Circle, Clockwise, Counter-Clockwise

Spinoza's definition of Circle--"the figure described by any line whereof one end is fixed and the other free"--is flawed in two respects.  First, it also defines a mere Arc.  Second, it leaves indeterminate the 'freedom'--clockwise or counter-clockwise--and, so, as a "proximate cause", it lacks Necessity.  Now, the first can be rectified by the qualification of "figure" as 'closed'.  But the second continues a deeper, pervasive problem.  Even those, e. g. Aristotle, who attribute divinity to celestial bodies because of their Circular motion cannot deny that such motion--spinning, orbiting--consists in one of two possible directions, of which there is no actual uniformity among those bodies in direction of motion, i. e. some are clockwise, some are counter-clockwise.  Now, according to the most highly developed theorizing of astrophysicists, these motions have been determined arbitrarily--e. g. by the happenstance of direction of spin at a very early stage of the development of the Universe.  Thus, even if subsequent motion does follow uniformly, the purported Determinism of the Physical Universe is falsified very early, and, with no guarantee of a recurrence of such happenstance any time thereafter, as Lucretius proposes.  In any case, specification of Circular motion as either Clockwise or Counter-Clockwise is not at all helpful to either astrophysicists or Spinoza.  For, while the distinction might be decisive for a Plane figure, it is only arbitrary for a Solid, e. g. from the perspective of the North Pole, the spin of the Earth is Counter-Clockwise, but from the South Pole, Clockwise.

Emanation, Evolution, Geometry

If, as Wolfson interprets it, Spinoza's doctrine is Emanationist, then his God/Nature/Substance begins as absolutely simple, and infinitely develops in complexity.  Thus, the doctrine can also be classified as Evolutionist, e. g. one event in which is a development in the complexity of the thumbs of some simians.  Now, according to Spinoza's Parallelism, all Ideas are fundamentally actual, i. e. correspond to some Body or another.  But, if so, his concept of Eternity is problematic when attributed to Ideas, for that attribution entails that an Idea might pre-exist its corporeal actualization.  More specifically, insofar as he defines a Circle in terms of how to draw one, the Idea of a Circle cannot be eternal, since a corporeal Circle cannot pre-date the evolution of the Mode with the bodily capacity to execute the definition.  Instead, on the basis of the premises of his doctrine, the Circle, and all figures that must be drawn, i. e. Geometry, must be a concomitant of Evolution.  In other words, in an Emanationist doctrine, as an in an Evolutionist theory, the idea of Geometry is no more eternal than Species are fixed.

Circle, God, Human

In the Ethics, Spinoza characterizes the idea of a Circle as 'caused by God'.  Thus, according to his Parallelism, a bodily Circle is also caused by God.  In Improvement of the Understanding, he presents an operational definition of Circle, i. e. how to draw one.  So, given that only Humans seem to have the capability of drawing anything, and Spinoza offers no example of a Circle that is produced by non-Human agency, the attribution of the production of a Circle to an omnipresent deity seems trivial.  Instead significant is his perhaps unwitting alignment with Protagoras that Geometry is a strictly Human science, with the implication that the Aristotelian deity is an anthropomorphic creation.  This status in his Rationalist doctrine thus perhaps stands as a transition to its all-too-human status in Kantian Rationalism.

Line, Vector, Ordinality

In Geometry, a Vector signifies Direction from a point of origin.  But Direction implies Motion.  So, a Vector is often used to represent Motion.  But, more immediately, it exemplifies Motion, i. e. the Motion by which a Line with an arrow affixed is drawn.  Furthermore, all lines are drawn, and drawn from an origin to a terminal point.  Thus, all lines are produced by directed motions.  Hence, all lines are vectors, whether or not they are illustrated as such.  Furthermore, the main elements of Pythagorean-Euclidean Geometry are constructed out of lines--planes and solids.  Now, Direction connotes Order, i. e. the drawing of a Line is ordered from origin to terminus.  Hence, just as Cardinal Numerology is derived from Ordinal Numerology, Pythagorean-Euclidean Geometry is derived from Ordinal Geometry, i. e. Vector Geometry.  The possible priority of the latter is in front of Kant whenever he uses the drawing of a Line as an illustration, but he does not recognize it as such.

