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

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

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

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 probabilistic since there would exist a meaningful exception that has probability 1. This exception then must exist somewhere in a non-probabilistic fashion as a pre-defined ultimate mathematical truth.

But this ultimate truth cannot be a meta-truth about mathematics, since all of mathematics has been defined as being probabilistic. To circumvent this contradiction one could claim that the truth of mathematics' probabilistic nature can itself only be evaluated by some probabilistic measure. This would be consistent with the claim itself but also obviously undecidable in the limit of only finitely many trials.

I further ask myself where the notion of truth comes in with your approach. If there is no proof existent that could refute your claim and your claim can indeed be part of a well-formed formula (wff), then it seems to me that your claim is generically undecidable.

Now, in your framework the probability for something to be mathematically the case increases if there are many independent paths of reasoning that suggest the answer to the initial question to be what we assumed it to be in the first place. One danger of this kind of reasoning could be that what we assume to be independent paths may in the future turn out to be just different forms of the same starting assumption. Moreover, this already seems to me to be the case for your claim that mathematics is fundamentally probabilistic, since to avoid that the truth of that claim turns into a non-probabilistic (eternally fixed) truth (by a priori assigning to it the probability 1 for its truth), one must define your starting claim as forever being just a probabilistic "estimate" that never can reveal a decidable answer.

Now comes my main concern: if mathematics (and with it logics) would be infinitely probabilistic, the notion of truth would vanish. But without the notion of truth one cannot even rationally think about whether or not mathematics is fundamentally probabilistic or not - it simply wouldn't make any sense since there would be no such things like "truths" and also no such things like "falseness". This then would open the door to undermine the principle of non-contradiction and surely also your lines of reasoning (that surely depend on that principle).

I would be happy if you would write what your points of view are concerning my lines of reasoning!

Best wishes,

Stefan

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.

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.

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

=snpAttachment #1: JNS-1-109_-_OSP_Model_of_Universe.pdf

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!

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

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 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

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

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