What I think AI will do is open up mathematics to so much more people, because suddenly things will become easy that were hard to do, allowing you to focus on ideas rather than technical details, while at the same time preserving the technical details down to the lowest level, because ultimately, without the details, there is no mathematics. Of course the space in which to express these ideas must allow for errors. I think what it will do is bring together pure and applied mathematics again. I set out to build the theorem proving system of my dreams, clearly an applied pursuit, and what I found was the logic of my dreams, which I would rate as rather pure. And I made plenty of errors finding that logic, most of them have their own DOI. I am actually just now writing a book about this logic that corrects these past errors and over complications: http://abstractionlogic.com . Note that the subtitle of the book is "A New Foundation for Reasoning, Computing, and Understanding." At first the subtitle was "A New Foundation for Reasoning and Computing", but then I thought of this blog. :-)
Today is the third (and final) day of the "Second Cosmic Explorer Symposium" ... in which the community of gravitational wave observers tackles the problem of providing "a human face for astrophysical general relativity and cosmology" ... uhh ... along with the problems of ACTUALLY DESIGNING, FUNDING, CONSTRUCTING, and OPERATING the next generation of gravitational wave observatories.
These folks are centralizing their enterprise---centralizing it very effectively---around the question "What foreseeable problems are hard to fix, and tragic if we get them wrong?" (as Matthew J. Evans' presentation felicitously phrased it).
Perhaps it would be no bad thing if the folks pursuing the development of AI in Mathematics centralized more of their discussion(s) around this same question?
It might be worth mentioning as a sort of tangentially related comment, that the graduate students at Boston University are currently on strike and that the Dean of the College of Arts and Sciences suggested--somehow--using "AI" to replace them (in the context of running undergraduate classes).
In some star trek utopia where no one has to work at Walmart to live a nice life I'm all for those people who want to do mathematics for their joy, cultural expression etc continuing to do so glass bead game style. However, the only thing that justifies paying professors of mathematics for what they do while others have to work at Walmart is the fact that proving things, teaching math etc etc has some benefit for society more broadly.
Until and unless we get something like universal basic income that lets everyone just contribute in whatever artistic or cultural way they want it feels deeply selfish and unfair to try to protect mathematics jobs from being mechanized.
It may seem like there is something unique and important to be protected there but everyone feels that about their own career and, I think for the most part, most careers that have been mechanized away have been replaced with equally valuable stuff as a result (on average).
I think the real issue we need to address isn't how AI affects math but how do we ensure AI creates a world where everyone is free to do something they love rather than a world where we all scramble to work 40 hours a week to keep up with the jobes handcrafting some mechanical watch or other work whose only benefit is the status of having been done by hand [1]. Once we solve that problem then, and only then, does it make sense to worry about how it affects mathematics as a cultural and artistic endeavor.
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Also, regarding the AIs having feelings if that's the case of course you can make it pleasant for them to do the math you want done. We can breed dogs to love herding sheep and we don't even have direct low level ability to intervene in their brains.
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1: Handmaid mechanical watches work no better than mass produced electric ones -- they are almost a pure Veblen good. If you imagine that watch makers had always been half as efficient everyone would be just as happy spending the same amount they do now on fancy rolexes even if they no longer got the phases of the moon because the true demand is only driven by the desire to have something rare that took a long time to create.
IMO if we don't create something like UBI there is a serious possibility that rather than a world of relaxation and fufilling careers we all work hard and feel stress just to make each other trinkets to display relative status.
I am very sensitive to these issues and I recognize that many mathematicians are paralyzed by the questions in the first paragraph, in particular. My book "Mathematics without Apologies" was largely devoted to addressing such questions. If the question is posed in this way — why do mathematicians deserve to make a living doing something they enjoy — then other ways of looking at the situation are ruled out. My answer is that putting the question this way is a symptom of a deep sickness of contemporary civilization. This is how I put it in the book:
…why is it a matter of general interest, independently of the uncertain prospects of short- or long-term benefits to human welfare, to have a small group of people working at the limit of their creative powers on something they enjoy? If a government minister asked me that question, I could claim that mathematicians, like other academics, are needed in the universities to teach a specific population of students the skills needed for the development of a technological society, and to keep a somewhat broader population of students occupied with courses that serve to crush the dreams of superfluous applicants to particularly desirable professions (as freshman calculus used to be a formal requirement to enter medical school in the U.S.). The prospect of taking pleasure in the research we freely determine would then be the lure that gets us into the classroom (the assumption that not all of us are eager to be there is a gross misreading of the true state of affairs). Or I could revert to the Golden Goose argument. But if the question is taken at face value it answers itself. Indeed, if the notion of "general [or public] interest" means anything at all, it should be a matter of general interest that work be a source of pleasure for as many people as possible.
