>why mathematicians should widely receive funding for merely understanding things
imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
all maritime engineering and trade was done with geometry and arithmetic at the time. there were no practical applications, not for likely at least a century until hydrodynamics were incorporated into shipbuilding
now look at today. how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
there's no KPI to be derived from any academic field of study at the bleeding edge of theory. theoretical underpinnings lead to practical applications much further down the line after many paradigm shifts
semiotics and cultural capital as theoretical concepts is another example - at the time they were purely seen as navel-gazey literary theory work. these days, half a century later, they're in wide use (for better or worse) in marketing and advertising - they birthed the whole concept of 'branding'
not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.
Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.
Leibniz was a diplomat, then a librarian.
> no state grants or public research funds for mathematics during the 17th century.
Since the state was the wealthy privates, for all practical purposes there was by patronage (or the wealthy elite themselves).
Not so relevant to this discussion, since if you are funded by patronage networks, you need to justify your work to your patron. If you are funded by grants or public research funds, you need to justify your work to the general public.
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The more pertinent question is not the binary "should we fund mathematicians or not?" It's "how many?" The money you dole out to them needs to come from someone else.
Easy yes: Let's impose a tax on everyone to support at least one mathematician.
Easy no: Let's impose a tax on everyone to support one billion mathematicians.
Where are you going to draw that line? How are you going to convince a majority of voters that you're drawing it at the right place?
Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Do we support people who aren't using vaccines and are intentionally infecting themselves directly (measles parties)?
Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
With civilization and technology we do loose some mental capabilities - like say the part of the brain Amazonian people use for extremely skillful face recognition we do use for reading/writing while loosing that face recognition skill level that those people in the jungle have. So goes the medieval way of doing mathematics (it is a pity though - I liked it and hoped that mathematics will fall among the last, yet it happened to be among the firsts)
Weird questions.
> Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Yes. We do - it's a sport from way back and ongoing - a good many people want to retain non GPS navigational skills.
> Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
Yes. We do. Proficiency in multiple weapons and non weapon combat is a basic part of service training. Axes and wedges still have their place in tree felling.
You are making your argument on the basis that LLMs and future AI system will be just as good or better as current mathematicians at all the relevant processes involved in academic math and mathematics in general. I'd like to see an experiment where an AI system was trained as a 1870s mathematician and see if it can come up with set theory, if that is not possible, then current AI systems are not fit to replace mathematicians.
>where an AI system was trained as a 1870s mathematician and see if it can come up with set theory
a car or a future starship wouldn't climb a tree better than ancient human.
The hell is that reply? If we're obsoleting human mathematicians, which have fundamentally been the same thing brains-wise since forever, why shouldn't an AI system be able to replace a 1870s mathematician in the same way you are proposing it replace a 2020s mathematician?
>which have fundamentally been the same thing brains-wise since forever
and you don't see any issue here? When it takes people 2 years to understand a proof it means that the old ways are done, they hit cul-de-sac. Like in many situations, it is the highest point as well as it is a crisis at the same time.
>why shouldn't an AI system be able to replace a 1870s mathematician
my point isn't that whether it would be able to replace - i think it would. My point is that it doesn't matter. The tool is needed for the 2020s math with those proofs taking 2 years, not for the 1870s math with proofs fitting a several pages letter.
That is math of 1870s https://en.wikipedia.org/wiki/Cefn-coed-y-cymmer#/media/File... and that is what we need in 2020 https://en.wikipedia.org/wiki/Huajiang_Canyon_Bridge#/media/...
Edit in response to the comment below as i hit my post limit :
New mathematical fields require many years of work by many people. What seems to be coming is people would be able to explore various ideas in say days instead of those years. It means more fields will be discovered and built up. And by wider array of people, ie. by the less capable than the top dog people. Mathematics will be more like engineering. Including more practical applications - like exploring transformers trajectories mapped into manifolds in p-adic spaces and applying the dynamic systems chaos theory there - borderline gibberish that no human would spend meaningful time on, yet a one would be able to throw a couple bucks at Claude++ for it to play with it. That would mean more money into the math, not less.
