Mathematics is Physical by Sara Walker

Sara,

I'd like to remark only that such sort of ideas of Nature as Symmetry and Homochirality can be easy found even in pure number theory ( Riemann problem and problem of nonexistence of odd perfect number). It could be considered as additional arguments for your thesis?

Best

Michael Popov

Dear Sara,

I really enjoyed your essay! You got me thinking and changed my thinking, for which I am extremely grateful.

To be honest, I had a difficult time accepting your title. I think of mathematics as a precise description, a precise language. Later in your essay, you discuss this. But I couldn't accept mathematics as being physical.

But you really got me to think more carefully about what it means to be physical. Your focus on causality was brilliant as it was undeniable. Can non-physical things be causal? I would have to argue they couldn't be. Conclusion reached! Bravo!

Near the end of your essay, you discuss CPT and the second law, and note that the constraints placed by physics are not quite comparable to constraints placed by Godel, because "the laws of physics are formulated by us." I don't think that it is that simple. In some cases, the laws of physics are so constrained by mathematical symmetries (eg. associativity and distributivity) that they are essentially dictated by those symmetries. (see my essay for example as well as the paper by John Skilling and myself: https://arxiv.org/abs/1712.09725).

And with that understanding, it strikes me that perhaps it is possible that Godel's theorem could constrain our physical laws. This is a fascinating thought that had not really sunk in until after reading and thinking about your essay. I am surprised that it hadn't occurred to me at that level because I go as far in my essay to discuss how Tarskii's Theorem makes probability theory generally applicable. But understanding comes in degrees or levels, and after reading your essay, I feel that I have a deeper understanding.

Thank you again!

Kevin Knuth

Hi Sara,

A wonderful, thought-provoking essay. You describe two distinct descriptions of nature: 1) the mathematical formulation of laws (e.g. Newton's laws), and 2) the algorithmic view. For the mathematical formalism, laws apply to initial states. They are deterministic, they are fixed, and they are independent of physical reality. Mathematical formulations typically interpret physical change as deterministic. This applies to classical mechanics, relativity, and to isolated (and unobserved) quantum systems.

For the algorithmic formalism, in contrast, you state that "the laws often depend, in part, on the current state of the system." Algorithms update their description of physical state at each iteration and redefine the physical state at each time step. Updating a state for a probabilistic algorithm describes irreversible and random change at each time step. This leads to the "path-dependent, historical narratives characteristic of the object-dependent dynamics of the biosphere" that you describe.

You state that "unification of these two formalisms ... is essential for some of the most difficult frontiers in science such as the emergence of life..." However, both algorithmic and mathematical formalisms are fundamentally deterministic descriptions, whether their predictions are definite or probabilistic. As deterministic descriptions, both formalisms therefore apply to initial states, are fixed, and are independent of physical reality. The fundamental difference between the mathematical and algorithmic formulations is not in their formalisms; it is with their different interpretations of time and change. Unifying their formalisms is not the difficulty; the difficulty is unifying their interpretations.

In my essay, I argue that it is physical reality itself, not physical laws, that are contextually defined. Empirical observations of physics are best interpreted by invoking a contextual reality that is defined with respect to its positive-temperature ambient surroundings. Absolute zero is an unattainable idealization, as is its implication of fundamental determinism. Nature resets a system's contextual state every time the system interacts with its ambient surroundings. In the case of measurements, a system's actual ambient surroundings is the measurement apparatus (or observer), and this resets the system's state upon measurement or observation. This interpretation is compatible with the algorithmic interpretation, and it enables the unification of the two formalisms and their interpretations.

I would love to hear your thoughts on this.

Best,

Harrison

Dear Professor Walker,

Thank you for your thought provoking essay!

The line that really stuck with me was this:

"The power of mathematics is precisely in that it is information that can berobustly copied, meaning we can readily see its structure across very different systems, giving it'sumph in the scientific arena."

It left me wondering if this could be an explanation of the difficulties biologists encounter when applying mathematics to the living world (at least compared with the glaring success of mathematics in physics). For what one never sees in biology is precisely this copy fidelity. No child cell is an exact duplicate of its parent.

