Why is Maxwell's theory so hard to understand? (2007) [pdf]
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I think that's the modern fundamental difficulty in Maxwell's reworked equations - the 4 we all know and love, not the 20 or so he originally published. To even begin to get a true intuition for them, you have to get really really good at visualizing idealized objects with flows running through surfaces, and (if you're lucky) symmetries that cancel each other out. You can't be afraid of imagining the infinitely small and the continuous to really get the most out of it, even if you "know" on some deeper level that the continuity of spacetime is a convenient approximation.
14 years later I am still grappling with the beauty of saying "yeah yeah, this area of interest is technically discrete, but let's pretend it's continuous and see what kinds of stuff falls out." If you have examples of things like this in other areas like mathematical finance, I'd love to hear about them.
I disagree, and there's no evidence for this. This is computer science leaking out; physics has no formulation of spacetime in discrete terms, and indeed, all of physics presumes continuity.
In QM, the space of wavefns is infinite-dim continuous, and if wasnt, QM wouldnt be linear.
Cognition is discrete, but the world is continuous.
You're definitely correct about the math, i.e. the systems that we humans have invented to model reality. But I guess most of us don't really care about what mathematical model scientists like to use (especially not whether they're "exotic" or not), but rather what reality could be like.
And the quantum properties of QM do seem to suggest that there's some sort of fundamental discreteness in reality. And it seems to run contrary to the resolute claims that reality must be continuous as if it were a proven fact. What I understand is that the math most commonly used by scientists is definitely continuous, but whatever we can measure seems to have some kind of planck limitation.
So are we talking about empirical science or science-flavored theology here? Have we actually found empirical evidence or proven the continuousness of space/time?
Your penultimate paragraph suggests some confusion about ideas like Planck scale and quantisation.
Firstly, there is nothing special about the Planck length itself. It's just a unit of length. Around that sort of scale, though, our current theories of physics happen to break down because both quantum and gravitational effects become significant. That doesn't imply spacetime is discrete (or preclude it being discrete) at that scale. It's just a realm that our current theories don't work in.
Secondly, while describing aspects of nature that are quantised was a large part of why quantum mechanics was developed (and the source of its name), it in no sense says anything like "there's some sort of fundamental discreteness in reality". Quantum mechanics deals with both discrete and continuous observables in a single framework: functional analysis, essentially. The set of possible values for an observable is modelled as the spectrum of an operator, which can be either continuous or discrete. Which sort of observable is appropriate for a given physical theory is a choice made in constructing that theory. For things like charge and spin we use discrete (quantised) values because we have evidence that those things are quantised. For things like position we use continuous values and have no evidence that using discrete observables would better match nature.
Space could in reality be either discrete or continuous, or not even exist in any form we'd recognise as "space" on those scales. Quantum mechanics doesn't give us any hints one way or another.
Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic.
So there is no easy way to prove or disprove continuity of space.
However, what's missed here is that discrete is a necessary but not sufficient condition.
Once you give any sort of plausible account of how reality could be discrete, as you've done here, you end up with non-computable aspects (eg., typically randomness). So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable).
Though the meta-meta-game around "simulation" is probably internally incoherent in itself -- whether reality is a computer or not would really have nothing to do with whether any properties had by it (eg., mass) are simulated.
Since either you take reality to have this property and hence "simulation" doesn't make sense, or you take it to be faked. If it's faked, being computable or not is irrelevant. There's an infinite number of conceivable ways that, globally, all properties could be faked (eg., by a demon that is dreaming).
The "Nondeterministic" in NFA means its transition function goes from states to sets of states, instead of from states to states. Informally, it can explore multiple paths in parallel for the cost of one. They're not probabilistic.
Thus, this semantics implicitly encodes the notion that the machine is nondeterministically choosing the next state in each execution.
The decision problem of whether an NFA accepts a string w is what allows for the informal parallel interpretation, that it accepts iff you imagine the computation is forking off a new thread at each nondeterministic branch. But to say that this not nondeterministic or not probabilistic is like saying the Many-Worlds Interpretation means there is no real superposition, or something like that. It's like saying a throw of a dice does not really involve probability because of a symmetry argument that a dice has six equal sides. Mainly, I don't understand that, because I see probability as a way to implement nondeterminism: a system is probabilistic only because it is making nondeterministic choices according to some probability distribution. And checking Sipser 2nd ed. p.368: "A probabilistic Turing machine is a type of nondeterministic Turing machine in which each step is a coin-flip step".
Anyhow, my main issue was that the original commenter casually claimed that probability makes things (physics) uncomputable. But Turing computability has nothing to do with probability, since as I recall the closest concept is the Non-deterministic Turing Machine (NDTM) and with that it is a basic proof to show that NFAs vs. DFAs, as well as NDTMs vs. DTMs, are computationally equivalent and there are theorems for that.
Meaning either they are using an idiosyncratic definition of computability or are ignorant of an introductory course on theory of computing which explains formally what Turing/Church's theories were about when clarifying the concept of computability. Okay or maybe they have a deeper philosophical disagreement with computability and complexity theorists - maybe they reject Sipser's definition above - but these are standard undergraduate curricula in CS by now and it could be argued that perhaps it is the non-CS experts who haven't thought deeply enough about what computability really is and would benefit from actually learning from these subdisciplines. I don't know, as they did not reply.
Also continuous doesn't mean uncomputable either, because in many cases the infinite amount of computation for continuum does not add anything interesting and finite approximation works good enough.
> So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable).
I don't see any evidence for this. For now we do not have a proof for one way or another. If for instance it turns out that quantum computers really can run Shor's algorithm factoring very large numbers, it would be a good evidence for continuum, but we are not there yet.
But even that would not be an evidence for reality not being a computer, since it will still allow the possibility of reality being a computer that can perform operations on real numbers.
This characteristic is observable for metals as well. Steel becomes less flexible as it's worked because it's grains become smaller and more chaotic - A microscopic property with a macroscopic effect.
Everything we can see move on the grid is at least 20 orders of magnitude bigger than the grid spacing. Any macroscopic objects we can experiment with are more like 30+ orders of magnitude bigger than the grid spacing and consist of numerous atoms that will all be moving within the object due to thermal jiggling over distances orders of magnitude bigger than the grid spacing.
