THE INTEGRITY PAPERS Genre Group & UIU Group http://www.ceptualinstitute.com
International Society for the Systems Sciences
ISSS-TORONTO July 15~23, 2000
Consistent Spatio-Temporal Reasoning in a
Transfinite Cantorian Universe
Kevin Johnson
University of South Florida
Philosophy Department
Tampa Florida
starhawaii@worldnet.att.netJames N Rose
Ceptual Institute
1271 Bronco Circle
Minden NV 89423
integrity@ceptualinstitute.com
note: all footnotes/endnotes are actively hyperlinked ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Abstract
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~A long-standing difficulty in Logic has been determining whether the operation of a given system should be described in terms of ‘serial’ versus ‘parallel’. A particularly insidious aspect of this difficulty arrived when it was learned that a perfectly flexible parallel system can mimic any serial system. This paper explores the difficulty further and suggests that the spatio-temporal analytical reasoning systems used to address this question suffer from at least two intertwined and critical limitations: (1) they subsume a serial program by virtue of the formal viewpoint maintained; (2) they presume to address a complete spatio-temporal universe of discourse encompassing both serial and parallel programs when this is actually only a claim based in hypostasis.
This paper names the complete spatio-temporal universe of discourse ‘Cantorspace’ to both honor Georg Cantor’s seminal work in transfinite set theory and tacitly suggest its overwhelming pragmatic utility to properly address the needed expansion of our analytical systems. Imre Lakatos’ sophisticated falsificationism makes clear that the onus rests on the theoretician to demonstrate not only that her new theory can address the content of a previous theory but show that novel and interesting results are predicted. In this case, the novel and interesting results will be to indicate a reflexive and useful expansion of Imre Lakatos’ theoretical work itself by allowing it to properly address work by Benoit Mandelbrot in self-similarity or fractals. Characteristics and dynamical relations inherent in the dimensional architecture will be correlated.
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The Paper
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~The purpose of this paper is to explore a general architecture of dimensions, the relations and properties that information has within the architecture of dimensionality, the functions by which we access related information within a transfinite dimensional architecture, and therefore the methods through which a transfinite dimensional architecture enables and enacts human abstractions, enabling valid self-consistent deductions about the nature of existence. It is unavoidable to attempt considering the processes and relations of existence without placing them in context of the ‘forms’ of existence. That is, logical connectivity and geometry are inextricably bound and co-referential.
In an extension of Cartesian pragmatism, all of current science accepts as ‘fact’ the notion that ‘parallel’ domains, which themselves may not intersect (in any geometry: Euclidean, hyperbolic, etc.) and may in fact not be able to verify (be relevant to) each other’s existence because of lack of direct connection, can, by inductive reliance on intermediary connections, enact meaningful relation operations between them. When considered against an environment of infinite domains and compounded dimensions occurring in infinite densities, ‘Cantorian transfinites’, and even co-resident with symmetry breaking relations which occur in transfinite dimensional architectures, then it becomes requisite to consider how information is co-functionally represented via the concurrent mechanisms of parallel and serial operators.
The authors believe that a long-standing difficulty in Logic has been determination of whether the operation of a given system should be described in terms of ‘serial’ or ‘parallel’. Some work in cognitive science confronted this difficulty empirically when the choice of which analytical method was appropriate to deal with the question of "how does the brain operate?" arose recently. Woodman and Luck (1999) attempted to answer whether visual attention is serial or parallel in nature. Luck1 claimed "We are the first research group to show definitively that the human brain processes images serially - paying attention to only one object at a time and shifting rapidly from object to object." But in their research itself the authors were forced to attenuate this claim, writing "A completely flexible parallel model can emulate any serial model, and it is therefore impossible to rule out all parallel search models without also ruling out all serial search models." 2 Also, Woodman and Luck do not deny that the brain does carry out certain tasks in parallel in other areas, usually when multiple types of data are processed cognitively. Rose corresponded with them pointing out that saccadic eye jumps occur by parallel pre-processing or co-processing of the entire field of vision in order for the eye-brain to pre-select which optical pattern to fixate on next. Saccadic jumps are not Brownian, but selective to the imagery present.
In keeping with that co-relation event, Johnson developed the position that an entangled relationship exists in human reasoning which encompasses more than what the terms ‘parallel’ or ‘serial’ alone can describe. The paper briefly explores the entanglement via critical foundational ‘analysis’ and proposes that the spatio-temporal analytical reasoning systems used to address this type of question suffer from at least two intertwined and critical limitations: (1) they subsume a serial program by virtue of the formal viewpoint maintained; (2) they presume to address a complete spatio-temporal universe of discourse encompassing both serial and parallel programs when this is actually only a claim based in hypostasis. This root of this synergistic hybrid is called ‘parrial’ or a combination of ‘parallel’ and ‘serial’. It is suggested that parriality addresses useful points made by Nelson Goodman 3, i.e. that the function of a constructional system is not to recreate experience but rather to map it 4 and that many good qualities emerge from a map itself that would not otherwise be necessarily discernable, not the least of which is the action of extending meaning, by way of translating relevance from domain assemblies to whole meta-domain entities. The importance to human reasoning and communication is self evident.
Parenthetically it should be noted that entanglement and parriality share several characteristics, but there is an important difference. Entanglement essentially super- positions alternative states of some singular function. Parriality superpositions two alternate functions into a bound single state/process. It is the superpositioning of two heretofore distinct logic strings.
