7 ms·

The "applications" are speculative at best. Chemists and biologists don't seem to care. This is really just category theory for the sake of category theory.

ulrikrasmussen 6y ago
Petri nets are a quite general model of concurrency that has also found use in many places in computer science. They are used in model checking, and also as the foundation for giving a semantics to BPMN, for example. I think it's not unreasonable to consider Petri nets as a generalization of automata (finite or infinite) with concurrency. Just like automata, they find applications in many places depending on how you further extend or restrict them.
- ampdepolymerase 6y ago
  I honestly doubt it will have any practical applications in biology. Computation chemistry and synthetic biology run on differential equations, anything discrete is generally useless. Perhaps it will have some use in a couple of decades when synthetic biology progresses beyond brute force simulations but at the current level it is the wrong abstraction.
  - FabGenovese 6y ago
    You can turn Petri nets into stochastic nets using some formal procedures. Stochastic nets have a "master equation" which spits out a system of PDEs representing reactions where many many things happen at the same time, and so it makes more sense to think in terms of concentrations etc. We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)
    - ampdepolymerase 6y ago
      We may disagree on this point but thanks for taking the time to reply nonetheless :) I will take a look at your paper.
  - wires 6y ago
    you have to understand that the real power behind this work is not the Petri nets but that fact that they are described so abstractly that they can take on different shapes, such as stochastic nets, or coloured nets, or... It puts the different models on the same footing (and hence allowing you to unify tools and do more work) Also, I'd even argue that biology/chemistry needs to embrace categorical methods and we would see some deep discoveries.
  - akimball 6y ago
    The differential equations are themselves in many cases an abstraction over statistical numbers of discrete events. Arguably, in those cases, they are the wrong abstraction - a thesis which can only be proven by demonstrating the superiority of a different abstraction. Has one been demonstrated? Not yet, for most purposes. Is this a promising avenue of study? IMO, yes.
crawfordcomeaux 6y ago
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.
wires 6y ago
Not at all. We have been applying category theory for the past few years to software and systems design and this paper in particular came as a result of those implementations. Soon these methods will be usable by many people for all kinds of purposes. You can think of cryptographic contracts, business process execution, game theory, functional reactive programming, quantum protocols and digital or analogue electronics. The only difference is what category the diagrams live in, or put another way, which semantics are you assigning to thee boxes and wires. You can stay tuned on statebox.org and process.io

ulrikrasmussen 6y ago

Petri nets are a quite general model of concurrency that has also found use in many places in computer science. They are used in model checking, and also as the foundation for giving a semantics to BPMN, for example. I think it's not unreasonable to consider Petri nets as a generalization of automata (finite or infinite) with concurrency. Just like automata, they find applications in many places depending on how you further extend or restrict them.

ampdepolymerase 6y ago
I honestly doubt it will have any practical applications in biology. Computation chemistry and synthetic biology run on differential equations, anything discrete is generally useless. Perhaps it will have some use in a couple of decades when synthetic biology progresses beyond brute force simulations but at the current level it is the wrong abstraction.
- FabGenovese 6y ago
  You can turn Petri nets into stochastic nets using some formal procedures. Stochastic nets have a "master equation" which spits out a system of PDEs representing reactions where many many things happen at the same time, and so it makes more sense to think in terms of concentrations etc. We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)
  - ampdepolymerase 6y ago
    We may disagree on this point but thanks for taking the time to reply nonetheless :) I will take a look at your paper.
- wires 6y ago
  you have to understand that the real power behind this work is not the Petri nets but that fact that they are described so abstractly that they can take on different shapes, such as stochastic nets, or coloured nets, or... It puts the different models on the same footing (and hence allowing you to unify tools and do more work) Also, I'd even argue that biology/chemistry needs to embrace categorical methods and we would see some deep discoveries.
- akimball 6y ago
  The differential equations are themselves in many cases an abstraction over statistical numbers of discrete events. Arguably, in those cases, they are the wrong abstraction - a thesis which can only be proven by demonstrating the superiority of a different abstraction. Has one been demonstrated? Not yet, for most purposes. Is this a promising avenue of study? IMO, yes.

crawfordcomeaux 6y ago

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.

wires 6y ago

Not at all. We have been applying category theory for the past few years to software and systems design and this paper in particular came as a result of those implementations. Soon these methods will be usable by many people for all kinds of purposes. You can think of cryptographic contracts, business process execution, game theory, functional reactive programming, quantum protocols and digital or analogue electronics. The only difference is what category the diagrams live in, or put another way, which semantics are you assigning to thee boxes and wires. You can stay tuned on statebox.org and process.io