What matter is a subjective topic. What we all have in common is logistics constraints. So if some people set as a goal something that requires to settle if reality is more easily handled when modeled in continuous or discrete manner for logistical reasons, then it this scope it matters. But whatever you settle on, human brain is thus built that it can always assume that the perfectly fitting model is only valid in its scope which is built on top of an other more subtle level of reality which is on it’s part better modelized with an antithetic approach.
Now, on a very personal out of blue opinion, I fail to see how any causal series might happen without an underlying continuous flow of event. I mean, supposing causal discontinuity is to my mind as relevant as supposing that universe as it is right now, actually just appeared, without anything we can think about it being relevant, and in the next instant could be completely different or nonexistent since universe is not bound in any remote way to what we might expect on our delusional just created sense of causality.
You say you can't comprehend how something can move from 1 to 2 discretely. But the paradoxical notion of infinite continuous change has been known since antiquity. It's faith either way.
Discrete doesn't mean state changes are wholly globally arbitrary. Imagine a graph with nodes and edges, a state machine as computer sciences call it. I think it's easy to agree that the universe could be parsed by a regex ;-) Heck, imagine an integer on the number line that can go up or down.
Worlfram has written a ton about this. Despite all his issues, his math is solid. (Which is not to say his physics is true.)
Zeno didn't believe that the latter was possible. But he wasn't stupid, he obviously knew that motion was happening all the time in real life. His paradox really only makes sense in the context of Eleatic philosophy which assumes that reality is an illusion because change is fundamentally impossible (how can something come from nothing?).
If you want to reframe it in more modern terms, Zeno's paradox shows a contradiction in axioms. If you want to get rid of the contradiction, you have to change some of the axioms.
In real analysis, loosely speaking, we remove the axiom that an infinite process cannot result in a finite outcome - this way we are allowed to sum (some) infinite series, for example. But we don't "know" if reality behaves that way.
The atomists found a different solution: they argued that reality was fundamentally discrete. This way, Zeno's paradox also doesn't arise.
my mistake - that should have read "are both continuous".
I didn’t mean that, sorry if my words that induced you to believe so.
What I want to point out is that, to my mind, if I assume a discrete foundation of universe, on meta-cognitive level I must recognize it implies everything I experiment through my current attention might possibly be a just made up state without any compelling ontological relation to anything I can recall or think of. So, as far as I’m concerned, believing in continuity is just a lazy way to relax on a metaphysical Gordian knot.
> It's faith either way.
Yes and no. It’s probably easier to change scientific perspective to whatever model apply best for some purpose than to adhere to some philosophy about Nature.
I kind of suspect "is the universe continuous versus discrete" will come down to that. I don't know what a hybrid of such things looks like. With our current conceptions it seems impossible. But it always does, before the breakthrough comes and then in hindsight all the people of the future will get to look back at us going "How could they not see this obvious thing?", to which my only defense is that you, dear future reader, only think it's obvious because it was handed to you on a silver platter and you'd be as confused as we are if you were back here with us.
These are two very different questions. As for the former, I don't believe there is consensus at this point with good arguments for and against. As for the latter, if we can reliably prove that the reality is not analog but digital, it has consequences at various levels, and we might make better choices when using math to describe it/make approximations.
And 93 years since the first Solvay Conference. [2]
[1] https://en.wikipedia.org/wiki/History_of_quantum_mechanics [2] https://en.wikipedia.org/wiki/Solvay_Conference
Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824
"Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers.
It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence record an infinite amount of information. No one claims this, and the formulation of physics (entirely on real numbers) does not require it.
Rather to say, eg., space is continuous, is to say its unbroken. There is no physical quantity which is becoming infinite.
For example in E = 1/2mv^2, a particle has kinetic energy in virtue of being matter in motion -- it is motion and matter which are basic. Energy is just a system of accounting which tracks motion in the aggregate over time (with kinetic/potential just being the future/past in the accounting) hence why energy conservation is just a temporal invarience.
When making arguments about the physical properties reality has (eg., whether aspects are continuous) you need to be exceptionally clear what your terms mean, and terminology in physics isnt designed for this.
There are no "information saturated volumes", this is a series of abstractions piled on top of each other.
All the words in this area have quite complex formal definitions that are have quite difficult to unpack semantics, you cannot just go around saying "saturated volumes" -- it is this sort of language which breeds cranks, and pop sci does it with abandon.
This entire discussion is a matter of several PhDs, and to be done only well by people with PhDs in the matter (philosophy of physics), or equivalent research. It's not possible to scrap fragaments of what compusci bloggers say and derive much that's likely to be actually correct.
He's also written papers that are basically philosophy of physics. It would be interesting to go over what he has actually said on this topic.
Various Hawking-Bekenstein results about black holes relate to information density, especially, shockingly, that information is proportional to surface area, not volume. This makes perfect sense because a black hole has all its incoming matter and energy sprawled, flattened and red-shifted on its horizon (to a distant observer). It can never export its internal state to the outside world, so you might never expect a volume's-worth of states to be exposed.
The idea was generalized by 't Hooft to the Holographic Principle, for 2D screens encoding the state of 3D volumes on the other side.
However, the full AdS/CFT Correspondence only applies to a certain type of AdS space, not our actual dS space. At the moment, it seems half of theoretical physics doctoral students are trying to extend AdS/CFT to dS space (obviously - strings :) and half of observational astrophysics doctoral students are desperately hoping to show we live in AdS space - LOL
At first I was inclined to agree with you that this is an appeal to authority, with the caveat that such appeals do not always constitute a fallacy. For example, if we both agreed that such a Gisin is an expert whose opinion on this topic can be trusted, then his statements are valid evidence for one way or another.
But then I realised that the very claim being challenged is whether Gisin knows what he's talking about. Floating his credentials and experience feels like a valid contribution. For what it's worth, back in my PhD days I read several of his papers and saw a couple of his talks at conferences, and can confirm he's one of the leading researchers in the field and is particularly thoughtful and careful in his work.
There's a problem for people who think reality can be modelled by computable functions of finite inputs: this makes classical physics non-deterministic, because chaos requires infinite precision for determinism.
