Provisional formalism

The provisional formalism is a foundational formalism designed to observe models and theories through variations of diagonality.

To show how a formalism may be used we will offer some heuristics or interpretations; these heuristics are not proofs within the formalism; they are not predicative; they do not point toward a model. We will leave the reason for or value in using the formalism to the reader.

Diagonality
Recall that a diagonal argument is ... related to the notion of adjoints and the abelian distinction between operators and results that motivated this formalism. Fixed point theorems and diagonal lemmas. ==Example== Given $$R_3^9$$, we can write $$R: 3 \to 9$$ So we might interpret $$R$$ as $$+6$$, written $$R = +6$$. Alternately, $$R = \times 3$$. Or $$R = -6 \circ \times 5$$ with $$\circ$$ as the composition operator. Or $$R = write(9) \circ erase(3)$$. Or $$R = m^2$$. What we would like is some way to determine what interpretation of $R$ is `best' or `ideal'.

Let $R_3^9$ be the collection of all interpretations of $R : 3 \to 9$, i.e. $$R_3^9 = \{ +6, \times 3, \times 5 \circ -6, write(9) \circ erase(3), m^2, ...\}.$$ Then what is the best, ideal, or most representative member of this collection?

Clearly we have to appeal to some theory of what constitutes `best', `ideal', or `most representative'. Before that though, note that we gave for $$R_3^9 = ...$$ above an \emph{extensional} description. Instead we can write $$R_3^9 : R_3 \to R^9$$ as an \emph{intensional} way of describing $$R_3^9$$. it is the same strategy we use in number theory when we use ideals to represent 'numbers with a certain property'.