Gil Skillman wrote:
> Ravi writes, among other things:
>
>>russell attempted to avoid the circularity by introducing a theory of
>>types, but then came godel's incompleteness demonstration and the end of
>>the golden age of mathematics-logic ;-).
>
> Well, perhaps not. It's possible that Russell's attempt to "derive
> mathematics from logic" was doomed to fail because he got the problem
> exactly backward: logic is best understood as a province of mathematics,
> not vice-versa.
i have not heard of george spencer-brown, but i have definitely read
some material expressing this alternate view. i will be sure to look at
spencer-brown's writing. your post and examples were an interesting read!
that said, i would like to defend russell: i do not read him as
suggesting that mathematics is a province of logic. rather, in a sort of
euclidean sense, russell took a few logical rules and operations as
self-evident and inherently true (something that a layperson would agree
with, with the slightest of preparation required to understand the
formal representations), and tried to ground mathematics (which had
grown into a large disorganized graph) - make it rigorous - by making it
possible to derive its results from these logical truths. this is only
the culmination of attempts, such as iirc hilbert's work to represent
algebra in euclidean geometry. to a large extent, i would say, he
(russell) succeeded. which does not contradict your point that [formal]
logic could well be a province of mathematics.
it is true that sticking to boolean logic as the underlying formal
system led to the limitations exposed by the incompleteness theorem. it
is also possibly true that other formal systems, that subsume boolean
logic, might avoid such or similar limitations and to that extent
russell's adventure resulted in failure. but complex numbers, infinite
sequences, and such were objected to as not self-evident (and even
meaningless) by the mathematicians of the time, not just laypersons.
which leads to einstein: insofar as the expressions of mathematics refer
to reality, they are not certain, and insofar as they are certain, they
do not refer to reality (which i am sure begs the question of "reality"
- is reimann geometry, which ironically, iirc, einstein used to
represent "reality", any less real than euclidean geometry, just because
euclidean geometry makes "sense", is self-evident, to homo sapiens?).
--ravi
