By way of contradiction May 5, 2013
Posted by Ezra Resnick in Logic, Math.2 comments
An indirect proof (or proof by contradiction) establishes the truth of a proposition by showing that the proposition’s negation implies a contradiction. For example, we can indirectly prove that the square root of 2 is irrational (i.e., it cannot be expressed as a ratio a/b where a and b are integers) by assuming the opposite — that √2 can be expressed as a ratio of integers — and showing that such an assumption leads to a contradiction (e.g., that b is both even and odd).
Some people find indirect proofs unsatisfying, or even a bit suspicious: it feels like we’ve been cheated out of understanding why the proposition is true. Direct proofs seem more intuitive and dependable. This raises the question: Does every proposition that can be proved indirectly have a direct proof as well? Or are there propositions that can be proved indirectly, for which no direct proof exists?
Before attempting to answer that question, let us first consider this humble proposition:
(p) This proposition cannot be proved directly.
We can prove proposition p is true — indirectly. Start by assuming the opposite, that is, assume there exists a direct proof of p. In particular, that means p is true. But p states that there is no direct proof of p — contradicting our assumption. So our assumption must be false; hence p is true.
Let us now attempt to prove the following proposition:
(q) Not all propositions that can be proved indirectly can also be proved directly.
We shall prove the truth of proposition q (you guessed it) indirectly. Assume the opposite: that is, assume any proposition that can be proved indirectly can also be proved directly. Then, since p can be proved indirectly (as demonstrated above), there must also exist a direct proof of p. However, the existence of such a proof contradicts (the proven) proposition p! So our assumption must be false — and q is true.
Ah, but the question remains: Can q be proved directly?
The perils of reasonablism April 27, 2013
Posted by Ezra Resnick in Reason.4 comments
I’ll be the first to admit that reason can be a useful tool: logical thinking and honest evaluation of real-world evidence may come in handy if you want to cure disease, or build an airplane, or solve a crime. But some people just can’t stop there: they arrogantly insist that everything in life ought to be approached in a reasonable manner! As extreme and fundamentalist as it sounds, I have actually met those who will claim (with a straight face) that it’s impossible to be too reasonable.
It saddens me to see people whose worldview is so narrow and closed-minded. What kind of world would we live in, if everyone were constantly expected to provide good reasons for their beliefs and reasonable justification for their actions? If everything were open for discussion and reevaluation based on evidence and argument? The reasonablists need to understand that some people are deeply attached to so-called “non-reasonable” beliefs, and they might become offended or angry if forced to question those beliefs. And whose fault would that be?
Anyway, how come the militant reasonablists get to define what’s reasonable? They may proclaim the value of logical consistency and intellectual honesty, but that’s just their opinion! Others are free to define “reasonable” however they want: following tradition, obeying an authority, wishful thinking — who are we to judge? The reasonablists’ insistence on being undogmatic is just another dogma; their rejection of blind faith is itself a form of blind faith.
It seems to me that the reasonablists should learn a little humility. After all, just because logical, evidence-based thinking has proved immensely successful at understanding how our world works, doesn’t mean we should rely on it when deciding how to live our lives and build our societies. Just because your conclusions are supported by “evidence” and “logic” doesn’t mean I have to accept them.
One of the highest April 8, 2013
Posted by Ezra Resnick in Ethics, Law, Religion.add a comment
Suppose that in some community of your city, a newborn baby is taken by his parents to a tattoo parlor, where they have the family emblem tattooed on his backside. The tattoo subsequently becomes infected, causing the infant to suffer brain damage and, eventually, die.
What would be the appropriate response? Should we shrug our shoulders, maintaining that parents are free to do whatever they want with their children? Or should we hold the parents (and the tattoo artist) accountable?
And what kind of parents would perform such a procedure on an infant, anyway?
Two infants in the last three months in New York City’s ultra-Orthodox Jewish community have been infected with herpes following a ritual circumcision, according to the health department. The boys were not identified.
In the most controversial part of this version of the Jewish ritual, known as metzitzah b’peh, the practitioner, or mohel, places his mouth around the baby’s penis to suck the blood to “cleanse” the wound.
One of the two infected babies developed a fever and lesion on its scrotum seven days after the circumcision, and tests for HSV-1 were positive, according to the health department.
Last year, the New York City Board of Health voted to require parents to sign a written consent that warns them of the risks of this practice. None of the parents of the two boys who were recently infected signed the form, according Jay Varma, deputy commissioner for disease control at the New York City Department of Health and Mental Hygiene.
