Merging quantum mechanics and gravity is a famously hard physics problem. Explaining why merging quantum mechanics and gravity is hard is, in turn, a very hard science communication problem. The more popular descriptions tend to lead to misunderstandings, and I’ve posted many times over the years to chip away at those misunderstandings.
Merging quantum mechanics and gravity is hard…but despite that, there are proposed solutions. String Theory is supposed to be a theory of quantum gravity. Loop Quantum Gravity is supposed to be a theory of quantum gravity. Asymptotic Safety is supposed to be a theory of quantum gravity.
One of the great virtues of science and math is that we are, eventually, supposed to agree. Philosophers and theologians might argue to the end of time, but in math we can write down a proof, and in science we can do an experiment. If we don’t yet have the proof or the experiment, then we should reserve judgement. Either way, there’s no reason to get into an unproductive argument.
Despite that, string theorists and loop quantum gravity theorists and asymptotic safety theorists, famously, like to argue! There have been bitter, vicious, public arguments about the merits of these different theories, and decades of research doesn’t seem to have resolved them. To an outside observer, this makes quantum gravity seem much more like philosophy or theology than like science or math.
Why is there still controversy in quantum gravity? We can’t do quantum gravity experiments, sure, but if that were the problem physicists could just write down the possibilities and leave it at that. Why argue?
Some of the arguments are for silly aesthetic reasons, or motivated by academic politics. Some are arguments about which approaches are likely to succeed in future, which as always is something we can’t actually reliably judge. But the more justified arguments, the strongest and most durable ones, are about a technical challenge. They’re about something called non-perturbative physics.
Most of the time, when physicists use a theory, they’re working with an approximation. Instead of the full theory, they’re making an assumption that makes the theory easier to use. For example, if you assume that the velocity of an object is small, you can use Newtonian physics instead of special relativity. Often, physicists can systematically relax these assumptions, including more and more of the behavior of the full theory and getting a better and better approximation to the truth. This process is called perturbation theory.
Other times, this doesn’t work well. The full theory has some trait that isn’t captured by the approximations, something that hides away from these systematic tools. The theory has some important aspect that is non-perturbative.
Every proposed quantum gravity theory uses approximations like this. The theory’s proponents try to avoid these approximations when they can, but often they have to approximate and hope they don’t miss too much. The opponents, in turn, argue that the theory’s proponents are missing something important, some non-perturbative fact that would doom the theory altogether.
Asymptotic Safety is built on top of an approximation, one different from what other quantum gravity theorists typically use. To its proponents, work using their approximation suggests that gravity works without any special modifications, that the theory of quantum gravity is easier to find than it seems. Its opponents aren’t convinced, and think that the approximation is missing something important which shows that gravity needs to be modified.
In Loop Quantum Gravity, the critics think their approximation misses space-time itself. Proponents of Loop Quantum Gravity have been unable to prove that their theory, if you take all the non-perturbative corrections into account, doesn’t just roll up all of space and time into a tiny spiky ball. They expect that their theory should allow for a smooth space-time like we experience, but the critics aren’t convinced, and without being able to calculate the non-perturbative physics neither side can convince the other.
String Theory was founded and originally motivated by perturbative approximations. Later, String Theorists figured out how to calculate some things non-perturbatively, often using other simplifications like supersymmetry. But core questions, like whether or not the theory allows a positive cosmological constant, seem to depend on non-perturbative calculations that the theory gives no instructions for how to do. Some critics don’t think there is a consistent non-perturbative theory at all, that the approximations String Theorists use don’t actually approximate to anything. Even within String Theory, there are worries that the theory might try to resist approximation in odd ways, becoming more complicated whenever a parameter is small enough that you could use it to approximate something.
All of this would be less of a problem with real-world evidence. Many fields of science are happy to use approximations that aren’t completely rigorous, as long as those approximations have a good track record in the real world. In general though, we don’t expect evidence relevant to quantum gravity any time soon. Maybe we’ll get lucky, and studies of cosmology will reveal something, or an experiment on Earth will have a particularly strange result. But nature has no obligation to help us out.
Without evidence, though, we can still make mathematical progress. You could imagine someone proving that the various perturbative approaches to String Theory become inconsistent when stitched together into a full non-perturbative theory. Alternatively, you could imagine someone proving that a theory like String Theory is unique, that no other theory can do some key thing that it does. Either of these seems unlikely to come any time soon, and most researchers in these fields aren’t pursuing questions like that. But the fact the debate could be resolved means that it isn’t just about philosophy or theology. There’s a real scientific, mathematical controversy, one rooted in our inability to understand these theories beyond the perturbative methods their proponents use. And while I don’t expect it to be resolved any time soon, one can always hold out hope for a surprise.
4 gravitons 
If you are going to mention non-perturbative theories of quantum gravity, don’t you have to mention the “Causal Dynamical Triangulations” of Renate Loll and her colleagues? She certainly addresses how perturbative approaches can cease to approximate essential aspects of the phenomena they are trying to describe. (https://perimeterinstitute.ca/people/renate-loll) Why is CDT never mentioned? Because it is more geometrical than physical? Because its observables are non-local and don’t correspond to our classical observables?
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I left it out because I don’t know as much about it, but the impression I get is that the issues are similar to those of LQG: it’s an approach that’s nonperturbative in the coupling but where it isn’t clear that it has the right continuum limit.
