answers for George Greene

From: tuckpointer (aatuckpointing_at_sbcglobal.net)
Date: 10/04/04 

Date: Mon, 04 Oct 2004 23:13:56 GMT

When I introduced triple systems as relevant to the foundations of
mathematics, George Greene asked, "Where is this geometry coming from?" (or
something close to that paraphrase).

Perhaps (the real) Frege can shed some light...

"...But that is just a defect in the kindergarten-numbers. The more
I have thought the matter over, the more convinced I have become
that arithemetic and geometry have developed on the same basis--a
geometrical one in fact--so that mathematics in its entirety is really
geometry. Only on this view does mathematics present itself as
completely homogeneous in nature. Counting, which arose
psychologically out of the demands of practical life, has lead the
learned astray."

                                   --G. Frege, "Numbers and Arithmetic"

Another one of George Greene's questions went something like "Why Kant?"

Keeping the preceding quote in mind, consider the avenue of research not
taken by Frege as discussed Volume II of "Grundgesetze der Arithmetic,"

"If we were going to dispense classes from the law of
the excluded middle, we might think of regarding them
(and, in fact, value ranges generally) as improper objects.
These could then not be allowed as arguments for all
first-level functions. But, there would also be functions
that could have as arguments both proper and improper
objects. At least the relation of equality (identity) would
be a function of this sort. (An attempt might be made to
escape this by assuming a special sort of equality for
improper objects. But that is certainly ruled out. Identity
is a relation given to us in such specific form that it is
inconceivable that various kinds of it should occur."

Curiously, sci.logic and sci.math had two posters trying desperately to
discuss formalisms relating to non-standard intuitions associated with
identity ("mitchism" and Correy non-self-identicals).

Is Frege's assertion of inconceivability legitimate? If one actually reads
Frege, it is clear that he is enamored with Leibniz. His notion of identity
corresponds with the Leibnizian rules referred to as identity of
indiscernibles and the indiscernibility of identicals. Strangely, it was
Kant who challenged Leibniz with the statement,

"If an object is presented to us on several occasions but
always with the same inner determination, then if it be taken
as an object of pure understanding, it is always one and the
same, only one thing, not many. But if it is appearance, we
are not concerned to compare concepts, difference of spatial
position at one and the same time is still an adequate ground
for the numerical difference of the object, that is, the object of
the senses. Thus in the case of two drops of water we can
abstract altogether from all internal difference (of quality and
quantity), and the mere fact that they have been intuited
simultaneously in different spatial positions is sufficient justification
for holding them to be numerically different. Leibniz took
appearances for things-in-themselves, and so for intelligibilia,
i.e., objects of the pure understanding (although, on account
of the confused character of our representations of them, he
gave them the name of phenomena), and on that assumption
his principle of the identity of indiscernibles certainly could not
be disputed. But, since they are objects of sensibility, in relation
to which the employment of the understanding is not pure, but only
empirical, plurality and numerical difference are already given to
us by space itself, the condition of outer appearances.

Whatever one may think about Kant's theory of knowledge or "intutionism" in
the philosophy of mathematics, what cannot be disputed is that Kant's
distinction between mathematics and logic correlates this objection to
Leibniz with a *mathematical* notion of identity distinct from the identity
of Frege's early logicism.

For anyone interested in modern developments, Czelakowski attributes the
origin of non-Fregean logics to Suszko's investigations. Look up references
to Suszko identity. Apparently, Suszko was motivated by Wittgenstein's
"state of affairs." But, this notion is also prominent in Husserl, and, I
believe that the interested reader will find that the situation theory that
arose in connection with Suszko identity is more easily understood relative
to Husserl's ideas on objectification.

As for George Greene's question, "Why is mitch smith such an ***?"

It must be a birth defect.

:-)

mitch

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