The Objects of Meaning
from the Limits of Logic
It is perhaps the ultimate physical theory showing how collective effects transcend the mere addition of individual causes: for in the theory, the collective effects correspond to the very emergence of the universe that we experience from entities that have no well-defined existence apart from their relationships with one another. Dating only since 1997 and called process physics by its creators--Dr. Reginald T. Cahill and his PhD students, Christopher M. Klinger and Kirsty Kitto of the School of Chemistry, Physics & Earth Sciences at Flinders University in Adelaide, Australia[1]--the theory shows how the collective dynamics of proto-physical entities collectively called self-referential noise (SRN) could generate the three-dimensional space of our experience and, from what are called topological defects (to be explained) exhibited by this space, what are ultimately to correspond to the objects of our experience as well.
Clearly, the theory (more formally called, [a] "Heraclitean Process System [HPS] after Heraclitus of Ephesus [6 BCE] who, in western science, first emphasised the importance of process over object.") is very abstract. As University of Arizona physicist, Roy Frieden, has exclaimed: " ‘I have never heard of anyone working on such a fundamental level as this.’ " This should not be surprising as the seeds of the theory derive from a field bearing on the very foundations of mathematics: the field of algorithmic information theory. Principally created by the IBM computer theorist, Gregory Chaitin, algorithmic information theory extends the work of Kurt Goedel and Alan Turing at the foundations of mathematics[2].
To recall: Goedel had shown that if one assumed a formal axiomatic system (i.e. a system from which the proof of a proposition proceeds from premises called axioms and agreed rules of operating upon the axioms) that was consistent (i.e. free from contradictions) and was rich enough to contain arithmetic, then it could be proven that such a consistent, formal axiomatic system would always be incomplete: it would contain at least one true proposition which could neither be proven nor falsified from the axioms and rules of procedure of the system. The most that could be done with such so-called Goedel sentences are to incorporate them as additional axioms in enlarged axiomatic systems which would, in turn, contain their own Goedel sentences.
Turing then reinforced this finding by showing, in reflection of Goedel sentences, that there was no mechanical procedure or algorithm for deciding whether a proposition was true or false within any given formal system matching the required specifications. That is, given any algorithm, one could not say in advance whether or not it would halt in its implementation to yield a decision as to the truth or falsity of a proposition: If the proposition to which the algorithm is applied happens to be a Goedel sentence, then the algorithm will never stop in its implementation; but one can never know for sure even if one implements the program forever!
Chaitin subsequently extended Turing’s finding thus: Instead of asking whether or not a specific algorithm or computer program (for that is, after all, what a computer program is) halts, we consider the ensemble of all possible computer programs. Each computer program is assigned a probability that it will be chosen. The total probability that those programs will halt corresponds to the halting probability—which Chaitin termed Omega. If the program never halts, Omega is 0; if it always halts, Omega is 1 (or vice versa, depending upon the convention chosen).
To help in answering this question, Chaitin expressed Omega in terms of the binary digits (i.e. bits) of computer programs (that is, as a string of ones and zeros). Chaitin then inquired whether or not it would be possible to determine the Nth bit of Omega as either a zero or a one. Availing of algorithmic information theory (AIT), Chaitin was able to show that Omega was algorithmically incompressible. (AIT was created by Chaitin himself and distinguishes between sequences of information, such as strings of ones and zeros, according to whether or not they will admit of algorithmic compression or not; that is, whether the sequence can be replaced by a sequence of instructions substantially shorter than itself when expressed in the same language as the sequence itself for generating the sequence in question). That is, the sequence of ones and zeros in Omega’s binary expression might as well be generated by the toss of a fair coin. This result that Chaitin obtained to the effect that Omega is random corresponds, according to Chaitin himself, to Turing’s result that the halting of an arbitrary computer program is undecidable.
This result, along with earlier results from the application of algorithmic information theory to the measurement of the complexity of algorithms, also entitled Chaitin to assert that, except for an infinitesimal set, the preponderance of mathematical truths were random[3]. That is to say, their truth was not a matter of proof, but rather of sheer accident and therefore, experimentation! Seizing upon this assertion by Chaitin and noting that Goedel’s work was applied to self-referential systems (Goedel began his enquiry through examination of the paradoxes arising from the self-referential statement "This statement is unprovable"), Cahill and his students reasoned that since the universe was also self-referential (as Cahill puts it: " ‘The Universe is rich enough to be self-referencing—for instance I’m aware of myself ’ "), then as with mathematics in which most truths have been revealed to be, amazingly enough, random, it must also be the case that most facts in the physical universe are random. That is, they are true by accident and not because of any possible proof.