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Alternatively, we could take a more empiricist path and not make grand claims about geometric structure that we cannot measure. How does a description of a measurement that we think of as characteristically about quantum gravity differ from a description of a measurement that is only about gravity or only about quantum theory? Suppose we have two measurement descriptions f and g and call the corresponding measurements an f-measurement and a g-measurement, how do those two measurements affect each other?
The useful aspect of this perspective is that descriptions can be smoothly classical. Indeed for a QFT on Minkowski space we typically take f and g to be functions in a Schwartz space on Minkowski space. In that simple case we have for the 2-measurement Vacuum Expectation Value and writing M[f] for an f-measurement that 〈V|M[f]M[g]|V〉 is a nonlocal manifestly Poincaré invariant bilinear functional (f,g).
More generally, a description could be any formal structure or even natural language. In the latter case, and hence in the general case, we could not expect a generally covariant theory to be linear in the descriptions f and g if we asked ChatGPT 23.0.5 to tell us how the 2-measurement manifestly generally covariant VEV for the quantum gravity case, 〈qg|M[f]M[g]|qg〉=K(f,g), changes as we change f and g.
With apologies that this is a polemic that I know will not move anybody who is looking for no less than a ToE, nonetheless I have found it helpful to discuss progressively refined descriptions —what a computer scientist would call metadata or a computer programmer would call the ReadMe file, without which Terabytes of data is almost useless— instead of trying to discuss the possibly infinite complexity of whatever it is that causes measurement results to be as they are. That said, I do not know any answer to my first question above: How does a description of a measurement that we think of as characteristically about quantum gravity differ from a description of a measurement that is only about gravity or only about quantum theory?
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I’m not sure how you’re distinguishing descriptions of measurements from the measurements themselves. Is the S-matrix, for example, a description of a measurement or a measurement?
If we just talk about measurements of observables, and put aside whatever you’re up to with descriptions, then it’s easy to distinguish observables that involve gravity with ones that don’t due to the presence of an observation that transforms with spin 2 and has the right distance scaling to be massless. As to quantum vs. classical, I tend to think of this as just being a difference in whether or not you describe your measurements in a consistent way. If you do, then they’re quantum, if not they might be classical.
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I was sure I could rely on you for probing questions. Thank you.
For a scalar QFT on Minkowski space, what we usually call a ‘quantum field’ φ(x) is an operator-valued distribution, not a measurement operator: the standard formalism ‘smears’ that with a ‘test function’ to give us what I’ve called an f-measurement, M[f]=∫φ(x)f(x)d⁴x. So in that context, I would take a description to be a smooth function on Minkowski space that also has a smooth Fourier transform. A ‘test function’ is often called a ‘smearing function’; it’s also very similar to the idea of a ‘window function’ in signal analysis. The VEV for a particular f-measurement, g-measurement pair would give us an amplitude of the ‘overlap’ or ‘resonance’ between those two measurements for a particular theory.
I think of the S-matrix as a (sesquilinear) functional of the VEV for particular choices of abstract measurements. We can define and discuss algebraic relationships between abstract measurements without knowing anything about a state, then a state tells us what statistics of measurement results to expect for a particular choice of abstract measurements in that state. I hope that’s not too terse.
I think I’d prefer not to jump straight to spin-2 representations of the Lorentz group, though this might be an aspect of why I feel very stuck, so perhaps I should. I worry that teleparallelism or symmetric teleparallelism, torsion or non-metricity, might be a better choice for physical models than a metric connection, at least locally. In contrast, I think I might like the phrasing “has the right distance scaling to be massless”, though I feel it’s not very clear how to define distance except by the relative decay of massless and massive propagators.
For quantum vs classical, I’m not sure what you mean by ‘consistent’? I think of a QFT as having a noncommutative measurement algebra that satisfies microcausality (so measurements at space-like separation commute), whereas I think of a classical random field theory (an indexed set of random variables, with the index set being a Schwartz space of test functions, exactly as for QFT) as having a commutative measurement algebra (so measurements at both time-like and space-like separation commute.) We also have to worry about the difference between thermal noise and quantum noise, with the latter being Lorentz invariant for a QFT on Minkowski space, which thermal noise is not, but that is a difference that can reasonably be adopted for a classical random field theory on Minkowski space.
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Hmm. I’m probably missing something, but from what you’re saying about descriptions I still can’t really tell the difference between them and observables (as long as those observables are defined responsibly with the use of test functions or are otherwise “integrated observables” like a cross-section). But this may not matter very much.
In general, I hadn’t been keeping in mind the overall goal you’d told me about before, of trying to build up a “fully empirical” picture. Thinking more in those terms, it seems to me there are two different ways you could think about it, with different implications for your question.
If you’re doing that kind of thing, though, then I don’t think you actually want to distinguish quantum gravity/gravity/quantum. A virtue of a first-principles empirical approach is that you can end up with very different categories than the ones we’ve ended up with over the history of real-world physics. I think it’s worth thinking about how this kind of first-principles approach would interpret historical data and classic experiments, but coming up with a direct translation feels like an unnecessary dilution of the potential of the approach.
