While cleaning my room this evening, I came across a scrap of paper on which a friend of mine a couple years ago had written the longest word in Hungarian, namely, megszentségteleníthetetlenségeskedéseitekért. As you might imagine, this fact appears on various websites variably translated; according to this one, it means “for your constant mentioning of the fact that [a particular thing] is unprofanable,”  which I gather is correct, or at least more correct than the one I saw at Wikipedia that had something to do with “pretending to be strong”.

Unfortunately, it’s proving to be far more difficult to find out where this word is used in print.  My friend said something about it occurring in some medieval liturgy, which both sounds completely plausible and would account for the paucity of internet references.  Maybe I’ll run across it some day…

Given my experience with the Anglican church and the fact that several good friends of mine identify as members, I am particularly saddened by the current difficulties it is going through  and I hope that they can work things out in a fashion amenable to everyone at the upcoming Lambeth Conference.

Unfortunately, I’m not terribly optimistic, especially given the following description of the new discussion format to be applied this year:

Unlike the previous thirteen meetings, this year’s Lambeth conference will not feature large sessions ordered toward producing resolutions. It will instead consist of small discussion groups (called indaba, a Zulu term for “gathering”) aimed toward the eventual production of a communal “Reflections” document.

Some have been skeptical of the intent and effect of the new design. There may well be sound reasons for the change; it can be difficult for all voices to be heard in enormous parliamentary sessions. But some are concerned that the new design will actually have the effect of preventing any outcomes such as the ones at the 1998 meeting, where the numerical strength of Global South bishops led to the passage of resolutions that were unpalatable to Northern liberals.

Perhaps I’m being too uncharitably cynical, but when compared to what I’ve seen in the RC neighborhood regarding innovations (very) allegedly dictated by Vatican II, I’m not surprised to see this kind of thing happen when you let the Lib…er, religious progressives get their foot in the door.

I owe to Instapundit the following link to the book, All Known Metal Bands. Check out the product description:

Product Description

This volume contains the names of over 50,000 metal bands. Presuming that each of these bands had an average of four members, and multipling that by the number of bands, one might figure that at least a quarter of a million humans have pledged allegiance to one of them at some point is his or her lifetime. Never has a genre of music relegated to the underground of a civilization had so many devotees; no radio needs to transmit the power of this music, for it is sought out fiercely and freely by the doomed and the dispossessed, whose ears are never soiled by songs of love and weakness.

These names are invisible tokens to be spoken aloud, each representing a human quest for superhuman spectacle: shaking floorboards and quivering walls, split ears leaking blood, with faces painted and ornaments pointy, voices uttering eternal truths shunned by woman and man alike.

Theognis is the only writer represented in this volume whose poetry has come down to us by a regular manuscript tradition.  His works are to be found, in whole or in part, in more than forty manuscripts, the oldest and best of which belongs to the early 10th century.  We have almost 1400 lines of elegiacs, which are variously divided to form between 300 and 400 poems, most of them single couples, the longest two poems of 30 lines.  At last, the novice might think, the critic’s task is straightforward: he is dealing with compete poems instead of stray fragments and he can ply his trade in peace.  But alas!  the field of Theognidean studies is battle-scarred, strewn with theories dead or dying, the scene of bitter passions and blind partisanship. Welcker in1826 divided the poems into a small corpus of ‘genuine Theognis’ and a large mass of poetry by other writers, earlier and later.  Separatists of various shades of opinion held the field till 1902, when Harrison published a vigorous defence of the unity of the  corpus, and since then combat has been continuous, except for interruptions due to real wars. (emph. mine)

-D. A. Campbell, Greek Lyric Poetry. 343-344.

…it would probably look like this.

In other news, I’ve been pretty remiss in updating this, mostly because I haven’t felt like I’ve had anything interesting to say in a while (…as though that dwarf/gnome post a little while ago were exemplary of  my standards for “anything interesting”…but I digress).  But I’ll mark out for myself two points to follow up on before I go off abroad again:

1) A reasoned explanation of why I support John McCain and will vote for him.

2) A few comments on the supremely tragic historical similarities between the Muslim conquest of Spain and the Spanish conquest of Mexico.

Hmm…

Would it be too thickly ironic to suggest a plot to blow up Parliament if demands should not be met?

Why don’t I have a record contract?

http://www.youtube.com/watch?v=wvkHIZg_954

I once met a wise dwarf who was fond of making gnomic statements.

In an earlier post, I opined that a large problem for American society is that social space occupied by the concept of “Taste” has been usurped by Radical Free Speech and PC Speech.  I think I’ve hit upon a slightly more coherent formulation of that idea.

We can define “Taste” as “Variable Self-Censorship”, wherein the variability is a function of one’s awareness of one’s social circumstances.  Radical Free Speech advocacy seeks to nullify “Censorship” part, whereas PC Speech attacks the “Variable” part.  The latter might need an extra word of explanation: Political Correctness is all about the increased codification of speech, which in turn decreases the range of legitimate expression available to the individual(*).  Both attacks, however, decrease the space available to individual for expressing himself on the basis of his perceived relationship with the social network in which he operates.  In other words, the modes of personal expression advocated by Radical Free Speech and PC Speech encourage solipsism and mechanism, respectively.  The basis of Radical Free Speech is the isolated individual, and the basis of PC Speech is the universal impersonal code.  As a result, neither fosters actual interpersonal engagement or understanding.

