3 ms·

> math problems are really there to solve a real world problem I think this is totally wrong. Math problems are almost by definition problems with a particular

> math problems are really there to solve a real world problem I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.

GPerson 16d ago
Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post. I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
- aurareturn 16d ago
  Question: Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
  - GPerson 16d ago
    My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks. Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
    - aurareturn 16d ago
      Do we know that AI can't invent a conceptual framework? If we give it a real problem to solve, it may just have to invent a new form of math to solve.
      - GPerson 16d ago
        It might. It might not though. Would a 1300s superintelligence advocate for heliocentricism in the face of all institutional players advocating for geocentrism, or would it create the most refined epicycle model imaginable? I’d guess the latter.
    - mrngld 16d ago
      Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
      - NonHyloMorph 16d ago
        Maybe fix energy and heat on earth first. Fusion would be nifty. How's the AI progress there?
      - GPerson 16d ago
  - pixl97 16d ago
    >If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past? This depends on unanswered questions on what math actually is and it's causal connectivity. Imagine we have problem A that needs to connect to math solution Z. The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable. As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems. The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
  - j2kun 16d ago
    > Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: https://ee.stanford.edu/~hellman/publications/24.pdf https://ee.stanford.edu/~hellman/publications/24.pdf You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA. I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
  - NonHyloMorph 16d ago
    A lot indicates the answer to your latter question might just be 'no'
- NonHyloMorph 16d ago
  Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing.. don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels.. Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?
aurareturn 16d ago
Humans only invest in solving problems that matter one way or another. I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
- freehorse 16d ago
  If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere. Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
  - aurareturn 16d ago
    So my question is this: Would this slow technological progress? Or make it go faster? That math problems are found and solved as we run into real physical problems.
    - freehorse 16d ago
      Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner. In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about. So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
keeda 16d ago
As an aside, it's amusing that this conversation is a re-statement of a main point in TFA: > However, math problems are really there to solve a real world problem. vs > That theory might be inspired by the real world, but the problem itself is purely theoretical. From TFA: > In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"
- BalinKing 16d ago
  I only just skimmed the referenced essay, but a priori I don’t think “problem-solving” as Gowers uses it has anything to do with real-world practicality. The problems under consideration are entirely theoretical, regardless of which “culture” a mathematician belongs to.
  - keeda 16d ago
    You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture. The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect. I may have misinterpreted either or both of them though!
linhvn 16d ago
If this is the viewpoint of mathematics, why does it make a difference that human or AI solve it? And forget about "understanding", because "understanding" in mathematical sense means in a very narrow way: top experts of math in certain field would understand and accept it (my estimation is that ~100 people in the world would understand Fermat's Last Theorem proof). Mathematicians could spend the whole year digesting FLT and "convince" the public that this is correct, and for most people (including math PHDs and professors), FLT is correct because some smart people say it is.
kurtis_reed 15d ago
If solving a problem is "purely" theoretical it's of no use outside the math community's enjoyment. Otherwise it's not purely theoretical. > none of the millennium problems have anything to do with a "real" problem P vs NP has a lot to do with real problems.

GPerson 16d ago

Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post. I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…

aurareturn 16d ago
Question: Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
- GPerson 16d ago
  My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks. Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
  - aurareturn 16d ago
    Do we know that AI can't invent a conceptual framework? If we give it a real problem to solve, it may just have to invent a new form of math to solve.
    - GPerson 16d ago
      It might. It might not though. Would a 1300s superintelligence advocate for heliocentricism in the face of all institutional players advocating for geocentrism, or would it create the most refined epicycle model imaginable? I’d guess the latter.
  - mrngld 16d ago
    Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
    - NonHyloMorph 16d ago
      Maybe fix energy and heat on earth first. Fusion would be nifty. How's the AI progress there?
    - GPerson 16d ago
- pixl97 16d ago
  >If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past? This depends on unanswered questions on what math actually is and it's causal connectivity. Imagine we have problem A that needs to connect to math solution Z. The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable. As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems. The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
- j2kun 16d ago
  > Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: https://ee.stanford.edu/~hellman/publications/24.pdf https://ee.stanford.edu/~hellman/publications/24.pdf You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA. I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
- NonHyloMorph 16d ago
  A lot indicates the answer to your latter question might just be 'no'
NonHyloMorph 16d ago
Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing.. don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels.. Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?

aurareturn 16d ago

Humans only invest in solving problems that matter one way or another. I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.

freehorse 16d ago
If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere. Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
- aurareturn 16d ago
  So my question is this: Would this slow technological progress? Or make it go faster? That math problems are found and solved as we run into real physical problems.
  - freehorse 16d ago
    Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner. In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about. So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.

keeda 16d ago

As an aside, it's amusing that this conversation is a re-statement of a main point in TFA: > However, math problems are really there to solve a real world problem. vs > That theory might be inspired by the real world, but the problem itself is purely theoretical. From TFA: > In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"

BalinKing 16d ago
I only just skimmed the referenced essay, but a priori I don’t think “problem-solving” as Gowers uses it has anything to do with real-world practicality. The problems under consideration are entirely theoretical, regardless of which “culture” a mathematician belongs to.
- keeda 16d ago
  You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture. The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect. I may have misinterpreted either or both of them though!

linhvn 16d ago

If this is the viewpoint of mathematics, why does it make a difference that human or AI solve it? And forget about "understanding", because "understanding" in mathematical sense means in a very narrow way: top experts of math in certain field would understand and accept it (my estimation is that ~100 people in the world would understand Fermat's Last Theorem proof). Mathematicians could spend the whole year digesting FLT and "convince" the public that this is correct, and for most people (including math PHDs and professors), FLT is correct because some smart people say it is.

kurtis_reed 15d ago

If solving a problem is "purely" theoretical it's of no use outside the math community's enjoyment. Otherwise it's not purely theoretical. > none of the millennium problems have anything to do with a "real" problem P vs NP has a lot to do with real problems.