So either you go for "reality is deterministic and continuous, and not computable" or "reality is non-deterministic, and discrete, and not computable"
either option in this fork includes properties that offend the minds of the people who want everything to be discrete.
I lean towards a preference for determinism & continuity (via, in QM, superdeterminism) since that's trivial to justify on our best physics
> I argue that there is another theory, similar but different from classical mechanics, with precisely the same set of predictions, though this alternative theory is indeterministic
and in the footnote he describes indeterminism to mean:
> given the present and the laws of nature, there is more than one possible future
Out of curiosity, why do you lean towards superdeterminism and not other deterministic interpretations of QM such as Many-Worlds or Bohmian mechanics?
The correct setting is a rigged Hilbert space: given an algebra of operators A on a Hilbert space H, let S be the maximal subspace of H such that |sa| is finite for any s in S, a in A. These are your states. Operators in A don't necessarily have eigenvectors in H, but they do have eigenvectors in the space S* of all continuous linear functionals on S. So <x|, for instance, is just the map `psi -> delta_x(psi)`.
This has the advantage of not having any funky "rigged" states suddenly appearing in your calculations and is also exactly how we deal with non-projective measurements in finite dimensional quantum mechanics.
See here, for example
The spectral theorem, rather than decomposing X in terms of a sum of eigenvectors & eigenvalues instead decomposes it as an integral over the spectrum with respect to the (spectral) projection-valued measure.
Now it is fair to question whether this "observable" is really observable, but it certainly works out mathematically consistently in the normal way we do things in quantum mechanics.
There are many interesting ways to probe this problem.... here's one:
Say I tell you to imagine a circle, an ideal Platonic circle in a Cartesian coordinate system (real coordinates, first uneasiness). Let's ignore translation, so it is centered at (0,0). I tell you the radius. Can you imagine the circle with Plato? Model the circle? Reproduce the circle? Do you need pi? Does the circle include or encode pi? But pi is has infinite information.
Perhaps all you need is the square root function? But that's also an infinite Taylor series expansion. You can plot and recreate the circle to any precision if you have a square root function. The series will only need to run to the required precision. The circle will always be granular, depending on the number of terms you use in pi, or the square root function. Yeah, right, obvious, so why is that a problem?
What if I tell you the circle is the physical manifestation of equipotentials of a stationary charge (say, nucleus), or mass (say Earth), with inverse square law - so basically a geometric fall-off with range determined by spatial (circular, spherical) considerations. What is the force at some distant point? Do you need pi? Do you need square root function? Or reciprocals? How does the other charge or mass feel the 56,323rd decimal place of the force due to the potential?
Maybe it doesn't, because by the time it has felt the second decimal place, time has moved on, the charges/masses have moved on, and the nuance of what would've/should've been felt in a never changing universe are never experienced. There is a modified differential equation that relates various time derivatives to precision of experienced forces (this almost sounds like relativity :)
The discrete explanation with photons goes like this: the force is produced by radiating photons. They automatically encode the geometric expansion as inverse square law, because of their pathways, no need for pi, or sqrt functions. But that is statistical, the accuracy is only as good as the number of photons that can arrive from the source. The circular/spherical nature of the force only emerges over time, as photons arrive and act. The accuracy of smooth circularity and inverse square only establishes itself over time...
Elapsed time affects experienced precision - hmmm, interesting.
How would you quantify such a thing, where time changes the precision of what you feel? Well, the other obvious example is the Heisenberg Uncertainty Principle. This is just a simple and obvious example of Fourier Analysis for any theory based on a linear wave equation. It almost doesn't need stating, and if it must have a name, it is certainly Fourier, not Heisenberg. Anyway, any math/physics/engineering student knows Fourier to their core, and it gives a nice solution to the information problem: coordinates may be real-valued degrees of freedom, but there is no way to mathematically or physically resolve all coordinates and their derivatives to infinite precision. It's just not possible, even if the underlying equations/reality maintain the fiction of real-valuedness.
Fourier combines time, waves, amplitude, velocity (momentum, etc.) with a specific expression for possible information. A picture is worth a thousand words at this point, just look at a wave-packet, it's obvious. Fourier is a masterwork, and vastly underappreciated as a fundamental limit on knowledge, in a real world sitting on smooth continuous waves.
So Fourier sets limits on knowledge, even in the wavy world of the smooth continuum. Of course, I do not believe in the smooth continuum anyway, but Fourier is my wingman to fight the real-infinitists on their own smooth turf.
It does not, according to any sane way of defining its information content. For example the Kolomogrov complexity of pi is clearly finite - I can write down a program for a Turing machine which will run (forever) and keep writing down digits of pi as it does so.
That's also a different point than the parent's. Seems they're saying if you were to specify pi as the limit of some expansion that describes the physical process of photons arriving in some area, then that specification's information increases with more terms added. Pi, being almost random by every statistical measure, has as much information as a random string, in fact, in any normal conception of information. You cannot wave that away by machine manipulation tricks or by defining a new constant, and this is borne out also by the parent's physical argument that in reality there are no low-complexity universal constants, but that there may be limits to information density (in space and time).
Continuous physics can be a manipulation of limiting quantities without being literal.
I wrote "Kolmogorov complexity" not "original Kolmogorov complexity" so this isn't particularly relevant. The application of the concept to the infinite string which represents pi is essentially trivial.
> Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program.
I honestly can't parse this.
> Pi, being almost random by every statistical measure, has as much information as a random string
This is wrong. You can consider something like a simple communication task. Alice and Bob share a phone line and she is attempting to tell him a number. Every second the line allows her to send a bit to Bob. For a truly random number she has to use the line infinitely many times to tell him the number. To send pi she can send a finite number of bits which amount to a program to compute pi and he can do the computation on his end.
You're assuming Alice and Bob have already pre-synchronized what kind of computing machine is going to be used, one in which pi is the output of a relatively short program, as opposed to another type of machine where some other random-looking number has that property (random to you, pseudo-randomly generated via some machinery for all you know). You are assuming many things away.
Also it is absolutely not trivial to extend Kolmogorov complexity to infinite strings. There are multiple formulations and they are a lot more difficult than for finite strings. Not the computation part but the complexity assignment part.