Let me add that we DO have a lot of choice in our current society if you are willing to take it.
Rather than me or my wife leaving one of our academic careers we just decided to put her career first, live a decent but not luxurious lifestyle live on one salary and have me do the home management (and help with her course materials etc..works out pretty evenly) and it means we both get to keep doing the research we love.
But most people I know didn't choose that. Given the option of cutting back (maybe not having kids like us) they choose to leave math or whatever instead so they could have more income. That's fine, other people see raising children or just having a pleasant life as what they want.
But if AI means society no longer benefits in the same way from keeping mathematicians around then it would be odd to ask them to keep subsidizing mathematicians so they don't need to decide if they are willing to have less material goods to keep doing the job. In the long run with increasing automation I believe we should have ubi so no one has to do anything other than what they love but we aren't there yet.
Sure, I agree with most of what you said. But the key point is that **justifies** them being paid well to do the job is exactly the social benefits they offer (and honestly it's not teaching, if you just need basic calc/engineering math taught Kahn academy plus glorified hs teachers work).
If AI comes along and means that mathematicians no longer offer those benefits because AI can do it than there is no reason to treat mathematics differently than writing twilight fan fic online. Some people can do it (and you can do it at the limit of human ability -- some ppl write better erotica than others). Standard Rawlsian justification for inequality.
I'm all about us creating a society with sufficient abundance that everyone can just do what they love (art, math whatever). But given the existence of scarcity it makes sense to give it to the people who are doing work society as a whole benefits substantially from. If AI changes who that is (still will be someone performing at limit of human ability in something) then who gets the scarce resources should as well.
What I think AI will do is open up mathematics to so much more people, because suddenly things will become easy that were hard to do, allowing you to focus on ideas rather than technical details, while at the same time preserving the technical details down to the lowest level, because ultimately, without the details, there is no mathematics. Of course the space in which to express these ideas must allow for errors. I think what it will do is bring together pure and applied mathematics again. I set out to build the theorem proving system of my dreams, clearly an applied pursuit, and what I found was the logic of my dreams, which I would rate as rather pure. And I made plenty of errors finding that logic, most of them have their own DOI. I am actually just now writing a book about this logic that corrects these past errors and over complications: http://abstractionlogic.com . Note that the subtitle of the book is "A New Foundation for Reasoning, Computing, and Understanding." At first the subtitle was "A New Foundation for Reasoning and Computing", but then I thought of this blog. :-)
Today is the third (and final) day of the "Second Cosmic Explorer Symposium" ... in which the community of gravitational wave observers tackles the problem of providing "a human face for astrophysical general relativity and cosmology" ... uhh ... along with the problems of ACTUALLY DESIGNING, FUNDING, CONSTRUCTING, and OPERATING the next generation of gravitational wave observatories.
These folks are centralizing their enterprise---centralizing it very effectively---around the question "What foreseeable problems are hard to fix, and tragic if we get them wrong?" (as Matthew J. Evans' presentation felicitously phrased it).
Perhaps it would be no bad thing if the folks pursuing the development of AI in Mathematics centralized more of their discussion(s) around this same question?
It might be worth mentioning as a sort of tangentially related comment, that the graduate students at Boston University are currently on strike and that the Dean of the College of Arts and Sciences suggested--somehow--using "AI" to replace them (in the context of running undergraduate classes).
In some star trek utopia where no one has to work at Walmart to live a nice life I'm all for those people who want to do mathematics for their joy, cultural expression etc continuing to do so glass bead game style. However, the only thing that justifies paying professors of mathematics for what they do while others have to work at Walmart is the fact that proving things, teaching math etc etc has some benefit for society more broadly.
