The Descent of Math by Sara Imari Walker

Sara Walker

Essay Abstract

A perplexing problem in understanding physical reality is why the universe seems comprehensible and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather than being an odd coincidence arising due to our strange position as passive (and even more strangely, conscious) observers in the cosmos, these two problems might be intimately related and potentially explainable in terms of fundamental physics - that is, if we are willing to make the concession that information can physically influence the world. My premise begins by distinguishing physically possible states of the world, defined as everything permissible by the laws of physics, from physically accessible states of the world, defined as those that are achievable from a given initial state. For universes where the laws of physics are such that the number of accessible states is less than all that is possible, I argue that the most probable states among all possible states are those that include physical systems which contain information encodings - such as mathematics, language and art - because these are the most highly connected states in the state space of everything that is possible. Such physical systems include life and - of particular interest for the discussion of the place of math in physical reality - humans. Within this framework, the descent of math is a natural outcome of the evolution of the universe, which will tend toward states that are increasingly connected to other possible states of the universe, a process greatly facilitated if some physical systems know the rules of the game. I therefore conclude that our ability to use mathematics to describe, and more importantly manipulate, the natural world may not be an anomaly or trick, but instead a natural part of the structure of reality.

Author Bio

Sara Imari Walker is a theoretical physicist and astrobiologist and Assistant Professor in the Beyond Center for Fundamental Concepts in Science and the School of Earth and Space Exploration at Arizona State University. She received her PhD in Physics from Dartmouth College and has held postdoctoral appointments in the Center for Chemical Evolution at the Georgia Institute of Technology and as a NASA Postdoctoral Program Fellow.

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John Hodge

Perhaps, ``the Assent of math''?

adel sadeq

Hi Sara,

You seem to be talking like a philosopher. Sorry for being blunt, but I don't see any physics, however it is fine as a fun work.

Maybe if you look at my essay and see if there is any fundamental relationship between physics and math.

Essay

Thanks and good luck.

Sara Walker

Blunt is fine. Philosophy is important to the conceptual foundations of physics, especially in areas where we aren't sure the best way to ask a question. I will take a look at your essay.

KoGuan Leo

Dear Sara,

Yes, I agree with your conclusion stated below: "Our ability to use mathematics to describe, and more importantly manipulate, the natural world may not be an anomaly or "trick", but instead could be an intimate facet of the structure of physical reality." In my essay TianmingRen in the Cloud, I advanced that the initial state: everything is connected that bit is it, and the final state and all in between: everything is still connected that it is bit. Hope you can comment and rate my essay.

Best of luck,

Leo KoGuan

Sara Walker

Thanks Leo. Will take a look. Best of luck to you as well.

Jonathan Khanlian

Hi Sara,

Any essay that works in some soccer references is good in my book. I like the analysis of information encoding being essential for life, but I think this may still be possible in a deterministic system.

You talk about information being lost in your node system when there are many-to-one mappings. I intuitively agree, but there could be a way around this. I think if the many-to-one mapping happens below the level at which the universe emerges (or outside the system), this may not be an issue that violates what Leonard Susskind refers to as the Zeroth or Minus 1st law of physics, which is that information is never destroyed. Are you familiar with Wolfram's causal networks? They are a node based systems where different updating orders still correspond to the same causal networks. Maybe in that sense information outside the system is loss, but information viewed from within the system is not. I don't think your node model and his node model are trying to explain things at the same scale, but it may still be worth a look...Maybe there are some ideas there that will help you and Paul flesh out your theory some more.

I definitely think information is essential, and I like node models, so I'm interested to see how your work evolves.

Please check out my Digital Physics movie essay if you get the chance.

Thanks,

Jon

Sara Walker

Thanks Jon. I will certainly look into the causal network models in more depth. It sounds very related to some of the work I am doing now.

As a clarification on specifically what I mean by "information loss" as a result of many-to-one mappings is that for such cases the past is uncertain as there are multiple possible pasts consistent with the current state. The "information lost" is therefore with regard to your certainty that specific past events happened.

Looking forward to reading your essay.

Best,

Sara

Jonathan Khanlian

I understand what you are saying about not knowing the past from the current state because there are many previous states that could have lead to the same current state.

I remember being surprised when Leonard Susskind talked about even when you have an OR gate in a computer that outputs a yes, you would think that you don't know if you had a yes/no, yes/yes, or no/yes input after the input bits are erased (e.g. an electrical charge is changed) in the computer. Although that information can no longer be retrieve from the gates in the computer, it's erasing has affected the environment, and therefore it hasn't been lost, from his perspective. I'm not sure if this viewpoint, that information is never lost, is broadly accepted in the physics community or not.