(*) Ok…so I’m aware that this is a blunt over-generalization, esp. in light of the potential counter-examples from societies founded on a highly structured sociolinguistic apparatus (cf. levels of politeness in Japanese).  But even there, one could make the argument that such highly structured speech conventions have over time contributed to the minimization/marginalization of the role of the individual in Japanese society.

If I had to pin myself down to a general conceptual area, I would say definitions. Synthesis might come in second—but I suppose that might just be another way of saying I like to steal a whole bunch of other people’s ideas, throw them together with a few alterations, and put my name on it. But hey, it seems to have worked for some people (cf. Alcaeus fr. 332 (Lobel-Page)). I’ve never really considered myself a particularly inventive or original person, though, and that’s frequently been a stumbling block for me when engaging in recreational mathematics.

Creativity tends to be associated with disciplines in the humanities, lately more and more with the sciences (cf. Einstein’s ubiquitous quote that “Imagination is more important than knowledge.”), but almost never with mathematics, probably on account of the fact that few people pursue the discipline further than the socially pragmatic function of numeracy would require. But even apart from that, the sheer invention upon which a fair amount of pure mathematical problem solving depends is difficult. When working proofs, success often hinges upon the ability literally to “build” a mathematical object with the proper attributes (Kudos to Prof. Burger for teaching me the basis of this). Some people find this pretty easy; I don’t. But I do find it incredibly fascinating to see it in action, esp. once the object has been built and the desired properties seem to come tumbling out of it like magic.

The examples I’m most acquainted with come from Number Theory, so here are two number theoretical proofs from this book.

-Prove that there are infinitely many prime numbers.

This elegant proof goes back to Euclid and can usually be found in any introductory book on Number Theory:

1) Assume that the set of prime numbers, P, is finite: P = {p_1, …, p_x} for some integer x.

2) Build the integer M, the product of all primes plus one: M = (p₁p₂…p_x) + 1.

The integer M then has no prime factorization, since the division of M by any of our assumed finite primes leaves a remainder of one: For 1< n < x, M/p_n is not an integer. This violates the Fundamental Theorem of Arithmetic, which states that every integer greater than one must have a unique prime factorization. Therefore, our initial assumption is false, and the number of primes in infinite.

As you can see, the proof hinges upon the properties of the integer M, which was built in such a way as to demonstrate whether or not our assumption was false.

A related proof that is much more interesting comes from the exercise section:

-Modify the proof of infinite primes in order to show that there exist an infinite number of primes congruent to five modulo six.

As before, we start off by assuming that the number of primes congruent to five modulo six is finite: Q = {q₁, …, q_r}, where for 1< n < r, q_n ≡ 5 (mod 6).

Off the bat, I’ll admit that I had to look in the back to get a hint, and it gave me the proper object to investigate, namely, M = (6q₁…q_r) – 1. In hindsight, it seems pretty obvious. First, you need to account for all the primes modulo 6, which suggests dealing with their combined product as in the prior proof. Then, you need it to be congruent to 5 mod 6, so that suggests multiplication by 6 and subtraction of 1 for the purposes of factoring out the modulus. It all seems so simple! I guess I just need more practice at that kind of thinking.

Anyway, once you’ve got your object M, the rest just kind of falls out of it. We have built M specifically to be congruent to 5 (mod 6). Now, M > q_r , which means that M cannot be prime, since we have assumed that q_r is the largest prime congruent to 5 (mod 6). Our task is to prove that M cannot be composite, thereby violating the rule that an integer must be either prime or composite.

₁₎ Start off by considering the hypothetical prime factorization of M:

M = p₁^a p₂^b … p_s^g. Then, it’s clear that M = (6q₁…q_r) – 1 = p₁^a p₂^b … p_s^g_.

2) No element of Q can be a prime factor of M, since that division of M by any member of Q would leave a non-integer remainder. Therefore, for all p_m, such that 1< m < s, p_m is not congruent to 5 (mod 6).

3) But since M is odd, all of its prime factors must be odd primes, which means that for all p_m, 1< m < s, either p_m ≡ 1 (mod 6) or p_m ≡ 3 (mod 6).

4) Now, for all integers z > 1, 3^z ≡ 3 (mod 6).

5) Let ‘x’ be the sum of the exponents on the prime factors of M that are congruent to 1 (mod 6), and let ‘y’ be the sum of the exponents on the prime factors of M that are congruent to 3 (mod 6). Then,

M = p₁^a p₂^b … p_s^g ≡ 1^x3^y = 3^y ≡ 3 (mod 6).

This contradicts the previously established fact that M ≡ 5 (mod 6); therefore, we conclude that our assumption was false and that there is an infinite number of primes congruent to 5 (mod 6). Q.E.D.

I was pretty stupid and insecure in college.  I think this fairly sums up my experience on both ends of the spectrum.