Math Matters by Sabine Hossenfelder

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

Hey Sabine --

Thanks for the really fun essay.

A few years ago there was a fad for predicting critical collapse. Some of this was really good, but other pieces I thought had a big flaw: you could not predict a singularity using an analytic approximation! (More correctly, the a singularity means an inability to use prior data to "extrapolate through" the pole.) You can have signatures of a singularity to come, but only conditional on a good model.

I wonder how this might play out in your suggestion for steering clear of Lorentz horizons. In the plasma case I'm going to guess that our best models are phenomenological. The pattern recognition system they used had some even more phenomenological model yet. Knowing what patterns it was picking up on could tell us a lot about the underlying dynamics.

It reminds me of some simulations we did years ago to measure critical indices in phase transitions. Our systems were finite, so no real singularity. But one way to think about it was that there was a singularity, in the complex plane, and as N got larger (the simulation more accurate) it rotated around and got closer and closer to the real axis.

Simon

Simon

Dear Sabine Hossenfelder:

You note in the beginning of your essay that:

"Physicists use mathematics simply because it is useful. Reality may not be math, but it surely can be well described by math."

I must submit that math can and does obstruct -

especially when that math describes things not observed in real life that becomes the basis of physics - as in de Sitter's expanding space - Friedman's creation of the world from nothing - and Lemaître notation - "If the world has begun with a single quantum..." these all obscure applicable common 3D physics hiding the physics of the Big Bang.

It is proposed that any evidence describing the Big Bang is beyond science's reach and yet this essay of mine entered January 18th Common 3D Physics Depicts Universe Emerging From Chaos presents a plausible explanation with plenty of current replicable evidence describing 'Reality.' Check it out.

Regards

Charles Sven

Dear Sabine:

Excellent essay that clarifies: Physics is not math. But math matters.

Your essay is bold, powerful and correct:

(i) ".... physics isn't math. ....For this reason, the topic of this essay contest - undecidability, uncomputability, and unpredictability - sounds very academic indeed. Who cares whether a big-brained scientist proved that a certain mathematical problem is unsolvable if we can never know whether this math is fundamentally the correct description of nature?"

(ii) "Looking for patterns that allow us to express data in simpler ways is pretty much all that scientists have ever done." -So far.

In praise of math assisting the practical world of achieving inexhaustible energy:

(iii) Modeling and finding the onset of instability a few millisecond ahead of time, allows for the incorporation of corrective plasma parameters in a fusion plasma chamber. This helps scientists to prevent the instability by proactively readjusting the necessary parameters.

The discussion around "Reductionism" is excellent and succinct.

I have also read your book, "Lost in Math". I consider your book to be an extremely timely and valuable "social service" for the physics community.

Chandra

Dear Dr. Hossenfelder,

Thank you for your wonderful essay. The work was written in a style that was clear and approachable which is remarkable given the complexity of the subject.

If you claim there is no free will, then that is your choice.

If you claim that large-scale events are fully dependent on smaller scale events than the following points should be address:

A sound of 20 dB has an energy of 10^-10 Watts/m^3, which works out to around 3*10^13 J/m^3 = 1.9*10^6 J/m^3 which is far too low of an energy density for an atomic transition for an atomic volume around 10^-30 m^3. Sound within the human range of hearing exists, but cannot be explained on the atomic or molecular scales.

Laminar and turbulent flow of a liquid and the ordered and glass states of solids have the same wavelength and energy density issues.

Electrical resistance (mostly) following Ohm's law is difficult to explain with quantum mechanics.

Nuclear decay is independent of chemical state and temperature. If this smaller scale is the engine of atomic change, why does it not change rate as it powers or slows a reaction?

Perhaps the wavelength difference between a butterfly and a storm front are enough so that they are independent.

Sincerely,

Jeff Schmitz

I should first mention that I am "a fan" and that I am here and wrote an essay for this contest because you made me aware of this contest by mentioning it on your YouTube channel.

You wrote: "Nothing real is infinite, therefore the whole formulation of the problem is scientifically meaningless. In practice, we never need an algorithm that can correctly answer infinitely many questions."

