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\chapter{1966 Mathematical Studies: Introduction}
Pure mathematics is the one activity which is intrinsically formalistic. It
is the one activity which brings out the practical value of formal
manipulations. Abstract games fit in perfectly with the tradition and
rationale of pure mathematics; whereas they would not be appropriate in
any other discipline. Pure mathematics is the one activity which can
appropriately develop through innovations of a formalistic character.
Precisely because pure mathematics does not have to be immediately
practical, there is no intrinsic reason why it should adhere to the normal
concept of logical truth. No harm is done if the mathematician chooses to
play a game which is indeterminate by normal logical standards. All that
matters is that the mathematician clearly specify the rules of his game, and
that he not make claims for his results which are inconsistent with his rules.
Actually, my pure philosophical writings discredit the concept of
logical truth by showing that there are flaws inherent in all non-trivial
language. Thus, no mathematics has the logical validity which was once
claimed for mathematics. From the ultimate philosophical standpoint, all
mathematics is as "indeterminate" as the mathematics in this monograph.
All the more reason, then, not to limit mathematics to the normal concept
of logical truth.
Once it is realized that mathematics is intrinsically formalistic, and need
not adhere to the normal concept of logical truth, why hold back from
exploring the possibilities which are available? There is every reason to
search out the possibilities and present them. Such is the purpose of this
monograph.
The ultimate test of the non-triviality of pure mathematics is whether it
has practical applications. I believe that the approaches presented on a very
abstract level in this monograph will turn out to have such applications. In
order to be applied, the principles which are presented here have to be
developed intensively on a level which is compatible with applications. The
results will be found in my two subsequent essays, \essaytitle{Subjective Propositional
Vibration} and \essaytitle{The Logic of Admissible Contradictions}.
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