Thursday, November 09, 2006

Transitivity: doctrinal, dogmatic - and dull?

For a change today, I decided to talk about something nice and abstract: the mathematical property called transitivity. Of course, GKC does not actually state the definition:

The relation "RELATES-TO" is called transitive in a given set when, for any a, b, c in that set,
and also
means that we must have
But, as usual he hints about it: "Respect should be a transitive verb, not an intransitive; there should be an accusative after it; you cannot have respect without respectability. Reverence should be an act of homage, not a state of mind." [ILN Jun 15 1912 CW29:308]

And before you complain that these things are not the same, or that all this math is dull and useless and boring (is it?) let me go deeper into the matter.

For far more interesting is his comment on the philosophy of transitivity - and we must remember that transitivity is a dogmatic and a doctrinal distinction made about certain abstractions in mathematics:
the dogmas are not dull. Even what are called the fine doctrinal distinctions are not dull. They are like the finest operations of surgery; separating nerve from nerve, but giving life. It is easy enough to flatten out everything for miles round with dynamite, if our only object is to give death. But just as the physiologist is dealing with living tissues, so the theologian is dealing with living ideas; and if he draws a line between them it is naturally a very fine line.
[GKC The Thing CW3:303]
It is thrilling to read this juxtaposition of ideas, because (as I have very good reason to know) there is a powerful connection between transitivity and living things.

When you do a SEARCH on a computer, you may take advantage of something called the "Knuth-Morris-Pratt pattern-matching algorithm" which attempts to locate a search-string within a destination-string. It is a very good, and quite clever technique. Yet the algorithm works because it is based on the idea that the search string equals some part of the destination-string - AND for our usual character set (alphabet, digits, punctuation), the relation called "equality" is transitive!

Alas. Not all relations are transitive. In DNA sequence analysis, the relation called "matching" on the "wild card alphabet" is not transitive. Some computer scientists were appalled when they saw, amidst the hundreds and thousands of A, C, G, and T characters, other letters like S, W, M, K, R, Y, or B, D, H, V, or N. They asked if the biologists could omit these, or perhaps do their work more precisely. Well, needless to say, that is not possible. These wild card letters are real, and our work must take them into account. Just to illumine you further, here is the Hasse diagram for this curious alphabet:

[Above image is taken from my doctoral thesis]

In this DNA wildcard alphabet, A matches M, and M matches C, but A does NOT match C. Hence "matches" is not transitive.

And lacking transitivity, any searches with such wildcards must take this into account. (But don't try to do such searches with standard search tools!)

So, attention to such "fine doctrinal distinctions" as transitivity - as complex as it may seem to the outsider - is not only far from dull, is critically important and powerful when we are "dealing with living tissues" or the code which underlies them.


  1. Every week I read Dr. Thursday's posts and think to myself that Chesterton -- at least The Everlasting Man, and probably more -- should be required reading in all biology, chemistry, physics, and any other science, undergraduate and graduate program.

  2. Yes, thank God, GKC is a universal teacher - he both cross-pollenates and is cross-pollenated by the Field - thus we at Chesterton University abide by Cardinal Newman's Idea of a University. For us, there really can never be such a thing as a "different subject". [cf. ILN February 17, 1906] It would be just as wrong to overthrow science or engineering for philosophy or literature as it is to overthrow literature or philosophy for science or engineering. After all, we're about bridge-building here, which is such a Catholic thing to be! [See Summa Theologica II-II Q187 A5 resp.]

    You may already know that TEM is already required reading in at least one school in America: Christendom College, which I have had the privilege of visiting, and hope to visit again. They also have a complete collection of G. K.'s Weekly... (now you know one of the reasons why I hope to go back!)


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