In mathematics, there is no political correctness, just correctness. A mistake is unlikely to offend or mislead many and for long. Hardy writes as one states mathematical conjectures: clearly, confidently, irreverently.
Mathematics is a young men's game. Most games are. But the ageing of one's mind need not entail an abrupt withdrawal from creation, and hence from life. Even if having practised mathematics has initiated a now retired mathematician into the beauty that is unmatched in other pursuits, he can still devote the remainder of his life to cultivating beauty in the non-mathematical world. Literature differs from mathematics only in degree, not substance. Flaws in the argument are easier to detect in mathematics. Literature's value resembles that of a mathematical conjecture with a proof sketch: possibly false, definitely incomplete, but inspiring. (Poetry is a conjecture without a proof sketch.)
Hardy denies to pure mathematics any utility other than its aesthetic pleasure, shared with art. Even though once-pure mathematics eventually finds practical applications, these unforeseen applications do not motivate pure mathematicians. Beauty, escapism, and competition do. A beautiful theorem is beautiful in its statement and in its proof, which link simply hitherto disparate ideas. An escape into mathematics is more credible than an escape into art. In art, one invents a world. In mathematics, one dreams of a world and then proves its existence.
The lucid minds who seek inapplicable beauty are not wasted. One would not wish to arrest the mutation of genes just because the natural selection were not forward-looking. Similarly, one should not arrest the work of a pure mathematician just because he is unable to foresee its applications. Insistence on applications will discourage him, instead of convincing him to change his topic or occupation.
"The noblest ambition is that of leaving behind one something of permanent value." Hardy believes that one can leave something of value only by associating one's name with a discovery. But name-recognition is not necessary for immortality. One can contribute to a civilization by enriching the lives of others in trite, anonymous ways, which will enable others to discover.
Hardy derides expositors and critics. Yet, a mathematical proof is the exposition of a theorem's statement. Intuition for a proof is its criticism, which can be viewed unfavourably, as the proof's corruption for the sake of the illusion of understanding, or favourably, as a puzzle explaining some steps in the proof and leaving intermediate steps to the reader. Even if criticism is the work of second-rate minds, it enables third-rate minds glimpse the beauty that would have otherwise been restricted to first-rate minds.
29 April 2010
18 April 2010
The Glass Menagerie
(Cambridge Arts Theatre, 17 April 2010)
The comfort of resignation is deceptive. The emotionally fragile but sane should not be housed together, in order not to aggravate their condition. The housing prescription must be observed most forcefully in families, where the shared melancholy disposition and the distrust for the outside world lead to the preservation of the streak that otherwise would have either killed or made its lonely bearer stronger---which is the risk worth taking.
The comfort of resignation is deceptive. The emotionally fragile but sane should not be housed together, in order not to aggravate their condition. The housing prescription must be observed most forcefully in families, where the shared melancholy disposition and the distrust for the outside world lead to the preservation of the streak that otherwise would have either killed or made its lonely bearer stronger---which is the risk worth taking.
28 March 2010
"Logicomix: An Epic Search for Truth" by Apostolos Doxiadis and Christos H. Papadimitriou (2009)
Madness is perseverance misapplied. Necessary to discover logical insights, perseverance—when applied to emotions and not moderated by reason—magnifies the emotions into madness. Madness is also in holding the outside world to the standards of certitude satisfied only by the world of one's imagination (e.g., mathematics, literature, music). Madness is the inability to accept ambiguity. The fear of madness is the fear of getting discouraged by the chasm between the desired and the attainable.
Knowledge progresses in two directions. Logic derives new implications of existing axioms. Logic also deduces the existing axioms from more elementary ones---or challenges their appeal. In social matters, axioms are tastes, selected by assessing the appeal of their implications, by introspection, and by analysing the lives of others.
Education consists in alerting individuals to axioms' malleability, in teaching to derive inferences from axioms, and in showing the limit of these inferences by pointing out that even a well-defined language can generate truths unprovable in that language. The existence of unprovable truths invites one to live, not just derive, one's life. If even mathematics must be an experimental science (i.e., computer science), so must be every intellectual pursuit.
Knowledge progresses in two directions. Logic derives new implications of existing axioms. Logic also deduces the existing axioms from more elementary ones---or challenges their appeal. In social matters, axioms are tastes, selected by assessing the appeal of their implications, by introspection, and by analysing the lives of others.
Education consists in alerting individuals to axioms' malleability, in teaching to derive inferences from axioms, and in showing the limit of these inferences by pointing out that even a well-defined language can generate truths unprovable in that language. The existence of unprovable truths invites one to live, not just derive, one's life. If even mathematics must be an experimental science (i.e., computer science), so must be every intellectual pursuit.
18 March 2010
"The Catcher in the Rye" by J. D. Salinger (1951)
Money helps one economise on thinking just as it spares one from housework. The poor learn to distinguish the nuances of a good's quality and to analyse their own tastes so as to identify a bargain. By contrast, the rich exploit the association between higher prices and higher quality, and channel thus conserved mental exertion into more rewarding pursuits. The habit of relying on that association, however, can leave transactions insufficiently examined when prices are dominated by social conventions and status, which may be irrelevant for many. Holden Caulfield is benevolently neglected by his parents, who buy him expensive education without trying it on first and then overlook the inflicted blisters.
