What Is Ultimately Possible in Physics? by Stephen Wolfram

Now before anyone shouts, I want to confirm that "today" in the abstract refers to a day last week, not literally today. The author submitted his essay in a timely manner before the contest deadline.

Dear Prof Wolfram, you wrote:

"we can imagine transferring our experience to some simulated universe, and in a sense existing purely within it"

- isn't the "Second Life" Game doing something like that already?

Thnaks,

(when you find time a vote is appreciated)

Greetings Stephen,

I enjoyed reading your essay. After resolving to read only the 'official' version, I did end up downloading the expanded paper and skimming through the missing pieces, because I wanted to see what you had to say.

I agree with many of the points you make. In my contest essay, I echo your sentiment that many physicists tend to forget that an equation is just a model, and that its predictive capability is largely a reflection of how closely it models what is real. Too many confuse equations with reality. But if a computational process gives rise to what is real, one would expect the order inherent in Mathematics to both emerge from and shape that process.

I wonder, however, if some of the current perceived limits to knowledge arising from limits to computation are due to our failure to recognize or incorporate the hierarchal nature that arises in any process of asbstraction. If we could somehow encode the hierarchality of symbologies involved into a generalized heuristic computational algorithm, this might allow some of the limits to what is knowable to disappear.

For example; perhaps Gödel simply started in the wrong place, with Arithmetic and Number Theory. I am a constructivist, and I believe that advances in various branches of Math rest on certain fundamentals which must be constructed out of first principles. Had Gödel posited that the rudiments of Geometry were necessary to Topology, which brings us topological distinctions or boundaries, and that this was necessary to Set Theory, which is part of the picture needed for Number Theory to be formulated, a very different picture might emerge.

Now; I'm not saying I think Gödel is nescessarily wrong, but the decidability gets more complicated when procedural hierarchality is figured in.

I am glad I had some exposure to your work, prior to reading this essay, as it is a merry romp through most of the key concepts you introduce in NKS. But I am happy to see you have an essay entered in the contest, as I'll have an opportunity to dig into what you've written and pose some additional meaty questions. I've long been a fan of the Computational Universe hypothesis, as it links up with my work in Cosmology with the Mandelbrot Set. I like the extensions of Wheeler's concept "It from Qubit" of Zizzi and Deutsch (plus Lloyd and Ng). And I coined a phrase imitating Descartes "It computes therefore it is!" which sums up that view nicely.

All the Best,

Jonathan J. Dickau

Hi Stephen,

You write 'Is there a direct correpondence of mathematical impossibility with physical impossibility. The answer is that it depends of what physics is made of. If we can successfully reduce all physics to mathematics, then mathematical impossibility in a sense becomes physical impossibility.'

While the universe is dynamic, mathematics, at least on paper is formal and static, so one suspects that physics cannot be completely reduced to mathematics. On the other hand, the only things we can say about physics are those which are invariant with respect to time so can be written down in static formal form. This is the beauty of differential equations which capture a dynamic process in a static string of symbols.So perhaps we may think about the relationship between physics and mathematics in terms of fixed point theorems. Since the universal dynamics maps the universe onto itself (there being by definition nowhere to go outside) we can expect to find fixed points which can be satisfactorily encapsulated in the physical literature and remain true for at least long enough to get published.

On this picture, we may think of the dynamics within which we find the fixed points as guaranteeing the mutual consistency of the fixed points. . . .

Best regards,

Jeffrey Nicholls

Dear Stephen Wolfram,

i am happy to read your essay here and to learn more about your general thoughts about physical possibilities/impossibilities. Your paper is well written and has a nice rythm in exposing your lines of reasoning.

My personal view of the computational paradigm is, that it will change at some time in the future - maybe not so far away - to a mind/spiritual paradigm. That sounds weird at first glance, but i think the computational paradigm is yet too much deterministic and "bottom up" to grasp reality in its fundamental structures. It's a little bit like the clockwork-paradigm, invented by Newton.

What could it mean to assume a mind/spiritual paradigm? Firstly, it would surely mean that there come human values into play (you labeled this in your essay with "purposes"). Secondly, it would surely mean that there must be a conection between determinism and holism. There are some profound human experiences that suggest the possibility that our spacetime-reality is in some sense a filtered subset of another realm, where hypercomputation is indeed possible. Those human experiences are usually called "near-death experiences". About 40 years ago, the mainstream opinion on this issue was that those stories are purely abnormal hallucinations of the experiencers. But today it is widely accepted that these phenomenons are at least real in the sense that the experiencer must have experienced them in the time his/her brain/body wasn't active at all. Some of the made perceptions in those states of consciousness could be verified and are scientifically relevant. Because these experiences touches issues like timelessness, seing into the future, diving into ones own past, having a multimodal view on the surrounding environment, and perfect insight and knowledge into ultimate reality (and last but not least "love") and so on.

