Digital physics

Digital physics holds the basic premise that the entire universe is made up of information, is therefore computable and may be imagined either as the output of some computer program, or represents a vast digital computer itself.

In more detail, it involves one or more of the following hypotheses. That the universe or reality is:

• Essentially informational (although not every informational ontology needs to be digital).
• Essentially digital.
• A computer itself.
• Implemented on a computer (a simulated reality).

History

Every computer must obviously be compatible with the principles of information theory, statistical thermodynamics and quantum mechanics. A fundamental link between these areas was proposed by Edwin Jaynes in his seminal papers in 1957 . Moreover, Jaynes elaborated an interpretation of probability theory as generalized Aristotle's logic, a view very convenient for linking fundamental physics with digital computers based on logical operations (Boolean algebra). Under this general theoretical frame, the hypothesis of Universe as a digital computer was pioneered in Konrad Zuse's book Rechnender Raum (translated by MIT into English as Calculating Space, 1970). Its proponents include Edward Fredkin, Stephen Wolfram, Juergen Schmidhuber, and Nobel laureate Gerard 't Hooft. They hold that the apparently probabilistic nature of quantum physics is not necessarily incompatible with the notion of computability. Quantum versions of digital physics have recently been proposed by Seth Lloyd, David Deutsch, and Paola Zizzi.

Carl Friedrich von Weizsäcker's binary theory of ur-alternatives, Pancomputationalism, computational universe theory, John Archibald Wheeler's "It from bit", and Max Tegmark's Ultimate ensemble are examples of related ideas.

Digital physics

Overview

Suppose that a digital computer performs some number-theoretic operations. As customary in number theory, these may be captured by generating functions. Let the Universe be consistent with statistical thermodynamics, quantum mechanics, and possibly other physical theories we trust. Partition functions are customary objects to capture all physical information associated with statistical thermodynamics or quantum mechanics. This point of view establishes a fundamental link between the number-theoretic (computational) and associated physical description.

The theory of digital physics suggests that there exists a program for a universal computer which computes the dynamic evolution of our world. For example, the computer could be a huge cellular automaton, as suggested by Zuse (1967), or a universal Turing machine, as suggested by Schmidhuber (1997), who pointed out that there is a very short program that computes all possible computable universes in an asymptotically optimal way.

Some try to identify single physical particles with simple bits. For example, if one particle, such as an electron, is switching from one quantum state to another, it may be the same as if a bit is changed from one value (0) to another (1). There is nothing more required to describe a single quantum switch of a given particle than a single bit. And as the world is built up of the basic particles and their behavior can be completely described by the quantum switches they perform that also means that the world as a whole can be described by bits. Every state is information and every change is a change in information (one or a number of bit manipulations ). The known universe could, as a conclusion, be simulated by a computer capable of saving about 10⁹⁰ bits and manipulating them, and could very well be a simulation. Should this be the case, then hypercomputation would be impossible.

Loop quantum gravity could lend support to digital physics, in that it assumes space-time to be quantized. Paola Zizzi has formulated a realization of this concept in what has come to be called computational loop quantum gravity, or CLQG. Also combining aspects of Digital Physics with Loop Quantum Gravity are theories by Marzuoli and Rasetti, and more recently work by Girelli and Livine.

Theory of Ur-Alternatives

Physicist Carl Friedrich von Weizsäcker proposed the theory of ur-alternatives in his book "Einheit der Natur" (1971) and further developed it in his book "Zeit und Wissen" (1992). The theory is an example of digital physics as it axiomatically constructs quantum physics from the distinction between empirically observable, binary alternatives. Weizsäcker used it to derive the 3-dimensionality of space and to estimate the entropy of a proton falling into a black hole.

It from bit

Physicist John Archibald Wheeler followed Jaynes' and Weizsäcker's footsteps when he wrote "it is not unreasonable to imagine that information sits at the core of physics, just as it sits at the core of a computer". David Chalmers of the Australian National University summarised his views as:

"Wheeler (1990) has suggested that information is fundamental to the physics of the universe. According to this "it from bit" doctrine, the laws of physics can be cast in terms of information, postulating different states that give rise to different effects without actually saying what those states are. It is only their position in an information space that counts. If so, then information is a natural candidate to also play a role in a fundamental theory of consciousness. We are led to a conception of the world on which information is truly fundamental, and on which it has two basic aspects, corresponding to the physical and the phenomenal features of the world".

