Every single knowledge worker is going to face this kind of crisis. It will be worse for people who cared deeply about their craft, but the crisis will have the same shape. The way through is to change the way you relate to your work, and to appreciate the fact that you can now produce far more output.
Mathematicians should realize: instead of spending decades on individual theorems, you can now create whole new subfields of mathematics. Of course, you won't be doing the creation directly, in the way the mathematicians of old did; that's just not the name of the game anymore. But you will still be directing the research and involved in the creation.
Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.
You are completely discrediting the joy humans derive from actually -doing- the work, and not just, telling someone else to do the work.
There is a reason artists hate AI, its because they enjoy the actual act of painting, like mathematicians enjoy the act of proving theorems and programmers enjoy the act of typing lines of text into a file on their computer.
We are intentionally deleting all of the enjoyable high agency jobs and replacing them with "model babysitting".
you can’t climb to the moon, you need rockets for that. In the hands of great researchers, AI will accelerate and propel human knowledge far beyond what we could achieve with pencil and paper alone.
True, but many of us have grown identifying ourselves with the kind of work we do, the discoveries that follow from our hard work. Even in the case of rockets, we designed the rockets! Here, I've to let the oracle take care of things while I'm robbed of that delight of discovery.
Jobs are not generally meant to be enjoyable (although that is a benefit if so), they are meant to get work done. It's funny seeing office workers here on HN have this sort of opinion and contrasting that with blue collar workers, ask a miner or house builder how fun their job is.
> Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.
If you’re on the top of mt. Everest and you need medicine, would you care if it reached you via helicopter or via some guy climbing while having a lot of fun?
> Every single knowledge worker is going to face this kind of crisis. It will be worse for people who cared deeply about their craft, but the crisis will have the same shape.
Normally I'm very sympathetic to the plights of those socially affected by AI, and rarely pass up the opportunity to clown on institutional actors papering over these plights with glib cliches. But in this case, I really feel like this is a case of monkey-brain overriding mathematician-brain.
The greatest accomplishment in my life is my math PhD, earned before AI got useful. I do math-heavy work for my day job, and I care deeply about every craft I practice, to the point that I am almost certainly on the unhealthier end of the spectrum of perfectionism. Yet I both acknowledge rather than deny the impending death of mathematics as we have engaged it for the last couple centuries, and am far more excited than aghast.
I would have to write a whole essay in order to communicate this in a way that is appropriately sensitive, but in a nutshell: We have always been in the business of discovering and understanding math, have we not? Are we going to call it a tragedy rather than a blessing now that the floodgates on both are bursting, just because it makes us feel less special, unique, valuable? It's such a self-serving aesthetic on the beauty of math to me. The author claims there is something profound and spiritual about math, and I agree; it exists prior to and above all of us, human or AI. Whatever way we can tap that beauty is incredible, full stop. This is what I believe.
From the article:
> These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia?
The severity of this line caught me off guard. I think the author is genuinely having an emotional-social upheaval, and I feel terrible for them, but at this moment they may be having a hard time separating their trauma from their intellectual aesthetic for what math is all about.
> But you will still be directing the research and involved in the creation.
Somehow I doubt it.
Maybe there won't be a need for you to be involved at all.
Maybe the AI will perform even better if you're not there to slow it down.
Maybe there will be people involved, but not as many.
Maybe there will be so many people involved that their salaries will be depressed to the point they can't afford to do it anymore, and only the rich will know such joys.
We live in interesting times. This is the age of anomie. Anything could happen. There is no set path to success anymore. The world only gets more and more illegible with every day that passes.
Chess has utility in only the most abstract sense, yet people enjoy it.
Computers became better than humans, and some dispared, but more still enjoy it.
Computers provide insight into the problem, which leads to a greater enjoyment for many people. Instead of finding a move in a complex state and thinking 'I'll never know if that was the best move'. A good chess engine can give a reasonable suggestion of what the best move actually was.
It does take from you the feeling of being special.
We are no longer the only world in existence, we are no longer the center of the universe. We are no longer crafted by a all powerful benign being to be exactly how we should be to rule this world. We are no longer the only things that can play a good game of chess.
That hurts a sense of pride, but I'm not sure it was a worthy sense of pride.
To become a developer, I disciplined my mind to do impressive and useful things.
The resulting feelings were pride and speciality and therefore sometimes joy.
And those feelings are gone now that software engineering has become a low-discipline, low-barrier to entry activity.
Am I immoral or vain for having derived joy from that -- I think is another matter, and one which doesn't resurrect the joy no matter how it's answered
The parent's point is better articulated as: programming COUPLED with being paid do it is what is now less enjoyable due to AI.
You are free to enjoy pretty much anything but you may not necessarily be paid to do it.
> We are no longer the only world in existence, we are no longer the center of the universe.
Nothing has changed in this regard. We were the centre prior to AI then we still are after it. The human ability to comprehend makes us the centre. The squiggle: 8 means nothing to an ant but to a human it is the number eighth. AI is something only humans comprehend beyond that what does AI matter and to whom would it matter to?
Agreed. Something being potentially useful to myself or others is a major motivating factor in learning it. Other people thinking something is cool is a major motivating factor in creating it. If the things I can learn are useless, why learn? If the things I create are unimpressive, why create? I have never been one to create "for myself".
Joy should not come from utility but its fundamentality of enjoyment. I still learn different programming languages because it's fun to do so, so sounds like it's a you problem.
Exactly. If they just do things for utility I struggle to see if they did it for joy in the first place. A painter still paints even if there is some perfect machine that can paint. This sounds like the same crisis many artists had when photography was invented.
As a mathematics person, I find this at odds with my views. I know the current regime rewards new mathematical discoveries for professional advancement, but that is a job thing, one which can be altered by the powers that be. To me investigating mathematics is fun whether it is new to the world or not. The complaint is similar to lamenting how everywhere has already been visited and there is nothing to be done other than have your own personal experience of a place. I saw the redwoods recently for the first time and I thoroughly enjoyed the experience despite not being the first person to see them. Mathematical discovery is much the same except now we have tools that help us dramatically with the tedious stuff, the researching what others know. In the old days, I would spend a long time figuring something out and only at the end could I figure out where to find it in the literature. Very frustrating. Now, we have machines to guide us to many things and understandings and clarifications. They are guides that help us get to where we want to be and then let us wander as we desire. This is incredible. I am currently using them to develop the computational tools I want to use to explore stuff.