Your criticism in this regard is entirely misplaced. Although infinity is certainly a valid target of criticism, the concept isn't obviously wholly spurious. (Personally, I prefer Feferman's "unfolding infinity.") For example, addition is an algorithm that works for any infinite class (i.e., every n m). Clearly, the infinite class includes numbers for which addition has never actually been verified to hold. But we take it on faith that the numbers do not get so large that addition ceases to work.

Speaking of faith, you wrote: "Science shouldn't rest on faith."

But it does. Or are you one of those true believers that claims that it rests on Truth? If so, I would like to introduce you to the Liar! (pun intended)

These are minor criticisms. Overall, as usual, I enjoyed hearing your thoughts and I generally agree with your conclusion. I think it is obvious that, at the very least, the impossibility theorems represent a limitation on the tools that we use. However, if you are arguing that the three un's (as you call them) are absolutely nothing more than limitations on the tools we use, then I do not think you sufficiently presented that argument.

Hi Sabine!

Wonderful essay! I love your point that "We may simply want to avoid situations in which it becomes unpredictable for us" since this is a very true and practical approach to many physical problems. I also wonder if you've considered that, even if we DID have all accurate information about things like the weather, the physical load it would take to compute would be too much to bear. Since there is a real, physical cost to computation, it could be that such systems might not be capable of even performing the computation needed to make a prediction. I talk about this a bit in my own essay, and I'd be very curious to hear your thought on it!

Cheers!

Alyssa

I always enjoy your essays, Sabine. Please keep writing! This one inspired me to enter this contest too.

In your signature style, you continue to emphasize two points physicists need to be continually reminded of: physics is not math, and experiment is the ultimate arbiter of truth in physics.

Some great turns of phrase I liked:

"If it's not a deadline that sets an end to your hesitation, then the heat death of the universe certainly will."

"They have never been deterred by not knowing whether what they aspire to is even possible, and hopefully they never will."

"And, looking at the literature on black hole collapse, I fear we may not answer this question in finite time either."

Excellent point about mathematical problems crucially dependent upon some kind of infinity not being very relevant to science. Another essay I read here (by Michael Kewming) made the similar point that a computer trying to solve a halting problem would halt eventually, due to physical limitations, the omnipresence of noise, its eventual degradation, etc.

Probably the one place where I have any substantive disagreement with you is about chaos, or unpredictability, or whatever you want to call it. In particular, I'm not sure I agree that linear = predictable. Maybe in a limited sense, e.g. if Laplace's demon knows the wave function of the universe at one time, and can perfectly solve the Schrodinger equation, then linearity means small inaccuracies in its initial knowledge do not blow up out of control. But in practice, we can only learn about the universe by making measurements of observables. Taking those measurements can affect the outcomes of future measurements in unexpected ways (see some of the recent literature on out-of-time-ordered correlators...which you may already be familiar with). In this sense, I think unpredictability is also a feature of quantum mechanical theories.

Very beautiful point about unpredictability in phenomenological theories being important, and that it can signal that interesting things are happening (e.g. hot plasma instabilities). Makes me think of how all kinds of singularities appear in physics (like particle bumps), but we never actually measure an infinitely large bump. The singularity is just a useful approximation.

Dear Sabine,

Excellent essay. I loved for example the incisive one-line bluntness of "Nothing real is infinite, therefore the whole formulation of the [halting] problem is scientifically meaningless," Hah! So much for all of our wordy pontificating in ever-so-many other essays!

Cheers,

Terry

Dear Sabine,

I really enjoyed reading your essay. Thank you so much for the point of "real butterfly effect". I did not consider it. On the Lorenz perspective, what do you think to reduce the computational cost in quantum-computing era? How much universality of "real butterfly effect" can we discuss?

Best wishes,

Yutaka

Dear Sabine Hossenfelder.

Great economy of thought and words

You state:

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

I beg to differ. Even physics has its own no-go theorems. Plus you concur that they can be a guide.