Digital vs. Informational physics

Not every informational approach to physics (or ontology) is necessarily digital. According to Luciano Floridi, informational structural realism [1] is a version of structural realism that supports the ontological commitment to a view of the world as the totality of informational objects dynamically interacting with each other. Such informational objects are understood as constraining affordances. Digital ontology and pancomputationalism are also independent positions. Famously, Wheeler supported the former but not (or at least not explicitly) the latter. As he wrote: “It from bit. Otherwise put, every ‘it’ – every particle, every field of force, even the space-time continuum itself – derives its function, its meaning, its very existence entirely – even if in some contexts indirectly – from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. ‘It from bit’ symbolizes the idea that every item of the physical world has at bottom – a very deep bottom, in most instances – an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes–no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe” (John Archibald Wheeler [1990], 5). On the other hand, pancomputationalists like Lloyd [2006], who describes the universe not as a digital but as a quantum computer, can still hold an analogue or hybrid ontology. And informational ontologists like Sayre and Floridi do not have to embrace either a digital ontology or a pancomputationalist position. (src: Floridi talk on Informational Nature of Reality, abstract at the E-CAP conference 2006)

Computational foundations

Turing machines

Theoretical computer science is founded upon the concept of a Turing machine, a hypothetical computer described by Alan Turing in 1936. Although they are mechanically simple, it turns out, as stated in the Church-Turing thesis, that Turing machines are powerful enough to solve any "reasonable" problem. (For theoretical computer scientists, "power" is the ability to solve problems at all rather than solving them quickly). A Turing machine therefore sets the practical "ceiling" on computational power, apart from the hypothetical possibilities presented by hypercomputers.

The principle of computational equivalence, as Stephen Wolfram calls it, is a powerful motivation behind the digital approach. If correct, it means that everything can be computed by the same machine, and by an essentially simple machine, thus fulfilling the traditional requirement in physics to find simple underlying laws and mechanisms.

Digital physics is falsifiable: a less powerful class of computers cannot simulate a more powerful class. Therefore, if our universe is being simulated, a computer at least as powerful as a Turing machine is being used. If we find or build a hypercomputer, on the other hand, we cannot be simulated by a Turing machine.

The Church-Turing (Deutsch) thesis

The modest version of the Church-Turing thesis claims that any computer as powerful as a Turing machine can calculate anything a human can calculate, given enough time. A stronger version claims that a Universal Turing machine can compute anything whatsoever, i.e., it is not possible to build a hypercomputer, a super-Turing computer. But the limits of practical computation are imposed by physics, not by theoretical computer science:

"Turing did not show that his machines can solve any problem that can be solved "by instructions, explicitly stated rules, or procedures", nor did he prove that the universal Turing machine "can compute any function that any computer, with any architecture, can compute". He proved that his universal machine can compute any function that any Turing machine can compute; and he put forward, and advanced philosophical arguments in support of, the thesis here called Turing's thesis. But a thesis concerning the extent of effective methods -- which is to say, concerning the extent of procedures of a certain sort that a human being unaided by machinery is capable of carrying out -- carries no implication concerning the extent of the procedures that machines are capable of carrying out, even machines acting in accordance with ‘explicitly stated rules’. For among a machine's repertoire of atomic operations there may be those that no human being unaided by machinery can perform."

On the other hand, if two further conjectures are made, along the lines that:

1. that hypercomputation always involves actual infinities
2. that there are no actual infinities in physics

...the resulting compound principle does bring practical computation within Turing's limits.

As David Deutsch expresses it:

I can now state the physical version of the Church-Turing principle: "Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means." This formulation is both better defined and more physical than Turing's own way of expressing it. (Emphasis added)

This compound conjecture is sometimes called the strong Church-Turing thesis, or the Church–Turing–Deutsch principle.

Criticism

The critics — perhaps including some physicists who work with quantum mechanics — argue against digital physics in a number of ways.

Continuous Symmetries

One objection is that the models of digital physics are incompatible with the existence of continuous symmetries such as rotational symmetry, translational symmetry, Lorentz symmetry, electroweak symmetry, and many others. Proponents of digital physics, however, reject the very notion of the continuum, and claim that the existing continuous theories are just approximations of a true discrete theory (the Planck length, for example, as a minimum meaningful unit of distance, suggests that space is at some level quantized).