Theres a bunch of fine detail in getting it down to defining an actual number measuring complexity which I don't care about at all, all I care about (in the context of this discussion) is that the number is finite.
Yeah. As they said, it’s computer science leaking out.
It can be misleading to reason about entropy, which is the relevant physical concept, as if it were strictly equivalent to information as computer scientists understand it. Entropy works perfectly fine with continuous densities of states and real numbers.
That being said, Gisin's approach is still interesting and his results can still be valid. But he starts with the assumption that real (irrational) numbers are unphysical, i.e. that – in a sense – our observable from above can actually only take on certain (rational) values, and then he derives certain predictions from that.
¹) Putting "information" in quotation marks here because no one really knows what it is.
There are some attempts of working with discrete spacetime (e.g. causal set theory), but yeah, all our best descriptions so far very much assume smooth spacetime.
There's little evidence even of this, except in the trivial sense that language (minus prosody) is composed of discrete units.
What offends the minds of some people is the world might not be like their mind at all. They want always to analogise everything to Reason.
Everything should be countable, everything should be knowable, etc.
A word doesn't even have a discrete meaning, except locally in relation to other words.
Saying A = B + C looks discrete, just by hiding any potential non-discreteness inside B and C.
For example, a formal proof is a discrete process: it follows step-wise rules that you can assign natural numbers to (this is the first step, this is the second step, this is the third step). A non-discrete process, a continuous one, would have a smooth transition between these steps, which is hard to even imagine.
While I am not convinced it is correct to say that "human reasoning is discrete", human language is definitely discrete. Words don't blend smoothly into each other. If you don't believe me, try to define a function f:[0,1] -> Words, such that f(0) = "red" and f(1) = "blue" and tell me what is f(sqrt(2)/2), or what is df/dx.
There are aspects of cognition that are discrete: a language contains a finite set of phonemes and words, a human mind is capable of (painfully slowly) carrying out purely symbolic algorithms like those a computer performs, etc. My point was that these things are a small subset of cognition, and most of cognition we have no particular reason to think depends on discreteness, which I think is the same point you're making.
Personally I strongly suspect that the "discrete" aspects of cognition are things that have evolved on top of / within a system that is fundamentally continuous (analogue) in nature.
How do you convince yourself that you have thoughts that cannot be accurately written down no matter how many words you use?
> Personally I strongly suspect that the "discrete" aspects of cognition are things that have evolved on top of / within a system that is fundamentally continuous (analogue) in nature.
How do you tell whether things are really fundamentally continuous, or a really high definition pixel art?
https://www.energy.gov/science/doe-explainsquantum-mechanics
That said, our Turing Machine model of computation is discrete, and the Church-Turing thesis implies human thought is Turing Complete.
It's not empirical evidence, but it's something. (I really doubt an empirical test is possible at all, so it seems philosophizing is all we have, unfortunately.) I'm not aware of any (communicable) model of thought that actually can't be reduced to the Turing model (in fact, that AFAIK precisely the reason he proposed the model).
Analog signals can be approximated to arbitrary precision, so while we conventionally think of it as continuous, it doesn't imply our cognition really has infinite precision floats internally...
I think it's really unfair to only focus on half of the picture (saying there's no evidence for "Cognition is discrete") where in fact we actually have no evidence at all whether anything is fundamentally continuous or merely approximated as such with high precision.
Traditionally the math in physics is continuous, and the math in computing is mostly discrete. If people point to Hilbert space as some kind of justification for believing physics is continuous, then it seems equally valid (or invalid) to use the Turing model as justification to believe cognition is discrete. I think both approaches are misguided, but as I said, it's really unfair to point out only the convenient half of these invalid arguments.
Here are some quotes from "Covariant Loop Quantum Gravity", Rovelli and Vidotto (slightly redacted). I suggest the whole chapter 1, in particular 1.2 to get an idea of why fundamentally spacetime may be discrete.
"In general relativity, any form of energy E acts as a gravitational mass and distorts spacetime around itself. The distortion increases when energy is concentrated, to the point that a black hole forms when a mass M is concentrated in a sphere of radius R ∼ GM/c^2, where G is the Newton constant. If we take L arbitrary small, to get a sharper localization, the concentrated energy will grow to the point where R becomes larger than L. But in this case the region of size L that we wanted to mark will be hidden beyond a black hole horizon, and we lose localization. Therefore we can decrease L only up to a minimum value, which clearly is reached when the horizon radius reaches L, that is when R = L. Combining the relations above, [..] we find that it is not possible to localize anything with a precision better than the Planck length (~10^-35 m). Well above this length scale, we can treat spacetime as a smooth space. Below, it makes no sense to talk about distance. What happens at this scale is that the quantum fluctuations of the gravitational field, namely the metric, become wide, and spacetime can no longer be viewed as a smooth manifold: anything smaller than the Planck length is “hidden inside its own mini-black hole”."
"The existence of a minimal length scale gives quantum gravity universal character, analogous to special relativity and quantum mechanics: Special relativity can be seen as the discovery of the existence of a maximal local physical velocity, the speed of light c. Quantum mechanics can be interpreted as the discovery [..] that a compact region of phase space contains only a finite number of distinguishable quantum states, and therefore there is a minimal amount of information in the state of a system. Quantum gravity yields the discovery that there is a minimal length lo at the Planck scale. This leads to a fundamental finiteness and discreteness of the world."
It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous.
There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it.
Whenever people try to do philosophy of physics the inevitable place everyone lands at is a series of false equivocations, often caused by the language of physics being ambiguous and polysemous. But "minimum length" here does not mean a sort of grid length.
Also, for what is worth, in QM the space of wavefunctions can also be finite dimensional (for instance the Hilbert space of a spin 1/2 particle).
You have an object at position p, and the behaviors of the system are discretely different between P and P + h, without an intermediary at P+h/2.
Even with a "minimum length" (in this specific sense), you have an object at position p, and can (move/observe) it at any p+dx continuously.
importantly, the question is whether the best theories of physics in a world with a minimum extension-in-space require continuous mathematics, and there's nothing about this plank length to suggest they wouldnt
By contrast, discretness has various unintuitive mathematical properties that mean it's not easy to fit into some other theories (particularly those relying on differential equations).
