Seven misconceptions in the foundations of physics

[@oberham]

This is such a great video! Your channel should have way more subscribers, you're sharing awesome knowledge for free!

[@gcarcassi]

Thanks! I have no idea what KZbin likes to suggest... sometimes even the same video has periods in which it is suggested and then it stops. I suspect the algorithm is not optimized for this sort of work. All considered, I am still amazed I can have the subscribes I have. :-D

[@whatitmeans]

Agreeing with your presentation, I believe there is a problem in the math currently used in physics that is even more fundamental, since it also apply to classical physics: scientist have restricted models such they only use Lipschitz differential equations, but the problem is that these equations can only have never-ending solutions, leading to non-locallity if the variable is space, or the absence of a finite extinction time is the variable is time. Non-Lipschitz terms allow the existence of points where uniqueness could be broken, points where a decaying solutions could be "stitched" to the trivial zero solution. Also, as example, no non-piecewise classic power series could become after some point exactly zero since it would violate the Identity Theorem. Since every differential equation I saw in engineering stand power series solutions, I felt pretty scamed when I accidentally learned on a paper that they never stop moving, since my experience tell things do stops oscillating in finite time. I know It could look like a nuance, but since math is the microscope we use to analize reallity, it do could lead to missconeptions. As example, the requirement of having a power series with solutions that vanishes at infininity LEADS to the quantization of energy in the classic time-independent Schrödinger equation. Another one, every example I have saw about the absorption/dispersion phenomena born from the Kramers-Kronig relations uses functions like Gaussian or hyperbolic secants functions, which do work for enginnering, but accurately speaking neither of them are causal since are never-ending: I cannot create two pulses in time without having the second one interfering with the first one from the beginning of time. PS: I am not currently working in science, I do it just because I love it, but I am pretty confident I could make a toy model that could solve this issue as starting point, with closed-form solutions and real life model for experimentation. But I am too busy in not starving right now to start writing a paper as a hobby.

[@gloop7384]

I'm a bit confused by section 5 because it seems to me that there does exist a contingent of physicists and mathematicians who work to make physical theories mathematically precise. The theory of rigged hilbert spaces resolves the problem with unbounded operators, for example, and has been known for quite a long time. Of course, not everything has been made rigorous, esp. QFT path integrals and such, but not for lack of trying!

[@gcarcassi]

That's what I thought as well... 😕 It doesn't play out quite like that. If by mathematically precise you mean find some abstract mathematical structure within which you can make calculations... yes, that does happen. But that's not the issue. 😕 The issue is having mathematical definitions that capture (and only capture) physical requirements, so the mathematical objects within the theory map to some kind of experimentally definable object (however idealized). Rigged Hilbert spaces do not resolve infinite expectations... because a rigged Hilbert space is a Hilbert space "plus some other stuff" (rigged means "equipped" like a rigged ship). Therefore you still have your infinite expectation value, and you still have unitary operator that map finite to infinite expectations values in finite time. What rigged Hilbert spaces allow you to do is to use delta-dirac as a sort of basis for an observable (typically the Hamiltonian). Which sort of works, though for that there is a mathematically better way to do it: work in the space of distributions. At that point you have everything and you have a richer geometrical structure. In either case, the problem of infinite expectation is just made worse: now you have "probability distributions" that are not countably additive (e.g. uniform distribution over an infinite continuous range). 😕 BTW, algebraic approaches do not solve the problem either. As for QFT path integrals, I have a colleague (on the math side) that is working on the problem of the fact that the measure on path integrals is mathematically non-definable. Their paper has been in review for... two years? They can't get it published because the mathematicians don't care (it's a "solved problem": you can't do it) and the physicists don't care (it's not a physics problem: it's a math problem). There are a dozen of theories of differentiation/integration: which one's are physically relevant? I have a set of necessary/sufficient operational assumptions under which it makes sense to use real valued quantities... It was once rejected from a physics conference because it was not "mathematically sophisticated"... as if mathematical sophistication is something we should strive for in physics. So, yeah... I thought too that "surely there are people looking at these details"! I don't know what to tell you. I once talked with someone from the foundations of math that had a side interest in the foundations of physics... He was impressed that I knew what the well-ordering of the real is: he never found a physicist that did. Which, for him, was a polite way of saying "I have never met a physicists who knew the first thing of the foundations of math".🤣 Sorry, this came out as a bit of a rant. The fact of the matter is: I still very betrayed by all of this. I thought that by looking at the details closely, I would have found answers, not gigantic holes. Anyway, that's what I see from where I stand.

