Very interesting essay! I agree that it's important to interrogate the nature of certainty in mathematics, given that many of us feel that it is somehow the last bastion of 'absolute' certainty. The idea that math (or perhaps a branch of math) should embrace uncertainty reminds me of Horgan's famous "Death of Proof" article in Scientific American, wherein he discusses alternative standards of proof or methods of proof. There are computer-generated proofs, probabilistic proofs, zero-knowledge proofs, and so on. Then, of course, there is the fact that a lot of published mathematics doesn't 100% spell out the details of a proof, e.g. things that are in principle 'easily checked' are sketched or glossed over. Finally, there are lots of statements mathematicians believe are true, and even use to prove other results (e.g. the generalized Riemann hypothesis is often used in number theory), but that have not been shown rigorously to be true.

The (possibly apocryphal) story about Hippasus I may not have heard before. Very funny.

You're not alone in wondering how it is that abstract ideas like numbers and addition map so perfectly onto the real world. I've wondered about it for a long time. Did it have to be this way? Because of the 'meta' nature of the question, it's very hard to even begin to come up with a sensible answer. It just seems to be a brute fact about the world.

I did not totally follow the section 'deriving' probability theory and quantum amplitude manipulation rules, but maybe I just need to go back over it or check the relevant papers.

I was pleased to learn about Tarski's theorem from your essay, a striking result in mathematical logic that seems to have been largely overshadowed by Godel's incompleteness theorems. Very interesting point about relationship between Tarki's theorem and probability theory (probability theory cannot specify what's true, so truth must come from some other source).

In closing, I'll say that mathematicians seem to be more comfortable these days with using heuristics and alternative means (e.g. numerical work) to help judge whether or not some claim is true. So to some extent, this kind of approach seems accepted. But it does not seem to me that these standards of proof will ever fully replace a 'rigorous' proof, whatever the issues may be with foundations and defining what 'rigorous' means.

Dear Satyavarapu,

Thank you for your kind comments and question.

Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is undecidable.

In science, we only entertain testable hypotheses. These are hypotheses that make predictions. And it is by making predictions, using our theories, that allow us to test them against experimental data by applying the weakest of the logical syllogisms listed in my essay. This is the process of inductive inference, also known as Bayesian inference. And it is possible that one can get around Godel's theorem in a way by assigning probabilities to hypotheses rather than truth values. Of course, nothing will be known with certainty. But this is the situation that we are very familiar with in the physical sciences.

My essay claims that mathematicians threw up their hands and gave up, whereas they (especially now) could re-visit these issues and consider applying inductive inference rather than deduction.

As an amusing aside, I was watching the TV show, Sherlock, about the famous fictional detective Sherlock Holmes, and in the show, Sherlock claims that he arrived at his conclusions via deduction. However, this is not the case. Sherlock routinely uses inductive inference rather than logical deduction.

I think that mathematicians should give it a go and try it too!

Last, thank you for pointing me to your essay. I look forward to reading it.

Sincerely,

Kevin Knuth

Hi Kevin,

I really enjoyed reading your essay and heard your talk as well in Vaxjo.

What do you think about the non-local box in this context? This is in the general probability theory (GPT). In this context, is the GBT final to resolve the Godel incompleteness theorem?

On the context of the Turing machine, my essay was briefly discussed.

Best wishes,

Yutaka

    Dear Luca

    Thank you very much for your comments and kind words.

    You note some potential typos or mistakes. Let me look at these first.

    > Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.

    You are correct. It should be:

    Given: If A is true, then B is true.

    Learn: B is false.

    Deduce: A is false

    > And similarly WS-2: Learn A false, then probably B false.

    Again, you are correct. It should be:

    Given: If A is true, then B is true.

    Learn: A is false.

    Infer: B is less plausible.

    > Finally on page 8. If u is undecidable and u follows from x1 and x2 and x3, than at least one of these must be undecidable. But the other also could be true. If one is undecidable then the conjunction also must be undecidable. (They do not to be true).

    I am not quite following what you are saying here. If u is undecidable and u is the disjunction (OR) of three atomic statements: u = x1 or x2 or x3, then

    at least one of x1, x2, and x3 must be undecidable. That you seem to agree with based on your following comment. So let's say that it is x1 that is undecidable. Then x2 and x3 must be either undecidable or false. If one of x2 and x3 were true, then we could deduce that u was true, and u would not be undecidable.

    Similar arguments apply to the compliment of u. And the result is that the existence of an undecidable statement u in the hypothesis space implies that at least two of the atomic statements are undecidable. Now it is not assumed that the atomic statements are true. The atomic statements are mutually exclusive and exhaustive so that one and only one of them is true. However, that true atomic statement must be one of the undecidable ones.

