Monday, February 28, 2011

The Essence of the Circle

Spinoza argues that the Euclidean definition of Circle, 'a set of points equidistant from a single point', expresses a property, but not the essence, of it. However, he does not further consider that the essence of the Circle does not consist in its status as a Geometric figure. No doubt, the Circle is most familiar as the latter, but it is also both a representation of Rotation and a model of the Wheel. As a representation of Rotation, it is independent of geometrical features, i. e. Circumference, Center, Diameter, etc. are all superimposed and extrinsic. As a model of the Wheel, such features are abstractions from physical parts, e. g. a circumference represents a rim, a center represents an axle, etc. Clearly, this two-fold abstractness of the Circle does not detract from its usefulness. But, this two-foldedness demonstrates that its essence consists in its conveyance of Motion, i. e. from observed phenomenon to constructed instrument, rather than in its formulation of abstract static relations.

Sunday, February 27, 2011

Euclid's Definition of Circle

The traditional definition of Circle, derived from Euclid, is 'a set of points equidistant from a given point'. What is plainly inadequate about this definition is that from a given point, there are an infinite number of sets of equidistant points, i. e. a given point can be the center of an infinite number of circles. Spinoza's genetic variation on the definition does not escape the difficulty. For, a line pivoting on one of its ends does not describe merely one circle, i. e. the path of the free end of the line, as Spinoza intends--in the process, the motion of every point on the line describes a circle, as well. The problem with any such attempt to define Circle in terms of Center seems to be that it infers that because the points of a circle defines a unique center, a center defines a unique circle.

Saturday, February 26, 2011

Spinoza's Definition of Circle

Spinoza defines Circle as 'the figure described by the movement of a line, in which one end is fixed and the other is free'. In other words, he defines Circle in terms of Straight Line and Point. However, the difficulty with the definition is the meaning of 'fixed', for, if it means 'immobile', then not only one end, but the whole line is immobilized if that one end is. In actuality, what he is referring to as 'fixed' is a pivoting on one end, as the use of a compass to draw a circle demonstrates. But to 'pivot' means to 'rotate', the understanding of which presupposes that of the concept of Circularity. Hence, Spinoza's definition not only does not refute the hypothesis previously proposed here--that the Center of a Circle is itself a Circle, i. e. a relatively smaller one--it tends to support it.

Friday, February 25, 2011

Circle and Point

Insofar as a circle is defined in terms of a center, as is traditionally the case, the concept of Circle presupposes that of Point. However, as Whitehead has shown, Point can be defined in terms of the 'regions' that include it. Thus, Center can likewise be defined as a set of progressively included circles. Now, the distinction that Whitehead is able to draw between Region and Point depends on the latter being conceived as a limiting case, i. e. as an entity which includes no regions. So, without that condition, Point is only a special case of Region. Similarly, therefore, Center is nothing but a relatively smaller Circle, i. e. Point is a species of Circle.

Thursday, February 24, 2011

Circularity and Quantification

Aristotle asserts that an infinite circular movement has no magnitude, because magnitudes are always finite. Bergson shows that the representation of Continuity as an extensive magnitude involves the introduction of divisibility. Together, these theses suggest that the determination of the length of a Circumference entails a double falsification--it treats a circumference as a finite line, a line which is furthermore construed as the sum of rectilinear segments. Even Calculus exhibits the inadequacy of the latter phase of the procedure--while purporting to equalize Curve and Tangent via Integration, it can still only formulate the units of curvilinearity in awkward terms of those of rectilinearity, e. g. as 'distance per time squared'. In other words, Circularity is essentially unquantifiable, regardless of the undeniably useful superimpositions upon it by Geometry and Calculus. But, if so, then not only is ╥ not a constant, it is not even a number, which three futile millennia of attempting to calculate it tends to confirm.

