Monday, February 14, 2011

The Law of Conservation of Mathematical Symbols

One of the frustrating things about math is that some symbols are reused again and again, as if someone was afraid of running out of symbols.

For example:

1. ' can mean a derivative of a function, or the transpose of a matrix or vector. This is confusing when a book refers to the derivative of a vector. Hmm, which is it?
2. | | can mean the absolute value of something, or the determinant of a matrix. This is confusing when transforming random variables and you need to refer to the absolute value of a Jacobian determinant.
3. ~ can mean "scales as" or "is distributed as."
4. Σ can mean the operation "sum the following" or the item "covariance matrix."

I'm reminded of Demetri Martin's line about vitamins: "When they were naming vitamins they must have thought there were going to be way more vitamins than there ended up being. OK let's name these: Vitamin A, Vitamin B...ok man slow down we've got a lot to cover here. B2, B3, B4, B5, B6, B12. Then they got to E and they were like 'We're pretty much done. We've got all those damn B's. This is embarrassing. Let's just skip to K and get the hell out of here."


Saturday, February 05, 2011

Good article

My good friend and law school classmate Mark Rienzi (of Catholic Univ. School of Law) has a new article on the horizon. It argues as follows:
The Fourteenth Amendment rights of various parties in the abortion context – the pregnant woman, the fetus, the fetus’ father, the state – have been discussed at length by commentators and the courts. Surprisingly, the Fourteenth Amendment rights of the healthcare provider asked to provide the abortion have not. Roe and Casey establish a pregnant woman’s Fourteenth Amendment right to decide for herself whether to have an abortion. Do those same precedents also protect her doctor’s right to decide whether to participate in abortion procedures?

The Court’s substantive due process analysis typically looks for rights that are “deeply rooted” in our history and traditions. Accordingly, this article addresses the historical basis for finding that providers do indeed have a Fourteenth Amendment right to refuse to perform abortions. This historical analysis shows that the right to refuse passes the Court’s stated test for Fourteenth Amendment protection. In fact, the right to refuse actually has better historical support, and better satisfies the Court’s stated tests, than the abortion right itself.

Beyond this historical case, a healthcare provider’s right to make this decision also fits squarely within the zone of individual decision-making protected by the Court’s opinions in Casey and Lawrence v. Texas, and protects providers from the types of psychological harm that the Court recognized in Roe and Casey. For these reasons, under Roe and Casey, a healthcare provider has a Fourteenth Amendment right to refuse to participate in abortions.
Worth checking out.

Wednesday, February 02, 2011

C.S. Lewis on 16th Century English

I've been slogging my way through C.S. Lewis's book English Literature in the Sixteenth Century, which is the only Lewis book that I haven't yet managed to read. It's been slow going, and I now understand why Lewis sarcastically referred to the book as OHEL (i.e., the Oxford History of English Literature), pronounced "O Hell."

The book is, of course, immensely learned, as one might expect (Alan Jacobs notes that “Lewis read every single sixteenth-century book in Duke Humfrey’s Library, the oldest part of Oxford’s great Bodleian Library”). But that's what makes it so difficult. The book is written as if you, the reader, are already familiar with all of the authors, political movements, theological and philosophical developments, etc., from the 16th century, and therefore don't need much explanation. Thus, nearly every page has numerous sentences that throw out names and terms in passing that are left undefined and unexplained, as Lewis immediately moves on to something else. (An example: "Wilder and more 'eldritch' even than this is the Dreme of 'Lichtoun Minocus' (Bannatyne CLXV); a dream, which has, for once, no allegorical significance.")

The book is nonetheless enjoyable if only for those flashes of insight that I do understand.

For example, this passage:
It may or may not have been noticed that the word Renaissance has not yet occurred in this book. I hope that this abstinence, which is forced on me by necessity, will not have been attributed to affectation. The word has sometimes been used merely to mean the 'revival of learning,' the recovery of Greek, and the 'classicizing' of Latin. If it still bore that clear and useful sense, I should of course have employed it.

Unfortunately, it has, for many years, been widening its meaning, till now 'the Renaissance' can hardly be defined except as 'an imaginary entity responsible for everything the speaker likes in the fifteenth and sixteenth centuries'. If it were merely a chronological label, like 'pre-Dynastic' or 'Caroline' it might be harmless. But words, said Bacon, shoot back upon the understandings of the mightiest. Where we have a noun we tend to imagine a thing. The word Renaissance helps to impose a factitious unity on all the untidy and heterogeneous events which were going on in those centuries as in any others.

Thus the 'imaginary entity' creeps in. Renaissance becomes the name for some character or quality supposed to be immanent in all the events, and collects very serious emotional overtones in the process. Then, as every attempt to define this mysterious character or quality turns out to cover all sorts of things that were there before the chosen period, a curious procedure is adopted. Instead of admitting that our definition has broken down, we adopt the desperate expedient of saying that 'the Renaissance' must have begun earlier than we had thought. Thus Chaucer, Dante, and presently St. Francis of Assisi, became 'Renaissance' men. A word of such wide and fluctuating meaning is of no value. Meanwhile, it has been ruined for its proper purpose. No one can now use the word Renaissance to mean the recovery of Greek and the classicizing of Latin with any assurance that his hearers will understand him. Bad money drives out good.