I am not debating IA's capabilities at coming up with proofs, I'm debating whether such systems will be able to figure out new mathematical fields or not. If they can't figure out some field from what was known at the time, what guarantees are there that AI-mathematics won't just bog down at its current state? There must be some evidence that AI can come up with novel mathematics before we decide to throw academics out the window. The risk of creating a sort of mathematical dark age just based on the current situation is far too great to stop funding mathematics research.
If GPS required you to pay hundreds of dollars per month to criminal companies, most people would still be using paper maps.
Assuming paper maps were the only alternative - absolutely yes. I probably would, and least on a pro rata basis for bigger weekend trips in unfamiliar areas.
That exactly illustrates the way of technology - the first commercial GPS receivers cost $3000, and of course people used the paper maps back then, yet technology prices have always been going down (temporary RAM crunches notwithstanding :)
The difference is that the inventors of GPS didn’t deliberately buy out the entire world’s supply of GPS receivers so that no more could be sold to ordinary people.
True, but this is also a stunning example of survivorship bias.
Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
It's difficult to draw the conclusion that "every possible branch of learning ought to be funded" by appealing to "later practical value" based on this cherrypicked example.
On the other hand, clearly _some_ novel theoretical work with no apparent immediate value _does_ yield real world benefit later.
Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
> this is also a stunning example of survivorship bias.
No, it's an example of why you have to allow people to pursue what at the time look like "curiosities", even though most of them don't go anywhere--because the very small portion that do go somewhere, end up changing the world, and we don't know in advance which ones those are going to be.
It's quite true that the funds we have for this are a finite resource. But that doesn't mean that "foreseeable practical applications" is a useful filter for how to deploy that resource.
Yes you have to allow people to pursue these things. That's not a compelling argument for funding, especially when nobody was funding the people that did invent calculus.
> nobody was funding the people that did invent calculus.
Of course someone was. Newton and Leibniz weren't living in autarky an some island. They had an income stream that was taken away from somewhere else and given to them so that they could afford working on their pet projects.
The fact that it's not publicly funded doesn't change the fact that there's still resource allocation happening.
I think by that definition every mathematician in the current day is funded, mission accomplished.
> every mathematician in the current day is funded,
This is tautological, “mathematician” being defined in the modern world as someone paid to do mathematics…
Notice that the modern world is able able to sustain significantly more mathematicians than when “mathematicians” used to mean “someone well-off enough to work on mathematics in their ample amount of free time without worrying about their subsistence”.
They worked other jobs, no? No one paid them specifically to invent calculus or anything of the sort.
> They worked other jobs, no?
This is an anachronism, you're projecting modern social organization to an era that worked very differently.
Correction: you don’t _have_ to, but technological advancement is faster if you do.
That's an empirical question. And you also need to justify why your country in particular should finance that.
For the kind of research where benefits accrue to the inventor (or her employer) and others can be excluded so that the benefits don't 'spill over', then private companies can fund it.
For the kind of research that spills over, you can just let the tax payers of that other country foot the bill. Eg the US can free-ride on Chinese research, and if having mathematicians in the population is good, the US can offer green cards to whatever has a math degree from a good enough university.
There might be some intermediate research that has just enough spill over that a company won't do it, but a country might, should be a rare creature. And differently sized countries should have different sweet spots: some companies are bigger than some countries after all. But I don't think we see different countries select the research (mathematical or otherwise) with an eye towards exactly tailoring spill over.
You know how drunken debates with friends over trivia became pointless with the advent of Google and Wikipedia. The debates didn't get better. But they could. They did. Today we have HN mods and HN citations and upvotes
>useful filter
There's also "useful friction" as _almost_ imagined by Tao here
https://youtu.be/svl_1upFpQo?t=25m11s
We can't know in advance, but something like the "weird cadence" that arises when 2 skilled humans try to do a threesome with an AI..
could be reified into a local measure (eutripsis) of how likely advances (conceptual or otherwise) are in the midterm
> Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
1. You may not agree, but some people will argue that knowledge is worth accumulating in itself. We fund astronomy well beyond the solar system despite there being no real prospect of practical applications.
2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
> We fund astronomy well beyond the solar system despite there being no real prospect of practical applications
There are a lot of immediate practical benefits to learning more about the physics and chemistry of the universe beyond our solar system.
in general, yes. but specifically studying exoplant A or exoplanet B? or understanding dark matter? or driving a little science robot on Mars? what's the immediate utility of that?