Varma said it was “too early to tell” if the babies will suffer long-term health consequences from the infection.
Since 2000, there have been 13 cases of herpes associated with the ritual, including two deaths and two other babies with brain damage.
Neonatal herpes infections can cause death or disability among infants, according to the Centers for Disease Control and Prevention.
“First, these are serious infections in newborns and second, there is no safe way an individual can perform oral suction on an open wound,” said Dr. William Schaffner, chair of preventive medicine at Vanderbilt University. “Third, these terrible infections are completely preventable. They should not occur in the 21st century with our scientific knowledge.”
Some rabbis told ABCNews.com last year that they opposed on religious grounds the law requiring parents to sign a waiver, insisting it has been performed “tens of thousands of times a year” worldwide. They say safeguarding the life of a child is one of the religion’s highest principles.
“This is the government forcing a rabbi practicing a religious ritual to tell his congregants it could hurt their child,” Rabbi David Niederman, executive director of the Hasidic United Jewish Organization of Williamsburg, told ABCNews.com. “If, God forbid, there was a danger, we would be the first to stop the practice.”
We must not inform parents of the demonstrable dangers posed to their child, because safeguarding the life of a child is one of the religion’s highest principles, and if, God forbid, there was a danger, we rabbis would be the first to stop the ritual, and since we haven’t stopped, there must not be any danger. So mind your own business.
Still, perhaps we should identify the infected mohel and stop him from harming more children?
The health department could take no action against the rabbi who performed the circumcision because the parents would not reveal his identity.
Safeguarding the life of a child is one of the religion’s highest principles. Not, however, the highest.
Long enough to find it out April 4, 2013
Posted by Ezra Resnick in Ethics.2 comments
Film critic Roger Ebert died today, at the age of 70. He was diagnosed with thyroid cancer in 2002, and in 2006 his jaw was removed, leaving him unable to speak or eat; yet he remained a good-humored, prolific writer until the end.
In 2009, he wrote on his blog:
I know it is coming, and I do not fear it, because I believe there is nothing on the other side of death to fear. I hope to be spared as much pain as possible on the approach path. I was perfectly content before I was born, and I think of death as the same state. What I am grateful for is the gift of intelligence, and for life, love, wonder, and laughter. You can’t say it wasn’t interesting. My lifetime’s memories are what I have brought home from the trip. I will require them for eternity no more than that little souvenir of the Eiffel Tower I brought home from Paris…
… “Kindness” covers all of my political beliefs. No need to spell them out. I believe that if, at the end of it all, according to our abilities, we have done something to make others a little happier, and something to make ourselves a little happier, that is about the best we can do. To make others less happy is a crime. To make ourselves unhappy is where all crime starts. We must try to contribute joy to the world. That is true no matter what our problems, our health, our circumstances. We must try. I didn’t always know this, and am happy I lived long enough to find it out.
An Islamist lexicon March 17, 2013
Posted by Ezra Resnick in Equality, Language, Religion.add a comment
The Muslim Brotherhood is extremely concerned:
The 57th session of the UN Commission on the Status of Women (CSW), taking place from March 4 to 15 at UN headquarters, seeks to ratify a declaration euphemistically entitled ‘End Violence against Women’.
That title, however, is misleading and deceptive.
Does the Brotherhood mean to say that the UN declaration is not actually aimed at eliminating the disenfranchisement, maltreatment, and subjugation of women? Well, not exactly.
That title, however, is misleading and deceptive. The document includes articles that contradict established principles of Islam, undermine Islamic ethics and destroy the family, the basic building block of society, according to the Egyptian Constitution.
This declaration, if ratified, would lead to complete disintegration of society, and would certainly be the final step in the intellectual and cultural invasion of Muslim countries, eliminating the moral specificity that helps preserve cohesion of Islamic societies.
A closer look at these articles reveals what decadence awaits our world, if we sign this document:
1. Granting girls full sexual freedom, as well as the freedom to decide their own gender and the gender of their partners (ie, choose to have normal or homo- sexual relationships), while raising the age of marriage.
2. Providing contraceptives for adolescent girls and training them to use those, while legalizing abortion to get rid of unwanted pregnancies, in the name of sexual and reproductive rights.
3. Granting equal rights to adulterous wives and illegitimate sons resulting from adulterous relationships.
4. Granting equal rights to homosexuals, and providing protection and respect for prostitutes.
5. Giving wives full rights to file legal complaints against husbands accusing them of rape or sexual harassment, obliging competent authorities to deal husbands punishments similar to those prescribed for raping or sexually harassing a stranger.
