’s that the math we know guides what we end up looking for in science.

Sometimes there’s a notion that the things we discover in science are somehow unique and inevitable. But thinking in terms of the ruliad we realize that there are actually an infinite number of different slices of computational reducibility, each in effect defining their own take on science. And over and over again historically what seems to happen is that first a conceptual framework is developed—often from math—and only then does it become clear that there’s an associated regularity one can look at, that shows the way to some piece of science. In other words, it’s not a mysterious convergence between pure math and science; it’s that the science is developed because the pure math exists.

In a sense this makes the enterprise of pure mathematics seem less speculative. It’s not that we’re doing a piece of pure math in the hope that one day an application for it will be found. Rather, it’s that doing the piece of pure math develops a way of thinking (and in effect reveals a pocket of computational reducibility) that gives one the chance to come up with the application.

Could a piece of pure math just turn out to be “fundamentally useless”? In some sense the answer is no. Because any time there’s a piece of pure math that’s successfully been done, it must correspond to some pocket of computational reducibility. And any pocket of computational reducibility will be associated with regularities which are inevitably the “raw material” for some form of science and some form of technology.

The catch, though, is that those might be very alien forms of science and technology, far from anything we’ve so far considered. It’s not that the math can’t give us science and technology; it’s just that the arc of scientific and technological development may not yet have reached the point where we care about what it gives.

Back in antiquity the great contribution of math to our way of thinking was the idea of deductive reasoning. In more recent times came a multitude of ideas around continuity, real numbers, etc. Then, a bit more than a century ago came the idea of abstract functions and then the idea of computation. Once reached, this idea might seem almost obvious. But historically it took a surprisingly tall tower of abstract mathematical thought to reach it. And, yes, the general notion of computation is of great practical importance. But—as I have explored over the course of many decades—it’s also foundational for the general way we think about things.

One of the notable features of pure math is that it involves a certain style of structured thinking that has long been seen as important in education. In some ways I think that computation—and particularly computational language, with its way of describing the world—provides a powerful alternative for learning structured thinking. But the foundational study of computation is much younger than mathematics, and much more immediately subject to the limitations of computational irreducibility. And the result is that modern pure mathematics has developed a much taller tower of structured thinking and formalization—and indeed, its tower is taller than any other field.

It’s a big investment to learn that tower. And few people make it. Those that do have a variety of motivations. For some, it’s a matter of challenge. For others, more a matter of the “scenic view from the tower”. In some ways the automation provided by AI makes the challenge less appealing—though, as in something like chess, there can still be fulfillment in pure human achievement. But when it comes to the scenic view, the fulfillment is in the human aesthetic experience, which is quite unrelated to automation and AI.

Half a millennium ago, there were mathematical challenges set up as spectator sports. Today the challenge aspect of mathematics is not as successful at garnering public interest. Indeed, nothing about the process of doing mathematics is typically exposed to the public at all, except to a tiny extent through things like personality-based movies.

(And, yes, one might wonder if there’s any way the doing of mathematics could provide any form of public entertainment. I can report that I myself have done a fair number of successful live mathematical computer experiments. And during the pandemic I livestreamed a great many hours of highly technical working sessions about mathematical physics—which garnered a surprising number of views.)

So what about math as an essentially aesthetic activity? Yes, it’s fulfilling and enriching for the person in the middle of doing it. But what does society as a whole get out of it? Much as some tiny fraction of art is public art, so one might wonder if there can be public math. Occasionally there will be public visual or sculptural math, or perhaps architecture or music that embodies math. But realistically it’s not math at the level of the top of the tower of pure math.

One thing to understand about the upper reaches of pure mathematics is that to a surprising extent it’s been passed down as an oral tradition and a chain of human connections. Perhaps this is partly a consequence of the structure of academia and academic publishing. But the fact remains that to maintain the flame of pure mathematics seems to require a certain community of individuals continually and actively pursuing it, and talking about how it’s done. In other words, even if their personal motivations are quite internal, their collective effort has the long-term external value of allowing the progress of pure mathematics to continue.

One might ask if at some point pure mathematics will somehow be “finished”. And to this we can definitively answer that it will not. Computational irreducibility ensures that there will always be an infinite sequence of mathematical facts—and surprises—to be discovered. And there will also be an infinite number of pockets of computational reducibility, representing new mathematical concepts and structures.

Pure mathematics stands as the single largest intellectual edifice built by our civilization. And out of it have come some of the most impactful ideas in human intellectual history—like formalization, abstraction and computation. In modern times, there are some accelerators for pure mathematics. Practical computation has been one. AI is another. Experimental mathematics in the style of ruliology is yet another, still fairly undeveloped. And I’d like to think that the language for pure mathematics that we’re building will—particularly when combined with AI—be still another.

