Cusa's principled distinction between the "absolute maximum" and "number" comes within our reach with a pedagogical exercise involving the creation of a "circle of two". The reader is invited to create a circle of two.

With some effort, the reader might conclude that they may not create a "circle of two" without first defining unity. After having established that "one" necessarily must precede "two", the incredulous reader will often retort that quantitatively this proves nothing. The small is a component of the large. "Of course you need one in order to make two! One is LESS than two!"

At this point the trap snaps shut and the error in this manner of thinking is demonstrated, as the second challenge is leveled. Create a circle of one-half. Is one also "less" than one half? Not quantitatively. But you cannot create a circle of one half without first defining a unit of measurement. From this it is shown that the concept of unity differs from the concept of number, and must exist, metaphysically, prior to that of number.

"More" and "less" are quantitative notions, which have no bearing on the relationship between numbers and unity. What matters is priority. Not temporal priority in the causal sense, but metaphysical prioricity, which creates the possibility of a certain state of relations. For example, if the existence of a thing is made possible only by the existence of another thing, as is the case for falsehood and truth, respectively, it is said that the latter has prioricity. Unity establishes the possibility for a state of relations we call number. Unity must necessarily exist prior to the existence of number. Again, not temporily, but metaphysically.

Cusa developed this concept fully in his "De Docta Ignorantia", which Professor Jasper Hopkins here at the University of Minnesota has graciously published for all to read.

Posted by Alexander Saint Croix on 27 April 2007 at 07:31 AM | Permalink | Comments (0)

In my first treatment on contraction and protraction in philosophy, I used a graphical image in its contracted and protracted forms, and showed that contraction and protraction are two aspects of a single conceptual relationship between representations of an idea. In the same way that division and multiplication are co-related concepts, or integration and differentiation are co-related, contraction and protraction are co-related actions of the mind.

By the end of my first illustration, I had suggested that, while computers were particularly good at contracting information, such as graphical images, by means of compression algorithms, they were not capable of hypothesizing the lost data back into existence. A bitmap contracted to a GIF format can not be restored to its original data resolution.

In my second illustration of contraction and protraction in philosophy, I showed how the human mind can contract ideas into languages, themselves possessing both logical and semantic undercurrents. Having done this, I showed how our minds can again protract this data to reconstitute its original meaning. But my second illustration, being formal and mechanical, might raise questions as to the distinction between the human mind and computerized logical machines. I wish now to return to a familiar example in order to show this important distinction.

Consider the following passage:

Two hunters are out in the woods when one of them collapses. He doesn't seem to be breathing and his eyes are glazed. The other guy takes out his phone and calls the emergency services.

He gasps: "My friend is dead! What can I do?" The operator says: "Calm down, I can help. First, let's make sure he's dead." There is a silence, then a gunshot is heard. Back on the phone, the guy says: "OK, now what?"

How did you react when reading this? Did you laugh? Maybe just a little bit, or barring that perhaps you felt a bit of shock. The passage above is a joke, and many people consider it funny. But why? The piece is ironic--it contains elements of classical irony. Let me protract this explanation.

Irony hinges on ambiguity, misinterpretation, and difference. The joke above begins when the operator says "First, let's make sure he's dead" and the hunter interprets that as "shoot him so that he's dead for certain". But there are many layers to this joke. In addition to the pure misinterpretation of the operator's intent, the blind obedience of the average man (the hunter), to the expert or official (the operator) informs us about the danger of suspending common sense at the behest of authority. So there is at least one subtle and embedded moral in the joke as well. The misinterpretation makes the moral poignant. In the difference between what was said and what was heard, a man dies.

This is why, even if you find yourself laughing, your laughter is tinged at some level with horror at the man's blind obedience. And so, even in the irony there is tension between the comic and tragic. This is particularly fitting, because both classical comedy and classical tragedy are constructed from ironic elements. The difference between events in the joke and how you might hypothetically have interpreted the operator's instructions unveils the idea of the joke to you.

So, if irony is bound in misinterpretation and hypothesis, this explains something extremely important about computerized logical systems. Computers have no sense of humor. Why not? Because logical systems cannot produce hypotheses! Logical systems are hard-bound and literal contractions of ideas, but what connects the contraction with its protraction? A hypothesis.

Think back to my first illustration, the fractal contraction. When you consider the following image, can you imagine the internal spirals, each branching into additional internal spirals, each continuing in this manner ad infinitum?

Can you imagine placing this under a microscope to get a better look at it, and finding its internal motions to be as intricate as they appear here?