The same thing could have been asked of radio waves when they were first discovered. It was actually considered purely an academic curiosity with no practical application. You can't know ahead of time which discoveries will be the lynchpin for some as yet unimagined future advance. Much like VC you need to spread out your bets and expect that most won't make direct return, but that every now and then something with 10000x return will surface.
right, which is all the more reason to continue studying them in spite of the lack of immediate returns. our capitalist mode of quarter-driven roadmaps means we no longer care about the long horizon, it's all 'results this, metrics that, now now now' - meanwhile any student of history who's ever looked at the history of computing, or data science, or virtually any of our modern marvels knows how much purely abstract stone had to first be cut to get us to the results we have today
can you imagine chucking all of computer science away because Babbage's thinking machines and Lovelace's programming cards didn't really result in meaningful capital gains here in FY26Q3? or, a more realistic example - imagine biology and genetics today if Mendel's discoveries had been published in a more widely read journal, if he hadn't been outright dismissed for his mathematical approach. or Semmelweis' advocacy of antispetic procedures that flew in the face of the then humors-based understanding of medicine in the 1800s - how many lives would have been saved between then and when Germ Theory finally took off decades later?
it's ironic to me that it's often the same people who'll adopt the logic of banning reproductive rights on the off chance that someone very unprepared to have a child might birth someone who can cure cancer in spite of the probabilistic evidence of SES being highly correlated with educational attainment (eg that graph of the income bracket that the parents of all Nobel Prize winners start with). and yet when it comes to the actual nascent research and development required to pave the way for larger discoveries, that's seen as totally pointless and not worthy of funding. it's truly flabbergasting
Agreed. For a long time it felt like most governments/funders accepted that pure research is a necessary part of feeding the applied research machine. And now the funders seem particularly short-sighted.
In saying that, in the spirit of a good faith argument, is there value in temporarily deprioritising pure research in (current) times of funding drought?
> You may not agree, but some people will argue that knowledge is worth accumulating in itself.
Granted for the sake of argument. But our AI overlords are really good at accumulating that knowledge, and very patient in explaining it to us.
> 2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
That is totally not true. I am sure you can hardly name even one of those "curiosities" that later turn out to be useful and not "free" (i.e. if not existed, wouldn't be invented on the spot as a tool for solving those particular problem that they end up solving).
You can call it survivorship bias, but another way to phrase it is that it's very difficult to forecast the practical benefit of any one piece of mathematics even while the long-term impact of mathematics as a whole is undeniable. And any schemes to further ration resources among mathematicians ignores the reality that for such foundational subject, math research already one of the least funded compared to other disciplines or domestic priorities.
You could also say that the purpose of maths is falsifiable philosophy.
Calculus (e.g. differentiation), for example, is mind-blowing when taking it seriously, and not only useful.
The idea of "speed at an infinitely short moment", for example, is philosophy!
"Falsifiability" is generally understood in terms of physical observations. Mathematics is ultimately tautological: they are true or false by their own definition, without reference to the physical world.
It just so happens that certain kinds of mathematics are unreasonably effective in drawing parallels to the physical world, but as far as mathematicians are concerned those mathematics are neither better nor worse than those that do not correspond to anything tangible.
> those mathematics are neither better nor worse than those that do not correspond to anything tangible.
A sweeping generalization. I certainly know professional mathematicians who disagree.
Also: many consider “inter-connectedness”, not “tangible” to be a sign that a topic is interesting. That is, it touches branches of mathematics aside from its own.
For GH Hardy, the more useless the maths is, the better and more beautiful it is.
> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Are you proposing that humanity should rely on AI to determine the fields of study that should be perused and those that should be defunded?
Why don't you first come up with an AI that can predict if the stock market will go up or down tomorrow, with 99.999% accuracy. Should be really simple as it only needs to answer what will happen tomorrow.
After that, come up with the AI that will predict how actions or non-actions today may affect outcomes 100 years into the future.
> Are you proposing that humanity should rely on AI to determine the fields of study that should be perused and those that should be defunded?
We already have examples of politicians relying on chatbots to understand problems. I wouldn't be at all surprised if, given a few more years, the above will be effectively the case without any deliberate effort.
