Collaborative articulation of how abstraction and language is employed in the computational manifestation of numbers -- including analysis of the role of syntax, semantics, and meaning in the specification and use of software interfaces.
[update 2008-05-29T16:15Z: Added Petzold's 2008-05-28 post with its nice tie-in of logic and Turing.]
update 2008-05-27T16:03Z: A little tweaking in my comment on Petzold's 2008-05-25 essay.]
I received an amazon.com e-mail warning announcing that Charles Petzold's The Annotated Turing is delayed and I needed to approve the shipping delay from May 23 to June 23. A number of times, I will receive one of these announcements only to have it followed by my order shipping immediately. Since Petzold's site reports the book is scheduled to launch on June 16, I will be content to wait.
Meanwhile, Petzold has been posting interesting tid-bits that attracted his attention while working on the book. In many cases, there will be deeper coverage when the text is available. Here are some that I am particularly keen to dig further into:
Starting out with a look at logical puzzles, Petzold looks at questions about truth, how connected to logic, and related challenges that did not make it into his book, Code: The Hidden Language of Computer Hardware and Software (recommended). In this essay, Petzold illustrates Charles Dodgson's 1896 Symbolic Logic approach using George Boole's original 1854 notation and will tie it back to computability in Chapter 12 of The Annotated Turing.
There is an interesting tension between computations that do not stop but in the limit ( as we were inclined to say when I was a school student) the output approachesconverges to some computable real number. Turing allowed such cases, and then demonstrated that there are still far more real numbers that cannot be computed than those that can. The way this is arrived at leaves some questions about how many real numbers we think there should be and whether we are mistaken in where conventional set theory takes us in that regard. Although I am not ready to concede to Petzold that any kind of Platonic commitment is necessary in conception of the transfinite, I definitely find this accompanying statement worthy of careful appraisal: "We like to pretend that mathematics is the most 'objective' and least human-bound intellectual endeavor, but our view of the natural numbers reveals mathematics to be founded on a very human metaphysical conceit. The natural numbers are not, in fact, "natural" — that is, intrinsically part of nature — but arise out of human discourse." In this essay, Petzold explores Brian Rotman's effort to avoid conception of the completed natural numbers (and certainly, in that case, the completed reals). The focus is on tying Turing's work to philosophical issues regarding the foundation of mathematics, an unexpected connection and one that Turing did not explore over-much.
Not expecting to find a connection between the work of Turing and L.E.J Brouwer, Petzold was startled to find one in the short Correction that Turing made to his original paper. Accounting for this leads to a chapter on "Conceiving the Continuum" in The Annotated Turing and this delightful essay on how that all unfolded. This is more on the issue of infinitudes.
From the beginning, Petzold has intimated that he wants to illuminate the Turing Machine as Alan Turing conceived of it, with it being acceptable for the machine to run forever. This is not the ordinary formulation that has survived into contemporary computation theory. Turing wants to bring the full Turing characterization to our attention. This essay motivates the difference and the historical context at the time of Turing's formulation and the subsequent revisions.
Considering some of the later blog posts, it is amusing to see this use of "ideal." This is a wonderful essay on the quirky fascination with attempting to produce a 300-page book, since Petzold has never managed to keep one that short.
This is a touching essay on The Annotated Turing now being on its way to print, with no more opportunities for changes or misgivings. We are reminded that Petzold had been working on the book since 1999.
Petzold provides an account of what he went through to reset the Turing paper so that he could match Turing's symbols and typeface when elements of the Turing paper are discussed in the "annotations." This is something that Donald Knuth will love, since the desired result was obtained (as the editor confides in a comment) by using LaTeX. It is delightful how the reset Turing paper was proofed against the original printing.