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".... Is a science like geometry possible? How as a systematic, endlessly growing stratified structure of idealities, can it maintain its original meaningfulness through living reactivability, if its cognitive thinking is supposed to produce something new without being able to reactivate the previous levels of knowledge back to the first?"
What Husserl is asking is where does one begin, where does one start in the challenge of understanding geometry? How far back do you have to go to make geometry self evident? Husserl suggests to go to the origin of geometry, to find out how and for what purpose geometry developed, but in doing so, Husserl cautions us from the historicism that doesn't go beyond the usual facts. The reactivating of geometry must include in it a personal vision, because it "involves a lively productively advancing formation of meaning." Geometry, Husserl says, "is not a handed down ready made in the form of documented sentences." To be meaningful, geometry must have I think that this means that there is room in geometry for interpretation. Serres does what Husserl suggests, by realizing that "beneath the theorem of Thales (involving the pyramids) a corps was hurried." Serres connects two parallel lines by bringing ritual back in to the history of geometry. Husserl brings ritual back to the act of trying to understand geometry, in the present. He writes, "it is something special...to have the intention to explicate, to engage in the activity which articulates what has been read, extracting one by one, in separation from what has been vaguely, passively received as a unity, the elements of meaning, thus bringing the total validity to active performance in a new way on the basis of the individual validities. What was a passive meaning - pattern has now become one constructed through active production. This activity is a peculiar sort of self evidence; the structure arising out of it is in the mode of having been originally produced." (Later in this paper, this will become the definition for drawing, it also sound to me like a definition of abstracting (?)). So in order to reactivate geometry, one must go to the absolute origin of it but still somehow "originify" the past by a kind of invention. Geometry, I think, exists only if we are conscious of it, and for one to be wakefully conscious of geometry, geometry must touch on more levels of our lives than just logic. How does geometry mean more? Could the deaths of Hippasus of Metapontum, Parmenides, and Theaetetus (which Serres suggests were related to the crisis of irrational numbers) mean more to the development, and present existence of geometry than we are ready to admit? If so, should we document more closely the deaths of those flirting with the fifth postulate?
Lets go to Euclid's five postulates written in about 300 B.C.
Let the following be postulated:
In other words, through a point outside a given line it is possible to draw only one line that is parallel to the given line. Euclid wasn't satisfied with the fifth postulate because straight lines were infinite. Many struggled to prove the fifth postulate (the new crisis). The elder Bolyai, in a letter written in 1820, warned his son against any obsessive interest in postulate five. "You should detest it as much as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life." But young Bolyai did work on the fifth postulate and he wrote his father, "I was astounded...out of nothing I have created a strange new world." What led him to this strange new world was assuming that the fifth postulate was wrong. The way that I understand it is that Bolyai and those who followed him were reading geometry three dimensionally, and so those two lines that are on the side on which the angles are less than two right angles, don't have to meet (one can be above the other).
What does it take to visualize non-Euclidean geometry? We must try to visualize non-Euclidean geometry within the three dimensional domain (the two dimensional diagram doesn't allow for point of view (we are not talking about perspective), in the three dimensional, choosing a point of view is necessary). Reichenbach says that we need to adjust our conception of congruence because "congruence is a matter of definition." "The adjustment necessary for visualization of a curved space consists in projecting congruence differently into three dimensional space." Basically the process that I see that the Euclidean mind should go through to see non - Euclidean is as follows: say something "wrong" about a "fact" and now find a way to project three dimensionally in such a way that makes the "wrong" possible, with out having to change the original "fact", or diagram. Euclidean geometry assumes a certain definition of congruence - don't, and you're in the non-Euclidean.
Three Models:
1. Bolyai – Lobachevsky |
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2. Lines: Length | |
| 3. Torus: inbetweeness |  |
The postulate that now replaces Euclid’s fifth postulate is, that through any point not on a given straight line, an infinite number of parallels, to the given line can be drawn. That is, that these lines will never have to meet (later the mathematician Riemann showed these lines will always meet because there are no parallels).
It is possible to visualize non - Euclidean geometry, so the question that I have now is how can one draw, or can one draw non - Euclidean geometry? If Poincare write that non - Euclidean geometry is about "the movement of variable solids", and the putting together of different perspectives of the same body, seeing several points of view so that "the object and the sensory being considered ... form a single body", can this geometry be frozen, can/should it be drawn? In the true sense of the word to draw (as in water from a well, or as in Husserl's reactivation), yes, because to draw means to reactivate. So, it is possible to draw this geometry if the viewer reactivates the drawing. But, this drawing as an end in itself is completely meaningless. The drawing must be drawn.
I think that Cubism was an attempt at abandoning Euclidean geometry, but by reassembling fragments according to classical geometric principles. Still, in my opinion, the cubist painting assists us in acting non-Euclideanly.
What does it mean to act non-Euclideanly? Vertov wrote: " I am the cinema eye, I am a mechanical eye. I, a machine, I can show you the world as only I can see it ... I ascend with aeroplanes, I fall and rise together with falling bodies." This, to me is a description of non-Euclidean action - the "I" plays a crucial role. Engaging in non-Euclidean geometry, or in a drawing - activating these, requires a point of view, an "I". Point of view can't be completely objective. Euclidean geometry, as I understand it, doesn't require more than one "flat" point of view, there is only one way to read it. Lines that look like they meet, do meet. Non -Euclidean geometry, because it moves, demands of us to be "I"s, to be narcissistic. Merleau - Ponty says, "since the seer is caught up in what he sees: there is a fundamental narcissism of all vision. I am not sure that all vision is narcissistic, (or equally narcissistic) but, Drawing and reactivating has to be, with out "I", non - Euclidean geometry wouldn't exist.
? 1996 Neta Katz