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.