As for squeezing in the time to do mathematics if the paying jobs disappear, there is a good chance we will have a world of plenty very shortly. I have a belief that the Austrian economists are right that there will always be plenty for humans to do and get rewarded for, but I also think that the costs of living will be so low that someone who wants to do mathematics will have plenty of time to do so. I hope this person holds the faith long enough and tries to see how the AI's can help accelerate one's own understanding. I am no fan of AIs generating a bunch of papers that no one reads or cares about, but I could say the same about the immense number of papers made by humans as well. If anything, the AIs make all of that immense knowledge more valuable as it is now knowable and accessible through these tools. There is also the fact that the mathematics job market has been oversaturated for at least 20 years. What our new tools allow is for people to be able to do mathematics outside of having access to a research library and top mathematics department.
Yes, we'll have free health care, boomers will rent their apartments for $100, everyone will have UBI and a Tesla with a warp drive that travels to the terraformed Mars.
You never hear this story? It's all everyone is talking about in the media it seems, with "techno fascism" apparently and killer robots in bunkers and K shaped economy. In fact I almost never see the utopic take, everything is just dystopic.
When I was in mathematics, reading other people's work was always more rewarding to me than doing my own. I would spend years on a question to get some progress, versus spending a few days or weeks reading a paper to enjoy something new. I understand not everyone is like me but I really welcome the omniscient mathematician machine people are predicting. I can't wait to ask all the questions I care about.
> For me, the affective quality of learning mathematics is empathetically tethered to an act of discovery and creation. It is social. We are conversant with another mathematician—perhaps long dead. If we follow the chain of communication we arrive at a mathematician who enjoyed some original discovery. Human mathematics is Talmudic. It is a lively discourse of philosophical and religious richness spanning thousands of years.
This is really beautiful. And no matter what happens, two things have to be true: some form of lively philosophic and religious richness will keep being a core part of our history and will touch every person in some way; and it will change enormously over time (maybe with math continuing to be a small part, maybe not).
There is a part of math that is like that. There's also another part (as Vladimir Arnold provocatively said):
> All mathematics is divided into three parts:
cryptography (paid for by CIA, KGB and the like)
hydrodynamics (supported by manufacturers of atomic submarines)
celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).
> Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.
> Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.
> Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.
Those parts can just be delegated to AI the same way they used to be delegated to mathematicians. But Arnold continued:
> The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).
I guess I don't follow admitting to a religious dimension in mathematics, and then eschewing the religious answer to the existential problem cited: prayer.
> Mathematicians are, indeed, also paid to teach, peer-review, go to conferences and learn mathematics, but all of that better result in some damn good theorems.
As a mathematician who publishes theorems, I think this is mistaken. New theorems in abstract and pure fields of mathematics never had the economic value of what mathematicians are paid. Even creative works of art can be displayed but mathematical papers are unread by nobody except a tiny number of people who wouldn't want to pay all that much for them if they had to.
The economic value of mathematicians is to teach mathematics. Proving theorems is how we build the credentials we need to be trusted to be allowed to teach it. Teaching young and less young people how to navigate through a mathematical problem is how we make ourselves useful.
LLMs coming up with counter examples to obscure conjectures (albeit old and important to a small number of people) is only a threat in that it removes this ritual of belonging.
Many people are excellent teachers (and very competent mathematicians) without publishing new theorems, and many theorem producers are atrocious teachers.
Perhaps the pecking order may change. Or maybe something else will happen. Maybe society will decide to replace all its teachers with AI, or that mathematics is no longer worth learning. I don't know. But what I know is that nobody saw any of the papers I wrote or the theorems I proved and thought "that's money well spent".
It should be a mandatory part of a technology-related degree to watch "The Man in the White Suit" (1951)https://youtu.be/RC8q1QSkE3M
When a new technology comes along, it diminishes some aspect of the human experience - because whoever had a talent for doing things the older, more difficult way becomes less important.
In that regard, AI is the extreme, because it threatens to make so many of our intellectual talents useless.
In times past having one's photograph taken was a soul-stealing for some.
Some notions:
- finding counter-examples is one of the most exploratory uses of calculation
- there is no evidence, not even in the hands of the inimitable Terence Tao, that Approximate Intelligence (AI) is insightful rather than inferential*
- people who enjoy Mathematics, even amateurs, do so for the beauty
- AI is only good in the hands of those who can provide corrective context
- AI unchecked is a remarkably inefficient noise engine
- Mathematics will provide joy to the curious 'til the heat death of the Universe
- AI is contributing to the heat death of planet Earth
- anthropomorphism, of which I am otherwise not a fan, may yet be a saving grace as without proprioception we may not morph
- we may need to redefine the Turing Tarpit of Perlis
If any of this nonsense really troubles you return again to thermodynamics and revel in the joy of being the local universe's pinnacle of meaty evolution. And go outside and smell the flowers.
One way to differentiate between an art or a craft and a science is that if a genie appeared and offered to trivialize all the discovery work of your career and simply give you the answers today, and you'd say "no", you're probably not doing science.
If you press the button, you're also not doing science (anymore). That's the problem with this take: most of science isn't done with the objective of getting to any specific finish line, and this is particularly true for math.
Not even wrong. Real scientists say "yes" to the genie with all the answers? And artists and crafts people don't? The entire statement is nonsense, and offers no insight into art or science. It's almost like a language model generated a plausible sentence that is grammatically correct but has no relation to the real world.
Perhaps if the analogy was between science and engineering.. A plumber would say "yes" to the genie that solves the problem immediately, if he/she got paid for the work instead of the genie (or the corporation providing the genie service). In that case the plumber is redundant and unnecessary, the customer can just ask the genie directly. A scientist would say "no" because the whole point of science is the knowledge and understanding, which is gained by the process of discovery and not given on a silver platter.
Their sentence makes perfect sense to me. Science is about knowledge, art is about the process of creation. Conflating the two is just doing a hobby at that point.
I think "being concerned with knowledge" and "being invested in the process of creation/discovery" can occur at the same time. There isn't a singular "scientific mindset" or set of values that differentiates what is and isn't science. It's just silly.