6. Equal inheritance (between men and women).
7. Replacing guardianship with partnership, and full sharing of roles within the family between men and women such as: spending, child care and home chores.
8. Full equality in marriage legislation such as: allowing Muslim women to marry non-Muslim men, and abolition of polygamy, dowry, men taking charge of family spending, etc.
9. Removing the authority of divorce from husbands and placing it in the hands of judges, and sharing all property after divorce.
10. Cancelling the need for a husband’s consent in matters like: travel, work, or use of contraception.
These are destructive tools meant to undermine the family as an important institution; they would subvert the entire society, and drag it to pre-Islamic ignorance.
The Muslim Brotherhood urges the leaders of Muslim countries and their UN representatives to reject and condemn this document, and to call upon this organization to rise to the high morals and principles of family relations prescribed by Islam.
So, ‘End Violence against Women’ isn’t really a “misleading and deceptive” title for the UN declaration, after all. On the other hand, I think I might have spotted a euphemism or two creeping into the Brotherhood’s heartfelt protest (which could non-euphemistically be titled ‘More Violence against Women’). Here, then, is a handy lexicon listing some common Islamist code words along with their actual meanings:
undermine the family: make it harder for men to control their wives and daughters
complete disintegration of society: a society where women are free and equal members
intellectual and cultural invasion of Muslim countries: concern for the wellbeing of all inhabitants of Muslim countries
the moral specificity that helps preserve cohesion of Islamic societies: brainwashing, ignorance, and coercion
decadence: anything not prescribed in the worldview of a 7th-century tribal warlord
(via Butterflies & Wheels)
You know who you are March 5, 2013
Posted by Ezra Resnick in Democracy, Politics, Science.add a comment
Two world systems February 18, 2013
Posted by Ezra Resnick in Reason, Science.add a comment
Galileo Galilei’s Dialogue Concerning the Two Chief World Systems, published in 1632, compares the geocentric model of Aristotle and Ptolemy with the heliocentric model of Copernicus, in the form of a discussion between three friends: Salviati, Sagredo, and Simplicio. Salviati is a scientist arguing for the Copernican position; Sagredo is an intelligent and curious layman who becomes persuaded by Salviati’s case; while Simplicio is a faithful follower of the established Aristotelian tradition.
At one point in the dialogue, after Salviati puts forward some observational evidence supporting the heliocentric view, Simplicio expresses confidence in his ability to “once more succeed in reconciling what experience presents to us with what Aristotle teaches. For obviously two truths cannot contradict one another.” Sagredo remarks:
I can put myself in Simplicio’s place and see that he is deeply moved by the overwhelming force of these conclusive arguments. But seeing on the other hand the great authority that Aristotle has gained universally; considering the number of famous interpreters who have toiled to explain his meanings; and observing that the other sciences, so useful and necessary to mankind, base a large part of their value and reputation upon Aristotle’s credit; Simplicio is confused and perplexed, and I seem to hear him say, “Who would there be to settle our controversies if Aristotle were to be deposed? What other author should we follow in the schools, the academies, the universities? What philosopher has written the whole of natural philosophy, so well arranged, without omitting a single conclusion? Ought we to desert that structure under which so many travelers have recuperated? Should we destroy that haven, that Prytaneum where so many scholars have taken refuge so comfortably; where, without exposing themselves to the inclemencies of the air, they can acquire a complete knowledge of the universe by merely turning over a few pages? Should that fort be leveled where one may abide in safety against all enemy assaults?”
I pity him no less than I should some fine gentleman who, having built a magnificent palace at great trouble and expense, employing hundreds and hundreds of artisans, and then beholding it threatened with ruin because of poor foundations, should attempt, in order to avoid the grief of seeing the walls destroyed, adorned as they are with so many lovely murals; or the columns fall, which sustain the superb galleries, or the gilded beams; or the doors spoiled, or the pediments and the marble cornices, brought in at so much cost — should attempt, I say, to prevent the collapse with chains, props, iron bars, buttresses, and shores.
