Epistemic opacity in simulations

This post is the result of reading Wittgenstein and the philosophy of simulation literature in close temporal proximity.

Here is Paul Humphreys on epistemic opacity in computer simulations:

a process is epistemically opaque relative to a cognitive agent X at time t just in case X does not know at t all of the epistemically relevant elements of the process. A process is essentially epistemically opaque to X if and only if it is impossible, given the nature of X, for X to know all of the epistemically relevant elements of the process. For a mathematical proof, one agent may consider a particular step in the proof to be an epistemically relevant part of the justification of the theorem, whereas to another, the step is sufficiently trivial to be eliminable. In the case of scientific instruments, it is a long-standing issue in the philosophy of science whether the user needs to know details of the processes between input and output in order to know that what the instruments display accurately represents a real entity.

The charge is that simulations bring something new to philosophy of science because they are epistemically opaque, unlike, say, the process of solving an equation analytically.

However, I’m not sure I understand how simulations are any more epistemically opaque than physical experiments or non-automated calculations in mathematics. First, consider experiments. It seems to me that the checks we make to ensure that the results of experiments are reliable are almost completely analogous to those we make to ensure that the results of simulations are reliable. Allan Franklin has a good list of the kinds of checks we make to ensure that experiments produce reliable results. All the seven criteria he describes there seem to be used to validate simulations as well as physical experiments. We do check that the simulation reproduces known results and artifacts. We do try to eliminate plausible sources of error. If the simulation produces a striking pattern that can’t be explained by plausible sources of error, we do use that pattern itself to argue for the validity of that pattern as a legitimate result. If multiple independently corroborated theories account for the results of a simulation, that does add to the validity of the results. Simulations are often based on well-corroborated theories. Finally, statistical arguments are used to argue that patterns seen in simulations are real.

So what is epistemically relevant in simulations that humans cannot know, that can be known in the case of physical experiments and mental or pen-and-paper mathematical calculations? I’m guessing that what Humphreys takes to be epistemically relevant in simulations but inaccessible to human knowledge is something like the results of each computational step in the simulation, or whether the mechanistic workings of the simulating apparatus produces mathematically correct results. But are the results of each computational step epistemically relevant? Here is one reason to think not. In a physical experiment, one never has a complete working theory of the apparatus that tells us the exact consequences of every step in the experiment. It seems to me that demanding that the result of every computational step in the simulation be epistemically accessible to humans is analogous to demanding that every step in the experiment be justified by a theory that describes every aspect of the apparatus.

What if Humphreys considers the reliability of the simulating apparatus, that is, whether it is producing mathematically correct results, as the epistemically relevant aspect of simulations that is essentially inaccessible to humans? As noted above, the same way one can validate the reliability of experiments without having a complete theory of the experimental setup, we have ways of validating the reliability of simulations. But they are not foolproof of course. Suppose we take seriously the possibility that our methods of validation still leave out epistemically relevant information. It is possible that even though our checks show that the results are reliable in a large variety of situations, some hocus-pocus is going on which can be discovered only by going through every single step in the simulation, which humans cannot do. But there is an analogous “problem” when it comes to mental or pen-and-paper arithmetic. One’s belief that one is calculating 2098×98723 correctly, if one is doing it for the first time, is based on one’s past success in calculating various other things correctly. Of course some hocus-pocus could be going on just this time, for the new calculation, a kind of hocus-pocus which did not show itself in previous calculations. But this possibility does not lead us to say that there is something epistemically missing from the new calculation. If one really wants to be paranoid, one could always doubt the results of mental or pen-and-paper calculations, because after all we do not know, mechanistically, how the human mind consistently applies arithmetical rules, and whether it always correctly applies them. We act as though it always consistently applies them because of prior evidence of its reliability, but these do not suffice to ensure with certainty that it will always consistently apply them. How is this different from the case of simulations? In simulations, we also only have the prior results of simulations, and the backing of mathematics and physical theories relating to the mechanics of the simulation, to assure us that this time the simulation will also be reliable.