The first critical spatio-temporal analytical reasoning system limitation, that the systems subsume a serial program by virtue of the formal viewpoint maintained, is exemplified by a ‘logical’ argument. A ‘logical’ argument is a single chain of reasoning that depends upon strict and consistent order for its ‘formal’ validity. One could not, say, randomly mix the order of lines or symbols in that chain without sacrificing that validity. The formal viewpoint maintained evaluates each constitutive symbol and collective statement in its turn. This serial mode of applied reasoning has been proven to be an extremely useful tool to help in explaining the mysteries of the world but it is suggested that this applied reasoning does not exceed the bounds of the finite much less the completed ‘countably infinite’ or À0 5 as a universe of discourse. Crucially, only a parrial improvement of serial mode can accomplish the important achievement of trans-Gödel connectivity, which has a bearing on quantum non-locality. Justifications for introducing the concept of the completed infinite will be left aside as these are argued in many other places. 6 What will be commented on is Georg Cantor’s 7 important work in identifying different degrees or senses of ‘infinity’; infinities with different cardinalities, such as À 0 and À1 (‘uncountably infinite’). 8 ‘Countably infinite’ means that the set can be enumerated element by element. ‘Uncountably infinite’ means that this cannot be done as there are infinitely many elements missed for every element selected. It is an unusual and highly abstract concept that probably requires more than cursory reflection or study. Consider even À2, or the ‘uncountably uncountably infinite’ whose very description is considered a grammatical error. 9
Now a pair of very crucial points can be discussed. First, if one were to be exclusively ‘resident’ in an objectively known ‘uncountably infinite’ set, then that set would always be ‘countably infinite’. This is due to the lack of ‘outside’ or ‘external’ reference to the set, i.e. the internal observer is an element of that set, hence instantiates its cardinality as well. Second, that cardinality of residence need not necessarily contain ordinal subsets of lower cardinalities, i.e. lower cardinalities that would allow a manner of ‘internal but still external’ internal ordering of set elements. Essentially, local ‘triangulations’ 10 which establish the validity of relations on either side of arbitrarily designated bounded-sets are then employed in a tessellation process, extended infinitely and transfinitely. The authors will attempt an example of the difficulties addressed by these points.
At present, it would seem that using the integers or real numbers freely in theoretical work has no unforeseen consequences. The integers are understood (by stipulative definition) to be a subset of the real numbers. The caution here, however, is that the integers (cardinality À0 or ‘countably infinite’) could ‘objectively’ easily be of the same cardinality as that of the real numbers (cardinality À1 or ‘uncountably infinite’). There is a tendency to consider human reason as imposing a necessarily overarching pattern upon the world. There appears to be no way to escape the ‘iff’ or biconditional of a stipulated definition: this must exhaust logical possibilities (Which appears to further subsume that logical possibilities exhaust natural (or other) possibilities: another fascinating area of study, i.e. probability theory.). It is important to recall that the authors are challenging the scope of human reason; certainly the mathematical constructs, i.e. integers and real numbers, governed by rules of logic fall within this scope. In recompense for the loss of heretofore certainty of relation under stipulated definition comes a certain release of the borders of pure mathematics.
This release of mathematical borders will even allow the contemplation of what are considered ‘finite’ sets at present in terms of ‘infinite’ sets of varying or even indeterminate cardinality. It seems likely that human reasoning heuristically truncates and/or assigns/imposes set cardinalities and orderings for its processing. The authors recognize the philosophical dilemma of causal relation versus some manner of higher agency (God, homunculus, etc.) respectively in the previous sentence. The authors believe that this dilemma need not be decided (and may comfortably be left open as to whether it even can be decided) in order for pragmatic and scientific progress to be made in even highly abstract topics.
The authors identify the extant formal viewpoint maintained, the viewpoint from which arguments are presented in human reason, as a floating singular viewpoint or FSV. It is suggested that the conscious mind considers itself singular in identity or nature and then necessarily moves in its activities from perspective to perspective in a linear and serial manner. Ostensibly, this is also identifiable as Gaussian bubbles and as flexible horizons and ‘lifespace’ This self-consideration of singularity ignores scientific evidence from cognitive neuroscience of multiple areas in the brain working in concert during conscious mental activity and, specifically, reasoning, i.e. evidence as obtained by recent fMRI and ERP studies and correlations.11 In ignoring this evidence, the human mind takes what is generated from multiple extensions in space and time and focuses it through a singular viewpoint. That viewpoint, as a simple but insidious artifact of its singularity, then acts as a ‘mirror’. This ‘mirror’ ‘reflects’ or ‘deflects’ even reflexive analysis; that is, in analytically reasoning about itself formally the human mind uses a focal point that has been purged of any analytically accessible content ‘prior to’ and/or ‘anterior to’ that focal point, i.e. the ‘I’: minimally the most difficult inverse problem humans face because all problems, including itself, are individually and collectively within its purview. The FSV is thus the focused conscious solution to an inverse problem of what lies behind or before it.
In moving from perspective to perspective the FSV operates within an inherently spatio-temporal universe of discourse, the dimensions minimally needed for ‘motion’ or even a ‘block universe’. The FSV cannot, with absolute simultaneity, manipulate the two dimensions of its universe of discourse without losing reference to both dimensions. This is well known as the problem of arbitrary reference in a system or the difficulty of objectively determining ones absolute position in a system without metaphysically leaving that system such as through an Archimedean point. This means that one or the other dimension, either ‘space’ or ‘time’, needs to be held constant and operate as a ‘pivot’ for reason, again simply as an artifact of the singular perspective. It bears noting that whether one considers space-time as fixed or fluid the need for this ‘point’ remains.12 As an intuitive observation, it is suggested that ‘logic’ is heavily weighted towards holding ‘space’ constant (as a pivot) and allowing ‘time’ to vary while ‘information theory’ is heavily weighted towards holding ‘time’ constant (as a pivot) and allowing ‘space’ to vary; both generic ‘logic’ and ‘information theory’ can be considered asymptotes to be approached.13 ‘Constancy’ and ‘variance’ are not stipulatively defined but rather their definitions are allowed to be understood by abstraction from examples.14 A stipulated definition, in any event, would subsume an ‘iff’ operation on primitive logical concepts and this manner of reflexive ‘prior’ analysis is rejected by axiomatic formal logics (and information theories).