In this OR gate situation(or any situation), I think Wolfram would say that the system has become a little more encrypted. So the passage of time is in essence a calculation that leads to more encryption... so the information isn't lost, but it would be very difficult to deduce what the state was before... Or actually maybe impossible if you consider yourself in the system. One thought is that in order to do the decryption calculation to figure out whether you had a yes/no, yes/yes, or no/yes situation, you would have to freeze the system, step outside the system, take a perfect account of every bit of information in the system without disturbing it (avoiding the uncertainty principle), do the calculation outside the system, and then pop back into the system to report your findings... which I don't think is really achievable for someone in the system... unless of course some of those spiritual, Ayahuasca trips to higher systems of consciousness are real. Maybe you and Paul could devise some new experiments? :) Just some thoughts... Do institutions give out psychedelic grant money? Hey, FQXi seems like a pretty open-minded institution;)

If any of these ideas have stimulated thoughts (or amused you), please check out my movie trailer if you get a chance:)

Jon

p.s. After using the hyper-linking syntax to link my trailer, I wondered what issues I'd have if the web link I embedded ended with "][/link ]"(less that extra space)... Could we create something in our universe that would hack the code that runs outside our own universe?!? Sorry to get all "digital physics" on your page here; A (wo)man's got to promote:)

Dipak Bhunia

Dear Sara,

Thanks for the essay. But if it advocates merely to establish an anthropic connection between physics and mathematics would definitely be a confusing; as you mentioned, "why does the universe seem comprehensible and why should beings like us be here to comprehend it". Because, we are, whether "passive" or active (is that make any difference?) observers, the intrinsic parts of the universe.

In that same universe (or nature in broader senses), obviously there would be one most important role of "information" indeed. Without information one never could comprehend anything about anything comprehensible. But such information communicates only through any quantized signal. That is any of ours comprehensible to and fro information would be fundamentally quantized too. Therefore, from where no such comprehensible quantized information emerges out to reach to or received by us never could be comprehended. That is, we have always an intrinsic quantized information limit for the nature to comprehend. That limit not necessarily would be in any technological terms but natural as we are fundamentally sum of few quanta. Hence all our perception patterns are appear comfortable to begin with any initial state of casualty from which any such quantized information about the physical system received for its comprehension.

I think, the domain of "conscious" would start beyond that zone of zero quantized information. But being an intrinsic quantized-observer we can only imagine about the patterns of such conscious domain. Never could comprehend that pattern properly.

Otherwise your essay is enjoyable.

Best wishes

Dipak

Sara Walker

Dear Dipak, Thanks for the comments.

With regard to whether observers are "active" or "passive" - I absolutely think it does make a difference. If observers are active, there is potentially an explanation for their existence as they actually matter to the world of matter - they are physical systems. If they are "passive" I think the existence of observers is much harder to explain, why would they be there in the first place?

Best,

Sara

Lawrence Crowell

Sara,

Nice essay. I have pondered something similar to this. The idea that N_A = N_P is similar to Gel Mann's totalitarian principle, "Anything that is not absolutely impossible is mandatory." This ergodic principle is in some ways a frequentist interpretation of probability. A phenomenological perspective is more Bayesian, which says that probabilities are assigned only on the basis of some data and a prior estimate. I am in some ways more disposed to this, though the ergodic principle is something I appreciate, and I have a book by Sinai on this.

I am working on trying to develop a principle which indicates that the fundamental Hilbert space for the universe, by which I also mean multiverse, is 512 dimensions. I am making progress on this part. What I want to demonstrate as a result is there exists only one electron, one up quark, one photon, one ... , and what we think of as a vast ensemble of is due to a massive partitioning of the path integral or partition function by the occurrence of event horizons. The path for the electron winds everywhere, but the occurrence of event horizons, such as the event horizon of the universe, freezes this path into a partitioned ensemble of paths. As a result this is, or I should say it should if I am right, indicate that the partitioning of vacua into bubbles with inflation is associated with a vast ensemble of possible world with a set of particles associated with the fundamental electron, up quark, T-quark, Z, states.

In the setting of eternal inflation this would mean that anything that is absolutely not forbidden is then mandatory. So where might life fit into this? It is likely as you say some extremal principle on the accessibility of physical states. However, these states as life or consciousness generate a phenomenology of what can be observed about the world. I can see the prospect that this has a basis in how observers emerge in a cosmology, say a cosmology within the vast landscape of cosmologies in the so called multiverse. The set of states N_A-obs that are accessible in the context of an IGUS or observer is mostly likely far smaller than N_P. Many of these clearly we also have N_A-obs < N_A, and the question is whether the phenomenology of the observer is of greater importance, say in a Bayesian perspective.

Cheers LC

Sara Walker

Dear Lawrence, These are some interesting point. I do think of this much like a cosmology of observers, so I am happy to see you made that connection!

To clarify your last paragraph - do you mean "obs" to represent the number of states with observers, or the number of observable states?

Chidi Idika

Hi Sara,

I very much appreciate your essay because it is not often that one comes across a physicist trying boldly to connect with biology at a theoretical level.