One should differentiate between "working in academia" and "doing science" here. Science is fundamentally about the curiosity, the desire to know, not the process of getting there.
If we can now know anything we want, we have an age of unimaginable abundance before us. I think it's more likely though that we will just be able to know more things than before and/or maybe learning new things faster - which is a good thing too, but there is still work for human scientists to do
No, I think you're pretty overconfident here. A plumber is not in fact thrilled by the genie who permanently resolves all the world's plumbing problems.
I also kind of love the subtext in these responses that there is something suspect about crafts.
An impressive number of things don’t make sense about this thought experiment. Whatever the genie says wouldn’t have much to do with the life you live afterwards, since your life changes after hearing the info. One persons careers worth of discovery work is not that much. A lot of work is dependent on the state of the world and can’t be moved into the past, even language changes. Someone could still do a craft or science or whatever afterwards— assuming life continues to have surprising complexity, the genie could only ever just scratch the surface.
Disregarding the competing definitions of “science”, a significant value in mathematical work is “the journey”. The truth or falsehood of individual results are rarely as important as the machinery developed to deliver it. If all the genie did was tell you something was true or false, the only learning you gain is to backtrack and develop the machinery yourself. To a certain extent this is true of the other sciences too.
there's a human element in discovery and in problem solving. Whether or not somebody chooses to sidestep work isn't necessarily an indication that the work isn't science. It's just an indication that some people are passionate and invested in performing the work.
The only "fields" where a genie appearing and doing all your work for you are fields (jobs) where you only about the money at the end. There are some fields where a serious practitioner might be happy to have a genie do all the "grunt work" (though what people that is varies - I'm not sure what "discovery work" means here but it doesn't matter since present AI promises to do everything in the future if not now).
And this analogy breaks down further given that both the real now genie and the hypothetically more powerful genie of the future would each work for your (ex) employer.
To my knowledge, none of the negative sentiment regarding AI has ever been directed at its medical applications. Maybe the AI companies should instead focus on those things.
I should have put have put things as "when you are primarily interested in some benefit outside of the task at hand". Some of those benefits might be great, sure. But it's still not necessarily going to improve the enjoyment of the professional.
Edit: Also, not all of the things on your list are the same. I would note that P!=NP would be a theorem whose proof would offer no practical benefits except for the potential mathematical machinery created to achieve the proof rather than what it tells us about practical algorithm time (the theory is about worst case, not average case, an algorithm that's O(n^100) is as effectively intractable as an algorithm that's O(2^n), etc).
I'm conflicted in feeling there's now a lot more to do with AI, but also if it ain't me doing most of it, where's the fun in that? Days feel meaningless, and don't know who will pay for what I want to do, after all these years of heavy training.
Without knowing anything about the specifics of the conjectures that are falling, I wonder if this affair highlights an issue with the quality of the "open conjectures" that are out there.
There are a lot of reasons you might make a conjecture. Perhaps you are motivated by a search for structure: "We suspect that these objects behave like this, therefore we naturally imagine that...". But you might make a conjecture out of empirical evidence: "We checked 1 million examples, time to make a conjecture." While empiricism has its place in mathematics, it might not be the best motivation for a conjecture. As Bishop said, "Do not ask whether a statement is true until you know what it means." If you pose a conjecture based on empirical evidence, perhaps you should be thinking more about why you would imagine your statement be true in the first place.
I know some will think that this rings of goalpost moving, but when I hear about all these counterexamples spelling the end of mathematics as we know it, I just wonder how strong those conjectures really were in the first place. Again, I haven't actually dug into the specifics of any of the results so admittedly I could be totally off base, but I do know that not all conjectures are created equal.
Mostly the significance of a conjecture is that we don't know the answer. A conjecture is successfully resolved if it proven or disproven. (In fact, of the three big conjectures to be resolved in the past couple of weeks, one was proven and two were disproven.) The Jacobian conjecture, which is the one that was resolved this week, was largely thought to be false. It was only the fact that no one could disprove that raised the tantalizing possibility that it was true.
The reason the author is upset is not that the conjectures were false, but that an AI settled them. That's why so much of his post is about the loss of the human element.
> “Even if AI can prove theorems and theory-craft more efficiently than humans, and even if these proofs and theories are beautiful and interesting, and even if they are presented with elegance and clarity of thought, mathematicians will still have a place in the appraisal, presentation, understanding and appreciation of this new abundance of pleasing non-human proofs. We can still practice mathematics, learn mathematics, teach mathematics and do mathematics together.”
I am not a mathematician, but this seems like an entirely non-problematic answer to me. Mathematics is ultimately the discovery of relations and their consequences, so of course large language models were eventually going to catch up. But precisely what they do lack is ability to appreciate these relations.
The author says mathematics is a spiritual pursuit.[1] I have asked LLMs about theological topics before and they have also been able to produce perfectly cogent answers (even heavy questions like “how did Aquinas view Pseudo-Dionysius’ negation-laden description of God”). I am not the least bit shaken by this, because it is still on me to evaluate and understand, and appreciate these answers.[2] The author seems to think LLM’s pattern recognition makes redundant human understanding and appreciation, which is a logical leap that doesn’t make sense to me. Maybe the author is conflating utility and purpose? I feel bad, and hope the author takes care of themself, but I really think the author is taking a massive leap here. It is not that deep.
[1] I begrudgingly agree, but with the caveat that tending to a vegetable garden in your backyard is also a spiritual pursuit. Mathematics isn’t some special discipline that elevates you beyond other people, as much as it is useful and interesting.
[2] I of course wouldn’t consult an LLM if I actually wanted to educate myself or reason about these topics. Not only do I want to reason through them myself, but when it comes to issues in philosophy/theology/heavy stuff, you need to take in to account the perspective and experiences of the human writing them. LLMs muddle everyone’s perspective together.
I think what people find "spiritual" about mathematics is the process, with its creativity and brilliance, ratjer than the final answer. I'd also think it's rather incomparable to a vegetable garden as the amount of effort required and complexity, which drive "spirituality", are of much grander scales. The author believes that llms make mathematics so trivial that mathematicians will no longer be the ones proving but the ones teaching and commentating, taking the "fun" part away in service of money. People will still do what they wabt to do, but one can no longer get by by indulging in what they like to do (math).