By allowing constancy and/or variancy of two parameters to act as pivots, a 2 X 2 matrix is generated. Note, once again, that as an artifact of the inherent nature of the FSV, there is the implication that absolutely simul- taneous constancy in both space and time cannot be differentiated from absolutely simultaneous variance in both space and time. Differentiability would require that the FSV be able to maintain absolutely simultaneous residency in two dimensions, which is a direct analytic contradiction of the FSV, i.e. floating singular viewpoint.15
Dimensional matrix:
Variant time: Constant time: Variant space: Constant space:
Sv Tv Sv Tc Sc Tv Sc Tc
Only Sc Tv and Sv Tc can be maintained by the FSV.
The second and related critical limitation of the spatio-temporal analytical reasoning systems is that they presume to address a complete spatio-temporal universe of discourse encompassing both serial and parallel programs when this is actually only a claim based in hypostasis. When the FSV presumes that it is addressing the complete matrix it is hypostatizing or ‘projecting forth’, at best, a partially addressed universe of discourse into a completely addressed one. But, carefully note that it is not claimed that the FSV is completely and categorically faulty in what it does address, rather merely limited. This type of move in hypostasis is commonly exemplified in the treatment of the terms ‘each’, ‘every’, and ‘all’ as equivalent quantifiers; it is suggested that this is plainly mistaken. Furthermore, the rejection by many mathematicians and logicians of the ‘completed’ infinity, such as Gauss historically, and Quine and Goodman in modern times16, is a tacit acknowledgment of this mistake by virtue of the qualitative difference between the ability of the present argumentative viewpoint under discussion (FSV) to maintain successive unending serial growth (mathematical infinity) versus completed parallel infinite affirmation (metaphysical infinity)17. The ‘move’ typically accepted in the matrix is basically the same move accepted in and between the quantifiers mentioned, as well as essentially the same spatial move commonly rejected in contemplation of the infinite: a singular ‘entity’ cannot be in more than one place at a single ‘moment’ of time. As the title of this paper suggests, recognition of the matrix as superpositioned provides a pragmatic consistency through the acknowledgment of their separate limitations rather than the inherent inconsistency evidenced by conventional disregard.. Moreover, unifying serial and parallel functions through their compatibilities, opens new computation and relevance horizons currently unexplored by science and perception.
It is important to recognize the paradoxical dilemma the FSV sets forth and so the earlier discussion regarding the how and why of the FSV is pointed to here. Parallel systems cannot be ruled out because a perfectly flexible parallel system can emulate any serial system. But consider that the serial FSV is ‘arguing’ on behalf of those perfectly flexible parallel systems. The authors believe the dilemma dissolves if one merely pragmatically accepts the consistent anterior insertion of this argument extension to the locus of the FSV. That is to say that the FSV is itself put forth by a perfectly flexible parallel system. Much research in cognitive science appears to support this notion; many systems of the brain operate in parallel outside conscious awareness. The FSV cannot reflexively analyze itself to determine its constituent structure without surrendering to that structure, i.e. become parallel in nature18. I.e., establish an open trans-Gödel architecture and process domain. This limitation in holding primary or axiomatic structures as objects of analysis has long been both explicitly and tacitly recognized by logicians and historians of science. For example, it is suggested that it is the primary argument used by Paul Feyerabend19 in his sophisticated elaboration of Thomas Kuhn’s 20 argument for incommensurability of paradigms21. That Feyerabend considers different paradigms to have an underlying basis means that some part of the dimensional structure, the ‘system’, needs to be held constant: a ‘pivot’ for reason. This pivot then can be either ‘space’ or ‘time’.
It is further suggested that the recognition of ‘logic’ and ‘information theory’ as equipotent modes of reasoning, differing merely in their designated pivots of reason, will allow even Feyerabend’s conceptions of paradigms to be directly compared or ‘commensurate’. This recognition exposes a limitation in current reasoning, again the limitation inherent in not recognizing commensurability-by-common-basis. In proleptic anticipation of this recognition, the authors believe their subsequent philosophical engagement should move to a transfinite arena where new forms of incommensurability can once again be postulated. This is extremely important and cannot be over stressed. It is as crucial and important a concept leap as the inclusion and use of ‘zero’ was in arithmetic computation for mathematics.
The intertwined relationship described above between the FSV and its results on conscious human reasoning encompass much more than what parallel or serial alone can describe. This synergistic hybrid should be described as ‘parrial’ in nature or a combination of ‘parallel’ and ‘serial’ that is not currently hierarchically describable. This has the reflexive implication that Hilbert space22, for example, is an artifact of the reflexively unaware FSV and to utilize Hilbert space as understood by the FSV in describing or organizing multiple dimensions is also a hypostatization (For the same general argument extends from a bi-dimensional spatio-temporal universe of discourse to an iterated dimensional universe of discourse.). Therefore, parriality operates in a complete spatio-temporal universe of discourse described by the previous matrix which will be named ‘Cantorspace’23 to both honor Cantor’s previously mentioned seminal work in transfinite set theory and tacitly suggest its overwhelming pragmatic utility to properly address the needed expansion of our analytical systems.