Wednesday, February 23, 2011

One of the main characteristics of any circle is the incommensurability between its curvilinear and its rectilinear features, i. e. between its circumference and its radius and diameter, a relation represented by the number ╥. In some contexts, one or other of the terms of this relation is accorded priority. For example, because of the ease of construction of a circle in terms of an integral radius, in basic Geometry, the rectilinear features are primary and rational, while the curvilinear one is derived and irrational. On the other hand, modern Calculus connotes the rectilinear term of the curve-tangent relation as the one that is 'derivative'. Regardless, what is irrational is ╥ itself, not necessarily one or the other of its relata, e. g. both a radius and a circumference can, in principle, be irrational quantities. Now, since there seems to be neither a perfect circle nor a straight line to be found in nature, the stronger case can be made that the concept of each is a priori, rather than a posteriori. Hence, ╥ expresses an irrationality within the a priori realm, contrary to the contention of some Rationalist doctrines, and in the structure of the Circle, a symbol of Perfection in e. g. Platonism and Aristotelianism.

Tuesday, February 22, 2011

Eternal Recurrence, Circularity, Linearity

A serious flaw in the concept of Eternal Recurrence as a cosmological doctrine is that it entails as fact a component that is nowhere in evidence. The concept is construed as a linear course of events attaining cyclicity, but neither personal biography, human history, nor natural phenomena evinces such one-dimensional sequentiality, except as the product of abstraction and simplification. Accordingly, the only efficacy of the concept as an Ethical or Psychological doctrine, as becomes clear from Nietzsche's treatment, is derived from the inescapable repeatability of specific experiences, not from the cyclicity of their totality. Hence, if the concept qua circular has had any meaningful exemplification, it has been as an Aesthetic paradigm. Two notable illustrations of such Circularity have been Finnegan's Wake and Ascending and Descending. The Circularity of the former is to be distinguished from that of Ulysses--while Ulysses only represents it, i. e. a 24-hour cycle, the Finnegan's Wake exemplifies it, by ending where it begins. As is the case with Escher's print, its Circularity entails Linearity, i. e. the sequentiality of its sentences, and Escher's single-file of ascenders and descenders, attains cyclicity.

Monday, February 21, 2011

Philosophy and Circularity

Circularity seems to have rarely been more than a peripheral explicit topic in Philosophy. Typical of its treatment is its status in the legacy of Pythagoreanism--while triangularity and musical tones are the best-known features of that doctrine, both, for Pythagoras himself, are grounded in circular patterns. Plato recognizes the perfection of the Circle, without entertaining it as the Form of the Good. For Aristotle, the self-sufficiency of the Circle informs his Ethical ideal, i. e. thought thinking itself. In modern Philosophy, the Circle serves Hegel as a symbol of Infinity, but the 'revolutions' of neither Kant's Copernican turn nor Marxism seem to have much rotational about them. It does become more prominent for Nietzsche, but his interest in Eternal Recurrence is primarily its repetition and unboundedness, rather than its circularity per se. In fact, if there is one Philosophical context in which Circularity is an explicit theme, it is in negative terms, i. e. in Logic.

Sunday, February 20, 2011

Exponential Polyrhythms

What has previously been called here 'exponential polyrhythms'--cycles of heterogeneous cycles--are pervasive in human experience. Some are based on natural cycles, e. g. days and years. Some have theological or mythological origins, e. g. weeks. Others are artificial, e. g. minutes and hours. Some are integral to cultural constructs, e. g. the four-year presidential term, the nine-inning baseball game. Perhaps the most fundamental of these cycles is the month, i. e. the propagation of the human race is a function of monthly ovulation and of nine-month gestation. However, despite the inarguable pervasiveness of Exponential Polyrhythm, it is not a topic that has received much Philosophical attention, under that or any other rubric. To the contrary, the Skepticism and the Atomism that underpins modern Philosophy are attempts to undermine that structure, i. e. if, as they hold, it is no more than a mental construct, then conceiving experience in its terms is an acquired habit that can be broken, as well. Still, such Disassociationism--a more accurate characterization of such theories than 'Associationism'--lacks the resources to deny that it is no more than the deconstructionist phase of a Reconstructionist cycle.

