(co-author: Alexis Clancy)
Alan N. Shapiro: I believe that the invention of a new computer science, one more powerful than that which presently exists, is possible; a more powerful computer science that often goes by the name of Artificial Intelligence. Shapiro Technologies will go beyond the digital or binary computing paradigm that has persisted since the seminal work of the Second World War generation of information theorists such as Alan Turing, John von Neumann, Norbert Wiener, and Claude Shannon, so as to achieve quantum computing.
The goal of quantum computing has been clearly and explicitly defined by computer scientists, but the mathematics of how to implement qubits and superposition states does not yet exist. It should be noted right away that most efforts to realize quantum computing are, in my view, too one-sidedly hardware-centric.
A crucial characteristic of quantum mechanics known as entanglement occurs under certain experimental conditions. Subatomic particles become ‘inextricably linked’ in such a way that a change to one of them is instantly ‘reflected in its counterpart’, no matter how physically separated they are. Quantum theory postulates a superposition of states that destabilizes the intuitive sensorial notion of spatial separation. Entangled particles transcend space and remoteness. They belong to a ‘shared’ system that acts as a single entity. The distance that divides the particles no longer plays any influencing role that would lead them to be regarded as having distinct identities. Once the entanglement state is established, the subatomic duo stays forever bonded. The two particles will always have either precisely opposing or ‘elegantly complementing’ relative values of key quantum properties such as polarization direction, regardless of how far apart they travel from one another.
Quantum mechanical phenomena, such as superposition and entanglement, are made use of to perform operations on what are called quantum bits, or qubits. Instead of the classical binary or digital bit, which has the discrete value of 0 or 1, there is a qubit, which may have a third state, an in-between-state, the momentary value of which is determined by the superposition of the state of many other bits in the system.
Entanglement and superpositioning enable this third state, which can be cultivated to correspond with the anticipated choice space of the ‘user’.
EXISTING METAPHORS FOR QUANTUM COMPUTING
In a landmark article called “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,” MIT mathematician and computer scientist Peter W. Shor defines algorithmic sequences for quantum computing in software. Shor asserts that digital computing, contrary to common belief and to the famous statements in information theory of Alan Turing (“On computable numbers, with an application to the Entscheidungsproblem”) and Alonzo Church (“An unsolvable problem of elementary number theory”), is not an efficient universal computing device. “It is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor,” he writes. “But this may not be true when quantum mechanics is taken into consideration.”
Shor considers two mathematical problems in cryptography, factoring integers and finding discrete logarithms, which are highly challenging to implement on a digital computer. He formalizes efficient randomized algorithms for these two problems but still leaves a crucial difficulty remaining to be solved by the hypothetical quantum computer. “To compute the period of a function f, we evaluate the function at all points simultaneously.” But quantum physics imposes on us the limitation that this information is never available to us. Since the mid-twentieth century, physicists have discovered that there is a reality of quantum physics, but have had trouble observing that reality. It is up to the designers of the quantum computer now to implement the quantum property of the ‘superposition of states’.
A measurement of superpositions yields only one value, and at the same time destroys all the others. Computer scientists working on quantum computers therefore rely heavily on the Fourier transform, a mathematical operation that transforms one function of a real variable into another, called the frequency domain representation of the first function, as the hypothesized way to solve the problem. The quantum Fourier transform is primarily thought of as being implemented in hardware. A hypothetical quantum computing device would have so-called ‘reversible logic gates’ which continuously allow sequences of reversible decompositions into mathematical unitary matrices.
In January 2007, I attended the conference “Consciousness and Quantum Computers” in Lucerne, Switzerland, organized by the Swiss Biennial on Science, Technics & Aesthetics (SBSTA). In his opening remarks, René Stettler, Founder and Director of the SBSTA, talked about the trans-disciplinary work that would be involved in the project of bringing to fruition quantum computing. It is especially an expanded understanding of consciousness that would be required to gain a real grasp of quantum physics. Yet, as Stettler pointed out, universities do not even seem to be striving for this trans-disciplinary knowledge. Hans-Peter Dürr, former executive Director of the Max Planck Institute for Physics and Astrophysics, and former collaborator of Werner Heisenberg, emphasized in his keynote address that physicists do not have the philosophical training necessary to understand what quantum physics really means. The celebrated mid-twentieth century physicists who discovered quantum mechanics did not understand it, they only spoke about it in metaphors. They settled on the practice of using applied quantum physics statistically without understanding what quantum physics means.
But quantum physics, according to Dürr, is the most profound rational knowledge that we have gained about the world. The necessary expanded understanding of consciousness and action would have to come from engagement with philosophical traditions like phenomenology, Buddhism, and Hindu cosmic perspectives like Vedanta. Excellent talks on the relationship between Buddhism and the philosophy of science were given at the conference by Geshe Obsang Tenzin, a Tibetan Buddhist psychologist living in America and working on mind/body medicine, and German philosopher Christian Thomas Kohl.