You say: "biological systems [may be] uniquely defined as a class of physical systems where information plays a causal role."

In fact in my essay I adopt a particular quantity to define man/life as my "initial condition"; something no more than what mathematics calls the number bases (Peano's "constant") and which analogously physics calls the "quantum" of observables. Initial condition is what I define to be "the observer" state (the "natural unit").

And to illustrate man as such, I adopt the quantity threshold potential of the action potential; it signifying say man's sensory threshold. And dare I say here I have found in my calculations that this quantity actually does make clear quantitative and qualitative connections with accepted physical constants such as Boltzmann entropy constant.

This is bearing in mind that any initial condition must be itself a state either of complete or null information (or indeed both?). This will mean that we are talking here of a state space as a Godel's "formal system" (Peano's "successor function" or "constant"). So a proof is any connection between systems (your "nodes").

For me, ultimately, initial condition is either a "completeness" or "incompleteness" of proof (essentially both i.e. really an "un-decidable" of Godel's and "uncertainty" of Heisenberg's). And information is simply negative entropy.

I find it exhilarating that you have thought along this line at all and I hope you will find the time to read and comment on my argument.

Regards,

Chidi

Sara Walker

Hello Chidi, Thanks for the thoughtful comments. I haven't had the chance to read your essay yet, but will try to do so. It does indeed sound related.

One important distinction I might make is that one of my motivations is that I would very much like to see a formalism that removes dependence on the initial condition. If you really think you can get "anywhere from anywhere" the initial condition is arbitrary. This is a particularly important point because under this case, if you start with an arbitrary initial condition, you are likely to eventually end up in states that contain life because they make the most states accessible (there may however be bottlenecks, so this I suppose could also be consistent with "life is rare" hypotheses). So there is very little dependence of the observed complexity on the initial condition under this formalism.

Chidi Idika

Dear Sara,

Question is: what exactly should we understand by "initial condition"? If we take initial condition to mean simply threshold as in "sensory threshold" or "quantum of observables" or even mathematically "number bases" or indeed Einstein's "inertial frame" then every initial condition is also unique.

I see initial condition as generalizing the intuitive notion "observer", or vice versa. Thus initial condition imply also a unique "path", more precisely a completeness; picture a geodesic or a spherical coordinate system or a phase space (indeed in thermodynamics an "isolated system").

This formalism brings us straight to an Everett sort "many worlds" but which I personally interpret in the sense of qualia i.e. observer specific reality. And it is my concept of quantum gravity.

I crave your indulgence to find the time to read and comment frankly on my basic axioms/formalism.

All the best,

Chidi

Martin Seltsam

Dear Sara,

WOW! I think this essay is my personal favorite/winner! The ideas are really interesting and actually reminded me of a book I read a year ago: "The Beginning of Infinity" by David Deutsch. The reach of explanations and transformative powers of understanding agents are explored in this book and its conclusions share some similarities with you key points. Your view that states containing information encoding are more probable/accessible certainly align nicely with the thoughts of Deutsch and I am very inclined to agree with your perspective.

As you touch upon the question of which physical states are realized, I thought I dare to point to my little opera "Map = Territory" where I try to address similar issues (what it really means for states to be physically realized) and ponder the possibility of an actual merger of the description and the described in fundamental physics.

I would be honoured by your feedback.

With deep respect and best wishes for your success,

Martin

Sara Walker

Dear Martin, Thanks very much for the enthusiastic response to my essay! You are right to spot the connection to David Deutsch's thinking - I am very inspired by his work, particularly some of the recent work on constructor theory. Interested to read your essay.

Best,

Sara

Rick Searle

Hi Sara,

I really loved your essay. It should be doing much better in this contest.

I am really looking forward to your upcoming work with Paul Davies.

Will this be a paper? A book?

Please take the time to check out my own essay, tell me what you think, and give me your vote.

http://fqxi.org/community/forum/topic/2391

All the best,

Rick Searle

Sara Walker

Dear Rick, Thank you. I am glad you enjoyed it! Right now I am working on a few papers related to the topics presented - a book perhaps, but down the road.

Best,

Sara

James Hoover

Sara,

This is an impressive essay that states truths usually not mentioned about the problems with "initial states" in physics studies.

You mention "a pattern in how the world works" reminds me of an example I use in quantum biology of the European robin. A British physicist works with an molecular genetics professor to determine how the robin navigates South to Africa (a pattern regarding how the robin's world works) . They present strong evidence of the connection of the macro and quantum worlds to produce a avian chemical compass. A combination of mind, math, and physics connect to make this possible.

Like you I believe the "ability to use math to describe, and more importantly manipulate, the natural world" uncovers physical reality.

Jim

Sara Walker

Thanks Jim. This is a great example.

Sujatha Jagannathan

Great art in your writing.

Smiles

- Sincerely,

Miss. Sujatha Jagannathan

Sara Walker

Thank you :)