The Simple Truth about Physics
blogs.scientificamerican.com
blogs.scientificamerican.com
What we should value is not simplicity, but power of explanation.
To maximize the chance of choosing a model which corresponds to reality, we must involve as few unchecked assumptions as possible when deciding between theories that agree with the data; and this matters more than any particular assumption having a relatively high chance of being right, because of this exponential sensitivity.
So if we want theories that are correct, yes, we should value simplicity.
I'm reminded of a problem I was given on a math exam: a hand-drawn polynomial on a 5x5 grid, where the curve intersected the grid in 5 points. "Find the equation of this cubic polynomial."
The given integral points weren't possible for any cubic; if the function was a polynomial, it was a quartic at a minimum.
Sadly, my teacher was not trying to trick us. Simplicity, itself, was an unchecked assumption which rendered correctness impossible.
Under the frequentist interpretation of probability, 95% of having a "correct assumption" means that as you observe iid experiments of this assumption, the long-run average of them will have 95/100 of such experiments exhibit the assumption as true. But "experiment" here isn't a physics experiment, it's a _universe_ in which this assumption could be true. We're in some _fixed_ universe. This assumption holds in ours with probability 0 or 1 (we don't know which).
So, I think the 95% you're referring to here is a Bayesian belief level. This is coherent in that it doesn't require some ambient set of multiple universes to meaningfully describe what is meant by "probability", but the prior required for this interpretation is a bit of weird beast. That is, we have some prior over the space of all possible assumptions for models of physics and update our beliefs based on evidence (we can collect evidence from our single universe multiple times and update our beliefs in a coherent way). In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question.
Within specific settings for statistical learning theory, we find that simple models generalize well. But that's an implication: _if_ you have a class of simple models and it fits the data well, _then_ you'll generalize well. When it comes to answering the question, "Which theories generalize well?", such analysis is incomplete.
A sketch: Suppose we have two theories, one ("A") with assumptions (a, b); the other ("B") with assumptions (a, p, q, r, s, t); and our evidence 'e'.
Pr(A) ~= Pr(a|e) * Pr(b|e)
Pr(B) ~= Pr(a|e) * Pr(p|e) * Pr(q|e) * Pr(r|e) * Pr(s|e) * Pr(t|e)
That is, the probability of each theory being right is the probability that each of its assumptions are simultaneously true. It might be difficult to come up with a specific number for each of those component terms, but we'd do well to estimate the total probability of each theory simply by counting the terms, since in the limit that will matter more than the probability of each term (assuming we think none of them are obviously low).
Also note that when counting, we can trivially factor out the common assumptions "a" (a.k.a. the things we don't wish to doubt or vary between theories, e.g. QM in "ordinary" regimes, GR, the Newtonian approximation to GR, etc.)
Yes, there is a "ground truth" theory which is absolutely true or false, but we don't have access to it. And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different?
Example stress tests of this idea:
e = the motion of the planets; A := "G = m1*m2/r^2"; B := [long list of epicycle parameters] --> pick "A".
e = the varied appearance/adaptations of animals/species; A := [reproduction, inheritance, variation, selection]; B := foreach animal x {"God zotted $x into existence like that because just_so_story($x)" --> pick "A".
e = my empty garage; A := "there's nothing in it"; B := "there's a dragon in it; the dragon is invisible; the dragon dodges your touch; the dragon has no heat signature; the dragon floats and leaves no footprints; ..." --> pick "A".
e = the behavior of the universe; A := [the standard model]; B := [the standard model; also it's a simulation; there are intelligent beings who set up the simulation; there is an external universe in which the simulation is occurring; the number of simulations happening in this universe is large; ...] --> pick "A".
First off, why are assumptions independent? Why are you allowed to factor p(a|e) * p(b|e) = p(a,b|e)? Assumptions aren't just independent binary variables -- it's not necessarily true that you can have some product measure over the set of assumptions (a, b, p, q, r, s, t) simultaneously.
More importantly, you dived write into some notation (Pr) without telling me what it _means_, which is what my OP was about.
> And I don't see how it's more problematic to use probability here than on any other classical event on which we have imperfect information, like a specific dice roll, or a baseball game, or the outcome of an election— One specific thing will happen, and no other outcome was possible, but we can still use probability to model our incomplete knowledge. How is physics different?
Here's a crucial difference. We can roll a die 100 times, 1000 times, 10K times, and the frequency of a six landing as we increase the number of trials will tend to 1/6. That's what we mean (if we're frequentists) when we say the probability of a six landing is 1/6. We can't "roll" universes with physics models.
