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There is an important point you are not addressing: it is one thing to make a statement about one single mathematical construct (it is cute, but not a big deal)

There is an important point you are not addressing: it is one thing to make a statement about one single mathematical construct (it is cute, but not a big deal). It is another thing to make a statement that constrains a whole family of seemingly disconnected mathematical constructs (this is now a serious constrain on what the laws of the universe can plausibly be). From a sibling comment: > There are some results around QG, such as holography, which seem to be universal. That is, they make sense with strings, loop quantum gravity or any of about ten other lines of work that people can do quantum gravity calculations with today. > If you can prove that all of those have to say the same thing, that is a powerful result and probably means something about our world.

pmiller2 8y ago
I'm not exactly sure which version of "If you can prove that all of those have to say the same thing...." you're arguing for. It doesn't scientifically matter what statements you can make about any number of mathematical constructs if none of those constructs make testable predictions. Remember, falsifiability is a necessary condition for a construct to be a valid scientific theory. OTOH, it does matter if you are able to make a statement of the form "Any mathematical construct that accurately describes X must have property Y." It is not clear that quantum gravity (or the linked article) meets this latter standard.
- krastanov 8y ago
  Ah, I think we have a deep "philosophy of science" disagreement. Basically, it is not just "falsifiability", but also "probability" of being correct. Probability that can be informed by things like formal versions of Occam's razor (e.g. Kolmogorov complexity, Akaike information criterion, etc). Here are a few premises on which I based my statement (I will pick one particular venue of scientific research, so I will not speak in all generality. You can of course disagree and say that it does not generalize at all, but I think this would be a philosophical discussion, i.e. it does not matter at all that we have different opinions, as both are equally valid ways of pursuing scientific truth): 1. Computational complexity is a science on its own. Computational complexity also makes useful statements about the limits of the physical laws in our universe (a la "extended Church-Turing thesis" or Aaronsons "NP-complete problems and physical reality"). 2. NP=P vs NP!=P is a question that has a definite answer within the relm of science but we do not know the answer yet. The answer will constrain what the permitted laws of physics are due to point 1 above. 3. There are a lot of clues that NP!=P coming from very separate disjoint fields of math. Mainly because we already know there are a lot of "phase transition"-like phenomena when we apply approximate algorithms to problems that can be parameterized to be NP-complete only if some ε is larger than some critical value. 4. These "phase transitions" are purely mathematical constructs, but due to the previous points they constrain what is permitted in the universe as strongly as any 5-sigma measurement in a particle accelerator. To bring this back to the discussion of speculative theories of quantum gravity: if all of the competing theories of quantum gravity (all of them being very speculative and unproven) have a handful universally agreed on predictions (although derived in completely separate ways), this is as strong of a constraint as any 5-sigma measurement in a particle accelerator or cosmological observation. Or if you permit me to attempt to phrase is it in yet another way: there are a ton of theory assumption behind any 5-sigma (or 10-sigma) experimental measurement. These assumptions are the same ones that inform the purely mathematical constraints. If both the math constraints and the interpretation of an experiment are based on the same assumption, why are you taking the interpretation of the experiment any more seriously than the pure math. Sorry for the length of this text - I am finishing my dissertation this week (doctorate in physics), so this is very much on my mind at the moment.
  - thereisnospork 8y ago
    > If both the math constraints and the interpretation of an experiment are based on the same assumption, why are you taking the interpretation of the experiment any more seriously than the pure math. Because math models the experiment, it does not define it's results. Even math as fundamental as the laws of thermodynamics does not constrain what is permitted; they are axioms which describe the universe as best we are able to view it. If all competing theories of quantum gravity agree on a point, it would mean almost[1] nothing until that point is observed. From the outside it feels like there is a fallacy common in at least some branches of physics (string theory...) where over-fitting math is equated with insight. [1]I suppose you could break out Bayes' theorem and try to work out a prior for comparable theories being wrong, degree of independence between theories, etc...
    - krastanov 8y ago
      Maybe a 100 years ago. With today's fundamental physics experiments (accelerators, non EM telescopes, and especially quantum foundations tests) the mathematical constructs are an inseparable part of the interpretation of the experiment. And there it is hardly over fitting when the same results come from vastly different mathematical constructs.
  - pmiller2 8y ago
    Yes, I think we do have a deep, philosophical disagreement here. You seem to be taking a more "coherentist" approach (very roughly, if theories X, Y, and Z all explain observed phenomena P_1, P_2, ... P_n, then there's probably something to all of them.") OTOH, I would counter that if theories Y and Z offer no testable predictions beyond that which established theory X does, then they are of limited value. In this way, I would follow Lakatos and Kuhn most closely. The irony here is that, although I am a professional software engineer, the entirety of my post-secondary education is in mathematics. I did some research in graph theory when I was in grad school, so I am actually quite familiar with this notion of "phase transition" that you're referring to. But, I also don't consider mathematics to be a science, because I define "science" in the same way as Wikipedia: "...a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe." There are many things mathematics can describe that are inherently unphysical, which, IMO, makes mathematics itself not a science, however excellent a tool for science it is. And, it's on this basis that I'd come at all of your points 1-4. Regarding QG theories and their explanatory power, I would say that the fact that they have a handful of universally agreed upon predictions does not make any one of them correct in the least. The value in any one theory of QG is not in the areas where it agrees with other theories of QG; it's in where that theory disagrees with established theory in a way that we can see in a lab, a telescope, or an accelerator. What we're trusting here is the physical behavior of the universe, and we're hoping that it's correctly predicted by the mathematical construct that is the theory.