    > But aren't the atomic statement usually the axioms of a formal language? But aren't the axioms by definition true?

    The atomic statements are not necessarily the axioms of a formal language.

    I look forward to reading your essay. And that will probably help me to better understand your question.

    Later you ask:

    > Under which physical conditions is measure theory applicable? And the sum rule?

    It would be best for me to point you to one of our most recent papers on this topic:

    https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800057

    In short, one needs to have closure so that if you combine one set of pencils with another set of pencils, you get a set of pencils. The combination operation must be commutative and associative so that shuffling does not matter. And last, there can be no problem with continuing to combine things.

    But now, looking at your question, it appears that you are interested in physical properties. In terms of closure, this would come down to how we choose to classify things. Combining a set of pens with a set of pencils does not result in a set of pens. But if I choose to think of them as writing implements, then I have closure. So part of the applicability has to do with the choices we make when we classify things. I hope that this helps. Although, I expect that I will understand your question better after I read your essay.

    Thank you very much for pointing out my two typos. I really appreciate it.

    And I hope that I have, at least begun, to answer your questions.

    Sincerely,

    Kevin Knuth

    Hi Kevin,

    you are right. In the context of lattice theory, atomic statements are of the kind: this particle is at location x and so on. Atomic statements are composed through disjunction.

    I falsely have mistaken atomic statements as axioms in the context of undecidability. There theorems are derived from axioms by conjunction. If now a theorem is undecidable one of the axioms must be undecidable. In Gödel's case this would be the axiom non contradiction. Which is the second Gödel's theorem, that consistency is not decidable. But I don't know if my simplified argument applies here, since what is an axiom and a theorem is not uniquely defined. Contrary to atomic statements.

    I certainly will look up your recent papers, as your approach to derive probability from the underlying symmetries and logic is really interesting.

    Luca

    Dear Prof. Kevin Knuth

    Thank you for nice reply. Your excellent Knowledge cleared this doubt.

    You are the only person WHO CLEARED MY DOUBT!!!

    REQUEST YOU TO PLEASE LOOK AT MY ESSAY AND RATE IT...

    Best

    =snp

    Dear Professor Kevin nuth. Had a great insight in your well crafted essay,you raise pertinent philosophical issues well blended into mathematics. in particular your statistical analysis diagram and simplification via the Boolean lattice.very impressive.Rated you well. How about you see something simple I have submitted on cognitive bias-https://fqxi.org/community/forum/topic/3525. Thanks and all the Best in the essay contest.

      Greetings Professor Knuth,

      I have downloaded and begun to read your essay. Looks interesting. In my essay; I take almost the polar opposite tack. Recently; Giulio Tiozzo proved some very broad connections between the Mandelbrot Set and entropy, completing some of the last work of Thurston (with whom he collaborated). I have been exploring some specific connections of M to Physics for more than 30 years now, and the location I focus on in my essay has relevance for entropic theories of gravity.

      I'd appreciate your comments on my essay. I will be back once I have read your paper in its entirety.

      Warm Regards,

      Jonathan

        Dear Satyavarapu,

        You are very welcome!

        Sincerely,

        Kevin Knuth

        Thank you, Luca.

        It is not clear that the axioms are the atomic statements. So I do not think that one can use my arguments to say something, in general, about the axioms. In fact, it could very well be that the undecidable statement is an atomic statement.

        Thank you for your kind words about my approach to deriving probability from symmetries. If you have any questions about that work, please feel free to email me.

        Sincerely,

        Kevin Knuth

        Dear Cristi,

        Thank you for your kind words about my essay.

        Luca did indeed find some typos. Specifically, I wrote the two syllogisms wrong. It was a transcription error that I did not catch. Very unfortunate. :(

        I wish you the best in this contest as well!

        I have printed out your essay and I am looking forward to reading it.

        Thank you, again!

        Kevin

        Dear Michael,

        Thank you for your kind words.

        I am very glad that you appreciated my approach, especially the collapse of the Boolean lattice under the truth-falsity equivalence relation.

        I hope to get to reading your essay as well, especially since I am curious how you relate cognitive bias to undecidability and such.

        I wish you all the best in this contest.

        Sincerely,

        Kevin Knuth

        Dear Yutaka,

        I am very glad to see you here in this essay contest.

        It is a shame that the Vaxjo meeting had to be delayed this year. I do hope to go again next year.

        I do not quite understand what you are asking:

        >

        Perhaps you can reply with a more detailed explanation of your question.

