Commentary on The Finite Nature Hypothesis of Edward FredkinRoss Rhodes RhodesR@BottomLayer.com ver. 1.0 2/6/2000 Contents
If we assume that all things in nature are discrete, it follows that all things can be described by a finite set of information. This set of information must be accompanied by a process causing it to evolve from one "frame" of time to the next. The most compelling model for meaningful transformation of one finite set of information to another is the computer, and in particular the computer architecture known as "Cellular Automata."
Finite Nature: The hypothesis that all things are discrete rather than continuous; "grainy" rather than "smooth"; step-wise rather than flowing. Exploring finite quantities A finite amount of information cannot evolve without a processing mechanism. Information + Process implies a mechanism of computation. Recapitulation The Cellular Automaton computer architecture
Exploring physics as though it were the product of programming run on a computer requires that we explain how the physics of the natural world can be produced by a computer program. Two aspects of physics as we know it -- quantum randomness and reversibility -- are shown to be possible within the framework of familiar computer architecture.
The Universal Computer Unknowable determinism as an effective source of quantum randomness Reversibility of the programming.
The Consequences of Finite Nature
A. The fundamental process that we know as the physics of the natural world is an informational process. B. Transformation of information is accomplished by a mechanism of computation which is not essentially different from the familiar computers of our technology. C. The physics of the natural world must be analyzed according to what is necessary and possible for computers to accomplish. D. There is not, and there need not be, access to the hardware of the Engine from within the program.
Information + Process = Computation
The Scale of the Simulation
[Top] ForewordIn 1992, Edward Fredkin published two papers which are the
indispensable shots-across-the bow in understanding the universe in which we live as an artifact produced by programming on a computer. An earlier profile of Fredkin published in The Atlantic Monthly in 1988 had stated the gist of Fredkin's thinking on the subject. The article had noted that many of Fredkin's ideas about physics and computing were acceptable to the scientific community, but his conclusions were not considered to be within the realm of science. "If Fredkin ... stopped at saying that the universe operates as if it were a computer, he could improve his stature among physicists...."^{[1]} Fortunately for the rest of us, he did not stop there: the universe operates as if it were a computer because it is a computer.This paper is an attempt to restate the first of Fredkin's papers, "Finite Nature," as I have understood and appreciated it. I encourage the reader to refer to Fredkin's paper, which is available at http://www.im.lcs.mit.edu/poc/fredkin/Finite-Nature [or http://www.digitalphilosophy.org/finite_nature.htm].
Ross Rhodes
[Top] Part I
Scientific investigation repeatedly has revealed the discrete qualities of nature. We know from high school that all things are made of individual atoms. That is, regardless of whether a substance appears to be solid steel or flowing water or chilly wind, it turns out that all things are made up of discrete, individual molecules and atoms. Similarly, the insubstantial energy of electricity and light
has been shown to consist of discrete "chunks," or
After many years of such investigations, it is now difficult to imagine that any phenomenon of our universe will prove to be continuous in nature.[2] Accordingly, Edward Fredkin's hypothesis of Finite Nature begins with the assumption that
In the world of physics, Exploring finite quantities
Given Finite Nature, there are no approximations, no subjective values. A collection of three coins, in our example, can be arranged and rearranged in only a finite number of ways before one runs out of possible combinations. Flipping one or another will yield different state-of-flip arrangements with exactly 8 possibilities, no more and no fewer:[3]
Similarly, the possible place-of-minting values will be exactly 27, no more and no fewer:
Choosing to describe both of these properties for our collection of three coins (Ph-Ph-Ph, Ph-Ph-Pt, etc.) increases the number of possibilities in a tidy mathematical way, so that resulting number of possibilities, while large, remains exact and decidedly finite.[4]
Fredkin's hypothesis of Finite Nature sees this restriction on the amount of information needed to provide a full and complete description of any aspect of physics as the prime implication of the assumption that all things are discrete, step-wise, or grainy. If all of space-time and all physical processes are fundamentally discrete, then for any given unit of space-time (i.e., for any volume of anything, or even a volume of nothing) there will be a limited, finite number of combinations that will describe all of the possible states of everything contained within that volume. Put another way, this is to say that for every volume of space-time there will be a definite and finite amount of
When Fredkin invokes the concept of "information," he is speaking of a "scalar quantity," that is, something which is "capable of being represented by a point on a scale," or "a quantity that has a magnitude describable by a real number and no direction." These are properties that can be quantified, properties that can be expressed as numbers. How fast? 60 m.p.h. (as opposed to the alternative non-scalar description, "rapidly"). Finite Nature supposes that all properties can be