[@gloop7384]

@@gcarcassi thank you for your detailed answer. I understand better what you mean, though I still disagree somewhat. The rigging of a Hilbert space consists of defining out a smaller space of states on which all physical operators *are* well-defined. That this arises as a subset (the topology is different) of a larger space does not seem to me a reasonable objection as long as there is a way to define the subset coherently. You cannot possibly disallow the formation of subsets in physical theories, otherwise you could not even define what a function is! (A similar (dual) situation which might strike one as even stranger at first blush is quotient spaces: The physical state space of the electromagnetic field is the space of potentials modulo gauge transformations) As for QFT, I can absolutely guarantee you that there is a tremendous amount of interest among mathematicians; And how could there not be? QFT has given us an almost unreasonable amount of mathematical marvels that have since been verified rigorously. Mirror symmetry, monstrous moonshine, Gromov-Witten invariants, Seiberg-Witten invariants, and the list goes on. Quite literally no major area of mathematics has remained untouched. Of course mathematicians want to get their hands on a rigorous path integral! For the simpler QM path integral, there are several ways to make it rigorous for physically relevant Lagrangians, and to my understanding the same is basically true for many quantum field theories. There is a great deal of work that has been done and continues to be done there! For some discussion see for instance: mathoverflow.net/questions/19495/mathematics-of-path-integral-state-of-the-art, and mathoverflow.net/questions/260854/a-roadmap-to-hairers-theory-for-taming-infinities for a more recent example of a research program in this direction. As a broader point, however, I don't think theoretical physicists *should* work mathematically more rigorously. Don't get me wrong, I am a big fan of the work above which seeks to establish rigorously the frameworks of theoretical physics and strongly believe in its importance. However, the success of the non-rigorous 'frontier', so to say, speaks for itself. I really doubt any of the (now in most cases mathematically rigorous!) wonders I listed above could have been discovered anytime soon if such a constraint had been in force.

[@philipoakley5498]

~29:27 - The matched pair arguments are a retrospective view. We already know the generics of the answer. The unit's choice is simply making sure the arithmetic works. I see similar cases in the generation of requirements in systems engineering where the example is almost always the car/automobile, because we know the desired answer. We never chose to define the requirements for the horseless carriage pulled by faster horses. It's an easy trap. In many ways there is the conjugate pair of 'must be comprehensible' by the average person/scientist (Occam's Razor, and Hume's Guillotine) to the generation of the laws and choice of measurement fundamentals. Lately we've decided the universe is fixed and we can extract constants from it (SI), but many see them merely as stepping stones to greater discoveries in their chosen area of study ;-)

[@entcraft44]

There is order in the universe, which means there must be some kind of laws. If those laws change, then there must be some law that governs the change of the other laws, otherwise the laws would change arbitrarily destroying all order. I can see no sensible way in which major parts of the universe are not fixed. This does of course not imply that all parts we believe are fixed are truly fixed. Do we have the right constants? There are precision measurements going on to determine limits on the variation of "constants". An example is the fine-structure constant, which has been tested for variations both in the lab (small timescales, few assumptions, very high accuracy) and in cosmology (huge timescales, more assumptions, high accuracy).