The fact that we don't have already a full system using discrete maths doesn't mean it is impossible, because our current system is based on a long tradition of belief in real numbers, and assuming physical space is continuous.
I'd argue (admittedly unhelpfully) that unless we have actually tried to formulate physics using discrete mathematics and found a barrier that we prove unequivocally that it is impossible to overcome, we can't claim that physics must be formulated using real numbers/continuous math. There's a difference between "we don't know how to do this" vs "we know we can't do this".
Planck constant would like to have a word with you. But it is true that CS shines a light on the matter of mapping the infinite into bounded spaces.
This matter of ‘cognition’ is the entire matter (npi) of contention. What is the actual relationship between number and perceived phenomena? What is the deeper meaning of the concordance of mathematics and physics? Where do these magical constants come from and what does it all mean?
It seems we bring the ‘world’ into being by partitioning. See Genesis 1 for details.
> the world is continuous
Reality is actually a unified undivided unity without form and timeless & eternal - that is all we can say with certainty. The “world” is our perception of this reality. Our cognitive machinary is discreet, and a mapping of this reality into metric & temporal spaces of the mind.
Do you want to flesh this out? Are you suggesting that because phase space is quantized, position space must be quantized as well?
(My actual point is that Reality is neither continuous nor discreet - it is an infinitesimal point and it is our mind — that likes to name and number things and relies on duality to make ‘distinctions’ — that creates the universe, the subjective reality that we perceive as inhabiting.)
The unnamable is the eternally real. Naming is the origin of all particular things.Free from desire, you realize the mystery. Caught in desire, you see only the manifestations.
Yet mystery and manifestations arise from the same source. This source is called darkness.
Darkness within darkness. The gateway to all understanding.
There's so many videos on it that you begin to feel like you really understand it after half a dozen or so repeat the same words at you. But the thing you don't realize is that those are literally the only words they could possibly communicate, and that's an infinitesimal fraction of the nuance of the real thing. Plus, there's a selection bias in that the videos that make you feel good get more views, so you're more likely to stumble upon videos that make you _think_ you understand it that videos that actually do, partly because the only videos that could make you understand are a graduate degrees worth of 80-100 hour lecture courses that you're gonna have to take notes on.
It makes me wonder if this is true for literally every subject. I fancy myself are least politically versed in modern events but is my entire understanding based on an entertainment-first version of an actual education? How much do i walk around using jargon that i don't really understand to make points whose true depth I'm completely unaware of?
No one knows what the universe of truly made of.
Reality is measured up to certain error tolerances.
Don't confuse the map (math and physics) with the territory (reality).
Also, Stephen Wolfram would like a word with you.
If you want continuity, then shun collapse, and believe in Many Worlds. You know you should.
Wolfram is another conversation, and one that does not fit in this margin.
If you're from Copenhagen every measurement is a lossy discontinuity that resets the wavefunction.
This is not an abstraction, it's directly observable.
As for discrete formulations:
https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
Also, your link is not a formulation of QM, it is a different theory which makes different predictions (it is a quantum gravity theory). And, per the sounds of the Wikipedia article at least, it is not actually proven equivalent to QM in the regimes where it needs to be ("There is evidence [1] that, at large scales, CDT approximates the familiar 4-dimensional spacetime", or in other words, it is not fully worked out if this is the case).
Rewriting the equations in vector form reduces the number to the modern number.
Moreover, the original equations are the complete system.
The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum.
The complete system of equations has around 6 or 7 or even more equations, depending on whether one chooses to have distinct notations for various physical quantities, such as electric polarization, magnetization, current of free electricity carriers etc., or not.
The variants with less equations are simplifications that are valid only in linear media, because only there you have proportionality relationships between quantities like electric field and electric polarization.
Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell.
The many simplified variants, for media without discontinuities where differential forms are valid, for stationary media, for linear media, for vacuum and so on, can be easily derived from the general form, while the reverse is not true.
> Rewriting the equations in vector form reduces the number to the modern number.
And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form.
> The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum.
> Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell.
Here I have to strongly disagree. The version of Maxwell's equations that is fundamental and exactly correct [1] is the vacuum version. The ones with magnetization and displacement vectors are only approximations where you assume continuous materials that respond to fields in simple way. In truth, materials are made of atoms and are mostly vacuum: there is no actual displacement vector if you look close enough.
Also the vacuum Maxwell equations are useful in many scenarios. For instance, that's how you compute the energy levels of Hydrogen atom or how you derive QED. Also, you have to start from them to derive the macroscopic versions with magnetization and displacement that you seem to like.
[1] Well, up to non-linear quantum mechanic effects.
Writing Maxwell's as 1 equation or 4 or more is just esthetic choice where you decide what to accentuate.
20 might be too much because three dimensions are not really different from each other so the notation that maps over them wholesale is probably a good idea.
4 equations seem perfect if you want to differentiate between classical effects of the electric field and relativistic effects (magnetism).
I don't know if single equation really shows that they really have the same source and the relativity is involved or is it just a matrix mashup of the 4 separate equations that doesn't really provide any insights.
However, the notation that lets you do this in this specific case is very natural and not specific to Maxwell's equations. Differential forms are very natural objects in differential geometry, mathematicians would have likely introduced them and studied without inspiration from physics. The fact that Maxwell's equations are very simple in this natural geometrical language does say something meaningful about their nature and elegance, I think.
Even today, there exists no consensus about which is the correct expression for the electromagnetic force. Most people are happy to use approximate expressions that are known to be valid only in restricted circumstances (like when the forces are caused by interactions with closed currents, or the forces are between stationary charges).
Moreover, when the vacuum equations are written in the simplified form present in most manuals, it is impossible to deduce how they should be applied to systems in motion, without adding extra assumptions, which usually are not listed together with the simple form of the equations (e.g. the curl and the divergence are written as depending on a system of coordinates, so it is not obvious how these coordinates can be defined, i.e. to which bodies they are attached).
While the vacuum equations are fundamental, they may be used as such only in few applications like quantum mechanics, where much more is needed beyond them.
