[FOM] Question on the number line
H Z
hzenilc at gmail.com
Wed Nov 16 14:35:41 EST 2005
> But Suarez says there is still a problem, however, because there seems
> no reason why the boundary should lie on the one half rather than the
> other.
If I am not wrong was Aristotle who found the first paradox of this
kind? It was about the way in which a line is constructed by points,
if a point has no extension and a line is a set of ordered points how
are they connected? The point cannot be connected with another by its
frontier because it has no parts and then there is no frontier or
limit, then if those points are together they must be the same! Thus
the line does not exist!
That maybe means that effectively there is no something like physical
mathematical lines, just because a point cannot be mapped to any
physical entity.
As an abstract line I think the dense order and those methods in
modern mathematical analysis avoid such kind of paradox
(not necessarily solving the philosophical or epistemological question,
if they need to be solved).
Even if the cut procedure to break a line in half sends such parts
into all possible cases mixing open and closed segments;
all them will have the same length and cardinality. If
both measures (one saying that they are going to match if are
superposed, and the other concerning the number of elements in each)
they coincide and the only possible way in which two parts of a whole
be the same is that they are in fact geometrically halves, even when a
segment seems to have an additional point, it can be
asked what do we mean by "same" or "equals" naively speaking. If the
question is topological the story is quite different, but finally
depends on those chosen naïve or formal meanings of "cut", "equals" or
"same" and what the abstract domain is (or your believes if you are asking for
physical continuity or open sets in nature).
Those paradoxes lies finally, I think, with all those constructions in which
infinity (mainly non-enumerable infinity) plays the main role.
The philosophical question could be then if those constructions are
just mental entities or if there exist a possible physical entity or
possible physical construction with such bizarre properties.
I mean the eternal so called platonism/naturalism problem.
I think science deals in some way with such matters in form of
claims as the Turing Thesis or Wolfram's Principle of
Computational Equivalence, in which avoiding non-countable
features seems to be the clue for their true. It is easy to see that
the limit is just between what is or effectively computable
and what are just mind abstractions like those models involving real numbers
(even non-computable), accelerated Turing machines or super-tasks.
Hector Zenil
Universite de Paris
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