[@philipoakley5498]

@@entcraft44 There is no guarantee of that "order in the universe" (a priori) that we are already aware of. As we look at more 'stuff' (science-y) we see that our old "order" isn't sufficient and lots of folks come up with speculative idea, most of which end up on the scrap heap. We also create new 'maths' to represent the ideas. The idea that there is an, as yet unknown, order that we could know is one one of the beliefs of science and engineering. Who knows how well it will pan out. [in terms of the current "order"] What are the units of "Intelligence"? (see AI, see science..?)

[@entcraft44]

@@philipoakley5498 I am talking in way more general terms here. If there were no laws whatsoever, no order whatsoever, then we wouldn't exist. There would be no structure, no periodic table of elements, no crystals, no rocks, no planets. Motion would be erratic and could not be approximated with Newton's axioms nor any other laws. I am not even sure the concept of motion would make sense.

[@lnm3221]

@@entcraft44 universe is predictable and this predictability is the laws of nature. This is a very deep point.

[@topos100]

This is why I left the fundations of physics while at Kansas State University....I just knew that sitting at a table doing Topos Theory and Cohomologies had NOTHING to do with the Universe...Finally...After reading ..thinking and studying Richard McEachern's writings aboit initial constants...I saw this as bunk....Its really freeing and at the same time depressing...I wasted sooooo much time. Its like looking at many photos about the universe...and the auther tells me that these photos are artistic renditions...WAF let down...

[@gcarcassi]

@topos100 Thanks for sharing! I don't want to impose, but, if you don't mind, would you feel up to reaching out to me (my e-mail is easy enough to find) and having a chat? I'd be interested to know more about your experience! (This is where it would be nice to have private messages on KZbin!)

[@areyoushitting5]

Waiting for a long time for these kind of topics ！👍👍👍 Thanks ……❤

[@gcarcassi]

Thanks for the appreciation!

[@lnm3221]

No general algorithm can determine whether a program will terminate or not. While technicaly true this is misleading. Contrary to what most phylosophers seem to believe no program is in some kind of quantum superposition of halting or not. You run a program, when it halts you've proven it halts. What you can't prove in general is whether a program doesn't halt. Halting problem's impact on metaphysics is marginal at best because they all either halt or not. Nature doesn't seem concerned with how difficult it is to compute something. You're spot on with the axiom of choice. Aoc is like Euclid's fifth axiom.

[@astonishinghypothesis]

Hm. If theories exist *outside* of metaphysical reality (i.e., what really exists), how can they exist at all?

[@gcarcassi]

Ah! I meant them to put them apart as they are not the possible subject of science... I didn't mean to make any ontological statement on the theories themselves!!! 🤣🤣🤣It didn't occur to me that it could be interpreted that way. Though I guess I do not even want to address in what sense mathematics "exists": that's another can of worms! 🤣

[@mikoajmetelski18]