If you’re doing that, then I think you should think of gravity not as a particular set of symmetries or scaling or anything like that, but in terms of a correspondence principle. That is, regardless of how your observables look when examined with sufficient detail, there should be some amount of coarse-graining where your observables look like the observables you get from classical Einstein gravity. You want to be careful to define that coarse-graining in a way that leaves open existing loopholes (so it’s not literally just an energy scale but has other restrictions based on the kinds of experiments people can do).
Similarly, for quantum, you need something that ends up with the right Bell inequality violation and things like that.
My comment about consistency was mostly snark. It was partly me being vaguely QBist (and thus something like “QM is just what happens when you take epistemology seriously”), and partly me being skeptical (probably unfairly at this point) of Jonathan Oppenheim-style mixed quantum-classical approaches.
While you can rule out classical physics with Bell tests/CHSH, I don’t have the impression that’s sufficient to “rule in” quantum. Quantum thus seems much more like an “internal” thing, a particular assumption about how theories work on the inside, than an external thing definable purely in terms of observables. So I think if you want to know whether a given set of observables are quantum you should just be doing a correspondence principle-esque thing, and making sure that they respect existing observations that are attributed to quantum effects.
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The difference between measurements as I’m taking them and as they’re usually taken in QM is, I think, that I take a measurement operator to generate expected statistics for a dataset, some number of Gigabytes stored in computer memory by an apparatus, instead of being somehow associated with a ‘particle property’ (or ‘system property’ with a ‘system’ constructed as a compound object, using a tensor product). I think many arguments about interpretations of QM rest on details of that ‘somehow’: put simplistically, mind or matter, et cetera, QBism or Bohmian mechanics, …. Axiomatic QM often begins with “There are systems and each one has a Hilbert space”, but all I see in the first instance are datasets for which the statistics are consequences of every detail of how an apparatus has been set up. I want not to say anything like “every event in an apparatus is caused by one system” as axiom #1. If it turns out that it’s consistent and useful for a particular set of experiments to say there are systems causing events, then I’ll be right there, but I don’t want to say that as axiom #1. QFT does not, so I’m much more comfortable with QFT than I am with QM.
Equally, I have become struck by classical signal analysis and classical data analysis because they are less constrained than classical mechanics, partly because they also do not introduce ‘systems’ a priori. It’s just been confirmed that I’ll be giving the Oxford Philosophy of Physics Seminar on October 24th, for which the title hopes to channel these ideas, “A Dataset & Signal Analysis Interpretation of Quantum Field Theory”. [At a theory level, I think in terms of signal analysis and almost ignore data analysis even though I think the difference can be critical when it comes to an empiricist take.]
I think your two points above about how to approach empiricism are well-taken: I think a ‘completely’ empiricist approach would be incoherent. I think we try out taking different historical aspects more or less seriously and succeed or fail on that choice. I think we are theory-laden. Part of what I’ve taken to be important is that how we describe measurement devices and measurements as part of our experimental apparatuses defines how we imagine what happens behind the scenes.
We can rule out traditional CM on phase space by Bell and other no-go theorems, but classical signal analysis is naturally developed as a Hilbert space formalism because of its love of Fourier and other integral transforms. In full generality, the measurement theory for signal analysis is exactly the same noncommutative algebraic structure as we have for QFT. Indeed, there are many videos on YouTube that say, in effect, “look, time-frequency analysis of sound waves is just like QM”. They fall short, of course, but I think they are partly right. What I think they lack is a coherent approach to why generalized probability theory is appropriate for signal analysis and data analysis instead of just probability as the Kolmogorov axioms would have it.
A classical signal analysis interpretation of QFT avoids the classical-quantum separation that plagues Oppenheim’s and any SDE approach. Instead of waving hands and the CorrespondencePrinciple-EhrenfestTheorem not-quite-inverse-pair, there can be isomorphisms. I find it instructive how what I’m doing parts ways with, for example, geometric quantization, by using the Poisson bracket differently to construct what I call CM+, noncommutative CM with quantum noise as well as thermal noise, instead of trying to construct QM. CM+ is effectively CM without the constraints that makes it different from signal analysis (and quantum noise requires signal analysis to be generalized to at least 1+1-dimensions because it requires the Lorentz group.) You could have a glance at the slides for a talk I gave to members of a research group in Berlin in May, which contains most of the mathematics I will present in Oxford (Dropbox PDF here.) My apologies if I’ve already linked to this and you were underwhelmed: I think it’s better than I had last year, but it’s still missing some secret sauce.
My take on why we can’t “rule in” quantum is that there are other generalized probability theories, including some with nontrivial nonassociative algebraic structure, however I think both signal analysis and quantum are sufficient to model any finite collection of finite datasets. I think the question is only that a more general structure might be more useful or more intuitively helpful in some circumstances. So for the time being I intend not to worry about more exotic structure.
I would love to find a Principle that would let me build a principle theory approach to gravity, but so far I think I haven’t seen even a hint of a candidate. Perhaps tomorrow I will have an AHA! moment, but I suspect it will come from someone else.
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I always liked Sati and Schreiber’s infographic on non-perturbative qfts: https://ncatlab.org/nlab/show/non-perturbative+quantum+field+theory#Idea. Even if you have no idea about the technicalities, you immediately get an impression how far away perturbative approaches are from the full non-perturbative picture.
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Which category of quantum gravity theory does your own work fit into?
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So, I can understand why you might have this misconception, but I’ve never actually published anything on quantum gravity!