Parriality can be used to efficiently facilitate the discussion of Goodman’s program of mapping24. To achieve this goal, parriality can meta-theoretically extend Imre Lakatos’ epistemological device/theory of sophisticated falsificationism. This extended device can then be used to both reflexively and consistently address itself through a form of reflective equilibrium25 as well as ‘map’ experience expressed in sequentially theoretic form. Before presenting this extension an outline will be sketched of extant sophisticated falsificationism26. The purpose of mentioning Lakatos’ work is to show that it can be extended by parriality and this extended or meta-theory fulfills its own inherited admonition to contain certain characteristics while at the same time pragmatically increasing its utility as a tool of analysis.
Lakatos describes the sophisticated sense of falsificationism as being the case when a theory T is considered falsified if and only if another theory T’ with specific characteristics has been proposed. These specific characteristics include: "T’ has excess empirical content over T: that is, it predicts novel facts, that is, facts improbable in the light of, or even forbidden, by T; T’ explains the previous success of T, that is, all the unrefuted content of T is included (within the limits of observational error) in the content of T’; and some of the excess content of T’ is corroborated." 27
He then explains that series of individual theories can then be identified that are ‘progressive’ due to each their members prediction of and confirmed excess empirical content over its predecessor. A theory is accepted as ‘progressive’ insofar as it both predicts and enjoys corroboration of excess content and ‘degenerating’ if it does not. To be accepted as ‘scientific’ a theory must minimally evince prediction of new content; lacking even this it is rejected as ‘pseudoscientific’. This schema allows otherwise nominal data (‘theories’) to be evaluated in a metric manner. Notably, consequences of this include: that "a given fact is explained scientifically only if a new fact is also explained with it"; it is a category mistake to "apply the term ‘scientific’ to one single theory" rather than the appropriate series of theories; "contrary to naïve falsificationism, no experiment, experimental report, observational statement or well-corroborated low-level falsifying hypothesis alone can lead to falsification. There is no falsification before the emergence of a better theory." 28
It is believed that both reflective equilibrium and sophisticated falsificationism lack the ability to warrant any sort of objective ‘progressive’ character; reflective equilibrium lacking this to a greater extent because it does not possess any historical reference29 whereas sophisticated falsificationism does, by maintaining an origin, i.e. the original theory. Chaos theory’s ‘self similarity’30 provides some examples that may be useful in visually picturing, and later mapping, this lack of objective progressiveness. Benoit Mandelbrot has referenced the Koch curve or ‘snowflake’ as an excellent abstract illustration of the difficulties that he faced in considering the actual length of coastlines31. The area of the ‘island’ (the snowflake) is always ultimately bounded by a circle described by the vertices of the original triangle, i.e. it has finite volume. This is an example of infinite32 (unbounded) progression through increasing length which can be abstractly compared to a state of reflective equilibrium: there is no origin from which to judge progress as every progressive iteration of the snowflake or ‘state’ is considered as a type of uniquely emergent entity and there is no historical ‘memory’ from which to compare one state to its predecessors. It is appealing to note that the Koch curve grows in a countable manner and so would be encompassed by À0.33
Interestingly, both reflective equilibrium and sophisticated falsificationism can be shown to be finitely bounded despite ‘progress’ on every available site. In other words, recursive internal research is important but certainly not to the exclusion of pragmatically directed external research; external research being that research which pierces the boundary circle of the snowflake or, in the case of sophisticated falsification, extends off the surface volume boundary of, say, a Menger sponge.
The Menger sponge34 presents a mapping which illustrates that Lakatos’ sophisticated falsificationism is subject to an abstract recursive ‘trap’; a series of theories could be counted as progressive even if they simply explored one infinitely long path (of infinitely many35 available) in the sponge. This example of infinite36 (unbounded) progression through increasing length then seems to be as unsatisfying an indicator of progress as that demonstrated by the evolving states of reflective equilibrium. Indeed what it really indicates is a limitation on the cardinalities that sophisticated falsificationism can map for Goodman. This can be easily remedied by appeal to the anterior parallel viewpoints that constitute the FSV. This appeal will be called the simultaneous multiple viewpoint affirmation perspective (SMVAP).
It is believed that SMVAP will allow the imposition of an ordering in Cantorspace over the transfinite sets37 revealed in both the coastline (Koch curve) and infinite surface area per zero volume problems (Menger sponge). This will allow sophisticated falsificationism to have a pragmatically and logically ordered structure to measure external progress versus the previously mentioned deceptive and spiraling internally recursive progress. The term ‘pragmatic’ is particularly appropriate because the matrix of viewpoint affirmations available indicates that only the barest scratch is being made on the surface of this theoretical construct, i.e. already illustrated but not utilized in challenging the commensurability of the paradigms proposed is the ability to affirm either constancy or variancy in a spatio-temporal universe of discourse.
Some headway is being made however. Of note in electronics is the use of the Menger sponge (orthogonal fractal skeins) and Sierpinski triangles (triangular fractal skeins) to construct six-fold and greater compacted miniaturized antennas38. What is different about these arrangements is that they generate capacitance and inductance - they self tune encountered signals and enact a wider and a finer sensitivity. Essentially, the fractal form establishes parrial co-relations which coordinate several interdimensional processes and effectively lase the signal streams, bounding and amplifying them in one unified process.