Saturday, February 19, 2011

Polyrhythm

Experience is not merely rhythmic, but polyrhythmic. The pervasiveness of polyrhythms can be gleaned from a simple example: hopping forward on one foot is constituted by a rhythm, which, when conjoined with the rhythmic movement of the other foot, produces walking. Likewise, all basic motility is polyrhythmic. Furthermore, a daily walk is polyrhythmic--e. g. a combination of Monday's walking rhythms with Tuesday's--which, therefore, demonstrates a poly-rhythm of polyrhythms. However, such an example exhibits a distinction of types of polyrhythm, which might be classified as 'arithmetic' and 'exponential'. In general. arithmetic polyrhythms are concurrent, while exponential ones entail some sequential juxtapositioning. The distinction presents a challenge to Associationism, which recognizes only arithmetic combinations. So, the example of polyrhythms exposes its inadequacy--while it explains the conjunction of two elements, it does not explain the closure of that conjunction, which any sequential conjunction would entail. Hence, the only way that it can conjoin Tuesday's steps with Monday's is as a continuation of the former. In other words, Associationism cannot fully explain Polyrhythm.

Friday, February 18, 2011

The Rhythm of Parallelism

According to Parallelism, the rhythm of mental events and the rhythm of physical events is one and the same rhythm, considered from two different perspectives. The significant intra-experiential distinction, for Spinoza, is between active and passive rhythms, i. e. between ones internally generated and those with external sources. One fundamental premise of Parallelism is the simultaneity of the mental and physical sequences. However, Spinoza is likely unaware of more recent neurological theories that demonstrate time lapses between both afferential and efferential physiological processes. In other words, they show that while the two rhythms may be parallel, they are out of phase with one another, just as the rhythm of echoes can be parallel to, but not coincident with, that of their source. So, Parallelism misses the rhythmic interplay between Body and Mind, or, more properly, between the Extending and the Retaining that, for Spinoza, is the idea of that Extending.

Thursday, February 17, 2011

The Rhythm of Social Experience

Though social interaction seems to be constituted by give-and-take, the Doing-Undergoing model of experiential rhythm is ultimately inadequate to it. For, that model represents only a detached perspective on such interaction, thereby abstracting from the oriented nature of involved participation. From the latter perspective, socialization is constituted by an interior-exterior dichotomy, in which its primary rhythm is Exteriorization-Interiorization, i. e. Extending-Retaining. While Sartre and Levinas, notably, recognize the experiential fundamentality of the interior-exterior distinction, they each fail to appreciate the active Exteriorization phase of the rhythm of experience--for Sartre, Being-for-others is passive, while for Levinas, the interior suffers an invasion by the Other. Neither entails active Exteriorization, i. e. Being-towards-others, or a reaching out to the Other, respectively.

Wednesday, February 16, 2011

Rhythm of Learning

One prevalent traditional model of the Rhythm of Experience is Stimulus-Response, a popular post-Darwinian instance of which is Destabilization-Adaptation. Dewey, for example, espouses, for the most part, the latter, which means that his Doing-Undergoing model is, more accurately, Undergoing-Doing, since Doing is a response to some environmental disruption. As the term 'destabilization' connotes, the response to a stimulus is a recovery of a previous condition. Hence, the Stimulus-Response model does not accommodate motion that is a surpassing of an already attained status, i. e. it does not explain Growth or Learning. Rather, the latter two exemplify the Extending-Retaining pattern, which is especially significant to a theory of Experience, given the frequently accepted synonymy of 'experience' and 'learning'.

Tuesday, February 15, 2011

Experience, Extending, Retaining

As previously proposed, the basic rhythm of Experience is Extending-Retaining. So, the concepts of Leibniz, Bergson, Whitehead, and Deleuze, share the inadequacy that, each, in its own way, does not recognize the Extending phase of experience. Leibniz' Monad always remains in itself. For Bergson, disembodied intuited Duration includes no datum beyond the most recently given. Likewise, for Whitehead, Concrescence terminates in internal satisfaction. And, Deleuze privileges contraction, folding, and intensity, with respect to which expansion, unfolding, and extensity, are, at best, derivative and subordinate. Even his earlier seeming commitment to a concept of open subjectivity eventuates in, as he affirms at the end of his Leibniz book, his Nomad remaining within a closed Monadic locus. Hence, none of these can fully appreciate the basic Extending-Retaining rhythm of Experience--doing; awareness of that doing; subsequent further doing; etc.