Later on, after Simplicio again mentions his reverence for the great authors of the past, Sagredo recalls an incident he witnessed:
One day I was at the home of a very famous doctor in Venice, where many persons came on account of their studies, and others occasionally came out of curiosity to see some anatomical dissection performed by a man who was truly no less learned than he was a careful and expert anatomist. It happened on this day that he was investigating the source and origin of the nerves, about which there exists a notorious controversy between the Galenist and Peripatetic doctors. The anatomist showed that the great trunk of nerves, leaving the brain and passing through the nape, extended on down the spine and then branched out through the whole body, and that only a single strand as fine as a thread arrived at the heart. Turning to a gentleman whom he knew to be a Peripatetic philosopher, and on whose account he had been exhibiting and demonstrating everything with unusual care, he asked this man whether he was at last satisfied and convinced that the nerves originated in the brain and not in the heart. The philosopher, after considering for awhile, answered: “You have made me see this matter so plainly and palpably that if Aristotle’s text were not contrary to it, stating clearly that the nerves originate in the heart, I should be forced to admit it to be true.”
Simplicio responds:
Aristotle acquired his great authority only because of the strength of his proofs and the profundity of his arguments. Yet one must understand him; and not merely understand him, but have such thorough familiarity with his books that the most complete idea of them may be formed, in such a manner that every saying of his is always before the mind. He did not write for the common people, nor was he obliged to thread his syllogisms together by the trivial ordinary method; rather, making use of the permuted method, he has sometimes put the proof of a proposition among texts that seem to deal with other things. Therefore one must have a grasp of the whole grand scheme, and be able to combine this passage with that, collecting together one text here and another very distant from it. There is no doubt that whoever has this skill will be able to draw from his books demonstrations of all that can be known; for every single thing is in them.
Furthermore, Simplicio asks,
… if Aristotle is to be abandoned, whom shall we have for a guide in philosophy?
Salviati replies:
We need guides in forests and in unknown lands, but on plains and in open places only the blind need guides. It is better for such people to stay at home, but anyone with eyes in his head and his wits about him could serve as a guide for them. In saying this, I do not mean that a person should not listen to Aristotle; indeed, I applaud the reading and careful study of his works, and I reproach only those who give themselves up as slaves to him in such a way as to subscribe blindly to everything he says and take it as an inviolable decree without looking for any other reasons. This abuse carries with it another profound disorder, that other people do not try harder to comprehend the strength of his demonstrations. And what is more revolting in a public dispute, when someone is dealing with demonstrable conclusions, than to hear him interrupted by a text (often written to some quite different purpose) thrown into his teeth by an opponent?
… So put forward the arguments and demonstrations, Simplicio — either yours or Aristotle’s — but not just texts and bare authorities, because our discourses must relate to the sensible world and not to one on paper.
The next day, before Simplicio arrives, Salviati shares with Sagredo his opinion of geocentrism’s defenders:
I have heard such things put forth as I should blush to repeat — not so much to avoid discrediting their authors (whose names could always be withheld) as to refrain from detracting so greatly from the honor of the human race. In the long run my observations have convinced me that some men, reasoning preposterously, first establish some conclusion in their minds which, either because of its being their own or because of their having received it from some person who has their entire confidence, impresses them so deeply that one finds it impossible ever to get it out of their heads. Such arguments in support of their fixed idea as they hit upon themselves or hear set forth by others, no matter how simple and stupid these may be, gain their instant acceptance and applause. On the other hand, whatever is brought forward against it, however ingenious and conclusive, they receive with disdain or with hot rage — if indeed it does not make them ill. Beside themselves with passion, some of them would not be backward even about scheming to suppress and silence their adversaries.
In 1633, Galileo was convicted of suspected heresy by the Roman Inquisition. He was forced to recant Copernicanism under threat of torture, and spent the rest of his life under house arrest.
The Dialogue was placed on the Church’s Index of Forbidden Books — where it remained until 1835.
The Emperor’s method February 3, 2013
Posted by Ezra Resnick in Science.add a comment
Once upon a time there lived a vain Emperor, who loved to dress in elegant clothes. One day, two tailors presented themselves before the Emperor with an unusual offer:
“We are two highly skilled tailors, and after many years of research we have invented an extraordinary method to weave a cloth of exquisite beauty — which is invisible to anyone who is too stupid to appreciate its quality.”
“Your proposition sounds very interesting,” said the Emperor, “and I would very much like to own a suit made of such an amazing fabric.”
The tailors rubbed their hands and smiled.
“But before I commission your services,” continued the Emperor, “I’m sure you wouldn’t mind taking a little time to demonstrate your product’s extraordinary properties.”