Geometry, Ordinality, Space

Kant misses an opportunity to Ordinally ground not only Mathematics, previously discussed, but Geometry, as well.  As is the case with his treatment of Mathematics, his concept of Geometry is standard--Pythagorean-Euclidean--and its connection to his concept of Intuition is only via vague allusion.  However, an alternative derivation begins with one of the prominent innovations of the era--Cartesian Analytic Geometry--and its representation as a centered grid from which arrowed axes emanate.  From there, it is easy to argue that Geometry is grounded in oriented Space, one of the two Forms of Intuition, according to Kant, on the basis of which can be constructed the Vector alternative to Pythagorean-Euclidean Geometry.  But oriented Space is Ordinal Space.  So, he also misses this Ordinal dimension of Intuition, as well, and implications for Geometry.

Mathematics, Ordinality, Time

The influential Pythagorean concept of Number is not only Cardinal, as has been previously discussed, but Ontological, i. e. a fundamental constituent of objective Reality.  Protagoras is an early critic of the latter, denying the objectivity of Numbers.  He thus anticipates Berkeley's thesis that Number is a Secondary Quality.  Kant further systematizes the counter to Pythagoreanism, by associating Mathematics with the Forms of Intuition.  However, the association is only vaguely posited, grounded only by the brief suggestion that it is to specifically Space, perhaps because a Line helps him explain how Addition is a Synthetic operation.  He thus misses a more rigorous derivation, but from Time, and in terms of Ordinal Numerology.  For, his concept of Time as Succession is Ordinal, i. e. constituted by moments following one after another, and, hence, always the possible ground of a sequence of First, Second, etc.  He further misses that the fundamental Mathematical operation--Addition--is nothing but a representation of Counting, which is an Ordinal process, even if the terms of the process are Cardinal Numbers.  In other words, what is commonly expressed as a sequence of Cardinal Numbers--One, Two, Three, etc.--is, more precisely, Ordinal--First, One; Second, Two; Third, Three, etc.--in which the Temporal Form of Intuition is made explicit.  But Counting is also the basis of Measurement.  Thus, the Kantian concept of the fundamental structure of Intuition as Ordinal completes the Protagorean critique of Pythagoreanism expressed by 'Man is the measure of all things'.

Indefinite Dyad and Ordinal Axiology

For some neo-Platonists, and perhaps Plato himself, e. g. in The Laws, The One vs. Indefinite Dyad contrast has Theological and Moral implications that have been influential.  For example, The One signifies God, and the Good, while the Indefinite Dyad signifies the Demiurge, and Evil.  Consequently, the doctrine has been a factor in the long tradition of Axiological dualism.  However, as has been previously discussed, the neo-Platonist concept of the two principles is informed by Cardinal Numerology, to which an Ordinal alternative is possible.  According to that alternative, the Indefinite Dyad is the infinite sequence of First, Second, etc., from which the concept of The One is derived via hypostasization and abstraction.  Likewise, an Ordinal Axiology is available, beginning with the Best, perhaps ending with the Worst, within which the fundamental evaluative contrast is Better vs. Worse.  One prominent example of Ordinal Axiology, though rarely recognized as such, is Utilitarianism, but the more significant one has been obscured.  That is Nietzsche's concept of an Order of Rank, an important feature of his repudiation of the long history of Theological and Moral Dualism.  However, that Axiological concept has gotten overshadowed by his own apparent commitment to an inversion of the predominant Good vs. Evil dualism, which is actually not a commitment at all, but only an unearthing of an alternative dualism, i. e. Master Morality, in the genealogy of the predominant dualism.  Instead, his actual commitment to the more radical Ordinal alternative remains affirmed but underdeveloped in the projected Will to Power collection.  So, while Utilitarianism carries on the tradition of Ordinal Axiology, Nietzsche's more explicit version has gotten lost in the attention to the more controversial parts of his oeuvre.

