Primary Algebra as Pattern Calculus (concl.)

Speaking of algebra, and having just encountered one example of an algebraic law, we might as well introduce the axioms of the primary algebra, once again deriving their substance and their name from the works of Charles Sanders Peirce and George Spencer Brown, respectively.

The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example of an algebraic law we noticed first, “a (  ) = (  )”, as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification we have naturally if tacitly adopted for their respective sets of axioms.

The arithmetic axioms were introduced by fiat, in a quasi‑apriori fashion, though it is of course only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi‑primal move.  The algebraic axioms, in contrast, can be seen to derive both their motive and their justification from the observation and summarization of patterns which are visible in the arithmetic spectrum.

Resources

• Logical Graphs
• Survey of Animated Logical Graphs

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