Re: Wittgenstein and the philosophy of mathematics
- From: ross.finlayson@xxxxxxxxx
- Date: Sat, 14 Feb 2009 19:22:33 -0800 (PST)
On Feb 9, 4:13 pm, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
Marko Amnell <marko.amn...@xxxxxxxxxxx> writes:
Crossposted to sci.math and sci.logic in case someone might be
interested in Mathieu Marion's book. I recall that the late Torkel
Franzen, a regular sci.math poster, also said that Wittgenstein's
philosophy of mathematics is uniquely conventionalist, setting it
apart from most other philosophies of mathematics.
Wittgenstein's willingness to accept thoroughgoing and consistent
anti-realism in mathematics is indeed quite unique, and Torkel did at
times comment on that. (The observation has of course been made by a
number of people, such as Michael Dummett.)
A standard conventionalist line would have it that there is no more
truth or falsity to axioms, in mathematics, that our agreeing to work
with them, or refusing to work with them. Usually, however, whether
this or that follows from these or those axioms is not taken to be a
matter of convention, but rather something determined as a matter of
(logical?) fact. Wittgenstein would have none of this, and famously
held that even if we have laid down rules of inference and axioms, we
make genuine decisions each time when presented with a purported
application, as to whether to accept it as being in accord to the
rules, and that there is nothing more to something following from the
axioms by the rules than this; that, for example, it is not possible
for a hidden contradiction to be lurking in the adopted axioms, just
waiting for us to find it, like a tuberculosis germ -- rather, unless
we have actually derived a contradiction, accepting the derivation to
be in accord with the rules, there is no matter of fact whether a
contradiction follows or not.
Such a view is hard to swallow, of course, which just goes to show
that thoroughgoing and consistent anti-realism is a heroic
undertaking, if indeed humanly possible at all.
--
Aatu Koskensilta (aatu.koskensi...@xxxxxx)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
"Games have one more feature in common with pure mathematics. It is
subtle but important. It is that the objects with which a game is
played have no meaning outside the context of the game.
Chessmen, for example, are significant only in reference to chess.
They have no necessary correspondence with anything external to the
game. Of course, we could _interpret_ the pieces as regiments at the
First Battle of the Marne, and the board as French countryside. Or an
interpretation could be brought about by a wager, say each piece
represents $5.00 and losing it means paying that amount. But no such
correspondence between the game and things outside is necessary."
-- Trudeau, Introduction to Graph Theory
Flattening outward graphs to tubes, spiral and expanding square lineal
progressions are space-filling and vice versa. That is about looking
at the cover of that book. Simply represented in small data
structures, verbose relations are preserved. That is about simple N-d
analysis, in linear discrimination in multispatial analysis, within
three or four short pages I can describe why it's so (in simple
metrics).
No theory is A theory. A theory, simply, has no axioms. Thus I
describe incompleteness theoretically as inconsistency.
You can affect trees with your will and not numbers. That the dialog
is mostly thinking numbers obey them seems one-sided. There's nothing
anyone can do to claim 1+1 =/= 2. One plus one equals two.
One plus one equals two, in terms of the numbers.
Hey Aatu that's an awesome post. I have not been following much the
discussions here on sci.logic since last I was posting. I don't
particularly suspect much o anything novel in terms of what I say but
there has probably been some good stuff in terms of what basically
everybody says. Basically it seems Albrecht is claiming finitism
shows infinitary reasoning finitary, which would be inconsistent,
while the Zermelo-Fraenkel Cantorians with their ZFC set theory with
classes call him stupid. At least Albrecht mostly actually does seem
correct him in his responses.
Get some!
Ross
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