Emanation, Indefinite Dyad, Ordinality

The neo-Platonist concept of Emanation has two main principles, each Numerological--The One, signifying the origin of Emanation, and the Indefinite Dyad, signifying the ensuing unlimited manifold.  The pair are thus a variation on the Pythagorean pair of the One and the Two, the latter of which is sometimes termed Dyad, though, because it signifies a determinate multiplicity, is more accurately a Definite Dyad.  But despite the variation, the acceptance of the One by Emanationists expresses an acceptance of Cardinal Numerology, with the structural incoherence of an implied lacuna between the origin and the subsequent manifold, incoherent because the concept of Emanation connotes continuity from the outset.  In contrast, conceiving the Indefinite Dyad as Ordinal preserves that continuity, i. e. from First to Second, etc.  Furthermore, that the One can be derived from the First, i. e. by abstracting it from the inherent Ordinal sequence, shows that the One is not necessarily a principle that is independent of the Indefinite Dyad, nor one that precedes it in any respect.  So, Emanationism is one system that implicitly elevates the Indefinite Dyad from Philosophical marginality, and Ordinal Numerology from Mathematical marginality.

Counting, Cardinal Numbers, Ordinal Numbers

A common confusion between Cardinal Numbers and Ordinal Numbers is evident in the phrase 'learning how to count'.  Since one has likely already learned how to count in one's native language, 'learning how to count' in a foreign language cannot consist in the development of a new skill.  Rather, it means, more precisely, learning what the names of Numbers are in a foreign language.  But the process is no different in one's native language--what a child who knows 'how to count up to Seven' needs to learn next is not a repetition of the process, but that the name of the next number is 'Eight'.  And likewise with all the Numbers, starting with 'One'.  Thus, 'how to count' is distinct from 'naming the Numbers in order'.  Now, regardless of whether or not the structure is to be classified as 'a priori', the process of Counting has a subjective Ordinal structure--a starting point, a next, etc., which Kant calls 'succession'--that governs any act of naming an sequence of terms, such as Cardinal Numbers.  So, even if a child has not learned the names 'First', 'Second', etc., implicit knowledge of Ordinality is presupposed in any 'learning to count'.  In other words, 'learning how to count' in English means learning that first is 'One', second is 'Two', etc., which not only distinguishes Cardinal Numbers from Ordinal Numbers, but illustrates that the latter are prior to former, even functioning as such only implicitly.  Indeed, the concept of Priority is itself Ordinal, and, hence, the assertion 'Cardinality is prior to Ordinality' is contradictory.  Kant's exposition of his concept of Mathematics as a priori might have been clearer with an emphatic characterization of its basis as Ordinal Numerology.

Logic, Mathematics, Quantity

As has been previously discussed, the standard attempt to unite Logic and Mathematics, via the equating of Conjunction and Disjunction with Multiplication and Addition, respectively, depends on the problematic reduction, within Logic, of Inference to either Conjunction or Disjunction.  In contrast, without that reduction, a immediate unification is possible, via not Propositional Connectives, but via Predicate Quantification, which is the medium of Aristotelian original Logic.  Specifically, Inference, qua Entailment, consists in the relation of Greater-Than, as is plainly illustrated by Venn diagrams, and exemplified by the Universal-Particular relation of Aristotelian Logic.  But, one reason why the standard approach to uniting Logic and Mathematics is not via the apparently common concept of Quantity may be that the concept is not necessarily common to them.  For, insofar as the Quantity entails an inherent comparison of quantities, it is not a property of Pythagorean, i. e. Cardinal, Numbers, which are mutually independent.  So, if Pure Mathematics is conceived on the basis of Cardinal Numerology, then Quantity is not 'Pure', and, hence, cannot mediate the unification of Mathematics with Predicate, i. e. Quantification, Logic.

