Math Matters by Sabine Hossenfelder

"Any proof is only as good as its assumptions" All too true.

And as Max Planck observed a century ago, the problem is, theoretical physicists are not particularly adept at identifying that some things, even are assumptions; with the result that "self evidently true" facts, lead to long periods of stagnation, until those "facts" are eventually shown to be just idealistic, false assumptions. The seldom stated assumptions underlying Bell's theorem are a case in point.

Rob McEachern

"... Einstein's Field Equations are widely believed to break down near the singularity because quantum gravitational effects should become important. And since the singularity is hidden behind the event horizon ..." In physical reality, is there such a thing as an event horizon? Consider the following question: After quantum averaging, are Einstein's field equations 100% correct?

Consider the following:

Einstein's Field Equations: 3 Criticisms

Can you cite empirical evidence that shows that any of 3 suggested modifications are wrong? I claim the following: Merely on the basis of mathematics, the alleged Rañada-Fernández-Milgrom effect is approximately equivalent to Milgrom's MOND -- whenever and wherever the MOND approximation is empirically valid. Do you agree or disagree with the preceding claim?

Dear Sabine;

I have read with great interest your essay and concluded that I agree with most of the deductions and conclusions. Only we are using different "words". Some remarks while reading your essay.

"A theory may have given us the most extraordinarily accurate predictions until today, and still tomorrow we could discover that it doesn't explain the next measurement." You are so right, my explanation (that may be different from yours) is that the next experiment may take place in the future, it is only the past that is deterministic and not the future.

Your paragraph 2 reminds me of the rules: rule1.The boss is always right. rule 2 If the boss is wrong, rule 1 is still valid. (your added axiom). The rules are mathematics, ignoring reality, so both are different things.

"Nothing real is infinite", is the same expression as "reality is finite", which does not mean that there is an infinity of realities. In my perception, this infinity of probable realities is "outside" our emerged reality.

"Lorentz "butterfly effect" is I think not only valid for predictions in the future, but also works for the past. The further we are reaching out in the past the lesser the chance that you will exist, the amount of IF's that are influencing your existence (meeting of ancestors, the female egg receiving the specific needed sperm etc, etc) is the butterfly effect in the past that leads to your existence. If all the IF's are fulfilled they amount to 10^2,685,007 !!!

"And since the singularity is hidden behind the event horizon, we can't just go and measure what is happening. ". This "event horizon" is in my perception behind the Planck length and time. It is the border of our reality.

"Given a sufficiently large and powerful computer, even human decisions could be calculated in finite-time" Human decisions are made as you say in "finite" time. Finite-time is deterministic because essentially it is already the past. The REAL choices are made in the seemingly future of the timeless unmeasurable entity outside our reality (Total Simultaneity).

I understand that you are a busy bird, but anyhow maybe you can find some time to read my essay .

Thank you

Wilhelmus

Dear Prof. Hossenfelder,

congratulations on a beautiful essay indeed! (like many of your writings with which I am familiar). I agree with most of your arguments and overall with your program against the (arrogant) reductionist view of physicists.

You might find some resonance with my essay, based on a work carried out with Nicolas Gisin, where we further developed the argument against the physical significance of real numbers. This seems to entail this almost Platonistic standpoint on mathematics, having its really existing entities and physics relies on them at an ontic level. As you also say, of course nobody questions the great power of mathematics in modelling physics, but math is not physics, nevertheless.

I wish you to get to the prize range, top rate from my side!

Dear Prof. Hossenfelder, congratulations on a beautiful essay indeed! (like many of your writings with which I am familiar). I agree with most of your arguments and overall with your program against the (arrogant) reductionistic view of physicists.

You might find some resonance with my essay, based on a work carried out with Nicolas Gisin, where we further developed the argument against the physical significance of real numbers. This seems to entail this almost Platonistic standpoint on mathematics, having its really existing entities and physics relies on them at an ontic level. As you also say, of course nobody questions the great power of mathematics in modelling physics, but math is not physics, nevertheless.

I wish you to get to the prize range, top rate from my side!

I think it is one thing to say the the grounds/the foundations of various mathematical branches such as arithmetic, geometry, anylysis, algebra, topology, probability, ststistics, etc and even physics are incomplete/logically inconsistent and a totally different thing to say that mathematics, in general, if there is such a thing, is incomplete, wouldn't you say?,,,I doubt that Godel ever stated the latter even as a 'gist' of his celebrated theorems, As their gist more accurately reflects the former, in my opion.

As to Platonism as a mere philosophical position and not a scientific one,let's just pose for a minute and ask ourselves the following question: would for instance be a science today called 'Cosmology ' if the mankind didn't have such a profound dialogue written by Plato, viz. Timaeus handed down through time to us?...Moreover, where would our science and culture of today with its main branches such as Physics, Mathematics,Logic, Ethics,Aesthetics, Metaphysics, etc be without the ancient Greeks?...

Sabine,

while I agree that the three 'uns' aren't physical problems, they yet pose a problem to physics. Gödel's theorems don't say that mathematics is either incomplete or inconsistent, they say that logic cannot deal even with traces of mathematics. In plain words they say that logic doesn't belong to mathematics and hence to physics.

Heinz

That is true Heinz: "logic doesn't belong to mathematics and hence to physics".

Physics represents "laws of nature" with equations, but you can't derive logic from equations. This is where Sabine is missing something in her otherwise excellent essay: physics can't explain the existence of logic. If physics can't explain the existence of logic then physics cannot explain the nature of the world.

Dear Sabine Hossenfelder,

Your views are very clear and helpful. They certainly do not fall into the phraseology that is common and leads nowhere.

I fully agree with your statement:

"Nothing realis infinite [4], therefore the whole formulation of the problem is scientifically meaningless. Inpractice, we never need an algorithm that can correctly answer infinitely many questions."

