2449 Uinta St.
Denver, CO 80238

We are traveling out of the country until June 27, but after that date, we will both have new mobile phone numbers:

Michael’s mobile: (303) 242-1086
Emmy’s mobile: (303) 550-5669

1. Timothy Gowers, “The Importance of Mathematics”, Part 1
2. Timothy Gowers, “The Importance of Mathematics”, Part 2
3. Timothy Gowers, “The Importance of Mathematics”, Part 3
4. Timothy Gowers, “The Importance of Mathematics”, Part 4
5. Timothy Gowers, “The Importance of Mathematics”, Part 5
6. Timothy Gowers, “The Importance of Mathematics”, Part 6
7. Timothy Gowers, “The Importance of Mathematics”, Part 7
8. Timothy Gowers, “The Importance of Mathematics”, Part 8

The second video I recently watched and enjoyed is a short documentary about Andrew Wiles’ proof of Fermat’s Last Theorem. Watch the introduction, see Professor Wiles almost break down in describing his discovery, and be hooked. This video is also available on YouTube in five parts:

1. UKTV Documentary, “Fermat’s Last Theorem”, Part 1
2. UKTV Documentary, “Fermat’s Last Theorem”, Part 2
3. UKTV Documentary, “Fermat’s Last Theorem”, Part 3
4. UKTV Documentary, “Fermat’s Last Theorem”, Part 4
5. UKTV Documentary, “Fermat’s Last Theorem”, Part 5

NP: Modest Mouse, Neverending Math Equations

The line is the motto of a secret organization of noble volunteers in the Series of Unfortunate Events by Lemony Snicket. It is also, incidentally, eerily reminiscent of the first line of a poem by Swinburne that Snicket quoted in the tenth book.

It seems that my girlfriend Emmy is also not (entirely) ashamed of my reading habits. About two weeks ago, we went to paint pottery at a place here in Cambridge. I painted a thoroughly unartistic and unimpressive teacup, but Emmy made me the loveliest mug I’ve ever seen:

Doesn’t she rock?

NP: The Gothic Archies, Smile! No One Cares How You Feel

Athens was a city in decline. Its hegemony had been eclipsed by that of Sparta, and its spirit was flagging. It had suffered a terrible defeat at Aegospotami by the Lacedaemonians, and its democratic ideals had been severely circumscribed by thirty tyrants who controlled the city for almost a year. Many of Athens’ heroes had been disgraced, perhaps most notably Alcibiades.[1] Athenian supremacy was no longer certain, and the traditional ideals were everywhere being challenged. Sacrifices were necessary to regain the graces of the gods.

At the beginning of the fourth century B.C.E., Socrates was charged with impiety and corruption of the youth. Socrates was an ugly,[2] stubborn sage who walked without shoes, fought bravely in battle and taught students in the marketplace. He was offensive in nearly every aspect. He was, by his own admission, unpopular among the major public figures of the time,[3] principally because he adopted the practice of demonstrating publicly that the reputedly wise knew far less than they claimed. He embarrassed politicians, craftsmen and poets, and degraded Athens’ most prominent citizens. Socrates was a novelty, an entirely new type of common man who defied the vestiges of nobility among the aristocracy.[4] Socrates was an archetype of the new Athenian, a convenient symbol of the ascendancy of reason over charm and force. Indeed, Aristophanes’ famous lampoon of Socrates had little of the actual man in it—with the exception, perhaps, of his unsightliness—but used him as a symbol of the philosopher, ascribing to him (remarkably reasonable) naturalistic philosophies and suggesting he collected fees for his teaching.[5] But why would a citizen of Athens unknown to Socrates bring a public suit against him, seeking his death as punishment, and how could a jury of over 500 citizens decide to impose such a penalty?

In answering this question, we are faced with a problem: most of our primary evidence of the trial comes from two students of Socrates, Plato and Xenophon; the former was the nephew of Critias, the head of the Thirty, and the latter fought against Athens. Xenophon’s account of the trial aftermath is a report of someone else’s report; the record before us is hearsay composed of hearsay, and the principal authors had an interest in the dispute. We cannot put to test the testimonial capacities of the witnesses, we cannot question their memory of the events, we cannot effectively challenge their sincerity or honesty. And indeed, Socrates faced similar problems at his trial. He had been regularly “accused” in the public sphere for many years before charges were brought against him. Aristophanes’ play Clouds was produced almost twenty-five years before the trial. As Socrates says of his old public accusers in Plato’s account of the trial, “one cannot bring one of them into court or refute him; one must simply fight with shadows, as it were, in making one’s defence [sic], and cross-examine when no one answers.”[6] If Plato may be given credence, Socrates simply denied the truth of these public slanders, and explained how his reputation might have arisen. Socrates suggests that the very people who have been embarrassed by his questioning sling out of bitterness insults that are available against all philosophers.

