[lit-ideas] Grice's Infinity, Popper's Infinity
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Fri, 7 Jun 2013 22:49:42 -0400 (EDT)
McEvoy is considering Euclid's Theorem in Popperian terms.
Then there's this link:
http://www.cs.joensuu.fi/~whamalai/skc/popper.html
that uses some interesting abbreviations that McEvoy may find useful.
Notably, World 3.3 is that bit of world 3 which has no exact correlate (or
is not part of any intersection) of World 3 with World 1 or World 2.
As the author notes:
"the existence of World 3.3 is crucial for World 3's independence -
otherwise we could reduce it to worlds 1 and 2."
The author goes on to ask:
"But are there such objects, which have not yet taken any form in World 1
or World 2?"
"Popper answers "Yes"".
"He believes that there are objects in World 3, which have not yet been
realized by people, but which still exist. They can be e.g. logical
consequences of the mentally recognized theories, or unknown relationships
between
World 3 objects. Still this "shade world" is real, because it already has
influence on World 1 by World 2. The problems and theories of World 3 can be
our products, but still they are not only our constructions: their truth or
falsity depends on both the stucture of World 3 and World 1. E.g. whether
a human being can solve some mathematical problem, depends on the fact,
whether it has a solution in World 3 and whether the solver has all the
required knowledge items of World 3 (and has understood them correctly)."
Also, talking of a different type of intersection:
"E.g. a human being can understand the idea of an infinite series of
integers without any physical representation of it (which would be
impossible)."
I think there are various readings of this above:
"A human being CAN understand" -- but does HE (or she) UNDERSTAND? It
seems that when we use 'can' we sometimes mean 'do not' ("He can play the
flute
but he won't").
----
I believe Dummett saw part of this -- in the Oxonian context -- when he
found that Oxonians had undervalued the intuitionistic side to infinity.
Or not.
Among the different formulations of intuitionism, such as it was NOT
popular in Grice's time, there are several different positions on the meaning
and reality of infinity.
In fact, the term "POTENTIAL infinity" is used to refer to a mathematical
procedure in which there is an unending series of steps.
After each step has been completed, there is always another step to be
performed.
For example, consider the process of counting:
Grice (to student): Count!
Student: One, Two, Three, Four...
Grice (to student): Stop!
Student stops. (Note that had Grice NOT uttered, "Stop!", the Student,
theoretically, would have proceeded "ad infinitum").
1, 2, 3, …
The term, "actual infinity", as it is hardly used in Oxford, refers to a
completed mathematical object which contains an infinite number of elements.
An example is the set of natural numbers, N = {1, 2, …}.
----
Mathematicians use "..." ambiguously, when what they mean, strictly, is
"∞"
Note that, as Grice notes, "∞" should be distinguished from "א" (aleph).
In set theory, a discipline within mathematics, the aleph numbers are a
sequence of numbers used to represent the cardinality (or size) of infinite
sets.
They are named after the symbol used to denote them, the Hebrew letter
aleph א.
The cardinality of the natural numbers is \aleph_0 (read aleph-naught,
aleph-null, or aleph-zero), the next larger cardinality is aleph-one \aleph_1,
then \aleph_2 and so on.
Continuing in this manner, it is possible to define a cardinal number
\aleph_\alpha for every ordinal number α.
The concept goes back to Georg Cantor, who defined the notion of
cardinality and realized that infinite sets can have different cardinalities.
The aleph numbers differ from the infinity (∞) commonly found in algebra
and calculus, and Grice knew this -- and Popper ignored this.
In Cantor's formulation of set theory, there are many different infinite
sets, some of which are larger than others.
For example, the set of all real numbers R is larger than N, because any
procedure that you attempt to use to put the natural numbers into one-to-one
correspondence with the real numbers will always fail.
There will always be an infinite number of real numbers "left over".
Any infinite set that can be placed in one-to-one correspondence with the
natural numbers is said to be "countable" or "denumerable".
Infinite sets larger than this are said to be "uncountable".
Cantor's set theory led to the axiomatic system of ZFC, now the most common
foundation of modern mathematics.
----
Note that Dummett's Intuitionism (as was influential in Oxford) was
created, in part, as a reaction to Cantor's set theory.
Modern constructive set theory does include the axiom of infinity from
Zermelo-Fraenkel set theory (or a revised version of this axiom), and includes
the set N of natural numbers.
Most modern constructive mathematicians accept the reality of countably
infinite sets (however, see Alexander Esenin-Volpin for a counter-example --
or Popper for another unconvincing one).
Brouwer rejected the concept of actual infinity, but admitted the idea of
potential infinity.
Brouwer made it clear, as I think beyond any doubt, that there is no
evidence supporting the belief in the existential character of the totality of
all natural numbers.
The sequence of numbers which grows beyond any stage already reached by
passing to the next number, is a manifold of possibilities open towards
infinity.
It remains forever in the status of creation, but is not a closed realm of
things existing in themselves.
That we blindly converted one into the other is the true source of our
difficulties, including the antinomies – a source of more fundamental nature
than Russell's vicious circle principle indicated.
Brouwer opened our eyes and made us see how far classical mathematics,
nourished by a belief in the 'absolute' that transcends all human possibilities
of realization, goes beyond such statements as can claim real meaning and
truth founded on evidence."
Grice's Finitism is an extreme version of Intuitionism that rejects the
idea of potential infinity.
Grice's example of an absurd claim:
So far as I know there are infinitely many stars.
(Unintended "implicature": I do NOT know [cfr. "Grice's Implicature,
Speranza's Implicanza").
According to Finitism, a mathematical object does not exist unless it can
be constructed from the natural numbers in a finite number of steps.
"Surely a star is NOT constructed, which poses an extra problem to the
absurd claim."
Cheers,
Speranza
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