expressed by numbers because all properties are discrete and step-wise. In this sense, Fredkin is referring to the data itself as the relevant information, rather than the meaning associated with the data. Accordingly, the information can as easily be expressed by numbers as by characters, words or sentences. Thus, heads could be represented by "0" and tails by "1", so that heads-heads-tails (or hht) could be represented by "001". The To assert that all information in nature consists of scalar quantities is a radical statement. We have become accustomed in mathematics and in the natural world to contemplating the infinite -- a series that goes on forever without end -- and its converse, the infinitesimal -- a hypothetical smallness that has no minimum. The invention of calculus by Newton and Leibnitz provided the mathematical tools for taming the philosophical concepts of the infinite and the infinitesimal. Engineers and scientists, for the most part, have stopped thinking about these concepts as problems because they no longer present any barrier to satisfactory calculation.[6] Consequently, the areas in which these twin concepts are most in conflict with our sensibilities -- the realm of time and space, or simply space-time -- are commonly thought of as continuous because, first, time and space appear to be continuous; second, the mind reels at the contemplation of any alternative; and third, space-time may safely be assumed to be continuous with no loss of function. However, to say that it is convenient to credit the appearance of continuity is not at all to say that space and time actually are continuous. The examples given from atomic theory in chemistry, and quantum theory in physics, serve as reminders that close examination can reveal step-wise, discrete properties at any (and perhaps every) turn. Fredkin supposes that investigation will show that there is no infinite and no infinitesimal; all things in nature and physics are finite. A finite amount of information cannot evolve without a processing mechanism. Information + Process implies a mechanism of computation.
We may ask how this change can be accomplished. In light of our assumption that time itself is discrete and step-wise, we are not permitted to imagine the information sliding or morphing from one state to the next; no, the information must be in one state in the This exercise recalls Zeno's paradoxes of motion involving space and time. Zeno argued that motion (i.e., an evolution from one position to the next) is not possible when space and time are discrete qualities.[7] Space is divided into some type of blocks or cells; time is divided into steps marked by ticks of the clock. Accordingly, at any given time step 1, the block or cell exists (by definition) wholly and completely in state 1. The finite amount of information necessary to make a complete description of the space-time unit will not change -- it cannot change, because any change would involve intermediate time steps which, according to our fundamental assumption of Finite Nature, do not exist. Nevertheless, at time step 2, the same block or cell exists in a different state 2! How does such a transformation occur?
The fact that we observe motion, and other aspects of a system which evolves over time, was the unsolvable problem for Zeno.[8] Fredkin, however, resolves the paradox neatly by reference to a system of apparent motion unknown to the ancient Greeks -- the common operations of a computer program. By reducing the volume of space-time to a symbolic representation of the
In order to better understand Finite Nature, Fredkin suggests that we look to examples of simple systems with similar properties. "A digital computer, exclusive of its input and output facilities (with just a processor and memory),[10] has many of the same properties that we find in Finite Nature." Like Finite Nature, a computer begins with one set of information and proceeds to another, different set of information. The computer's progress is always step-wise, running through a series of ritualistic modifications of the information according to a strict set of programming rules. It is the application of the programming rules that effects the change from the The computer's information is represented symbolically by the arrangement of binary switches in the computer's memory. The "state" of the computer is the aggregate arrangement of these memory bits. Consequently, when we consider the state of the computer, we must look at the arrangement in static, fixed form as it exists in its initial state and at the end of a step of programming. The arrangement cannot be changing as we consider it, because the "meaning" represented by the state of the computer depends on the relationship of each bit to every other bit. The computer considers the information coded in its "state" and applies its programming rules to that information, changing the arrangement of the memory bits according to the rules. As the memory bits are being changed, the internal arrangement of the computer is in a state of transition. If the process were stopped in the middle of this transitional phase (before all of the rules for this "step" were fully carried out), an observer looking at the arrangement would not be able to extract any meaning. The overall arrangement would be "wrong" because the application of the rules had not been finished. This would be a computer crash. To illustrate, suppose the memory bits were soldiers lined up in a row, and the rule was "take two steps forward, then take one step back." The sergeant must give these orders, one at a time, to each individual soldier, and each soldier then must carry out the orders. If we halted the process mid-application, we might see a very ragged line of soldiers because some of them had taken two steps forward and one step back, some of them had only taken two steps forward, and some had not yet moved. On the other hand, if we allowed the rule to be applied fully, we would see a tidy row of soldiers which had neatly advanced one step. A computer achieves this tidy progression through a hardware technique called single clock. As Fredkin explains, "Single clock means that when the computer is in a state, a single clock pulse initiates the transition to the next state. After the clock pulse happens, various changes propagate through the computer until things settle down again. Just after each clock pulse the machine lives for a short time in the messy real world of electrical transients and noise. At the points in time just before each clock pulse the computer corresponds exactly to the workings of a theoretical automaton or Finite State Machine." The single clock synchronizes the information processing in a computer, so that the programming can be applied in steps. At the tick of the computer's "clock," the programming is applied and the arrangement of the memory bits begins to change. After all memory bits have been affected, the clock tick is finished and the programming rule has been fully applied.