Humphreys’ ascription of epistemic opacity to machine calculations but not human calculations is an interesting inversion of one point of view that Wittgenstein discusses at various points in his philosophy of mathematics. Wittgenstein identifies the philosopher of mathematics’ love for axiomatic reductions of mathematics due to the idea of “mechanical insurance against contradiction” (RFM, p. 107e, his emphasis). The idea is that by reducing mathematics to a set of rules that even a machine can follow, one excludes mistakes from mathematics:

We may trust ‘mechanical’ means of calculating or counting more than our memories. Why? — Need it be like this? I may have miscounted, but the machine, once constructed by us in such-and-such a way, cannot have miscounted. Must I adopt this point of view? — “Well, experience has taught us that calculating my machine is more trustworthy than by memory. It has taught us that our life goes smoother when we calculate with machines.” But must smoothness necessarily be our ideal (must it be our ideal to have everything wrapped in cellophane? (RFM, 106e)

Humphreys, P. (2008). The philosophical novelty of computer simulation methods Synthese, 169 (3), 615-626 DOI: 10.1007/s11229-008-9435-2
Wittgenstein, L. (1967). Remarks on the Foundations of Mathematics, ed. G. H. von Wright, R. Rhees, and G. E. M. Anscombe, trans. G. E. M. Anscombe. MIT Press.

Geometry Movie

This is just brilliant (HT: Douglas Kutach). Funny how the comments at Google Video mostly display perplexity as to the point of the exercise.

Learning General Relativity

Before I started on my post-graduation attempt to learn GR, I’d had only some grounding in the fundamentals of differential geometry. In my undergrad course where this was covered, we only got as far as vectors, covectors and (a bit of) tangent bundles. No physics involved, although there was a somewhat awkward attempt to do thermodynamics on manifolds.

I began this journey with a study group partner, whom I shall call A, whose main objective was to learn differential geometry rather than GR. At his recommendation, we started with Reyer Sjamaar’s freely available notes on manifolds and differential forms. I quite disliked this from the start; it introduces manifolds very late in and for the first few chapters you’re faffing about shifting the indices on differential forms! I could get no intuitive understanding at all from them, so after a while I gave up and suggested Schutz’s Geometrical Methods of Mathematical Physics. I actually quite liked this book. Schutz is pretty good at explaining things in intuitive terms. I like this kind of approach when I’m starting on a new subject, because I tend to learn better with an ‘examples first‘ approach than with an ‘axioms first’ approach. The book is clearly catered to physicists, though — mathematicians would find it unrigorous. A is a physicist himself, but he wasn’t happy with the lack of rigour. Three chapters in, A wanted to change books, in particular to Analysis, Manifolds and Physics. Now this book is clearly written by mathematicians, for mathematicians. In fact, some of the early parts introducing manifolds are hard to get through unless you’re already reasonably familiar with topology. The authors had an annoying habit of using theorems in topology to prove propositions in differential geometry. A was quite happy with the rigour in this when we started out, but we got lost so quickly (me having only the most cursory knowledge of topology and him having none) that he was only too happy to let me suggest the next route. While the book itself has a section summarising topology, it is a typical mathematician’s summary — exceedingly terse and, though possibly useful as a refresher to someone who has studied topology before, quite unhelpful to novices.

I had heard many good things about Wald’s General Relativity, the classic graduate text, so we tried that next. I think we both really liked it from the beginning, and perhaps only became frustrated in the last month or so. The first two chapters were great — good prose explanations, and the math is about the right level for those who have some experience with differential geometry. I definitely would not recommend it to a complete novice, though. At this point, we’re slogging through the problems at the end of Chapter 3. Chapter 3 itself was a long, hard slog for us — it’s where Wald introduces most of the apparatus needed to determine curvature, geodesics and such, and the proofs are involved. The problems, now, the problems. Along the way we acquired another study group participant, someone who’d done a GR course before but had forgotten most of it. He had little formal math background but had a great intuitive feel for just shifting indices around until they fell into an arrangement suitable for solving the problem. So he didn’t really understand the formal mathematical characteristics of manifolds and tangent vectors and such, and we had to clear up his confusions about those a few times, but he’s by far the most efficient at doing the problems. Indeed, you can do many of them by just shifting indices around in the right way. Even so, we’ve frustratingly spent about three hours per problem on average for this chapter, and the things we prove don’t even seem particularly physically or mathematically interesting! There’s a lot of counting of symmetries for very special, low-dimensional cases. Since we meet once a week for a few hours, we’ve basically spent the last month doing the first four problems at the end of Chapter 3. Depressing. We’ll press on, though, and hope we get better with practice.

At the same time, I am working my way alone through David Malament’s freely available notes (pdf) for his Foundations of GR course. I like them, perhaps because they closely follow the prose style and theoretical approach of the person who first taught me differential geometry. Lots of nicely worded intuition pumps, like Schutz, but more mathematically rigorous. The problems also seem easier and more interesting than those in Wald. (Since there’s no external evaluator to judge that I’ve mastered the subject, I’m not convinced that I understand it adequately until I can do all or at least most of the problems in whatever book or notes I choose to follow.) I’m only up to the chapter on Lie derivatives now, though, so for all I know they may turn tedious and unrewarding once we hit the Riemann tensor.