Ordinal Rationality and Cardinal Rationality

Often each conceived as synonymous with Rationalism, Logic and Mathematics have sometimes been made interchangeable by virtue of equating Conjunction with Multiplication, and Disjunction with Addition.  Thus, the essential Logical operation of Inference is translatable into Mathematics by virtue of the representation within Logic of it in terms of Conjunction or Disjunction.  However, that representation is susceptible to the same challenge that Kant levels at Hume's concept of Causality.  According to that challenge, Hume's rendering of Causal connection as Conjunction abstracts from the temporal ordering of Cause and Effect, i. e. the terms of a Conjunction, but not those of Causality, are interchangeable.  Likewise, a Conjunction or a Disjunction is symmetrical, and thus abstracts from the ordering of Antecedent and Consequent.  So, the attempt to unify Logic and Mathematics via that abstraction is also similar to the common reduction of Ordinal Numerology to Cardinal Numerology, as has been previously discussed, i. e. Antecedent and Consequent are essentially in a First-Second relation, irreducible to a mere juxtaposition such as that of Pythagorean Cardinal Numbers.  Thus, the preservation of Ordinality also preserves the essential distinction between Logic and Mathematics, one that is the distinction between what can be rendered as Ordinal Rationality vs. Cardinal Rationality.

Measurement and Ordinal Numbers

A Ratio and a Measurement are each a kind of comparison, and are perhaps intuitable as such.  But the significant difference between the two is that the terms of a Measurement are ordered, while those of a Ratio are not.  For, in the former, one term is the measurer and the other is the measured, whereas in a Ratio, both terms are Numbers.  Now, the Measurer-Measured relation is ordered because the former is prior to the latter in some respect, e. g. a timer precedes the timed.  But, being ordered means that one term is First, and the other is Second.  In other words, Measurement is based on what can be called Ordinal Numerology, in distinction from the Cardinal Numerology of Pythagoreanism.  But Ordinal Numerology cannot be reduced to Cardinal Numerology.  For, since the Cardinal Numbers, according to Pythagoreanism, are each a self-subsistent entity, they are also independent of one another.  But Ordinal Numbers are mutually implicative.  So, to the contrary, Cardinal Numbers are perhaps abstracted from Ordinal Numbers. In any case, even if it is not Man who is the measure of things, Protagoreanism presents a radical alternative to Pythagoreanism simply by virtue of introducing Measurement as its fundamental concept.

Numerology, Intuition, Anthropomorphism

As has been previously discussed, Pythagoreanism is fundamentally Numerology, from which Mathematics and Geometry are derived.  Correspondingly, insofar as the Cognition involved with Mathematics and Geometry is Reason, that involved with Numerology is Intuition.  Thus, each of the Numbers, as well as a relation between them such as a Ratio, is intuited, not calculated.  Now, Kant, an exemplary Rationalist, agrees that Mathematics and Geometry are based in Intuition.  However, in his system, Intuition is a Human faculty, not shared by a divine Mind, the sole faculty of which is Reason, because it is the faculty of a finite being.  So, Kant also agrees with Protagoras, rather than Pythagoras, that Man is indeed the measure of all things Mathematical and Geometrical.  Thus, in the case of Numerology, Protagoras' famous thesis expresses less Relativism than Anthropomorphism.

Pythagoreanism and Numerology

The fundamental principle of Pythagoreanism is that Numbers are Real self-subsistent causally efficacious entities with properties beyond mere quantity, e. g. One = Unity; Two = Multiplicity; Three = Balance, etc.  So, Pythagoreanism is a forerunner of Platonism, and, thus, of the entire tradition that follows.  Furthermore, relations obtain between Numbers, which are what Mathematical operations represent.  One such relation is Ratio, the basis, thus, of Pythagorean Rationalism.  Likewise, the spatial representation of the Numbers and some of their relations is what constitutes Geometry in Pythagoreanism.  Hence, while Mathematics, Geometry, and Rationalism area common characterizations Pythagoreanism, they each refer to one of its derivative features.  Instead, fundamentally, Pythagoreanism is Numerology, just a more rigorous version than some of its more popular varieties.

