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Topic: For the readers of WM: Resources on empirical time to consider as
a foundation for mathematics

Replies: 10   Last Post: Mar 21, 2013 11:34 PM

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Posts: 1,968
Registered: 12/4/12
Re: For the readers of WM: Resources on empirical time to consider
as a foundation for mathematics

Posted: Mar 21, 2013 11:08 PM
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On 3/21/2013 8:26 AM, AMeiwes wrote:
> "fom" <> wrote in message

>>>> "Reality is the ultimate arbiter since
>>>> mathematics has been abstracted from
>>>> reality."

>>> circular thinking since one is thinking in circles.

>> I, personally, have no problem with circularity.
>> At issue, however, is the correctness of its application,
>> and "thinking in circles" is disqualified.

> if disqualification leads to disagreement, I agree

It is actually a precise statement in a historical
sense. Admonition against all circularity is a recent
phenomenon arising from Russell's treatment of viscous
circles. If one takes the reference to "thinking" in
the sense of Boole's "Laws of Thought" then the relationship
to logic is loosely established. And, since "thinking"
is perceived as an activity it can be compared with the
activity of the transformation rules within the deductive
calculi of a logic. With this in mind, one can consider
the following remarks.

This is what Aristotle says of circular demonstration:

"Unqualified demonstration clearly cannot be circular,
if it must be derived from what is prior and better
known. For the same things cannot be both prior and
posterior to the same things at the same time, except
in different ways (so that, for example, some things
are prior relative to us, and others are prior without
qualification -- this is the way induction makes
something known [because it argues from what is prior
relative to us -- that is, our generalizations of
particulars]) If this is so, our definition of
unqualified knowledge will be faulty, and there will
be two sorts of knowledge; or, rather, perhaps the
second sort of demonstration is not unqualified
demonstration, since it is derived from what is merely
better known."

To give you some context, the example Aristotle
uses before he defines "unqualified knowledge"
is better than the definition itself. Concerning
certain arguments from Plato's Meno he writes:

"... Do you or do you not know that every pair
is even? When you say you do, the produce a
pair that you did not think existed and hence
did not think was even. They solve this puzzle
by saying that one does not know that every pair
is even, but rather one knows that what one knows
to be a pair is even. In fact, however, and
contrary to this solution, one knows that of which
one has grasped and still possesses the demonstration,
and the demonstration one has grasped is not about
whatever one knows to be a triangle or a number,
but about every number or triangle without qualification;
for in a demonstration a premise is not taken to
say that what you know to be a triangle or rectangle
is so and so, but, on the contrary, it is taken
to apply in every case."

So, Aristotle's notion of unqualified knowledge
is knowledge expressed through universal quantifiers.

If the interpretation of the universal quantifier
relies on a course-of-values determination of
existing instances, then "the same things" are
both prior and posterior to "the same things at
the same time" and his systems of definitions
do not support his notion of unqualified demonstration.
They fail because a counter-example reduces the
epistemic strength of an assertion from a universal
that applies generally to what is known only

Thus, "circular thinking" is disqualified.

But with regard to the principles of a demonstrative
science, Aristotle acknowledges the fact the
immediacy of the principles forces certain circular
relations in their expression. Examples of this
interpretation can be found in Leibniz.

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