Re: Ontolog Forum • Alex Shkotin

AS:

Looking at “categories, or types” in Precursors Of Category Theory • Hilbert and Ackermann what do you think of to say “Precursors Of Type Theory” as Category Theory is a math discipline?   […]   It seems you collect for three topics:  phil‑cat, type theory, math cat‑theory.

Dear Alex,

When it comes to math, computer science, and their applications to logic and linguistics I see categories and types as pretty much the same things.  No doubt the words are used differently in other contexts but I am concerned with the above contexts at the moment.

The diversity of categorical systems across different disciplines and theorists is obvious to all observers.  But when we examine how systems of categories operate in grammatical, logical, or more generally semiotic frameworks we can detect a common function all the more useful systems share.  The semiotic framework is already well marked in Aristotle’s founding text on interpretation and the function of category references as go-betweens from unruly language to the rule of logic is clearly delineated in his treatise on categories.  It is that order of function which is preserved from Aristotle’s categories to our current mathematical variety.

References

• Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
• Collection Of Source Materials • Determination • Peirce (CP 5.447) (CP 5.448)
• Foundations Of Mathematics List • C.S. Peirce on “General” and “Vague” • (1) (2) (3)

Resources

• Precursors Of Category Theory
• Survey of Precursors Of Category Theory
• Lane, R. (2001), “Principles of Excluded Middle and Contradiction”, in M. Bergman and J. Queiroz (eds.), The Commens Encyclopedia : The Digital Encyclopedia of Peirce Studies New Edition, Pub. 140731-0107a.  Online.

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science