Socrates was able to cross-examine his new accuser Meletus, but he did not test his capacities as a witness. Socrates had been charged with impiety and corruption of the youth, and his examination of Meletus suggested merely that he did not understand the charges.[7] Meletus was never questioned about his recollection of particular events, or about his sincerity in recounting them. His honesty was impugned, but not for telling lies about the facts; Socrates claimed that Meletus brought the charges either to test the jury or because he couldn’t come up with any true wrongdoing with which to accuse Socrates. Socrates did point out that Meletus failed to bring witnesses to attest to any harmful teachings, but whatever evidence was introduced against Socrates seems to have gone largely untested.

Unfortunately, we have little indication of what evidence might have been presented. Still, we might surmise what could have persuaded a jury of Athenian citizens to convict Socrates. Voltaire suggests that Socrates’ death was orchestrated by a couple of scheming public officials who feared that Socrates could expose their corruption;[8] this idea is not without support in Plato’s account. But this explanation of the prosecution is insufficient, as it fails to suggest why the jury would uphold such charges. The contention must be either that many in the jury were themselves fearful of Socrates, or that there was evidence that charges had some legal merit to them.[9] Most likely, members of the jury perceived Socrates as a real danger, or perhaps as a symptom of worrisome cultural change. Few could ignore that two of Socrates’ students (Critias and Charmides) were among the chief oligarchs placed in power by the Spartans, or the defection of Socrates’ favorite Alcibiades, or Xenophon’s opposition to Athens during the war. But even more than this, Socrates may have been seen as a repudiation of Athenian honor and tradition.[10] Through his dialectic, Socrates challenged Athenian beliefs about the gods,[11] about the very nature of piety and honor, and left them with no counter-argument except to silence him. Socrates posed a danger to tradition; conveniently, he was also a very public exemplar of the dialectician, and made for a fitting sacrifice to the gods, to Athenian tradition. As to the actual charges: it is an easy equivocation between viewing Socratic dialectic as a danger to spiritual traditions and believing that Socrates was himself impious, whatever that might have meant. Moreover, Socrates himself admitted that he influenced the youth to follow his example;[12] if his example is anathema, there is little question of his corruption of the youth.

What defense could Socrates possibly mount in the face of such charges? Surprisingly, he didn’t directly challenge either charge. He admitted that he experienced a divine sign, essentially affirming the charge that he introduced new gods. His only real defense to the charge of corrupting the youth was the claim that he himself would suffer if he made his neighbors worse, and so would never intentionally do so. He never refuted the claim that his conduct was corrupting, except to point to a lack of evidence. Even worse, he repeatedly reminded the jury that few people would complain if he were eliminated.[13] If the jury wanted to render an acceptable verdict, it could hardly go wrong by convicting him. But he did give two reasons supporting acquittal. First, he argued that his accusers brought the charges in bad faith. Second, he offered an alternative narrative—rather than being a symptom of Athens’ decline, he put himself forward as a cure. He argued that his challenge to tradition was good and healthy for the city. He said, “Indeed, gentlemen, I am far from making a defence [sic] now on my own behalf, as might be thought, but on yours, to prevent you from wrongdoing by mistreating the god’s gift to you in condemning me; for if you kill me you will not easily find another like me.”[14] He compared his role in questioning the citizens to a gadfly that stirs a sluggish horse to action. He spurred his interlocutors to recognize their faults and to exercise some humility with respect to them. This, he argued, would make them better citizens. Thus, he put before the jury the question of whether they recognized the service he performed for the city’s benefit.

In a sense, the trial had been going on for decades, and Athens finally had occasion to render a decision. The question put before the jury was whether Athens wanted to encourage this new mode of examined life, or to cling to the old ways. The dikastic courts were not well suited to weigh arguments—no time was allotted for deliberation—but it could render the judgment of the Athenian people. They voted for death. Plato later doubted the wisdom of putting such a question to the jury: “[H]e may be the physician who is tried by a jury of children. He cannot say that he has procured the citizens any pleasure, and if any one charges him with perplexing them, or with reviling their elders, he will not be able to make them understand that he has only been actuated by a desire for their good.”[15] But Socrates knew going into the trial that he faced such a problem. He intentionally avoided saying things he knew would persuade the jury to find for him. For Socrates, death was preferable to any judgment that left him unable to continue his mission to challenge the reputedly wise. The jury told him that it would be unacceptable for him to continue, so Socrates drank the hemlock.

Emmy gave me tulips Tuesday because… well, because she’s wonderful like that.

