Monday, June 07, 2010

Represent!

If I were a Springer-Verlag Graduate Text in Mathematics, I would be J.-P. Serre's Linear Representations of Finite Groups.

My creator is a Professor at the College de France. He has previously published a number of books, including Groupes Algebriques et Corps de Classes, Corps Locaux, and Cours d'Arithmetique (A Course in Arithmetic, published by Springer-Verlag as Vol. 7 in the Graduate Texts in Mathematics).

Which Springer GTM would you be? The Springer GTM Test


(ht)

My knowledge of this field is so slight that the sum total of what I know is that it's part of a more general field, representation theory for groups, and that it's the part that represents finite groups using linear algebra. Or something like that.

Thursday, June 03, 2010

Links and Notes

I meant to mention it; yesterday Siris finished its sixth year.

* Stabilize the debt. An online simulator that allows you to play around with various proposals that have made for making the U.S. federal debt more manageable in order to see if you can bring the U.S. debt down to an estimated 60% of the GDP by 2018. It's interesting to try out different alternatives, and things often look somewhat different in final overview than they do when you take things one by one.

* A castle is being built in France using only methods available to the 13th century. (ht)

* Virtual tour of the Sistine Chapel. (ht)

* Soviet illustrations of Tolkien's The Hobbit. The Russian fairy-tale style is charming, and fits the tale quite well. (ht)

* Moon Zoo is a site where you can assist lunar researchers in their survey of craters on the moon. I love it when scientists do this. This sort of thing was advocated in the nineteenth century by the likes of John Herschel and William Whewell (Herschel recommended a low-tech version of precisely this sort of program for meteorology); there still were dreams of a world in which science was done not by scattered individuals but by societies, nations, civilizations. This is not what we actually got; for instance, by the end of the nineteenth century brilliant amateur natural historians like Beatrix Potter were being steadily frozen out of increasingly specialized scientific circles. One advantage of our age is that the increasing ease with which one can gather data on truly massive scales has forced sciences to reopen projects of this kind.

* Tim O'Neill reviews Agora.

* Christopher Donohue discusses Fustel de Coulanges. I'm reminded of an anecdote from Duhem's German Science, in which students were amused that whenever they would make a claim about some historical matter, Fustel de Coulanges would reply, "Do you have a text to support that?"

* Jimmy Akin discusses the Old Brass Wagon and how people dance in a culture that frowns on dance.

* Rod Bennett on Justin Martyr.

* Stephen Barr discusses neutrinos.

* Charles Lwanda and the Ugandan martyrs.

I will be busy attending a cousin's wedding this weekend, so there will probably be no more posts until Mondary or Tuesday.

Corpus Christi

Bread is broken on the table;
Cup now overflows with wine;
by this word the Word our Savior
is the substance of the sign.

Adam's flesh from fleshly Adam
freed from sinful flesh once more:
we are made, by blood and body,
flesh and blood with Christ our Lord.

Speak, my tongue, of His scourged body,
blessed and broken for our race.
This so priceless blood now flowing
pays our price and grants us grace.

Sing, my voice, the song of angels
wondering at God-filled tomb,
which, with side-sprung water flowing,
compassed us to be our womb.

Love, my heart, the changeless ancient
who descends from God above,
made a babe and passion's patient;
He is God, for God is Love.

Trust, my soul, in Truth most holy:
Truth is true and does not lie.
Free from lie, from lies He freed us;
see the sign Truth truly died!

Hope, my spirit in your Savior,
who is life, in dying lives,
who is given by the Father
as this bread that true life gives.

Shout, my sisters; shout, my brothers!
From the housetops make it known.
Tell the tale on every mountain;
own this well: you are His own!

Wednesday, June 02, 2010

Wounds of the Sad Uncomprehending Dark

Still Falls the Rain
by Edith Sitwell


The Raids, 1940. Night and Dawn.

Still falls the Rain---
Dark as the world of man, black as our loss---
Blind as the nineteen hundred and forty nails
Upon the Cross.

Still falls the Rain
With a sound like the pulse of the heart that is changed to the hammer-beat
In the Potter's Field, and the sound of the impious feet
On the Tomb:

Still falls the Rain

In the Field of Blood where the small hopes breed and the human brain
Nurtures its greed, that worm with the brow of Cain.

Still falls the Rain
At the feet of the Starved Man hung upon the Cross.
Christ that each day, each night, nails there, have mercy on us---
On Dives and on Lazarus:
Under the Rain the sore and the gold are as one.

Still falls the Rain---
Still falls the Blood from the Starved Man's wounded Side:
He bears in His Heart all wounds,---those of the light that died,
The last faint spark
In the self-murdered heart, the wounds of the sad uncomprehending dark,
The wounds of the baited bear---
The blind and weeping bear whom the keepers beat
On his helpless flesh... the tears of the hunted hare.

Still falls the Rain---
Then--- O Ile leape up to my God: who pulles me doune---
See, see where Christ's blood streames in the firmament:
It flows from the Brow we nailed upon the tree

Deep to the dying, to the thirsting heart
That holds the fires of the world,---dark-smirched with pain
As Caesar's laurel crown.