It is suggested, in the spirit of Lakatos’ sophisticated falsificationalism, that a way to utilize a deductive chain of inference across infinite sets derived by induction, of the same or differing cardinality, is to have the chain of deductive inference ‘orthogonally’ oriented to an arbitrary ordering of those sets (by axiom of choice), namely in the order in which they are addressed in the inferential chain. If every set crossed in this orthogonal path of inference is ordinally and ontologically separate from the other sets also crossed, then a deductively valid chain of inference can be constructed using sentential propositional elements occupied by the residence-taking orthogonal deductive element. These chains of inference should have the same warrant of deductive ‘conclusiveness’ as more pedestrian deductive reasoning, i.e. dimensionally concurrent, not orthogonal39. Most importantly, they should provide both a means of connecting reflective equilibrium states, which are given propositional status, and ordering ‘progress’ across those states. A mapping can then be made between the states of reflective equilibrium and elements/theories, i.e. links, in the chains of theories ordered by sophisticated falsificationism. In other words, that sets of inferential statements be allowed to become ‘discrete’ in the form of sets of propositions; sets of propositions which are themselves discrete elements when considered internal to the set. These discrete internal elements can then become proper elements of discourse for a chain of deductive reasoning orthogonal to that set. Note - that the described orthogonal method allows inclusion of transfinite sets such as the ‘snowflake’ border and ‘sponge’ area while Lakatos’ original method develops new theories that are necessarily of the same cardinality as those they replace: a cardinality limitation. Again, the purpose of mentioning Lakatos’ work is to show that it can be extended by parriality, and this extended or meta-theory fulfills its own inherited admonition to contain certain characteristics while at the same time pragmatically increasing its utility as a tool of analysis.
Lastly, Lakatos’ methodology, as well as pragmatism itself, demands that at least one novel and useful fact is predicted. Two that spring immediately to mind are predictions that the speed of light and absolute zero will be seen as artifacts of FSV affirmation. Both are observationally and theoretically determined by the FSV internal to an infinite set; just the sort of limitation this pragmatic expansion of logic is intended to overcome.
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Footnotes / Endnotes
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~1. Johnson, 1999
2. Townsend, 2000
3. Nelson, Goodman, 1906-1998, Professor Emeritus of Philosophy, Harvard
4. Goodman, 1972. p.15
5. Cantor, 1899, pp. 113-117
6. See, for example, Moore, pages 1-3. Also in Moore, on pages 51-52, there is the excellent discussion of the terms introduced by Peter of Spain (c. 1220-1277), who became Pope John XXI, of the terms ‘categorematically’ and ‘syncategorematically’ corresponding roughly to ‘metaphysical’ and ‘mathematical’ infinity respectively. It is the authors’ opinion that these distinctions between infinities have their genesis in using ‘space’ and/or ‘time’ as pivots of reason. Very briefly, the achievement of completed or ‘metaphysical’ infinity depends on allowing ‘time’ to vary; this is exemplified in the use of the logical quantifiers ‘all’, ‘each’, and ‘every’ as equivalent.
7. Georg Ferdinand Ludwig Philipp Cantor, 1845-1918. See Bell’s excellent description of his life and career, pp. 555-579.
8. Cantor, pages 113-117
9. Of course, an inductive inference made by a singular viewpoint can readily postulate a set of statements of cardinality ‘countably infinite’. Hempel (page 17), for example, uses the idea of infinitely many statements created from English words. (See also Seuren page 116 for an excellent discussion of how Humboldt and Chomsky (among others) relate generally to this topic.) Cantor’s diagonal argument could be, and is, used to extend the cardinality of this set to ‘uncountably infinite’ but no further upwards. Many people object to the diagonal argument, but note, for the sake of discussion that the Cantorian argument can only point towards the higher cardinality, not claim ‘residency’ in it.
The authors believe that ‘uncountably infinite’ requires its cardinality to be measured with respect to a ‘countably infinite’ set (Indeed, by definition and construction it is the ‘power set’ of a ‘countably infinite’ set.). Even more esoteric is the understanding that a singular viewpoint cannot move to occupy the ‘uncountably infinite’ set (and simultaneously allow that set to remain ‘uncountably infinite’ in cardinality), i.e. be resident in it where residency describes being an element of the set. This understanding is obscure because the typical examples of transfinite set theory that one encounters are the number systems ‘the integers’ and ‘the reals’.
The integers are ‘countably infinite’ and the reals are ‘uncountably infinite’ but the integers are commonly ‘understood’ to be necessarily a proper subset of the reals; this ‘understanding’ originates from an ordinally biased conception of the two respective sets. From generalizing this ‘ordinal’ example of infinity we forget that it is not required that a lower cardinality of infinity be ‘resident’ in a higher one. A relationship exists between the two but this relationship does not require ‘residency’.
Another example of how an element of a set may not require ordinal underpinnings is to consider the set {5}. The element ‘5’ does not ordinarily imply that ‘1’, ‘2’, or ‘4’ are tacit members of the set, despite their ordinal inclusion in ‘5’. It is the cardinality of ‘5’ that is used to mark membership in this particular case. Note as well that {1, 1, 1, 1, 1} is not considered a proper subset of {5} despite the fact that ‘5’ can be constructed with the summation of these elements. In a like manner, the power set origin of a higher level need not require the presence of the lower level used in its construction. Mathematics and its expressions in set theory may give the outward impression of complete or exhaustive order but extant knowledge is quite biased towards the convention of happenstance, albeit often pragmatically.
The specific question as to why the combination of adverbs ‘uncountably uncountably’ is considered a grammatical error in violation of a ‘rule’ is subtle in nature. The causal basis for any particular syntactical rule is currently unknown: While it may be possible to write a rule that describes the usage of language it must be remembered that this rule is empirically generated; one or more native speakers of the language serve as the final judges as to whether a particular combination of words is sensical and/or appropriate or not. Many constructions are possible ‘logically’ that are in fact absent from actual usage in one dialect or language (but then present in others). Why, for example, is it more ‘correct’ (or ‘pleasing’ to the ear) to say "tall, dark, and handsome" versus "dark, tall, and handsome"?