NEW TOPOLOGIES, STRANGE ATTRACTORS
Alexis Clancy: * Third Space mechanics: I consider a model to be a dynamic series of frames. In modeling a universe, I consider two sets. First, the set F of everything that I know. Second, the set D of everything that I do not know. Something can either be known to me or unknown to me. It cannot be both. [”_F_eicte” is the Irish word for “seen.” _D_ofhiecte is the Irish word for “unseen.”]
The set F of everything that I know is characterised by collapsed wave form Kroneker Delta functions which are finite, bounded and measured. [A Kroneker Delta function is a function whose value is one at a unique instance, zero everywhere else. It best describes the collapse of a waveform on measurement, the wave collapsing to an absolute negation of probability at a certain point on this measurement.]
The set D of everything that I do not know is characterised by Schrödinger type equations, spacewise infinite and unbounded. However, the perimeter of the set F of everything that I know presents a problem, as a point on this perimeter exists in both spaces F and D [Imagine someone standing on the border of Belgium and The Netherlands – essentially, they are standing in both countries at the same time]. This contradicts the first Rule. I correct this model by introducing a small cleft about the perimeter, small yet big enough to exist. Epsilon small. I call this cleft the Epsilon Cleft. This is the Third Space.
Locally and superficially, the dimensionality of F strictly does not go beyond 2D, and it is Euclidean. The dimensionality of D is a function of time; as time progresses, symmetry breaks [i.e the character of an absolute law dictating the character of D is no longer a given. See ] and as many dimensions as are needed to patch the model are used. Ignoring the first term, the sequence (as stated previously) is 4, 11, 26, 57… The Epsilon Cleft is the source of these dimensions. My assertion that symmetry will always break (as long as there is time) dictates that the Epsilon Cleft will have an inexhaustible supply of dimensions. [This assertion is taken as a direct inference of Gödel’s Incompleteness Theorems.] It is therefore countably infinite. Adopting this attitude towards a model renders the ‘problem’ of innumerable infinites not a problem, but rather an actual contributor to an overall dynamic and evolving model.
I like to view spaces like the Epsilon Cleft as a “novelty” space. I find them to be analogous to the “No Mind” structure referred to in the Samurai Creed (“I have no sword. I make No Mind my sword.”) and the characteristic consciousness produced by Samadhi practices of Buddhist and Hindu Yogic meditation; I place my faith in the Epsilon Cleft to provide a space for novelty to emerge. In this case, we design the solution space such that the novelty that emerges is Artificial Life.
- Faith: The interesting thing to me about a probability spike, as derived and described in Shor’s paper, is that even if it hits, say, a 99.9999999999th percentile of certainty, 0.0000000001 must be taken on faith. Faith is a qualitative rather than a quantitative construct – once it is there, it is there and it becomes a fundamental aspect of the overall paradigm, contributing to the overall efficiency of the paradigm. To negate it would take an infinite amount of time. I feel that faith is critical to any sort of AI paradigm, quantum or otherwise. I remember speaking about faith with a friend of mine who is a Jesuit priest. He said that what a lot of people forget is how practical a construct faith is. Indeed, faith is a large part of the Bushido creed: faith in discipline, faith in training, faith in “No Mind,” faith in the Way or Path. Faith is a real time saver.
To have gaps in the spacetime model provides many advantages: the possibility of motion, for one, and more to the point, the capability to creatively evolve and provide the model with as many dimensions as needed. Since it exists outside spacetime mechanics, the “novelty space” is fast. Thus we have the qualities of dance: motion, creative capacity for change, speed; this is the kind of dance Choreographer Michael Klien describes as ‘a state of excitement in a system whereby change becomes possible’ .
- My theory of mutation relates to stochastic (a stochastic method is a method whereby a “guess” is made as to the operation of an observed phenomena and then the “guess” is “tweaked” into rigor) methods, more precisely genetic algorithms. It is an example as to how the behaviour of quantum geometries can be used in developing solutions strategies for the macro world. While I have only observed the probability of mutation as being a constant in contemporary theories, my preliminary theory tries to state otherwise. As stochastic modelling methods rely on their closeness of adherence to natural processes and phenomena, viewing mutation as a multi-variable function improves the algorithm.
