Math Matters by Sabine Hossenfelder

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

All rational deterministic thinkers understand this simple fact, we have limits in knowledges and even the best theories are limited , we cannot explain several things , that is why a TOE cannot exist, not need to discourse about this truth I beleive, it is the same with our assumptions, we cannot affirm them without proofs simply. The maths I repeat is a tool permitting sometimes to prove an assumption, and when it is proved, so it is accepted, it is only simple than this. All is not predictable and computable simply because we dont know several things, like the foundamental objects or others, even the philosophy is not proved generally about our main causes. Maths and physics matter indeed if and only if they are proved with their laws, equations, axioms. It is even easier to predict and prove the Vanity of thinkers inside the sciences Community than other unknowns lol spherically yours

That was precisely my point above too Heinz. Frege used to refer to logic as "the third realm" as distinct from maths/ideal and physics/real. If Physics can't explain Logic then physics can't explain the human mind and its thinking structures which is an integral part of nature/Universe as a whole, yes Lorraine, I would agree with that. Interesting, in this context, is of course Heidegger's attempt to ground logic on ontology in Being and Time.

Hello Dr. Hossenfelder,

I will admit that I am humbled to write a response to your essay:

"...Reality may not be math, but it surely can be well described by math..." I am unsure if you have read Tegmark's Our Mathematical Universe (2014)- he addressed this already.

You went on to state "...You are free to believe that reality is math, but since this belief is scientifically indefensible I will not defend it. We also don't need it: Physicists use mathematics simply because it is useful. Reality may not be math, but it surely can be well described by math..."

I respectfully must disagree with "...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..." If I'm not mistaken (please correct me if I'm incorrect): Gödel's famous work implies that mathematics, itself is intrinsically unknowable.

Overall, your essay presents the "three uns" in a highly practical manner. This is not unexpected as you are a professional physicist.

"...Physics isn't math, and Gödel's theorem is irrelevant for scientific practice..." Despite the intimate relationship between the [hard] sciences and maths- true, they are not one in the same.

I urge you to visit work done after Gödel. The Fields Medal was given to Paul Cohen for forcing. I am embarrassed to cite Wikipedia, however, I am going to anyway "...By Gödel's second incompleteness theorem, one cannot prove the consistency of any sufficiently strong formal theory, such as using only the axioms of the theory itself, unless the theory is inconsistent..."

I liked how you compartmentalized the essay appropriately- you put relevant information together well.

"...Nothing real is infinite [4], therefore the whole formulation of the problem is scientifically meaningless..." I'm sure that you're well aware that it is for this reason that the foundations of mathematics and theoretical physics are so utterly different despite the "unreasonable effectiveness of mathematics" (Wigner). Your quote is the distinguished and consequential and I think it's worth stating that you wonderfully addressed the following in your essay too. Cantor's hierarchical infinity means nothing in physics but any в€ћ indicates some failure on the part of the equation or our understanding of the equation(s) as exemplified in an integral of 1/r in which r is the radius of a black hole and r = 0. You would know better than I do.

Penrose (1974) contributed to mathematical tiling "The role of aesthetics in pure and applied mathematical research", Bulletin of the Institute of Mathematics and its Applications, 10: 266ff. I also recommend Berger, R. (1966), The undecidability of the domino problem, Memoirs of the American Mathematical Society, 66. (retrieved on March 15, 2020 from Wikipedia). I see that although you noted the fact that Wang's Domino Problem was undecidable. I suggest citing Berger (a student of Wang) who proved this.

You described an individual falling into a black hole (in which anything that falls in and is viewed by an observer stops at the surface of the black hole and the individual/ or creature falling in the hole perceives something completely different). There is still an unaddressed issue of subjectivity which is currently an open question when we describe proper time as well as a satisfying philosophy of time which too is currently open as well.