Not only are there no endless physical phenomena in reality, but thanks to Planck, many of the limits of these phenomena are well known to us.

My comment on:

"Again, we conclude that impossibility-theorems are mathematical curiosities without scientific relevance."

If I understand you well, we should not waste our time solving difficult problems that are scientifically meaningless. In my essay, I have pointed out three trivial problems that are great obstacles to understanding reality, and can be easily solved.

I cannot fully agree with your position:

"The major difficulty we face in making predictions is that we either don't have sufficient data or don't have the math for handling the data, not that there's a mathematical theorem preventing us from making predictions."

The fact is that the giants of natural philosophy in the past, came to a remarkable predictions that did not even have the information about the existence of other galaxies than the Milky Way. I even argue that the abundance of data has greatly confused scientists, who have failed to understand some phenomena. I particularly emphasize here what is related to the works of Hubble and Lemaitre.

As a meteorologist I read with interest your rational explanations of the butterfly effect and chaos theory. Thank you for that.

I can agree with your point of view "Physics Isn't Math", but contrary to you I claim: Reality is math, and surely can be well described by math. I claim that because I got it by compiling the results of Newton, Kepler, Planck, partly Einstein ...

Regards Branko

Hi, It is a beautuful general essay , I see the maths and physics like this. The maths are very important indeed, they permit to prove assumptions and this tool is foundamental when they respect the logic. We need these maths to prove simply.That said we must be prudent about the mathematical extrapolations, sometimes they imply confusions , it seems that all is a question of limits in the interpretations, the maths indeed imply many assumptions, like the multiverses, the whormholes or the time travel or this or that. But after all only the proved works are accepted inside the sciences Community. The physics and maths converge when the pure determinism, rationalism or logic are respected. About the predictions and computabilities, we are limited I beleive, we know so few still and we have still many secrets to add. We cannot predict all of course , just because we are not sure about what are our foundamental mathematical and physical objects, are they points , or strings or in my model 3D coded spheres ? we don t know and we cannot affirm, the sanme for the philosophy of main causes, have we a 1D main field like in the strings or this or that, we don t know, all what we have are just assumptions and they must be proved simply by experiments or mathematical proofs.

Regards

Gödel's theorem does require an infinite number of predicates and Gödel numbers that are the subject of a predicate. The diagonal or self-reference and the Cantor trick leads to the lack of enumerability and unprovability. The first of Gödel's theorem then explicitly appeals to infinity. Physics of course avoids infinity the way most now try to avoid Covid-19. Clearly, we do not measure infinite quantities.

Invoking Gödel's theorems in quantum mechanics is not easy. As I have argued decoherence is a process where quantum phase is taken up by auxiliary quantum states, and a needle state in a measurement is then as a result at least analogous to a Gödel numbering of quantum states.

Quantum mechanics in principle has an infinite number of states in Hilbert space. This then means at least in a formal sense this can be done. Physically this is a bit more problematic, and often I think of quantum systems as finite, accessing some finite subset of Hilbert space, but where there is no fundamental upper bound.

Physics has plenty of instances where infinity is appealed. Calculus with its lim_{x-->∞} has been used for centuries since the time Leibnitz struggled with this. We learn this and then teach this without batting an eye. Yet I, and I think many of us, regard this as a sort of idealization cast in symbolic form. The application of Gödel's theorems in physics, in particular quantum mechanical foundations is a similar move.

A benefit of doing this is to end this nonsensical kerfuffle over quantum interpretations. If these are really just unprovable auxiliary postulates, analogous to axioms, then these are really just formal statements we impose, and that nature ignores. We could put all of this trash over quantum interpretations where they below, the basura.

I wrote this in MS Word and did not copy all of it.

On balance your paper is reasonable, at least precautionary. We do all to often frame physics according to mathematical systems, which can lead to expectations about nature that are not correct.

Dr. Hossenfelder:

Historically, when physics models have limits in predictability, changing the fundamental physics postulates results in better predictability. For example, your weather analogy with all the sensors on Earth would still be limited as you suggest. But some model of the Sun activity would be required and could increase predictability more effectively than a model of unpredictability. Adding the fusion reactor at the center of the Earth would do more still. A model of unpredictability could only indicate a problem, it couldn't be a better model. Is the solution to the Lorenz problem different postulates rather than models of unpredictably?

Are there some mathematical rules that can help decide the next viable model? You excluded infinity as a physics concept. The 3un's use ordinal numbers in there proof. Is the conclusion that ordinal numbers should not be part of physics? Or, that math using ordinal numbers is not physical?

Are imaginary numbers part of the real universe or an indication of flawed physics models?

You seem to reject irrational and transcendental numbers (those real numbers that are not rational) in science but these can also have quantifiable error-estimates. pi seems everywhere in science. Does the use of irrational and transcendental numbers indicate flawed physics?

Should physics be predictable?

Large things being made of small things sounds more like emergence. Electron, protons, and neutrons form many more atoms. Atoms form many more molecules. When Newton identified gravity and applied it to objects falling on Earth and the Moon, that was reduction-ism. Later his aether traveling faster than corpuscles (photons) and directing photons and matter seems better reduction-ism for light interference which can explain the interference experiments that reject wave models. And aether directing matter sounds close to General Relativity. Copernicus vs Ptolemy. Emergent: simple to complex. Reduction-ism: complex to simple. So, it seems simplicity rather than complexity such as super-symmetry develop better models (my lesson from your book "Lost in Math").

I think we already have enough data to form the Theory of Everything. There are many observations that are ad hoc or unexplained in astronomy. The core of Quantum Mechanics (the small) is the yet unexplained interference of light (Many experiments reject wave models, the Afshar experiment rejects Copenhagen).

Hodge