I am beginning to think that this theory of Mutation is much more important than I originally surmised, as it bridges Darwinistic theories of evolution and assertions of intelligent design. Once the system comes into a bind, mutating seems like the intelligent thing to do. A shrinking probability interval and the existence of choice is key. So there is a randomness (albeit a shrinking one) and a free will paradigm at play. In terms of theology, I do not feel there is any need to delve any more to further the understanding of the model; a choice exists, that is all. So the mutation theory can stand out not just as stochastic proposal, it is also a bold illustration as to how Möbius (symmetry breaking) Incompleteness in a Riemann geometry can give rise to what can be deemed intelligent behaviour.
Consider any object in a Riemann Geometry. It is a property, a mathematical truth of this object that any line section of it is Möbius (i.e. contains a 180° half twist. See in structure -¬ the Riemann object can be described as a pinched S^3 sphere and, by examining it as a Clifford (Fibre) bundle of a Riemannian manifold, we can say that all sections are Möbius in character, as there exists a ‘choice of sign’ with respect to the vectors therein. The Möbius twist itself – the “interval of inflexion” – leaves a gap in the model – this concept is expounded on shortly.
In a Riemann type geometry, a conic represents a pinch of some sort. An unmolested bounded space can be taken to be a sphere but some stress on the system will render it not so – the most basic morphing will be hyperbolically conical. I state gravity to be a constraint simply due to its universality with respect to binding a system. With respect to separating the time and space factors, I feel that, as we are dealing with a spacetime metric, the mutation function is a coupled bivariable function. It is almost a rule of thumb that nature will not use a simple linear function to do anything – a simple non-linear function is generally the case. The geometry can be taken to be a quantum geometry, but I believe that most of what we experience has its origin in these kinds of spaces. I feel that the solution space metric we will design should embody these qualities and also be breathable (my term) and elastic – a mathematical weave as opposed to a mathematical covering (6). I am inspired by Goethe’s quote: Search nothing beyond the phenomena, they themselves are the theory.
Take the interval of inflexion and call it the Aleph Point. This is the point where symmetry breaks (down) in the overall section and the system is called on to evolve to a higher dimension. Now consider a closed space under some sort of constraint, gravitational or otherwise, and represent it as a conic, with a time interval T operating. Note that as we travel down the conic, taking sections at points a, b, c, and d, we can deduce, as we travel ‘down’ the conic, that the probability of an “Aleph Point” being called on increases for two reasons. First, because the section interval is shorter (|a|>|b|>|c|>|d|), so the “aleph” point is more likely to be “chosen.” Second, because the time interval is operating faster as spacetime is getting denser toward the bottom of the conic. The overall conclusion is that the greater the strain/constrain(t) on a system, the greater the probability of symmetry breaking.
With respect to genetic algorithms, I propose that symmetry breaking is analogous to mutation. The challenge lies in fabricating an appropriate metric for the solution space so that a suitable variable mutation function can be applied and a more efficient algorithm developed. It must be a Möbius weave, as opposed to a covering. It must be non-Euclidean. It must not be decimal or binary in base foundation, as, to my mind and acumen, there are no universal harmonics bound to 2 or 10. Right now, 360 seems to be the most appropriate base.
This is a very important demonstration as to how Möbius topologies, incompleteness and Riemann geometry combine in a sense that seems intelligent.
- One of the fundamental challenges with respect to our quantum computing/A-Life project surrounds waveform collapse. There is the choice space of the mind of the user and the solution space of the A-Life device. The probability space of the range of choices presented to the user collapses into a decision and the superposition of states offered by quantum computing/A-Life must collapse into the same decision.
My investigations of waveform collapse suggest that it happens due to a “well” of incomplete spaces, and collapse happens in pairs of wave potentials. This is due to the anticipation of Newton’s third law – every action has an equal and opposite reaction. So there is an “instance” (action) vector and a “shadow” (reaction) vector, one anticipating the other. In terms of the dimensionality of the event (i.e. waveform collapse), it is infinite in potential and finite in eventual unfolding – this finitude following the sequence 1, 4, 11, 26, 57, 120, 247… [see the figures ‘the symmetry breaking of aleph’] (based on dynamic patching of incomplete spaces generated by Möbius inflections). In my modelling, the only way that I could stop the well being infinite (and the event taking an eternity to happen) is by observing the origin (the origin being the ordained source of vectors of a given framework) being sucked into the event as all vectors are brought to the event’s location. There is a “kiss” of origin and event, the dimensionality of the collapse reaches a finitude, a moment of absolute parity is achieved and then the collapsed waveform unwinds.
It is this moment of parity that we must strive for – the equality of what is in the mind of the operator and the equivalent member of the solution space. I have meditated on the ‘=’ symbol for many years and the result of these contemplations is that the symbol is actually quite special and not to be used lightly. I regard it as a ‘parity license’ and, like all licenses, it must be applied for.