NP: Colin Hay, Beautiful World

The theory begins with the daring question of what it means for a proposition to be true. Philosophers, of course, have never shied from the difficult questions (except, of course, that they almost always shy away from them—by refusing to acknowledge them as questions, or by answering slightly different questions, or by attacking the questioner’s integrity, or by offering asinine answers to them, in earnest, with straight and serious mien—see, e.g., Socrates’ definition of “justice” in Plato’s Republic). Many people trace the Correspondence Theory as far back as Aristotle, who brilliantly stated that stating what is, is, is “true”, and stating that what isn’t, is, isn’t. Metaphysics 1011b25 (my translation). Presumably, Aristotle would have stated that his statement is true, which would have been, if his theory were true, a true statement.

We would have to wait more than a thousand years for a more intelligible articulation of the theory. It was the indomitable and infallible Aquinas who wrote: “Veritas est adaequatio rei et intellectus” [truth is the equation of the thing and the intellect]. De Veritate Q.1, A.1&3. That is, a statement is true precisely when it corresponds to the fact of the matter. To put it still more bluntly, a statement is true if and only if it is not false. Aquinas’ accomplishment lay chiefly in shifting the discourse from the outré Greek to the far more civilized (and easier to translate) Latin tongue. Clearly we have made progress.

There was, however, still much progress to be made. In 1921, Wittgenstein published his groundbreaking Tractatus Logico-Philosophicus, which consists of a mere seven statements (with explanatory notes). The principal statements were as follows:

1. The world is all that is the case.

2. What is the case—a fact—is the existence of states of affairs.

3. A logical picture of facts is a thought.

4. A thought is a proposition with a sense.

5. A proposition is a truth-function of elementary propositions.
   (An elementary proposition is a truth-function of itself.)

6. The general form of a truth-function is [p, ξ, N(ξ)]. This is the general form of a proposition.

7. What we cannot speak about we must pass over in silence.

Wittgenstein, Tractatus Logico-Philosophicus (tr. D.F. Pears and B.F. McGuinness). One might well wish that Wittgenstein had heeded his final statement, and passed over the Correspondence Theory with a more determined silence, but he advanced the field considerably, as one can see when one perceives how the constituent statements are related to each other. [I note in passing that Wittgenstein referred to the Necker cube in explaining his fifth statement, writing: "To perceive a complex means to perceive that its constituents are related to one another in such and such a way. This no doubt also explains why there are two possible ways of seeing [the Necker cube] as a cube; and all similar phenomena. For we really see two different facts.” Id. at 5.5423.] Taken as a complex, the Tractatus suggests that a truth-function with a sense is a logical picture of the world, or of a fact, or of a state of affairs. Propositions are true when they constitute a true picture of the world. We owe Wittgenstein a very great debt indeed for pointing out that only propositions with a sense may be deemed true or false; senseless propositions are merely publishable.

Modern philosophers, most recently led by David Armstrong, have refined Wittgenstein’s advancement still further. Now, rather than saying that propositions present a “picture” of the world, they talk about propositions “corresponding” to the world, or the facts, or the states of affairs. With this development, we were finally able to assign a name to the theory, an absolutely essential step for the fruitful continuation of a discourse. Unfortunately, it also highlighted the problem that propositions are not facts, facts are not propositions, and that it isn’t clear what it means for th’un to “correspond” to t’other. Armstrong saved us all from certain doom with the assertion of the almighty “truth-maker” (sadly, talk of “falsity-makers” is far less common). A “truth-maker” is whatever ensures that a certain proposition is true, i.e., that it corresponds to a certain fact or state of affairs. Thus, the new Correspondence Theory of Truth holds that a proposition (with a sense) is true precisely when there exists a “truth-maker” that makes it correspond to the fact of the matter. Obviously.

What is, perhaps, the most surprising development in all of this is that the Theory is not accepted by all who encounter its imponderable brilliance. The Theory has critics. Quite clearly, anyone who rejects the Theory on the idea that a proposition might be true even when it is false isn’t worth listening to (but, again, publication in Philosophy journals is hardly ruled out). Some philosophers have, however, begun to notice that facts and states of affairs may not be the same thing, and great battles are being fought over whether propositions are true when they correspond to facts, or whether correspondence to states of affairs is necessary. Many good people have died in the fight. Furthermore, Armstrong is not without his dissenters. While Armstrong suggests that truth and falsity are appropriate predicates only of propositions, for only propositions admit of internal truth-making relations, many philosophers believe that a theory of truth is also required, not just for propositions, but for assertions, beliefs, thoughts, ideas, judgments, statements, utterances, and sentences as well. It is doubtful whether anyone at all will survive if the fact/state battle is fought anew on each new battleground. It may simply be best for the species to abandon the Theory altogether.

NP: Screeching Weasel, My Brain Hurts

NP: Röyksopp, What Else Is There?

Update [9/6/2005]: I just switched servers, and I’m redoing all my photos, so many of the links here are broken. They will be for awhile. Apologies to anyone who clicks and gets the 404, but I figure you’re a pretty small audience, and school’s starting up, so I’m a bit busy now.

NP: The Alkaline Trio, Stupid Kid

NP: Palace, New Partner