“Of course,” replied the tailors. “It would be our privilege to help your Highness try on our—”
“That’s not what I had in mind,” the Emperor interrupted. “I have a method that I employ in such situations, and it has served me well. Here is what you must do: We shall summon one hundred of my wisest subjects, and divide them into two groups of equal size. (We shall let chance determine who joins which group, by the flipping of a golden coin.) One group shall be presented with a steward clothed in your fabulous fabric; while the other group shall be presented with a steward wearing nothing at all. We shall then see how many members of each group claim to have seen any clothes.”
The tailors exchanged glances, dismayed. “We would love to oblige your Highness,” they said, “but we fear the proposed method is flawed: in our experience, true wisdom is very rare; so it is quite possible that none of the summoned subjects will be capable of seeing our wondrous fabric.”
“I see,” said the Emperor coldly. “Tell me, then: Is there some set of questions, some test we can administer, in order to determine in advance whether a person is wise enough to be able to detect your amazing cloth?”
“Actually, your Highness,” replied the tailors, “we’ve found that the only reliable indicator that a person is wise enough to see our material is that he does, in fact, see it.”
“That is quite unfortunate,” said the Emperor. “I wonder, then, how one could ever possibly tell the difference between your fine product and that of a scoundrel, who offered the same story but no actual fabric at all?”
The tailors looked insulted. “Begging your forgiveness; we are but poor, humble craftsmen, and cannot match your Highness’s intellect. But if your Highness — who is wise indeed — would only be willing to try on our clothes for himself, I’m sure we could demonstrate the quality of our wares to his utmost satisfaction. Surely your Highness would trust the testimony of his own eyes, and the word of his closest advisers? Surely one should follow one’s personal intuitions on such matters, rather than sterile methods and tests?”
The Emperor raised his eyebrows ever so slightly.
No one knows what became of the two tailors, but they were never heard from again.
Angels December 16, 2012
Posted by Ezra Resnick in Belief, Superstition.add a comment
There are many things we still don’t understand about the Connecticut school shooting that left twenty small children dead; and some questions may go forever unanswered. Dealing with such a tragedy, and consoling those who lost loved ones, is one of the hardest things any of us could ever have to do. But one thing we should not do is pretend to know things we do not know.
Olivia Engel had a part in a nativity play at St. Rose of Lima Catholic Church. “She was supposed to be an angel in the play. Now she’s an angel up in heaven,” Monsignor Robert Weiss told a standing-room-only crowd at the church before the play on Saturday.
I’m sure some grieving people are comforted by that idea (without thinking through its implications) — but there is absolutely no reason to think it’s actually true. Tempting as it may be, false consolation is the easy way out: instead of dealing with reality and teaching our children (and ourselves) how to grieve, we imply that it’s OK to deny the facts and believe whatever makes you feel better. This is not a harmless “white” lie: disconnecting from reality has a price. Specifically, believing that people go to a better place when they die cheapens our lives here on Earth. Beliefs have consequences, and beliefs that take the “sting” out of death are especially dangerous. In fact, such beliefs do a lot of work for those who wish to rationalize killing children.
Darkness and light December 8, 2012
Posted by Ezra Resnick in Reason, Religion.5 comments
I will not light a candle for miracles: I will not celebrate gullibility, ignorance, self-deception, wishful thinking. I will light a candle for skepticism, for intellectual honesty, for experimentation, for evidence-based thinking.
I will not light a candle for tribalism: I will not celebrate sectarian, parochial worldviews that arbitrarily divide humanity into separate categories. I will light a candle for equality, for empathy, for solidarity.
I will not light a candle for worship: I will not celebrate submission, propitiation, servility. I will light a candle for self-respect, for independence, for human dignity.
I will not light a candle for militarism: I will not celebrate violence, aggression, retribution, revenge. I will light a candle for those who put themselves in harm’s way, who defend those that cannot defend themselves.
I will not light a candle in yearning for some idealized past: we have struggled long and hard to overcome many historical errors and injustices. I will light a candle for progress, for learning from past mistakes, for continuing to expand our knowledge and raise our standards and improve our society.
I will not light a candle and pray for some all-powerful being to save us in our hour of need: the universe doesn’t care about us and wouldn’t notice if our planet went dark. I will light a candle for rational decision-making, for responsible public policy, for building a sustainable future — so that the lights may stay on a bit longer.
I will not light a candle just because I am commanded to: I will not celebrate dogmatism and blind obedience. If I choose to, I will light a candle for individuality, for critical thinking, for personal liberty.
Also, I will not light a candle for fried food — that stuff will kill you. Go eat an apple.
