As the map is not the world, so neither is the text of a work the idea which created it. And saying it is does not make it so. None could refute this. Text is no more the idea of a work than the limestone shaped as prehistoric trilobites under the Washington avenue bridge are the creatures themselves. Text is, then, the fossil of an idea, a sort of map of the world of ideas which resided the living mind of its creator.

If our purpose in study of literature is to reassemble the original living ideas of those esteemed men and women who fashioned western civilization from the rude tribalism of Europe, then we must not satisfy ourselves with the mere possession and transmission of fossils, but we must rather view them as clues, indicators, notes on a page from which living music may be reassembled.

How to do this? How can a modern observer reassemble the living idea of a dead mind? How can a student today make their mind the mind of Shakespeare or Aquinas?

We can do this by placing ourselves into a situation where we understand the forces that brought these ideas to urgent and hurried life. We can do this by understanding the context that drove the necessity of these ideas, the seeds and needs of the mind who made the thing. Whom did Shakespeare read? What were the ongoing relationships between England and Venice during the time he wrote Merchant of Venice? What contemporary elements colored his literary palate?

Certainly the Greek classics. Certainly the Latin and old English classics. Dream of the Rood, Beowulf, Spencer. But also certainly Petrarch, and who knows whom else? There are elements of these and more scattered throughout his corpus. Piecing together these contextual references would surely help reawaken the mind of the author.

So, context. But context is given such short shrift on the Internet today, it isn't provided for in the structure of HTML and HTTP, which focus solely on presentation, internal structure and transmission. Work moving forward must establish context in a more thorough manner than a simple binary "linked-or-not-linked" sense.

This is the origin of my work in building a hypercontext system for textual analysis. More on that soon.

Posted by Alexander Saint Croix on 16 May 2006 at 07:00 AM | Permalink | Comments (1) | TrackBacks (0)

In his De Docta Ignorantia, the German Cardinal Nicholas of Cusa carefully educed the relationship between unity and number as infinite in nature. The idea of an "infinite relationship" is paradoxical in that the two concepts of number and unity are irrelative. Since unity must necessarily exist prior to number, it is superlative and beyond relation to number, the nature of which is relation itself. Unity does not, in fact, relate to anything, since its nature is primary to that of relation.

For example, one could assert that the "relationship" between 2 and 1 is 2. 2 / 1 = 2. Similarly said, 1 * 2 = 2. All numbers describe relationships to other numbers and are bound by unity, but nothing new about them not contained in their identity is expressed or revealed when one compares them to unity. In that sense, one cannot in fact divide or multiply a number by 1. To say otherwise is a subtle sophistry that masks the important distinction between number and unity. For convenience, teachers and mathematicians tell their students that 1 divides 2 twice. But this is a meaningless statement, a vacuuous tautology, a short-circuit that avoids fascinating questions about the nature of multiplicative and divisive relation on an abstract level independent of measure.

For example, if you were to draw two circles of differing indeterminate measure and I were to ask you to "multiply the smaller of the circles by 3 and show me where on the perimeter of the larger circle that value lies, relative to any given starting point", you couldn't solve the problem. You cannot multiply by 3 without first defining a unit circle. Neither could you simply "multiply the larger circle by the smaller" if I asked you to do so, without first determining some unit of measurement to assign each circle.

However, the infinite relationship between unity and number can be expressed geometrically in a manner that reveals a singularity in one of the most elusive geometrical shapes in the history of mathematics--the straight line, which is an arc segment of an infinite circle.

The straight line lies at the heart of the controversy between Euclidean and non-Euclidean or anti-Euclidean geometries. How can one, for example create a cartesian manifold without them? How can one find parallels for curved lines? Euclid's parallel postulate falls short of being able to accurately describe and capture these relations in nature, since we've no basis upon which to physically construct straight lines. We take Euclid's nominalist a priori construction for granted, that a line is the shortest distance between two points, but it isn't necessarily so.