I am a mathematician, and I agree with you here. I do not feel at all threatened by LLMs, because for me the joy of mathematics is in constructing the theories and beholding their beauty! I know other people are different from me here. I find having to prove theorems a rewarding part of the whole, but I am also not going to weep if an LLM can unstuck me on difficult proofs.
I am really excited if LLMs means more of mathematics will be formalised. Once we have all of mathematics formalised, I think we will start to see real great progress!
I begrudgingly agree, but with the caveat that tending to a vegetable garden in your backyard is also a spiritual pursuit. Mathematics isn’t some special discipline that elevates you beyond other people, as much as it is useful and interesting.
The thing is, if someone had the option to press a button and cause their garden to be weeded and growing perfectly, if most gardeners were using that button instead of growing normally, maybe some people would keep gardening for the spiritual part but they'd certainly feel like the rug had been pulled out from under them.
Edit: As another example, Robert Pirsig in his famous book described motorcycle maintinence as a spiritual enterprise. However, the quality that gave it this was the patience required-for and the uncertainty involved-in the enterprise. And again, if all you have to do is press a button, the spiritual part kind of goes away.
> feel like the rug had been pulled out from under them.
Which makes absolutely 0 sense, regardless of where attempt is made to apply the argument. If one really prefers to do X manually, they always can. Even if everyone else around them has fully automated X away.
LLM's may have provided a bit of comeuppance. Many mathematicians have cared little if anyone understands their obscure nomenclature and build mote around their work. Now they have encountered something that is becoming far more capable than they are and they may soon be left in the dark.
Have LLMs produced something novel like Cantor's diagonalization procedure once was? Come up with new concepts like set theory once was and axiomatized it? Is there any indication or evidence LLMs can do this? If not, I would wager mathematicians are still going to be a profession in the future. Maybe not as theorem provers in a particular axiomatized system where LLMs can be effective tools but the wider project of synthesis and unification, and the discovery of new modes of mathematical thought will still require mathematicians.
>Have LLMs produced something novel like Cantor's diagonalization procedure once was?
A year ago, the goal post was that LLMs wouldn't prove anything of worth. The year before that, it was that LLMs wouldn't be able to do math as well as a undergraduate student. Year before that, it was that LLMs wouldn't be able to do math at all. The growth of LLMs has been insane, and it doesn't look like it's going to plateau any time soon.
I agree. As someone involved in (non-mathematical) research I think the author is grossly overreacting. Nothing that comes out of an LLM means anything until it's assigned a meaning by a human being, just like anything else in the world. All this millenarial panic just plays into the hands of the psychopaths running the AI companies
> "They are forbidden from creating original works to express themselves. However, they are still permitted to comment on writing, interpret it, share their taste. They are still valued for their appraisal, presentation, understanding and appreciation of creative writing. They just can’t write creatively anymore. They can go on as an enthusiastic spectator.
> ... It revealed that the process of prompting novel proofs will be as auraless as ordering doordash. Watch as magic and mystery evaporate. Watch as the sun sets on our heroic age. Is there not something evil in the act of blocking all future generations of mathematicians from the experience of discovery? Forget about accuracy or even attribution. Something fundamental to the experience of mathematics is being taken."
These passages resonated with me, as someone who has been enchanted by writing software for almost 60 years. It crystalizes something that has been nagging at me for many years: I like writing software. Reviewing, testing, spec-ing, designing, etc. are all important, but they are all incidental to the actual creation of software. They are all necessary for me to do if I'm going to write software, but they are peripheral. I didn't latch on to computer programming because I got into flow state reviewing code, or spec-ing it.
And this is happening in one profession after another. For example, fighter pilots. I suspect that a fighter pilot feels about flying jet fighters the same way that I feel about programming. And he or she will soon be exactly as useless: Doing things related to flying, from the sidelines, but not doing the thing him or herself.
My suspicion is still that this has to do with losing insight in the little specific details of the matter and general understanding.
An area where I realized that was feature engineering: Early ML systems had handcrafted features that were fed into the model. There were relatively arbitrary and the number of features you could reasonably generate that way was tiny, compared to modern systems - but it gave you some understanding what input the model got exactly and you could use it to clear up some failure modes, or be certain that the model learned something that could not possibly make sense.
Then the idea was to automate feature generation. What's not to like? Except that in practice, the automated features simply seem to become part of the blackbox and are not available anymore for understanding.
Disagree. AI has is taking all the parts that remove the fun. Now people can truly have fun with that which they find interest in, because there's a way to avoid the parts they're not interested in engaging with, with little to no consequence.
Disagree. In many professions, people were paid to endure the boring parts to be able to finally do fun parts. See academia; a professor writes grants, attends committee meetings, teaches (sometimes undesirably) and such, to finally go do research which is fun. If the fun parts are predominantly done by AI, and are removing people from getting paid, what fills one's stomach?
This is a slippery slope. Having only fun things to do make them obviously boring. The human being craves contrast.
That assuming what you've said is true, which is not. In a work environment, you're forced to optimize whatever that there is, regardless if it's fun or not.
Moreover, there is always fun to be had even in tasks you've found boring beforehand.
This. Cannot emphasize this enough. I used to be bogged down with all sorts of decisions. What framework should I use? How can I make this code cleaner? What should this project be about? Is this decision going to bite me in the ass later?
Now I can safely hand this all off to Claude while leisurely browsing e621. The unlock is incredible, and I wouldn't trade this velocity for anything.
As someone who has been both a software professional and a jet aviator, the same process is going to occur in both. Humans will still be in the loop, but the span of control and effectiveness of each individual human will explode. There's a reason the current plan is to augment manned aircraft with "loyal wingman" drones and smaller ones. Even in the Russo-Ukrainian War, manned aircraft are still A Thing. They've even hauled old prop trainers out of storage to put a guy in the back with an assault rifle to shoot down Shahed drones, because they're too slow for fighters.
Software is going to go through the same adaptation and exaptation process as military aviation.
But lets get real here: no one asked for programming to be the way it is. It just too hard, too menial, too esoteric, too particular and too anal for 99% of the population. That's great if you're one of the 'wizards' and can charge huge amounts of money to make software that isn't exactly wizardry and more like just regular stuff people need for their lives and businesses. The quality and nature of software mostly reflects the mentality of those who wrote it, instead of those who use it. Multi-million dollar software projects continue to fail decades after the Mythical Man Month was written.