The problem of the limits of computability is linked with complexity. That is the stage where in my opinion "top-down"-causes come into play and new phenomena "emerge" at the very top of this irreducible complexity. My assumption is, that this is only possible because "emergence" can only be possible with a cetain top-down dynnamics as natural feature of ultimate reality. This means at the same time that the very notion of cause-and-effect isn't universally valid *without* purpose and subjective values. The subjective "spiritual" world isn't build up of only one subject's imagination, but through the power of a multitude of subject's values and imaginations, all in agreement with each other and fixed on the same purpose. That lead to the emergence of physical laws and the borders of irreducability.

I am strongly convinced that this "panpsychism", "super-natural" view will sooner or later develop out of the computational paradigm, because the very fact that the search for ultimate reality is build into ultimate reality via humans/consciousness is a hint in this direction and at the same time maybe its deeper purpose. See therefore my own essay here in to contest for a possible mechanism of ultimate reality to achieve all this.

Best,

Stefan Weckbach

Hi Dr. Wolfram,

I hate to be one of those guys who says "Nice paper, now please read mine," but in this case, since I titled my paper in homage to your book "A New Kind of Science," I couldn't resist piling on with exactly such a request.

Your paper seems to be a great distillation of many of the themes emerging from different contestant's submissions. Plus, it has the benefit of being written conversationally and in a natural, engaging tone. Bravo.

The very earliest spark of my paper was, right after I finished "A New Kind of Science," I read Seth Lloyd's book "Programming the Universe," and thought to myself, "This is the first time I've purchased a book with the word 'programming' in the title that hasn't contained a single line of source code." To co-opt the terminology of pure mathematics, the digital physics community seems to have contented itself with producing existence proofs, but not constructions. That community seems to agree that "Yup, the entire universe could indeed be software," but nobody seems to have taken the next logical step, to say "OK, what might that software look like? How might its source code be constructed?" My paper offers up a starting point for exploring such possible constructions. That starting point is, as is yours, fundamentally graph-theoretic in structure. More specifically, it is a fractal that operates within graph-theoretic space.

Traditional fractals like the Mandelbrot set, Menger sponge (indeed anything listed at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_d

imension) consume a subset of traditional n-dimensional space; part of the reason a Sierpinski triangle's Hausdorff dimension is less than 2 is that a complete representation of it can fit inside a 2-dimensional plane. The same reasoning ensures that the Menger sponge's Hausdorff dimension is less than 3. No fractal at the wikipedia page has a Hausdorff dimension greater than 3.

I wonder if part of the reason this is the case is that traditional fractals, because they work by "claiming" points within a larger predefined space, can only _consume_ space. The fractal I propose in my paper (the "Object" class) simultaneously consumes _and_generates_ space (by "claiming" points/nodes and then also having a mechanism for creating new points/nodes). This means, if we were to find a suitable generalization of Hausdorff dimension that can take graph-theoretic fractals into account, then the "Object" class's Hausdorff dimension could be something greater than 3. It could even be, for instance, pi.

Anyhoo, the other day I was lamenting to Ray Munroe Jr that "I wish someone who had as much history as I do with computation, and also as much history as you do with physics, would read the paper and comment on it." Seems to me you're just the right man for the job -- or, at worst, overqualified.

Thanks,

Owen Cunningham

Hello again,

I like some of Stefan W's comments a lot, and would like to point out that they relate to my statement about the need for hierarchality in an observational procedure. One needs to rise to a higher level of abstraction, sometimes, in order to see the answer to a problem, or see that what is being viewed is part of a larger whole. When we look at illustrations in a Math text, we are viewing the idealized figures from a point off the page. Only then can we see the 'true' nature of a circle.

I tend to believe that we see both bottom-up and top-down procedures at work in nature, arising from the very fact that the levels of abstraction required to create or observe anything do have a natural hierarchy. And this is easily linked up with the mind/spirit paradigm. But as Stephen has pointed out, a lot of this sort of behavioral complexity can arise from very simple computational systems. So we are left to wonder if perhaps the universality of their emergence is the result of natural order inherent in Math. I tend to believe that what's out there, in the land of mathematical abstractions, has an influence on what happens here and that this reflects the very mind/spiritual element of which Stefan speaks.

Regards,

Jonathan

Dear Dr. Stephen Wolfram,

"No doubt, though, we will one day master the construction of atomic-scale replicas from pure information. But more significant, perhaps our very human existence will increasingly become purely informational~at which point the notion of transportation changes, so that just transporting information can potentially entirely achieve our human purposes. ...

...But consider a time in the future when the essence of the human condition has been transferred to purely informational form."

Is there more you can say within this forum to clarify what you mean by becoming purely informational?