But these are different images entirely! The first is a different file on the computer filesystem. If we were to actually zoom in on the first image, we'd find this pixelated garbage:

In other words, our hypothesis about the first image is more expressive of its internal parts than the image itself. Why? The image is contracted. Making the picture larger doesn't change this. The same is true of logic. Logic is contracted reason. What is lost in the contraction? Hypothesis.

People understand jokes but computerized conversational agents do not, because logical systems are less expressive contractions of the human mind, which employs reason and hypotheses to form judgements. Logic is nothing more than Reason's footprint.

In consideration of this, might it not be possible that ideas themselves are mere contractions of objective truth? As I continue to explore the concept of contraction, I believe that this hypothesis will be lent additional weight and salience.

Enquiries on Contraction and Protraction in Philosophy

• Part 01
• Part 02
• Part 03

Posted by Alexander Saint Croix on 9 December 2006 at 12:44 PM | Permalink | Comments (1)

Revisiting my earlier explanation about the relationship between contraction and protraction in philosophy, I thought to give a much more rigorous and less interpretive example. In logic, the phrase "x and y" is a conjunction. It is sometimes symbolized as:

Similarly, the phrase "not x" is a negation. It is sometimes expressed this way:

In like fashion, the phrase "x or y, but not both" is referred to as an exclusive disjunction. It is also often symbolized as "XOR" in logical languages. So, if I were to write:

I would, in more basic terms (using only negation and conjunction) mean:

This, in essence, reads "it is not the case that both not x and not y can be true, and it is not the case that both x and y can be true". This is quite a mouthful, however logically correct, so we often just say "x or y, but not both".

The first statement is a protraction of the second. The second statement is a contraction of the first. Stated another way,

is a contraction of:

which is a protraction of:

So far, so good, right? You can see why we contract ideas--they're tough to communicate otherwise. But our minds can unpack a contracted idea and reconstitute its original meaning and implications. Our minds can also contract a protracted idea so that it can be more easily communicated.

But, does the contraction really save us that much effort? Maybe not with the simple example above, with the XOR statement. But there's a problem with the word "or". Exclusive OR is only one possible interpretation of the word "or". There's a similar concept "x or y, or both" that's called an "inclusive disjunction" or "inclusive or", which in logical languages is sometimes written like so:

Now, this seems pretty straightforward, right? But how do we define this in terms of negations and conjunctions? Well, we really want to be specific, so we'd have to write:

This reads "either x and not y, or y and not x, or both". It's pretty simple to understand now that we have a concept of exclusive disjunction established. But, recalling that XOR is itself a contraction of other conjunctions and negations, we'd have to unpack all of the XOR statements as well. If we were to do this, we would end up with the following:

And that is a protracted version of the expression "x OR y". In review, our mind can contract ideas to more communicable forms, such as language or symbols. Conversely, it can protract these contracted forms back to their original ideas.

Now, here's a challenge. If you have a language "x" with a finite number of words and grammatical constructions, how do you logically express an idea for which there has never been a word or expression? Mull over it.

Enquiries on Contraction and Protraction in Philosophy

• Part 01
• Part 02
• Part 03

Posted by Alexander Saint Croix on 8 December 2006 at 09:29 PM | Permalink | Comments (0)

Descartes fails in his “MEDITATIONS ON FIRST PHILOSOPHY” to establish a method whereby his opinions, as well as those of his readers, can possibly be demolished. The method he employs in his attempt to establish doubt in our senses does not achieve his stated goal, specifically to “go straight for the basic principles on which all [his] former beliefs rested” (p.76). Descartes' method for establishing the possibility for doubt, far from accomplishing his end, only succeeds in entangling itself and illustrating its inability to even possibly accomplish that goal.

In his “FIRST MEDITATION”, Descartes' first of three stages of doubt relates specifically to the trust of his senses. He reasons that "from time to time . . . the senses deceive, and it is prudent never to trust completely those who have deceived us even once." In other words, if we are lied to by a person and fooled by our trust in their lies, then we should refrain from completely trusting this individual in the future, in case their words are untruthful.

An example of this is when we trust the time shown by a broken or unwound clock. It might be early morning, but the clock might indicate mid-afternoon. Once we know that the clock is deficient, we should never completely trust its reports on the time. From this Descartes resolves never to trust his sensory perceptions, and sets about the task of building a framework for doubt in all of his senses.

This first stage of skepticism or doubt anticipates the two which follow in that it establishes the possibility for suspicion in and consequent disregard for all information coming from our senses. It differs from the latter two in that it eliminates sense-certainty only with regards to size and distance, and basic comparative judgments made about objects which we perceive. It does not provide for the possibility, for example, of ruling out the existence of hands and feet, the existence of our immediate surroundings, the existence and identity of basic colors, the existence or function of simple and composite mathematical principles, or any other information which is subject to the careful scrutiny of our senses. Possible means by which we might cast those other matters under suspicion are provided in the two subsequent stages of doubt, namely "Dream Skepticism" and "Malicious Demon Skepticism".