Recapitulation
The Cellular Automaton computer architecture
There is no difference in principle between the individual cellular automaton and any other computer. Both consist of a block of memory which is acted upon by a set of programming instructions. The difference in practical terms is that the cellular automaton operates according to a severely restricted set of instructions (programming), and so requires a comparatively modest amount of memory. With these limitations, we can create a vast number of cellular automata using our finite memory and programming resources. The key to the utility of the cellular automata ("CA") computer architecture is that when we assemble this vast number of simple, independent computing units, they can interact among themselves in breathtakingly complicated ways. To illustrate, let us consider two simple cellular automata, side by side, whose only function in life is to display a color -- either blue or green or red. The one and only rule each must follow is that it should ponder the color of the automaton next to it, and take the color which is next in line alphabetically. Thus, if its neighbor is green, it should take the color red; if red, it should take the color blue; if blue, it should take the color green. We can then watch as the situation changes over time (having arbitrarily assigned colors to each automaton to begin with):
. . . and so on. We have a pattern which, in this case, repeats itself every six steps. Exactly the same rule is applied at each time step, but the pattern is slightly complex. It is the interaction among the cellular automata which causes their states to evolve, and to evolve in a way that bears a relationship between past, present and future such that a pattern emerges.
Cellular Automata theory considers each automaton as a "cell" surrounded by other cells.
The simplicity of CA architecture gives rise to a vast complexity of interaction which, in turn, produces pleasing patterns of
Fredkin concludes: "Given Finite Nature, what we have at the bottom is a Cellular Automaton of some kind." And, because "Automata, Cellular Automata and Finite State Machines are all forms of computers," this is to say that at the bottom of the physics of the natural world, we have a computer of some kind. [Top]
Part II
If the ultimate computer is universal, as it must be, then we should be able to simulate or, more properly, emulate its operations by the correct programming of our own feeble computers (in the way that a good programmer should be able to emulate a Cray supercomputer using a desktop PC). All of physics as we know it should be expressible as a computer program. As Fredkin puts it, "The difficulty here is that there are features of our world which have long been thought to be impossible to duplicate by computer. If these feats are truly impossible for a computer, then Finite Nature must fail. However, Fredkin shows that the most problematical of these tasks -- true quantum randomness and the reversibility of computation -- are not the insurmountable barriers they were once thought to be. Unknowable determinism as an effective source of quantum randomness
"Uncertainty is at the heart of quantum mechanics. Finite Nature requires that we rule out true, locally generated randomness because such numbers would not, in this context, be considered finite. The reason is that there is no way to create, within a computer, a truly random number that is orthogonal to everything in the computer.""Orthogonal to everything in the computer" means completely unrelated to anything in the computer. True randomness has defied computer scientists because a number -- even a "random" number -- must be produced by the application of some rule embodied in the programming code, and the rule must by its nature refer to something in the computer (such as the value stored in a particular memory location). Quantum Mechanics, in its standard interpretations, requires something even more random than this: an element of complete and utter randomness.To solve the problem of computer-generated randomness necessary for a satisfactory model of Quantum Mechanics, Fredkin invokes the concept of "unknowable determinism," a term that is at first blush an oxymoron. Determinism implies a set of rules that can be applied to obtain the eventual result; unknowable implies the absence of any method for obtaining the eventual result. Nevertheless, Fredkin posits a situation exhibiting just these two apparently contradictory facets.
A peculiar aspect of the operations of Quantum Mechanics is that its operations may be influenced not so much by what
Given these limitations, running the simulation would not obtain the eventual result until after that result had already occurred on the "original" cellular automaton (or, in the very best hypothetical case, at the same time as the result occurs on the original). Predicting a situation which has already occurred (or presently exists) would not satisfy the Quantum Mechanical definition of a state of affairs which Reversibility of the programming.
If the physics of our universe is
In 1982, Fredkin developed a model of a computer which [Top] Part III
In his paper, Fredkin lists "some of the more obvious" consequences of the hypothesis that all things in nature are discrete, quantifiable and describable within finite limits. In part, these reiterate the major points of the paper, with some of the statements going into further details of digital physics and how it can model different aspects of the natural world.[21]Fredkin's final point anticipates the second of his seminal papers on this subject, "A New Cosmogony," by introducing the concept of the parts of the model which by their nature are not themselves contained within physics, and so are not to be found in this universe.