Monday, February 14, 2011

Rhythm of Experience

Dewey proposes that the basic rhythm of experience is Doing-Undergoing, i. e. an interaction between an entity and its environment. A familiar example of such give-and-take is exhalation and inhalation, and a less obvious one is stepping, which is constituted by both a foot exerting a force against the ground, as well as the converse. However, cardiac systole-diastole is vital but intra-organistic. Likewise internal is what can be called Extending-Retaining, which refers to a motion, and the preserving of it, potentially to be further extended-upon. For example, a step becomes a walk only by virtue of its being preserved and then extended-upon by subsequent steps, each, in its turn, a new Retained Extending. In other words, Doing-Undergoing constitutes only the Extending phase of an Extending-Retaining rhythm, and, hence, is only a component of that more fundamental rhythm of experience.

Sunday, February 13, 2011

Rhythm, Contraction, Expansion

Two intimate examples of biological Rhythm are the contraction-expansion patterns of both breathing and the heartbeat. Deleuze distinguishes that biological sense of 'contraction' from a mental one, which, as he argues, is not complemented by a corresponding 'expansion'. On his analysis, Repetition is the product of mental contraction, i. e. of a synthesis that generalizes the multiplicity of elements occurring in a variety of cases, as one formulation, e. g. Hume's Constant Conjunction as Habit. Hence, Repetition, for him, is always, and only, a mental construct. He thus overlooks that to repeat is to enact the product of that mental construct, not merely to entertain it, in the process missing the opportunity to consider that such an enactment functions as an 'expansion' corresponding to the 'contraction' that produces it. In other words, his concept of Repetition inadequately represents the fundamental rhythms of life.

Saturday, February 12, 2011

Rhythm, Circle, Wheel

Deleuze begins his analysis of Repetition with the observation that a repetitive pattern is never given as such as an experiential datum, but is the product of a mental synthesis. However, if he had kept to his insight that Repetition is Rhythm, and had recalled his declaration, at the end of his book on Hume, that Philosophy is a "theory of what we are doing", his examination of Repetition might have instead begun with a study of a producer of Rhythm, i. e. of a drummer. By observing the act of drumming, he might have noticed that what the drummer produces is an expression of variable cyclical physiological motions, not the sameness, or the difference, between beats. He then might have realized that while the Circle might be a symbol of Heterogeneity, it is also the prototype of both human creativity and human-made motion, i. e. of the Wheel.

Friday, February 11, 2011

Cycle, Turn, Return

Deleuze interprets 'Cycle' as 'return of the Same', so, therefore, as inadequate to dynamic Repetition, which he conceives as 'return of Difference'. However, those characterizations are both classifications of 're-cycle', which presupposes an independent definition of 'Cycle'. Deleuze seems to overlook that 'return' entails 'turn', which may be one reason that he does not recognize that a cycle consists in an active turning, independent of the character of multiple turnings. Hence, rather than Cycle being inadequate to his dynamic concept of Repetition, the latter derives its dynamic character from the Cycle that constitutes it.

Thursday, February 10, 2011

Rhythm and Cyclicity

Dewey's definition of Rhythm as "ordered variation of changes" seems to challenge his own proposal that "rhythm is a universal scheme of existence". For, if in a "uniformly even flow . . . there is no rhythm", as he explains it, then Rhythm is not universal. In contrast, if Rhythm is defined as entailing periodicity, then, as Whitehead observes, it does apply to all motion--the subatomic constituents of all phenomena consist in orbits, and even a uniform flow consists in a regular variation with respect to time. Hence, the revised formulation, that 'rhythm is the periodicity of all existence', seems to resolve Dewey's ambivalence. Furthermore, it sharpens the distinction between Rhythm and Repetition, for, while, the traditional concept of Repetition expresses Identity, and Deleuze's expresses Difference, Rhythm expresses Cyclicity.