Sabine, you also wrote about (what you described as) "...a superficially entirely different system that, however, has many parallels to plasma blow-ups and weather forecast: The stock market..." (p.5). Please notify me if I'm uninformed- indicative of one of the unique niceties of human beings is that we can make social sciences (ergo have an elaborate economic system). Consciousness can be described by mathematics just as an economic system can. One could surely apply some of the "uns" to consciousness in of itself.

Lastly, will some (potentially futuristic) theory of quantum gravity not be a mathematical based theory (e.g. some version of either loop quantum gravity or string theory)? Indeed, math matters.

Aside from few grammatical errors, you wrote a wonderful essay. I enjoyed your essay.

Many thanks,

Dale Carl Gillman

You say: "the future is already determined, up to the occasional random interference from quantum events."

This is a very strange view. It sounds as if you believe the true laws of physics as non-quantum, and quantum mechanics is just making it sloppy. Your footnote says that this refutes any sensible notion of free will. So do you think humans are governed by physical laws that do not include quantum mechanics?

I know from your blog that you don't even believe in quantum randomness, because you subscribe to superdeterminism. So why even bring it up? Why not stick to your superdeterminism beliefs?

Hi Sabine

Your wonderful words..........And yet, physics isn't math. Physics is science and as such has the purpose of describing observations of natural phenomena. Yes, we use mathematics in physics, and plenty of that, as I'm sure you have noticed. But we do this not because we know the world is truly mathematics.It may be mathematics, but Platonism is a philosophical position, not a scientific one. You are free to believe that reality is math, but since this belief is scientifically indefensible I will not defend it. We also don't need it: Physicists use mathematics simply because it is useful. Reality may not be math, but it surely can be well described by math......

Same thing i expressed in my essay to.......

Best

=snp

The Navier-Stokes equation has already been mentioned. Thus we have the hydrodynamic regime: low energy excitations, long wavelengths. The matter density is a continuous function of space, which is of course not true. But in the hydrodynamic regime this is good enough; when we sit in the bath tub then we do not see molecules. Foundational physics should look at condensed matter physics; these systems are usually so complicated (mainly due to many particle interactions) that all models are only effective.

Hi Sabine,

I very much enjoyed reading your essay. I love love LOVE how you clarify the nature of physics (relating specifically to experiments and observations), as well as why mathematical theorems must be "taken with a grain of salt." Further, your discussion of the irrelevance of infinity (and even real numbers) to the real world was apt and led me to read your cited reference ("The physics of Infinity") which was fantastic. Also, your analysis and ultimate conclusion that math can indeed be helpful and relevant for figuring out aspects of physics was spot on.

Where I strongly disagree with you is your notion that the universe is ultimately predictable "given a sufficiently large and powerful computer." In fact, you seem to immediately contradict yourself in the very next sentence in which you admit that the world is not deterministic, but only to the extent of "occasional random interference from quantum events." I would argue that quantum superpositions, which govern the extent to which the universe is deterministic, are not pesky occasional features of the world, but rather ubiquitous and controlling. They are absolutely everywhere.

In my essay, on which I would be very honored by your comments, I argue that the universe is essentially a consistent set of facts whose history is embedded in correlations among entangled objects, and that a superposition is essentially the lack of a relevant fact. In that sense everything in the universe is constantly in superposition to some degree, and new facts constantly emerge to change or reduce those superpositions.

You identified chaos in your essay, but discounted the impact of quantum events. I would argue that not only are quantum events happening constantly, but natural/random amplifications cause chaotic systems to become chaotic much faster than without amplifications. If there was ever a case in which the weather on a particular day was correlated directly to a quantum event via chaotic amplification - a case that you must certainly agree has nonzero probability - then how can we discount quantum events in an analysis of determinism or predictability?

Your essay is excellent and thought-provoking and I wish you the best of luck on this contest!

Andrew Knight

Dear Sabine,

A novel, beautifully written and slightly irreverent set of valid insights as usual. But am I right feeling low ambition pervades it? Do you really think that aiming high has the same chance of hitting heights than aiming low?