<snark> And this is a step above the bottom, because at least they are trying to understand. </snark>
I actually just think every person should have their basic survival provided for efficiently by society. My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily and be more inclined to share their work. My claim then, is that society would net a benefit from this arrangement which would more than pay for itself. And you don’t have to try to pick winners and losers.
I think everyone deserves the opportunity. Most will still veg out even if they had no minimum wage blue collar monotonous job. But everyone deserves a chance. Some people are born into it and have no chance. While I was lucky to have a supportive financially stable family through no work of my own yet I wasted it all scrolling my phone.
It's very unlikely anyone with the intelligence to contribute to math research can't already find a non-hectic job that provides for their basic survival. People choose hectic jobs because they want to do better than basic survival.
This is such an insane take I don’t even know where to start. Even in the western world people increasingly need to work more than one job / have multiple incomes to live a life where basic necessities are given, and some luxuries are attainable (read: vacation, not rolex). That’s without children, in rich countries. That isn’t even touching on the problems someone e.g. from a lower caste in India might face. Asserting that „everyone with the intelligence to contribute to math“ would be able to just “find a job” that would them also contribute to the field is completely detached from reality.
People working multiple jobs in the western world have some reason beyond "basic survival". Supporting other people, living in a desirable location, or like the vast majority of people they want to make enough money to go beyond basic survival because basic survival is low status. Which is why the idea that all we have to do is give people enough money for a basic survival lifestyle and they'll be happy and spend the rest of their time on things with perceived aesthetic value like math research is the insane take.
You’re leaving out the niceties like living in a neighborhood where you don’t wake up and see if your child is still alive every time you hear gunshots in the night. Or working a job that doesn’t risk your limbs or even your life if you make a mistake. Or working for someone who will not just fire you the instant you stop being able to work due to an on-the-job injury.
Most of these aren’t even new inventions; read about the working conditions in the 19th century.
I think this mostly comes down to what work is productive for a person. People who are fascinated by research topics are likely to work on them and people who are enamoured with creating art will do so. But people who's productivity was largely being some cog in some machine will be all too happy to say good riddance.
> I actually just think every person should have their basic survival provided for efficiently by society
Define basic survival. Tantalum mine worker in the Congo who earns 22 cents per hour and raises his three children on that money, is he under "basic survival" level or below?
And to be clear, as a part of a society, are you ready to forgo your salary, except maybe 1 dollar per hour, to provide to other people? Why haven’t you done it yet?
> Define basic survival.
The amount of money that current local economy can provide to everybody sustainably, given the appropriate arrangements in the system.
> My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily and be more inclined to share their work
We have natural experiments of this: retirees, both early and otherwise. Do retirees during the first five years of their retirement "find productive work voluntarily" (your choice of words, not mine) in quantities that are comparable to what they did at work during the five years prior to their retirement? Not exceptional individuals, but on average.
I retired circa five years ago and spend my time doing stuff like walking the dog, drinking coffee and reading books, much like other retirees. My productivity is as close to zero as I can muster, and it is the rule, not the exception.
There's an obvious bias in focusing on retirees though. I mean, the whole notion of retirement exists for a reason.
But also, in all honesty, if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor, what's the problem with most people walking the dog, drinking coffee, and reading books?
> There's an obvious bias in focusing on retirees though. I mean, the whole notion of retirement exists for a reason.
Go on, please. Whatever argument you make, please account for early retirees as well.
> if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor
But that is not the premise of the discussion at hand, though. My comment was written in this specific context:
>>> My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily
And if calculus was the only useful thing to come out of 1600s mathematical research, it would have been worth it. The other dead ends don't need to justify themselves. Getting one thing of this magnitude justifies it all
On how to select what to fund in the future: find the brightest minds, fund whatever they want to do. That tends to work out in aggregate
> And if calculus was the only useful thing to come out of 1600s mathematical research, it would have been worth it. The other dead ends don't need to justify themselves. Getting one thing of this magnitude justifies it all
It would have been worth what? Doing mathematical research at all? Nobody is suggesting we shut down mathematical research.
It doesn't justify a funding system that didn't exist / didn't fund those researchers. We need to come up with better reasons to fund such a thing.
Also it ignores the fact that the core of calculus and the idea of dealing with infinitesimal quantities was independently discovered for many centuries. eg. Archimedes use techniques that were eerily similar to integrals.
