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There's no speculation about the usefulness of having a mathematical foundation for understanding. Chemists and biologists not caring can have a lot to do with

There's no speculation about the usefulness of having a mathematical foundation for understanding. Chemists and biologists not caring can have a lot to do with hyperspecialization and fear of math. Category theory is still relatively esoteric.

spekcular 6y ago
"Mathematical foundations" are not always useful to practitioners – or even at all. Algebraic quantum field theory is a good example of the latter, to a large degree [0]. The fact that workers in the field with the presumed "applications" don't seem to care is pretty telling. My guess is that working at a level of generality where subject-specific insights are suppressed automatically limits the usefulness of the work. [0]: http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf
- FabGenovese 6y ago
  It's funny, this is exactly what people have been saying of Categorical Quantum Mechanics and ZX calculus for 14 years, until they didn't, and now ZX calculus is used to draft quantum protocols and businesses like Cambridge Quantum Computing are investing heavily in it. The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.
  - spekcular 6y ago
    Your characterization of the development of coordinate-free differential geometry is incorrect. Intrinsic geometry was not a seemingly "useless" generalization; it was motivated by concrete and specific problems about surfaces and mathematical physics. For example, one of Riemann's papers was literally titled "A mathematical work that seeks to answer the question posed by the most distinguished academy of Paris." (Perhaps not literally, given that's a translation, but you get the point.) I don't know anything about ZX calculus, but if people are using it to solve real-world problems, that sounds good to me. What I object to is pure mathematicians giving "applications" of their work that aren't useful or real. And when the authors allude to applications in chemistry and biology, and no chemists or biologists are doing anything with category-theoretical analyses of Petri nets, I think it's reasonable to point this out.
    - crawfordcomeaux 6y ago
      Do you expect something that's useful to be immediately acknowledged as such in the field, much less adopted? Human cultures are more complex than that. Capitalistic pressures to produce results have had a noted impact on research and development. There will likely need to be a practitioner willing to do the field work to bring about the big "Aha!" for that field. Based on the silo'd history of so many fields, it's a safe bet to say that may be necessary in every field where CT can be applied.
      - spekcular 6y ago
        Yes, I expect practitioners to care about something useful, assuming the result has been properly written up and explained in a way they can understand. Scientists want good results and will adopt new tools if you can make a case that they will help them publish impactful papers. Again, it's pretty telling specific applications aren't mentioned, and instead there's just vague gesturing. How can we expect scientists to "do the field work" necessary to bring the applications to fruition if we can't even tell them what those applications are? Also, regarding your first sentence, the vast majority of influential mathematics was indeed "immediately acknowledged," because it made progress on some important problem. Perfectoid spaces are a recent example.
        FabGenovese 6y ago
        Nice, if everyone thought like you do, number theory wouldn't have existed, elliptic curves along with it, and hence much of public key cryptography as well. Indeed, number theory has been exquisitely useless for a couple of millennia, give or take. I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.
- crawfordcomeaux 6y ago
  I am a practitioner of life sciences. I use category theory to understand what I'm looking at around me in the world. I don't know if I'm even applying it properly and it's still been incredibly useful to me. I've developed basic principles for myself I use in daily life, to handle my emotions, and for raising my child. Practitioner apathy and willingness to dismiss things as not useful can reflect a lack of intellectual humility in their cultures and lives. The generality is useful if one practices seeking subject-specific insights from the generality. It sometimes leads to new options I haven't even thought of or leads me to reconsider decisions. "Useful" is a judgment and most judgments like this are made on too short of a timeline. What's actually happening is someone hasn't found a use for the thing labeled as "useless" and is blaming it on the thing, themselves, or where the thing came from. Instead, a growth mindset focused on learning, accountability, and responsibility might be "I did this thing and the outcome I wanted didn't occur. I haven't yet learned how to use it for said outcome."

spekcular 6y ago

"Mathematical foundations" are not always useful to practitioners – or even at all. Algebraic quantum field theory is a good example of the latter, to a large degree [0]. The fact that workers in the field with the presumed "applications" don't seem to care is pretty telling. My guess is that working at a level of generality where subject-specific insights are suppressed automatically limits the usefulness of the work. [0]: http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf

FabGenovese 6y ago
It's funny, this is exactly what people have been saying of Categorical Quantum Mechanics and ZX calculus for 14 years, until they didn't, and now ZX calculus is used to draft quantum protocols and businesses like Cambridge Quantum Computing are investing heavily in it. The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.
- spekcular 6y ago
  Your characterization of the development of coordinate-free differential geometry is incorrect. Intrinsic geometry was not a seemingly "useless" generalization; it was motivated by concrete and specific problems about surfaces and mathematical physics. For example, one of Riemann's papers was literally titled "A mathematical work that seeks to answer the question posed by the most distinguished academy of Paris." (Perhaps not literally, given that's a translation, but you get the point.) I don't know anything about ZX calculus, but if people are using it to solve real-world problems, that sounds good to me. What I object to is pure mathematicians giving "applications" of their work that aren't useful or real. And when the authors allude to applications in chemistry and biology, and no chemists or biologists are doing anything with category-theoretical analyses of Petri nets, I think it's reasonable to point this out.
  - crawfordcomeaux 6y ago
    Do you expect something that's useful to be immediately acknowledged as such in the field, much less adopted? Human cultures are more complex than that. Capitalistic pressures to produce results have had a noted impact on research and development. There will likely need to be a practitioner willing to do the field work to bring about the big "Aha!" for that field. Based on the silo'd history of so many fields, it's a safe bet to say that may be necessary in every field where CT can be applied.
    - spekcular 6y ago
      Yes, I expect practitioners to care about something useful, assuming the result has been properly written up and explained in a way they can understand. Scientists want good results and will adopt new tools if you can make a case that they will help them publish impactful papers. Again, it's pretty telling specific applications aren't mentioned, and instead there's just vague gesturing. How can we expect scientists to "do the field work" necessary to bring the applications to fruition if we can't even tell them what those applications are? Also, regarding your first sentence, the vast majority of influential mathematics was indeed "immediately acknowledged," because it made progress on some important problem. Perfectoid spaces are a recent example.
      - FabGenovese 6y ago
        Nice, if everyone thought like you do, number theory wouldn't have existed, elliptic curves along with it, and hence much of public key cryptography as well. Indeed, number theory has been exquisitely useless for a couple of millennia, give or take. I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.
crawfordcomeaux 6y ago
I am a practitioner of life sciences. I use category theory to understand what I'm looking at around me in the world. I don't know if I'm even applying it properly and it's still been incredibly useful to me. I've developed basic principles for myself I use in daily life, to handle my emotions, and for raising my child. Practitioner apathy and willingness to dismiss things as not useful can reflect a lack of intellectual humility in their cultures and lives. The generality is useful if one practices seeking subject-specific insights from the generality. It sometimes leads to new options I haven't even thought of or leads me to reconsider decisions. "Useful" is a judgment and most judgments like this are made on too short of a timeline. What's actually happening is someone hasn't found a use for the thing labeled as "useless" and is blaming it on the thing, themselves, or where the thing came from. Instead, a growth mindset focused on learning, accountability, and responsibility might be "I did this thing and the outcome I wanted didn't occur. I haven't yet learned how to use it for said outcome."