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FQXi FORUM

October 29, 2020

CATEGORY:
Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020)
[back]

TOPIC: Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes by David H. Wolpert [refresh]

TOPIC: Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes by David H. Wolpert [refresh]

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the role of abductive reasoning in mathematics, 3) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, 4) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes, and 5) the prior distribution that a Bayes rational mathematician ought to have over possible mathematical systems. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

David Wolpert is a professor at the Santa Fe Institute, external faculty at the Complexity Science Hub in Vienna, and adjunct professor at ASU. He is the author of three books (and co-editor of several more), over 200 papers, has three patents, is an associate editor at over half a dozen journals, has received numerous awards, and is a fellow of the IEEE. David Kinney is an Omidyar Postdoctoral Fellow at the Santa Fe Institute. He received his PhD in Philosophy in 2019 from the London School of Economics. His work focuses on formal epistemology and philosophy of science.

Professors Wolpert and Kinney:

I’m pleased to read your summation of the possible weakness of math ending with:

“Following in that spirit of weakening assumptions, here we have aimed to demonstrate the potential fruitfulness of weakening the assumption that mathematics itself is fully deterministic. We believe that this reveals a rich landscape of novel results and subtleties, many still waiting to be uncovered.”

Current common 3D physics observed without resorting to mathematical assumptions as a Richard Feynman study is all one needs to understand creation. See my essay entered January 18th.–*Common 3D Physics Depicts Universe Emerging From Chaos*.

Regards

Charles Sven

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I’m pleased to read your summation of the possible weakness of math ending with:

“Following in that spirit of weakening assumptions, here we have aimed to demonstrate the potential fruitfulness of weakening the assumption that mathematics itself is fully deterministic. We believe that this reveals a rich landscape of novel results and subtleties, many still waiting to be uncovered.”

Current common 3D physics observed without resorting to mathematical assumptions as a Richard Feynman study is all one needs to understand creation. See my essay entered January 18th.–

Regards

Charles Sven

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Dear prof. Wolpert

I ask for your opinion.

Is the solution for describing the universe in discovered mathematics or invented mathematics or both?

Regards,

Branko

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I ask for your opinion.

Is the solution for describing the universe in discovered mathematics or invented mathematics or both?

Regards,

Branko

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Hi Branko,

As the phrase goes, we "have no dog in that fight".

Certainly not without more precise definitions of terms than philosophers of mathematics have managed to construct in thousands of years of trying.

David W.

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As the phrase goes, we "have no dog in that fight".

Certainly not without more precise definitions of terms than philosophers of mathematics have managed to construct in thousands of years of trying.

David W.

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Dear David & David,

thanks for a thought-provoking essay.

Nonetheless, probabilistic mathematics seems to pose the question of how probable it is that mathematics is indeed probabilistic, independent of some human considerations or claims about that question. If your claim is true, then this claim has a probability of 1 for being true and, thus, not all of mathematics can be...

view entire post

thanks for a thought-provoking essay.

Nonetheless, probabilistic mathematics seems to pose the question of how probable it is that mathematics is indeed probabilistic, independent of some human considerations or claims about that question. If your claim is true, then this claim has a probability of 1 for being true and, thus, not all of mathematics can be...

view entire post

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Dear Stefan,

Thanks for your thoughtful comment.

1) Just to emphasize, we are careful NOT to make any claim. We just raise a possibility.

2) In particular, we do not claim to "prove" that mathematics is inherently stochastic. So any stochasticity that is in mathematics would not somehow cause our paper to "self-destruct in a poof of logic", in the famous of phrase,

3) The issue you raise is actually endemic to foundational results in all of mathematics. For example, Godel uses standard arithmetic to prove his incompleteness theorems, and therefore (very loosely speaking) may be using logically inconsistent reasoning.

4) Less profoundly, current human beings *are* subject to mistakes. There is both nonzero probability of a flaw in every equation we write (e.g., the ones concerning abduction), and nonzero probability of a flaw in every equation in mathematics textbooks. No paradox.

David W.

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Thanks for your thoughtful comment.

1) Just to emphasize, we are careful NOT to make any claim. We just raise a possibility.

2) In particular, we do not claim to "prove" that mathematics is inherently stochastic. So any stochasticity that is in mathematics would not somehow cause our paper to "self-destruct in a poof of logic", in the famous of phrase,

3) The issue you raise is actually endemic to foundational results in all of mathematics. For example, Godel uses standard arithmetic to prove his incompleteness theorems, and therefore (very loosely speaking) may be using logically inconsistent reasoning.