The dual suffix ‘ly’ of ‘uncountably uncountably’ is an example of a homoeoteleuton or a rhyme having like endings. This type of construction with adverbs is considered clumsy, unpleasant sounding, and (in the case of the dual suffix ‘ly’) especially enfeebling. (Perrin, page 414) The authors of this paper suggest that the antipathy for this usage is related to the difficulty of its, i.e. the usage’s, unambiguously communicating a SMVAP, i.e. there is no fixed reference for the second ‘uncountably’ since the first ‘uncountably’ is deliberately descriptive of an abstract state beyond a one-to-one correspondence.
As an aside, the history of linguistic analysis should doubly emphasize the concern researchers should have when applying a serial system such as logic to conclusively investigate entangled natural systems. See, for example, the subject-predicate debate that extended from Aristotle to modern times (and still remains, though is largely ignored). (Seuren, 1998, pages 120-139).
10. Gödel's logic says that a larger frame of reference is needed in order to evaluate and completely categorize the integrated completeness and/or self-consistency of a designatable/bounded system.
However, since such designations are sub-regions less than the whole (which is transfinitely and totally open in the extreme) -and - since all sub-boundary designations are essentially variable and arbitrary in what is excluded/included and defined, then even if we are inside a system that is bounded and can't seemingly validate notions because of lack of juxtaposable 'surrounding context', there is a way to get around this and accomplish a valid determination.
It is possible to engage in a process of choosing regions INSIDE any 'outer bound' and examining if these smaller models remain consistent. Doing this randomly throughout the knowledge-set is like surveying a geography by 'triangulation' ... establishing where a third locus is relative to two others that constitute a known 'baseline'. In the current application, the 'baseline' is an arbitrarily designated information 'environment' and the third 'locus' is any small subset region inside the larger environment.
The concept leap is a get-out-of-Flatland jump. Instead of loci, the process is applied to spaces/volume ... "content". The process is scaled dimensionally.
By 'scaling' .. and by designating that the 'triangulation' process is valid for all spaces and all content no matter where or how chosen or defined. Then Gödel is surmounted by noting that all 'external' information must be in some way compatible with all current 'inside' information of current 'arbitrary' size/content, if it is to be included at all under any conditions or requirements. There are eventually no limits, no disconnects and no discontinuities .. even and especially anywhere/anytime of the total infinitely open transfinite architecture. The process is scaled transfinitely (Rose. 1973, 1992). This technique is identical with Leibnitz’s and Newton’s ‘partitioning to infinity’ process used in the invention of the Calculus.
11. See Dehaene, 1998 and 1999, for example
12. Simply adding further dimensions (from which to note the topological morphability of space-time, say) merely increases the initial complexity of the universe of discourse. Space-time is referenced purely out of recognition to its historical importance and practicality. It would do little to facilitate initial understanding if immediately the universe of discourse were to include countably infinitely many dimensions and/or other measures. The ‘Archimedean point’ is simply another way of stating ‘God’s eye’ or absolutely objective view.
13. Weighting should be understood in the sense that logical and information theoretic modes of reasoning, as they currently stand, are blends of one another in varying proportions. It is suggested that foundationally both ‘logic’ and ‘information theory’ subsume one another. That is to say that the variance of ‘time’ or ‘space’ is intuitively appealed to as ‘prime’ or ‘simple’.
14. Nor are ‘space’ and ‘time’ excluded from holding one another as predicates. By subsuming ‘time’ as variant, logic is supported by information theory (and vice versa): ‘variance of time’ becomes a ‘spatial form’ which in information theory is allowed to vary.
15. These two positions in the matrix are an avenue for future interpretations of ‘incommensurable’. In other words, an ‘out’ for future bearers of Feyerabend’s torch but only if a SMVAP is utilized (discussed later in this paper).
16. Goodman, 1972, pp.173-177
17. Moore, 1993, pp1-2
18. Then note that the cponception of 'parallel is likewise limited. The terms can remain the sme but the 'prior' interpretation is a figurative shadow of the 'subsequent'.
19. Paul Feyerabend, 1924-1994. Philosopher of Science.
20. Kuhn, page 10.
21. Feyerabend, 1993, p. 212. When Feyerabend argues that a paradigm is incommensurate with another, he does so by appealing to the inability of formal reasoning systems to reflexively analyze their ‘primitive’ terms. The primitive term is essentially considered ‘defined’ via a separate meta-theoretic-language; it is appealed to intuitively where intuition is expressed via this ur-system. This means that Feyerabend holds the ‘form’ of the primitive term constant through any series of comparisons over time; comparisons meant to establish commensurability. This is the asymptotic form of logic as described above. What Feyerabend does not take into account is the asymptotic form of information theoretic reasoning also described above. This is the mode of reasoning that allows ‘space’ or the ‘form’ to vary while holding time as a constant. This is a fairly abstract notion that it is hoped can be illustrated with a real world example in mathematics.
The fraction ‘1/2’ is considered the ‘same’ as the fraction ’23/46’. That they are considered equivalent is information theoretically based. ‘That’ they are demonstrably ‘different’ fractions, i.e. ‘1’ is not ‘23’ etc., is logically based. Both modes of reasoning manipulate one or the other parameters of the spatio-temporal universe of discourse; neither parameter manipulated can be considered pre-eminent in ‘warrant’ so the two asymptotic modes of reasoning are equipotent in warrant. Thus, again, it is a flaw even in current reasoning to hold that two or more paradigms can be considered incommensurate.