One may easily respond to the first of Descartes' three stages of doubt by distinguishing between necessary and contingent truths. Descartes' assertion that "it is prudent never to trust completely those who have deceived us even once" (p.76) does not necessitate that it is prudent always to completely distrust those who have deceived us even once. His method of complete distrust in all sense perceptions therefore must not necessarily follow from this assertion. From time to time, we know with absolute certainty the maxim that "even a stopped clock tells the right time twice a day." If we interject the a priori axiom never to trust the stopped clock, then the exercise of distrust itself becomes the source of deception "from time to time", and if we are to remain consistent with Descartes' method, can itself never be completely trusted and should be done away with.

This necessarily unwinds the utility of his entire method and not only the assertions, but also the course and conclusions of his argument. The subsequent two stages of doubt fall prey to the very same paradox. The dreamer may experience a hybrid of dream-senses and waking-senses, and as has been shown, some of the latter may possibly be true, such as pain from recent dental surgery that is constantly with the dreamer in their sleeping state. No doubt in dreamt-senses may therefore ever be described as necessary. Doubt here again is only possibly justified. Since, once again, doubt itself sometimes can deceive us, we must never completely trust it if we are to remain consistent with Descartes' earlier reasonings. Even were our senses nothing more than illusions owing their existence to the machinations of a malicious demon, we cannot by this method know that all of our senses must necessarily be illusory. We once again know that they may only possibly be illusory. Since doubt therefore exists about the scope of the demon's phantasm, consistency with Descartes' earlier reasoning demands that our doubt in our senses must never be completely trusted.

We therefore arrive at the end of the first meditation no more or less certain about any matter, nor any more or less ignorant about any matter. Descartes has not proven the necessity of doubt, only provided for its possibility. And if his argument in favor of the use of doubt is to be trusted, it illustrates only that his argument in favor of the use of doubt cannot be trusted. The method of doubt, therefore, cannot possibly demolish our earlier opinions or free us from them.

Descartes might respond to this paradox by determining that it were somewhat better for him to begin not with doubt, nor dream, nor the invocation of malicious demons to assist his reasoning, but to establish possible truths as he has done here, and then to examine with discipline their implications, in each step determining whether or not his assertions can possibly contradict themselves, and if so, what factors might contribute to his blindness. I would agree with his amended method, for it is true both that ultimately false statements may contain elements of truth, and that ultimately true statements may not contain elements of falsehood.

Descartes' method fails to distinguish between necessary and contingent truths, and in doing so confuses the possibility for doubt in one's senses with the necessity of doing so. In recognizing this, the reader may easily free themselves from the shadow of Cartesian Skepticism and regard the state of confusion and doubt in which he concludes the end of his first meditation as unjustifiable. His assertion “never to trust completely” (p.76) in no way necessitates “always to distrust completely”, which is the basis of his method. If this argument for complete distrust is applied consistently, Descartes' method is either false from the onset, or self-contradictory and paradoxical by its implications. The method, then, is by itself necessarily not capable of establishing even the possibility for doubt. Consequently, Descartes fails in his “MEDITATIONS ON FIRST PHILOSOPHY” to establish a method whereby his opinions, as well as those of his readers, can possibly be demolished.

Posted by Alexander Saint Croix on 20 November 2006 at 07:00 AM | Permalink | Comments (0)

My philosophical investigations have revealed to me the concept of contraction, or contracted representation of information. These can be understood in comparision with abstraction and protraction which are themselves distinct. Today I mean to clarify what I mean by "contraction" and "protraction", and the distinction between them.

I will do this by examining various representations of an action vector. In the image below, an action pathway is represented by interlocking grey and white fields. The perimeter of the interior grey field is the fractal, or "recursively self-similar" action. I mean action in the sense of that transitory state of being which would arise from moving an infinitely precise laser along the boundary between the two colored fields.

Observe the action pathway in its entirety:

The image above shows an abstract view of the fractal action in its entirety. This represents the entire field of possible actions. Sensing the intricacies of detail in the action pathway, we select a small internal portion of the whole:

We next isolate this portion of the whole and show it by itself below:

This image is a selection, a portion of the unity above. If we examine this very same image file more carefully, by expanding its width and height by a factor of 13, we notice a distinct lack of resolution:

This close examination reveals to us that the small image we have considered previously is not at all clear, it is actually a rather careful optical illusion of clarity which purports to represent the action described above, but in truth is no more than a grid of colored boxes which reveal no more detail when subjected to closer examination. This is the nature of our present medium, which is the Graphics Image Format (GIF) file type. The GIF images do not store in memory any further information about this particular mathematical action, except that necessary to present the above likeness. However, in the original program, a great deal of memory is expended in storing the data represented in each particular vantage point.