Wednesday, February 9, 2011

Duration and Rhythm

As Deleuze notes, Bergson seems committed to each of two, apparently irreconcilable, theses--that Duration is private and simple, and that Elan Vital possesses universal Duration--i. e. that there are a plurality of Durations, and that there is one Duration. However, Bergson's occasional characterization of a Duration as a Rhythm suggests an easy resolution to the apparent dichotomy. For, a given rhythm can be either composite, or part of a composite, or both, without loss of integrity. Thus, that Duration is immediately given as undivided does not preclude that it is composite as well, e. g. that it is a composite of a multiplicity of physiological rhythms, as Spinoza holds, or that it is part of some more comprehensive rhythm. On the other hand, it also follows from the interpretation of Duration as Rhythm that the distinction between Intensity and Extensity is only provisional and perspectival. Now, whether or not Deleuze or Bergson would abandon that distinction is unclear, but insistence upon it would indicate that the Inner vs. Outer distinction is more fundamental than that of Many vs. One.

Tuesday, February 8, 2011

Leibniz, Harmony, Rhythm

As Deleuze shows, for Leibniz, musical harmony is an expression of pre-established inter-Monadic harmony. That is, the togetherness of two discrete tones expresses, according to Leibniz, a togetherness of two opaque Monads which obtains prior to, and independently of, the act of expression. Hence, Leibniz denies that what might be called 'rhythmic coordination' plays any part in the expression of musical togetherness. Rhythmic coordination entails at least some of the following--a time-signature common to each participant; physiological coordination between the perception of a sonic element and the production, e. g. via voice, of a subsequent one; and, the physiological inter-participant imparting of rhythm. In other words, Leibniz denies, for example, that toe-tapping can be caused by the irresistible beat of a drummer, a common and palpable phenomenon. In contrast, to acknowledge the actual effectiveness of rhythmic coordination is to recognize that musical harmony, and, in general, interpersonal togetherness, can be creatively established without the presupposition of its being pre-established

Monday, February 7, 2011

Habit, Repetition, Rhythm

As previously discussed, Deleuze at one point recognizes the synonymy of Repetition and Rhythm, though the former remains the exclusive focus of his attention. However, the advantages of the latter can be observed in the context of his discussion of Humean Habit. In following the traditional analysis of Hume's concept of Habit as entailing constant conjunction, Deleuze likewise glosses over a nuance that he himself occasionally recognizes--that the conjunction involved is not merely a transition from one term to another, but an "easy" transition. Furthermore, it seems to be generally under-appreciated that upon the completion of an acquisition of a habit, the transition is transformed into a continuity. Now, 'Rhythm' seems to express 'fluid segue' more vividly than does 'Repetition, just as it seems better suited than the latter to express, in general, a principle that is conceived as vital and dynamic.

Sunday, February 6, 2011

Deleuze, Repetition, Rhythm

A primary aim of Difference and Repetition is to distinguish fundamental, dynamic, Repetition from derivative, static, 'repetition'. One way that Deleuze attempts to contrast them is as 'rhythm' vs. 'cadence', respectively, though given the accepted synonymy of those two, this effort is less than clarifying. Another way is his characterizing the static version as 'repetition of the Same', thereby implying that the dynamic version be understood as 'repetition of the Different'. However, this contrast can be interpreted in two, not necessarily consistent ways. On the one hand, 'repetition of the Same' can be interpreted as a relation between two identical terms, i. e. A and A, whereas 'repetition of the Different' requires a minimum of three terms, at least two of which must be distinct, e. g. A and B, and B and C. On the other hand, a 'repetition of the Same' could denote two terms qua identical, with a 'repetition of the Different' accordingly denoting those same terms qua numerically distinct. So, the distinction that Deleuze means to draw is not immediately clear, but what he does successfully make explicit in the relevant passage is his recognition that 'Repetition' and 'Rhythm' are synonymous.

