Math Matters by Sabine Hossenfelder

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)

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

Dear Sabine,

Thank you for writing this enjoyable essay.

I agree with you that physics isn't math. This implies many things. One is that all uses of infinities in physics are mathematical constructs. This applies to so many domains in physics: the definition of pure states in quantum mechanics, these exercises where we are given two infinite plates and have to compute the electric field among them, particles coming from infinity to compute the scattering matrix, the very definition of a ground state as occuring at zero temperature (1/T --> infinity), phase transitions occuring at the thermodynamic limit in statistical mechanics... And the entire link between physical problems and computational complexity problems. This includes the "physical" problems which have recently been proven undecidable, like the spectral gap and others. (In particular, (the model of) a physical problem can be undecidable only if the number of instances is infinte.) But it also applies to essentially all computational complexity statements that are made of (models of) physical problems.

Namely, what is value of proving that the ground state energy problem of the Ising model is NP-complete? This result shows that finding an algorithm that solves this problem *in infinitely many cases* is among the hardest problem for a non-deterministic polynomial time machine. From a physical point of view, this is rather irrelevant, because there are no infinite number of cases. Yet, this also shows that finding an efficient algorithm to solve this problem is equivalent to solving one of the biggest open questions of our time, namely P=NP. So this is, first, a warning of the difficulty of the enterprise ahead - if one were to attempt to solve this. And second, it says something about the many facets of the conjecture P=/=NP. But I don't think it says something very deep about the ground state problem itself, only about the model (with infinitely many cases) of this problem.

Sorry for the long text. I just wanted to ask whether you share this point of view.

Thanks again for the essay, and best regards,

Gemma

Hi Sabine,

Very nice reading experience. I had a niggling issue that followed me throughout:

The essay seemed to present a view of science as essentially a black box; we don't know why it works, but it often keeps giving us numbers that match our experiments.

But a philosopher might wonder why it is that "maths works" and why reality can be so well described by it. Why do our "observations of natural phenomena" fit? You don't give any clues to this and on the whole it reads as quite a stark instrumentalism. Obviously, one of the reasons behind treating physics as math in some sense is precisely this (unreasonable) effectiveness: there seems to be be a structural mapping, so facts about one apply to the other. The fit seems to give us a *reason* that "this math is fundamentally the correct description of nature".

Moreover, when you write "mathematical theorems are not of much use for the messy science of real-world systems", this ignores the fact that much (if not most) of the science we have of real-world systems, originated in some mathematical theorems - e.g. the steps from differential geometry to GPS or analytical mechanics to holonomic robotics. So, even if only to discredit it, I think you have to say something about the business of why math works when considering the kinds of issue you discuss in order to show that such things as mathematical theorems are relevant or not.

Best

Dean

Sabine,

Turing had different ideas on Undecidability, in particularly in his computer search for counterexample for Riemann problem ,he used generalized definition where Undecidability is merely a part of some UUU - computational construction.

My essay attempts to reconstruct such sort of approach,also. May be it could be interesting for further development of your provocative idea.

Respectfully

Michael Popov

I very much enjoyed your essay (as indeed I enjoy all of your writing!). Godel's incompleteness theorem and other similar results do tend to get brought up in many contexts in physics, often in fairly dubious ways, and so what you have done here by taking a step back and examining the actual practical consequences of these results for physics is extremely useful, thank you!

I did have one minor question - at one point you suggest that '(if the renormalisation flow ran into a singular point) reductionism would be demonstrably false, and other scientific disciplines would count as equally fundamental as physics.' Is it your intention here to suggest that physics is inextricably tied to reductionism, and that non-reductionist approaches (e.g. top-down causation) wouldn't count as physics at all? Or is this just a reference to the fact that our current best physical theories are reductionist in character?

"And really all of science is just a sloppy version of physics."

That was my favorite line. You remain one of the best contrarians in the business. Excellent essay.

I'm purely speculating here, but I would bet that Sabine is simply referencing the fact that our current best physical theories are reductionist. I think if she were presented with a sufficiently well-supported downwardly causal theory, she would probably accept it. But that's just my gut feeling...

I'm often torn on this. I do think that this "unreasonable effectiveness" of mathematics in the sciences is something that is worthy of an explanation. And as someone whose PhD is in pure mathematics, I am inclined to be sympathetic to mathematical explanations. That said, the fact that I also possess a degree in engineering tells me that there's an awful lot that we can do without questioning why it works.