        I wish you all the best in this essay contest and I hope to read your essay on the Turing machine.

        Sincerely,

        Kevin

        Dear Jonathan,

        Thank you for your comments. I look forward to your return and further comments.

        I am not aware of these connections between the Mandelbrot set and entropy. I will have to look into this as you have piqued my interest. And entropic theories of gravity are interesting in their own right, so I am eager to look at your essay as well.

        I wish you the best of luck in this essay contest.

        Thank you again,

        Kevin

        Thsnks Kevin,

        Tiozzo was a PhD student of C.T. McMullen at Harvard...

        He was set to work on a problem by Tan Lei, to make general a result involving local symmetry at Misiurewicz points against global asymmetry, which I think applies broadly to Physics. This led to his work with Thurston beofre Bill's demise.

        More later,

        Jonathan

        This essay is beautiful work Kevin...

        I am not in full agreement however. Some statements are only absolutely true if you rule out the possibility of hyper-dimensional super-determinism - an avenue I have been exploring of late. I have a paper in peer-review on "Painting, Baking, and non-Associative Algebra" that talks about forced ordering in higher-order Maths. Things must be done in proper order and sequence to work. The epic statement by Connes "noncommutative measure spaces evolve with time!" is amplified in non-associative geometry. But I digress.

        I loved what you wrote about counting and the 'why?' of addition. Children are seldom taught the profound difference between none and one of something. Then it naturally follows that as you add more units of the same item, the additive rule comes into play. That worked since ancient times, but has been forgotten. The result of Turing was also well-known for years in indigenous cultures, as documented by my friend Evan Pritchard in "No Word for Time." So I regard the halting problem and its generalizations as established fact. But I think perhaps Gödel's work is a near-miss.

        Still; I think it is remarkable that you reproduce some things that would be true even in a hyper-d super-d generalization, in what I see as a flattened form. This reminds me of comments in a lecture by John Klauder where he talked about QM as a projection from infinite-d Hilbert space onto a specific 2-d surface, and the possibility to lift the target space. So a lot of the evolutive properties from higher-d would automatically carry over as rules of inference in lower-d projections. So your paper is a tremendously fun read, when I put on my "Griff" cap (see MIB 3). I am wearing it now.

        For the record; I loved the idea on softening logical statements by Jaynes, and how you worked that in. I also greatly love the works by Polya which spell out a wonderful roadmap for learning about most anything. But I'll have to return with additional comments.

        More later,

        Jonathan

          Dear Prof. Kevin Knuth

          Hope you will read and rate my essay after getting your comments cleared by me soon....

          Let me also reciprocate later...

          best regards

          =snp

          Hope I didn't gush too much Kevin...

          But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might. I think I'll probably read this essay again for detail, even though I've already given my rating. Good luck in the contest.

          Best,

          JJD

          Dear Jonathan,

          Thank you so much for your kind words and the detail with which you discussed my essay.

          > But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might.

          I think that things are fundamentally simple. And that we add layers of complication, which is not always necessary.

          Thank you again!

          Kevin

          Dear Prof Kevin H Knuth

          This post is a discussion post after well knowledged reply given on May. 4, 2020 @ 07:55 GMT above ... Your nice words.........

          Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is undecidable.

          ............... Well said!

          Your nice words.........

          In science, we only entertain testable hypotheses. These are hypotheses that make predictions. And it is by making predictions, using our theories, that allow us to test them against experimental data by applying the weakest of the logical syllogisms listed in my essay. This is the process of inductive inference, also known as Bayesian inference. And it is possible that one can get around Godel's theorem in a way by assigning probabilities to hypotheses rather than truth values. Of course, nothing will be known with certainty. But this is the situation that we are very familiar with in the physical sciences.......................

          This process is different in Dynamic Universe Model, there Linear tensors are used, no differential and integral equations,.....

          Your nice words.........

          My essay claims that mathematicians threw up their hands and gave up, whereas they (especially now) could re-visit these issues and consider applying inductive inference rather than deduction.

          As an amusing aside, I was watching the TV show, Sherlock, about the famous fictional detective Sherlock Holmes, and in the show, Sherlock claims that he arrived at his conclusions via deduction. However, this is not the case. Sherlock routinely uses inductive inference rather than logical deduction.

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

          Wow very good, How to apply inductive inference..???

          Your nice words.........

          I think that mathematicians should give it a go and try it too!

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

          You can get the mathematics of my Model, in my blog

          " https://vaksdynamicuniversemodel.blogspot.com/ "

          You can have a general outlook of this model at my essay....

          " https://fqxi.org/community/forum/topic/3416 "

          Best Regards

          =snp