Yikes

I submitted the ‘kinematic’ solution to the Cambridge University Press blog’s 5th Martin Gardner puzzle contest. I’d assumed that the requirement of not using calculus simply meant not using any differentiation or integration or limits, but on hindsight using distance = velocity * time probably counts as using calculus. Oh well, at least my answer was quoted.

The real non-calculus answer is depressingly simple.

Physics and the Riemann Hypothesis

I was skimming through this recent PRL paper [preprint here], which purports to show that the non-trivial zeroes of the Riemann zeta function correspond to the energy levels of a quantum mechanical system. My initial reaction was that it’s a cute result, but not one that’s going to help prove/disprove the Riemann Hypothesis. The final paragraph of the paper caught my eye, though:

Apart from providing a new tool in the spectral approach to the Riemann hypothesis, there is also the possibility that it will allow a laboratory construction of a system for which the physics is described by the ‘‘Riemann Hamiltonian’’, the existence of which would prove the Riemann hypothesis.

The mathematical existence (whatever that means) of that Hamiltonian will prove the Riemann hypothesis (RH from now on). But I wonder how physically constructing a system with that Hamiltonian will have any relevance to RH. Firstly, the model is an approximation anyway (to get the correct energy levels, you have to consider the ‘semiclassical limit’), so if we think that quantum mechanics is the ‘true’ theory, then in the real world, that system will not have energy levels giving us the nontrivial solutions of the Riemann zeta function. But more importantly, it seems that the existence of a mathematical operator should be independent of whether there is a physical system whose Hamiltonian is that operator.

So why would one think that the existence of such a physical system would be relevant to RH? Here’s one possible line of thought:
P1. The existence of a self-adjoint operator H having such-and-such mathematical properties implies RH.
P2. If a physical system whose Hamiltonian is a self-adjoint operator H having such-and-such mathematical properties could exist (in our world),* then there exists a self-adjoint operator H having such-and-such mathematical properties.
C1. If a physical system whose Hamiltonian is a self-adjoint operator H having such-and-such mathematical properties could exist (in our world), then RH is true.

I phrased P2 in terms of the possible existence instead of the actual existence of a physical system having that kind of Hamiltonian, since I am assuming that it cannot be that RH is false before physicists construct a certain physical system and then true thereafter — that the truth value of RH is independent of what happens in the physical world, and insofar as the construction of the physical system in question is relevant to the mathematical problem, it can only help us discover if RH is true.

Even if we accept the above argument, it’s not clear how the construction (in the laboratory) of the physical system in question is relevant to RH. One might think that the construction of such a system would show that yes, that system could exist. That would provide us with the antecedent of P2 and hence (if we accept the argument leading to C1) prove RH. But there is an ambiguity in our imagining the construction of ‘such a system’. The authors derive the Hamiltonian they need from a system consisting of a charged particle on the xy plane in a constant uniform perpendicular magnetic field and an electric potential . Suppose we do construct a system with a charged particle in a plane and the requisite electromagnetic field. How are we to determine if its Hamiltonian really is the one derived by the authors? We could simply say that we know it is so theoretically, but in that case, we need not have physically constructed the system. If we want to say more than that, we can try to find experimental evidence for its Hamiltonian being what we suspect it is. We can measure and count the system’s energy levels. We might then ‘prove’ RH as follows: those energy levels are best explained by the fact that in the semiclassical limit the system has a Hamiltonian with such-and-such properties, so we infer that the system has a Hamiltonian with such-and-such properties, so a Hamiltonian with such-and-such properties exists, and hence RH is correct. That is, the only motivation for physically constructing such a system would be to check that it in fact has the energy levels we expect it to have, otherwise we could just sit pretty on the existing theoretical derivations. But since the inference outlined above is essentially an inference to the best explanation, it won’t be a proof of RH — if someone disproved it by mathematical deduction, then we would discount the ‘experimental’ proof and conclude that the explanation for the energy levels we observed is not the existence of a certain type of self-adjoint operator.

So much for physically constructing the system. What about the theoretical derivation itself? Would the mere theoretical possibility of the system suffice to prove RH? Here’s one way it might. If we think the authors’ derivation of the Hamiltonian for the system in question implies that a system with such a Hamiltonian could exist in our world, and we accept the argument from P1 and P2 to C1, then yes, the theoretical possibility of the system would prove the Riemann Hypothesis. But I think it’s very difficult to think that the theoretical derivation alone implies that the system could exist. As a matter of scientific practice, it isn’t the case that all physical systems that are consistent with fundamental laws are considered systems that could exist — many physicists would reject systems that have backwards causation, for example. This is to say that there are factors other than consistency with fundamental theories that we use to determine if a system ‘could’ exist. So if RH was disproved by mathematical deduction, I suspect many would take that as a reason to view the derivation of the Hamiltonian in question as suspect — perhaps to the point of wanting to revise the theories used in the derivation. That is, we would reject quantum mechanics if it implies a mathematical impossibility, but we would not reject mathematics if quantum mechanics says something inconsistent with it.

