Hi Stefan,

We never use the term "flaw". The concept you are getting at may be the (very well understood) concept of "consistency" in formal systems.

We are very careful; we only (start to) investigate what happens if the probability distribution of mathematics is not a delta function.

Hi

Interesting, thought provoking essay. The very idea, that mathematics could be wrong in some way, is very unusual and disturbing to me. Also Roman V Yampolskiy in his interesting essay in this contest has the view of mathematics as an experimental science.

What is not completely clear to me in your essay is what the 'laws of mathematics' are. Are these merely the for building WWFs or the ones for building Theorems or both? Are the axioms themselves noise free? True by definition. But might the rules maybe wrong or merely the applications of the rules? By humans or machines.

I also have some epistemological difficulties: if the proof of a theorem might be wrong (because human or machines make mistakes or because mathematics itself is stochastic) then there is no way to ascertain, whether the proof is wrong or not, because we should know, whether the proof was successful (proofed a true theorem) which we don't know, because we need the proof to be true to know, that the theorem is true.

The point is, we have learnt that mathematics is true a priori. We cannot think a world, where it is not true.

Contradicting myself I can imagine a few situations, where math might be noisy:

1. Brouwers intuitionism: all is build basically build from the operation of counting. But this operation might fail, if I start to count to many marbles so that they collapse in a black hole. Or I have to little memory to store the past counted numbers, so that different higher numbers become indistinguishable. The point here is, mathematics might be the structure of possible operations, but that these operations might fail under specific physical conditions.

2. Similarly in my essay I explore the possibility, that the universe might be the realization of specific MUHs, which I call Semantically Closed Theories. In these theories all quantities are operationally definable within the theory under specific physical conditions. One of these condition is that objects and subsystems can be defined, which are separable from the environment. But these condition is in fact always only an approximation. Hence there are limits on the accuracy under which operations/measurements can successfully be made.

Sorry for the lengthy reply, but I found your essay inspiring and I needed to advertise my essay a bit.

Luca

    Hi Luca,

    Thankful for your probing comments on our essay! A few brief responses.

    1) I would not want to say that we explore the possibility of mathematics being wrong in our essay so much as we explore the possibility of it being stochastic, though this interpretation is admittedly invited by our calling worlds in which there is no stochasticity in math "mistake-free". I also note that disturbing as it may be, we are careful to say that we are just exploring the consequences of the idea that mathematics may be fundamentally stochastic; we are not arguing for the view that this is correct.

    2) I would say that there is no exact analog to "the laws of mathematics" in our framework. There are just formal systems that partially define a particular NDR-world, and are applied stochastically to generate particular assignments of syntactic values to particular strings. There are also the answer distributions that partially define an NDR world. Neither of these can have the property of being correct or incorrect, although one such NDR world is actual.

    3) I believe that it is indeed an implication of our view that one could not have the kind of mathematical knowledge that you describe above; one can't know for sure which NDR world one is in.

    4) Your claim "we have learnt that mathematics is true a priori. We cannot think a world, where it is not true" begs the question against the view that we explore here. We are trying to see what happens when one drops this assumption, and considers a space of worlds in which there are mathematical facts other than those that hold in the actual world.

    I will read the essays that you link to with interest.

    Best wishes,

    David

    Hi David,

    thanks for your reply.

    I think your investigation is generically analogous to the framework of the many-world interpretation in QM. If mathematics is stochastic, its truth values aren't pre-determined, but somehow emerge from an underlying "math foam". If the MUH is taken seriously, "math foam" may be identical with "quantum foam". In any case, whatever is driving that stochastic dynamics, it is analogous to QM interpretations that do not embrace local realism, since "measurements" of mathematical truths are fundamentally stochastic.

    One step further the question arises to me if that whole NDR machinery was there for eternity or had some origins in time. In other words, the interpretational question whether mathematical systems - and with it in the framework of the MUH also all physical appearances - are just time-dependent phenomena or are eternally existing, ever-changing patterns. In my last two essays I argue for the former and I would be happy if you had the time and interest to read them and leave an opinion about it at my current essay page - that would be great! Especially your opinion about my use of abduction in these two essays would be interesting to me.

    Best wishes

    Stefan

    Very interesting model! It's a good point to think about math as a "fundamentally stochastic enterprise". I wonder what your thoughts are on a few things:

    1. How do you think the "stochastic-ness" of this process relates to an incomplete view of the world? Would it be possible to "bake-in" a partial view of the mechanisms that generate data?

    2. Do you think the mechanisms that generate data (data that is used to build mathematical laws) are inherently noisy on all levels of organization within a system? How would this degree of noisiness attribute to a mathematician's ability to make a Bayesian update?

    3. Do you think mathematics is inherently stochastic because the world is stochastic, or because our limited view of it is? And to what degree do you think these have on our ability to create claims that are mistake-free?

    A very excellent read! I'd be really interested to see what your thoughts are on the ideas I presented in my essay, since it centers more on utilizing state spaces for a Turing Machine to operate in, rather than the mechanisms of Turing machines.

    Cheers!