You might be thinking about N=8 supergravity, which is the source of the research question the blog is named after. It’s a question I’ve always been curious about, but I’ve never actually been part of the team working on it. That question is a purely perturbative one, about what happens when the loop order gets high. In this case, I don’t think anybody thinks of N=8 supergravity by itself as a solution to the problem of quantum gravity, in part because it’s known to not have the right non-perturbative behavior. The hope is more that it might give useful insights for another approach.
The other, less likely, possibility is that you’re thinking about N=4 super Yang-Mills. This is of course not a theory of quantum gravity…except that it is, because it’s dual to string theory on AdS_5/CFT_5. This is one of those rare cases where string theory has a complete non-perturbative theory, but only because you can define that theory to be the thing dual to N=4 super Yang-Mills (which one is motivated to do because it matches a lot of perturbative calculations and other approximations). My work in N=4 super Yang-Mills isn’t really relevant to that situation, but it is a big motivation for other people researching the theory.
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“To an outside observer, this makes quantum gravity seem much more like philosophy or theology than like science or math.”
There’s plenty of confusion between science and natural philosophy among many people today, who wish to push the boundaries of science far beyond where they should be into what would traditionally have been regarded as natural philosophy, and who call themselves scientists but who really should be called natural philosophers.
The whole point of science is the scientific method, which is that you have some hypothesis (gravity is quantum) and then you take the hypothesis, come up with some kind of experiment to test the hypothesis (i.e. use a large enough object so that gravity matters but small enough that the object can be placed in superposition, then see if the gravitational field is in superposition), and then perform the experiment to see if the hypothesis holds or is falsified.
However, currently, there have been no actual experiments to see whether gravity itself is quantum. Thus, we don’t actually know whether gravity is actually quantum, and all this theorising about quantum gravity by string theorists or loop quantum gravity theorists or asymptotic safety theorists or entropic gravity researchers is just a very mathematics heavy natural philosophy. And that isn’t to say that the mathematics and the natural philosophy of quantum gravity isn’t important – it is very important for making progress in other related fields of physics such as particle physics and condensed matter physics, and to provide some starting point for the science of quantum gravity whenever the experiments do arrive, but it isn’t science.
In so far as where quantum gravity theories do end up being experimentally tested, and thus enter the realm of science – it is usually for some condensed matter physics system where the quantum gravity theory is an effective field theory for some emergent phenomenon. But this has nothing to do with the original question of whether gravity itself is quantum; which is currently beyond scientific experiment and so still very much remains the realm of natural philosophy.
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I should say something about string theory. Unlike the other proposed theories of quantum gravity, string theory was a scientific theory, a scientific theory of hadrons and the strong force that was abandoned as a theory for the strong force when quantum chromodynamics came on the scene and showed itself to be superior than string theory at explaining and predicting hadronic dynamics.
Then string theory was discovered to have a graviton in it, and it was repurposed as a theory of quantum gravity, but since there wasn’t and still isn’t any scientific experiments done to show that gravity is quantum, string theory then left the realm of science and became part of natural philosophy.
However, there are still some string theorists who are working on string theory as an approximation for the strong force – those string theorists are the ones who are able to make testable predictions which can be and have been tested by experiment, and so string theory in that sense is still part of science.
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I think instead of trying to apply concise definitions to messy human activities like research fields, it’s better to focus on the concrete question one is interested in. In this case, there is something characteristic of fields like philosophy (namely, that you can have long-standing disagreements with no resolution) that is not typical of the sciences. The question is, why is quantum gravity an area where there are long-standing disagreements?
The lack of experiments is then part of the explanation, but it can’t be the whole explanation, because mathematics is also a field that in general does not have these kinds of long-standing disagreements.
Part of what’s missing there is rigor. Mathematics works very hard to express claims unambiguously and only insist on claims that uncontroversially follow from their premises. You don’t have to focus as hard on this if you can do experiments, because you still have the real world’s input to force you to agree.
But I don’t think you need the full power of mathematical rigor either. Here’s a thought experiment: suppose we could only test the perturbative regime of QCD. Would there be controversy about confinement?
I don’t think there would be. Yes, we can’t rigorously prove confinement. But between high-quality numerical simulations in Lattice QCD and various heuristic arguments like the QCD string, we could still be pretty confident that confinement would exist.
I don’t think we have anything quite that robust for any quantum gravity proposal (even though several are based on techniques that resemble Lattice QCD!) And so while experiments would change things, and mathematical rigor would change things, I think the most proximate cause of the long-standing disagreements is this, that there is a lack of sufficiently satisfying non-perturbative understanding.
As an aside, the proposal that gravity might not be quantum after all is itself a quantum gravity proposal, not some kind of automatic default., and despite recent advances (Oppenheim et al), it is still very non-obvious that it can be upheld consistently. I certainly don’t think it’s ahead of the proposals I listed in the post, and I think even its proponents would admit it’s a bit behind at this point.
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Please consider restricting yourself from making broad claims about mathematics. Your statement about mathematics not having long-standing disagreements is simply false. It is less apparent because the sociology surrounding applied mathematics leads people to relegate whatever is inconvenient to “philosophy” or “history.” And this is done by invoking unwarrantable rhetoric — one often sees argumentum ad populum, appeal to authority, and ad hominem where beliefs about mathematics intersect with applied methods.