In all practical applications of the Maxwell equations you must use the approximation of continuous media that can be characterized by averaged physical quantities that describe the free and bound carriers of electric charge. The useful form of the Maxwell equations is that complete with electric polarization, magnetization, electric current of the free carriers and electric charge of the free carriers. It is trivial to set all those quantities to zero, to retrieve the vacuum form of the equations.
This is just a matter of taste, but OTOH I would not include descriptions of how some materials respond to the fields in the continuous limit as part of a definition of EM.
It is true that for most terrestrial applications you do need those to do anything useful with EM. But if you want to study plasmas you need to add Navier-Stokes to EM, doesn't mean hydrodynamics is part of EM. To study charged black holes you need EM + GR, but it still makes sense to treat them as mostly separate theories.
I actually basically agree with your viewpoint, I studied Plasma Physics in graduate school in a regime where we did _not_ use Navier-Stokes or constitutive relations and everything was in fact just little smeared-out packets of charge moving according to the Lorentz Force Law and radiating.
But of course you're absolutely correct that continuous and discrete systems are approximately equal.
We use a trick called renormalization which, in this case, is recognizing that the mass of the electron has a term from the EM field. We’d assume that the EM theory is not completely true but that below some distance the theory breaks down. Working in momentum space there is a certain momentum that corresponds to the cutoff distance so we just don’t integrate beyond that. You can vary the cutoff and also vary the other parameters of the theory (such as the bare mass of the electron) so the theory gives the same answers at macroscopic distances so it doesn’t matter where you put the cutoff.
Thus it does not matter much what the “true” theory is whether space is discrete or the electron really is a little ball or the EM field merges with the other forces at high energy to make some different force that (slowly) eats protons or quantum gravity or whatever.
Discretization is problematic in a relativistic world because it breaks Lorenz invariance. That is, if I am moving quickly I would see the gap between the “pixels” get smaller. Now maybe the pixels can be non-Lorenz invariant but can “fake it” at low energies and large sizes but when the energy gets large you’d expect to see some evidence of the grain. Even if the gap was the Planck length you’d probably see things get weird at much lower energies, such as those of the highest energy cosmic rays. There has been a lot of research on that and there is no clear evidence of relativity being broken but it is still highly mysterious
https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80...
for instance Lorenz violation might allow particles to bypass that GZK limit.
There are tons of examples in mathematical finance but the obvious one is the Black/Scholes [1] paper where one of the key assumptions they make (which they know to be untrue but helpful) is that you can replicate a portfolio in continuous time. This allows them to use a constructed portfolio of a risk-free intrest-bearing instrument and the underlying to replicate the price of an option, and the process is a Brownian motion. Everyone knows that actual trading (and thus price processes) are discrete in real markets, but continuous time is much easier to model. Much later on people like Heston and Matytsyn(?sp) came up with stochastic vol models with jumps to replicate discrete price discontinuities, but they're a lot harder to work with in many ways.
[1] https://www.cs.princeton.edu/courses/archive/fall09/cos323/p...
Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations:
It's a little distracting how it looks like an ad for an adult themed video game, but it's very well thought out.
Well-put: Does that demon-squid in the intro look like a Chihuly glass installation to anybody else?
Also I swear I've heard that song long ago on OCRemix.
>the strength of an electric field depends on the number of electric field lines.
the number of electric field lines, depends on the strength of an electric field. ,
At some point the definitions become almost circular and opinions about what it fundamental have shifted a bit over the centuries. The cgs system of units -- which differs profoundly from SI in the treatment of electromagnetism -- was associated with those who viewed D and H rather than E and B the most fundamental. I'm quite happy with the level of theory used being appropriate to solve the problem at hand. There's always a bit of wiggle room around exactly what that problem is, however ;-)
It doesn't matter in terms of the math (in the vast majority of situations), so while the conventional idea of electric flow is incorrect, we keep it anyway.
Current moves far faster than electrons. it is more similar to a wave in the ocean with the electrons being the water molecule.
As a result, and counterintuitively for most, the speed of electrons will give you a completely wrong answer for when a light will turn on after you flip a switch.
Current is the movement of charges. It cannot “move” faster than said charges. (Or, perhaps, you meant the electomotive force that makes the electrons move along the wire, then sure, that thing spreads pretty quickly.)
Your comment could be reduced to "lines arent real. show me a perfect line"
Did I sense mild sarcasm here?
e.g.
Imagine a 4-dimensional hyperbolic surface defined by the lightcone of a particular point in space-time, now imagine that this surface is stretched/compressed by the distortions of gravity. Now let's talk about equations which are only loosely tied to this surface.
vs.
Consider the metric tensor defined by this 4x4 matrix g_xz. Distance is computed as a^x b^z g_xz, now consider all possible walks from point a to c to b. Now let's show the relation between these walks and a quantity we'll call the stress-energy tensor which represents the energy/momentum density at any particular point in space, and it's flux towards any other direction in space.
The latter is a very algebraic description, which does not rely on the audience having to visualize the constructs involved. Practically, even if you get a feel for what a Riemanian manifold looks like in 4-dimensions - you'll struggle to visualize the Riemann tensor, or Christophel symbols.
One chapters deals with Maxwell. After this it was easy to understand
https://www.cambridge.org/core/books/applied-differential-ge...
According to wikipedia [1], Heaviside significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death.
The integral equations of Maxwell, which few know today, are much more generally applicable and actually easier to understand.
The differential equations of Heaviside are valid only when certain restrictions about continuity are true. Moreover, the meanings of curl and divergence are hard to understand otherwise than by deriving them from the integrals over curves and surfaces used in the original equations of Maxwell, which are also necessary to determine how to handle discontinuities.
The differential form of the equations looks prettier on paper due to a simpler notation, but it is less helpful for understanding and for solving practical problems than the integral form.
In my opinion, it is a serious mistake that almost all manuals show the equations of Maxwell in the Heaviside form, instead of showing them in their original form. This is one of the main reasons why they are hard to understand for many.
I'm not sure that's true. What kind of model of causality are you using here?
Other people were also on the cusp of doing (most of) what Einstein did. Ultimately, without Einstein most likely progress would have been delayed by a few years, and the laurels would be spread amongst more people. (Eg Lorentz' work was on the way to formulating special relativity.)