Operating on this premise, Cahill and his students have devised what is perhaps the most fundamental physical theory to date: the theory doesn’t even posit the existence of objects (which is assumed in all other physical theories) but rather begins with entities collectively called self-referential noise (SRN)—the proto-physical analogue of the random truths of mathematics. This resort to SRN is complemented by resort to a particular dynamic called self-organizing criticality (SOC; to be explained) to achieve what Cahill and students call an axiom-less model of reality.
To elaborate: Cahill and students inform us that all present physical models of the universe assume that the notion of objects is operative and such objects are then labeled by numbers in the theoretical models. Involving numbers in physical theories inescapably leads to involvement of arithmetic, its generalisation in set theory, and therefore the full panoply of abstract mathematics as needed. The result is what Cahill and students term axiom-based modeling [axioms are premises for reasoning] that assumes a starting set of objects and the rules that operate upon them. Such axiom-based modeling has the apparent defect, as Cahill and students point out, that they lead to a hierarchy of models with an infinite regress of nested objects and their rules sustained by the object-based logic of axiom-based modeling.
The astrophysicist, John D. Barrow (whose well-deserved renown is associated with seminal contributions to the marriage of nonlinear dynamics and general relativity), amplifies this point admirably[4]. He tells us that whatever ultimate Theory of Everything we may formulate that can be shown to be logically consistent, such a theory would still not prevent the possibility, nay, probability of an unending regress of "finding more and more elementary particles of matter at every level one probes." Such particles would be associated with "additional forces of Nature that are intrinsically very weak, or highly selective in the things they act upon, or which have a minute range." He tells us that such "ghostly forces" would not need to "play any great role in the structure of the everyday world, or even the world of the present-day high-energy physicist, but their presence totally determines the form of the ultimate Theory of everything we seek."
It is precisely to avoid this infinite regress that SRN and SOC are invoked. SOC is a dynamic first studied in regard to digitally simulated sand piles[5]. It is a dynamic that causes the systems in which it operates to spontaneously move towards a fractal description, i.e. that is, a scale-invariant description at which all scales of events are permissible. For example, in the case of the sand pile being formed by the steady dropping of sand grains, the pile eventually assumes, after experiencing avalanches of increasing size as the slope of the sand pile increases, what is termed a critical slope at which avalanches of all sizes are experienced as well as long periods of static behavior or stasis. Such scale invariance of avalanches, it should be noted, implies that the grains of sand are effectively behaving in a "cooperative" manner: a single grain may suffice to "persuade" others to cooperate. Generalizing, we can say that "in the critical self-organized state, two events are equally likely to act together, whether or not they occur close to each other in space and irrespective of how much time has elapsed between their individual occurrence."
In contrast to but further informing our understanding of SOC systems, non-SOC systems display the fractal or universality property of permitting events of all scales only through appropriate adjustment of parameters (i.e. variable interfaces with the external world, such as temperature or pressure) that require no adjustment at all for SOC systems. It is only at what are called critical values of parameters that non-SOC systems lose their individuality and join a so-called universality class of other systems behaving in the same way.
It is precisely because of the universality displayed by SOC systems that an axiom-less model of reality is afforded: for with SOC in effect, it does not matter if there is an infinite regress of axiom-based models of reality, each with their object-based logic arising from nested objects and the rules governing them: they all belong within the same universality class within the embrace of SOC. In effect, what HPS modeling achieves (as Cahill and Klinger explain) is to "apply object-based logic and mathematics to an individual realisation of a, hopefully, SOC relational process, but then confirm and extract the universal emergent behaviour which will be independent of the realisation used…The key idea is that a truly bootstrapped model of reality must self-consistently bootstrap logic itself, as well as the laws of physics. Further, only by constraining our modelling to such a complete bootstrap do we believe we can arrive at complete comprehension of the nature of reality."