The program used to create the original abstract image, however, is not limited to this contraction or compression of data. Using the very same selection in the fractal viewer, we are able to protract, or magnify the selection:

This is a protracted, or amplified portrayal of the same selection. It is not derived in any way from the pixelated selection above, but is a protracted representation of the same part of the original fractal action. The selection above had been contracted to a GIF image format, which explicitly and necessarily results in loss of data. This data loss is clearly shown when we expand the width and height as above.

The contraction of data in this manner is necessary, given the nature of its own manifold of representation, which is an electronic portrayal of the original fractal vector. Because of the finite resolution of computer displays, all visual representations are contractions necessarily--computer displays can represent at maximum 72 dpi, or "dots per inch" of data. The GIF algorithm is useful precisely because, through contraction, it dispenses with all data except that needed to portay a recognizeable likeness of an original, more protracted signification. This reduces the size of the "memory footprint" of each image, allowing faster data delivery over the Internet. Given the infinite complexity of the fractal vector, the size of the image file could scale infinitely, far outstripping the physical limitations of the computer hardware.

The fractal browser is not limited to the size constraints imposed by the Internet, and can allocate a large sum of memory to recording data about the fractal action pathway. This means that the representation of the fractal signifies a much greater resolution of information. The information present in this high-fidelity portrayal is "contracted" into the GIF format. Conversely, the information present in each small portion of the GIF format is "protracted" or magnified in the fractal viewer.

The process detailed above continues further "down" into the fractal. We can isolate another selection from our last vantage point:

Which we show here:

This GIF image, upon closer examination, is itself a contraction of the original action vector:

The pixelation shown in this close examination of the GIF is a contraction of the following, protracted signification:

We can see clearly here the relationship between contraction and protraction. It is important to note that computers can not protract an image, thereby increasing the resolution of that image and the amount of clarity or data which is stored in the image. This is shown frequently in popular films, but has no basis in reality. Compression is final for computers. Once they discard data, it is lost.

The human mind, however, does not suffer from this limitation. It freely contracts and protracts on a regular basis. Language is no less a contraction of meaning than the above images were contractions of an action pathway. However, the mind can "reconstitute" language, protracting it in order to derive the original meaning of its author, with varying degrees of success.

Another way to think about contraction and protraction is as a projection screen. If you were to project the image of a circle upon a flat, curved, and textured surface, you would get three different visual representations or contractions of the original, protracted form. Similarly, if you print a circle with varying qualities of equipment and material, ranging from dot matrix on green paper to color laserjet on glossy card stock, you get varying contractions of the original action. An action may be contracted multiple times, each time reducing the clarity or somehow distorting the original until it is no longer recognizeable by the mind.

In my own research, I have concluded that logic is a discrete and mechanized contraction of reason. Logical processes can not generate hypotheses as rational processes can. They are not able to resolve their own paradoxes or even prove all of their own axioms. They are severely limited in this regard. This is not to say that they do not have their uses. They may easily be automated, for example.

I suspect also that the mind is a manifold of action which admits certain contractions of truth, and that the nature of the mind admits of alterations which increase its resolution or expressive capacity, its ability to accurately reflect truth. This might be tied to our apparent ability to protract meaning from a contracted manifold.

I hope that my illustrations above help convey the meaning of my use of "contraction" and "protraction' as I continue my investigations. I will likely return to this entry several times in the future as an example of my meaning.

Please feel free to republish this document in its entirety, so long as proper attribution is given to its author. All fractal images were created with Tierazon and Photoshop CS2.

Enquiries on Contraction and Protraction in Philosophy

• Part 01
• Part 02
• Part 03

Posted by Alexander Saint Croix on 19 November 2006 at 12:24 PM | Permalink | Comments (0)

Abdulkader Tayob wrote an analytical piece today concerning recent controversies surrounding Pope Benedict XVI’s recent lecture at the University of Regensburg. Mr. Tayob's analysis included the following observation:

"The central argument of the lecture was devoted to the urgency of reinserting the positive role of religion in modern culture. But the pope proposed to do this by turning attention to a medieval synthesis of reason and religion that has been lost in modern society through what he called the de-Hellenisation of Christianity. The Hellenic roots were lost through the Reformation, liberal theologies and, last but not least, de-Europeanisation. Boiled down, Benedict’s argument is that most of the modern legacy of Christianity has to be jettisoned."