Your description of physics purpose as "describing" observations seems just a mathematical view, but do you not consider "explaining" or "rationalizing" may lead to more advancement of understanding?

You say; "without scientific relevance", but do you see Godel etc as limiting ontology in any way as well as mathematics?

You write QM is "unpredictable by assumption not by theorem." I agree with Bell that some flawed assumption LED to that! I hope you may look and comment on the one I identify; That OAM has two REAL momenta states (linear, and, orthogonally, polar 'curl') which change inversely by Cos Lat over 90 degrees. Bells 'theorem' is then bypassed, as he predicted. Or is that aiming to high for you?!

You say "we can't just go and measure what's happening" behind event horizons, Yes in astronomy we've long studied AGN toroid dynamics (from emissions) and consistent hypotheses emerge; i.e. the atomic 'Mexican Hat' profile and polar jets replacing the mathematical 'singularity' Einstein agreed couldn't physically exist. Blackholes may then just run out of fuel as observations suggest (in FINITE time!). Does that aim to high?

Your comments on plasma instabilities are pertinent. Do they reveal some belief that unpredictability can be reduced by understanding?

I like that you agree reductionism should work in principle, and that "Practical use in not the only thing we care about."

Also your last line, maths matters as long as we rely on it to "understand" nature.

Good food for thought. But I hope you'll get to read and comment on my essay which suggests corrections to flaws in our deepest foundations, and identifies a Dirac "cut off" limit.

An enjoyable read as always Sabine. I hope it scores better this year.

Very best

Peter

Dear Sabina, it is difficult to argue with your mathematical essay. With the statement that they cannot control the thermonuclear reaction because of the inability to predict its progress at all stages large and small, I completely agree. We cannot do this because of the lack of a deeper understanding of the essence of matter at all stages of its behavior. I invite you to discuss my essay, in which I show the successes of the neocartesian generalization of modern physics, based on the identity of Descartes' space and matter: "The transformation of uncertainty into certainty. The relationship of the Lorentz factor with the probability density of states. And more from a new Cartesian generalization of modern physics. by Dizhechko Boris Semyonovich ". At the very beginning of the essay, I repeat twice the idea that rectilinear motion, in essence, is a motion around a circle of infinitely large radius and, if this radius is reduced, then in infinitesimal, the laws of motion of the theory of relativity will go over to the laws of quantum mechanics.

Next come mathematical formulas that only spoil my essay, but without them in any way. I will be pleased if you catch their main meaning and bless me for the further generalization of modern physics.

Sincerely, Dizhechko Boris Semyonovich.

Sabine:

Very well argued. Having studied CS Peirce for 30 years, you seem to be converging to many of his views. Now, you like many might not be very aware of his work but it's been spreading for a while.

Regardless, what you are getting across is very important. Thanks.

Sabine I do believe that math matters and I appreciate your interest in reduction and unpredictability. In my revised essay, I describe the lowest level of Successful Self Creation and how it converts chaos to order. This level is also the lowest level of reduction and it converts unpredictability to predictability. You may find this "level" interesting to consider in your research. Also, in the revised essay, I describe how the SSC processing produces and "incorporates" its own mathematics and algorithms into its processing and results. My essay gives credence to Tegmark's statement-" The Universe is Mathematics" and Lloyd's statement- "The Universe is a (Quantum) Computer. It also provides a "why" to your statement- "Mathematics has worked incredibly well for physicists". It shows why mathematics is a reliable tool to describe physics. I feel that my essay is pertinent to both your research and your essay. I would appreciate your comments. John Crowell

Dear Professor Hossenfelder,

Thank you for this highly original contribution to this contest, which reminded me of Wittgenstein's last philosophical thoughts, collected in "On Certainty", to the effect that most discussions take place within a 'container', but some discuss the container (which he calls the set of hinge propositions) itself. Your essay seems to belong to the latter category, since you challenge, quite rightfully, the whole subject of this contest.