In regard of that comprehension, SOC also emphasizes that SRN is not to be understood as a thing but rather as "
a realisation-independent characterisation of the self-referencing." That is to say the SRN, because of the SOC dynamic operating on it, may (as I understand it) dispense with clearly identifiable identities of what exactly constitutes it because they are embraced by universality. This qualification of SRN by SOC is a favor returned by SRN to SOC in an example of the self-consistent bootstrapping adverted to above for a truly viable bootstrapped model which HPS aspires to be: SRN affords the possibility of SOC because SRN, being the analogue of Goedel sentences in the proto-physical sense, implies an open universe even though, by definition, the universe is all there is[6]. This effective rendition of the universe into an open system is required for SOC systems because all such systems are open systems (i.e. systems that permit the exchange of matter and energy and presumably, their proto-physical analogues, with an inclusive or complementary environment).The definition of an all-inclusive universe stated above is extremely important for HPS: an all-inclusive universe implies, if the universe is qualitatively rich enough, that that universe is necessarily self-referential, thus permitting the possibility of SRN; moreover, SRN that is nonlocal since it is prior to the three-dimensional space of our universe. Since that SRN continues to be nonlocal even after, as we shall see, the emergence of three-dimensional space[7] through SOC, the SRN then affords basis for the nonlocal effects known to definitely operate in quantum mechanics ever since the work of Bell.
Operating within such an analytic and synthetic framework described above, Cahill and students’ effort to date at HPS have produced a model of physical reality in which SRN participate in collective dynamics to produce pseudo-objects called monads, which monads then engage in collective dynamics of their own (both dynamics within the compass of SOC). It is from the collective dynamics of these monads driven by the SRN—a dynamic that is non-geometric (i.e. because SOC and SRN permit nonlocal effects beyond confinement by any demarcated geometry and because the HPS model simply assumes there is no geometry to begin with); non-quantum (i.e. because the objects of quantum mechanics are not yet in existence and only emerge as secondary effects of the dynamic); nonlinear (i.e. because of collective, emergent, qualitative departures in behavior); stochastic (i.e. because parameter values of the dynamic involved are free to change); and consisting of iterative or repetitive, discrete transformations called mappings (i.e. to produce the fractal structure needed to invoke universality feature from SOC)—that the three-dimensional space of our experience emerges.
Elaborating, in a paper by Cahill and Klinger involving the computer simulation of an HPS model consisting of 100 monads and 10 iterative mappings, although they freely confess that such a model was quite insufficient to produce the necessary fractal structure to give evidence that SOC was operating, they did find evidence suggesting such fractal structure: Monads first linked to form reactive gebits (gebits being a contraction of geometric bits) which, under the influence of the iterative mappings, began cross-linking themselves to generate the expanding three-dimensional space of our experience through self-feeding involvement of free monads forming new linkages with reactive gebits. The fractal structure in this dynamic is suggested, according to Cahill and Klinger, by the decay of older gebits ensconced in the steadily expanding 3-space [thus suggesting a scale-invariant structure for the emergent space in terms of SRN]. To establish genuine fractal structure in the 3-space generated will probably require improved analytical techniques [i.e. the equivalent in terms of mathematical equations of Goedel’s logical proof of his incompleteness theorems], according to Cahill[8], since numerical simulations by computer would probably be technically infeasible.
The 3-space generated also exhibits what are called topological defects (i.e. defects in the structure of the space having to do solely with connection, irrespective of size, angle, or curvature) that are, as a matter of course, also replenished by the same process replenishing the space they mar. The significance of these topological defects (which are nonlocal in character with respect to the 3-space and are described in their properties by a nonlinear functional Schrodinger equation) is that they are able to yield quantum phenomena and therefore the objects and object-based logic of our experience.
Aside from 3-space and objectification, another emergent effect expected of HPS is the experience of a "contingent present moment" as distinct from a "recordable past" and an "unknowable future"—that is the irreversible time of our experience. In this regard, Cahill and Klinger inform us that "(experiential) time is only predicted in this model if there is an emergent ordered sequencing of events at the level of universality, i.e. above the details which are purely incidental to any particular realisation." Since, as we have seen above, there is only suggestive evidence of a fractal nature of the 3-space generated and therefore inconclusive evidence of SOC to afford appeal to universality, it really cannot be concluded that the HPS model has generated definitive evidence of irreversible time emerging.