[Top] cells and bits. It is the result of the fundamental process run in the cells with the bits. 2. [7.] Information must have a means of its representation. If we had the right kind of magic microscope, we should always be able to see the digits that represent whatever information is present. Information is never "just there".[22]
3. [16.] Physics is a consequence of the
[Top]
5. [1.] The 6. [8.] What cannot be programmed cannot be physics. This is a very important idea. If a process cannot be programmed on a particular universal computer, despite having enough memory and time, then it cannot be programmed on any computer. If we can't program it on an ordinary computer, Finite Nature implies it can't be a part of physics because physics runs on a kind of computer.
7. [17.] The
8. [18.] The
9. [19.] Any universal
10. [20.] Physics is totally independent of the characteristics of the
[Top]
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[Top]
[Top] Part IVInformation + Process = Computation Fredkin illustrates the fundamental properties of systems which evolve -- that is, systems which undergo changes in an ordered way such that recognizable and meaningful properties emerge. From the realm of biological systems, Fredkin comments on the wonder of cloning trees (or anything else, for that matter). Beginning with a single cell undergoing successive divisions, we eventually arrive at a tree -- with all of its structured root system, and protective bark, and chlorophyll factories for leaves, and everything that makes a recognizable, functioning tree.Cloning a tree (as growing a tree from a seed) depends on the pre-existence of a store of information constituting a blueprint or map of the eventual tree. The design of the whole tree exists in the genetic material of the single cell -- no matter that the cell may come from the root or the bark or the leaf. Thus, the first fundamental property of an evolving system is structured information. To explore the concept further, Fredkin asks why one cannot "clone" a Boeing 747 from scrapings taken from one of the passenger seats. The first answer is that a scraping of vinyl does not contain the blueprint of a 747; it might as well have been scraped from a Chevy Nova or a sofa in the den or any other bit of vinyl. It is not connected, in any informational sense, with a 747. There is no way, even in principle, to discern a Boeing 747 in a scraping of vinyl.
Suppose, then, that we begin with the blueprint of a Boeing 747 (or a microchip or any other manufactured product) contained on a floppy disk. There We see that an evolving system must begin with information -- whether in DNA strings or on a floppy disk or otherwise -- and that information must be matched to a process suitable for transforming the information into the final structure. For seeds, the process is self-replication, requiring an environment containing the elements found within the cell itself. For blueprints of mechanical systems, stored on floppy disks, the far clumsier process is a computer capable of reading and understanding the stored information, and acting upon it with all manner of ancillary manufacturing systems. Accordingly, we begin with information in a digital state. (The genetic material of a tree or other living cell satisfies this definition through its molecular arrangement, which is not different in principle from any other data array.) We must then match the information to a process which will act upon the information and transform it into its eventual, future state. Information matched to a transformative process. This is the recipe for development. It is also a fair description of the workings of a computer, in particular a cellular automaton. Given Finite Nature, it is ultimately a description of nature as physics has come to understand it. To Fredkin, this convergence is more than a curiosity: it is an insight into the fundamental workings of our world. Finite Nature means that our world operates as though it were the product of a computing system, and Fredkin sees that this is because our universe is an artifact produced by a computer of some sort.
Canton, Ohio
[Top]
Appendix
We might imagine asking, "How small (or large) is Windows98(TM)?" It would be nonsensical to respond, "Windows98(TM) is 17.7205875 sq. in.," which is approximately the surface area of a standard CD-ROM disk. The true measure of the size of Windows98(TM) is its information content, i.e., a certain number of megabytes. Properly speaking, then, we are asking how much information is contained in the fundamental units of physics, and how many smaller blocks of information must be cobbled together to produce that level of complexity. Nevertheless, there are ways to relate the scale of the automata to the scale of the phenomena with which we are familiar. One proposed "natural" unit of length is the Fermi, which is the range of the strong nuclear force. Another much smaller unit is the Planck length, which is derived from the three fundamental physical constants of gravity (G), the speed of light (c), and the quantum of action (h).
In the final analysis, the scale does not impact on the features of the cellular automaton computer being investigated. Whatever the scale, we may think of the fundamental units of length, mass, time, and all other properties as "1", at which point the scale is immaterial. Without choosing among these possible "fundamental" units of length, Fredkin notes that the number of such units contained in the area occupied by, say, an electron (considering the size of an electron as its wavelength, and not as a point particle), is enormous. At any plausible scale, such a vast number of cells, considered as bits of data, would be sufficient to represent a huge amount of information. |