    Alyssa

      Dear Professor David Wolpert and Professor David Kinney,

      Given my knowledge of mathematics, logic and related subjects is confined to undergraduate level certain technical aspects of your essay was beyond me.

      However, the idea of actually modelling a community of mathematicians as a special type of probabilistic Turing Machine is deeply creative!

      In particular, the following conclusion, though I openly admit I could not follow every step of your reasoning, struck me as profound: "Thus, our augmented version of the MUH allows for the possibility that mathematical and physical reality are both fundamentally stochastic".

      In my essay I resorted to the use of MUH ( with a twist, MUH was not the ensemble of all universes, but all the possible mathematical models we have at our disposal), and since MUH according to Tegmark did not permit intrinsic stochastic laws of physics, I ( along with my co-author) came to the conclusion if Nature is truly random it posits a fundamental barrier to satisfactory mathematical modeling and representation.

      One thing in which your essay did not provide comment on what noise reduction. If mathematics suffers from noise due to the physical bounds of the system producing can we engage in noise reduction? Do you have in mind any analogues process to something like Noisey-channel coding theorem Shanon has for information theory?

      Kind Regards,

      Raiyan Reza

        Very interesting essay! It's pretty mind-bendy to try to mathematically model the totality of human mathematical modeling. Neat to see a stab at formalizing the error-prone, human-driven practice of mathematics.

        A couple technical nitpicks. I'm not sure how justifiable "sequential information source" property is, but it may be that I just misunderstand. The net effect of it looks like it makes your stochastic process into a Markov/'memoryless' process, which is a pretty strong assumption.

        Also, is it safe to assume that the claims probability distribution converges in the many iteration limit? In noisy gene regulatory biology, where probability distributions completely characterize stochastic systems, it is common to observe oscillatory behavior in the long-time limit that prevents P_inf from being mathematically well-defined.

        Here's an admittedly kind of dumb situation where I think P_inf wouldn't be well-defined, just to illustrate that it's conceivable. Imagine a universe (perhaps a desert island) consisting of people that hatch out of eggs, one at a time. Only one human/mathematician lives at a time. When they die, the next egg hatches, and another human/mathematician pops out. On this island is a computer whose screen displays, forever, a single mathematical claim: "P is ____". At one time, it says "P is true". The human alive at this time unreservedly believes the claim, so the mathematical knowledge of the world at this time is just that "P is true" with 100% belief. Unfortunately, the computer is glitchy, and happens to switch "true" for "false" and vice versa every time a mathematician dies. The next mathematician believes "P is false" unreservedly. The next the opposite, and so on. Here, there is no convergence, except possibly in the long time limit where everyone is dead, and there is no mathematical knowledge anymore.

        Now for some more philosophical questions. I think there's an underlying semantic issue about what you 'mean' by mathematics. Full disclosure, I'm on the side that mathematical truth is independent of human mathematicians, and can thought to 'exist' in some Platonic sense. I think well-formed claims within well-defined axiomatic systems are either true, false, or undecidable, and that this is completely independent of our reasoning abilities. If you believe this to be true, mathematics is not stochastic, but our confidence in results (especially controversial results with very complicated proofs, like the classification of finite simple groups or more recently the abc conjecture) and choice of what problems to work on certainly is.

        I find it hard to make the leap that, because human mathematical reasoners are fallible, and cannot be completely confident in every "proved" mathematical result, physical universes equivalent to formal systems must be stochastic. Again, while our reasoning about these systems is certainly prone to error, it's much harder to imagine the universe itself is prone to error when computing the consequences of its own rules. What would that even look like? Would it be possible to 'observe' the universe making a mistake?

        Still, regardless of where my beliefs might differ, I admit that it's logically possible that some scenario like this could be true, even if it's hard to imagine. Very thought-provoking essay overall.

        Dear David,

        This is very interesting approach. I would like to clarify the relationship or the difference to the concept of "Approximate Bayesian computation (ABC)". Is there any relevance?

        Also, on the philosophy of the Baysianism, we assume the probabilistic description. However, is this natural? On the computational viewpoint, the probabilistic description is too difficult to be implemented as seen in my essay for the reference. What do you think about the philosophy of the natural computing?

        Best wishes,

        Yutaka

        Hi Alyssa,

        Thank you for taking the time to read our essay and to ask such interesting questions. Here are some initial thoughts on each of your comments in turn:

        1. One of the foundational assumptions of our paper is that the noise in an NDR world cannot be fully explained by any agent's partial view of the mathematical universe. Rather, said noisiness is intended to be an observer-independent feature of that NDR world.

        2. You will note that we don't provide any mechanistic explanation for why a given NDR world is stochastic (i.e. not mistake free). As such, I don't think that what we say in the essay can vitiate as to whether mathematical data is necessarily noisy at every level of abstraction, if said levels are to be defined in terms of data-generating mechanisms. However, in extensions of our framework it might be possible to say interesting things about what kinds of answer distributions permit mistake-free coarsenings, and which ones do not.

        3. I'd say my answer on this point is very similar to my answer to my answer on point 1. However, one caveat: we do not do anything in our essay to argue that mathematics is fundamentally stochastic in the way that we describe. Rather, we try to explore the implications of assuming that mathematics has this property.