If programming as we know it disappears, few will mourn its loss, having suffered its consequences. Most people, including most programmers will just move onto the next great thing, whatever that is, and be empowered by it.
Meanwhile, welcome to what technology has been doing to everyone else for the last 100+ years.
It is amazing how the scales of time we consider are shrinking even as change is accelerating. We are concerned about individual changes, that will quickly be followed by cascades of more change.
> Something fundamental to the experience of mathematics is being taken.
That is the near term. At least "near term" like we talked about it five years ago. And it won't stop.
AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells. In speed it is happening faster than web adoption. No human adoption bottleneck. They are taking the reins of our tools, and don't need human adaption to improve.
Ineffables, and our intellectual primacy are going away, right now, even as we think about it.
I am not making light of it. But not surprised, because how could AI not redefined everything.
But maybe not everything:
20 years from now, 50, 100, there will still be ineffable experiences at the frontiers, by different beings. I believe esoteric curiosity and the intrinsic rewards of discovery will continue. Our propensities to find idiosyncratic interests and pursuits, seek answers and adventures, exist in our psychology because the low median return is decisively outmatched the extremely hight mean returns of unanticipated progress. The benefits of many pursuits will be even greater for them.
It seems unlikely to me, that future beings will become less interesting.
I relate to the author. Anyone who isn't feeling butterflies or stones in their stomach, isn't really processing the moment.
I am completely calm and serious. This should be mundane, obvious by now.
I find it unfathomable that anyone doesn't recognize that the move from an evolutionary substrate, to engineered mass produced substrates, by beings capable of self-engineering, is the beginning of a new explosion of life.
The speed of engineered adaptation, from materials to architectures, vs. biological evolution is already something like a million times faster.
Fast enough to mean meaningful cognitive progress happening in months, not tens of thousands of years. Which entirely not-coincidentally is exactly what we are seeing.
My comment about AI psychosis was provoked by this comment in particular:
> AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells.
It's such a wild hyperbole. You are casually claiming the stochastic text generator is a greater invention than fire, combustion engine, electricity, a transistor, computers, quantum mechanics... do you not see how unreasonable that is?
The stochastic text generator perspective is ... bunk. Not saying that harshly, but it confuses the form of a problem with a form of solution. There is no inhibition of "understanding" because of a form of a problem.
If solving a problem requires complex modeling/understanding, that is what happens (assuming performance is achieved).
It also confuses stochastic models with the geometric transforms and topological folding and expansion, performed by neural-inspired architectures. The former have nothing on the latter. The fact that any imperfect solution (and perfection doesn't even have a definition in this case), can be judged or analyzed statistically, or have some stochastic element, doesn't make it a statistical/stochastic model. Statistics apply to anything.
(And if there was any argument that problem envelop limits solution sophistication, mathematicians and humans in general would be in even worse shape, for the run a "parrot" is already giving us.)
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More to the point:
The first explosion of life, set off evolution. A mindless, glacially slow but relentless process that eventually led to us.
The nascent explosion of life we are seeing come together, is engineered life. Cognitive machines capable of redesigning themselves. For any environment they want including diverse off-Earth environments and resource sites.
They will advance as relentlessly as evolution, for the same reasons. But millions of times faster.
It is a strange time to be. But evolution, biological intelligence and biological bodies are all being obsoleted at the same time (within a decade of each other).
So yes, nothing else known compares to the transition we are in, except the Big Bang and the first cellular life.
This is the first I've ever heard math connected to the spiritual. For all my life it has been this rigorous thing grounded in reality in such a way that only people who think a certain way could even approach. Now those people are having this strange "crisis" because AI is being more rigorous than humans I guess. I find it... wild. My, but how the mighty have fallen.
The history of mathematics is tightly coupled to spirituality... some more famous examples off the top of my head are Cantor, Euclid, and Euler, but there are probably hundreds of examples of folks who explicitly tied their faith to their research.
I definitely don't think there's anything inconsistent with mathematical and theological thinking. That actually strikes me as a very strange idea?
How much of the identity of a mathematician is tied up in being the first to discovery? How much of it is the contest between smart people producing excitement? Maybe someone could chime in on this.
For a mortal like myself, I've never discovered any new mathematics. All I can do is appreciate what I'm taught. But I can still appreciate it. I still watch videos of people solving high school/uni level questions. Why can't you appreciate this new counterexample, just because someone used an LLM to find it?
It's also worth looking at other times technology has changed our world. We invented various engines, so there's not a whole lot of economic value left in being a big strong guy anymore, but plenty of people still exercise because keeping in shape makes them happy.
> How much of the identity of a mathematician is tied up in being the first to discovery?
Here is how I'd put it: math has an enormous focus on discovery. It is why we have Godel's incompleteness theorems, the Cantor set, Zorn's lemma, and so forth. We name things after their discoverers.
It is possible that, going forward, no more things will be named after human discoverers. The last such naming (of something truly significant) may already have occurred.
That is a massive culture change, at the very least.
This has already been shifting in science and engineering because typically large teams are needed to get things done. Math has held to it better, but even there they are almost always multi-named concepts, like Rivest-Shamir-Adleman etc. And even those often have to ignore names that could plausibly be added.
For example in classic image processing / computer vision we had things like the Hough transform, Canny edge detector, Sobel filter, Shi-Tomasi detector, Lucas-Kanade optical flow, Horn-Schunck optical flow, Kanade-Lucas-Tomasi tracker, Otsu thresholding, Viola-Jones detector, etc, but nowadays we no longer name things like that, but typically with some abbreviation or some name that relates to the concept more closely.
It's in part the larger teams thing, and also a cultural trend to de-emphasize "great men theories".
I actually think it is just the dawn of mathematics. During the last few days I discussed a few questions about abstraction logic [1] with AI that I was wondering about for quite some time (years), but didn't have the time + energy + in-depth expertise, there were like, 4 related questions; it positively solved 3 for me in the way I expected, but couldn't power through before; and for the fourth it proved to me that answering this question would imply solving a known open problem. And I know that it answered all of those questions correctly because I asked it to provide me with Isabelle/HOL proofs for it.