1. [Back] | R. Wright, "Did the Universe Just Happen?" The Atlantic Monthly, April 1988,
29, 41. |

2. [Back] | On the one hand, such an experimental proof would be problematical because no matter how closely we look, we can always imagine that if we looked even more closely the continuity would resolve into a finely-grained discreteness; therefore, the appearance of continuity in nature, as in the motion picture theater, can never itself be proof of actual continuity. |

3. [Back] | The reader can see why this limitation applies only to discrete properties. If we were to allow intermediate states such as the coin standing on its edge, and all of the instantaneous positions through which it can pass while rotating as it is being flipped, then there would be an infinite number of possibilities for the "state" of the coin. The difference between eight possible states and an infinite number of possible states is our hypothesis that the coin can only exist in a complete state of being "heads," or alternatively in a complete state of being "tails," but never in between. Oddly, the scientific proofs that natural phenomena are fundamentally discrete rely on just this type of puzzling limitation. |

4. [Back] | The possibilities would be the cube of 2 (three coins with two possibilities for each coin) times the cube of 3 (three coins with three possibilities for each coin), i.e., 2 ^{3} x 3^{3} = (2 x 2 x 2) x (3 x 3 x 3) = 216 possibilities. |

5. [Back] | The meaning is anything the programmer wants it to be. As David Eck puts it, "Suppose I were to point to some particular sequence of bits inside a computer and ask what it represents. Without further information, the answer could be almost anything--the current date, the color of some particular pixel on the screen, the board position in a game of computer chess, Joe DiMaggio's batting average in 1939 .. .. What it actually means is determined not just by the sequence of bits but also by the physical structure of the computer itself, by the overall structure of the data encoded in the computer, by the program that is running, and by the intentions of the person using the computer." D. Eck, The Most Complex Machine, at 11 (A.K. Peters, Ltd., Wellesley, MA 1995). |

6. [Back] | See, xx Scientific American; xx Scientific American. |

7. [Back] | By a different logic, Zeno argued with equal fervor that motion also is not possible when space and time are continuous qualities. |

8. [Back] | Actually, we know little of Zeno's preferred solution, if any, because his paradoxes come to us through Aristotle's tracts ridiculing the arguments themselves. Perhaps Zeno was a Fredkin ahead of his time. |

9. [Back] | To those who might object that a symbolic representation of a coin would not substitute for a real coin in one's pocket, consider whether there is any difference in purchasing power between the change in your pocket and the digital information held by your bank, representing your account balance. Either will suffice to obtain a new toy at the store, and the cashier does not care whether you hand over a fistful of change or swipe your debit card through the validation machine. |