Best of luck in the contest!

Rick Searle

Hope you will reply your comments and have a look at my essay please

Best

=snp

Hi Sara,

As a follow-up to my previous comment, a logical consequence of the contextual interpretation that I described is a definition of functional complexity and a principle of spontaneous self-organization. See my Medium essay, The Arrow of Functional Complexity. I use the term homeostate in the essay, but it is identical to David Deutsch's constructor.

Harrison

'Whereas I cannot take charge and separate it from an electron.' The theory of weak quantum measurements can lead to so-called Cheshire cat states, where the electron is over here and the charge over there. But only in a statistical post-selection context. Not sure if this affects your argument. Second comment: the concept of infinity has an uncertain status. Does it have causal power? Example: Hawking predicted black hole radiance with a Bogoliubov transformation between in and out modes of a quantum field. To get his famous thermal result, he had to integrate to infinity. If the integral is truncated at, say, the Planck frequency, the black hole just emits a puff of radiation and not a steady flux. So all the paraphernalia of black hole entropy and the associated research flowed from the use of infinity as if it is a real thing operating in the real world. Is this an act of faith, a pragmatic assumption or a comment on the deep nature of physical reality?

Dear Sara,

I greatly appreciated your work and discussion. I am very glad that you are not thinking in abstract patterns.

"While unification of these two formalisms is not necessary for the domains of science where each is independently valid, it is essential for some of the most difficult frontiers in science such as the emergence of life, which arguably occurs when our traditional Newtonian approach to physics based on initial states and fixed, deterministic "laws" of physics must yield the to path-dependent, historical narratives characteristic of the object-dependent dynamics of the biosphere. We need to explain a transition from a physics where abstractions and relations are descriptive to one where they are also causal. She expounds on all of this to me and then asks "When can you do the math and explain this to me?""

While the discussion lasted, I wrote an article: "Practical guidance on calculating resonant frequencies at four levels of diagnosis and inactivation of COVID-19 coronavirus", due to the high relevance of this topic. The work is based on the practical solution of problems in quantum mechanics, presented in the essay FQXi 2019-2020 "Universal quantum laws of the universe to solve the problems of unsolvability, computability and unpredictability".

I hope that my modest results of work will provide you with information for thought.

Warm Regards, `

Vladimir

Hi Sara,

Good story, nice idea. My comments are partly covered in your essay, yet I wish to extend it for further clarification.

> .. that information, or abstractions have causal consequences. It is however difficult to explain this beyond mere anecdote.

No, it is not so difficult to see why information has causal consequence, only that we have to understand reality of information differently. From the first principles of constancy of causal relation in the nature of change, if an interaction among physical systems results in an observable state S of a physical system P, then S of P must remain congruent with, or correlate with the information of the causal context effecting the change. Otherwise, measurements do not have an interpretation relating to the cause. Causal context includes precursor state descriptions of interacting systems. For example, mass of a physical system Q denotes its causal power in an interaction, which constitutes Q's function or the basis of its relation with other systems. If a system P interacts with Q and gains a state S due to this causal function of Q, then S of P is said to correlate with this information; `mass' is mere label for the causal function of Q, which constitutes primitive of semantics, upon which more structured and abstract semantics can be constructed.

For the same reasons of natural causal dependence, S of P also must correlate with what the observed precursor states of interacting systems correlate with. This is a second order correlation which inductively takes into account all causal descriptions responsible for S of P. It is the second order correlation that allows construction of structured and abstract semantics as shown in Fundamentals of Natural Representation Would you like if I state that reality of all information arises only from causal correlation of states of matter? Therefore, all information has causal consequence. Of course, most of them could be limited to the neural system of the brain without external action.

> What separates information from other physical attributes such as charge or mass? Take for example the charge of an electron. It is often considered to be a physical attribute because every electron in the universe has a charge and because the charge cannot be separated from the electron.

