Continuity of a line
I have just finished reading Professor Richard Dedekind's monumental essay on the theory of numbers, "Contunuity and Irrational Numbers", published in 1872.
In the essay the author defined continuity of a line, call L, as follows:
"I find the essence of continuity in the...following principle:
If all points of the straight line fall into two classes such that every point of the first class (call A1) lies to the left of every point of the second class (call A2), then there exists
one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions."
The author claimed that in making this statement, he was "utterly unable to adduce any proof of its correctness, nor has any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line."
It would be much appreciated if any one could let me know whether this statement is the formal definition of continuity (in fact, Dedekind derived the familiar epsilon-delta definition of continuity from the aforementioned concept of "cutting", famously known as the Dedekind cut); whether it could/should be treated as an axiom in geometry; and if not, whether it is provable as a proposition.
Pictures from Hong Kong
I have never been a hyper-nationalistic person like some of the flag-wavers out there. Nevertheless, I have to day that my hometown Hong Kong looks really great at night. Here's my proof:
http://www.pbase.com/accl/hong_kong
Banality of Evil
I talked to my friend Rachel tonight and in our conversation I mentioned Hannah Arendt's observation of the Eichmann trial in 1961 (for those of you who do not know, Eichmann was a Nazi war criminal who was tried and hanged in Israel. He was the first and only person executed in Israel).
I mentioned Arendt's thesis that Eichmann was evil not because of his anti-semitic fanaticism, but because of his banality as a bureaucrat. (i.e. He valued his promotion thru the bureaucratic ranks and subsequently the "meeting of the target" more than everything else) Consequently, he could have killed his own parents, if that means a wage-raise or promotion.
However, I failed to define banality during my conversation. What is banality? According to Arendt,
evil is a function of banality, which is the failure of normal people to take risk", or more simply, the failure of individuals to be introspective. There is, of course, different degrees of banality. Eichmann could always have resigned his job, and the risk of doing so was low. He could have remained silent. And it is
his banality that morally justified his death sentence, not his
Nazi fanaticism. (In fact, according to Arendt, his evil has nothing to do with Nazism: he would definitely have become an NKVD officer istead were he born in 1900 Russia, or an inquisitor under some medieval Pope should he find himself in 15th century
Spain...)
Many people, especially the leaders of Jewish communities, criticized Arendt, claiming that she unwittingly supported Eichmann's own plea, that he "merely followed the orders". I disagree.
Eichmann's evil, the evil of "following orders", is even more menacing than that of many of his more "enthusiastic" colleagues, and Arendt showed us why. If evil is merely the result of a form of radicalism, say, Nazism, it is context-specific. However, if it is the result of banality, it is universal. With the aid of
thousands of Eichmann, the Holocaust could easily replay itself at any time in any place. This has been proved throughout human history: Native Americans, Romanis, Armenians...
People were discomforted by Arendt's report because they see the shadow of Eichmann in themselves. When one condemn a man as evil as Eichmann, it is natural for one to hope that he is a horned monster who is so different from the rest of humanity that there is no way one would one day repeat what he had committed. Unfortunately, this might not be the case.
Eichmann was a regular, spineless functionary with a pathetic character, just like many civil servants we see. The first impression he gave Arendy was "not even sinister". We consider those regular joes we see good guys because so far they've been serving under a republican government (small "r") that is somewhat accountable to the "masses". But what if some day a Hitler-like maniac take over our world. Arendt claimed that
there would then be thousands of Eichmanns emerging among ourselves, competing for the favor of the "fuehrer".
This is, indeed, terrorizing. Functionaries today, war criminals tomorrow. It is really our duty to spread the virtue of
introspection, so that there will be more Karl Jasper and Martin Niemoller instead of Adolf Eichmann, or most Germans and Japanese during that dark era, for that matter.
Socrates of Chicago
I found an interesting article online, which is about Dr. George Anastaplo, a prominent legal scholar who graduated first in his class from Chicago. He was extraordinarily courageous and outspoken, and his teachings has inspired far more people than he realizes.
Among his classmates whom he outshined in class, are Abner Mikva, judge, congressman and White House counsel; Ramsey Clark, one-time attorney general under LBJ; and many others. Had he decided to comply to the will of the authorities of his day, he could have been sitting in a Supreme Court chambers today, rather than his dingy office in Loyola University Chicago. However, as the saying goes, "For what shall it profit a man, if he shall gain the whole world, and lose his own soul?" Anastaplo could have had the world, yet he valued his own soul much more...
It was he who first confronted the irrational McCrathyite mania of his day. It wasn't a hard thing to do: he simply had to tell his Bar Admission officers that it was not proper for them to ask him, "Are you a communist?"
It cost him 10 years of his youth, to appeal against the decision of the Illinois State Bar to deny him the law license, and to lose that appeal.
Like Socrates, he sacrificed a prosperous life, and like Socrates, he continues to highlight the follies of his fellow citizens.
The link is: http://www.realuofc.org/anastaplo/tribune.html
A great start
I have decided to start this weblog as an online journal where my friends and I could post our thoughts on any topic that interests us.
Personally, I am interested in many different things: from Egyptology to classical analysis, few area fails to attract my attention. The friends I invited to join me in creating this blog have very diverse interests as well.
We will post our essays, notes, reviews and random thoughts from time to time, reflecting our understanding of U.S. and international politics, American law, philosophy, mathematics, religion, history, music, movies and ourselves.
Hope you'll enjoy it!