22. David Hilbert, 1862-1943, pre-eminent mathematician.
23. A term suggested by James Neil Rose
24. But Goodman and Quines’ joint nominalistic rejection of the infinite should at least be mentioned in passing. (Goodman, 1972, pages 173-177) Briefly, the authors hold that even the isolated weight of the previous remarks concerning quantifiers should lead to a reconsideration of this stance. It is thought that some of the abstract entities so objected to by these logicians are likely not abstract whatsoever but find instantiation in the very sentience of the concrete entities doing the objecting. The roots of their objections in the paradoxes of type theory bear re-examining as well. (See Quine’s introduction to Russell, 1908a, pp. 150-152)
25. Rawls, 1999, pp46-53. But note that Rawls also acknowledges debts to Goodman.
26. The following is a more extensive outline of Lakatos’ description of dogmatic falsificationism and movement through to sophisticated falsificationism; the latter he took to be an extension of Popper’s position of naïve falsificationism. He describes dogmatic (or ‘naturalistic’) falsification as admitting "the fallibility of all scientific theories without qualification, "(Lakatos, page 95) whilst reserving unto itself some manner of infallible empirical basis. It is strictly empiricist without being inductivist, i.e. it does not allow the certainty of its basis to hereditarily distribute to theories in its purview. By blocking this distribution, dogmatic falsificationism has been described as "the weakest brand of justificationism."(Lakatos, page 96)
Lakatos stresses that the best indicator of dogmatic falsificationism is the recognition that every theory is equally conjectural. Theories cannot be proven by science, but they can be disproven. Integrity in science then rests in proleptically specifying an experiment that has the power to decisively refute a theory. To specify a falsifying experiment is not a difficulty for reflective equilibrium; any refutation would simply announce the need for further reflection and adjustment. Thus, growth in science is the result of cyclic refutations of new theories with empirical counterevidence. Lakatos states that dogmatic falsificationism rests on two assumptions: "that there is a natural, psychological borderline between theoretical or speculative propositions on the one hand and factual or observational (or basic) propositions on the other"; "that if a proposition satisfies the psychological criterion of being factual or observational (or basic) then it is true; one may say it was proved from facts." (Lakatos, page 97)
The methodological falsificationism described by Lakatos should be understood as a brand of conventionalism, which itself is viewed in terms of ‘activist’ and ‘passivist’ theories of knowledge. A ‘passivist’ theory of knowledge asserts that true knowledge is the result of Nature’s script on a tabula rasa or perfectly inert mind. An ‘activist’ theory of knowledge asserts that Nature’s script is only available via mental activity involving our expectations and theories. (Lakatos, page 104)
‘Activists’ may be subdivided into at least two categories: ‘conservative activists’ who feel that our basic expectations and conceptual framework are present at birth; ‘revolutionary activists’ suggest that conceptual frameworks can be developed and superceded by better ones. Lakatos suggests that, clearly, Popper can be characterized as a ‘revolutionary activist’. (Lakatos, page 105)
Conservative conventionalism is the position that when a theory enjoys substantial initial empirical success a decision may be made to protect the theory from refutation; clearly this is the case with reflective equilibrium. This protection is afforded by explaining (away) apparent anomalies through the use of auxiliary hypotheses or other ‘conventionalist’ approaches; troublesome intuitive judgments are adjusted through reflection. Both conservative conventionalism and reflective equilibrium then have the quizzical side effect of diminishing the power of empirical evidence in science in direct proportion to science’s growth. (Lakatos, page 105)
Popper criticized the idea that "conceptual frameworks turn into prisons which cannot be demolished."(Lakatos, page 105) He established one of two rival schools of revolutionary conventionalism. The first school is known as Duhem’s simplicism, which holds that "no physical theory ever crumbles merely under the weight of ‘refutations’."(Lakatos, page 105) Simplicism claims that a physical theory may crumble under the influence of continual repairs, eventually leading to a loss of its original simplicity and necessitating its replacement. (Lakatos, page 105)
Popper’s methodological falsificationism is the second school of revolutionary conventionalism. This school is a hybrid, being both conventionalist and falsificationist. However, it "differs from the [conservative] conventionalists in holding that the statements decided by agreement are not [spatio-temporally] universal but [spatio-temporally] singular". (Lakatos, page 106) It differs from dogmatic falsificationism by "holding that the truth-value of such statements cannot be proved by facts but, in some cases, may be decided by agreement." (Lakatos, page 106)
This latter mentioned "decision by agreement" in the Popperian system involves two levels, which correspond to the two assumptions of dogmatic falsificationism but differ in important respects: foremost, the Popperian is not a justificationist in that ‘experimental proofs’ are rejected and the fallibility of the decisions and incumbent risks are acknowledged. The methodological falsificationist knowingly ‘applies’ fallible theories in a given context as unproblematic background knowledge to be ‘tentatively’ accepted, as with reflective equilibrium, while ‘the’ or ‘a’ theory is tested (the theory being tested must be demarcated from the unproblematic background knowledge). These theories, and those statements they adjudicate, can be called ‘observational’ but only as an artifact of speech inherited from naturalistic falsificationism. In this usage the most successful theories are considered extensions of our senses thereby far exceeding the "dogmatic falsificationist’s range of strictly observational theories." The scientific community at large is expected to agree on a standard of safety control by which stray observational results can be filtered out. This filtering process is still akin to reflective intuitive adjustments.
