Insight and Principle of Sufficient Reason

As has been previously discussed, Insight is cognitive access to a substratum, and to how the stratum is generated out of the substratum.  In other words, it is the basis of an explanation of the Causality of the substratum.  Often, the object of Philosophical Insight is a Principle, e. g. the Water of Thales, the account of which includes an explanation of how what is apparently not Water is derived from Water.  Now, such explanations have not always stood up to scrutiny, prompting the new Insights and new Principles that have constituted the history of Philosophy.  Thus, the object of Philosophical Insight can be generalized from these efforts as a Principle of Sufficient Reason, i. e. a Principle that explains both itself and everything else.  But, with Parmenides, the Philosophical project becomes truncated.  For, he exempts his Principle, The One, from having to either recognize or explain Multiplicity.  The subsequent tradition is not as extreme, generally recognizing Multiplicity, as an inferior realm, but  offering no explanation of why the superior realm generates it.  Notable in this tradition is Schopenhauer, who resuscitates the Principle of Sufficient Reason, only to consign it to the world of Representation, without recognizing that he needs to apply it to his Principle, Will, in order to explain how Representation comes about, regardless of whether or not Representation is real, irreal, or unreal.  As a result, Philosophical Insight is reduced to functioning as a vehicle to presumed Otherworldliness, rather than as determining the nature of the given, as a prelude to modifying it.

Insight and Causality

With considerable baggage having accrued to the term 'Intuition' over the centuries, a fresher alternative is available in common parlance--'Insight'.  Insight can be the means of access to any of the substrata of Philosophy--Form, Essence, Noumenon, Reality, etc., but without the specifications that have rendered them incompatible, undermining the usefulness of the term.  But the object of Insight is more than such a substratum--also cognized is how the stratum is generated from the substratum, an account that is often lacking in Philosophical systems, e. g. how a Phenomenon develops from a Noumenon.  Hence, the proper object of Insight is Causality.  Thus, Insight qualifies as what Spinoza calls Adequate Knowledge, but it does not reduce to either Intuition or Reason.  For Causality that is the object of Insight is primarily a singular but complex concrete event, e. g. how a murder occurred in a mystery, fiction or non-fiction, i. e. neither homogeneous nor a pattern.

Intuition and Geometry

Intuition is usually distinguished from Reason as non-discursive, and from Sensation as non-Empirical.  Usually, its Object is conceived as simple, and contact with it as immediate.  But, there have been a wide variety of concepts of that Object--for Platonists, a Form; for Spinoza, God; for Bergson, Motion; and, in ordinary parlance, an ulterior motive of another.  So, Euclidean Geometry, as a Deductive system, is clearly non-Intuitive.  And Pythagorean Numerology is likely Intuitive.  But, the classification of the Cognition of Pythagorean Geometry is less clear.  On the one hand, as a Form, a geometrical figure might be Intuited.  But, once an Angle is involved, the object becomes more complex.  For, as has been previously discussed, an Angle cannot be reduced to a mere Vertex, and, instead, must be derived from a concept of Circulinearity, to which mere Rectilinearity is inadequate, and, indeed, with which it is incommensurable.  So, plainly, the Pythagorean Theorem, which entails the measures of both Lines and Angles, is not simple, and, hence, cannot be Intuited, even at a moment of nascent inception.  Thus, even if not as formalized as Euclidean Geometry, Pythagorean Geometry is too complex for Intuition, and, so, must incorporate Reason at some moment.

Triangle, Trilateral, Measurement

If Triangle and Trilateral are equivalent, it is only by virtue of Angle being conceived as an intersection of two sides.  But once measurement becomes a factor, there is no interchangeability, e. g. in the famous Pythagorean theorem, which relates the measurements of three sides, there can be no linear substitution for the concept of Right Angle that qualifies it.  Now, Trigonometry establishes a correspondence between linear measurement and angular measurement, but the correspondence is not grounded in a mathematically systematic translation.  Furthermore, there can be no such ground.  For, linear measurement is extensive, i. e. the counting of units that are progressively appended, while angular measurement is intensive, i. e. a subdivision of a whole, usually 360 degrees, an arbitrary quantity.  But that whole is a rotation.  Hence, the angle between any two sides of a Triangle is that between two radii of a Circle, abstracted from the context of the latter.  So, if an Angle is something more that a mere intersection of Lines, Triangle and Trilateral are not equivalent.

Geometry, Intuition, Angle

As has been previously discussed, the lack of any Empirical evidence of geometrical figures in the non-Human world supports the Empiricist thesis that Geometry is the product of abstraction from the cognition of that world.  One standard response to the Empiricist is that the elements of Geometry compose the noumenal substratum of that world, accessible via a type of Intuition.  However, challenging for that response is to explain how perhaps the most fundamental of the elements, the Line, is at all even intuited without some phenomenal content.  So, less challenging in that regard is for another element--the Angle--an in-between that has no phenomenal content.  Of course, the Empiricist can argue that an Angle is nothing but an abstraction from two Lines, but the Intuitionist can respond that the Empiricist reduction fails to account for how the two Lines are arranged.  Still, another shortcoming of the thesis that the Angle is a noumenal existent is that as existing where there are two Lines, it is restricted to two-dimensionality, and, hence, is inadequate as a substratum of the three-dimensions of the empirical world.

Circle, Point, Sphere

It might be argued that Circular motion, whether that of a celestial body, thought-thinking-itself, or entailed in Spinoza's definition of Circle, is still only an approximation to the idea of a Circle, so its imperfections, e. g. the directionality of the motion, are not that of the idea.  However, the definition that Spinoza rejects--a set of points equidistant from a point--has its own shortcomings, even if it does not entail directionality.  For one, the set of such points is infinite, which compromises clarity, and for another, the concept is dependent on a point that is not part of the set--the center point.  So, this definition comes no closer to an illuminating cognition of a self-sufficient perfection.  But, it does suggest a problem from a different direction.  For, plainly, this formulation is hindered by the attempt to define a Circle in terms of Points.  But, once a difference, and a possible incommensurability, in dimensionality is introduced, so, too, is, implicitly, another one--between Circle and Sphere.  Then, the challenge becomes to establish the independence of the former from the latter, which seems daunting, given that the Sphere, even if itself not in immediate evidence in Reality, is at least closer to it than a Circle is in terms of dimensionality.  Likewise, in general, for the Pythagorean-Euclidean tradition of Geometry--its elements diminish concrete Reality, a strong indication that it is abstract Knowledge, and not an object of esoteric cognition, rather the basis of useful human Technical Knowledge.

Geometry, Theory, Techne

On the one hand, Kant's treatment of Geometry in the context of Human Intuition continues Spinoza's anthropomorphizing of the topic.  But, on the other, by classifying it as an object of Cognition, not as a product of Construction, he seems to regress to the Pythagorean tradition.  However, that apparent regression is actually due to an underdevelopment of his own system.  For, entailed in his eventual subordination of Theory to Practice is a revision of the role of Intuition in Experience--from mere observation, to integration into Action.  Accordingly, Space and Time, revised, are Forms of Action.  Similarly, Geometry and Mathematics, revised, are Forms of Construction, i. e. grounds of the creation of human artifacts, just as is implied by Spinoza's operational definition of a Circle. Tending to confirm this status is the plain fact that while there are no apparent regular Geometrical figures in the non-Human world, they are in abundant evidence in the products of Human creativity--wheels, buildings, etc.  So, the status in Human history of Geometry as a Theoretical Knowledge is provisional and anticipatory--a first stage of its development as a Technical Knowledge.