## the cone of democritus # logic wars ##
galathaea
recently
from people who freely admit
they do not know the history he refers to
he is an old warrior
in a new world
where belittlement now has debate validity
democritus
the great atomist
gave a thought experiment
he took a cone
/\
/ \
/ \
/ \
--------
and with a flash of mental swordplay
he disected it
/\
/ \
----
A
B
----
/ \
/ \
--------
and asked:
is the new face A created on the top half
equal in area to the new face B
created on the other half?
-+-+-
of course
i can imagine the reasoned response
from the average sci.math denizen interested in such discourse
it would go something like:
" Of course they are the same area, you stoopid _idiot_!
Were you born mentally handicapped, or do you just enjoy
portraying one on usenet. Learn math. Come back. "
but...
democritus continued and asked:
if they are always the same
if every cut parallel the base
separates faces of the same extent
then do we not have a cylinder?
how can the circles of a cone diminish to the apex
if every slice separates faces that are the same size?
democritus had the answer of an atomist
his solution was that matter is not infinitely divisible
and that a separation divides atoms from atoms
the faces are then unequal
and the slopes of the cone are not "smooth"
but instead incremental at the atomic level
-+-+-
note the irony that this solution presents
the currently accepted physical solution to the problem
that the math of democritus
atomism
is accepted in science
and derided in math
such is the blasphemy of the word...
-+-
but also note the important point that arises
concerning the nature of a point
the solution of continuous math
of modern geometry and the analytics of limits
still asserts these things:
- that a point may be divided or duplicated in disection
- that there exists a property to a point on a curve
its differential
that does not exist for property of isolated points
now the solution becomes rigorous
to modern mathematics
when we note that the disection structure is that of a covector
and the differential gives meaning to the limit process
of infinitessimal planar separation
but still
and this is important
points in curves may have properties
they do not have when isolated
this has always been a foundational point
for those who would distinguish
the continuous from the countable
beyond cardinality
the continuum
curves
are not just atomistic points existing in some collection
they are a process for generating points
a process which carries properties beyond the points themselves
precisely because it is
a process that cannot be finitely completed
the modern definitions of a real number
the dedekind cuts and cauchy sequences
have structure quite different from the defintion of any natural
but even still
they do not have the properties
of points in a continuum...
-+-+-+-+-
chryssipus
the stoic logician whose words have been destroyed
by time and ignorance
gave another kind of answer
he said that:
the surfaces are _neither_ equal nor unequal
but the bodies are unequal in that
the surfaces are neither equal nor unequal
this type of answer is foreign
to those who believe all meanings have relation to existence
and thus all things described either true or false
but it also seems to be expressing
an important quality of the modern solution
it is the relation the surfaces have
to the bodies they are attached
that gives meaning to the dilemma
that the equality of surfaces for a cylinder
does not have the same meaning
or consequence
as that of the cone
and that the terms
equal and unequal
both do not apply for these inferences
-+-+-
but the real answer these days
seems to be that mathematicians do not care
what does it mean to cut a volume in this way?
is one surface open and the other closed?
which one?
these are not questions for the mathematician
who cannot be bothered to question what he _means_
they are instead the questions of those imbeciles
who have not yet learned
all answers lie in
measure theory
limits
and the infallible symbolism of the religion of mathematics
( ignore the differences in the physical solutions... )
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
politician...@yahoo.com
Man!, you sooo hilarious, I can't help laughing reading your post,
BHWAWAWAWAWAWAWWAWAWAHAHAHAH!