Date: Jan 1, 2013 5:56 PM
Author: kirby urner
Subject: Re: A Point on Understanding
On Sun, Dec 30, 2012 at 4:57 PM, Joe Niederberger
> R. Hansen says:
>>I was surprised that my son understood that statements involving infinity involved an ongoing process that never completes.
> That's the way Gauss understood it. However, prevailing sentiment against actual infinities started to change with Cantor (not that there were not earlier proponents) and its now status quo to accept both kinds.
> Here are some links for reading should anyone desire:
> * http://www-history.mcs.st-and.ac.uk/HistTopics/Infinity.html
> * http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html
> * http://en.wikipedia.org/wiki/Actual_infinity
I committed some time to the study of the above readings. No regrets.
Poincaire also had thoughts on "infinity" that are most apropos. I
found this paragraph illuminating:
Poincare's prohibition of impredicative definitions is also connected
with his point of view on infinity. According to Poincare, there are
two different schools of thought about infinite sets; he called these
schools Cantorian and Pragmatist. Cantorians are realists with respect
to mathematical entities; these entities have a reality that is
independent of human conceptions. The mathematician discovers them but
does not create them. Pragmatists believe that a thing exists only
when it is the object of an act of thinking, and infinity is nothing
but the possibility of the mind's generating an endless series of
finite objects. Practicing mathematicians tend to be realists, not
pragmatists or intuitionists.
[ http://www.iep.utm.edu/poincare/ ]
As a practicing Wittgensteinian I would say I am not a mathematical
realist so much as a pragmatist and intuitionist.
Both Platonism and nominalism share the same view of meaning as a kind
of pointing and think "perfect circles", if not evident in nature,
must mean something as "objects in the mind's eye" i.e. their meaning
must be somehow witnessed in an observational moment or act.
The Platonist (called a "realist" by such as the Catholic
Encyclopedia) thinks mathematical objects such as "perfect spheres"
and "perfectly flat infinite planes" really exist in some "realm"
accessible to the mind as a kind of private witness.
An observer (also Platonic) ostensively defines "the meaning" of these
ideas by means of some private act, not unlike a private act of
worship people might say "points to" their deity (who might be a
perfect circle for all we know, some secret beetle in some private box
we never get to peer into).
The nominalist is typically portrayed as sharing the Platonist /
realists notion that words (nouns especially, not so sure about
prepositions) are "pointers" with their meanings being objects in the
shared and/or private vista.
The nominalist just refuses to believe in the Platonic Realm and
considers "generalizations" such as the perfect circle or perfect
sphere to be an extrapolation, perhaps an averaging, of actual
experiences with material objects.
As I shared with the physics teachers recently -- on a list with
partially overlapping membership from here -- referring to my own
reading program in philo:
Ryle's ridicule of the 'private ostensive definition' -- as if we
could "see" what we mean by [some mental phenomenon ] rolls into this,
as early 1900s Oxford - Cambridge wrestled with its metaphors for
mentation, pre-computer (back then, a "computer" was a type of paid
employee, so no help there, in terms of providing an analogy, Von
Neumann's PC e.g. a 386 running Windows, still in the future).
This training (reading course) removes many temptations to introspect
in search of what cognitive terms such as "understanding" might mean,
i.e. there's no closing the eyes to study "my understanding process"
as if some "process" were in need of witnessing (observer / observed)
for the term itself to have meaning.
[ http://groups.yahoo.com/group/WittrsEX/message/4949 -- copy of
complete post, archived to Wittgenstein list ]
Message was edited by: kirby urner