Until and unless we get something like universal basic income that lets everyone just contribute in whatever artistic or cultural way they want it feels deeply selfish and unfair to try to protect mathematics jobs from being mechanized.
It may seem like there is something unique and important to be protected there but everyone feels that about their own career and, I think for the most part, most careers that have been mechanized away have been replaced with equally valuable stuff as a result (on average).
I think the real issue we need to address isn't how AI affects math but how do we ensure AI creates a world where everyone is free to do something they love rather than a world where we all scramble to work 40 hours a week to keep up with the jobes handcrafting some mechanical watch or other work whose only benefit is the status of having been done by hand [1]. Once we solve that problem then, and only then, does it make sense to worry about how it affects mathematics as a cultural and artistic endeavor.
--
Also, regarding the AIs having feelings if that's the case of course you can make it pleasant for them to do the math you want done. We can breed dogs to love herding sheep and we don't even have direct low level ability to intervene in their brains.
--
1: Handmaid mechanical watches work no better than mass produced electric ones -- they are almost a pure Veblen good. If you imagine that watch makers had always been half as efficient everyone would be just as happy spending the same amount they do now on fancy rolexes even if they no longer got the phases of the moon because the true demand is only driven by the desire to have something rare that took a long time to create.
IMO if we don't create something like UBI there is a serious possibility that rather than a world of relaxation and fufilling careers we all work hard and feel stress just to make each other trinkets to display relative status.
I am very sensitive to these issues and I recognize that many mathematicians are paralyzed by the questions in the first paragraph, in particular. My book "Mathematics without Apologies" was largely devoted to addressing such questions. If the question is posed in this way — why do mathematicians deserve to make a living doing something they enjoy — then other ways of looking at the situation are ruled out. My answer is that putting the question this way is a symptom of a deep sickness of contemporary civilization. This is how I put it in the book:
…why is it a matter of general interest, independently of the uncertain prospects of short- or long-term benefits to human welfare, to have a small group of people working at the limit of their creative powers on something they enjoy? If a government minister asked me that question, I could claim that mathematicians, like other academics, are needed in the universities to teach a specific population of students the skills needed for the development of a technological society, and to keep a somewhat broader population of students occupied with courses that serve to crush the dreams of superfluous applicants to particularly desirable professions (as freshman calculus used to be a formal requirement to enter medical school in the U.S.). The prospect of taking pleasure in the research we freely determine would then be the lure that gets us into the classroom (the assumption that not all of us are eager to be there is a gross misreading of the true state of affairs). Or I could revert to the Golden Goose argument. But if the question is taken at face value it answers itself. Indeed, if the notion of "general [or public] interest" means anything at all, it should be a matter of general interest that work be a source of pleasure for as many people as possible.
Let me add that we DO have a lot of choice in our current society if you are willing to take it.
Rather than me or my wife leaving one of our academic careers we just decided to put her career first, live a decent but not luxurious lifestyle live on one salary and have me do the home management (and help with her course materials etc..works out pretty evenly) and it means we both get to keep doing the research we love.
But most people I know didn't choose that. Given the option of cutting back (maybe not having kids like us) they choose to leave math or whatever instead so they could have more income. That's fine, other people see raising children or just having a pleasant life as what they want.
But if AI means society no longer benefits in the same way from keeping mathematicians around then it would be odd to ask them to keep subsidizing mathematicians so they don't need to decide if they are willing to have less material goods to keep doing the job. In the long run with increasing automation I believe we should have ubi so no one has to do anything other than what they love but we aren't there yet.
Sure, I agree with most of what you said. But the key point is that **justifies** them being paid well to do the job is exactly the social benefits they offer (and honestly it's not teaching, if you just need basic calc/engineering math taught Kahn academy plus glorified hs teachers work).
If AI comes along and means that mathematicians no longer offer those benefits because AI can do it than there is no reason to treat mathematics differently than writing twilight fan fic online. Some people can do it (and you can do it at the limit of human ability -- some ppl write better erotica than others). Standard Rawlsian justification for inequality.
I'm all about us creating a society with sufficient abundance that everyone can just do what they love (art, math whatever). But given the existence of scarcity it makes sense to give it to the people who are doing work society as a whole benefits substantially from. If AI changes who that is (still will be someone performing at limit of human ability in something) then who gets the scarce resources should as well.