Then sounds the voice of One who like the heart of man
Was once a child who among beasts has lain---
"Still do I love, still shed my innocent light, my Blood, for thee."

You can hear Dame Edith read this poem about the Blitz at Poetry Archive. Benjamin Britten also did a famous musical setting for the piece; you can find Part One and Part Two at YouTube. (I don't think Britten's piece gets either the rhythm or the mood of the poem right.)

Every F is G, with Fewer G's than F's

I notice at Dale Tuggy's "Trinities" blog that the claim came up that "if every F is a G, then there can’t be fewer Gs than Fs." This claim goes back to Richard Cartwright's old paper on the Trinity, and is often presented as if it were self-evident.

It's notable, though, that it's not difficult to find apparent counterexamples. Suppose I have a list:

1. Mary Ann Evans
2. Samuel Clemens
3. George Eliot
4. Mark Twain

If you ask me how many things are itemized on my list, it's entirely reasonable to say that there are 4 things itemized on my list. I can also say, "Everything itemized on my list is a real person." But how many real people are on my list? Only two; each one shows up twice. The F's here are things itemized on my list; the G's are real people; every F is a G; but there are fewer G's than F's.

Let's take an example that's closer to home. On a typical statistics counter for a webpage or blog, there is a counter for what are called "Unique Visitors." Unique visitors are counted by determining what number of visits (which are different from the number of views) has been made by way of that particular computer. Now it's true that every unique visitor is someone visiting your site, at least, if one sets aside bots (which many statistics counters do). But the number of unique visitors has no particular relation to the number of people visiting your site; if a single person visits your site from two different computers, they will be counted as two unique visitors. On the other hand several different people using the same computer to visit your site will register as the same unique visitor. The former characteristic, the ability of one person to be more than one unique visitor, is the one of interest here. Setting aside bots again, every unique visitor is a person visiting your website. But you cannot conclude from this that there can't be fewer F's than G's.

When faced by this people will usually reply by saying something like, "Well, look, the four things on the list are really not people; they are names. And the unique visitors are really not people; they are identifiable computers." This is an attempt to remove all intensional and modal factors in counting; but it needs to be noted that this is already a restriction of the original claim. In practice we do not treat counting in purely extensional terms; how we describe or know things is important. I know my list counts people; but I may not know that it fails to count each person uniquely. And it is entirely possible, as with the unique visitor count, to count things in ways that do not count them uniquely. Indeed, there are many cases where one might do this quite deliberately. Setting aside a few odd cases, every airline passenger is a person, but the same person can get counted by airlines as different passengers. This makes a lot of sense, because the thing that primarily matters for airlines is not the number of people but the number of passengers.

Thus the original claim really boils down to the claim, "When every uniquely identifiable and separate F is a uniquely identifiable and separate G, then there cannot be fewer G's than F's." Every dog is a mammal; but, more than this, this can be glossed in straightforwardly extensional terms: each separate, individual thing that is a dog is exactly the same as a separate, individual thing that is a mammal. Given this, there cannot be fewer mammals in a room than there are dogs, because each separate dog will count as a separate mammal. In such a case the claim really is self-evident. But once we are in territory where different ways of counting have different intensional and modal features, it not only is no longer self-evident; it's provably false. The claim can only be saved by restricting it to cases where there is either only one way of counting or, if there are several ways of counting, if their results are directly convertible into each other (i.e., their results map onto each other perfectly). But there are plenty of areas of life where we are interested in how ways of counting that are not directly convertible relate to each other.

Tuesday, June 01, 2010

St. Justin, Martyr

Around about 90 BC Antiochus of Ascalon reacted against the skepticisms of the New Academy by founding the Old Academy. The name 'Old Academy' and the peculiarity of its being newer than the New Academy is quite deliberate; Antiochus regarded it as intolerable that the School of Plato was so very un-Platonic, and thus advocated a return to Plato. Such reform movements are never merely reactionary; rediscovery in the midst of opposition to one's project is inevitably innovative. The new movement drew from Peripatetic and Stoic sources in order to defend itself, attack Academic skepticism, and form alliances with other positions. Like any reform movement, it simultaneously identified itself with a past purity -- the Old Academy of Plato and his first three successors -- and used this as a framework for developing new solutions to more contemporary problems. The resulting project was Platonic and Peripatetic and Stoic, which while eclectic is less eclectic than it is usually made to sound, because returning to the root made it possible to find genuine commonalities. The Peripatetics were an early offshoot of the Academy, through Aristotle; and many of the ideas we associate with Stoicism first found their basic form in the successors of Plato.

We don't have much on Antiochus, although what we do have suggests that he was better at articulating a vision than developing it. But the challenge was taking up by many others over the next several centuries, the most important of whom were Eudorus of Alexandria, Plutarch of Chaeronea (the best known of the three), and Philo of Alexandria. The third of these is especially intriguing and important, for Philo was a Jew (hence the other name he is known by, Philo Judaeus). Philo's integration of Jewish belief and Middle Platonism would set the stage for centuries to come. For Christianity partly grew up in the crucible of Middle Platonism, of the Philonic type.