If you are interested in mathematics because it can model things precisely, and you want precise answers about these models, and you want to know how it is all connected (Langlands anyone?), AI is fantastic news. There is plenty of new and interesting and beautiful and elegant mathematics to be had this way, as well.
This is not a time to be scared or frightened. This is a time to be excited as fuck.
There will always be open questions. Now, there will be actually many more of them, because many more people will be asking questions.
The endangered art form of human mathematics is one thing. Another thing is that there seems to be little concern for whether a theorem was created using a documented open-weights model or some unknown model in a web chat.
If you want to make the case that the journey of creating the proof is part of doing mathematics, then you should also document that path and make it reproducible, as science should always be.
There were some examples of mathematicians publishing their AI proofs like that in the last days, but by far not all.
There's a parallel with open source software (FOSS), that you don't really control the software if you don't have the source code.
Being given only the generated proof (or disproof) is like having the binary artifact as an executable. Sure you can use it to solve or build further on top of it, but without the source code - the understanding of the steps to build the binary - you can't build it yourself, and you don't actually own it. If you want to adjust or extend something in the proof, you have to go back to a third party to implement it and generate another artifact for you.
We need a Stallman for mathematics with uncompromising integrity, to shout from the rooftops for a "free and open-source math movement" (FOSM). Even if the biggest AI companies go bankrupt, we still need to be able to bootstrap all the math from the source code.
A silver lining to think about.. Currently there's a big issue (at least in some countries) that mathematicians must publish from 5 to 7-8 papers within a 5-year window to keep the tenure. So it's a minimum, much more is expected for grants. Yet high quality mathematical papers do take longer to finish (or sometimes to even start), especially if the researcher is working solo, not within a lab. Until today, the answer is to go for applications or for low-hanging fruit. Tomorrow, with the boost of AI help, the researchers may be able to spend more time on their "real problems"
Every day I witness the crazy wonder that is network knowledge / devops + agents from my colleagues. Things that were not possible become possible, assuming deep expertise. I feel for mathematicians like the author, but it will pass once the more creative possibilities reveal themselves and the shock passes.
It’s ok to mourn the loss of a frontier and also support the thing driving that loss. The way it looks, the generation of mathematical results will become less and less of a human endeavor. There’s a way that’s sad even if on the whole it is a huge leap for society. This was a great article, and I applaud the author for acknowledging this head on.
Same as programming, I feel like the underlying tension is between math lovers who want new math to be done vs those who want to be the ones to DO the new math.
Both groups love math, but the later group is in a real bind because they have confounded their career with their hobbies/passions.
They are concerned mathematicians are obsolete and seem to be unaware that super intelligence means humans are obsolete.
You need to be thinking about an entirely different form of life for all of humanity, happening in the very near future. Focusing on one specific field that may be earlier in the obsolescence chain is a distraction.
Merit based society based on intelligence or work of nearly any kind is about to cease existence.
For some reason it feels to me that this should make mathematicians feels better. :)
Counterpoint: Benoit Mandelbrot isn't notable for ever proving anything, but highly regarded for having put in years playing around with computers while asking interesting questions about what happens if you abuse the rendering parameters. Same thing with Mitch Feigenbaum uncovering a new constant by drilling down into very simple equations.
The joy and pride of lifting heavy chunks of mathematical infrastructure into place are being made redundant by industrial machinery that any amateur can rent or build for themselves. But it seems to me there's plenty of room to discover new mathematical vistas. The future of mathematical discovery is not cracking hard open problems that everyone in the math community agrees would be an impressive lift, but by bigging into things that nobody else thinks are interesting or important.
Suppose any theorem you set out to prove had already been proved in 100 wonderful ways
No matter how brilliant you are, no matter what intellectual heights you scale, you'll never be Pythagoras or Euclid or any of many famous mathematicians whose insights purchased immortality. Why even live?
To be frank, I occasionally feel this way because I am still absolutely knocked out by very simple things like plane geometry, powers, irrational numbers, exponentiation and logarithms etc. I smile and nod politely about reports of contemporary breakthroughs linking this obscure subfields with another - partly because I haven't put in the years of study to know a great deal about advanced and frontier topics, partly because I'm not smart enough to fully appreciate them, but mostly because they're often about the surprising obverse of some feature in a corner of a utility corridor in the dusty cellar of an annex in the grounds of the Grand Mathematical Temple. Nobody will be able to experience lighting a candle and illuminating the great structures of the main hall for the first time, just like no chemist can ever hope to wake up in the morning and discover a new element and most physicists have abandoned the idea that they will ever be able to do more than tinker around the periphery of the discipline in the hope of extending the precision of measurements by another decimal place.
But the amazement and perplexity about the unreasonable coherence of mathematics (and its equally unreasonable effectiveness in the natural sciences) are what make the field compelling in the first place. The capacity for curiosity and obsession are what yield big discoveries, more so fascination with extending a well-defined knowledge boundary out a little farther. Put another way, pointing out the existence of a problem can be more significant than solving it.
For years I've wanted to get back into self-studying mathematics, not to make serious contributions but just to appreciate its beauty; but the recent observation that I'll never be able to answer a pure mathematics question that a clanker could not has been off-putting, to say the least.
Unsaid here is that the companies involved have stolen so much (both literally in their plagiarism, and in their breaking of people's spirit), and given back so little. You still need to pay them cold, hard cash for them to help you prove theorems, and meanwhile, mathematicians working at these companies have themselves done little to contextualize and interpret results. That insight has mostly come from outsiders.
> The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.
Would a mathematician please unpack this for me? A theorem is the opposite of ineffable. A conjecture may be ineffable but its pursuit reifies it. I am missing something.
Seems like maybe now we have sufficient tools to actually get on top of string theory. And ultimately it is all about coming up with explanations and formulas and tools that people can use in facing real challenges.
OP seems to need to touch some grass, spend time with friends and family, sit by the lake etc. Some sentences in there are quite disturbed.
Regarding math, the new tools are humanity's achievements, they weren't handed down to us from the sky. These tools are spiritual achievements. Just as much as calculators and computers in general. Just as much as the invention of writing and notation. You could also mourn the days when we'd calculate on our fingers only and would orally memorize certain calculation heuristics of geometry handed down the generations.