You acknowledge the “hard work” of formulating unambiguous definitions and axioms to obtain uncontroversial claims using deductive methods. It is precisely because of such discipline that the very nature of “definitions,” “axioms,” and “the character of deduction” become ambiguous. As such, the mathematical controversies become useless to “science,” whence pragmatic considerations among scientists discount the existence of these controversies.
I am not “an enemy of science.” But, I do think that the work of mathematicians ought to be respected. It is not. Mathematics is often portrayed as contributing to problems created within the scientific community by the scientists, themselves.
I have studied a single Hilbert problem for most of my life, and that is why I am familiar with these controversies. I have had to sort through translations of historical documents because of more appeals to authority than I can count. That mathematicians are said to have “followers” should indicate just how fragmented mathematics has become. And, it has taken this form because of a discipline not shared by people who learn mathematics as an adjunct to topics about which they have beliefs.
The relationship between science and mathematics is difficult to discern correctly. More than a handful of philosophers and mathematicians have tried to demarcate mathematics for the express purpose of justifying the knowledge claims of physics. Then there are the indispensability arguments which declare that mathematical objects must exist because science is true and truth is philosophically bound to existence.
(sigh)
Empiricists want to invoke statistical inference to discount all of this “philosophy.” But, if one follows the definitions involved with assumption 7 in the link,
https://en.m.wikipedia.org/wiki/Naturalism_(philosophy)#Providing_assumptions_required_for_science
one finds that the definitions by which random sampling is a meaningful concept are grounded in unempirical mathematics.
So, the next rhetorical tactic is to deny “meaningfulness” as understood by mathematical logicians (Never mind that this undermines assumptions taken for granted by Turing computability and Markov’s constructive mathematics — essential to the actual construction of material computers). Eliminative materialism classifies the received view of meaningfulness under “folk psychology” with the promise that “science” will someday replace folk psychology with a “correct” scientific basis.
“Because science is true,…”
(sigh)
If you wish to get a sense of just how much unresolved controversies have been part of mathematics, just peruse threads from the first 10 to 15 years of the archive at
https://cs.nyu.edu/pipermail/fom/
As for how a Hilbert problem could force one to sort out confusion surrounding statistical inference, random populations, and probability, look at the link,
https://en.m.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
For what this is worth, I am very appreciative of the views you express on this blog. And, I agree that scientists ought to pursue their questions (but, maybe on their own dime instead of relying upon the authoritarian power of governments to tax others — everyone else seems to be funding their passions by asking people to donate $19.95 per month on their credit cards).
And, should you ever run into an uncontroversial completion of the expression,
“Mathematics is …”
please write about it.
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You’re right that I made too broad a statement. There are debates in mathematics as un-resolvable as the most unproductive debates in philosophy.
That said, I think it’s fair to say that those debates are un-characteristic. They pop up in relatively small areas like the foundations of mathematics, and they seem to me to exist precisely because the usual tools of mathematics are insufficient to address the root of the disagreement. So while I agree with you that it isn’t useful to demarcate away those debates as philosophy, I expect that they are relevantly similar to philosophy in that they can only be addressed by methodologies that are characteristic of philosophy (and uncharacteristic of mathematics).
Fundamentally, I’m not just getting this impression as a scientist “user of mathematics”, but as someone who interacts socially with mathematicians a fair amount. When mathematicians disagree with others’ research, it’s rarely for these kinds of foundational reasons. More typically, they think that others are pursuing problems that they aren’t equipped to address, or are focused on irrelevant details of something that could be pursued more interestingly in a different way. At most, you get friendly discussions over beer of whether the empty graph is connected or not. Wholesale disagreements about whether the core arguments of another sub-field even work at all are much rarer. And those kinds of disagreements are quite typical when you ask quantum gravity people about other approaches to quantum gravity. That suggests that there is something significantly different between quantum gravity and (typical) mathematics (even if it turns out there are areas of mathematics that work like that too). That difference demands an explanation, and I think this post is a plausible one.
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Thank you for your moderate reply, and, it is one of your characteristics for which I have great appreciation. As is somewhat obvious, I am somewhat frustrated by the circumstances.
Without question there is amicable collaboration between applied mathematicians and disciplines within science. Your own experience with “periods” being transformed into a discrete alphabet is one such example.
But, because of your successes (and I am sorry about the choice you had to make because of the immigration difficulties) you are far removed from the issues of which I speak. It is not that you are unaware — I like your acknowledgement of the relationship between QBism and epistemology. Yet, for example, epistemic logics are a form of modal logics. Years ago, Tarski established that topology serves as a semantics for S_4 modal logic. Later, Kripke would formulate a semantics acceptable to symbolic logicians. Lastly, using methods aligned with category theory, Awodey would show that naive concepts of continuity only yield modal necessity for S_4, not determinism. And, the pretense behind “computation” and “information theory” involves S_5 modal logic.
Your mention of epistemology, although meaningful, is based upon a debate ignoring the work of mathematicians.
A similar situation exists with trigonometric functions. Tarski had shown that the theory of closed real fields is decidable. Richardson had shown that arbitrary extensions of that theory with analytic functions introduces undecidability. The mathematics of infinitesimal rotations is invoked because it is algebraically useful to use vector algebra. Do we live in a “quantum universe” or are we inventing an ontology because we are applying mathematical tools recklessly?
I cannot answer that question. It is doubtful that an empirical answer is available without data. And, I would not wish to suspend scientific inquiry because of what I know about mathematics.