10. [Back] | The question of the input and output facilities is a fascinating one which, to this author, completes the model of the hows and whys of human experience. See R. Rhodes, "A Cybernetic Interpretation of Quantum Mechanics" at 5 (1999). However, the input and output facilities are somewhat arbitrary in the sense that the information as processed by the computer can be read in many different ways according to the convenience of the user. Accordingly, Fredkin focuses on the internal operations of the computer, without analyzing how this information might be put into usable form. |

11. [Back] | G. H. Whitrow, The Natural Philosophy of Time, Clarendon Press, Oxford (2nd ed.) 1980. |

12. [Back] | In his paper "Finite Nature," Fredkin does not address the ramifications of the E-P-R effect, whereby a quantum unit can be "influenced" by other quantum units far removed from its immediate neighborhood. However, other aspects of computer architecture may provide the most plausible explanation for this phenomenon. See R. Rhodes, "A Cybernetic Interpretation of Quantum Mechanics" at 13 (1999). |

13. [Back] | This is the case for General Relativity theory, as well. |

14. [Back] | See R. Rhodes, "A Cybernetic Interpretation of Quantum Mechanics" at 15 (1999). |

15. [Back] | Determinism in nature has been a holy grail for physicists of the neo-reality school since Einstein first expressed his intellectual dissatisfaction with true quantum indeterminacy ("God does not play dice"). From one perspective, quantum indeterminacy implies that there is nothing "real" beneath the mathematical operations of quantum mechanics, and many theorists have attempted to imagine some type of "gears and wheels" that would produce the quantum effects we observe. Suffice it to say that the goal of discovering an underlying determinacy in quantum mechanicsm -- a satisfying "reality" at some fundamental level of our universe -- has proven elusive. It is ironic that Digital Mechanics approaches the problem by attempting to discover the programming producing the effects. Query whether this would satisfy those longing for something "real"? |

16. [Back] | See R. Feynman, QED, The Strange Theory of Light and Matter at 82 (Princeton Science Library, Princeton, NJ 1985). |

17. [Back] | E.g., S. Hawking, A Brief History of Time, 143-53 (Bantam, New York, 1988). |

18. [Back] | See R. Feynman, QED, The Strange Theory of Light and Matter, at 97-99 (Princeton Science Library, Princeton, NJ 1985). |

19. [Back] | E. Fredkin and T. Toffoli, "Conservative Logic," International Journal of Theoretical Physics, vol. 21, nos. 3/4, 1982, pp. 219-253. |

20. [Back] | Caveat that we are not describing "particles" in the sense of atoms or electrons or anything else in the particle zoo of the Standard Model. Instead, we are describing a method of computing which can substitute for the transistors of the desktop computer chips, but accomplish the same task of processing information. The information contained in the position, momentum and direction of the billiard balls is simply substituting for the billions of on-off switches in conventional transistors. |

21. [Back] | For the most part, these short statements are set forth verbatim from Fredkin's original paper. They have been only rearranged, according to this commentator's sense of a logical sorting. Their original order is indicated in brackets. |

22. [Back] | This is a metaphorical statement which seems at odds with the model being proposed. The key here is Fredkin's use of the term "magic," because a being residing inside the physics produced by the engine cannot view the actual source code which resides, as Fredkin later asserts, outside of the universe.However, upon further discussion with Prof. Fredkin, it appears that he intends the concept literally, the only "magic" being the extremity of the magnification. In Fredkin's concept of a completely closed universe, there does not appear to be any difference in principle between observation of a collection of billions of cells (i.e., a macroscopic object) and observation of a single cell, i.e., a single bit of information. Some background in his views on the nature of the observer may be found in his draft manuscript, "On the Soul." |

23. [Back] | Although Fredkin asserts this as an "exception," it is not clear why the calculations on a larger cosmic scale are different in principle from calculations on the smaller quantum scale. The "dividing line" between quantum and classical physics has been the topic of much debate (i.e., why do we not notice bizarre quantum behaviors at the relatively large scale of our five senses?). By some views, the disappearance of quantum effects at increasing scales involves the cancellation of microscopic effects on the macroscopic scale. By other views, it is a simple matter of approximation whereby the unpredictability of quantum effects is declared "negligible" at some point. General relativity is a case in point: it's equations operate at all times and in all situations; however at low energies the Newtonian equations are deemed "good enough." |