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?...

I am sorry, wrong link :) I meant my essay: https://fqxi.org/community/forum/topic/3436

(Although, Eric's essay is a good one!)

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

          John Hodge,

          Whether physics is fundamentally predictable or not is a question we will never be able to answer. The only thing we can tell is whether our currently best theories are or aren't predictable. What I argue in my essay is that even though the impossibility theorems cannot tell us anything about nature per se -- because we will never know whether the math is ultimately the right one -- they are important in practice simply because we use specific types of math.

          Thanks for your interest,

          Sabine

          Hi Flavio,

          Thanks for pointing out, I will have a look at your essay! With best regards,

          Sabine

          Wilhelmus,

          Thanks for your thoughtful comment. I largely agree except on your remark about the black hole horizon. A black hole horizon is a global property of space-time. It's a macroscopic feature and has nothing to do with the Planck scale. For large black holes the curvature at the horizon can be arbitrarily small. I will have a look at your essay. With best regards,

          Sabine

          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

          I saw above the question whether nature avails itself of a formalism. This is a misunderstanding. An electron does not calculate anything. A theory May be a representation of a data collection, in the algebraic sense. The objective is to obtain a sufficient level of isomorphism. Thus an effective model may represent the data, even when there is no one-to-one correspondence between the terms of the model with elements of reality. See e.g. the Bogoliubov and Froehlich transformations or the old fashioned normal mode transformations. Is a normal mode a physical object?

          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