The work of pure mathematics is long and hard (yes, a consequence of computational irreducibility). But over the course of millennia its importance to the core intellectual development of our civilization has been very great. And there is no doubt that it has more to give. And that even after the issues of today are long forgotten pure mathematics will continue to shine as a beacon of human achievement. So, yes, there’s every reason to expect a bright future—now with some additional help from AI—for that most rarefied of human pursuits: research in pure mathematics.

Further Reading

“Can AI Solve Science?” (2024)

“The Physicalization of Metamathematics and Its Implications for the Foundations of Mathematics” (2022)

“Who Can Understand the Proof? A Window on Formalized Mathematics” (2025)

“The Empirical Metamathematics of Euclid and Beyond” (2020)

“Computational Knowledge and the Future of Pure Mathematics” (2014)

“What Is ChatGPT Doing … and Why Does It Work?” (2023)

“Will AIs Take All Our Jobs and End Human History—or Not? Well, It’s Complicated…” (2023)

Note

Some of the ideas in this piece I discussed over the course of more than three decades with my wife Elise Cawley, who sadly died shortly before the piece was written, and who would no doubt have had many valuable and incisive insights about it, now lost forever.

Andrei Tarkovsky’s Stalker (1979)

Charlie Munger said: “Show me the incentive and I will show you the outcome”. The issue with Academic Research, in my opinion, is that the outcomes have drifted very far away from the original goal, which I believe to be the advancement of scientific understanding, and training of the new generation of researchers. Academic research was meant to be about breaking new ground, keeping to honesty and good scientific conduct, being clear and upfront about uncertainties, errors, and mistakes, and improving our common understanding of science, all the while training the new generation to follow these goals and principles.

Recently, I have bumped into multiple cases where I believe the correctness of the results, the honesty of the people writing them, or the lack of curiosity once they are told that their results are wrong, incorrect, faulty, or misleading, has been unsatisfying. Simply put, researchers are not too interested in learning that their papers or reports are wrong, and/or misleading others who don’t happen to know that the results are — known to the authors, and a few select others to be partially — incorrect.

The issue is, once you published the paper, and got the promotions and fame, it doesn’t matter that the results are wrong, and known to be wrong or misleading, and potentially doing harm to the advancement of science. It’s not your problem. It’s someone else’s problem. In fact, when I challenged the authors of one such paper, considered state-of-the-art, and known to the authors to have incorrect evaluation, one of the author’s response was along the lines of acknowledging the issues, but refusing to retract the paper, and instead asking if it’s bothering me in publishing my paper.

The incentives are wrong. In my opinion, it’s not all about the papers — mine or others’. It’s about scientific integrity, about being honest with each other, it’s about caring about the results, it’s about advancement of our common scientific understanding through seeking of truth and correctness. It’s about communicating when something is wildly wrong, and either retracting the relevant incorrect claims, or notifying the community of the known serious issues. It’s about caring for what we all consider to be the common understanding of what is the truth, and cultivating an environment where the new generation grows up to learn what the proper scientific conduct is, and that it is not acceptable to seriously deviate from it.

Unfortunately, I am seeing more and more PhD students who are less and less interested in correctness, precision, and what I’d consider proper scientific conduct. They have learned from those successful in the field what does and does not matter. I remember when someone once told me that my approach for a particular algorithm was wrong (about strongly connected components discovery), and how upset I was that I had no idea. I went home that day and I immediately fixed my tool to use the right approach (i.e. to use Tarjan’s algorithm). I felt deep shame that I had no clue what I was doing, and that I messed up. In contrast, recently talked with a PhD student who wrote a tool that was meant to perform well in certain contexts. When I explained the student that the approach to the evaluation was incorrect, they didn’t follow up at all. I think they were surprised that I later followed up, demonstrating that indeed the evaluation was not careful enough, and the tool is less useful than it seems from the paper’s evaluation. It was a strange experience — when I was a PhD student and someone sat down and showed me that what I was doing was potentially sloppy, I was super worried and very curious to find out what’s going on. I still remember this moment when a reviewer of my dissertation challenged a graph and I was stressed for a week before I figured out they misread the graph, because I didn’t label it clearly. Or when in a presentation I accidentally left out performing so-called ‘restarts’ as a major advancement in the history of SAT solvers, and someone in the audience rightfully pointed it out. I felt embarrassed for having made such a mistake.

I am not sure how to fix this problem. Seemingly, researchers are less and less keen on correctness, precision, and scientific curiosity. They don’t seem to be incentivised to do so. I sometimes wonder if the system has become so damaged, so many PhD students have grown up to be professors in this environment, that much of what I wrote here seems alien, even repulsive, to many. It makes me sad.