Hi Gabriele, I subscribed some time ago seeing that there is some interesting content going on far away from crackpottery! I found one of your statements in one of the previous videos very insightful: "Study of foundations of physics is not the study of nature, but the study of our description of it". I also see that this content is targeted at non-experts to which I belong, so take my comments as originating from ignorance. I have a few critical comments on the contents of the presentation. 1) At 19:15 you present a diagram in which physical theories are distinct from the "physical reality", or even the "metaphysical reality". I disagree with that classification on a similar basis on which David Deutsch (I think) based his pursuits: "computation is a physical process, and limits of computation are set by physical laws alone, not mathematics or computer science". In the same manner I would be ultra-realist about physics, and say that it takes physical work (a concept from within the blue circle) to produce physical theories (also a concept from blue circle) but the blue circle as a whole should sit within the green circle. "Doing physics is a physical process, and limits of physics are set by physical laws alone, not mathematics or computer science". I think it would be fantastic have the tools to to dig at the fundamental (entropic) cost of physics or mathematics. At the same time I agree that your viewpoint is "fruitful", and my comment is a pie-in-the-sky idea. 2) I generally agree with the idea that we can arrive at scientific conlusions by reason alone, but I think it is fundamentally because what we call reason is not "pure" but tainted by our memories, which could be viewed as records of a previous measurement. That is to say, a "reasoning" is an idealised process (in your blue circle :) ) that we only perform to a level of approximation, and such idealised process cannot conclude scientific truths because it lacks the truth-values of some of the premises. In every place of Galileo's stone argument, whenever you placed "suppose that", you introduce an indeterminate. At the end of the day after the argument is valid, but not necessarily sound, and only after inserting thuth values to all the "suppose" statement can the argument follow. Which is to say that an experiment had to be performed either way. 3) I think it is a bit self-contradictory to say "there is many ways to do mathematics" and then list the limit on mathematics based on provability of the statements. After all, if you claim there are other ways, they may not be based on proofs at all (albeit it would be hard to call that mathematics). I similarily noticed that the way that physicists do mathematics is very different from that of the mathematicians, but perhaps this differences should be enshrined into a disjoint system and it is to the physicists to think hard about that system, and not for the mathematicians to fix the details. For example: when I studied, I was always bothered my professors never wrote what set some things belong to. Is it a number? Is it an operator? Is it a function? The success and applicability of Mathematica to theoretical physics and idea of symbolic manipulation detached from set-theoretic aspects in a is a practial reality. A full lecture of theoretical physics goes by and set inclusion symbol is never written down. I imagine that a project on foundations of physics would be interested in exploring why is that and what that really says about applicability of mathematics as done by mathematicians to physics. 4) I have a minor comment on verifiable statements. You said that e.g. mass of fundamental particles can never be known exactly. I would just point out that sometimes that exactness is possible - for example, when specific parameter regime is not adiabatically connected to other regimes. In the photon case I will allow myself to make a little false story: if the mass of the photon was not zero, we could be in a different phase of the universe, which is not adiabatically connected to the zero mass case, and the two cases can be distinguished by a measurement of the order parameter which is of the "yes/no" type rather than "how much" type. Of course in the yes/no case, hypothesis testing applies so this would hold only to a certain confidence level, but I this case feels (without evidence) fundamentally different from the determination of a real constant. 5) At 32:40 you speak of unbounded operators having undefined expectation values. I agree that problems at this are great hints and must eventy be dealt with. But also they are a by-product of the model (the blue region). You can always add extra statements to the model that take them out, by just postulate: "unbounded operators have the following expectation values (blabla)". It is Occam's razor that produces this stuff. On the other side - if physics is ok with gauge theories as long as the extra degrees of freedom are removed because then they become experimentally inaccessible, why not be okay with experimentally inaccessible quantities like expectations over all space that do not get removed?

[@gcarcassi]

@mikoajmetelski18 Thanks for in depth comment! I hope the reply make sense: it's easy to talk past each other... 1) I think you are talking about something different. IIUC, you are pointing out that the activity of creating physical models is also a physical process (which you could model and then the model the modelling of the model, etc..). Your question is more along the line of epistemology, as you are asking how it is what we can make these models. I am simply saying that the physical model itself (i.e. the ideal pendulum, the ideal gas, ...) are not systems that literally exist. Note that this, to my understanding, was one of the big steps made by Galileo (he would imagine pendulum and falling bodies in a vacuum). 2) I thought that goes without saying. I can reason about "whether the heavier body falls faster than the lighter one" because I have an intuitive sense of what "heaver", "body", "falls", "faster" and "ligther" mean. So, agreed. 3) "it would be hard to call that mathematics", right. The "choice" in mathematics are the starting axioms, not whether you have proofs or not. As for physicists "not using mathematics well", that's something we are actively working on... and it turns out that when you do you are forced to make the physics clear. If you are interested more in the details, I have put an overview of our project on the channel, the second part "Physical Mathematics" shows some details of the approach. 4) The mass of the photons is measured to be 10^-18 eV. en.wikipedia.org/wiki/Photon has a whole section on the measurements. You'll see that they typically measure it from some symmetry we expect to exist... but then the symmetry is only verified up to some level of precision. Even the type of argument you give, will only put a bound. The physics community did actually make a blunder along those lines: if you looked at old books or an old particle databook, the mass of the neutrino was zero (no error bound)... turns out they were wrong. 5) Two reasons: first, because you don't know what is physical and what is not anymore (this is the case in quantum foundations). Second, because "extra things" in math are not always good. For example, the sets to which we assign "length" are the "Borel" sets (which correspond to the one connected with an experimental test). The subsets of the real number is a much bigger set. You could argue: why don't we assign a "length" to all subsets? Turns out: mathematically you can't. Note that the path integral formulation of quantum field theory has a similar problem: the measure (in terms of math) that you'd need to write the integral can be shown not to exists. That problem should be fixed, though it is not being fixed... and effectively it's almost impossible to work on it (i.e. you don't get published). In software terms: it's code smell. You are hiding a problem.