pmiller2 8y ago

I'm not exactly sure which version of "If you can prove that all of those have to say the same thing...." you're arguing for. It doesn't scientifically matter what statements you can make about any number of mathematical constructs if none of those constructs make testable predictions. Remember, falsifiability is a necessary condition for a construct to be a valid scientific theory. OTOH, it does matter if you are able to make a statement of the form "Any mathematical construct that accurately describes X must have property Y." It is not clear that quantum gravity (or the linked article) meets this latter standard.

krastanov 8y ago
Ah, I think we have a deep "philosophy of science" disagreement. Basically, it is not just "falsifiability", but also "probability" of being correct. Probability that can be informed by things like formal versions of Occam's razor (e.g. Kolmogorov complexity, Akaike information criterion, etc). Here are a few premises on which I based my statement (I will pick one particular venue of scientific research, so I will not speak in all generality. You can of course disagree and say that it does not generalize at all, but I think this would be a philosophical discussion, i.e. it does not matter at all that we have different opinions, as both are equally valid ways of pursuing scientific truth): 1. Computational complexity is a science on its own. Computational complexity also makes useful statements about the limits of the physical laws in our universe (a la "extended Church-Turing thesis" or Aaronsons "NP-complete problems and physical reality"). 2. NP=P vs NP!=P is a question that has a definite answer within the relm of science but we do not know the answer yet. The answer will constrain what the permitted laws of physics are due to point 1 above. 3. There are a lot of clues that NP!=P coming from very separate disjoint fields of math. Mainly because we already know there are a lot of "phase transition"-like phenomena when we apply approximate algorithms to problems that can be parameterized to be NP-complete only if some ε is larger than some critical value. 4. These "phase transitions" are purely mathematical constructs, but due to the previous points they constrain what is permitted in the universe as strongly as any 5-sigma measurement in a particle accelerator. To bring this back to the discussion of speculative theories of quantum gravity: if all of the competing theories of quantum gravity (all of them being very speculative and unproven) have a handful universally agreed on predictions (although derived in completely separate ways), this is as strong of a constraint as any 5-sigma measurement in a particle accelerator or cosmological observation. Or if you permit me to attempt to phrase is it in yet another way: there are a ton of theory assumption behind any 5-sigma (or 10-sigma) experimental measurement. These assumptions are the same ones that inform the purely mathematical constraints. If both the math constraints and the interpretation of an experiment are based on the same assumption, why are you taking the interpretation of the experiment any more seriously than the pure math. Sorry for the length of this text - I am finishing my dissertation this week (doctorate in physics), so this is very much on my mind at the moment.
- thereisnospork 8y ago
  > If both the math constraints and the interpretation of an experiment are based on the same assumption, why are you taking the interpretation of the experiment any more seriously than the pure math. Because math models the experiment, it does not define it's results. Even math as fundamental as the laws of thermodynamics does not constrain what is permitted; they are axioms which describe the universe as best we are able to view it. If all competing theories of quantum gravity agree on a point, it would mean almost[1] nothing until that point is observed. From the outside it feels like there is a fallacy common in at least some branches of physics (string theory...) where over-fitting math is equated with insight. [1]I suppose you could break out Bayes' theorem and try to work out a prior for comparable theories being wrong, degree of independence between theories, etc...
  - krastanov 8y ago
    Maybe a 100 years ago. With today's fundamental physics experiments (accelerators, non EM telescopes, and especially quantum foundations tests) the mathematical constructs are an inseparable part of the interpretation of the experiment. And there it is hardly over fitting when the same results come from vastly different mathematical constructs.
- pmiller2 8y ago
  Yes, I think we do have a deep, philosophical disagreement here. You seem to be taking a more "coherentist" approach (very roughly, if theories X, Y, and Z all explain observed phenomena P_1, P_2, ... P_n, then there's probably something to all of them.") OTOH, I would counter that if theories Y and Z offer no testable predictions beyond that which established theory X does, then they are of limited value. In this way, I would follow Lakatos and Kuhn most closely. The irony here is that, although I am a professional software engineer, the entirety of my post-secondary education is in mathematics. I did some research in graph theory when I was in grad school, so I am actually quite familiar with this notion of "phase transition" that you're referring to. But, I also don't consider mathematics to be a science, because I define "science" in the same way as Wikipedia: "...a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe." There are many things mathematics can describe that are inherently unphysical, which, IMO, makes mathematics itself not a science, however excellent a tool for science it is. And, it's on this basis that I'd come at all of your points 1-4. Regarding QG theories and their explanatory power, I would say that the fact that they have a handful of universally agreed upon predictions does not make any one of them correct in the least. The value in any one theory of QG is not in the areas where it agrees with other theories of QG; it's in where that theory disagrees with established theory in a way that we can see in a lab, a telescope, or an accelerator. What we're trusting here is the physical behavior of the universe, and we're hoping that it's correctly predicted by the mathematical construct that is the theory.