All this without considering whether the Hamiltonian of the system should be considered as an instance of a mathematical operator that exists, or whether the Hamiltonian itself even exists, etc.

—–
*The ‘in our world’ provision is to say that the system could exist in a world with the laws of nature as this world.

Germán Sierra, Paul K. Townsend (2008). Landau Levels and Riemann Zeros Physical Review Letters, 101 (11) DOI: 10.1103/PhysRevLett.101.110201

Mobius Battle

One of the best xkcd strips I’ve seen.

Found via the 63rd Philosophers’ Carnival.

Familiarity and Understanding

I’ve long suspected that much of what I take to “understand” in mathematics or the mathematical sciences is simply familiarity. This is confirmed when I discover that concepts I was previously uncertain about seem to crystallize immediately when I re-read explanations of them. I may completely fail to grasp a concept the first time I read about it. But as I re-read definitions of it, written in myriad forms, I start losing that feeling of unnamable fear I get upon first encountering an abstract definition. Perhaps surrounding a concept with related pictures, which I acquire as I read more widely on the subject, helps me to make connections. But there is also a component due to sheer familiarity — I’ve lost count of the number of times I’ve said to myself ‘oh, it’s just that other thing I read about elsewhere’, and subsequently felt more confident that I understood the concept.

Thus, as I go through the Ehrenfests’ seminal summary of the foundations of statistical mechanics for the second time, I find myself easily skimming through some parts, even though reading it the first time was a torturous process. Parallel experiences can be found in my six-month-long attempt at learning differential geometry — it was only after reading many different approaches to the subject that a picture finally formed in my mind of what a tangent vector really was (that is, a more accurate, abstract picture than that of an arrow tangent to a surface).

My feeling is that this relation between familiarity and understanding is stronger in mathematics than in other disciplines. Perhaps because it is more true in mathematics than anywhere else that knowing how to use a concept is definitive of understanding.

Predigested Formalisms, Spoonfeeding of

Finally, the van Kampen paper which is not available in the Premier Institute of Social Engineering has arrived from Oxford (thanks to a friend there who got her friend in the college that had the book in its library to photocopy it — yes, socialising comes in useful sometimes). I’m lucky enough to be in a job where I actually have access to whatever passes for a university library here, but the thought that I could have gotten the paper in ten minutes rather ten days if I was still in Chicago makes me a little unappreciative of my situation. I will probably write a bit on van Kampen’s take on the Gibbs Paradox soon, but for now, van Kampen’s opening message to Dirk ter Haar is worth quoting:

Beste Dik,

It is a long time since we both attended the lectures by Mrs de Haas-Lorentz on thermodynamics. They were excellent from a paedagogical point of view since they forced you to figure out almost everything yourself. That is a much better way of learning physics than the spoonfeeding of predigested formalisms which is nowadays regarded as the highest wisdom in education. I remember how puzzled I was by the sudden appearance of the term in the entropy. After I figured it out I found that there is still much confusion about it in the literature. That is my excuse for bothering you with such a time-worn subject.

“spoonfeeding of predigested formalisms” — how appropriate that this description should also be so undigested.

And, just because it’s also about pedagogy and because I have said similar things: Timothy Gowers explains why “examples first” is his favourite pedagogical principle. I couldn’t agree more.

The Mathematical Universe Redux

Max Tegmark’s revised Mathematical Universe paper,¹ accepted by Foundations of Physics, is out. The “initial conditions” section I’d criticized for being unclear earlier has been ironed out, but I still think the main thesis is full of holes. What follows is a more detailed take-down of the paper than my previous spiel.

To refresh our memories:

• The External Reality Hypothesis (ERH) states that there exists an external reality completely independent of humans.
• The Mathematical Universe Hypothesis (MUH) states that our external physical reality is a mathematical structure.

Proofs Demystified!

Via James Cook, I find out that Timothy Gowers is blogging. Amongst other things, he has explanations of two mathematical theorems that proceed by analogy from a lower-dimensional case. The first is on how to discover the formula for solutions to cubic equations by analogy with the formula for quadratic equations. The second is on how to get to the essentially two-dimensional Cauchy integral theorem from the one-dimensional theorem that if and is everywhere 0, then is constant. Clear, intuitive explanations in fluent prose — a refreshing change from the usual math-textbook style.