James

Dear Dr. Wolfram,

I much enjoyed your essay and your books, in particular modeling using Turing state machines.

In my FQXi essay I suggest a computer model of the universe with atomic systems being bimodal Truing machines which alternate between modes. One mode conducts classical computation and the other mode quantum computation as a network. Have you or to your knowledge anyone else thought along those lines?

Sincerely,

George Schoenfelder

Dear Dr. Wolfram,

I just finished reading the long version of your paper - It looks like the correct length! It is odd that so many authors want you to read their papers, and yet your community score is relatively low.

It sounds like you are applying the Principle of Computational Equivalence to solve unknown problems (such as TOE) computationally. There are other papers here that attempt similar tasks (Abhijnan Rej and Owen Cunningham), but it is obvious that more simplifications or approximations need to be applied.

I think that my Geometrical Approach Towards A TOE might be the type of idea that could simplify these computations. Any feedback would be appreciated!

Sincerely,

Ray B Munroe - Author of "A Geometrical Approach Towards A TOE"

p.s. - I love the Wolfram Research site. I use it and Wikipedia quite often. In fact, my essay references your research site.

To Uncle Al: Are we to infer from your comment here that you view the computational approach to physics as fundamentally at odds with the experimental ethos that has driven science forward? If so, that is unfortunate, because introducing an experimental angle to the existing, and very young, subspecies of physics known as "digital physics," which has heretofore been a purely theoretical genre, is precisely what my paper is attempting to do. In the digital physics world, the best way to conduct experiments is to write some code, run it, and see whether its behavior at all reflects that of the universe.

To George Schoenfelder: Your paper sounds extremely interesting. I'm going to read it in detail and post a comment on its thread at some point in the next few days.

Stephen

A very thought provoking essay with a welcome conclusion. Some comments:

1. I think you you place too much some reliance on Godel for the incompleteness of mathematics. I refer to him slightly in my essay but I still doubt his theorem. It only applies to structures with countable = separable & Hausdorf etc. sets of axioms and theorems. It is therefore appliable to digital computers / Turing machines - but not analogue computers. My essay focuses on the incompleteness of mathematics for pragmatic scientific reasons. One aspect of that incompleteness is that the sciences of Computation & Information have not yet been coherently incorporated into main stream mathematics. There is stlll scientific apartheid. Some measures via model theory and category theory are being taken to address this but results so far are ineffective.

2. You write "But traditional mathematical models of physics tend to have parameters that are specified in terms of real numbers." This is wrong. Input from physics can only be Rational numbers. The relevance of Real numbers is that by assuming their existence they enable continuity of representation of the physical variables between measured Rational values. This requirement is why the Topology of Open Spaces is fundamental to PHYSICAL measurement. (see page 9 of "An Introduction to Geometrical Physics" Aldrovandi & Pereira). Note that Turing Machines rely on this assumption of topological continuity between each step.

3. There is a hidden assumption that Logic is complete. I doubt it.

4. You attribute the Ultimate limit on physics to Teleology. Purpose implies causality. It is interesting to note that conventional physics does incorporate causality - but its reductionist structure back to explaining everything in terms of elementary particles is a process that eliminates causality at the elementary level. Reverse the process direction and at the Top Level cause we get Purpose. Your conclusion is predicated on emergence, and purpose on consciousness. These are the 2 of the things my essay requires for the Ultimate completion of mathematical science.

Stephen,

The idea of computational equivalency is related to something which occurred to me some years ago. As I might pour cream in my coffee the flow of the cream becomes complex with tentacles and filigree that has a growing amount of complexity. This is a common observation of course that we don't often think of as having some complexity to it in the sense of a Connection Machine. However, that flow of cream is effectively computing something, say the hydrodynamic evolution according to the Navier-Stokes equation. The foldings of tendrils of cream often have a measure of recurrence and these are at least comparable to the iterative computation of "something." We just do no ordinarily couple a cup of coffee to output devices to register its output in an alpha-numeric form. This also seems to strike a sense with the growing field of virtual reality computation. The Pixar movies, and those made by similar computer animation groups, use physics and algorithms to simulate reality. So in approximating what we observe in the ordinary world ends up requiring a large amount of computational complexity. These can involve how a sheet or item of clothing will fold as it is dropped on a surface, or the intertwining of leaves in a pile that has been brought together by wind. The latter of these has always intrigued me by how this exhibits a noticeable pattern that differs from a pile raked up by a person.