But, at this point, I feel justified at pointing out that the ‘M’ in STEM is being manipulated by the ‘STE’ components.
I seem to have started a wider discussion that I must now read. As always, though, thank you for your blog.
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I think it wouldn’t quite be fair to refer to the people I had in mind as applied mathematicians. Certainly it would be a stretch call a structural matroid theorist or structural graph theorist an applied mathematician, or to call most algebraic geometers applied mathematicians.
What they are not, and I think this gets at the root of what you seem to have in mind, are people who study the foundations of mathematics. And I do think that is an important difference, and one with variants that crop up in many different fields. Foundational work, no matter the field, tends to be the most controversial, and it also tends to be the most likely to yield the kinds of examples you’re giving here, of arguments that things the rest of the field/other fields have taken for granted may be flawed. Foundations of mathematics is like that, but so is the sub-field of quantum foundations. I think one could conceivably argue that that is part of the problem with quantum gravity, at least for certain parts of the area (parts of the Swampland program, black hole information theory).
And yeah, the fun thing about delving into those foundations is that one runs into things that make one want to doubt the whole scientific edifice on top of them! Your examples are things I never would have thought to question, and will probably motivate me to go on a wikipedia binge at some point to get an idea of what’s going on.
Looking back through your list though, I find it interesting how much of it consists of “so and so established X”. Not “so and so claims to have established X, others think the argument doesn’t work”. I can appreciate that there probably are controversies of that kind you aren’t mentioning here, but it’s striking how even in this context there are things you can uncontroversially state as fact. I think it would be hard to write a similar paragraph on foundational issues in quantum gravity.
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“In this case, there is something characteristic of fields like philosophy (namely, that you can have long-standing disagreements with no resolution) that is not typical of the sciences.”
This is actually typical of the sciences. For example, the hypothesis that the world was made of atoms was a long standing hypothesis in natural philosophy and science dating back to Democritus in ancient Greece but only became testable using the scientific method (and thus part of science from my point of view) in the early 1900s. Quantum gravity in 2024 is in the same position as the atomic hypothesis was before the 1900s – lots of theorising but no readily available experiments.
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Atomism is an interesting example, but I think a qualitatively different one. Atomists and anti-atomists used their beliefs as inspiration for their science, but in general almost up until the evidence was available, the explanations they proposed were compatible. The models proposed by one group could be explained in the other’s terms. Atomism was much more like Bayesianism vs frequentism or the debate between different interpretations of quantum mechanics. It was resolved by experiment, and those probably will not be, which is an important difference. But a lot of the quantum gravity disagreement can be resolved through pure theory, provided we become a heck of a lot better at non-perturbative physics and at characterizing whether lattice models have continuum limits, and that’s a relevant difference too!
(Also, as I pointed out to mls, a characteristic trait doesn’t have to hold universally! The point is, one can observe that the sciences have a method for resolving disagreements in many cases that other fields lack, then think about what parts of that method hold in different circumstances)
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“As an aside, the proposal that gravity might not be quantum after all is itself a quantum gravity proposal, not some kind of automatic default.”
Yes, the automatic default for science would simply be saying that “we don’t know that gravity is quantum or not” and either setting up scientific experiments to test whether gravity is actually quantum or not or leaving it to the philosophers to speculate about the true nature of gravity and moving on to some other topic more tractable to the scientific method.
That isn’t to say that the philosophical work of quantum gravity isn’t important – it is very important, as you mentioned, to get mathematically rigourous theories of quantum gravity or to figure out non-quantum versions of gravity like Oppenheimer’s theories so that we can theorise what behaviour gravity might have when it interacts with the quantum realm. But those are ultimately philosophical questions about quantum gravity, rather than scientific questions about quantum gravity – the scientific question for these theories being simply, “does this theory hold in the real world, and how can we test this theory in the real world?”.
“Here’s a thought experiment: suppose we could only test the perturbative regime of QCD. Would there be controversy about confinement?
I don’t think there would be. Yes, we can’t rigorously prove confinement. But between high-quality numerical simulations in Lattice QCD and various heuristic arguments like the QCD string, we could still be pretty confident that confinement would exist.”
Yes, but QCD confinement would in that case just be a philosophical hypothesis that happens to be widely held among physicists, since it would never be amenable to the scientific method. Numerical simulations and heuristic arguments from QCD are great philosophical justifications for why somebody might believe that confinement would exist in the absence of any evidence from the real world on this topic, but they are no substitute for scientific evidence from the real world. Physicists were once confident that a lot of things existed in the past based on the numerical calculations/simulations and heuristic arguments that existed at their time that were later shown not to exist at all by evidence in the real world.
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Some of these claims are much more mathematical than philosophical. “Does LQG have a smooth continuum limit?” and “is there a consistent non-perturbative theory that has perturbative limits that match on to the various string theories?” are mathematical questions, not philosophical ones. (Or to be more pedantic, questions amenable to methods characteristic of mathematics rather than of philosophy.) If we had the answers to those questions, we would be faced with some number of competing hypotheses. Only if there was only one option at the end would we be faced with what you’re characterizing as a philosophical question, about how much we can generalized what we know about physics beyond the realm of existing experiments. But (as I think you get into in your next comment, I haven’t read it yet 😉 ), this is a pretty inevitable type of philosophical question, and maybe not one to be all that worried about.