Locality

Some argue that the models of digital physics violate various postulates of quantum physics. For example, if these models are not based on Hilbert spaces and probabilities, they belong to the class of theories with local hidden variables that some think have been ruled out experimentally using Bell's theorem. This criticism has two possible answers. First, any notion of locality in the 'digital' model doesn't necessarily have to correspond to locality formulated in the usual way in the emergent spacetime. A concrete example of this case was recently given by Lee Smolin. Another possibility is a well known loophole in Bell's theorem, known as superdeterminism (sometimes referred to as predeterminism). In a completely deterministic model, the experimenter's decision to measure certain components of the spins is predetermined. Thus, the assumption that the experimenter could have decided to measure different components of the spins than he actually did is, strictly speaking, not true.

Real numbers

It can be argued that any physical theory involving real numbers (and all major theories do, at the time of writing) poses problems. Known physics is held to be computable, but that statement needs to be qualified in various ways. A number — thinking particularly of a real number, one with an infinite number of digits -- is said to be computable if a Turing machine will continue to spit out digits endlessly. In other words, there is no question of getting to the "last digit". But this sits uncomfortably with the idea of simulating physics in real time (or any plausible kind of time). Known physical laws (including those of quantum mechanics) are very much infused with real numbers and continua.

"So ordinary computational descriptions do not have a cardinality of states and state space trajectories that is sufficient for them to map onto ordinary mathematical descriptions of natural systems. Thus, from the point of view of strict mathematical description, the thesis that everything is a computing system in this second sense cannot be supported".

Moreover, the universe seems to be able decide on their values on a moment-by-moment basis. As Richard Feynman put it:

"It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?

However, he went on to say:

So I have often made the hypotheses that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities. But this speculation is of the same nature as those other people make – ‘I like it’, ‘I don’t like it’, – and it is not good to be prejudiced about these things".

Criticism rebutted

However, most of the above criticism is obviously in error. To see this, it is sufficient to realize the following facts: The generating functions, which are fundamental objects of number theory, and partition functions, which are fundamental objects of statistical thermodynamics, quantum mechanics and even string theory, are intrinsically the same objects. As the partition functions are manifestly invariant with respect to all symmetry operations required, also all number-theoretic consequences of the generating functions are inevitably invariant.

It is also well-known that Möbius group and its subgroups (modular group) feature as the symmetry operations and fundamental tools in number theory. However, it is well-known that the Möbius transformation is just the Lorentz transformation of special theory of relativity in disguise. Consequently, all logical and number-theoretic facts remain Lorentz-invariant as well. Taken also into account that probability theory may be treated as a unique extension of Aristotle's logic, as proven by E. T. Jaynes and R. T. Cox, the link between the discrete world of logic (digital computers) and possibly continuous world of probabilities may be established, including all the underlying symmetry operations required by physics. The interpretation of probability theory, as an extended logic, is compatible with locality requirements and particularly convenient for the model.

Consequently, the model of digital physics is principally consistent with other fundamental theories of physics, including their underlying symmetries. A generalization to continuous and probabilistic alternatives of the model, rather than discrete and logical alternatives, might encompass quantum computers as well.

See also

• A New Kind of Science by Stephen Wolfram
• Cellular automata
• Church-Turing thesis
• Continuous spatial automata
• Digital philosophy
• Digital probabilistic physics
• Ed Fredkin
• Holographic principle
• Hypercomputation
• Physical information
• Quantum computation
• Seth Lloyd
• The Fabric of Reality by David Deutsch
• Paola Zizzi
• Konrad Zuse

External links

• Paolo Gontigno: Hypercomputation and the Physical Church-Turing thesis
• Petrov, Plamen, and Joel Dobrzelewski, "Digital Physics". 1998.
• "Digital physics". Mountain Math Software.
• Schmidhuber, Juergen "Algorithmic Theory of Everything, 1997-2002".
• Scan of Zuse's paper in PDF
• Translation of Zuse's Rechnender Raum in English (MIT): "ftp://ftp.idsia.ch/pub/juergen/zuserechnenderraum.pdf"
• The Oxford Advanced Seminar on Informatic Structures
• Wired: God is the Machine
• The World as Evolving Information, C. Gershenson, New England Complex Systems Institute and C.L. Apostel, Vrije Universiteit Brussels