In the first three parts of this series, I have tried to establish what I mean by "contraction" and "protraction" of ideas. I have likened the process to data compression and logical representation, but in seeking to demonstrate my meaning, I have overlooked a simpler means of conveying my point: physical action.

Consider a spyglass. Hold one end up to your eye and you can see distant objects as though they were near. This amplification of your natural sense of vision is "protraction". It stretches, amplifies, and "makes large" what before appeared small.

However, the spyglass has another use. Were you to turn it around and look through it the other way, everything which was before very close and distinct would appear small, distant, and indistinct. This is "contraction". It compresses and "makes small" what before loomed large. Contraction also makes indistinct what before was clear.

Now, consider what actions the spyglass performs. Does it physically affect the actual material that is represented to the eye? No. The objects in the distant scene are not physically affected at all. However, their representation to the eye is "bent" or "warped". From one direction, the representation is contracted and the objects become less distinct. All of the detail that existed before is still there, though our eyes lack the ability to clearly process the contracted information. When turned around and used as intended, these indistinct representations are bent in such a way that our eye can better distinguish their parts and we are thus able to process a greater amount of information.

Protraction and contraction are simply inverse relationships, best considered as light passing through a lens from one and the other direction. As a spyglass bends light in contraction and protraction, so does the mind bend ideas. I will explain how and why at a later time. For now, I ask the reader only to consider "what if" this were the case, and regard my hypothesis as plausible if not true.

This notion of "contraction" comes to me from Nicholas of Cusa, and was a principal consideration in his study of the infinite arc segment, which I have reconstituted in my study of The Identity of Unity, Nullity and Infinity and further developed in How Unity Relates to Number. I illustrated the Tusi Couple, which is a "two-cusped hypocycloid obtained by rolling a circle of radius α inside of a circle of radius 2α. The curvature of the hypocycloid is commonly regarded in mathematical circles as a "line segment". That is, straight, but better thought of as flat. Armed with the Nicholas of Cusa's notion of contraction, I will now show that this is not true, and that the Tusi Couple is not a straight line at all!

What creates a line?

Is, as many mathematicians assert, a line nothing more than a "set of points" that fulfill a certain set of conditions in line with our expectations? I hold that this Euclidean conception is flawed because it places our definitions and axioms as the basis of the Idea. If the Idea of a circle is "that thing which meets the requirements of how I define a circle" then I might be begging the question and leading myself into a prison of my own subjectivity. What if I were to project an image of a circle (with, say, an overhead projector) onto a piece of rumpled white fabric? Are the points described by this new shape on the cloth "the sum of all points equidistant from a given point on a plane"? Certainly not.

But what if we take another definition of a circle--that physical action which encloses the greatest area with the least perimeter? The "isoperimetric" action, as this is called, actually translates to the new, more complex surface, so that the wavy and complicated line which passes through the peaks and valleys of the new curved landscape is in fact a circle, though you won't be able to prove it with a compass and straight edge. However, the curvature that defines the scope of possible actions on this new curved landscape precludes the possibility of the "set of all points equidistant from a given point" definition. The assumptions that underlie the Euclidean definition of "circle" just do not translate to the curved surface. However, the Idea expressed by physical action does.

So lines aren't "sets of points" that obey axiomatic assumptions. They are representations of physical action, or, better put, perceptive contractions of physical action. This realization casts the common understanding of the "Tusi Couple", which describes through physical action the relationship between number and unity, with a "flat line", into question. Allow me to illustrate.

First, the Tusi Couple from my entry on How Unity Relates to Number:

Consider this supposed "line segment" as the result of a physical action. Is this really an action that describes a constant and uniform motion, as would befit a constant and uniform surface? I mean, if this "line segment" were "flat", like a groove in a flat plane, and you rolled a marble along it, would the marble behave in the same way that the smaller circle's point of interaction on this supposedly "flat" line does? Would the marble, as the intersection of the smaller circle and the "line segment" it traces does, speed up and slow down near the end points, if the groove were on a "flat" surface? It would not. After all, uniform manifolds of action only permit uniform motions--it's implicit in their nature.

So, what is expressed by the variation in the rate of action witnessed in the Tusi Couple above? What is it we are seeing? A hint can be found in our earlier consideration of the projection of the circle onto the curved surface.

Recall in our earlier example that I suggested the image of the circle could be projected onto the curved surface using something like an overhead projector. If the surface of projection were "flat", we would perceive a circle that comformed with the Euclidean definition of what a circle ought to be. However, if the surface were anything other than flat, this was no longer so. Imagine now that instead of a projector, we used a laser to trace a circular path on a flat wall. The circle is now, more than ever, the product of a physical action upon the "flat" surface of the wall. Let us further suppose that were we to measure the rate of motion that the laser travelled along the wall, that we would find it uniform in every part of its pathway.

Now interject the curved surface of wrinkled fabric between the laser and the "flat" wall. The pathway of the laser on the surface of the cloth must travel up and down through the hills and valleys of this new surface, but it completes its cycle in the same amount of time. In effect, because the laser must now travel over a greater amount of survace area, punctuated for brief moments by "flat" areas, its action along the surface must "speed up" and "slow down". Though the motion of the actual laser pointer device has changed not at all, the effect witnessed--that of the laser traveling across the curved surface--is one of nonuniform speed. If viewed from the perspective of the laser pointer device, there is no distinction at all in the rate of speed. However, if viewed from the perspective of an observer on the curved surface--the dot which is the laser's intersection with the surface speeds up whenever it goes up or down hills, and slows down at the peaks and valleys.

Now, reconsider the physical action described by the Tusi Couple below. I have plotted equal-time markers on the previously considered "flat" line in order to show that, in fact, there is a nonuniform rate of action.

We can now, freed from our Euclidean axiomatic assumptions, and armed with Cusa's idea of "contraction", clearly see that the so-called "line segment" described by the Tusi Couple is not flat at all, but a contracted arc segment! Just when we thought that we had found the ever-elusive flat line as an infinitesimal relationship between unity and number, it again slips from our grasp. My next step will be to create visualizations which clearly protract this "flat line", which is not flat at all, and determine its true nature, if possible.

Posted by Alexander Saint Croix on 20 December 2006 at 07:00 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:

x && y

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

!x

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:

x XOR y

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

(!(!x & !y) && !(x && y))

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,

x XOR y

is a contraction of:

(!(!x & !y) && !(x && y))

which is a protraction of:

x XOR y

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:

x OR y

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:

((x && !y) XOR (!x && y)) XOR (x && y)

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:

!( !( !( !( x && !y ) && ( x && !y ) ) && !( ( x && !y ) && !( x && !y ) ) && !( x && y ) ) && !( ( !( !( x && !y ) && ( x && !y ) ) && !( ( x && !y ) && !( x && !y ) ) && ( x && y ) ) .

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)

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)