## Friday Jan 18, 2008

### How Many

How many non-standards models of Peano Arithmetics exist? (Hint: The Gödel construction gives you ${\\aleph_0}$ many models. One can also prove the existence of $\\mathbf{c} = \\aleph_1$ many such models using the fact that there are  ${\\aleph_0}$ many primes.)

## Wednesday Jun 06, 2007

### Truth -- What's Consistency Got To Do With It?

 Frege used a 2-D notation for his logic. Many years ago, while with the Graduate Group in Logic and Methodology of Science at Berkeley, I had written a little paper with a tease of a title: "Consistency--What Is Logic Got To Do With It?" After a recent discussion, I realized more clearly how that paper might have been making too far of a leap in logic. Perhaps, it should have been named "Consistency--What Is Truth Got To Do With It?"  By "consistency," all along, I had meant "consistency" as understood by mathematicians and scientists, not what we understand "consistency" to be when we speak of, say, "consistent" behavior, which is a totally different concept when compared to a "consistent" theory.

In the mathematicians' definition of truth, as expounded by Alfred Tarski, one can only speak of "truth" within the confines of a mathematically "consistent" theory. Thus, one arrives, in mathematical logic, at incomplete theories that meet mathematical "truth" criteria even as they remain incomplete. In a sense, their incompleteness is more true about these theories than their "truth." Their "truth" exists within their limited, incomplete domain, which, always remains positively finite or countable or of much lower cardinality than the continuum of claims that they can neither prove nor disprove.

## Saturday Apr 07, 2007

### Lessons from the Persian Gulf

Abbas Edalat, professor of computer science and mathematics at Imperial College of London, draws some lessons from a recent incident in the Persian Gulf.

Check out, also, his inaugural lecture which begins with some important historical facts on computational arts and has a wonderful slide on the "failure of floating point computation." He gives an example of a floating point arithmetic failure in the "First Gulf War" to demonstrate the importance of the more exact "real arithmetic" based on Kashani's technique for estimating π.

## 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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