Saturday, February 5, 2011

Adamism, Platonism, Repetition

The two predominant Cosmological traditions, in at least 'the West', can be called 'Adamism' and 'Platonism'. According to the former, all events are copies of some privileged original, while according to latter, they are instances of some eternal archetype. Part of the originality of Whitehead's system is to combine the two, with 'God' as both primordial existent and an embodiment of all the eternal archetypes. In contrast, a theory of Eternal Recurrence, regardless of Nietzsche's intentions for it, is neither Adamist nor Platonist--it recognizes neither some privileged original nor the existence of any entity that transcends the cycle of events. Now, the meaning of 'repetition' is a function of fundamental principle--in Adamism, it denotes any copying of the original, while in Platonism, it refers to any instantiation of a universal. In contrast, in a theory of Eternal Recurrence, it is itself the fundamental principle, not a derivative one. One of Deleuze's primary aims in Difference and Repetition is to establish 'Repetition' as such a fundamental principle.

Friday, February 4, 2011

Deleuze and Rhythm

The main theme of Deleuze's book on Leibniz is how the latter's system reflects Baroque characteristics, and, indeed, Deleuze eventually shows the relation between Leibniz' conceptual and the musical senses of 'harmony'. However, the presentation includes no treatment of Rhythm. Furthermore, despite the etymological root of 'rhythm' in 'flow', there is little attention to Rhythm in either the texts of recognized philosophers of Flux, such as Bergson and Whitehead, or in Deleuze's interpretations of them. Deleuze does marginally refer to it in Difference and Repetition, but only as a brief illustration of his concept of Repetition. In contrast, Rhythm is a central topic in, say, Dewey's Art as Experience. Hence, on the basis of Deleuze's thesis that Philosophy reflects Culture, the scarcity of consideration of Rhythm in the works of eminent pioneering European thinkers perhaps expresses a lack of exposure to African musical traditions.

Thursday, February 3, 2011

Deleuze, Leibniz, Vinculum

Deleuze proposes that one of the virtues of his interpretation of Leibniz' Monad as a Fold is its clarification of one of the features of the Monad--the Vinculum. The function of the latter for Leibniz is analogous to that of the Pineal Gland is for Descartes, i. e. as a mediator of a dualistic structure. Because of its topological characteristics, the Fold illustrates how the Vinculum unifies the interiority and the exteriority of a Monad. However, as previously argued here, the Fold is an hypostasization of two more fundamental processes--Enfolding and Unfolding--the distinction between which is not reducible to a topographical feature. Hence, despite Deleuze's effort, as a reconciliation of a posited fundamental dichotomy, the Vinculum is an ineffective as is the Pineal Gland.

Wednesday, February 2, 2011

Deleuze, Leibniz, 'Best Possible'

Deleuze offers "richest compossible" as a synonym of Leibniz' formulation 'best possible'. But, while by 'compossible', both men mean 'non-contradictory', the qualification "richest" remains uncertain. Almost surely, it is less likely that Deleuze means by it 'wealthiest' rather than 'greatest number'. But the latter is still unsatisfactory, since it entails, for example, that a world consisting of four people indifferently co-existing is a better world than one consisting of three productively cooperating. Furthermore, both Deleuze and Leibniz occasionally seem to suggest that 'best'='most harmonious'. However, 'harmonious' is as undetermined as is 'richest'. So, in the absence of any further clarification, Leibniz' assertion that 'this is the best of all possible worlds' is more illuminating as a Theological than as a Cosmological thesis, i. e. it is more informative about Leibniz' concept of God's creative process than that of the objective character of the world that is the product of that process.

Tuesday, February 1, 2011

Deleuze, Leibniz, The Best

While Leibniz' thesis 'This is the best of all possible worlds' is most frequently associated with either Modal Logic or Voltaire's ridicule, Deleuze notes its significant departure from Platonism. Instead of the absolute Good-Evil dualism of the latter, 'the Best' entails comparative evaluative degrees. Deleuze further considers Leibniz' application of this Axiology to personal conduct, which can, accordingly, be judged in terms of its degree of perfection, rather than of 'Good' vs. 'Evil'. On the other hand, in some passages, what Leibniz promotes is not perfection in conduct, but progress with no definite upper limit, which Deleuze presents without either noticing the discrepancy, or questioning the implications of the possibility of unlimited progress to a closed entity such as a Monad. So, as he characterizes Leibniz as promoting both the perfection as well as an "augmentation" of a Monad's personal traits, he glosses over a possible basis for the distinction that he attempts to draw between Monad and Nomad.