## Saturday Mar 17, 2007

### Axiomatization of Transactions -- A Fable for Relational Algebra

David Hilbert had a program for axiomatizing science, in particular, and all the rest of thought, in general:

David Hilbert, Axiomatisches Denken, Math. Ann., 78 (1918) pp. 405– 415. English translation in: William Ewald (ed.), From Kant to Hilbert: A Source Book in Mathematics, Oxford 1996

Hilbert's program of axiomatization faced challenges even in mathematics, as Jan Brouwer unfolded his approach to doing mathematics

What were axioms? Axioms were simply a set of consistent statements written in a language composed of symbols representing variables, constants and relationships. "Models" were then constructed to give meaning to the symbols in a manner consistent with the Axioms. In this sense, "natural numbers" composed a model for some arithmetic axioms, say the Peano Arithmetic Axioms. However, models were rarely "minimal" to the axioms unless constructed from the axioms in particular ways, all of which produced isomorphic models of a certain kind. Not all models of the same axioms were equivalent or isomorphic. So, a theory of models had to be developed to explore the relationship among models.

In the world of business and economics, and in the social milieu of transactions, digitization of these transactions, i.e. the tendency towards demarcating sharp boundaries for transactions, led to the axiomatization of "rules" governing these transactions while at the same time managerial hierarchies built continuity into vertical integrations of such transactions within large organizations.

As the number of transactions vertically integrated within an organization increased in proportion to the volume of business activities, the managerial hierarchy faced  growing coordination challenges. However, axiomatization of many "business processes" (read "out-of-market transactions") had already begun to work its magic to make the managerial hierarchies independent of particular men or women. What was needed was a mechanics to propagate the axioms through individual transactions.

The mechanics was relational algebra and the machine that operated according to its principles was the relational database. The rest is history -- of modern business enterprises' use of IT technologies.

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