Measure, Sophistry, Geometry

Protagoras' saying, "Man is the measure of all things", is, strongly influenced by Plato's treatment of it as 'Sophistry', usually interpreted as a formulation of Individual Moral Relativism, even if the surviving textual evidence is less than conclusive in that regard.  Regardless, some of that evidence suggests its relevance in other respects.  For example, in conjunction with his dismissal of Geometry as non-Empirical, i. e. because the elements of Geometry, points, lines, regular figures, etc. are not found in nature, the thesis anticipates the Kantian principle that Geometry is a product of Human cognitive structures.  Accordingly, it coincides with the shift in emphasis in the study of Geometry signified by Eudoxus, as has been previously discussed--to measurability, a plainly Human activity.  So, at least some of what Plato dismisses as Sophistry is consistent with Rationalism that both precedes and succeeds his.

Rational and Measurable

Pi illustrates the distinction between 'rational' and 'commensurable' suggested by Eudoxus.  It is a Ratio--between the Circumference and the square of the Radius of a Circle.  Hence, it is not literally 'irrational' qua not expressible as Ratio'. But it does consist in an incommensurability between those two, often obscured by the glossing of it as 22 : 7.  Furthermore, the incommensurability is not a characteristic of one of the two terms, but of the relation between the two.  On the standard basis that it is a characteristic of a term, if R = 1, C = Pi, and, hence, C, alone is the presumably 'irrational' or the 'incommensurable' of the two terms.  But the Ratio of 2Pi : Pi is an example of Commensurability as a relation.  Now, 'commensurability' means 'co-measurable', so the shift in terminology shifts focus from some features of Reality to the measuring of those features.  Hence, Pi mediates the measurement of R and the measurement of C, not R and C themselves.  Now, a sharper illustration of the concept of Commensurability involves a contrast of R with a alternative type of measurement of C, in which the latter is, indeed, like the former, a Natural Number, yet, the two are plainly not commensurable.  This is when the span of C is given as 360--degrees.  In this case, Incommensurability is due to two absolutely independent systems of Measurement being involved.  So, the shift from Rational to Commensurable signifies a further, more radical, perhaps earliest critique of Rationalism--that of Protagoras, i. e. that it signifies not Reality but the measuring of Reality.

Swerve, Acceleration, Rationalism

Lucretius' concept of Swerve signifies a deviation of direction.  But it can also signify a change of Motion in which direction remains constant--Acceleration.  Still, the original concept is adopted by Modern Physics in the standard illustration of the relation of Acceleration and Velocity--as a curve with respect to a straight line.  The standard analysis of the illustration thus also reveals the essential moment of Swerve--an infinitesimal deviation.  But, Modern Physics does illustrate that moment with precision--it merely approximates it as 'approaches zero'.  It further glosses over the lacuna by neutralizing it, e. g. via Newton's Third Law, Conservation of Energy and Motion theses, etc.  In other words, Modern Physics is a Rationalist system that achieves universal commensurability by suppressing the incommensurability that constitutes much of what it represents.

Rationalism and Incommensurability

The term 'irrational number' is a misnomer.  For, no less than a 'rational number', an 'irrational number' can be expressed as a Ratio.  Instead, as the later Pythagorean Eudoxus observes, the distinction is between the commensurability and the incommensurability of the terms of a Ratio.  A terminological adjustment is also relevant to Modern Rationalism.  That variety, the basic operation of which is Inference, seems to have little to do with a Ratio.  However, the concept that is the basis of its fundamental principle--Contradiction--is derived from Incommensurability.  For, Contradiction is a special case of Incommensurability.  Others are not opposites, just radically different in some respect.  Likewise, there is a more comprehensive system of which what is generally conceived as Modern Rationalism is a special case.  Since that system is governed by correspondingly more comprehensive concept of Incommensurability, and Incommensurability is a characteristic of a Ratio, this more comprehensive system is a variety of Rationalism, perhaps identifiable as Super-Rationalism.