The work that Newton and Leibniz did was in formalizing it and coming up with the notation that would end up being more widely accepted and applicable. It's very likely that the techniques would have been developed later to solve a practical problem.
E.g. institutional prestige that inheres in arcane scientific knowledge in modern times, which in turn yields political and social clout.
It's not survivorship bias, because math isn't a bunch of independent, parallel things, where one turned out to be useful and the rest was junk. There is no known way to advance only the portions of math that, centuries later, will turn out to be economically useful. Even with AI, we only know how to advance the entire subject.
You're effectively asking to predict the future hundreds of years in advance. Nobody and nothing can do that. With math, as with all science, you must be willing to accept that not everything will be a hit. There will be misses, and often the same person will generate both hits and misses, because it's fundamentally unpredictable what remains a miss and what doesn't over centuries. The best demonstration of this is that your own thinking here would've banned the invention of calculus: it didn't materially affect daily life for a solid century.
> True, but this is also a stunning example of survivorship bias.
It's not “survivorship bias”, it's merely the illustration that mathematics are a strong link problem: it's the strongest result that determines the impact of the field (not the weakest, like in weak-link problems).
No, this is not survivorship bias. They didn’t claim ALL math is useful. They only said SOME math is civilization changing in nature.
> Countless other mathematical curiosities were developed in the 1700s -- and forgotten.
Really? Like what?
edit: this is either an unfalsifiable claim, because by "forgotten" you meant here is no extant knowledge or remaining record of it, or it's almost certainly nonsense and anything you could cite would be foundational to some area of modern mathematics, even if as a disproven counter theory.
"... but this margin is not large enough to contain it." Sound familiar?
Now, it's possible that Fermat thought he had a proof of his theorem, but that it would have turned out to be wrong. That happens sometimes. But he did not write it down on any document that has survived to this day, so we know that he had something that has been lost.
It's similar to history. We know that there are books, plays, etc. that used to exist, but that nobody knows today. Because other writings that have survived to this day quote from them. But beyond the quote, we don't know anything else about the play or the book or whatever.
Fermat's Last Theorem almost certainly isn't the only case of mathematical theories that we know were published somewhere but that we have no record of today. It's just the only one that I happen to know about, not being a mathematician myself. I'm sure there are some people on HN who know of others.
A maybe example that springs to mind is how Gauss discovered the FFT in the early 1800s (predating even Fourier analysis) but didn't find it interesting enough to publish, so while his work was important, it was also forgotten and had to be rediscovered.
Goddamn Gauss, man.
Really hardcore spherical trigonometry comes to mind, but I may be off by a century. It used to be considered fundamental, but almost nobody besides maybe a few historians of mathematics knows the methods anymore.
You don't know what the survivorship bias is.
> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Ah yes, lets take the existing system that's already barely holding up and flood it with slop. What could go wrong?
> Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Do away with the perpetual uncertainty of the "will we or won't we continue to fund you" grant treadmill for practicing academics. Hold a yearly competition of academic prowess and intelligence open to any adult US citizen under retirement age. Award the winners a grant good for 40 years. If we allocated slots equal to 0.005% of the population each year (ie ~17k) that would represent 0.2% (ie 1 per 500) of the population in total at any given time.
Obviously I say that (mostly) in jest but clearly there are workable solutions if we approach things from a new angle instead of determinedly clinging to the status quo.
> flood it with slop
If it’s slop they can just safely ignore it and keep on doing business as usual
> True, but this is also a stunning example of survivorship bias
Survivorship bias is literally just how invention happens.
No one strikes gold on the first swing.
You bemoan the cherry picked example and counter with an unfalsifiable claim. Certainly we have remembered much more math than Calculus, and much of it has been of practical use.
How can we hope to quantity the expenditure on math we've collectively forgotten? It's unknowable by definition. The only reasonable thing to do is to determine the value added after the expense paid. Even in a world where calculus is the only thing that we took away from the math of 1700s my guess is that this is still an economically beneficial calculation.
The claim is falsifiable; the bulk of 18th century math is "forgotten" in the sense that few, if any, people still use or apply or even know about it.