Because I've defined them that way. I mean them to be independent choices you could make when designing your model that could be varied to fit the data. If two aspects of the model are not independent; i.e. they are covariant in some way, then there is some common parameter that explains them both, and that parameter is the one that should be seen as an input to the model.
> We can roll a die 100 times, 1000 times, 10K times [...] That's what we mean (if we're frequentists)
We're not frequentists.
You can't "re-roll" the 2016 election 10K times, either. There was only one, and there was only one way it could come out; we just didn't know enough to say what it would be before it happened. All the particles in all the voters were obeying the laws of physics at every moment; never was there any freedom for a different outcome. Nonetheless, even though there was/is only one "ground truth" that could ever be, we assigned probabilities to each possible outcome, given our incomplete knowledge.
This is a pretty standard application of probability. State estimators (e.g. the Kalman filter) are doing the same thing— you have some noisy readings of reality, and you use Bayesian logic on some assumed probability distributions to pick the estimate from the space of possible "ground truths" that has the highest probability of being the right one.
Concretely: I'm measuring roll rate, local acceleration, compass heading, barometric pressure, and GPS, all with significant error, and I want to know where my quadcopter is most likely to be at the current moment. There is only one true answer to that question, the quadcopter is in one place, not 10,000 places (or 10,000 flights), there is a single ground truth. But Bayes will give me a probability, given my readings, that any given estimate is the true ground truth (and some math will help me solve for the highest one).
In this case, instead of assigning probabilities to possible election outcomes or system state "ground truths", the "configuration space" is models of reality. But all we've changed is the domain of our probability distribution; the math doesn't care what kind of thing our "ground truth" represents. And it doesn't matter if reality contains only one "ground truth" or many; the fact is that we are choosing between many options (and we are ranking them by likelihood).
Re probability, I'm glad you committed to the Bayesian interpretation. Bayes gives you a _degree of belief_, based on your priors.
It's quite fortuitous that you mention the 2016 election. As you say, there's only one instance here. Which is why the prior matters a lot. We can incorporate (partial) evidence from past elections, but it's going to be very sensitive to the priors that we place, since the net amount of evidence we're working with is very small.
As we found out in 2016, that means these beliefs aren't worth much in such low data scenarios, since the prior has a large impact! https://projects.fivethirtyeight.com/2016-election-forecast/
This brings me to my original point:
> In the limit of evidence, this prior matters less, but constants matter here! How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore? We don't really have a good sense for what the structure over this space of "assumptions of physical models" is, so we can't really answer this question.
But still tractable, I'd say. My core claim is that counting independently-variable assumptions will be a highly performant way to select between theories which agree with the data. Or put another way, it's the best Fermi approximation calculation for measuring "how good is your theory". How you do that for any given theory, while important to do correctly, is an implementation detail which I think is secondary to the discussion of whether doing it at all is a good idea. :) (seems like we might agree that it is?)
> We can incorporate (partial) evidence from past elections, but it's going to be very sensitive to the priors
To the extent that election forecasts are unreliable, I think that's because they are forced to involve a lot of assumptions (e.g. similarity to past elections) that turn out not to correspond well to reality. Models which make fewer such assumptions will do likely do better! (and IMO fivethirtyeight's forecasts did the best job of this out of any; most of the rest put Hillary at around 97%).
Unfortunately with elections, there is a comparatively high lower bound on the number of assumptions we must make, thanks to the complexity of their dynamics and sparsity of data/knowledge we have about each. I think this is much less the case with physics, where we are varying comparatively small physical assumptions to explain mountains of data. But in either case, I contend that the most performant models will make fewer (unmeasured) independent assumptions.
> How much evidence do we need before we can be confident the "belief levels" we're throwing around aren't that subjective anymore?
The point I'm trying to make is bigger-picture than the above level of detail: Counting independent assumptions, in the limit, matters more than the specific constants of each assumption (assuming they're not low/zero), precisely because it's so hard to come up with "accurate" numbers for each.
That is to say, the probability is not sensitive to those belief levels: We could choose widely varying distributions for the probabilities of our assumptions, including choosing probabilities very close to 1, and it will hardly ever matter to the total probability as much as the absolute number of independent assumptions we make.
General relativity has a couple postulates, my theory has one. I guess I'm more likely to be right.