Reality, Rationality, Irrationality

Probably the oldest version of Rationalism, and still influential, is Pythagoreanism, according to which the latent structure of Reality is Mathematical, i. e. constructed in terms of Natural Numbers.  Now, the standard challenge to Rationalism is based on the premise that Mathematical relations are merely mental constructs, and only projected into the external world.  But, a sharper challenge originates with a little-known early Pythagorean--Hippasus--who is credited with the discovery of Irrational Numbers, the implication of which to Pythagoreanism is that the latent structure of Reality is just as Irrational as it is Rational.  The focus of Hippasus' discovery reportedly is the square root of 2, previously believed by Pythagoreans to be ultimately analyzable as a ratio of two Natural Numbers, even if not immediately obviously so.  On that basis, the Irrational component is correspondingly obscure.  But, once his discovery is extended to Pi, the example of the Irrationality of Reality is plainer--any circular motion, notably the orbits of celestial bodies. In other words, long preceding the Modern Empiricist subjectivization of Rationalism, there is presented the stronger refutation of it--a counterexample.

Indefinite Dyad, Reflection, Irrationality

As has been previously discussed, Swerve begins as an instantaneous inclination, which can emerge as a curve, which can develop into infinite circular motion.  So, a number that expresses the Indefinite Dyad is that which mediates rectilinearity and circularity: Pi.  Now, Swerve can also be conceived as Flex.  Likewise, therefore, Reflection is constituted by circular motion, as Aristotle is well aware.  However, he seems less aware that, as Pi signifies, this paradigm of Rationality is thus, in fact, an Irrational process.  Furthermore, since Reflection can be reiterated ad infinitum, it is doubly Irrational.

Indefinite Dyad and Circle

Lucretian Swerve is an example of the Indefinite Dyad, since it consists in a deviation from some determinate motion.  Now, as is clearer in Epicurus' original term Clinamen, from 'incline', the deviation may be only momentary, since the new motion might subsequently revert to determinacy.  So, Swerve is more patently illustrated by continuous deviation, i. e. by a curve, and, hence, ultimately,  by perpetual curvature.  But the figure that illustrates perpetual curvature is the Circle, doubly so, because perpetual motion is infinite.  So, if there is a symbol of the Indefinite Dyad, it is the Circle.

Matter and Multiplicity

The neo-Platonists conceive the Indefinite Dyad as the process of producing, out of The One, both Multiplicity and Matter.  However, they offer no clear concept of the relation between the two, other than accepting the Aristotelian concept of the latter as, in itself, characterless 'stuff', and of Multiplicity as an attribute of this stuff.  Hence, following Aristotle, they fail to notice that Matter is also the complement of Form, content that is unified by the latter.  But that means that Multiplicity and Matter are one and the same.  The difficulty in recognizing this identity is rooted in the failure to distinguish the Definite Dyad from the Indefinite Dyad, i. e. between a quantified manifold and an unquantified manifold, the consequence of which is an equivocal use of 'multiplicity', one of which is attributed to the other.  Compounding the difficulty is the traditional association of 'matter' with 'solid particle', an association that persists with the premise that there exists an ultimate particle, i. e. an Atomist premise.

Indefinite Dyad and Invisible Hand

The influence of The One on the Indefinite Dyad transforms an indeterminate manifold into a determinate multiplicity.  One way that it can accomplish that is by structuring the manifold.  Hence, one example of the influence of The One on the Indefinite Dyad is Smith's concept of Division of Labor.  Another is his concept of the Invisible Hand, by virtue of which the manifold of Market transactions is organized into a Just distribution of Wealth.  Accordingly, if, as critics of his system argue, there is in fact no such Invisible Hand, nor any of its equivalents, e. g. some inherent Market tendency towards equilibrium, then Capitalism is an example of the Indefinite Dyad as is.

