July 30, 2014
Old family photo posts have been moved to a new page.
Old family photo posts have been moved to a new page.
July 19, 2014
A Curious Way to Represent Numbers: Ternary Factor Tree Representation
This post is only likely to be of interest to those interested in the hidden backwaters of math, and maybe not too many of them. It also has little to do with the other posts here so far.
The conventional system of number representation cannot exactly represent most numbers. Fractions that have factors in the denominator that are not in the number system's base have infinite decimal representations (e.g. other than 2 and 5 for base 10). Square roots and other irrational numbers \ have infinite, non-repeating representations.
In the late 1990s I came up with an alternative system that can exactly represent any rational or irrational number as well as most transcendental numbers using only a finite number of bits. This type of representation is based upon the idea of factored representations of integers extended in a logical way to a nearly universal system of describing number structure.
The conventional system of number representation cannot exactly represent most numbers. Fractions that have factors in the denominator that are not in the number system's base have infinite decimal representations (e.g. other than 2 and 5 for base 10). Square roots and other irrational numbers \ have infinite, non-repeating representations.
In the late 1990s I came up with an alternative system that can exactly represent any rational or irrational number as well as most transcendental numbers using only a finite number of bits. This type of representation is based upon the idea of factored representations of integers extended in a logical way to a nearly universal system of describing number structure.
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| Pretty but unrelated |
July 2, 2014
Universe, Physics and Simulation
[Another post from the archives, this time from late March 2014. I overlooked it or I would have posted it earlier.]
The chief business of science, particularly physics, is modeling of the universe's phenomena. Modeling phenomena is also a definition of simulation. The universe may not be information, but all we can know of it is information. The universe may not be simulation but all the theories that we can make about it can only be tested through simulation and comparison of the information from the simulation with that from the universe.
The universe may in fact have characteristics of a simulation; it seems likely that models designed to resemble the universe will do so, and therefore that the universe will resemble the models just as well as the reverse – sometimes in unforeseen ways. Some of the characteristics of models that are commonly thought to be artificial or mere approximations may be capable of telling us secrets of how the universe really works.
Physicists spend a great deal of time with equations, but it is only when actual numbers representing a given situation are plugged in that they can these equations be said to be a representation of anything in the physical world; in fact, to represent any kind of fields in any but the simplest situations demands iterating the equations with numbers for every point in space and all velocities or wavelengths, which means plugging in astronomical numbers of coefficients to the equations even for a crude approximation.
How to minimize the number of computations for a given level of accuracy and complexity is the central concern of simulation. For instance, often space is divided into a mesh which is sparse where the situation is simple and dense in more complicated regions. Another commonly-used technique is rather than storing enormous matrices with most entries zero, only storing the few entries containing informative numbers. Other types of compression are also used whenever possible, and compression itself is a rich subject.
If the universe resembles a simulation, it too should show evidence of compression techniques. One obvious one is to only store one copy of identical items, and just use a pointer to that copy wherever another such item appears. A bit more advanced is to only store the differences between near-identical items, together with a single prototype copy as above. This is essentially a programmer's sort of platonism. Even beyond that is compression of analogous structures more generally, which can quickly become quite complex.
This compression effect would also potentially seem able to account for some of the observations that led Rupert Sheldrake to propose the existence of morphic fields and morphic resonance. Once a prototype structure exists, it takes much less computation and storage for the universe to support similar structures, assuming the universe is simulation-like in compressing form. Crystals of new substances should be easier to create again once they have first formed elsewhere. Biological structures, behaviors, and for want of better words what I'll call “plots” and “tropes” should be similarly primed if the compression algorithm can handle such subtle and complex analogies.
More speculatively, if the universe's resemblance to a simulation is not mere appearance, then teleological questions of who is running the simulation and for what purpose arise. The biggest potential increment of efficiency for such an entity in simulating a universe would come from only accurately simulating the regions and events of interest, with sparse and approximate methods used in other regions. This could lead to glitches in the simulation such as are often seen in video-games: lag and other time discontinuities, failure to load sections of the simulation properly, changes in item prototypes, non-player characters not performing when the simulation does not register that the NPCs non-action is perceptible to the player characters, conflicts between versions of the simulation when different PCs high-detail regions come into contact, more radically different simulations coming into contact, continuity errors, objects and characters failing to load or loading twice, violations of physical law, cheat codes, hacking, ... All these, or similar effects have been reported numerous times by different people. (See accounts on the “Glitch in the Matrrix” sub-Reddit) It is often the case that their brains are glitching rather than the exterior simulation, but this sometimes seems to be ruled out by corroboration from other witnesses or by physical evidence. Sometimes these glitches seem purposeful, as when avoiding certain death or when missing items reappear in response to a request. Often, though, they seem to be true glitches, either mistakes or with no apparent purpose other than perhaps revealing the simulated nature of things.