This "medieval synthesis of reason and religion" is not, in fact, medieval at all. The synthesis of reason and faith is one of the keystone elements of Christianity, dating not only to Augustine, but to the scriptures themselves, as is evidenced in John 1:1 (consider the greek translation for "word").

Siger of Brabant and Boetius of Dacia promoted a philosophical movement beginning in the thirteenth century generally referred to as "Latin Averroism". The Averroists promoted, among other things, belief in the existence of twin truths--one reached through reason and the other through faith. The fonts of these two truths--God and the Laws of Nature--were coeternal. This view, which opposed scholastic principles, was carefully refuted by Thomas Aquinas, and the teachings of Siger were condemned by the church in 1270 A.D. Further dismantling of the Averroistic legacy was completed by the scholars of the Florentine academy, most notably Nicholas of Cusa in his "De Docta Ignorantia", which further cultivates the teachings of the Eleatic philosopher Parmenides, which come to us through Plato. Cusa's articulation of the maximum-minimum principle provided a solid foundation for the works of Leibniz, and especially the works of many 19th century mathematicians, most notably Karl Gauss and his intellectual lineage. Johannes Kepler treats the unity of his faith and reason explicitly in the opening chapters of his "Astronomia Nova".

The Averroistic impulse, on the other hand, can be seen in the works of Locke, Newton, Berkeley and their empiricist, materialist, and positivist derivations. Newton's assertion that space was the "sensorium" of Deity, an organ through which Deity perceived creation, is one famous example of an Averroistic assertion, because it attempts implicitly to establish a boundary to the capabilities of Deity. For this reason it was roundly condemned by Leibniz (see the Leibniz-Clarke Correspondences for more information).

In essence, the thesis that the enlightenment was brought about by virtue of an innovative distinction between reason and faith or religion is patently false. It was rather by virtue of the consistent refuations of the averroistic impulse to isolate reason from faith, or (more commonly) to conflate logic with reason, made by the Florentine academics, and later by Leibniz and his intellectual descendants, among countless others that the enlightenment and renaissance flourished in Europe and were transplanted to the New World.

Pope Benedict XVI’s determination to dispel this illusionary distinction from contemporary western culture is, therefore, perfectly concordant with over two millenia of theological and philosophical tradition. Guarding against the impulse to "de-Hellenization of Christianity" is perhaps the greatest challenge that western culture has ever faced. That reason (logos) and faith are ultimately one, joined by right action in a triad which reflects the unity of the Christian trinity is an idea as ancient as it is scarce in today's philosophical discourse.

Courses of action which implicitly destroy the ability and opportunity for reason (read: violence), are therefore, in his thesis, contrary to and incompatible with the nature of faith itself.

Posted by Alexander Saint Croix on 26 September 2006 at 07:00 AM | Permalink | Comments (0)

During a discussion in a Philosophy course this week on campus, several students saw fit to rail against the writings of Gottfried Leibniz, whom it is doubtful they have carefully read. The passage thrown carelessly before them and out of context, to defend itself alone, was that we reside in what Leibniz called "the best of all possible worlds". Immediately, the question came under fire by the students. How, one asked, does his assertion mesh with a situation like Hurricane Katrina? How could this be the "best of all possible worlds"? The consensus at the end of lecture was that "this world sucks on a regular basis, and God must be to blame, and either this ISN'T the best of all possible worlds, or else God can't create a perfect world."

Before I begin, I'd like to assert that criticism of analysis is no substitute for reading and critiquing primary source texts. If my analysis is flawed, I welcome constructive criticism of it. If you've a desire to criticize the ideas, take up the argument with the primary source text, not with me. For your time, you'll have a much larger impact by sticking to the principal texts of Leibniz and Cusanus.

There are several things which must be taken into account before one can truly appreciate Leibniz' argument. First, the argument is both an ontological and theological one. Leibniz believed in God. It's no secret. He didn't merely believe in God, but understood the existence of God to be a matter of metaphysical and ontological necessity, and his writings were almost certainly informed by the German Cardinal Nicholas Cusanus' "De Docta Ignorantia", which has been translated and published online by the University of Minnesota's Doctor Jasper Hopkins. Early in his career, Leibniz was a Cartesian (by his admission in his New Mechanics). However, he later would recant his Cartesian beliefs and publish a series of refutations on the metaphysical paradoxes inherent in Descartes' "Meditations". Among other things, Leibniz sought to dispel Descartes' Scholastic prejudice toward soul-body dualism.

Second, Leibniz' understanding of Deity differed from his British Empiricist counterparts in that he attributed three infinite qualities to Deity. In addition to the quality of "omnipotence" which the Empiricists agreed upon (and even wholly relied upon for the metaphysical underpinnings of their mechanics, as is shown in the Leibniz-Clarke Correspondences), Leibniz asserted the qualities of All-Wisdom and All-Goodness. This follows directly from the writings of the Florentine Cardinal Cusanus.

