On 8/5/2019 6:22 PM, Jon Alan Schmidt wrote:
JFS: Peirce developed outstanding theories of logic and semeiotic
with just a topological version of continuity. There is no evidence
that anything more would be significantly better. I agree with MEM.
JAS: I concur with the first two sentences, but I am puzzled
by the third... The thesis of Moore's paper is that Peirce's
"supermultitudinous" conception of continuity was inadequate to
do the specific work for which he tried to deploy it throughout
the Harvard Lectures in 1903, and therefore inessential to his
late philosophical system as a whole.
For the topological version, I'll quote Havelnel's summary in
https://www.academia.edu/25930950/Peirces_Topological_Concepts_Jerome_Havenel
:
Peirce's topology considers as similar two objects that can be
continuously deformed into one another, provided that during
such a deformation "no parts are separated which were at first
continuously connected" (MS 1170, article "topical")
I believe that this definition is adequate for Peirce's uses
of the term 'continuous'. I agree with Moore that Peirce never
stated a definition of the term 'supermultitudinous' that is
mathematically adequate (i.e., stated with sufficient detail and
precision that another mathematician could use the definition in
a formal proof).
JAS
With that in mind, I would like to call attention to a passage
from one of the Harvard Lectures that Moore did not specifically
address, because I believe that it can helpfully illustrate the
difference between the two conceptions of continuity.
None of the examples cited depend on the supermultitudinous claim.
CP 5.119 and CP 5.132 can be interpreted with just a finite number
of parts. Whether the others are finite or infinite is not obvious,
but there is no need to consider anything that the topological
methods can't handle -- no need for anything supermultitudinous.
John
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