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Why is Maxwell's theory so hard to understand? (2007) [pdf]

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Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#171

Post by waveBidder »

rini17 wrote: post #61
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation 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.

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. 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

But we never know any quantity to full precision, so it's not like we get infinite bits about any given quantity.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#172

Post by devmor »

steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. 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: https://www…

Wow this video is actually great at imparting the concepts with animation.

It's a little distracting how it looks like an ad for an adult themed video game, but it's very well thought out.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#173

Post by mikhailfranco »

lupire wrote: post #126
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation 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.

Is wave function collapse continuous? Is photon absorption and emission continuous? 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.

Collapse is not continuous, by definition. Copenhagen is discontinuous.

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.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#174

Post by hnfong »

omnicognate wrote: post #66
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation 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.

> Cognition is discrete There's little evidence even of this, except in the trivial sense that language (minus prosody) is composed of discrete units.

You're technically correct.

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.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#175

Post by hnfong »

omnicognate wrote: post #162
lupire wrote: post #138

Earlier quoted context omitted.

I don't see how what we know is discrete. 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.

I entirely agree. I wasn't making a statement that "what we know is discrete". I was referring to a particular subset of cognition as "what [i.e. the things that] we know are discrete". 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 po…

> 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.

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?

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#176

Post by jerf »

lll-o-lll wrote: post #56
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation 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.

> the world is continuous 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?

There are many times in physics where people have thought they've had to choose between either one thing or its opposite, where both choices had clear deficiencies. The ultimate solution ended up being a new hybrid that nobody thought of for a long time.

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.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#177

Post by omnicognate »

hnfong wrote: post #175
omnicognate wrote: post #162

Earlier quoted context omitted.

I entirely agree. I wasn't making a statement that "what we know is discrete". I was referring to a particular subset of cognition as "what [i.e. the things that] we know are discrete". 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 po…

> 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. 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 evo…

I can't. That's why said "I personally strongly suspect" rather than presenting my partially-informed intuitions as facts.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#178

Post by eigenket »

mr_mitm wrote: post #115
rini17 wrote: post #61

Earlier quoted context omitted.

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. 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

All observable quantities are eigenvalues of some operator, which are real numbers but discrete. How can they contain infinite amount of information?

Consider the Hilbert L^2([0,1]) associated with a physical particle that has position somewhere between 0 and 1, the corresponding multiplication operator X which takes a wavefunction f and maps it to Xf where (Xf)(x) = x f(x). Then X is a bounded self-adjoint operator. It doesn't have any eigenvalues or eigenvectors but it's spectrum is exactly the set of numbers [0,1] as you'd expect (prefect measurements of position return real numbers in [0,1]).

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.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#179

Post by n4r9 »

mjburgess wrote: post #68
rini17 wrote: post #61

Earlier quoted context omitted.

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. 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

All the people who use thinks like the word "information" in this context are confusing thermodynamic, logical, probabilistic, (+ many others) and equivocating. "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 re…

I suspect that Gisin has a very clear idea of what he means by "information" in this context, having worked for over 40 years at the forefront of theoretical physics with a specialisation in quantum information theory.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#180

Post by eigenket »

mikhailfranco wrote: post #165
rini17 wrote: post #61

Earlier quoted context omitted.

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. 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

I am very sympathetic to Gisin and his cause, but he does not propose any sensible resolution. By the way, not a fault, and no blame for him. Pointing out logical deficiencies always comes before a satisfying solution, and he is to be praised for his insight. 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 sys…

> But pi is has infinite information

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.

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