Leibniz' argument for all-wisdom proceeded something like this: if the laws which govern reality at any point contradict themselves, the entire system would be a paradox, and would not function. This creation were unbefitting a perfect Creator. Therefore, the laws which govern reality do not contradict themselves by implication or relation to each other. They are all seamlessly interjoined (example: sine and cosine). The creation of such a system require not only All-Power, but All-Wisdom as well.

For All-Goodness, Leibniz makes use of the metaphysical and ontological argument raised by Cardinal Cusanus, that Goodness must necessarily exist prior to the existence of Evil. An analogue of this argument can be shown as follows:

1. There are three types of logical statements:
2. Ones which include only true conditional statements.
3. Ones which include a mixture of true and false conditional statements.
4. Ones which include only false conditional statements.

If a logical sentence-part only contains true conditional statements, it resolves to true. For example, "IF a AND b AND NOT c" contains three sentence parts. "a", "b", and "NOT c" If all three of these resolve to TRUE, the statement as a whole resolves to true. If all or any of the conditional statements resolve to FALSE, the statement as a whole resolves to false.

Now, imagine that you have a logical statement with hundreds of millions of conditions. If even a single one of them resolves to FALSE, the entire statement is false. It only takes one. What does this mean?

It means that ultimately TRUE statements may only be composed of other true statements. On the other hand, ultimately FALSE statements may contain bits of truth and falsehood. The proportion of truth and falsehood, however, matters not at all. If even a single part of a statement, no matter how large, is false, the statement as a whole is ultimately false.

Truth, then, neither relies upon, nor admits of falsehood. In fact, truth demonstrably knows nothing of falsehood, for no true statement contains false parts. It is possible then, that truth existed prior to falsehood. Falsehood, on the other hand, is well acquainted with truth, and often admits of true parts (which themselves do not admit of false parts). It is not possible that falsehood could have existed prior to truth, since falsehood admits portions of truth in itself.

Since no other possibilities can possibly exist (there are no other combinations of truth an falsehood than those three above), we conclude that the only possibly true option is, in fact, a necessary truth. So many false statements can be shown to contain elements of truth, that we must acknowledge the metaphysical prioricity of truth to falsehood. This isn't a causal or temporal relationship, but one of metaphysical prioricity. Truth does not cause falsehood any more than unity causes number. And time, well time isn't a prerequisite to either truth OR falsehood, and we needn't trouble with it here. A similar argument is that one cannot admit of organic molecules existing prior to the existence of the carbon atom. One may not admit of atoms existing prior to the existence of electrons. Electrons are not made of atoms. Carbon atoms are not composed of organic molecules.

This argument is analogous to the one made for Good and Evil. Leibniz treats the question at great length in his "Theodicy", and Cusanus treats the question as well in his "De Docta Ignorantia". Their two accounts of this principle are harmonic. I will leave its exploration to the industry of the reader.

The end result is that Good exists, necessarily, prior to the existence of evil. And All-Goodness is a quality which is only possessed by God, for no other possibilities can claim axiomatic, ontological, metaphysical or any other form of prioricity. That which forms the basis for existence is All-Good then, in addition to being All-Wise and All-Powerful. This forms a trinity of principles which necessarily imply each other.

In Leibniz' view, then, a view he consistently promoted against the Empiricists, the laws which govern the universe do not contradict each other, since for them to do so would contradict the principles of All-Wisdom and All-Goodness, and relegate God to the image of the doddering old man on the mountain, Zeus, rather than the Triune described by Cusanus. Not only were an imperfect creation unbefitting an all-powerful Deity, but it would require that Deity constantly perform "miracles" to resolve the innate problems of the system. Think of Microsoft rolling out weekly patches to its operating system, and then raise the complexity to an infinite degree, and you'll have an idea of the tremendous amount of labor required. The empiricist Clarke promoted this view as proof of Deity's Omnipotence. Leibniz, however, promoted his alternative as proof of All-Goodness and All-Wisdom.

That alternative was that we live in the best of all possible worlds. He attributes the existence of evil not to Deity, but to "sin", or the necessarily innate imperfection of all finite beings. Deity is neither finite, nor a being, but being itself, or what Cusanus called both the "Absolute Maximum" and "Absolute Minimum", something beyond comparative relation. Leibniz argues that sin and evil are neither caused by God nor necessary in Creation, but merely permitted. Why are they permitted?