A number of nineteenth century geometricians Bolyai, Lobachevsky and especially Riemann contributed revolutionary advances to geometric science, opening the door for breakthroughs in physics and mechanics in the 20th century.¹ They did this by abandoning the irrational portions of Euclid's "Elements", and drafting physical and rational constructions for geometric relationships independent of measure. Riemann's elliptical geometry adopted Gaussian cyclotomic number theory in order to draw the concept of the infinite into a finite contraction.

In this system, a straight line is a singularity or boundary condition in species of curvature. It conforms with an arc segment of an infinite circle. But how do we produce an arc segment of a circle of infinite diameter? Were we to draw it on your screen, wouldn't we need an infinite number of pixels?

That's a silly question. Infinity isn't a number, and cannot be contracted to number. Infinity, unity and even nullity are shared aspects of the same concept, as I've already shown. You can't "divide by infinity" because infinity exists prior to the concept of division itself. Division is wholly reliant on the existence of infinity-unity-nullity for even the possibility of its own existence. As the German Cardinal Nicolaus of Cusa was careful to show, infinity is beyond the scope of the concept of relation itself, and cannot relate to anything other than itself, which would normally be to say nothing, except that infinity, the absolute maximum, is three as well as one. And so the infinite can be shown to relate to unity, in a different manner than 2 relates to 3. It can likewise be shown to relate to nullity, with which it shares identity.

And so if number cannot relate to infinity, are we lost for showing the superlative curve? No. We can exploit the union of the maximally great (infinity) and the maximally small (unity), which are at once the same and yet relate. We can geometrically capture the infinite in a finite number of pixels by inverting the problem. A fascinating aspect of the line of maximal curvature is that it is the same for each species of curvature, whether it be a hypocycloid, a circle, a parabola, a catenary, or any other species of curvature.

Consider this: a curve of the second degree x², and a curve of the third degree x³ are the same if x == infinity, just as they are the same should x == unity or x == nullity. Within the context of circular relation, then, we can capture the curve by establishing a maximum hypocycloidal curvature. I've done so here:

Here, within the boundaries of possible relation from the enclosing unit circle, we have established the infinitude of the relationship between the first number, which is 2, and the unity that gave it life. The resulting relationship is expressed in a hypocycloidal line of maximum curvature, the arc-segment of the infinite circle. This maximal curvature conveys the singular relation between its own infinite nature and the infinite nature of unity.

And so, the "difference" or "relation" between 1 and 2 is infinite. It is the difference between unity and number.

What are some philosophical implications of this geometric construction? What underlying assumptions about localized spacial curvature must be made for this evidence to hold? Can we invert those assumptions and learn something new about the nature of spatial relation itself? What truths can we educe from this simple and beautiful concept?

We shall investigate these paths on a later occasion.

See also

Posted by Alexander Saint Croix on 14 May 2006 at 07:00 AM | Permalink | Comments (1) | TrackBacks (0)

There is no better authority on the subject of Natural Law philosophy whom I have yet had the pleasure to read than John Wild. Were he still alive today, I would offer him my heartfelt thanks for his strident defense of the Platonic and Socratic school. His work on Plato is without a doubt the best analysis on the subject I've seen come out of the 20th century.

I hope he'll be remembered as such.

His 1952 book, Plato's Modern Enemies and the Theory of Natural Law, is no longer in print. It should be required reading for every American high-school senior, a prerequisite before one comes of age to vote and assume their duties as citizens of our Republic.

I look forward in the next three months to finding the owners of its copyright and republishing it if possible. I've also received word from the most admirable Catharine Tierney that there have been a number of texts written about the late John Wild in the last few years. One was published last autumn, likely when I was first reading his text on Plato on the fourth floor of Wilson Library.

Foremost in my thoughts as I turn to the last chapter of this remarkable text are the source and role of what Wild calls noësis and noëtic action. His analysis of human nature in light not only of platonic text, but also the work of moral realist philosophers dating back to the dark ages, lays the "oration on the dignity of man" by Pico della Mirandola either to waste or to rest.