- Where the challenge lies is in accessing a Schrödinger waveform to “play” with. It may be of use to draw on a conjecture that I developed regarding Schrödinger’s Equations and Parametric Normal Distributions. The question I pose is this: Do statistics imitate life, or does life imitate statistics? The conjecture is based on the meditation that, because Gauss’ rigorous definition of the Normal Distribution [the ubiquitous “Bell Curve” (because it looks like a bell) seen in most statistical models, particularly in models whose elements have the possibility to chose their state] predated the development of Quantum Theory, the results of experimentation and thought experiments were mathematically retrofitted into Gauss’ model and taken to be a system of “statistical aggregates.” However, it is my view that Gauss’ Normal Distribution is a trans-dimensional fractal, mimicking in form and behaviour its quantum origins on a macro scale.
This conjecture is supported by David Bohm’s first postulate of his highly regarded quantum theory: that the Schrödinger equation is not only a mathematical object – it is also an object of form. This expansion of consciousness permits the accessing of a Schrödinger waveform through parametric data. There must be an analogue input at some instance, but the scale is not important. I feel that this search for the appropriate input could be like Edison’s search for the appropriate filament for his lightbulb.
- numbers
3, or threeness, is very important with respect to escaping the tyrannies of binary/digital. Indeed, the ‘trick’ with respect to our macro q-device will be to build a device that goes beyond Turing’s definition in ‘On Computable Numbers’ of a universal computing device while using components which conform to that definition.
12 has many strengths. Its combination of three and four make it a very musical number, and it has almost self-organizing properties. This is also the case in a strong way for things divided into 360 parts. It is well to remember that base 2 and base 10 are to be seen as some sort of enemy to our thinking regarding this project.
64 is important with respect to partitioning a vector space. My research has led me to conclude that anything after a 64th part partition is meaningless. It represents the gauge of the ‘vector net’ that we are to establish. I believe that complete coverings cannot be applied to real-world scenarios as they fail to incorporate concepts of incompleteness. *Breathable metrics are what is called for*. In order to fabricate these metrics, there is a requirement for a given, acceptable tolerance to this metric and its least element. I propose that 64 be this tolerance. Although it is a classic number in binary computing, it does have a nice twelveness to it in that there are 4 parts of 8 and 8 parts of 4, and this twelveness is crucial in modelling a nexus of any given spacetime scenario.
There must be some analogue input somewhere along the way. Resonance is, as far as I know, one of the few quantum phenomena that can be experienced on a macro level. It is a way to access a Schrödinger waveform in a fractal, macro sense.
Incompleteness is almost treated as a dirty word in modern physics – I am of a polar attitude. I find it to be critical, and, if harnessed properly, the way forward with respect to the development of true Artificial Intelligence. I cannot stress enough how important Gödel’s work is to my overall thesis. It throws “wobbles” into any proposition. Furthermore, on examining the epistemology of axiomatic reasoning, a rigorous examination of any axiom-based theory will inevitably reduce to an examination of the word itself. Axiom: that which can be taken as “self-evident truth.” Well, what is “truth”? It is certainly no objective matter, so, indeed, it is a matter of faith that it is taken to be “true.” Though the faith element may be considered epsilon small in dimension, it still exists, but is habitually glossed over. I have come to see axioms as totems, as opposed to hardline written-in-stone truths. In view of the incompleteness theorem, it behoves one to have mutable, evolving, breathable axioms or else the theory will be crushed by incompleteness at some later point in time and space.
Alan N. Shapiro : Here is the answer to the riddle of quantum physics: not measure, but perceive. And an expansion of consciousness supports an expanded perception. Quantum behaviour is a reality. Physicists thought that they could not observe or measure this reality without destroying the information therein. But they conceptualized the methodology of observation conventionally. The space from which one can observe the reality of quantum behaviour without destroying the information therein is also a reality, a fact of nature. We do not have to invent this space, we only have to perceive it. This space of non-destructive observation really exists, just as quantum behaviour really exists, and we will get it working in software. To perceive this space, we have to change our consciousness. That’s all that we have to do! We have to recognize as being scientific some ways of perceiving that belong to other traditions that Western science has so far small-mindedly regarded as non-scientific. This expanded perceiving includes creative mathematics, the deconstruction of classical spacetime mechanics, Buddhist and Hinduist meditation/ontologies, Aboriginal-sacred-mystical-expanded consciousness thinking, and Continental semiotics/grammatology.
Your idea of the Epsilon Cleft is a very good representation of that protected space which provides an extra framing dimension enabling observation of the set of bits (in a body of real numbers that is beyond Turing’s idea of what is computable) which are in a 0 state, and the set of bits which are in a 1 state. The Epsilon Cleft is safe and protected, non-destructive, a breathable space based on breathable axioms, outside spacetime mechanics.