Still, there is also suggestive evidence: The iterative mapping employed by the HPS model precludes even any definition of an inverse-mapping procedure because of the noise term involved. So in this fact alone, there is already a divergence from the time-symmetric differential equations of traditional physics, a divergence that is only made more meaningful by the observable dynamic yielding the following effects: 1.) some persistence in the reactive gebits, thus giving rise to a partial-memory effect corresponding to a recordable past; 2.) the contingency on noise of the outcome of the next iteration, thus discriminating a present moment; 3.) the inability of the spatial structure of the far future from being known without performing the iterations. Furthermore, the sequencing of spatial and other structures is never repeated with each iteration.
So the HPS model described does give at least suggestive evidence that the universe we experience could have arisen from pure randomness in the form of SRN. It can therefore be asserted, as Cahill and students have, that "[s]pace, quantum phenomena, objects and process-time [may be] seen to be the logical consequences of the limitations of logic" as reflected in a dynamic that yields effects greater than the mere sum of its constituent causes. Elsewhere, I have written (see my article "The Disadvantages of Omniscience") that because meaningful information—as revealed by information theory and nonlinear symbolic dynamics—requires predictive fallibility, then there is something that even omniscient beings could envy non-omniscient beings such as ourselves: the possibility of meaningful lives. Now Cahill and students have shown that non-omniscience may yield something more fundamental than even meaning: nothing less than the very objects of meaning. This astonishing conclusion reveals the astuteness of the assertion made by that sagacious science fiction and science writer, Arthur C. Clarke, when he wrote[9]: "It is as important to discover what cannot be done as what can be done; and it is sometimes considerably more amusing."
In this connection, it should be noted that Chaitin himself admits, with glee[10], "that the most interesting thing about [my] field of program-size complexity is that…it proves that it cannot be applied [i.e. "[b]ecause you can’t calculate the size of the smallest program"]. But that’s…why program-size complexity has epistemological significance;" and apparently, as Cahill and students have chillingly shown, why it has physical significance as well. In so doing, Cahill and students afford the most poignant expression yet, it seems to me, of the pronouncement of the theoretical physicist and Nobel Laureate, Steven Weinberg,when he wrote[11]: "The effort to understand the universe is one of the very few things that lifts life above the level of farce, and gives it some of the grace of tragedy."
Acknowledgement:
The author is grateful for clarifications regarding certain points, both technical and non-technical, provided by Dr. Reginald Cahill in the composition of the article above. Dr. Cahill is not to be held responsible for any errors that remain in the article.
Notes and References:
February 2000). "Random Reality." New Scientist; Reginald T. Cahill and
Christopher M. Klinger. (2000). "Self-Referential Noise and the Synthesis of Three-
Dimensional Space." General Relativity and Gravitation 32(3),
529-540. (I am unable to cite the precise pages of the articles in the publications named
from which information was derived because my copies of the articles therein are
downloads from the web.)
within which the physical analogue of Goedel’s Incompleteness Theorem operated would
be an open one of limitless possibilities was worked out as early as 1979 by the
iconoclastic Princeton mathematician, Freeman Dyson. In a seminal paper [entitled
TIME WITHOUT END: PHYSICS AND BIOLOGY IN AN OPEN UNIVERSE.
Reviews of Modern Physics, Vol. 51, No. 3, July 1979] that established the
study of the ultimate future as an intellectually respectable discipline, Dyson wrote: "If,
as I hope, my answers turn out to be right, what does it mean? It means that we
have discovered in physics and astronomy an analog to the theorem of Godel
(1931) in pure mathematics. Godel proved [see Nagel and Newman (1956)]that
the world of pure mathematics is inexhaustible; no finite set of axioms and rules of
inference can ever encompass the whole of mathematics; given any finite set of
axioms, we can find meaningful mathematical questions which the axioms leave
unanswered. I hope that an analogous situation exists in the physical world. If my
view of the future is correct, it means that the world of physics and astronomy is
also inexhaustible; no matter how far we go into the future, there will always be
new things happening, new information coming in, new worlds to explore, a
constantly expanding domain of life, consciousness, and memory.")