So it's hard to say that there was any single 'only reason' for any discovery here.
Albert Einstein was smart, and he definitely was smarter than me. But his achievements weren't beyond all the other smart people. Especially individually and with more time.
See also how Newton and Leibniz came up with calculus at the same time. Or how public key cryptography was invented independently multiple times.
"Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith."
However the general relativity has no relationship with the equations of Maxwell discussed here.
While general relativity is the most original work of Einstein and the one best known, the second most original work of Einstein is the one that had the greatest impact on practical technology: the discovery that for computing the properties of electromagnetic radiation one must take into account also the stimulated emission (like in lasers), not only the spontaneous emission and the absorption.
For me, Einstein's paper on stimulated emission is the most important of his work. Even if more than a century has passed, it is not yet clear if Einstein's mathematical model is the best for gravity and inertia or if there exists another model that would be more comprehensive and which could relegate Einstein's model to be an approximation that would be no longer useful.
Einstein reportedly ran thought experiments in his imagination.
His intuition was visual and dynamic.
That is images that change - animations.
A natural language description of relativity is one step removed from the moving images of Einstein's intuition.
Fields are not grounded in everyday experience but most modern movie-goers are happy with rapid and grand changes of scale and thanks to wizard and superhero movies there exists a sophisticated visual grammar of rapidly propagating fields, strange paradoxes and simultanous weakly interacting realities.
Many of the concepts of quantum field theory can be grasped by a wide audience when presented as animations.
I recently read "A Student's Guide to Maxwell's Equations", and it was perfect for me - it explained enough of the maths to understand the equations, without having to first learn differential geometry. https://www.cambridge.org/highereducation/books/a-students-g...
But like with other theories, people find ways to simplify the notation and formalism and explain it better. Quantum mechanics, special and general relativity are similar in that regard.
I really liked Kathy Joseph's historical reviews of vector physics and the people who developed it, which explain some of the reason's for how it's taught. Most texts don't even develop electrodynamics from relativistic electrostatics as a demonstration.
I think I fell down the rabbit hole from Freya Holmer's "why you can't multiply vectors".
The key being that all of the Hamiltonian fields can be found in a single quaternion equation, which is just what happens when you start multiplying vectors together.
On the other hand he did derive the electric and magnetic fields from a scalar and potential field. In that sense Heaviside made a step backwards.
While Maxwell’s work was inspired by Faraday's, it was also built upon the contributions of many other scientists (Coulomb, Ampère, Thomson, Neumann, Lenz, ...)
This is a nice remark, but very difficult to implement in practice. In reality, many non-modest people will overrate their contributions, while the few modest people will have a hard time to act non-modest in certain situations. We are in a world where modesty is even rarer than 100 years ago. I am sure many important discoveries are hidden in the myriad peer reviewed publications published just for quantitative reasons.
https://m.youtube.com/playlist?list=PLVV0r6CmEsFzDA6mtmKQEgW...
"This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly."
Electric field-strength is measurable no less directly than energy, it is a force experienced by a unit charge placed within the electric field.
What? No you can't. The fields are invariant under gauge transformations.
That, had it been true, would have made the gauge observable.
Of course, the context matters. Often if one compares potential and field, field would be the one directly measured. It is just semantics really.
> The reason for these arguments is that the various interpreters are trying to describe the quantum world in the words of everyday language, and the language is inappropriate for the purpose. Everyday language describes the world as human beings encounter it. Our experience of the world is entirely concerned with macroscopic objects which behave according to the rules of classical physics. All the concepts that appear in our language are classical. [...] The battles between the rival interpretations [of quantum dynamics] continue unabated and no end is in sight.
Replace 'quantum dynamics' with metaphysics (or post-Kantian metaphysics) and the statement seems true as well.
Very important to people like me, because I really struggle with advanced math. I dropped an EE degree because while I could do the math, it was incredibly hard and in no way intuitive to me.
It's a common misconception that electrons or current transfer energy. In reality it's the electric field that exists between the wires that is doing the heavy lifting, the electrons in the wires are just controlling the field.
This has always confused me and I was very irritated when I first learned electromagnetics about how rote all the initial learnings are. I wish more work was put earlier into making everything relate back to Maxwell's equations to make it make sense.
It's not about the misconception about "AC is vibrating so how can electrons be delivering their energy from the power plant to the light bulb far away?"
They are talking about how the electric field is outside the wires almost entirely.
Their argument is that in the water wave analogy, the wave wouldn't be in the water at all, because it's "actually" transmitted via an invisible field in the space above the water, which pushes back on the water farther away.
Most respected electricity/physics YouTubers disagree with Veritasium's emphasis on this perspective, by the way. The think he conflated the first misconception I mentioned with the second idea, which is about how you model electric circuits.
It's the same argument as "Einstein corrected the misconception that Newtonian mechanics is how bodies interact, and it's irritating how rote mechanical engineering of a car is."
Feynman explained this nicely. He said essentially, you ask me to explain what is electmagnetism. Is it like two hands pushing on other? Well, if it is, then what is "pushing"? Pushing is just the result of electomagnetism in your hands! It is impossible explain electromagnetism to you in terms of anything simpler that you already understand. It is a fundamental force.
https://i.etsystatic.com/16048150/c/500/397/254/275/il/4c656...
With an emphasis on basis functions and computing we could do it in a single semester.
The more interesting thing was that most students had practically no useful previous knowledge despite formally having some years of eduction on the subject
My insight was the biggest issue was the eduction system failing the students and not anything specific to Maxwell’s equations.
“Here you are, a speck of thinking matter. Oh and by the way you just so happen to have a special capacity for mathematics, the secret language of the universe.”