It could also be possible that the universe is natural (more-or-less), with the simulation-like aspects being not artifacts but implicit in the universe's necessary informational self-consistency. Nevertheless, conscious beings arising in the natural universe could learn to hack it from the inside, causing glitches and intimations of purposefulness for other, less adept residents of the universe. The general rule of self-consistency is likely only relative to a given branch of implications of occurrences; inconsistencies define other branches of possibilities. (Perhaps in the beginning was the inconsistency 0=1 : the big bang followed because all propositions and their opposites can be derived from a single contradiction -- but there are branching patterns in the successive derivations of implications from that initial seed.)
Also see Daniel Burfoot's quite readable book on ArXiv, “Notes on a New Philosophy of Empirical Science”, particularly pages 8 to 29. (arXiv:1104.5466 [cs.LG], version 1 April 2011)
The chief business of science, particularly physics, is modeling of the universe's phenomena. Modeling phenomena is also a definition of simulation. The universe may not be information, but all we can know of it is information. The universe may not be simulation but all the theories that we can make about it can only be tested through simulation and comparison of the information from the simulation with that from the universe.
The universe may in fact have characteristics of a simulation; it seems likely that models designed to resemble the universe will do so, and therefore that the universe will resemble the models just as well as the reverse – sometimes in unforeseen ways. Some of the characteristics of models that are commonly thought to be artificial or mere approximations may be capable of telling us secrets of how the universe really works.
Physicists spend a great deal of time with equations, but it is only when actual numbers representing a given situation are plugged in that they can these equations be said to be a representation of anything in the physical world; in fact, to represent any kind of fields in any but the simplest situations demands iterating the equations with numbers for every point in space and all velocities or wavelengths, which means plugging in astronomical numbers of coefficients to the equations even for a crude approximation.
How to minimize the number of computations for a given level of accuracy and complexity is the central concern of simulation. For instance, often space is divided into a mesh which is sparse where the situation is simple and dense in more complicated regions. Another commonly-used technique is rather than storing enormous matrices with most entries zero, only storing the few entries containing informative numbers. Other types of compression are also used whenever possible, and compression itself is a rich subject.
If the universe resembles a simulation, it too should show evidence of compression techniques. One obvious one is to only store one copy of identical items, and just use a pointer to that copy wherever another such item appears. A bit more advanced is to only store the differences between near-identical items, together with a single prototype copy as above. This is essentially a programmer's sort of platonism. Even beyond that is compression of analogous structures more generally, which can quickly become quite complex.
This compression effect would also potentially seem able to account for some of the observations that led Rupert Sheldrake to propose the existence of morphic fields and morphic resonance. Once a prototype structure exists, it takes much less computation and storage for the universe to support similar structures, assuming the universe is simulation-like in compressing form. Crystals of new substances should be easier to create again once they have first formed elsewhere. Biological structures, behaviors, and for want of better words what I'll call “plots” and “tropes” should be similarly primed if the compression algorithm can handle such subtle and complex analogies.
More speculatively, if the universe's resemblance to a simulation is not mere appearance, then teleological questions of who is running the simulation and for what purpose arise. The biggest potential increment of efficiency for such an entity in simulating a universe would come from only accurately simulating the regions and events of interest, with sparse and approximate methods used in other regions. This could lead to glitches in the simulation such as are often seen in video-games: lag and other time discontinuities, failure to load sections of the simulation properly, changes in item prototypes, non-player characters not performing when the simulation does not register that the NPCs non-action is perceptible to the player characters, conflicts between versions of the simulation when different PCs high-detail regions come into contact, more radically different simulations coming into contact, continuity errors, objects and characters failing to load or loading twice, violations of physical law, cheat codes, hacking, ... All these, or similar effects have been reported numerous times by different people. (See accounts on the “Glitch in the Matrrix” sub-Reddit) It is often the case that their brains are glitching rather than the exterior simulation, but this sometimes seems to be ruled out by corroboration from other witnesses or by physical evidence. Sometimes these glitches seem purposeful, as when avoiding certain death or when missing items reappear in response to a request. Often, though, they seem to be true glitches, either mistakes or with no apparent purpose other than perhaps revealing the simulated nature of things.
It could also be possible that the universe is natural (more-or-less), with the simulation-like aspects being not artifacts but implicit in the universe's necessary informational self-consistency. Nevertheless, conscious beings arising in the natural universe could learn to hack it from the inside, causing glitches and intimations of purposefulness for other, less adept residents of the universe. The general rule of self-consistency is likely only relative to a given branch of implications of occurrences; inconsistencies define other branches of possibilities. (Perhaps in the beginning was the inconsistency 0=1 : the big bang followed because all propositions and their opposites can be derived from a single contradiction -- but there are branching patterns in the successive derivations of implications from that initial seed.)
Also see Daniel Burfoot's quite readable book on ArXiv, “Notes on a New Philosophy of Empirical Science”, particularly pages 8 to 29. (arXiv:1104.5466 [cs.LG], version 1 April 2011)
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