        I will read your essay with interest!

        Best wishes,

        David

        Hi proffesor.i admire your line of thought on how humans build maths. very incisive rated you accordingly.is it all emergent from cognitive bias as I have discussed in my simple essay here https://fqxi.org/community/forum/topic/3525.pls read/rate.all the best in the contest.

        Dear Sir,

        You say: "Humans are imperfect reasoners". Since no one is perfect - having all the required or desirable elements, qualities, or characteristics; as good as it is possible to be, your statement is correct. But it does not prove that everything about human reasoning is imperfect. If that is so, then your essay itself is imperfect and need not be taken seriously.

        You further say: "In particular, humans are imperfect mathematical reasoners". This implies, all of mathematics is wrong. And this begs the question: "What is mathematics"? The validity of a mathematical statement is judged by its logical consistency. When the equations in dynamical systems predict something, innumerable experiments show that it is correct. Do you mean to say, the landing of space crafts on Moon or Mars, which was based on these equations, a hoax? If "They are fallible, with a non-zero probability of making a mistake in any step of their reasoning", then the space crafts cannot land. Because even a slightest mistake would take it miles apart. If "This means that there is a nonzero probability that any conclusion that they come to is mistaken. This is true no matter how convinced they are of that conclusion", then nothing based on mathematical modelling should work. But this is contrary to observation.

        If "Since individual mathematicians are imperfect reasoners, the entire community of working mathematicians must also be one big, imperfect reasoner. This implies that there must be nonzero probability of a mistake in every conclusion that mathematicians have ever reached", then all that you teach or the essays or papers you publish, are not worth reading, as they are imperfect. And if something imperfect works, then the definition of your "imperfect" needs to change.

        Stochastic refers to a randomly determined process. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. In physics as well as in mathematics, there is nothing as random. There is a deeper order behind the seemingly chaotic system. If you could not see that, it is your fault - not that of the system. If one person is blind, it does not mean that the human race is blind.

        Your example of "a collection of human brains" is a laughable proposition. It is "a collection of ideas" and not human brains, which are like computer hardware. Without the right software, it can't function. Since as per your principle, your paper is imperfect, I need not read more.

        Please do not take it personally,

        With regards,

        basudeba

          7 days later

          Dear David H. Wolpert and David Kinney!

          Thank you for your interesting essay. We have specific questions. Whether the metric of space-time is non-ideal? Can we say that the distances between points are variable and change stochastically? Is a perfect ball possible in mathematics? Or it has bumps in random places.

          Pavel Poluian and Dmitry Lichargin,

          Siberian Federal University.

            Dear David and David,

            Thank you for writing this enjoyable and original essay. I was wondering how your approach relates to that of Intuitionism -- if it does at all. In particular I wonder whether one could construct some map between the two approaches. If this were the case, this might helpful for your formalism (I believe), as you may be able to import results from this other more-studied field of logic. I don't know if there is such a map, but I feel it may be the case.

            Thanks again for your inspiring essay, and best regards,

            Gemma

              Hi Raiyan,

              Thank you for taking the time to read our paper, and for your response. We have not considered, as far as I am aware, whether noice-reduction techniques could be brought to bear on what we're doing here, but it's certainly an interesting question!

              Best wishes,

              David

              Dear Pavel and Dmitry,

              Thank you for reading our essay. I believe that, on our approach, the particular form of the metric of spacetime could be thought of as being generated via sampling from a probability distribution over possible metrics, rather than as being metaphysically necessary. Similarly, in the actual world, a perfect ball could be possible or impossible depending on the outcome of sampling from a probability distribution.

              However, once we have fixed a given world as the actual one, the mathematical fact of that world, although they may be generated via random sampling, are not subject to change.

              Best wishes,

              David

              Dear Gemma,

              Thank you for your kind words about our essay. Indeed, we are very interested in connections between our approach here and various approaches to philosophical logic, and hope to develop those connections more fully in future work.

              For now, let me say that one aspect of our approach is that whether the law of the excluded middle is a theorem can, on our approach, be a stochastic matter. So whether logic is intuitionistic or not is determined by sampling from a probability distribution.

              Best wishes,

              David

              Dear Basudeba,

              Thank you for taking the time to read our essay. Certainly I do not take your criticism of its contents personally, but I do believe that at least some of what you say is based on misunderstandings.

              For instance, I do not believe that the success of any feat of science or engineering implies that there was a non-zero probability of any reasoning step used to achieve that feat being mistaken to some degree. Further, I do not believe that a publication's having positive probability of being wrong (a property, I would argue, that is possessed by all such publications) means that it should be treated as epistemically worthless, unless one is willing to accept a nihilistic epistemology.

              Best wishes,

              David

              I always encounter something new whenever I read David Wolpert.

              Who would have thought of considering the field of mathematics itself as stochastic except for the two Davids?

              What I was left wondering were what were the implications of such a view for science more generally? Might this lens of collective computation be applicable to other disciplines? Might we in some way be able to map and measure the "shape" of a particular field and even identify where breakthrough might lie because some question was not deemed important by larger communities?

              Great essay!

              Thank you for writing it!

              Rick Searle