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

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