There are five decisions made by the methodological falsificationist in the course of testing a theory: (Lakatos, pages 106-112)It appears that Lakatos’ extension of Popper’s work appears to begin at about points 4 and 5. He remarks that Popper did not seem to understand some points clearly, i.e. their consequences. (Lakatos, page 109) Lakatos then, his statement to the contrary notwithstanding, clearly moves away from Popper’s work in differentiating ‘naïve’ versus ‘sophisticated’ methodological falsificationism; Lakatos supporting the latter. (Lakatos, page 116)1. As mentioned above, certain spatio-temporally singular statements are arbitrarily made unfalsifiable in accordance with a ‘relevant technique’ that
would allow disparate individuals to concur in this decision. These would correspond to unproblematic intuitive judgments under reflective equilibrium.
2. A decision is then made in sorting out accepted basic statements from all basic statements; the process of reflection, i.e. comparing the tentative principles with the ‘basic’ intuitive judgments.
3. Certain rejection rules are identified which classify statistically interpreted evidence as being ‘inconsistent’ with probabilistic theories, otherwise not ‘falsifiable’. This is a particular principle held by the methodological falsificationist.
4. When a theory is tested together with a ceteris paribus clause, and the conjunction of the two is refuted, a decision must be made as to whether this refutation constitutes a rejection of the theory; this is an example of establishing reflective equilibrium.
5. Whether to eliminate ‘syntactically metaphysical’ theories out-of-hand must be decided. That is to say, theories that "because of their logical form cannot have spatio-temporally singular potential falsifiers." This is another principle to be tentatively decided in the course of reflective equilibrium.
Lakatos demarcates the two versions thusly: in the naïve sense, a falsificationist will consider any theory capable of being experimentally falsifiable as ‘acceptable’ or ‘scientific’; in the sophisticated sense, a falsificationist will consider a theory T falsified if and only if another theory T’ with specific characteristics has been proposed. These specific characteristics include: "T’ has excess empirical content over T: that is, it predicts novel facts, that is, facts improbable in the light of, or even forbidden, by T; T’ explains the previous success of T, that is, all the unrefuted content of T is included (within the limits of observational error) in the content of T’; and some of the excess content of T’ is corroborated." (Lakatos, page 116)
Lakatos then explains that series of individual theories can then be identified that are ‘progressive’ due to each their members prediction of and confirmed excess empirical content over its predecessor. A theory is accepted as ‘progressive’ insofar as it both predicts and enjoys corroboration of excess content and ‘degenerating’ if it does not. To be accepted as ‘scientific’ a theory must minimally evince prediction of new content; lacking even this it is rejected as ‘pseudoscientific’. This schema allows otherwise nominal data (‘theories’) to be evaluated in a metric manner. Notably, consequences of this include: that "a given fact is explained scientifically only if a new fact is also explained with it"; it is a category mistake to "apply the term ‘scientific’ to one single theory" rather than the appropriate series of theories; "contrary to naïve falsificationism, no experiment, experimental report, observational statement or well-corroborated low-level falsifying hypothesis alone can lead to falsification. There is no falsification before the emergence of a better theory." (Lakatos, page 119)
27. Lakatos, 1970, p.116
28. Lakatos, 1970, p.119
29. Reflective equilibrium can be described as the net result of attempts to provide a warrant for general principles; particular cases provide evidence for our intuitive judgments. When these intuitive judgments are examined collectively for internal consistency it is often lacking. To obtain consistent general principles, new tentative principles are forwarded that address most of our intuitions. Those intuitions that conflict with these tentative principles are re-examined, but from the perspective of these new principles. By repeating this recursive process it is suggested that a position can be approached, but perhaps not reached, of ‘reflective equilibrium’, where our examined intuitions are harmonious with our examined principles. (Lowe, page 753)
The idea of reflective equilibrium has been challenged by disputing the justificatory status of the evolving series of principles. (Lowe, page 753) The point of contention is clearly visible with respect to Lakatos’ string of linked theories in sophisticated falsificationism. To mix some metaphors, the terra firma of reflective equilibrium is a ‘floating island’ whose progress in an ‘ocean of uncertainty’ is determined by a ‘current of tentative principles’. The entire island must move because recalcitrant intuitive judgments are not allowed to remain anchored but are rewritten in terms of the island moving in the current, i.e. these troublesome intuitive judgments only temporarily describe where the island has been and then are adjusted to fit where it is headed. There is historical information that is lost; the path the island has taken has left no trail markers. Certainly the point of origin is left unknown. Modern information theory manipulates vectors and despite the inherent inverse problem with constructing vectors, namely that an infinite number of paths could lead from the tail to the head, the origin is known. This external knowledge is sacrificed in reflective equilibrium for the sake of internal consistency and that loss gives rise to questions of justification. As it stands, it appears that reflective equilibrium, as the ‘approached’ concept, would fall under Lakatos’ description of dogmatic falsificationism.
30. Gleick, p.103
31. Gleick, p. 99
32. In the sense of infinity of 'growth without limit' or 'mathematical' infinity.
33. À 0 is a metaphysical or ‘completed’ infinity. A mathematical or ‘potential’ infinity is a finite set at any given moment.
34. Gleick, 1988, p.101
35. Both uses of 'infinitely' in this sentence should be interpreted as potential infinities since sophiticated falsificationism is a method associated with the 'growth' of knowledge.
36. In the sense of infinity of 'growth without limit' or 'mathematical' infinity.
37. Collection of 'infinte' paths of varying cardinality.
38. Scientific American. July, 1999. Practical Fractals, p.38.
39. It appears that this is what our imagination does do, and this gives an additional pragmatic impetus to considering this line of reasoning. Human imaginations, however, likely operate in parallel so that the required orthogonality is supplied by multiple simultaneous viewpoints taking up residency of the ordinally ontologically separated sets. The simultaneous multiple viewpoints also allow those same sets to be ontologically available for this residency.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Bell, E.T. (1937). Men of mathematics. New York: Touchstone.
References
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