On 3/21/2013 8:26 AM, AMeiwes wrote: > "fom" <fomJUNK@nyms.net> wrote in message > news:1ZudnajAj7TF09fMnZ2dnUVZ_vydnZ2d@giganews.com... >>> >>>> "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 subjectively.
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.