I would take some departure from the last paragraph on about this diverges from physics. The computations are ultimately the evolution of physical states. The universe may well be fundamentally a quantum computer, or a quantum gravitational computer. The occurrence of a macroscopic world may be a signature of exiomatic incompleness. The underlying algorithm, say a quantum error correction code --- maybe even deeper the monster (Fischer-Greiss) group, is subject to decoherence or some chaotic process which results in a set of all possible "computations," or physical algorithms. So the universe (or multi-verse) might contain all possible Turing machines or algorithmic processors, where most have an undecidable halting status, or a Chaitan halting probability. Then the universe is a Universal Turing Machine is not able to compute all its outcomes from a bottom up sense. Yet through this the computational states are ultimately the physical states of atoms, particles, quanta or a tiny volume of fluid in a flow.

On balance this is a well written and interesting essay.

Cheers LC

Dear Stephen,

i read your essay and it appears you are living with the computational world. Some others live in the mathematical world and some like me are experimetalists and empirical people who prefer to live in their own world of making. But the maker if the universe lies beyond us and still has generated we all individually in this universe which got created far earlier than when the humansm first appeared on the scene. Yes, it is achallenge. But when we look at whatever we have been able to decipher about the universe evolution, the entire Physics got born with the primordial matter ( we don't know yet what it was). Ths in turn created the visible and dark matter world along with dark energy that is attributedly repelling the minor component called the visible world. The visible componennt is baryonic in nature while dark matter is non-baryonic. They do not have means to interact with each other, except that the non-baryonic one repells the baryonic component gravitationally. Within the visible world the gravity is attractive force field.

From this description it become clear that nature governs Phsyics and not vice versa. Humans have a mind of their own. It is known to be complex, wandering entity. t is what we use to study nature. It is simple but our means are complex and that is we have made the universe a complex subject. As one works towards one theory in Physics that may explain all processes, we go towards the simplicity of approach. But our mind comes in the way to complicate the matter. Thus we need to quiten the mind and discipline it further. That is we need improvement in ourselves before we can improve our computational, physical and mathematical tools. The experiments are governed by the technolgy and so it will follow physical machinenaries that we develop. Thus we are in a vicious circle of possibilties that accompany immpossibilities. Enjoy ht egame but keep your mind wide open, calm it down and discipline it to see the simplicity of nature , rather than complicate its simplicity.

Dear Stephen

We search universe in two ways: with the mind and with the consciousness. When we integrate both approaches we have optimal results.

I publish about the subject on

http://vixra.org/pdf/0910.0018v1.pdf

yours amrit

Dear Stephen-

Thank your for your interesting essay, although I have a different opinion about the suitability of a computational approach to fundamental physics. Basically, I have similar concerns with many other posted essays, which have a computational, philosophical or formal character. This leads to an unbound search for, and speculative discussions about, a unified theory of physics.

In your essay, the suggested 'discrete machine' or computational approach relies on formal axiomatic theory. This is, in my opinion, unlikely to provide an accurate representation of nature. In principle, an infinite number of machines can be imagined. Without guiding principle to reduce this number, this leads to an unbound search problem. In addition, certain machines may produce output that, deceptively, resembles observed natural behavior, although the output may never a good representation of nature when observing it at a small enough scale. Within the computational approach, it also remains to be explained how machines are created.

From within a 'discrete machine world' you'll not be able to comprehend nor completely accurately represent the possibilities of the 'analog' world, which nature appears to be. Theorems applicable to the world of discrete machines are in general not generalizable to the analog world. In addition, time is implicitly present in the operation of a machine, which makes it impossible to explain physical time from within a machine.

From my perspective, it is far more efficient to start a search for a unified theory directly with already observed 'known to be true' physical properties. This dramatically limits the search effort. These physical properties must be represented in terms of a minimal coherent formalism. From this theory it should be demonstrated that all observed quantum and relativistic effects appear and correspond to observed behavior within currently measured accuracy. This is essentially what Quantum Field Mechanics (QFM) attempts to do (see my essay). In the 'analog world' of QFM, interaction gives rise to unceasing pulsating phenomena. These quantum beat processes exhibit wave-corpuscular behavior (as Louis de Broglie suggested) and can be interpreted as massive particles. The mentioned pulsation has a finite duration (10^(-20) sec for electrons) and short spatial jumps (with a size equal to the Compton wavelength). This results in dynamically emerging discrete space and time, although the two attracting fundamental fields from which this behavior arises are 'analog and continuous'. In this analog world, dynamically emerging behavior has no relation to discrete machines and does not require multiple universes.

I have provided an -overview- of this theory and some of its extensions in my essay. The references provide the in depth rationalization. A slide deck on my website provides an alternative description.

Thanks again for your essay.

Regards,

Ben Baten

Dear Stephen Wolfram,

While analog computers cannot compare with digital ones with respect to reproducibility and performance, they were in an important sense closer to reality. For more details you might have a look at the attached files and 527.

Kind regards,

EckardAttachment #1: 1_Ritz09.pdfAttachment #2: 1_M283.html