Indeed, what should worry us in the moment is the uncanny similarity if not parallelism between physics and math. Take for instance the Gödel second incompleteness theorem and what we may call its parallel in physics the Heisenberg uncertainty principle or the Landauer limit. For me, importance of the no-go theorems are not so much about their stated or otherwise tacit limitations to human knowledge (in math as well as in physics) as it is about making the unreasonable effectiveness between math and physics reasonable instead.

Secondly, you assert that "Nothing real is infinite"

But I consider that we may actually live in an infinite world. To tame this infinity we must then presume ourselves as minds to represent a particular norm (axioms) within the infinity. This will be analogous to how the ZF Axiomatic set theory must tame Russell's Paradox or how physically the Planck constant must tame the ultra-violet catastrophe.

Taming infinity remains a persistent problem. Modern physics has, for instance, a clear and present danger in the so-called vacuum catastrophe and then mathematics has, among others, declared what it terms the mass gap existence problem.

Doesnt it upset your argument that ironical modern mathematics is worrying about explaining some actual existence (the mass gap) while modern physics is worrying about specifying correctly the idea of "nothing" (the vacuum)?

Chidi Idika (forum topic: 3531)

* ironically (not ironical). Its a typo pls.

Dear Sabine,

Your essay, thoughts and conclusions are extremely important for finding a way to overcome the crisis of understanding in the philosophical basis of fundamental science. I have some questions and comments for discussion:

"Physics Isn't Math"

Yes, it's true. But mathematics is the "language of Nature." Physics without mathematics is dumb and blind. In no case should we oppose Nature (physics) and "the language of Nature" (Mathematics). Their ontological structure is identical.

"... but Platonism is a philosophical position, not a scientific one."

But Philosophy is the "mother of all sciences." The platonistic position of mathematicians intuitively connects mathematicians with the ontological roots of mathematics. Physicists must also "dig" to these single roots in order to "grasp" "matter" into the network.

"We still wouldn't know whether one day we'd find a theory more fundamental than the Standard Model, in which case the pendulum could swing back from unpredictable to predictable."

A more fundamental model can only be built on the basis of breakthrough ontological ideas. Therefore, Carlo Rovelli is right: Physics Needs Philosophy / Philosophy Needs Physics Carlo Rovelli among the list of questions posed the question "What is space." I believe that it is appropriate to recall the philosophical covenant of Paul Florensky: "We repeat: worldunderstanding is spaceunderstanding."

"Equations aren't everything, but as long as we rely on them to understand nature, math matters."

With the help of "equations" Nature cannot be understood. To understand is to "grasp the structure" (G. Gutner, "Ontology of Mathematical Discourse"). Any "formula" is "clippings" from the existence of the Universum as an holistic generating process. The holistic paradigm should come to the aid of the paradigm of the part dominating in fundamental science. Then the support in achieving an understanding of Nature and its language to the full extent will be included not in "equations" and "numbers", but in absolute (unconditional) forms of existence of matter (absolute, ultimate states). Physics and the modern information revolution are pushing this. Unfortunately, ontology has also been experiencing a crisis for a long time, therefore, new "crazy" ontological and dialectical ideas are needed.

You are doing a very big job in modern fundamental science. I wish you success!

With kind regards,

Vladimir Rogozhin

Dear Prof. Hossenfelder,

Your essay is sharply written which makes it pleasure to read. I would like to focus on the 'uncomputability' aspect since my essay has this word in the title. In connection to it, you say among others that in science we really don't deal with real numbers which of course is true. But the problem is that real numbers are indispensable for theory, QM breaks up without them, QM based only on rationals is impossible. One can brush this away as an artifical issue which would be unscientific, or treat it as a signal that something deeper might be lurking there. Thinking probably along these lines Tegmark proposed radical thesis that physics is just a mathematical structure but this is seen more as belief since there are no constructive arguments for supporting it. In my essay I am sketching a constructive way via uncomputability in the form of uncomputable sequences which are giving rise to the emergence of mathematical structures due to the action of permutation groups. The groups are enormous, uncountably and countably infinite ones which provides headroom for extremely complex structures and physics could be one among them.

Best regards,

Irek Defée