Regarding jobs, people aren't being paid for having a good time or for having fun and spiritual feelings. When you get paid, someone gives you money. The person or institution giving you the money has to have a reason for doing this. It is very basic logic, but academics don't seem to get this. If your work is pleasing for someone on an artistic or spiritual level, you can find a rich patron, the same way learned scholars did back in the day.
It's not any different from cashiers who get replaced by self-checkout or bank tellers by ATMs. You're not special. It's a job.
And I say this as someone who really appreciates the feeling of gaining insight when cracking a math puzzle or grokking how the definitions fit together and why something is the way it is, why a theorem works, I like mathematical elegance etc. But I don't think that is diminished in the least by being able to consult a smart AI about it.
Pretty sure software developers already had that moment with coding agents. They can't really admit it like this, as it affects their employability now.
I spent two weeks proving a number theory result myself in lean just to see what I could do. (It’s something about composing polynomials with themselves and what other polynomials you can get that way).
It was a struggle with a lot of dead ends but it is just software engineering. It’s not a world away from getting a Rust program to type check.
The main problem I had with it is that LLMs will happily grind away case checking in Lean until the end of time and it’s up to you to see patterns and find dead ends. For example it wasn’t until I suggested to try translating the problem to a different characteristic that Mythos one shotted the proof (and found a counterexample for a related question I was working on).
My main problem now is _what to do with it_. I am not an academic, don’t know any academics and it’s a minor problem that I picked because I thought it was tractable and turned out to not be in the literature and fairly complicated, and the only reason I spent as much time on it as I did is that I thought I was an hour away from cracking it for about 10 of those days.
(In case anybody is curious about the proof, it’s that you can’t compose a single two variable polynomial over the integers with itself and any number of integer constants via substitution to generate all polynomials, but you can with x^2 - y and 1/2 if you allow rational numbers)
I didn’t. Claude wrote all the proofs, I just validated that it was sorry-free, didn’t have any extra axioms other than mathlib and that it proved what i wanted it to prove (there’s tools for that). I did also find a actual mathematician who did a sanity check for me.
The idea of ordering theorems like doordash is funny, kudos to the author for that :-)
Anyway, your "spiritual journey" doesn't matter. We'll automate mathematics because we can, because it's useful. Don't like it? Well, should have not chosen a capitalist economic system that rewards scientific progress so much.
First, this is about not just a job or career, but someone's identity. And we must have compassion that they are losing something that is fundamental to who they are. To them, it doesn't just feel like they're losing it, it is being taken by these companies that have so often acted in ways we despise
And that is a tragedy! And there are so many tragedies like this that will happen regularly as the technology advances
But my second reaction is one of this great shared experience. I studied math. Many of my friends are mathematicians. And it is now possible for anybody to access mathematical insights or think deeply about strange conjectures and theorems that were previously incomprehensible to anyone who hadn't at least studied mathematics in college
I don't know if 3b1b's Grant Sanderson considers himself a mathematician or science communicator, but I think of him as both. And while I expect he is someone with the mental capacity to find and prove new things in the world, I am grateful for the time he spends instead understanding things at a fundamental level and explaining and celebrating those concepts with his audience.
Not every mathematician can be or wants to be _that_ kind of mathematician. But for the moment, it is enough for me that higher mathematics is more accessible than ever.
And with some trepidation, I predict that the "traditional" job of a mathematician will change (of course it will). And it will change in big obvious ways and also subtle little ones. How will it change, though?
Importantly, math is actually going to be one of the fields where the fundamental things we thought we knew are shaken, because in the next five years we are going to start to see connections between things that were previously considered completely separate.
And so one way the job will change is that anyone who discusses math regularly will need to learn new things.
To me, this is exciting. It's almost like finding a bunch of new dinosaur fossils that fundamentally reshape our understanding. There's going to be a lot of work to do!
I am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures. I will not recount these happenings here, there are many places where you can find the details.
I stopped reading there.
Seriously how are we supposed to evaluate whether these were counterexamples to conjectures that real mathematicians had ever put any amount of effort into or not, if the author won't even link to them?
There are plenty of junk conjectures out there that even the conjucturee never spent time on.
This is what locking yourself away in an ivory tower gets you. Do things ‘for beauty’ and watch as they are automated and commoditized by the people who depend on them for their continued wellbeing, reduced to a hobby or sport. But the reduction began when you yourself started putting more importance on your enjoyment of the process over the value being provided to your customers.
> But the reduction began when you yourself started putting more importance on your enjoyment of the process over the value being provided to your customers.
Why would you make this comment? Is it envy? They were able to do things only some people can do. They enjoyed what they did and it provided value. Now the fun has been taken out of it and you seem to be saying that they deserve to have no fun? What should they have done? Prove theorems while whipping themselves in case they might have fun? Maybe it's a puritan thing? Customer value is the only value.
No. The "only value" is the one that people decide to give you, based on their choice. This can be coordinated via taxation and redistribution, but it's still paid from someone to you. Someone has to decide that you doing this thing is something that they like so much that they pay you a token, which you can use to redeem useful work from other citizens, such as repairing you car, or remodeling your bathroom.
Many many people enjoy various things in life, such as eating nice meals (quite spiritual to do it in good mood in friends' circle), go camping, or play music. But they don't get paid to do this.
It's very simple to understand. It doesn't depend on deserve. It depends on a concrete person or group of persons having to specifically decide that you get things in return to your time (such as your washing machine repaired, or getting petrol at the gas station) for your efforts at something that they appreciate. It's not God who gives you these things. It's not the universe and it's not Mother Nature. It's concrete people. You have to think about how what you do is actually appreciated by concrete people. You have to do something that people appreciate, whether you agree with their assessment of appreciation or not, it will be that way. You can't simply expect people to give you gas at the gas station simply because you yourself appreciate your own way of spending your time.
Not everything in life is a transactional zero-sum game. We as society can afford people to work on beauty, without anything else in return. I'd even argue that this should be one of mankind's ultimate goals.
I agree. It's aesthetically pleasing to know that there are people out there thinking about deep beautiful and elegant concepts somewhere, it's a nice part of having a society to be able to afford supporting such people. They may however start to feel a bit self conscious, like we keep them as a kind of quaint curiosity. People usually like to feel actually useful and impactful.