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I don’t think that many physicists, especially theoretical physics, realise just how much philosophy they actually do on a day to day basis as part of their research.
Take the example of gravitational waves. Sure, the existence of gravitational waves is mathematically derivable from Einstein’s theory of general relativity, but that didn’t mean that gravitational waves needed to exist in the real world – equally it could have been that general relativity (and likely also special relativity) was wrong and needed to be replaced with something else. The existence of gravitational waves only became part of science in my opinion when Joseph Weber started to construct the first gravitational wave detectors in the late 1960s, and it was eventually confirmed in 2017 by the LIGO team. But before the failed Weber bar experiments, gravitational waves were solely the realm of philosophical thought experiments based on the mathematics of general relativity. However, gravitational waves was part of physics ever since it was conjectured by Henri Poincaré in 1905 as a consequence of the speed of light limit in special relativity.
Physics itself as a whole was a well defined subject well before the scientific method came along in the 1600s. It is called Aristotelian physics by us in the modern world, and was a branch of philosophy largely consisted of a number of speculative philosophical theories and mathematical models of how objects moved with no relation to real world experimental evidence at all, quite like the situation with quantum gravity today. But physics didn’t become a science until physicists started subjecting their theories and models to the scientific method in the 1600s.
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Sure, but the sorts of extrapolations you’re labeling as philosophy here underlie everything we ever say in science. Engineers expect a device to work because they’ve built similar devices and think they know the laws behind them, but whatever new device they’re building might be different enough, for a totally unforeseen reason, that it doesn’t obey those laws. The sun might not rise tomorrow. Gravitational waves might only exist for the purposes of experiments performed by humans, and be undetectable to aliens, due to some complicated conspiracy by people simulating both groups on vastly more computational power. And so on. The problem of induction is still the problem of induction, and no amount of scientific evidence is ever a guarantee. We’re always generalizing, the question merely is “how much”?
In practice, though, we’ve settled on a degree of generalization that works. Not all the time, certainly. Physicists have definitely been utterly convinced of something and then, upon testing it, found out they were wrong. But we’ve been doing less of this over time, and that can’t just be chalked up to the scientific method, because it deals with how to generalize, which the scientific method doesn’t tell you. I suspect it has much more to do with the extent that we try to put mathematically consistent frameworks on top of those observations. Nothing can be ruled out “by pure thought”, but a lot more can be ruled out if one has both mathematics and experiment, than if one just has the latter.
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So, our exchange is becoming unreadable on my device. I am resetting the format issue with a new comment(s).
Yes, modern mathematics departments generally introduce a division between applied mathematics and pure mathematics. Typically, the topics in pure mathematics are algebraic in nature. And, one will certainly find accounts of “mathematical” as used in “mathematical logic” attributing this use as comparable to algebra. Such accounts, however, obfuscate some history.
The “algebraization of mathematics” actually occurs in the 20th century with criticisms of Russell (logicism) and Zermelo (axiomatic set theory) by Thoralf Skolem.
Structuralism is a philosophical position that arises in response to algebraization (I could find no particular explanation for the topics you listed, whence I assumed the modifier to be general). Structuralism has particular prominence in the debate between category theory and the first-order paradigm. It is used to characterize how one ought to think of the elementary theory of the category of sets as different from the “material” concept of set in the first-order paradigm.
Algebra is essential to applications of mathematics — after all, calculation is essential to these applications. But, the arguments used to reject hidden variable theories in quantum theory, for example, use inequalities. And, numerical integration suffices for engineering purposes. Both reflect reasoning typical of analysis rather than algebra.
Few people are aware that Cauchy had objected to the algebraization of mathematics just as such research had begun,
Click to access rnoti-p1334.pdf
Cauchy’s approach, in turn, suggests a re-assessment of what is meant by “the continuum.” The intermediate work exposing such a possibility is Robinson’s development of non-standard analysis based upon the first-order paradigm. It seems that the algebra of differentials may be incompatible with an Archimedean real number line. Katz and Borovik have named the non-Archimedean real numbers a Bernoullian continuum,
Click to access Borovik-Katz-2011.pdf
For my part, I think there is a problem if the mathematics I am taught is irrelevant to the scientific claims while I am told that I need to understand the mathematics to assess the scientific claims.
The SEP has an article on structuralism,
https://plato.stanford.edu/entries/structuralism-mathematics/
Vaughn Pratt is an advocate of algebraized mathematics. His SEP article is at the link
https://plato.stanford.edu/entries/algebra/
And, I should mention a bit of historical circularity involved here. Although algebraization becomes prominent after Skolem, British philosophers had pursued a logical calculus while Continental philosophers developed the Leibnizian calculus. To that end, Augustus de Morgan had realized that almost all algebra could be reduced to the mere stipulation of rules governing syntactic stipulation. He made exception for the sign of equality, as substitutions ought to be warranted.
The idea of language signatures, uninterpreted symbols, and rule-governed syntactic transformations are all found in de Morgan’s work. It had become prominent in modern debates with Hilbert’s introduction of a syntactic metamathematics. Category theory can subsume Hilbert’s metamathematics at the expense of obscuring its intuitive basis. People who advocate for computation as a foundation often use the “concreteness” of symbols as an advantage over platonism. Relation to the assumptions of the Hilbert school are being ignored.