My entries this summer will doubtless reflect my exposure to his work, a thorough review of my Loeb editions of Plato, and my proximity to completion of an 18-month thrust in software development. I'm closing in fast on a number of my software projects, which when combined should help me deliver much better analysis, integrated with primary source materials in a way that's never been done before. I don't want people to grope in the dark in search of these texts and their connections to each other anymore. I hope to end all that.

Posted by Alexander Saint Croix on 12 May 2006 at 02:09 AM | Permalink | Comments (1) | TrackBacks (0)

For a decade betweeen 1992 and 2002 I nursed an overly skeptical outlook on life. Disavowing, among other things, all forms of faith and all strong convictions, I tread water in the sea of agnosticism, while fish from the nihil depths nibbled at my churning heels.

Then something happened. In August of 2002 the U.S. entered into what I believed to be an illegal and unconstitutional war with Iraq. I had already begun studying political theory by that time in my life, and in my research I stumbled upon the court transcripts from the Nuremburg war tribunals. The defense counsel for Germany kept accusing the prosecuting nations of harboring "false sentiment". His assertion was that the trials were a political performance staged by hypocrites who in truth cared nothing for those who'd fallen beneath the boots of the Germans. The criticism was this: you who feign outrage over the spilling of blood, and thirst for blood in turn, are you more just than these men?

That was a powerful notion to me, who, having disavowed objective notions of morality, nevertheless felt outraged by political events I saw taking place in the world around me. I continued studying political philosophy and history, looking for a way out of my paradoxical state of mind. My studies eventually led me to a quest to read and comprehend everything ever written by Plato. I invested in a complete set of the Loeb Classical Library texts, which have been the irreplaceable iron core of my personal library ever since. I'm never far from a Loeb text.

I found in my studies that I'd taken too many things for granted, including my skepticism about morality. How can one profess outrage when one doesn't have a clear concept of right and wrong?

What is the difference between knowledge and belief? Between truth and opinion? Between the good and the pleasant? How can one feel injustice when one has no consistent notion of justice? Most importantly, how can I be so certain of the injustice I see in an aspect of the world with which nearly everyone else I meet approves?

My studies began in earnest that Autumn, just as I was about to complete my undergraduate studies. These were the October days leading up to Senator Paul Wellstone's death. In those weeks, trying to understand and oppose a war that nobody believed had already begun, I stumbled onto one of the most profound truths about American society. It regards civil rights. We didn't create them. We don't confer them on each other. We cannot by vote or majority strip a person of them. They are beyond us.

I realized that no matter what happens, no matter how the rest of the world feels about it, no matter what the polls say, right and wrong have nothing to do with opinion. They are fixed, permanent. Unyielding. They aren't supernatural (nothing is), but neither are they fashioned by human writ. They are absolute. Not because someone says so. Not because they're written in a piece of scripture or in the Constitution. They exist prior to our acknowledgement of them. That's what the Framers meant when they wrote about Natural Law.

I was forced to admit the possibility of laws, order which is beyond the purview of human law or action. I have since developed a keen interest in theological and philosophical treatises, but stepping into the domain of unalterable order also led me to geometry, music and aesthetics. The whole classical tradition stretched out before me.

I'm still distrustful of ritual and fable. I still deeply distrust tendencies towards hero-worship in academics. My respect for humanity is too great to permit unbridled appreciation for any of its members. But whatever else I believe I have made my peace with the absolute, permanent, universal laws that human hands did not write. I'm grateful for them, and for my ability to occasionally recognize them. I am comfortable in submitting myself to them, and I have satisfied myself rationally as to the necessity of their absolute goodness.

I could furnish any number of examples, many which are easily retrievable by spending a weekend with Euclid (though the reader should take care with his parallel postulate). But having a collection of principles in a book isn't to have them at all. The reader would do better to discover them independently.

As I continue my studies, I now turn my healthy skepticism toward irrational reliance on merely mechanical logic, formalism, positivism, materialism, idealism, aristotelian sense-dependency, hateful polemics, and other human errors.

Posted by Alexander Saint Croix on 2 May 2006 at 07:26 PM | Permalink | Comments (1) | TrackBacks (0)