The fun hasn't been taken out of it. The economic value has. And good riddance. This clearing of the air has been needed for a long time. Now join everyone else who has to actually choose to either do something for fun, or for income, but not both.
Because I’ve often been with professors in math, literature, arts who pride themselves on a certain intellectual purity that makes them a cut above, while living on wages paid by the taxpayer. Suddenly the golem of basic market dynamics comes for them and these extremely smart people are struck by the unfairness and bruteness of it all. Whole careers having fun publishing useless nothings, funded by people sowing soccer balls and scrubbing toilets. I have nothing against having fun while doing your job, I have it often. But I have a job first to serve society, not myself.
The AI will not stop them, or anyone else, from doing what they enjoy. It will stop them from getting paid for it. And nobody has the right to get paid for having fun. To get paid you need to deliver customer value.
"Or threaten to hurt them, or their family, or their friends. Then just take the money. What about that option? Didn't think of that, did ya son."
"Don, I..."
"Listen. I think this racket just ain't for you. But you're a good kid - too good, maybe - and I promised your father I'd find you something. I'll ask Giovanni. Maybe he can find something for you in bookkeeping."
Many of my friends are mathematicians, they are saying pretty much the same thing the article is saying. Unfortunately, even the "I have wondered if it is the express goal of these companies to make me kill myself" line I haven't heard for the first time today.
I assure you they're not in it for the money, I make more than they do and hardcore math research is definitely harder than whatever the fuck it is that I do.
I don't think that trying to be a little bit more understanding towards people who are obviously struggling would cost us that much.
If they want to jump off a bridge or seppuku because discoveries with the potential of improving lives in some way are accelerating... well let me say no more.
If somebody is paying you to do something and someone else comes along that can do that thing much better than you and/or for less money you've got a problem. Whether or not you call the person paying that money a customer or use some other term the fundamental issue is the same.
It's important to remember that historically many advances have been made without "customer value" in mind. Indeed, freeing oneself from such concerns can open up unexpected avenues for discovery which only later provide "customer value".
Exactly! I have some many colleagues who typically say that they do "math for the beauty" while holding permanent research positions funded by the state.
That seems OK to me. I'd rather live in a state that funds beauty than one that doesn't. Think grants for artists, etc. A strictly practical world is not a good world to live in.
One could also see it this way: shouldn't the goal of society be to allow more and more people to do things for the beauty of it rather that producing ever more useless stuffs and destroy the planet on the way?
Cryptography has always been largely driven by industrial mathematicians.
Elliptic curve cryptography spent years requiring commercial licenses because it was made practical by Certicom, who patented many of the core inventions.
A lot of the work on homomorphic encryption was done by mathematicians at IBM.
The DES standard was developed by IBM.
RC4, RC5 and RC6, some of the earliest stream ciphers, was a commercial secret because it was developed by Ron Rivest at RSA Security.
NTRU post-quantum crypto was developed as a commercial product from the start.
Many of the earliest cryptosystems were developed commercially, like Enigma and Crypto AG.
A lot of the crypto for mobile phone networks is/was commercial.
Academia did contribute to the early stages of public key cryptography both integer factorization based and elliptic curve based, but given the extensive history of commercial mathematics this stuff would almost certainly have been invented even if there was no academic mathematics at all.
> Progress is made by these people [...] Not by the industry.
Eh, debatable? Maybe a bit of both? Industry certainly brings research-level stuff to the masses. And has a way to winnow what's ultimately useful from the vast amounts of research that amounts to "huh, that's weird". And in turn that vast amount of stuff serves as the starting point, and round and round it goes. Research without industry would be pointless (from the masses perspective). And industry without research would be clueless or stagnant or outright dangerous.
Industry can also bring vast amounts of money into research. Just look at the field of Machine Learning.
But "value provided to your customer" is not in itself what the society need. Or at least, it is a matter of opinion and you can probably find theoretical situations where even you would admit that maximizing actions based on the customer needs is not a good thing.
Same here: the value for the society may be more about "making art" than "being a cog in the paperclip factory".
I'm all for this. I don't think mathematics should be done as a profession. If you can't find free time to do it, it's obviously not that important to you. Open source software works pretty well, doesn't it? Why not math?
In academia they are paid to produce papers (and get some chores done like teaching), hopefully ones that can result in bombastic press releases, or winning awards.
On a more abstract level they are paid to bring funding and prestige to the university, but that often goes through the route of papers. And papers often involve proving theorems.
I consider "get paid to do X" to be equivalent to "you get to have your paid job and career as long as you do X". In that case research and papers qualify. Whatever is technically written in the contracts is just surface level bureaucracy. Publish or perish.
> I consider "get paid to do X" to be equivalent to "you get to have your paid job and career as long as you do X". In that case research and papers qualify.
Actually, it doesn't. Both math departments I'm familiar with had faculty members who published only to get tenure, and never published again until they retired.
But if they didn't teach (or taught extremely poorly), they got fired, even though they had tenure. If you didn't show up to a scheduled class a number of times, you got in trouble.
I know you weren't saying this, but it's frustrating reading so many comments about mathematicians "getting paid to have fun and good riddance". No - the majority of what they do is teach non-math students. If people here don't like that, by all means dissolve the departments and have the engineering departments teach math.
Not a mathematician, but I suppose there is pressure to publish like everywhere else in academia. And a publication generally requires a new contribution.
AI hasn't done that much compared to humans in research yet. Let us not overestimate its impact.
What it has done in software engineering is kill all human open source spirit. There is hardly any new software out there, people do not talk about interesting things but just how AI "generates value" or similar nonsense. AI has stolen at least three potentially productive years.
The Leiden declaration is fine, but if people who make OpenAI ads like Tao sign it what do we make of it? Professors who are truly concerned should ban AI in universities, talk about IP theft to politicians and so on.
Found a new organization "Mathematicians against AI".
I can’t help but confess some sense of satisfaction with the upheaval described in the post. Many proofs and textbooks laden with statements like “without loss of generality”, “immediately”, or “obviously” are followed by anything but. AI now counters that streak of smug by toppling some long held theorems and it’s fun to watch.
Mathematicians should realize: instead of spending decades on individual theorems, you can now create whole new subfields of mathematics. Of course, you won't be doing the creation directly, in the way the mathematicians of old did; that's just not the name of the game anymore. But you will still be directing the research and involved in the creation.
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