The concept of an electron as an isolated entity is not driven from any ontological first principles, but rather it depends on what consequences are observable due to the associated causal qualities (charge, mass, spin). So, the electron is defined as an entity that has rest mass of 0.511 MeV, charge of 1 negative electronic charge, and spin of one half. That is, the very identity of an electron (or any physical system) is defined in terms of constancy of certain observable causal powers. As it is so, the charge cannot even be conceptualized to be separated from its identity. In fact, causal states (observable qualities) are fundamental (primary), identity of an entity emergent (secondary / dependent). This distinction is crucial in associating the causal reality of information with the observable states rather than with the entity as is mistakenly done in physics.

I agree with your assertion, "If mathematics were not copiable there would be no way to build mathematical models of any physical phenomenon". If we consider two extents of physical system, one that is limited to an individual system like brain, and another that involves many. Indeed, the neuronal system of a brain does exchange the semantic values of information that has consequence in the physical world due to the causal dependence of neurons on each other. But, for the lack of such causal dependence or constraints, inter-personal transmission must be coded. Mathematical coding is far more precise as you mentioned, yet it has it limits Kako, E.; Wagner, L. The semantics of syntactic structures.

In my essay, I take a dig on, "The unreasonable effectiveness of mathematics in describing physical reality", which might entertain you for a moment or longer depending on your tolerance.

> These mathematical descriptions have a different property than the systems they are intended to describe. They can readily be copied between different physical media. .. I can take Coulomb's law and formulate it in my mind, write it on a piece of paper.

Coulomb's law in the mind has a semantic content, structure, and function, it does not require an interpretation. But what we copy from one medium to the next is not the information, but the bits that encode the information under certain scheme. I strongly felt that you were talking about the causal reality of semantics of information, not the artificial mechanism of coding. This treatment of information that entails storing, copying, transmitting in bits, has come at a cost; the cost is the formation of a mental image that information may come into play only when encoded, interpreted, and stored in bits etc., in some manner. This mental frame has kept the generations of researchers away from dealing with the semantics of information, and how causally they come into existence without a need for an interpreter. We know a device (brain) exists that processes semantics of information, but we never attempted to shed the Shannon's idea of quantity of information, or statistical properties of information to get to the semantics of information. So much so that even when such a mechanism is presented from first principles, Fundamentals of Natural Representation , it remains ignored.

Since your awareness has a causal basis, it can cause consequences. Though, states of registers (bits) may also cause consequences in physical universe, but without human constructor they would not be relevant to the mathematical idea because of the artificial coding involved.

Rajiv

Paul Davies beat me to the punch on this. You wrote:

What separates information from other physical attributes such as charge or mass? Take for example the charge of an electron. It is often considered to be a physical attribute because every electron in the universe has a charge and because the charge cannot be separated from the electron. It is for this reason that charge is considered an intrinsic property in physics. Of course we cannot be sure that it is an intrinsic property because charge is only measured under interaction with another physical system, e.g. via measurement.

There are so called Cheshire Cat experiments which illustrates how the charge and spin of a particle can appear in different regions. They can be localized in separate regions. This is as I see it why elementary particles are not that different from so called quasi-particles in solid state physics. Elementary particles are I think entanglements of states corresponding to elementary observables, qubit etc, with topological content.

Your take on mathematics has some following out there. I can't think of the man's name, but he talks about how extremely large numbers simply do not exist. This is because they can never be represented in any realistic way. The idea is that mathematics is something which has a one to one correspondence with what is physically generated in a representation. This is a latter day form of a philosophy of mathematics similar to Brower's Constructivist interpretation of mathematics.

The funny thing is that we can't ever know which is the case. One can go around and around on this, from thinking mathematics is some objective Platonic, or Platonic-like, reality beyond physical reality, to thinking that is too mystical and mathematics only has content in what is represented in some explicit contextual sense. Which is is> I have no idea.

Your essay though makes some points along these lines that are worth consideration, at least if one were to pursue some particular direction with information and the contextual aspects of observers. Some of those observers are what we might call mathematicians. So I gave your essay a decent score.

Cheers LC

Dear Sara,

I would appreciate to learn if you happened to read my post on your essay. No, reading is not a requirement, only the information.

Rajiv