It should also be remembered that the word Renaissance is in a curiously different position from the general run of historical terms. Most of these, when not merely chronological, designate periods in the past by characteristics which we have come, in the course of our historical studies, to think distinctive or at least convenient. The ancients were not ancient, nor the men of the Middle Ages middle, from their own point of view. Gothic architecture was not 'Gothic' at the time, it was merely architecture. No one thought of himself as a Bronze Age man.

But the humanists were very conscious of living in a renascentia. They claimed vociferously to be restoring all good learning, liberating the world from barbarism, and breaking with the past. Our legend of the Renaissance is a Renaissance legend. We have not arrived at this conception as a result of our studies but simply inherited it from the very people we were studying.

If the earlier modern scholars had not themselves been bred in the humanist tradition it may be doubted whether they would have chosen so lofty a name as 'rebirth' to describe the humanist achievement. The event, objectively seen, would perhaps have appeared not quite so important nor so wholly beneficent.

But, once established, this glowing term inevitably linked itself in the minds of English scholars with those two other processes which they highly approved in the sixteenth century -- the birth of Protestantism and the birth of the physical sciences. Hence arose, as it seems to me, that strong prejudice, already more than once alluded to, which predisposed our fathers to see in this period almost nothing but liberation and enlightenment; hence, too, by reaction, and among scholars of anti-Protestant sympathies, the opposite tendency to see in it little else than the destruction of a humane and Christian culture by kill-joys and capitalists. Both views perhaps exaggerate the breach with the past; both are too simple and diagrammatic. Both thrust into the background things which were, at the time, important.
Better yet, consider this passage on discerning the "spirit" of historical periods:
Some think it the historian's business to penetrate beyond this apparent confusion and heterogeneity, and to grasp in a single intuition the 'spirit' or 'meaning' of his period. With some hesitation, and with much respect for the great men who have thought otherwise, I submit that this is exactly what we must refrain from doing. I cannot convince myself that such 'spirits' or 'meanings' have much more reality than the pictures we see in the fire.

Whether the actual content of the past or (less plausibly) of some artificially isolated period in the past has a significance is a question that need not here be raised. The point is that we can never know that content. The greater part of the life actually lived in any century, any week, or any day consists of minute particulars and uncommunicated, even incommunicable, experiences which escape all record. What survives, survives largely by chance. On such a basis it seems to me impossible to reach the sort of knowledge which is implied in the very idea of a 'philosophy' of history. There is also this to be said of all the 'spirits', 'meanings', or 'qualities' attributed to historical periods; they are always most visible in the periods we have studied least. The 'canals' on Mars vanished when we got stronger lenses.
And this was even better:
It is impossible not to wonder at this sudden extinction of a poetical literature which, for its technical brilliance, its vigour and variety, its equal mastery over homely fact and high imagination, seemed 'so fair, so fresshe, so liklie to endure'. . . . But however we explain the phenomenon, it forces on our minds a truth which the incurably evolutionary or developmental character of modern thought is always urging us to forget. What is vital and healthy does not necessarily survive. Higher organisms are often conquered by lower ones. Arts as well as men are subject to accident and violent death. . . .

We ask too often why cultures perish and too seldom why they survive; as though their conservation were the normal and obvious fact and their death the abnormality for which special causes must be found. It is not so. An art, a whole civilization, may at any time slip through men's fingers in a very few years and be gone beyond recovery. If we are alive when such a thing is happening we shall hardly notice it until too late; and it is most unlikely that we shall know its causes.
On a lighter note, some of the original sources that Lewis describes seem humorous, intentionally or unintentionally. Consider the following Scottish comic poetry from some time prior to 1501:
It relates (in rough four-beat couplets) the creation of the first Highlander. God and St. Peter were out for a walk in Argyll one day when St. Peter, observing a certain unsavoury object on the path, jokingly suggested that God might like to create something from it. One stir of the almighty 'Pykit staff' and "vp start a helandman, blak as ony draff'. Questioned about his plans, the new creature announced that he would be a cattle-thief. God laughed heartily, but even while He was doing so (it is like Mercury and Apollo in Horace's ode) the Highlander had contrived to steal His pen-knife.
Or this:
Indeed it is impossible to read Mulcaster long without smiling. Sometimes he meant us to smile. He recalls innumerable scenes in headmasters' studies when he writes of the father who will 'very carefully commend his silly poor boy at his first entry, to his maisters charge, not omitting euen how much his mother makes of him, if she come not her selfe and do her owne commendation' (Positions, IV). But he is quite serious when he recommends holding the breath as a beneficial physical exercise 'though all men can tell what a singular benefit breathing is' (ibid. XV) or writes 'Consider but the vse of our legges, how necessarie they be (ibid. XX).

In other Lewis news, I see that someone finally reissued Lewis's co-authored 1939 book The Personal Heresy: A Controversy, which had long been out of print. In fact, I corresponded a few times with Douglas Gresham (Lewis's stepson) several years ago to lament that very fact.