[@mikoajmetelski18]

​ @gcarcassi Thank you for your response back! Sorry I did not phrase my comments as questions but I see you replied to them as such, talking past each other is a real possibility in those topics so focusing on the (implied) questions is the way to go. Regarding 1) yes that is precisely what I had in mind. I just thought it is a cool idea worth sharing. But this also ties in to misconception (4) as follows. For example (making stuff up, just to give an example): "if you produce a physical model, you should be able to compute all observables in a finite time". Otherwise those observables could not have been computed by the universe itself. This is a statement that limits the scope of possible physical teories in the sense you are talking about on the slide at 42:32. It is also a reasoning in the sense of misconception (4). How would one go about experimentally verifying a statement like that? Regarding 2) If it goes without saying that the reasoning must contain elements of experiments, then the 4th misconception in your video ("The laws of physics are found experimentally") is not a misconception, but reality? Regarding 3) - but surely the rules of inference are (tacitly) included in the axioms? This means that 1) what constitutes a proof is flexible since proof as a metamathematical object itself is axiomatically defined 2) mathematics with no proofs (or rather without logic) is still mathematics, even if not much can be said about statements in that axiomatic system because of lacking rules of inference. Replying to your response, I would be very hesitant with saying that physicist use of mathematics is "not well" and refer to the original comment: after all, perhaps symbolic manipulation can be formalized without recourse to set theory that underlines mathematics. When physicists write "let x be position" on the blackboard, perhaps there is no answer to the question "what set does x belong to?" because these words are not part of the universe of discourse. For the other points thanks for your response! Looking forward to seeing more about the foundations of physics project. But also some (very) concrete examples of how to apply the results of your work to physics practice (I'm a condensed matter physics student) are very welcome :)

[@Kram1032]

If you want to be strictly computable you also have to reject the Law of the Excluded Middle. Which can lead to very fun things such as a world where many shortcuts physicists tend to take in re: differentiability are just directly valid. (Note: Rejecting LEM doesn't mean LEM *never* holds, it only means it doesn't *always* hold and you have to prove/justify individually if it holds for whatever usecase. Rejecting LEM also means not every vector space has a basis, but I think that's fine. For the ones we care about we can still construct a basis.)

[@gcarcassi]

There is "something like that" from the non-termination of experimental tests, which is encoded into the lattice of open sets is a Heyting algebra. But it's definitely tamer...

