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Topic: Notation
Replies: 14   Last Post: May 27, 2013 1:46 AM

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Posts: 1,968
Registered: 12/4/12
Re: Notation
Posted: May 27, 2013 12:31 AM
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On 5/26/2013 10:01 PM, Ross A. Finlayson wrote:
> Nice of you to introduce Boolos, a 20th centory logician who may well
> have already covered the desirable features of theory with the
> acknowledgment of the limitations of either the purely synthetic or
> analytic, and the necessity of their combination of the total and
> wholistic.

Are you surprised that I actually read moderns? Well,
the description theory is a long list of 20th century
developments and relates to a significant body of
researches ignored by "mathematical logic". Its onset,
however, is significant in the works of Frege and Russell.

> Then, of note is a reference to Boolos' rehabilitation of Frege's
> Grundgesetze, here in terms of the action in philosophy to have the
> technically analytic result in the utility of the synthetic for
> philosophically logical and logically philosphic foundations for
> mathematics, there's an example of such technical underpinnings as
> might be of note and use to represent a suitably concise form for
> discussion on the general matter: general matters.

Zalta has done a nice introduction to Frege and the modern
day rehabilitation of his ideas,

Of course, set theoretically this would involve one with
Quine's New Foundations. That too has a resurgence. I
am interested, but the general mathematical community
dictates consideration of set theories evolving from
Zermelo as primary.

> Far from it from that to be said to accomplish this goal, along the
> lines of the Hilbert program is not to complete mathematics then to
> integrate for completion the theoretical foundations of mathematics in
> as to their completion, yet, that is the statement of what would see
> in the basically analytic and synthetic as to, for example, an
> axiomless system of natural deduction, for inference, induction, and
> reason. This is synthetically from null or void, and analytically:
> that the first principles are final causes, for mathematics as a
> whole.
> I'm reminded of your earliest posts, fom, and their plainly
> enumerative content: from what simplest principles and observation,
> is thus reason evidenced? In describing equality as tautology and
> identity, where is it that what are otherwise seen as the simplest
> logical primitives are themselves structured, and, how is it so that
> then: this truth-preserving theory encompasses "no paradox" and "all
> and none"?

Circularity is inherent to mathematical practice. Before solving
a problem, analysis dictates considering a plausible solution and
working backward or both backward and forward. The final presentation,
however, is synthetic and forward-directed from premises to conclusions.

The tautologous intepretation of "A is A" as "A=A" predates the logical
atomism required to make sense of the compositional language structure
associated with the modern paradigm of formal logic. But, the history
of mathematics leading to "number systems" as abstract axiomatic
systems with deductive calculi dictate treating identity in relation
to "systems". Thus, identity, diversity, and negation are intimately

The notion of "priority" begins with Aristotle and grounds his
epistemological argument for distinguishing between "demonstrative
science" and "dialectical argumentation". Late nineteenth century
and early twentieth century trends organized the natural sciences
into a hierarchy of priority. Logicism is a philosophical statement
of priority in this sense. So, when considering logicism critically,
one is confronted with priority as the fundamental issue.

However, the semantical notions of Frege and the philosophies that
most influenced the received paradigm rose to prominence at the
expense of Aristotle's epistemological arguments. Analysis of
belief took the form of propositional attitude reports rather than
the rhetorical argumentation of the Aristotelian dialectic. So,
Aristotle's epistemological justifications had been deprecated.

This minimal influence of Aristotle's "Posterior Analytics" is
actually documented in the link,

"In the seventeenth and eighteenth centuries, with the new
Platonisms, the anti-Aristotelian bias of the new science,
and developing empiricism, the theory of demonstration came
to be ignored in mainstream philosophy, although it persists
as an element of Thomism."

For my part, I eventually came to justify my use of circular
definition for the introduction of 'morphemic' predicates
in terms of Aristotle and Leibniz' interpretations of Aristotle.
Naturally, "objects" such as denotations of constants and
functions should not be introduced with such syntax. But, I
had to figure out how the use of "undefined language symbols"
came to be introduced. That is found in Bolzano, and, it is
related to notions of definition in relation to syllogistic
logic interpreted extensionally. Leibniz' logic is intensional
as are his explanations of individuation and naming. So
there is a mismatch in directionality.

What is novel in my construction is that I recognize the
role of both directions in justifying the axioms. A 'framework'
must be established intensionally (impredicatively). But,
the ontological foundation and concreta must be extensional
so that the model-theoretic utility of the axiom of regularity

At the core of this is recognizing the identity/diversity
dichotomy in relation to a system as opposed to the self-identity
of logical atomism.

> That would be of general interest, particularly in the general
> assignment of an integrated logic with general bibliography, in as to
> the parallel carriage of fact from the various to and from the
> specific and macro and micro and synthetic, and analytic.
> Basically, logic is reversible in: that the complement: is itself.
> And the infinite is infinite.

... unless you consider a theory with completed
infinities. Then the infinite is the absolute infinite.

(The singular is to the universal as the individual is
to the infinite -- a paraphrase of Kant)

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