The importance of wealth in helping isolate oneself from disagreeable strangers must not be underestimated. The wealth of the Western world accounts for its peace, which prevails not only because wealth educates and raises the losses from conflicts, but also because wealth makes it easier to be tolerant---by having insulated oneself from the tolerable.
For Caulfield, however, family wealth is a poor point of departure. His wealth compels to conform, expands choices but does not inform of their consequences, suggests anarchy as the sole cure where a less wealthy would apply hard work. Ascending the social ladder confers the responsibilities that can be overlooked by those born at the top. Caulfield squanders money in order to descend to the position from which he would feel comfortable starting.
Intelligent, perceptive, and immature, Caulfield is a critic who has not yet become an artist, which would require kindness, in addition to his tolerance of bores and beauties. He lacks kindness to those who are kind to him---whatever their motivation: politeness, unrequited love, or the expectation of profit. He needs the society of multiple generations and social classes---not just his peers---in order to motivate his creativity. His character requires nurturing that is informed by more than just the prices of various boarding schools.
The adult Caulfield would think as the teenage Caulfield does, but more abstractly. The task of a successful civilization is not to break a rebel, but to accommodate him profitably.
It takes maturity to appreciate the division of individual efforts contributing to the civilization and to realise that often one can help more individuals in more substantial ways by acting indirectly, sometimes impersonally. Mr Antolini, Caulfield's teacher: "But I do say that educated and scholarly men, if they're brilliant and creative to begin with---which, unfortunately, is rarely the case---tend to leave infinitely more valuable records behind them than men do who are merely brilliant and creative. They tend to express themselves more clearly, and they usually have a passion for following their thoughts through to the end. And---most important---nine times out of ten they have more humility than the unscholarly thinker."
The importance of wealth in helping isolate oneself from disagreeable strangers must not be underestimated. The wealth of the Western world accounts for its peace, which prevails not only because wealth educates and raises the losses from conflicts, but also because wealth makes it easier to be tolerant---by having insulated oneself from the tolerable.
For Caulfield, however, family wealth is a poor point of departure. His wealth compels to conform, expands choices but does not inform of their consequences, suggests anarchy as the sole cure where a less wealthy would apply hard work. Ascending the social ladder confers the responsibilities that can be overlooked by those born at the top. Caulfield squanders money in order to descend to the position from which he would feel comfortable starting.
Intelligent, perceptive, and immature, Caulfield is a critic who has not yet become an artist, which would require kindness, in addition to his tolerance of bores and beauties. He lacks kindness to those who are kind to him---whatever their motivation: politeness, unrequited love, or the expectation of profit. He needs the society of multiple generations and social classes---not just his peers---in order to motivate his creativity. His character requires nurturing that is informed by more than just the prices of various boarding schools.
The adult Caulfield would think as the teenage Caulfield does, but more abstractly. The task of a successful civilization is not to break a rebel, but to accommodate him profitably.
It takes maturity to appreciate the division of individual efforts contributing to the civilization and to realise that often one can help more individuals in more substantial ways by acting indirectly, sometimes impersonally. Mr Antolini, Caulfield's teacher: "But I do say that educated and scholarly men, if they're brilliant and creative to begin with---which, unfortunately, is rarely the case---tend to leave infinitely more valuable records behind them than men do who are merely brilliant and creative. They tend to express themselves more clearly, and they usually have a passion for following their thoughts through to the end. And---most important---nine times out of ten they have more humility than the unscholarly thinker."
21 February 2010
"Quantum Theory: A Very Short Introduction" by John Polkinghorne (2002)
John Polkinghorne asserts that the realism---as opposed to positivism---of science must be true because many talented researchers presumably must believe in it, or else they would not be motivated to do science. But the theories' realism, an ontological notion, must be unaffected by the researchers' beliefs. Besides, the largely universal demise of the derogatory view of positivism---the view that much pre-dated positivism---is responsible for providing many with the living conditions above the subsistence level. Technological and social progress must surely suffice to motivate researchers to perfect their ability to predict correlation.
Correlation is reality. To predict is to understand. Ontological questions not only do not merit answers---they do not exist.
Why must there be greater pleasure in discovering what "is" (suppose "is" were meaningful) than in discovering correlations? When one wins a lottery, one delights in having correlated his prediction with the lottery's realization. One's delight would not be amplified if one understood the mechanism of this correlation (here, a mere chance). One's enjoyment of music is unaffected by the knowledge of the physics of sound. (The knowledge of physics, however, can be a separate source of enjoyment, perhaps, because one has an intrinsic taste for predicting the future and deducing the past.)
The stories behind equations are mnemonics, not reality. The stories are elected pagan gods, employed by scientists to animate measurements and to amuse.