Indefinite Dyad and Pleasure

The primary topic in the Philebus is the superiority of the Good over Pleasure.  Thus, the equivalence in it of Pleasure and the Unlimited suggests a role of the Indefinite Dyad in Ethics that has had far-reaching implications.  For, the Definite Dyad is the Indefinite Dyad made determinate by The One via quantification.  Thus, the quantification of Pleasure produces an instance of the Definite Dyad.  Hence, the Definite Dyad is a cardinal factor in Aristotle's doctrine, which entails the calculation of the Mean in the pursuit of Pleasure, and in Utilitarianism, the calculus of which is based on the quantification of Pleasure.  But, one significant difference between the two applications is that for Aristotle, as it is for Plato, control of Pleasure-seeking is the goal, whereas for Bentham and Mill, maximization of Pleasure is the goal. So, implicit in Utilitarianism, as well as in Hedonism, in general, is the superiority of the Indefinite Dyad over The One in at least one instance.

The Indefinite Dyad and The One

Some neo-Platonists refer to Plato's concept of The Unlimited as his 'Second Principle'.  But, his concept of The Limited is for them not his 'First Principle'.  For, in their systems, the First Principle is The One, and the Second is The Indefinite Dyad, with The Limited, i. e. The Definite Dyad a synthesis of the two.  In other words, for them, it is by virtue of The One that an indeterminate manifold becomes determinate, i. e. becomes quantifiable.  However, an alternative arrangement is suggested by the doctrine of Spinoza, who is interpreted by some to be influenced by Plotinus, one of the leading neo-Platonists.  For, in Spinoza's doctrine, his version of The One--Substance, or God, or Nature--is constituted by two parallel Attributes, one of which extends, the other of which connects.  In other words, in his doctrine, The Unlimiting and The Limiting are not subsequent to The One, but are, in fact, two complementary constituents of The One.  Thus, according to Spinoza, in terms of neo-Platonism, The One is nothing other than a combination of The Indefinite Dyad and The Definite Dyad, and in terms of the Philebus, it is nothing other than a combination of The Unlimited and The Limited.  And, in terms of Aristotle, it is nothing other than a combination of Material Causality, as has been defined here, and Formal Causality.  Implicit in all these cases is a rejection of the Parmenidean supremacy of the simple The One.

Indefinite Dyad and Dionysian

If, as some scholars believe, Plato's concept of the Unlimited is what Aristotle signifies by the Indefinite Dyad, and if, as some conceive it, the Indefinite Dyad is a dynamic principle, then so, too, is the Unlimited dynamic.  On that basis, the Unlimited connotes not a static indeterminate manifold, but a process of overcoming limitations.  One way of eliminating a limitation is by dissolving it, and one limitation is one's concept of one's own Self.  Hence, one instance of the Indefinite Dyad is Dissolution of Self. Now, one prominent example of the process of Dissolution of Self is that effected by Nietzsche's Dionysian principle, though he never seems to recognize it as such.  Similarly, his Apollonian principle corresponds to a dynamic version of Plato's concept of the Limited.

Indefinite Dyad and Heliocentrism

One passage in Plato's recognized writings that perhaps exemplifies his purported otherwise apocryphal interest in the One-Indefinite Dyad contrast is in the Philebus, in which the contrast is rendered as Limited vs. Unlimited.  If the passage does so exemplify the topic, it is potentially instructive for Modern Philosophy.  For, a notable instance of the Limited-Unlimited contrast is that of Geocentrism vs. Heliocentrism, in which the concept of the Universe as closed is replaced with one of it as open.  In other words, Modern Astronomy, at least, exemplifies the Indefinite Dyad.  Nevertheless, Theology persists in retaining an upper bound to Human affairs, and Modern and Contemporary Philosophy has generally followed suit, e. g. the systems of Hegel, Marx, Bergson, Alexander, and Whitehead, each has an upper limit, as does that of Spinoza, insofar as Eternity and Perfection are criteria, though not insofar as Substance is dynamically Infinite.  In each of these, the Ancient depreciation of the Indefinite Dyad is expressed, with the difference that Modern Astronomy and Human history is making that depreciation obsolete.