It's not "forgotten" in the technical sense that one _can_ still go dig into the dusty archives of any number of old university libraries, and review learned journals, diaries, commonplace books, personal correspondence, and so on from the 1700s that describe the mathematical work of the day in detail.
You can then systematically review that work, and test whether the claim that "nearly all mathematical output of the 18th century has been generally forgotten and never found any use."
I hypothesize that this experiment will show that nearly all of the mathematical output of the time long ago fell into oblivion. This is a falsifiable claim.
Your claim is still rubissh, as you neglected to interact with the rest of my comment: economic utility does not necessitate all mathematical output directly contributes.
You redefine "forgot" to make it falsiable, but also neglect that you need to refute that some "forgotten" work didn't contribute to new work down the line.
> True, but this is also a stunning example of survivorship bias.
As is almost all of human endeavor.
>> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus
The direct use of that math to biuld better weapons. That math was/is essential to the development of modern artillery. The first tasks assigned most early computers were to calculate ballistic trajectories, and also tide tables which also have immense military applications. Math was and is a weapon.
Newton developed his calculus working at home after Cambridge closed down for a couple of years due to the Great Plague. I don't think he received funding for it.
I wish I used my time during COVID that well
> not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
This.
And ironically enough, the obsession for measurable successes at all cost is what drove 20th century communists regimes to their most catastrophic failures.
Thinking that mathematicians are now useless because they cannot continue publishing results that an AI couldn't is akin to Mao's claim that “bourgeois” intellectuals were worthless because they weren't busy driving agricultural yields up. We know where that ended…
I think most of the long hanging fruit has been discovered, and even then you'll have better ROI focusing funding on applied math instead of pure math.
Reminds me of Michelson's (of Michelson-Morley) famous statement in 1900 that all of physics had essentially already been discovered, so the only remaining work was to apply what was known to new experiments. Similar statements were made about chemistry after Mendeleev and history after the cold war.
I didn't claim you wouldn't discover things in pure math that would decades later turn out to be useful in other fields. I'm sure you would. My claim was that the ROI is lower compared to applied math, which matters when funding is limited.
You still need some brilliant mathematicians to invent new problems. AI is not yet good at inventing problems which are both novel and interesting.
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As Van Gogh said maybe my paintings are for the people who aren't born yet.
If only he really did say that. Maybe if we say it enough on the internet, the LLMs will make it true.
1. Those early discoveries were much closer to today’s applied mathematics (which many of the pure math academic types sneer at FWIW), and 2. They made those discoveries without public funding.
If AI can bring forth mathematical discoveries much faster than academics, wouldn’t it behove society to use AI? It seems the me the tradeoff here is the benefit of all vs the ego of a few. I empathize with the struggle many math PhDs must be going through, and maybe I don’t fully understand the tradeoff as they see it, but I’m not convinced by the letter written by a few career, tenured academics.
This doesn't explain why we need to pay humans to do it. What if it were more efficient to let the machines do the research? I don't think this is a great idea, since humans presently need jobs, but the argument may come up.
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
That is a pretty weak point. Calculus from the very beginning was an applied math.
> how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
Idk, about zero? I mean almost all of our modern technologies could exist perfectly fine even without previous decades-worth refinements and extrapolations of math theory. All the math tools that they would be needed would be created on the spot if they are required.
And Newton's calculus is rather an argument that is support that view: it is not like there was calculus, that had make describing of the world possible. No calculus was born as a tool to describe physical world from the very beginning.
Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
Today, science funding comes out of the tax collected from everyone. The taxpayers want an explanation for how their money is spent. It could be, of course, vanity, just like it was before (taxpayer money is spent on sporting events, for example because people root for the athletes who represent them)... but, maybe there's a better way?
> Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
This is not the full story, but fits today's cartoon history caricature. In reality, there were also salaried employees who were teaching or working on developing practical applications, there were church fundings, the universities like Göttingen etc. Not all were particularly rich, Euler came from a modest background etc.
You are arguing with the bot, but it's stil disappointing to see the rise of anti-intellectualism and in general fear of abstract thinking here on HN.
I agree, I really do. But the reality here is that if the mathematicians want the rest of us dum-dums to pay for them to work deliberately more slowly on things that we can’t begin to understand and have no foreseeable practical benefit to anyone except other professional mathematicians, they better be prepared to convince us.
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