At any rate, the reality imputed to quantum physics is not of the most attractive kind. "An electron does not all the time possess a definite position or a definite momentum, but rather possesses the potentiality for exhibiting one or other of these if a measurement turns the potentiality into actuality." This is too of an anthropocentric reality to be appealing. According to another interpretation, "one should acknowledge that everything that can happen does happen." (The emphasis is in the original.) This reality's observer is not self-effacing either; he multiplies with the reality in order to observe the electron in each of its possible states. Richard Feynman's interpretation is least unattractive: "one should picture a quantum particle as moving from A to B along all possible paths, direct or wriggly, fast or slow." (The emphasis is in the original.)
Believing in reality, however, is a harmless habit. If the belief pleases the scientist and makes him work free, then it should be disturbed only with the greatest of respects.
Correlation is reality. To predict is to understand. Ontological questions not only do not merit answers---they do not exist.
Why must there be greater pleasure in discovering what "is" (suppose "is" were meaningful) than in discovering correlations? When one wins a lottery, one delights in having correlated his prediction with the lottery's realization. One's delight would not be amplified if one understood the mechanism of this correlation (here, a mere chance). One's enjoyment of music is unaffected by the knowledge of the physics of sound. (The knowledge of physics, however, can be a separate source of enjoyment, perhaps, because one has an intrinsic taste for predicting the future and deducing the past.)
The stories behind equations are mnemonics, not reality. The stories are elected pagan gods, employed by scientists to animate measurements and to amuse.
At any rate, the reality imputed to quantum physics is not of the most attractive kind. "An electron does not all the time possess a definite position or a definite momentum, but rather possesses the potentiality for exhibiting one or other of these if a measurement turns the potentiality into actuality." This is too of an anthropocentric reality to be appealing. According to another interpretation, "one should acknowledge that everything that can happen does happen." (The emphasis is in the original.) This reality's observer is not self-effacing either; he multiplies with the reality in order to observe the electron in each of its possible states. Richard Feynman's interpretation is least unattractive: "one should picture a quantum particle as moving from A to B along all possible paths, direct or wriggly, fast or slow." (The emphasis is in the original.)
Believing in reality, however, is a harmless habit. If the belief pleases the scientist and makes him work free, then it should be disturbed only with the greatest of respects.
17 January 2010
"The Man Who Was Thursday" by G.K. Chesterton (1908)
The novel postulates an inherent merit in suffering, as if suffering entitles one to happiness, as if a universal law of the conservation of suffering holds. If such a law held, then the systematic (though not uniform) increase in the well-being of the world's poor would forebode the attrition of happiness reserved for the after-world---to the chagrin of the novel's characters.
If suffering earns one respect in the eyes of one's enemies---as the novel suggests---then why do not the pursuits of pleasure or of sin do so as well? Whence the asymmetry? Self-inflicted suffering is only excusable if it is an unwelcome by-product of aspiring to happiness or of an error. Otherwise, self-inflicted suffering is cowardice.
Luckily, the novel's characters appear to be more entertained than suffering---which is, perhaps, because the novel has been designed to be approved for all audiences.
If suffering earns one respect in the eyes of one's enemies---as the novel suggests---then why do not the pursuits of pleasure or of sin do so as well? Whence the asymmetry? Self-inflicted suffering is only excusable if it is an unwelcome by-product of aspiring to happiness or of an error. Otherwise, self-inflicted suffering is cowardice.
Luckily, the novel's characters appear to be more entertained than suffering---which is, perhaps, because the novel has been designed to be approved for all audiences.
2 January 2010
New York Philharmonic New Year’s Eve Concert with Thomas Hampson
(Avery Fisher Hall, 31 December 2009)
The soloist, Thomas Hampson, keeps a distance between himself, a baritone opera singer, and the characters that he voices. Even when the distance is minimal (as in Cole Porter's "Where is the Life that Late I Led?"), the voice comes across foremost as a musical instrument, only then as an individual.
In symphony orchestras, in contrast to jazz bands, players take no visible initiative in creating the mood; only the conductor innovates. Furthermore, bound by tradition, the conductor innovates minimally, without changing the arrangement. Hence, it may be too much to expect Alan Gilbert to improvise and swing it in every work, but more passion in his interpretations would be admissible. Adherence to the tradition was entirely appropriate, however, in the Yuletide hymn, conducted by Gilbert and sung by all present to conclude the night.
Porter and Gershwin, two of the three featured composers, have become most American by trying not to be such and failing at it. In the programme, their work outshines most of Copeland's, except his song "Simple Gifts."
The soloist, Thomas Hampson, keeps a distance between himself, a baritone opera singer, and the characters that he voices. Even when the distance is minimal (as in Cole Porter's "Where is the Life that Late I Led?"), the voice comes across foremost as a musical instrument, only then as an individual.
In symphony orchestras, in contrast to jazz bands, players take no visible initiative in creating the mood; only the conductor innovates. Furthermore, bound by tradition, the conductor innovates minimally, without changing the arrangement. Hence, it may be too much to expect Alan Gilbert to improvise and swing it in every work, but more passion in his interpretations would be admissible. Adherence to the tradition was entirely appropriate, however, in the Yuletide hymn, conducted by Gilbert and sung by all present to conclude the night.
Porter and Gershwin, two of the three featured composers, have become most American by trying not to be such and failing at it. In the programme, their work outshines most of Copeland's, except his song "Simple Gifts."
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