Material Causality and Indefinite Dyad

Because it is nearly apocryphal, e. g. Aristotle alludes to it explicitly only once, the so-called Indefinite Dyad, allegedly of Platonism, has not been systematically developed.  But, there is sufficient evidence that it closely approximates the concept of Material Cause that has been defined here.  For, it is conceived to be the source of indeterminate Multiplicity, and of the substratum of Formal Causality.  Hence, where recognized, it has been subordinated to the principle of Unity, and, accordingly, denigrated as the source of Evil insofar as it functions independently of that superior principle.  So, one reason why Material Causality has been generally ignored is not because it has little Philosophical significance, but because of an implicit valuation based on the Parmenideanism that continues to influence Philosophy.

Disjunction and Determinism

Determinism can be either Rational or Mechanical, i. e. development according to Logical laws, or according to mechanistic laws, with the latter derived from the former in some cases.  So, in the first case, Determinism is a function of what the Logical principles happen to be, and hence, distinguishable on that basis, e. g. Analytical Logic vs. Dialectical Logic.  Now, as has been previously discussed, a generally unrecognized Logical operative is the one-place Disjunction, i. e. Other-Than, which generates Alteriority, corresponding to which is Material Causality, as has been defined here.  A notable application of one-place Disjunction is to the process of Emanation, which consists in a spreading out from an original point, a principle subscribed to by Plotinus and, arguably, Spinoza.  But, as has been previously discussed, Disjunctive development is indeterminate.  So, one variety of Determinism, at minimum, incorporates an Indeterminist component.

Swerve, Indeterminacy, Disjunction

Swerve is an indeterminate concept--it can signify any of an indefinite number of motions, i. e. an indefinite number of angles of inclination from a rectilinear motion.  Likewise, Variation is an indeterminate concept--it can signify any of an indefinite number of alternatives to some given.  But, in these cases, indeterminacy is not an accidental characteristic, perhaps a liability as such.  Rather, indeterminacy is of their essence, or, indeed, they are all instances of Indeterminacy.  Now, one possible equivalent of Indeterminacy is Alteriority, the unwieldiness of which in traditional Logic is expressed by the standard reduction of Other-Than to Not, of either the Abstract or the Determinate variety.  But Other-Than is neither of these; rather it is an operation that does not exist in either Analytical or Dialectical Logic--Disjunction as a one-place Connective, or, more properly, Disconnective.  The significance of this alternative is that none of the standard inferences of either Logic apply, including, notably, that of the Thesis-Antithesis-Synthesis sequence of Dialectical Logic, regarding which Deleuze is insistent about.  Thus, Marx goes astray from the outset of his oeuvre, in his Dissertation, when he tries to reduce Swerve to Determinate Negation.

Flexibility, Free Will, Determinism

Swerve can be conceived as a variety of flex, and rigidity is equivalent to inflexibility.  So Lucretius' contrast of Clinamen and rectilinear motion can be conceived as a difference of degree, not of kind.  Likewise, Peirce's criterion of Probability implies that his distinction between Tyche and Determinacy is one of degree, not kind.  Now, if flexibility can be attributed to natural causality, it is applicable to human volition, as well.  Accordingly, the Voluntary-Involuntary distinction is, as Aristotle originally conceives it, before getting exaggerated to Eschatological proportions by Theological commitments, nuanced.  In other words, human behavior is more or less determined, or, equivalently, more or less free, or, equivalently, more or less flexible.  The persistence of the Philosophical reduction of it to mutually exclusive absolutes reflects a continued Theological influence more than empirical evidence.