[@entcraft44]

Wonderful talk, and I will have to look into the project when I have more time. The approach you introduce at the end sounds very interesting. That said I have some parts I disagree with, especially in section two. There is of course historical bias to give explanations "in terms of mechanisms" as referring to the interactions of constituents. This is also the reductionist view on physics. However the impression I get on quantum interpretations is that they don't seem very reductionist. I have not seen interpretations on the "measurement problem" that try to solve this with smaller parts. The explanations seem way more focused on knowledge about the system. But I don't work directly in this field, in fact I am still a student, so maybe my view is not accurate? Furthermore: Schrödinger wrote in 1935 that a measurement on a quantum system does not actually determine a property that the system had before the measurement. He then states that the only sensible definition must be along the lines of "an interaction between a system under test and a measurement device, itself a quantum system, which produces the same visible result when repeated multiple times" (paraphrased). That a measurement is a projection by definition is thus nothing new (provided I understand your point correctly). But as I understand it you do not explain at all the thing that most try to explain: Why is it sensible to pick the categories "unitary process" and "measurement process" specifically? Why can we describe the behavior of an isolated system as deterministic and reversible, even if it has internal interactions? While external interactions with a measurement device change the behavior?

[@gcarcassi]

>I have some parts I disagree with Good! It wouldn't be interesting otherwise! 😊 There are indeed very different approaches to the measurement problem... and the mechanism in the explanation is not always "smaller elements". It can be "world branching", "hidden variables", "information update", ... But you are still relying on other objects that are seen as more primitive. So you still have the problem of justifying those elements. >Why can we describe the behavior of an isolated system as deterministic and reversible, even if it has internal interactions? These are things we are working on... and the answer that these approximations are somewhat key to how we must do science. The idea is that, for example, we can define a physical object only insofar we are able to manipulate it independently from other objects. A lot of basic physics, in fact, is to find such objects that "decouple well" from the rest. We do not have yet an answer that we can encode precisely into math. But, yeah, these are types of questions we definitely interested in.

[@danielvarga_p]

THANK YOU VERY MUCH! This is AMAZING job, and explanation. I try to use similar principles!

[@davidhand9721]

Saying that you don't need to understand the measurement problem to understand quantum mechanics is just redefining "understand". This is no more convincing than any other of the thousand times I've heard "shut up and calculate". This isn't a misconception, it's a difference of semantics. When I hear "there's a contradiction in our most basic physical theories", that absolutely meets my definition of something we don't understand. I can understand why, for the purposes of your career and the progress of technology, you would stop worrying about this at some point, but be honest about it. You don't understand it. That's fine. No need to play word games.

[@gcarcassi]

If you want to criticize, that's fine, but maybe first understand the point one is making? 😉 The only point that is made is that a solution to the measurement problem based on an underlying mechanism is never going to work for all cases because QM is applied at many different level of description. So, it still needs to be understood, but in a different way. While you may disagree on the point, it is the opposite of "shut up and calculate". It's figuring out where one should look. 😁 But you are free to do whatever you want! If the approach is not useful to what you want to do, ignore and do whatever you want! 😁

[@davidhand9721]

@@gcarcassi Yes, I get the route you took to this point of view. It's a valid point that could potentially narrow the field of solutions to the measurement problem; I just don't think it solves the problem or makes it less important. Maybe I am misinterpreting the intent if you are not telling anyone to shut up and calculate. It _is_ an oddity that QM applies just as well to composites like buckminsterfullerene molecules as it does to fundamental particles like electrons. Now that I'm thinking about it, that makes objective collapse theories a little awkward. If every particle has, for example, some probability to collapse the whole wavefunction at any moment, then why wouldn't, e.g. a proton be 3 or more times as likely to collapse as an electron? It has to be a feature of the scale itself, not a feature of fundamental particles. That's an interesting idea.

[@gcarcassi]

@@davidhand9721 >a proton be 3 or more times as likely to collapse as an electron? Exactly!!! 😁A good solution to the measurement problem needs an answer to these sort of things. Thanks for the attention. 😊

[@hyperduality2838]

Union is dual to intersection. Conjunction (and) is dual to disjunction (or) -- Boolean algebra. "Always two there are" -- Yoda.

[@glcpit7797]

try to do experiments without thinking about it before ... you will discover that the idea that laws of physics are found no by "reasoning" is a big bullshit or you have expressed the concept very very bad.