I just don't think that your section 4 soes real justice to the reduction-emergence debate (you may deliberately avoid the latter term), which is so much wider than the context of effective QFT. For example, the emergence of the classical world from quantum theory (if only in theory) relies on the role of small perturbations, whose importance grows near the classical limit and/or in large systems and which make non-classical states collapse; this has nothing to do with RG or effective QFT arguments, and there are many other examples of emergence like that - if emergence seems to fail, you just have to look harder.

Apart from this, and the tiny criticism than in paraphrasing Gödel's Theorem you seem to conflate provability (which his theorems are about) with truth (which notion unnecessarily complicates matters, although admittedly your rendition is seen very often), I really enjoyed your essay, which will give me food for thought form some time to come. A relevant context which I like is "Earman's Principle" from the philosophy of science, which states that "`While idealizations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed.". Your claim seems to be that the three theorems about the "un's", which from the point of view of physics are indeed idealizations, do not satisfy this principle, but this remains to be seen. With the idealized assumptions weakened, these theorems may still make weaker but valid and interesting claims about the real world (or at least about theories thereof). It is often very challenging to find such weaker but de-idealized versions of mathematical theorems, but they typically exists, for example, in approximations to the law of large numbers. An example relevant to your essay in spontaneous symmetry breaking, which according to official idealized theory can only occur in infinite systems (which would be very puzzling since it is seen in many finite materials), but which in fact is foreshadowed already in finite systems, once treated correctly. I certainly believe that this also applies to Chaitin's Incompleteness Theorem which you mention (and I implicitly use this in my own contribution to this contest) and I presume there should also be weaker version's of Gödel's Theorems to the same effect.

Best wishes, Klaas Landsman

Hello Mr Crowell,

If I can, you cannot affirm this about a pure mathematical universe, but I respect your points of vue about this physicality and the philosophy correclated. Like Max Tegmark , you consider so a kind of pure mathematical universe.

I beleive that nobody can affirm the generality of this universe, we have just assumptions about the main cause. The problem is this one for me, the maths are very important and permit to prove our assumptions like experiments, so they are essential when we want to formalise, renormalise or quantize physical ideas if I can say, but these maths also can imply an ocean of odd extrapolations also due to mirrors or reversibilites of this or that, let s take for example these whormholes like mirrors of BHs in our GR, can we affirm they exists ? no, the same for the reversibility of this time or the multiverse, they are just mathematical conclusions not proved. We have many examples in fact about these maths. And if we go deeper in philosophy, what is the origin or main cause of this physicality ? the sciences Community is divided about this, a part consideres like you mathematical causes from a kind of probabilistic accident from a nothing or an energy ? others consider an infinite heat and so the BB and inflation and after they consider just photons oscillating with the strings and a 1D main Cosmic field and strings at this planck scale creating our geonetries, topologies, matters and properties with particles and fields, or others like me consider an infinite eternal consciousness creating a physicality with coded particles, 3D spheres for me and 3 E8 superimposed implying this reality.

But nobody can affirm in fact, we cannot reach these planck scales and this main philosophy, we have limitations implying so unpredictabilities, undecidabilities, uncomputabilities, maybe the only one wisdom is to recognise this. And in the same time we can be humble in respecting the philosophies of thinkers without insisting about things not proved.

Maybe in conclusion we must just consider the proved laws, axioms, equations and relativate our assumptions and extrapolations, mathematical, physical , philosophical.

Best Regards

Dear Sabine,

Thank you for your very good and well-written essay. I mostly agree with your main contributions. There is however one point that I have to disagree with which is expressed by your sentence: "Physics isn't math, and Godel's theorem is irrelevant for scientific practice". I agree that physics isn't math, but I think scientific practice, as opposed to a scientific, falsifiable theory, embodies more than things that have testable consequences. As a computer scientist, I think all my CS and math colleagues would agree that Gödel's theorem is indeed relevant for scientific practice in their respective fields, although not having any testable consequences. I think it would have been more correct to talk about "scientific practice in the physical sciences" instead of "scientific practice" in general.

Best regards,

Ruben