So, yes, one does not normally think of pure mathematics as applied mathematics. But, if Cauchy’s concerns have any substance, the generalities of algebra, taught as pure mathematics, deserve closer scrutiny.
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To clarify, I didn’t mean “structural graph theory” and “structural matroid theory” in the sense of structuralism as a paradigm within the philosophy of mathematics. What I’ve heard it used to refer to are things like Ramsey theory, where one proves things like “graphs of a certain size are guaranteed to include X”. (Or analogous claims for other combinatoric objects of study like matroids.)
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I actually expected as much. But when I looked for specific characterizations, I could not find them. One mathematics professor had referred to her dissertation as structural matroid theory. Yet, I found nothing in the dissertation identifying a classification of problem type which would distinguish it from the general theory of matroids.
I apologize for these long replies. Should you take some time to investigate, I hope it is fruitful for you.
And, for what this is worth, there are many practices in the pedagogy of physics which do not conform to what one might expect from mathematical pedagogy. I have simply found myself in the unenviable position of trying to sort through the mess. Meanwhile, a great many people are pointing to blackboards full of equations while insisting that they are explanatory.
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You asked about those who might dispute results I mentioned. Although I do not give credence to mathematics as demarcated by formal systems, all of my personal work respects formal systems. My personal disagreement lies simply with the idea that symbolic signatures are fully specified a priori. My personal deliberations begin with empty signatures extended as symbols are introduced with formulas.
But, my personal work and my objections are irrelevant to received views.
The work of Tarski and Richardson is done with respect to the first-order paradigm. To disagree with the results is to promote a distinct paradigm. This is certainly possible. Yet, people typically reason with a classical, bivalent logic. And, they typically reason as if their words have referents. Alternate paradigms have consequences for such assumptions.
Applications of mathematics often reason about real numbers as if the pictorial representation of a number line is uncontroversial. Real analysis is taught with respect to Hilbert’s formal axiomatics. So, the real question concerns the fidelity of results from first-order model theory with intuitive reasoning.
In foundations, it is not uncommon to read of a “crisis in geometry.” And, the response had been to trust numerical methods characteristic of analytical geometry. This is what becomes known as “the arithmetization of mathematics.” This led to Frege’s logicism and Cantor’s transfinite arithmetic. A different issue — one of which Cantor had been aware — motivated Hilbert’s formal axiomatics. Hilbert sought an identity for mathematics differentiated from applications of mathematics (hence, the source of your mention concerning “users of mathematics”). This is why the consistency question had been so essential to Hilbert.
Hilbert’s early work had been a treatise on geometry which demonstrated that the consistency of geometry reduced to the consistency of arithmetic. Although important to later developments recalling combinatorial methods in algebra, the work of other mathematicians became more important for applications. Instead, Klein’s Erlangen program and Poincare’s algebraic topolgy would prove more useful.
Neither Klein’s work nor Poincare’s address questions involving the structure of the real line.
Cantor and Dedekind had exchanged correspondence related to point-set topology. Hilbertian formalism, however, deprecates the importance of Dedekind cuts and equivalence classes of Cauchy sequences. Such concepts are deprecated to mere features of a formalist theory of real numbers — that is, a formal axiomatization of real numbers in the first-order paradigm is not a hierarchy of logical constructions. In this respect, then, it is arguably equivalent to the use of an intuitive real number line. And, this is why I make note of the results from Tarski and Richardson without concern for alternate views. What alternate view that preserves bivalent reasoning more closely aligns with an intuitive use of real numbers.
If I open a text on linear geometry describing the topological group theory underlying the continuous groups spoken of in quantum theory, the topological character of those groups is based upon the point-set topology of the real line. Presumably, then, one ought to be justified in asking if issues involving the real number line are simply propagated to the more complicated mathematics of matrix algebra.
An important aspect of the first-order paradigm is that equality statements are taken to be decidable. Hilbert had noted this explicitly, and, this is one aspect of why the deductive rules of first-order logic and logicism coincide. Brouwer’s intuitionism is one paradigm in which this assumption is not taken for granted. The Brouwerian paradigm emphasizes apartness over decidable equality. Statman and van Dalen wrote a paper about equality in the presence of apartness. The model theory for this is the modal semantics of Kripke frames. In so far as one may speak of intuitionistic logic as an alternative to the first-order results of Tarski and Richardson, it does not have a classically bivalent logic. And, so long as one insists upon conservative extensions to a theory, decidable equality is replaced by an infinitary hierarchy of statements.
With regard to computation, Markov had wished to reconstruct classical reasoning when constructive methods had been primarily intuitionistic (Brouwer-Heyting-Kolgomorov semantics). So, he introduced strengthened implication based upon “givenness.” This can actually be found in Russell’s works. But, it is problematic because the associated logic is not “realism” despite being a bivalent logic.
What is even more laughable is that the classical propositional logic is not even categorical. In 1999, a second model had been discovered by Pavicic and Megill for its algebraic formulation. In 2006, Schechter formulated a syntactic interpretation which he calls the hexagon interpretation. These results are essentially ignored by people researching logic in terms of formal systems and proof assistants. The reasoning behind classical formal proving systems has an essential reliance upon the distributivity of truth table semantics. The hexagon interpretation is non-distributive.
At the very least, I hope this response explains why I did not speak of alternative opinions with respect to the results by Tarski and Richardson.
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