The theory you paint would be composite of other assumptions (involving lighting, forces, or stars, e.g.?) which in turn are either in contradiction with observation (implying Pr=0%), or rely on further (very low probability) assumptions in order to avoid contradiction.
If that's not true, then you've just given a strange name to the empty set of assumptions, which is the null hypothesis; and the null hypothesis DOES get epistemological privilege.
There is no a priori reason why simple theories should be correct. The suggestion that any model can be correct is presumptuous - you might call it an assumption ;-) To borrow from George Box, all models are wrong, but some are useful...
Sad philosophically, but very likely correct.
Also, always remember: we can formulate laws of physics all we want, we can only conduct actual experiments to validate them in a very tiny corner of the universe. Who's to say that - say - Maxwell's law are valid a million light-year away from here?
The scientific method only works locally, unfortunately.
Astrophysicists, that's who.
Astrophysicists are constantly trying to cross-check both theories and constants with observed astronomical data to look for such violations.
Any such violations found would be a Big Deal(tm).
The simple results take centre stage, but science usually requires a lot of complicated work behind the scenes to get them.
On the contrary, relativistic mechanics is strictly simpler than Newtonian mechanics, because the symmetry group is enlarged by the inclusion of time on equal footing with space. A few extra squares/roots don’t make relativistic mechanics less simple than Newtonian mechanics. I cannot emphasize this technical point enough — it is extremely important. More symmetry means fewer parameters that need to be empirically fitted to data (or hand tuned), which means more predictive theories.
This is why symmetries are placed on a pedestal in fundamental physics, and taken as the organizing principle, as far as we know.
The differences boil down to the fact that the Lorentz group is O(1,3) whereas the (homogeneous) Galilean group is R^3 x O(3), both 6-dimensional.
The idea goes like this. If you want to talk about simplicity, beauty, or elegance of a physical law--maybe because you think that's more likely to be correct than a complicated law--then we've already got tools like Kolmogorov complexity to talk about such laws.
https://www.preposterousuniverse.com/podcast/2019/12/09/76-n...
All scientists should strive for parsimony.
In other words Occam's Razor: "It is futile to do with more things that which can be done with fewer", where things in scientific theories can be considered assumptions.
Parsimony, along with falsifiability, are the two most important features of any reasonable scientific theory.
We need to teach more philosphy of science at all levels of science education. I was quite surprised to read that a physics post-doc would openly seek to complicate their models just for the sake of appearance.
Plenty of beautiful, elegant, parsimonious, wonderful, understated, aesthetically pleasing models in physics turned out to be complete bunk when confronted with real data.
The other half appears to be an advert for the author's paper https://arxiv.org/abs/1910.13608 which advocates a particular measure they call "explanatory depth". It's not immediately obvious (to me) how this works, or how it relates to other bayesian & information-based measures. But it seems worth a look.
I also believe that we also value Sophistication, which is at least a partial driver of the interest in string theory, with less free parameters.
If we're talking about understanding the universe, strong/weak force (also Neutrinos have mass), dark matter/energy (and Cosmic Microwave Background), gravitational waves (and black holes) all seem pretty significant and happened after 1920.
What has happened is that the advances have gotten harder to explain and more ubiquitous in their effects. So maybe it feels like there's not advancement, because we're soaking in Moore's law and post WW2 economic growth?
Schrodinger equation - 1925
Theory is good when it explains observations, no application necessary. The latter is mostly engineering challenge, how do we refine uranium for example.
Now if you had said "half century" instead of "century", I would be more inclined to agree with you. Neutrino oscillations are a more recent phenomenon, but they fit relatively cleanly into the Standard Model and have little effect on the fundamentals. You say that "dark matter/energy" are problematic, but I would argue that no serious physicist thinks they are anything like "confirmed". (The DAMA/LIBRA collaboration aside, since no one else believes them.) They are just the best available ideas to explain things; the hope for a better explanation or experimental evidence for a particular direction remains strong.
Back when I worked in this area, I was fond of pointing out that there hadn't been much progress in fundamental theory since the '70s or so, then asking when LSD was banned....
There are not.
At energy scales up through electroweak unification and the Higgs mechanism (~10^2 GeV), the known forces of nature are basically completely understood at the fundamental level. There are many applications yet to be invented -- basically, solving the equations is really hard for anything more complicated than a hydrogen atom -- but the fundamental theory? It's done.
But what about quantum gravity? Dark energy? All those other weird things?