Convenient, isn’t it?
unrelatedly, i feel like i kind of understand maxwell's theory in terms of vector analysis, but the clifford algebra formulation is still beyond me. it sure looks a lot simpler
Special relativity includes classical mechanics when speeds are low. In the same way, any replacement for Maxwell's paradigm must reproduce the predictions of Maxwell's equations under the large swathe of conditions where they agree with reality.
for that matter i use the aristotelian paradigm of physics when i expect my bed to stop moving when i stop pushing it across the floor; i don't bother with calculating the deceleration due to the friction coefficient with the floor
Both Newtonian gravity and Maxwell's Equations are still very good approximations in their regimes of validity.
i thought it was too obvious to be worth saying that classical physics is still an excellent approximation to reality, but hopefully you've enlightened someone reading this thread
Are you commenting on the capitalization of "Maxwell"? It is a proper name, and it should be capitalized.
Maxwell wrote tons of equations during his life, but there's only one set of "Maxwell's Equations." Maxwell himself never even wrote down Maxwell's Equations in the form we now know them.
Does it, though? Sure, QED is a field theory, but it's perturbative where classical EM is exact, its fields are operator-valued distributions where classical fields are number-valued functions, its interactions are transition probabilities rather than forces - it's not clear to me that these are smaller jumps than introducing fields in the first place.
The problem is that because of trying to cram a degree into 4 years, you wind up having a class on electromagnetics without any understanding of vector fields.
Electrical engineering is particularly bad about this. You never get exposed to the Hamiltonian formulations of classical mechanics, and you never get exposed to vector analysis. Consequently, you are stuck with the Heaviside-Hertz pedagogy with silly things like "displacement current" and stupid, weird-ass integration contours (which don't work in motors, LOL)--and the attendant difficulty in understanding Maxwell's theory.
However, if you have vector analysis and fields, then you can understand formulations like Carver Mead's "Collective Electrodynamics": https://www.amazon.com/Collective-Electrodynamics-Quantum-Fo...
Suddenly, emag is a whole lot more straightforward to understand. It's not EASY as it's very math heavy, but it has a lot fewer weird things that are just "we say it works."
We did get stuck with Gibbs' vector calculus formulation as the canonical view unfortunately.
That's a really large conceptual jump and physicists did not make that jump lightly or easily.
Maxwell's paper was 1865. The Michelson–Morley experiment was 1887. Michelson himself couldn't make the shift. Quoting Wikipedia (https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper...): "The negative result led Michelson to the conclusion that there is no measurable aether drift.[1] However, he never accepted this on a personal level, and the negative result haunted him for the rest of his life (Source; The Mechanical Universe, episode 41[8])."
In an attempt to preserve "aether", Lorentz contraction then enters the picture as an ad hoc explanation for the Michelson-Morley result. It turns out Lorentz contraction is correct, but not because of the existence of "aether" but because of the constancy of the speed of light--c (Einstein Special Relativity--1905).
Once you finally give up on "aether" after 40 years of trying otherwise, you can finally just roll with the mathematical implications of Maxwell's equations.
[1] Michael Faraday's Thoughts on Ray Vibrations, 1846, cited by Maxwell in his paper. Faraday says: "The view which I am so bold to put forth considers, therefore, radiation as a kind of species of vibration in the lines of force which are known to connect particles and also masses of matter together. It endeavors to dismiss the aether, but not the vibration." (https://pwg.gsfc.nasa.gov/Education/wfarad1846.html)
Gerard ’t Hooft used to have a humongous list of textbooks for aspiring theoretical physicists. I'd sure look there for starters.
Perhaps somebody who has read this book can comment in more detail.
Geometric algebra can reduce Maxwell’s equations into a single, hilariously terse equation:
∇F = J
Disclaimer: maths degree, so my unease was not a plain lack of understanding.
Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM.
The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.
> They do not do a good job of tracking the physical structure of electromagnetism
What do you mean by tracking the physical structure?
By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?).
> and in fact end up baking in a lot of assumptions about the setting that fail to generalize.
Can you explain a bit more what you mean here please?
> Can you explain a bit more what you mean here please?
It's all downstream of geometric algebra leaving the metric implicit in its operations, basically
- The metric is an extremely important physical quantity: it doesn't necessarily look that way when everything is classical and flat, but you have to start caring about it in curved spacetimes.
- Even when the metric can be safely neglected, doing so makes it very easy to confuse degree (n-k) elements of your vector space V with degree k elements of its dual V. V and V are isomorphic but not canonically isomorphic - you have to choose a basis. And keeping track of where you introduce a choice of basis matters, because all physical quantities are basis-independent. Nature has no preferred coordinate system.
- Geometric algebra doesn't make sense on a general manifold: you have to embed it in a sufficiently large geometric algebra and inherit structure from the embedding. This is better than working in a particular coordinate chart but worse than doing differential geometry in a purely geometric way.
- Geometric algebra is not invariant under diffeomorphism, which means GR is dead in the water. You can build mostly equivalent theories by enforcing the equivalence principle in your dynamics instead, but it's more complicated and ironically far less geometric.
That’s the point! That’s the entire point!
Mathematicians want the most general, most abstract approach. They want to generalise to a wide range of problems and not be painted into any one specific example.
Physics theories have an opposite goal to this: the ideal theory ought to take no parameters, and produce “reality” as the one and only possible outcome. The ideal theory ought not generalise to un-physical models.
For example, the mathematics of general relativity have excess degrees of freedom that must be constrained through additional restrictions. Similar issues turn up almost anywhere matrices are used: they have too many degrees of freedom.
Geometric Algebra is typically a better fit for what actually goes on in physics.
For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. Rotations implemented using GA have none of these issues.
Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches.
> For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated.
No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is!
This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes
- Look at this horrible vector algebra with explicit charts and numbers all over the place
- Now look at this nice coordinate-free geometric algebra construction
- Therefore geometric algebra is the right setting for physics
but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.
It's a good thing Maxwell is not alive and that he did not follow Dyson's advice, lest Hacker News accuses him of attempting to abuse the attention economy and promote his research like a salesman.
https://news.ycombinator.com/item?id=33043945 https://news.ycombinator.com/item?id=22297855 https://news.ycombinator.com/item?id=39144845
But I think he's trying to make a slightly more general point: why are "parities" (2-fold symmetries) so common in nature and mathematics? Why not more 3-fold symmetries?
(Obviously I can't say whether that is true or not, but it might be a possible explanation of current observations).
What's concerning (aside from your callous disregard for people who have small screens) is that the PP created a new document, and didn't show it, suggested that we might find it more digestible.