[FOM] Advertising an approach to the philosophy ofmathematics(with some logical technical work)
drago at unina.it
Fri Sep 14 18:32:09 EDT 2007
About Feng statement :
1. This research does not assume or rely on the claim that the universe is
finite and discrete. The point is merely that our scientific theories
accurately describe things within some finite range only (from the Planck
scale up to the currently recognized cosmological scale), and infinity and
continuity in our mathematical models of nature are approximations to things
within that finite range.
I suggest to state the problem of finite mathematics vs. infinite
mathematics in a more convenient way, i.e. the way Goedel puts in the
summary of his celebrated theorems he gave in Konigsberg 1931:
According to the formalist view one adjoins to the meaningful propositions
of mathematics transfinite pseudo-assertions, which in themselves have no
meaning, but serve only to round out the system, just as in geometry one
rounds out a system by the introduction of point at infinity. This views
presupposes that, if one adjoins to the system S of meaningful propositions
the system T of transfinite propositions and axioms and then proves a
theorem of S by making a detour through theorems of T, this theorem is also
contentually [in its real content] correct, hence that through the
adjunction of the transfinite axioms no contentually false theorems become
provable. This requirement is customarily replaced by that of consistency.
Now I would like to point out that one cannot, without further ado, regard
these two demand as equivalent...
About Feng question
>The goal is to offer a logical explanation of how exactly
> infinite mathematics is applicable to physical things within that finite
The answer is furnished by the plain theory of geometrical optics, which
invokes the points at infinity for explaining the path followed by the image
given by a convex lens, of a point moving between the lens and a disatnce a
little more far than lens focus . This means to give meaning to 1/0 in the
formula 1/p + 1/q = 1/f.
To give meaning to geometrical intuition of the point at infinity was
exactly the basic assumption of the contemporary Cavalieri's mathematics of
indivisibles. Is easy to prove that it is equivalent to Weyl elementary
mathematics (1918), founded just upon the same postulate (the existence of
the l.u.b. of an approximating series).
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