Well, the simple fact that we do not have experimental evidence to guide the development of these theories is prima facie evidence that they are not important at everyday energy scales. If we could access the requisite domains, we would have that evidence! Yes, these theories are important cosmologically or for true "fundamental" understanding, but so was electroweak unification. They will never help us build better gadgets, full stop.
And you say there is nothing coming that could affect our day to day life...they might have said that in 1901.
Who knows what might unfold...
By 2019 we have exquisite verification that we do, in fact, understand things at "everyday" energy scales: https://en.wikipedia.org/wiki/Precision_tests_of_QED
I do not claim there is no new physics to be found.(In fact, the process of renormalization used in modern field theories explicitly parameterizes our ignorance of very high energy scales.) I merely claim there is no new physics to be found that can impact anything already constrained by a precision measurement, and that such precision measurements cover the entire spectrum of everyday phenomena.
(And just because you don't like some of the metaphysical interpretations of the double-slit experiment doesn't mean we don't know how to calculate it... very, very accurately. That, to me, suggests some level of understanding surpassing a Neanderthal!)
Quantized fields are the fundamental building blocks of our reality. Everything else builds from that. The entire concept of a particle is a useful but flawed model, similar to newtonian gravity.
Indeed - non-relativistic quantum mechanics does explain the double-slit experiment and does not use the notion of a field.
Nothing behaves like the classical concept of particle or wave.
There are edge cases where this is the case, and this is the reason the classical theories were and are still successful.
And then there are cases where this breaks down catastrophically, like the double slit experiment. Which are the very reason we had to look for new theories. In this case, it is Quantumelectrodynamics, which perfectly explains the double slit and everything else concerning electromagnetic force. It is the most precise theory in any scientific field, with theoretical predictions matching measurements to 13 significant digits.
Feynman, Schwinger and Tomanago got the Nobel in 1965 for this.
If we reason strictly by analogy (with the UV catastrophe being considered relevant to the observable word), quantum gravity has such a gaping loophole that it cannot explain how we can have simple empty space — the kind we see all over the place!
Otherwise largely agree with your previous point about the lack of easily measurable effects pointing to the low likelihood of technological consequences. It’s an under-appreciated intuition!
Oh, btw (as a fellow physicist) what did you work on? Inspire link? :-)
It doesn't help that there aren't all that many experiments left to do (in the fundamentals realm). Either they're likely to be uninteresting, or they're currently beyond our reach. I think there are going to be many lean years ahead for fundamental particle physics.
I used to work on neutrino experiments, but (as the username indicates), don't do that stuff any more. I could link my publication record, but all that's really there is a bunch of nitty-gritty detector characterization, plus a few blockbusters that my name got on because, well, I helped build the bloody detector :) Turns out that if you can build neutrino detectors that work, there are plenty of other things you can do with your life....
https://en.wikipedia.org/wiki/Ultraviolet_catastrophe#Soluti...
(Oddly, it looks like the term "ultraviolet catastrophe" maybe originated in 1911?)
The math behind the double slit experiment is pretty well understood, seemingly to the point where it can model or predict any desired experimental outcome. We get quantum physics. We don't like it, but we get it.
There is still some debate about how superconductivity really works, though, and how it's connected with other branches of the physical sciences. When/if we come to understand everything there is to understand about superconductivity, I'd venture to say "better gadgets" will be the least of the benefits.
There's this big tube in Switzerland and France that potentially could be worth looking into
So it's completely true there haven't been any equivalent game-changer advances since then.
GR and QM were philosophical revolutions which changed the kinds of thoughts that were possible in physics and engineering.
So I have a suspicion that the next revolution will be a similar game-changer. I.e. no grand unification of this with that, no more fundamental model of particles and fields, and no gentle expansion of GR and QM; more likely a new world view that completely tips over the table of existing descriptions, and makes everyone think very differently about how reality works.
If we changed that to 60 years, I'd make an even stronger statement. It seems that all the great theories and discoveries in all scientific fields happened more than 60 years ago:
Evolution, Big Bang, special relativity, general relativity, quantum mechanics, Godel's incompleteness theorem, continental drift, asteroid death of dinosaurs (K-T extinction event), fission/fusion, germ theory, etc.
Certainly there has been great advances in technology (semiconductors, superconductors, quantum computing, etc.) But the great theories on which those are based happened before most of us here were even born.
You'd think that with scientific knowledge supposedly doubling every 7 years that we'd have many new important discoveries and groundbreaking theories. We don't.