"...applying instead to wide classes of theorems." should have been "applying instead to wide classes of formalisms.", sorry.

Jochen,

Thanks for your feedback which is much appreciated. However, you are mixing together two separate things that I discussed separately. Goedel-like theorems are not relevant to scientific practice because of their axiomatic structure. Theorems that rely on something being infinite and so result in something being uncomputable or undedidable are not relevant for science because they don't apply to the real world.

"While it's true that, no matter how much we like a certain mathematical formalism, this doesn't tell us anything regarding whether nature actually avails herself of that formalism, perhaps much to our chagrin, the theorems of Gödel, Turing et al are not limited to some formalism, but rather, are metamathematical theorems applying instead to wide classes of theorems."

I don't know what you mean by that. These theorems apply to classes of axiomatic systems. I am saying that in physics we can never tell whether the axioms are actually the correct ones, so why worry about it.

"Furthermore, I'm not quite sure if I agree with you that these questions really only apply in infinitary contexts."

This isn't a matter of opinion. Just look at the derivation.

I am happy to hear you liked my essay. With best wishes,

Sabine

Sabine,

thanks for your quick reply. The point I was making with my example above, however, is that while we may never be able to 'tell which axioms are actually the correct ones', we may be able to tell what class of axiomatic system is best in line with the data we observe, and whether that is a system to which the limitative theorems apply.

As I showed, it may very well be the case that we observe finite data and find a law such that the most reasonable explanation is that the underlying function is noncomputable. Sure: it could be that the agent above is wrong, and that their success in predicting future observations is accidental, or that they have not yet found the true fundamental law; but such is all the certainty we get in science. Quantum mechanics' predictive successes could be accidental, or there could be an underlying deterministic theory; then again, there might not be, and as our failure to find one drags on, it becomes more and more reasonable to believe so.

Cheers

Jochen

Well Said Sabine,

...................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...........

What we do is describing Physics and nature by using mathematics. Some setup is more successful in explaining it mathematically, some setup cannot. It is all depends on true open thinking oppurtunity available to the researcher is it not?

I just elaborated what should be the freedom available to an author when the " real open thinking" is supported. Have a look at my essay please.

"A properly deciding, Computing and Predicting new theory's Philosophy"

=snp.gupta

    Dear Sabine Hossenfelder,

    i rate your essay as the best one in the contest so far. Perfect on all points.

    Regarding the weatherforecast and Navier Stokes: Golbal Temperatur is an intense property in physics. So a physicist knows we can't measure average temperature with many measurements at different places at same time or with many measurements at different times but a single place. Both does not tell us anything about the Global temperature.

    How to measure Global temperature in physics?

    We know that if earth gets hotter, within the CO2 circulation more of the distribution should be in light atmosphere than in water, as if oceans get warmer the CO2 level in water decreases while in atmosphere rises.

    So to measure the temperature of a living planet earth we measure with the Keeling curve at Mauna Loa : https://de.wikipedia.org/wiki/Messstation_Mauna_Loa

    So an intense property in Physics we can only measure indirect. As far as i identified the key problem of the Measurement-Problem it is the detailed reasoning for space, time, velocity and speed at what stage where and how we have to deal with intensive and extensive quantities of space-time plasma to fuse mathematics- with physics in a new way.

    Best regards and much thanks for the Essay.

    Manfred U.E. Pohl

      Sure. My example just shows that one can achieve the same degree of provisional certainty with respect to issues of uncomputability, even given finite observations, and that thus, it's not quite right to say that 'we will never know whether the three un's apply to nature on the most fundamental level'; we can know it, with exactly the same certainty we can know anything in science, at least in certain cases.

      Dear Sabine, I happen to disagree with most of your essay's statements right form the outset. For instance: 'Godel taught us that mathematics is incomplete' I would say that's an overstatement and that rather, his results proved unambiguously that mathematics (or rather its particular branches such as geometry, algebra, topology, etc) can never be grounded on formal, symbolic logic, as per Hilbert's' axiomatic set theory foundations program, successfully completed by him in Euclidean geometry in 1899. Of course Godel results did not stop Kolmogorov in 1933 (just 2 years after Godel's paper) to give theory of probability an axiomatic foundation, or Leonard Savage to mathematical statistics, for that matter, just to pick two example of the top of my head, right?...Why do you think that is?...

      Simply, in my opinion, because there is heuristic power in the formal axiomatic method, not because it's the ultimate principle of reality.

      Here's another statement in your essay that seems paradoxical to me: 'Yes, we use mathematics in physics, and plenty of that, as I'm sure you've 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.'

      In other words you're saying that Platonism, Aristotelianism, Cartesianism, Kantianism, Hegelianism, etc. played no role in the history of human mind and its quest for scientific knowledge and certainty?...not to mention other great Greek minds from Thales, Anaximander,Euclid, Democritus to Archimedes and Potelemues. Obviously, in your opinion, philosophy has nothing to do with science as we know it today even though, say, Descartes and Leibniz were primarily philosophers and only second, mathematicians or scientist, right?...

      Even closer to out time and in direct connection with our theme, Godel himself was a philosophical minded person (logic has been part of philosophy for millenia at least until Boole, Frege and Russel extended it into the mathematical real), a logical positivist and a member of Vienna Circle in his youth and a well-known Platonist later on in life so if Platonism is not science then his results might also appear to be non-scientific, or come totally against his philosophical beliefs, right?...Besides, is there such a thing as Platonism?...If so, what would that be?...Is it just a dusted label of a rich and powerful philosophy of the Greek culture at its peak that has been continuously influencing mankind ever since or just a non-scientific and irrelevant view of the world that must be abandoned to oblivion?...I think Godel will disagree with the latter part remark...Will you?

      All the best,

      Mihai Panoschi

        "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

          Dear Manfred Pohl,

          Thanks for this interesting remark, I will keep it in mind! With best regards,

          Sabine

          "In other words you're saying that Platonism, Aristotelianism, Cartesianism, Kantianism, Hegelianism, etc. played no role in the history of human mind and its quest for scientific knowledge and certainty?"

          I didn't say anything like that.

          As to your picking on my introductory sentence. I know what the theorem says. That's a brief summary of the gist of it and not a formal statement.

          Thanks for the kind words. I will have a look at your essay. With best wishes,

          Sabine

          "... 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?...

              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