The answer is Free Will. It better befits the Glory of Deity that we human beings, who are permitted to choose good or evil for ourselves, can choose good. In fact, it were not possible for us to love in the absence of Free Will. We would be driven, as contemporary positivists and materialists, behavioral scientists and other unsound minds still hold we are, by mechanical necessity. Since the possibility exists that even a single person amongst us would choose to love what is good, and since that which is perfectly good is Deity, for Deity, Unity, Absolute Maximum, Absolute Minimum alone admits of no imperfection, the alternative--a mechanically certain universe, Leibniz argues, is not possible.

And if a world were possible which were better than this, the All-Wisdom and All-Goodness of Deity would again be in question. Such a world could certainly not be driven by mechanical necessity, for it were better that Free Will govern human action.

In other words, the evils we see in this world are attributable to ourselves alone. We alone choose our fates. We alone bring evil to ourselves. Deity offers the option of goodness, but we alone are free to choose it.

So, to return to the question of Hurricane Katrina. It is a manifestly evil thing that a population of human beings living in flood plain beneath sea level should direct such a large percentage of its daily attention and energy to the pursuit of entertainment and pleasure, when the scientific and industrial demands of the geographic location in which they reside are so great. In doing so, they fail to learn from millenia of history. Entire societies are regularly washed out to sea as a result of their own poor choices. It were better that they wash out to sea. It were BETTER STILL that they choose to know nature, to use their faculties and employ their labors toward the protection of their neighbors, their children, and their posterity. That sickness follows a poor diet is instructive and therefore good, and none may show that it, however unpleasant, is evil.

If Deity alone could have stopped the tragedy in New Orleans from taking place, then why are so many people upset with President George W. Bush? It is manifestly true that human foresight and right action could have prevented the flood from taking place. The Army Core of Engineers are certainly capable of building a system of Levees which are capable of withstanding hurricanes and floods of mythic proportions. The fault lies not in the laws of nature, nor in the power, wisdom or goodness of Deity, but in the Will of human beings.

We are more than capable of defending ourselves against the wind and the wave. That we failed to do so is our fault. The responsibility to protect ourselves and the neighbors upon whom we rely belongs to every person.

That pain and misery exist, that wars are fought, that sickness claims the lives of our loved ones, that we remain here on Earth when a universe awaits our footprints, all these things are the necessary results of our choices. And all of the evil that human beings can concoct and implement are overshadowed by the possibility that even a single one of us might love our Creator.

Posted by Alexander Saint Croix on 20 September 2006 at 07:00 AM | Permalink | Comments (0)

What is the Empirical basis for the use of Hypothesis in modern science and positivistic philosophy? Is there one? What technique have we to determine the origin of hypothesis which does not make use of hypothesis and therefore rest upon it? What logical mechanism do we possess to create or fabricate the act of hypothesizing within a meaningful context? Does a logical system exist which can create meaningful hypotheses? That is, hypotheses which relate in a meaningful way to their subject matters?

Can a logical program ever answer the question why it itself works? Why its own methods can possibly be or must necessarily be true? No. Gödel proved this conclusively. Every logical system ultimately rests upon one final axiom, the principal axiom of all logic, which is the relationship between "if" and "then". Ultimately, all logic boils down to a relationship, but relationships cannot have an Empirical basis. They are infinite in nature. What is the relationship between an atom and the atom next to it? What field equations interject, what fluid forces compel them to dance when close and draws them together when they are separated? Through what medium do they relate, and if it indeed exists, is it truly a medium or merely another substance whose relationship must in turn be analyzed?

There is no true vacuum, no such thing as nothing. And so, relationships have themselves no physical basis but only a basis in concept or reason. Number is one expression of such a relationship. Differential another. Hypothesis a third, consequence, origin, all of these combine into a patterned whirlwind of relation. What empirical basis is there for any of it?

None. No logic or mechanism may capture it, as it exists prior to both. The faculty of apperception, which permits an infinite circle to share an identity with an infinite line or infinite triangle, the faculty of metaphor, which permits one thing to share in unity with another in spite of their differences, these are the domain of reason. Hypothesis binds them together, points out the possibility of their union as a determined imagination suggests the possible flow of a darkened path so that sense perception can flail thanklessly forward, dragging the body's feet through the sand of discovery, in fear of roots and deviation, and assuring itself of its dominance and safety when by luck no such roots happen to challenge its forward progress.

No, the imagination is a vital part of the process of discovery in that it suspends disbelief long enough to interject possibilities which might lie outside the scope of the senses. It can reveal paths of action which lie outside the domain of our current species of action. And even so-called "positivistic" or "empirical" science is caught in its thrall, for as logic binds these sciences together, mere hypotheses and axioms bind logic. Without hypothesis, science built on logic would stand as tall as towers built with bricks and stone but without mortar.

