Reactivating
Geometry
by Neta Katz
PART ONE:
".... 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?
PART TWO:
Lets go to Euclid's
five postulates written in about 300 B.C.
Let the following
be postulated:
- To draw a straight
line from any point to any point.
- To produce
a finite straight line continuously in a straight line.
- To describe
a circle with any center and any distance
- That all right
angles are equal to each other.
- That, if a
straight line falling on two straight lines makes the interior angles
on same line less than two right angles, the two straight lines,
if produced indefinitely, meet on that side on which are the angles
less than two right angles.
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).
PART THREE:
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:
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).
PART FOUR:
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.
PART FIVE:
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.
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