There are parts of Paul's letters and the Epistle to the Hebrews in which one might see such influence, but the influence is most marked in the Gospel of John (and it is notable that Hellenistic Jews come up explicitly a couple of times in the book, e.g., 7:35, 12:20ff). One of the most important means Philo had used to integrate Jewish and Greek thought was his developed account of the Logos. Logos, reason or word, was a common term in Greek philosophy, and was associated with the divine; it also occurs quite often in the Septuagint to refer to God's word. On Philo's account the Logos is a power in God; it is the divine Mind containing the Platonic Ideas and is active and effective as God's Word:

[I]n the one living and true God there were two supreme and primary powers, Goodness [or Creative Power] and Authority [or Regent Power]; and that by his Goodness he had created every thing; and that by his Authority he governed all that he had created; and that the third thing which was between the two, and had the effect of bringing them together was the Logos, for it was owing to the Logos that God was both a ruler and good.(Cher 1.27-28, qtd here)

It's the general Philonic idea of divine Logos that is in the background of the opening words of the Gospel of John:

In the beginning was the Logos
and the Logos was with God
and the Logos was God.
The same was in the beginning with God.
Through him all things were made;
without him nothing was made that has been made.
In him was life,
and that life was the light of men.
The light shines in the darkness,
but the darkness does not comprehend it.

And, he continues, the Logos became flesh and dwelt among us. Thus Christianity from its early days built on the vocabulary, or, rather, vocabularies, of Middle Platonism.

It is then unsurprising that many Middle Platonists found Christianity to have its attractions, and this brings us to the saint whose feast-day it is, Justin the Martyr. For Justin was born to a pagan Greek family in Samaria (as he calls himself, he was "Justin, the son of Priscos, son of Baccheios, of Flavia Neapolis, in Palestinian Syria") in the early second century. He became a philosopher of Middle Platonist stamp, drawing now from Stoicism, now from Pythagoreanism, now from Aristotelianism, eventually making his way to Ephesus. He converted at some point to Christianity, in part, it seems, because he found them to be less morally objectionable than people in other schools of philosophy (particularly in terms of their courage and charity to others); and much of the argument in his works is that philosophy, especially as conceived by Socrates and Plato, finds its natural culmination in Christianity. Both philosophy and Christianity share in their love of Logos, but with Christians holding that Logos so reciprocated as to become man; he explicitly mentions Socrates as a proto-Christian in his argument that it is unjust for the Empire to persecute Christians:

We have been taught that Christ is the first-born of God, and we have declared above that He is the Logos of whom every race of men were partakers; and those who lived reasonably are Christians, even though they have been thought atheists; as, among the Greeks, Socrates and Heraclitus, and men like them; and among the barbarians, Abraham, and Ananias, and Azarias, and Misael, and Elias, and many others whose actions and names we now decline to recount, because we know it would be tedious. So that even they who lived before Christ, and lived without Logos, were wicked and hostile to Christ, and slew those who lived according to Logos.

We don't know the full details of how it is that Justin became a martyr; certainly he was never hesitant to insist in public that Christians were right, although he is largely quite irenic about pagans. There is some indication that in his wandering about first Ephesus and then Rome engaging in philosophical arguments he may have made enemies; he explicitly says at one point that he expected it to happen, and even named one Crescens, a Cynic philosopher, as a likely person to cause problems for him. Thus Crescens perhaps did get to him, or someone like Crescens, although we don't really know for sure. We do know that he was tried with six others in a standard trial under the prefect Rusticus during the persecutions under Marcus Aurelius and was put to death. It is perhaps an irony that the parataxis, the obstinacy, that the philosopher Marcus Aurelius loathed in Christians, the philosopher Justin exalted as showing the paradoxical summit of a good life, a life that put truth before life. The emperor probably never knew the teacher, but for different reasons they would both have considered Justin's death appropriate to his views.

Justin lived in the evening of Middle Platonism. In the very next century Plotinus for the pagans and Origen for the Christians will begin to push for major innovations, and the Neoplatonic phase of philosophy will begin. Much of Middle Platonism will survive, but the doctrines will become less eclectic, more systematically integrated, at times reinterpreted; philosophical criticism of other positions will become more developed and thorough; and the questions about the relation between philosophy and religion will become a bit more complicated than they were in the days of Justin and Clement of Alexandria and others of their kind who took Christianity to be divine philosophy. But the role of Middle Platonism in early Christian life, as witnessed by people like Justin, would have significant consequences for centuries to come.

Surf of Fire

Love in a Hut
by Roy Campbell


Maternal Earth stirs redly from beneath
Her blue sea-blanket and her quilt of sky,
A giant Anadyomene from the sheath
And chrysalis of darkness; till we spy
Her vast barbaric haunches, furred with trees,
Stretched on the continents, and see her hair
Combed in a surf of fire along the breeze
To curl about the dim sierras, where
Faint snow-peaks catch the sun's far-swivelled beams:
And, tinder to his rays, the mountain-streams
Kindle, and volleying with a thunderstroke
Out of their roaring gullies, burst in smoke
To shred themselves as fine as women's hair,
And hoop gay rainbows on the sunlit air.