Posted by Alexander Saint Croix on 8 September 2006 at 03:17 PM | Permalink | Comments (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)

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)

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)

In the context of Gaussian number theory, it can be demonstrated that zero necessarily does not represent "nothing", unity or "one" is necessarily not a "number", and infinity necessarily shares its identity with them both.

These truths provoke important questions in many areas of mathematical research, from nonstandard analysis to revolutions in our conception of number itself. The deeper implications on all fields of human knowledge are likewise immense.

Continue reading "The Identity of Unity, Nullity and Infinity" »

Posted by Alexander Saint Croix on 23 April 2006 at 04:20 PM | Permalink | Comments (0)

An acquaintance of mine whose love for mathematics I hold in high regard suggested to me today that modal logic was not necessary for debate, and did not factor into debate. He went so far as to suggest that it wasn't even used in logical debate.

This acquaintance knows a man who is building a computer program wherein which two parties can enter the terms of their argument and the program will hold them logically to said arguments. I, interested and not without experience in these sorts of systems, inquired whether this program took modal and deontic systems of logic into consideration, or rather if it were robust enough to support the use of these types of logic. It was at this point that he denied the utility of modal logic in debates.

Through my work on the development of rule engine software, expert systems and fuzzy logic nets I became intimately familiar with the need for alternative and less-known schools of logic such as modal and deontic logic. Especially in expert systems, or in systems which purport to assist in the analysis of philosophical or theoretical research, which is the specific area in which I presently labor.

A modal logic expression might look like the following:

"If A is true, then it is possible that B is true."

If one were assigned to do all it could to prevent B from becoming true, one would need to determine whether A necessarily leads to B, or whether other presently unseen conditions contribute to the rise of B in the system/argument. One brings modal logic to bear on such cases.

In computer science, this becomes very useful in avoiding unecessary processing. If the negation of A necessarily makes B impossible, then one needn't examine the other preconditions leading to B to know that B is impossible. One may state definitively that B is not true without examining all of its preconditions.

In debate, modal logic is even more useful, so useful that it is nearly ever-present. Recognizing and avoiding the acknowledgement or concession of statements which might possibly contribute to undermining one's own argument by implication is fundamental to debate. Knowing that an assertion necessarily leads to a given conclusion is the subject of modal logic, and is why I pay it such high regard. No system for examining the implications and relationships between systems of theory or assertions of fact can afford to ignore this extremely human mode of thought.

I'm not alone in this belief, either.

Posted by Alexander Saint Croix on 19 January 2006 at 06:48 PM | Permalink | Comments (0)

One of the several ways in which physical (textual) artifacts of an idea are distinct from those of a perception is that noncontracted ideas relate to one another in a continuous manner, whereas noncontracted perceptions do not relate to one another. Idea is in this sense a manifold of continuous action, whereas perception is a discrete manifold of existing or not-existing. One perception, for instance, does not permit or lead to another perception. If one needs proof, one can just flip on the television and channel surf. The series of images you perceive while flipping from channel to channel do not cause each other to exist.

Idea, however, is the mental process of apperception or hypothesis, by which we come to understand the universal meaning that corresponds to those objects of our sense perceptions. While a perception merely is, idea becomes, in a continuous process. The discrete "snapshots" of this continuous process are manifold, but exist in definite relation to each other in a progression of evolving mental action. The continuity of idea necessitates, then, the existence of relationships between instances of itself when it is contracted to a discrete manifold. This distinction is further elucidated in the Parmenides dialogue.

Whether the relationships between contractions of a given idea are causal, modal, or temporal, is the subject of my present investigation. Does the process of idea flow like a pebble flows along currents of gravity toward the earth? Can multiple ideas come together in a higher idea to form a continuous action of a higher order, such as planetary orbit? The answer to both of these questions appears to be yes. Is this in fact a "coming together", or are the two ideas products, projections or contractions of the higher idea? In which direction does order flow? The cognitive process of discovery appears to be able to travel in either direction.

As such, any electromechanical system which aims to contract the continuous relationships that exist between discrete states of an apperceptive process should permit the existence of complex or compound ideas, which are characterized as ideas of a 'higher order', or those which exist on a 'higher' manifold of action. Vis, "governing dynamics".

Of course, any system capable of storing these contracted relations must itself be a contraction of a yet higher idea. The apperception of this contraction, indicates a continuum of higher ideas. Our apperception of this continuum itself indicates a yet higher continuum of continuums. The process repeats itself, becomes fractal, and resolves itself into an analogue of Cantor's Paradise.

Posted by Alexander Saint Croix on 1 December 2005 at 11:15 PM | Permalink | Comments (0)