4) Less profoundly, current human beings *are* subject to mistakes. There is both nonzero probability of a flaw in every equation we write (e.g., the ones concerning abduction), and nonzero probability of a flaw in every equation in mathematics textbooks. No paradox.

David W.

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Dear David Wolpert,

thanks also for your quick reply.

Reasoning about mathematical solutions for certain (more complex) mathematical questions is certainly not error-free in general.

When we assume such flaws to be present to a certain degree in every act of mathematical reasoning, then it seems to me that “flaw” indicates in every case the existence of a flawless answer – independent of whether or not machines or human beings are able to facilitate that answer. Please correct me if I am wrong.

If correct, it seems to me that what we call “mathematics” is then a kind of double-pendulum, a chaotic deterministic machinery that contains all correct mathematical answers, but the latter are hard or even impossible to deduce in most cases with probability 1. Therefore we are stuck with only probabilistic measures to set some limits to the area where the right answer might be most probably found. Again, please correct me if I misunderstood something here.

If correct, then “probabilistic mathematics” is a term for the high complexity of many interesting mathematical questions that one has to cross through to at all (if at all possible) correctly answer them with probability 1. Is this the correct understanding of what your essay says?

Best wishes,

Stefan

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thanks also for your quick reply.

Reasoning about mathematical solutions for certain (more complex) mathematical questions is certainly not error-free in general.

When we assume such flaws to be present to a certain degree in every act of mathematical reasoning, then it seems to me that “flaw” indicates in every case the existence of a flawless answer – independent of whether or not machines or human beings are able to facilitate that answer. Please correct me if I am wrong.

If correct, it seems to me that what we call “mathematics” is then a kind of double-pendulum, a chaotic deterministic machinery that contains all correct mathematical answers, but the latter are hard or even impossible to deduce in most cases with probability 1. Therefore we are stuck with only probabilistic measures to set some limits to the area where the right answer might be most probably found. Again, please correct me if I misunderstood something here.

If correct, then “probabilistic mathematics” is a term for the high complexity of many interesting mathematical questions that one has to cross through to at all (if at all possible) correctly answer them with probability 1. Is this the correct understanding of what your essay says?

Best wishes,

Stefan

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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.

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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.

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Dear Prof David B Kinney,

Wonderful Analysis please... Your frame work...

We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the role of abductive reasoning in mathematics, 3) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, 4) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes, and 5) the prior distribution that a Bayes rational mathematician ought to have over possible mathematical systems................ is extremely correct. I discussed some thing very similar in my essay

A properly deciding, Computing and Predicting new theory’s Philosophy also...

For example in Dynamic Universe model.... Your framework........

1. here mathematics dont generate research, Physics generates.

2. No abductive reasoning, Physics guides reasoning, not mathmatics

3. There were multiple proofs and predictions came true

4. Physically possible Universe Model is presented see attach

5. You may pleas go thro' paper and you can find your self it is rational or not....

Best

=snp

attachments: JNS-1-109_-_OSP_Model_of_Universe.pdf

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Wonderful Analysis please... Your frame work...

We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the role of abductive reasoning in mathematics, 3) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, 4) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes, and 5) the prior distribution that a Bayes rational mathematician ought to have over possible mathematical systems................ is extremely correct. I discussed some thing very similar in my essay

A properly deciding, Computing and Predicting new theory’s Philosophy

For example in Dynamic Universe model.... Your framework........

1. here mathematics dont generate research, Physics generates.

2. No abductive reasoning, Physics guides reasoning, not mathmatics

3. There were multiple proofs and predictions came true

4. Physically possible Universe Model is presented see attach

5. You may pleas go thro' paper and you can find your self it is rational or not....

Best

=snp

attachments: JNS-1-109_-_OSP_Model_of_Universe.pdf

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I enjoyed reading your essay which definitely made me think! You've both convinced me that maths is 'fuzzy' and that you can understand a lot about maths by looking at how people use it (or construct it). If you have the time, I would be interested in your opinion on my essay (posted March 31st) as it's all about trying to look at what maths is really saying about our Universe. Any feedback, good or bad, would be appreciated!

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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...

view entire post

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...

view entire post

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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

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

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

taking in to account what people say in interviwes, i've heard from Greg Chaitin that mathematics is not a formal system , also in my essay i think i justify fairly what is the problem with ( re ) presenting numbers

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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

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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

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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

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

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

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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

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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...

view entire post

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...

view entire post

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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

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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

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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.

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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...

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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...

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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

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

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.

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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.

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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

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 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

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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

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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

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

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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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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