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Proof that all integers are equal June 1, 2017

Posted by Ezra Resnick in Math.
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In particular, we prove that n = -1 for any integer n.

First, assume that n is odd. That means that n – 1 is even, so n – 1 = 2k for some integer k.

The following identity is trivially true for any n:

n2 – 1 = (n – 1)(n + 1)

We can substitute 2k for (n – 1):

n2 – 1 = 2k(n + 1) = 2kn + 2k

Next, we subtract (2knn – 1) from both sides of the equation, yielding:

n2 – 2kn – n = 2k – n + 1

Factoring out (n – 2k – 1) produces:

n(n – 2k – 1) = –(n – 2k – 1)

Now we simply divide both sides of the equation by (n – 2k – 1), and voila:

n = -1

Alternatively, assume that n is even. That means that n – 1 is odd, so n – 1 = 2k + 1 for some integer k.

Once again, we begin with the identity:

n2 – 1 = (n – 1)(n + 1)

This time we substitute (2k + 1) for (n – 1):

n2 – 1 = (2k + 1)(n + 1) = 2kn + 2k + n + 1

We subtract (2kn + 2n – 1) from both sides of the equation:

n2 – 2kn – 2n = 2k – n + 2

We factor out (n – 2k – 2):

n(n – 2k – 2) = –(n – 2k – 2)

And now we simply divide both sides of the equation by (n – 2k – 2) to complete the proof:

n = -1


Skeptic vs. imbecile May 3, 2017

Posted by Ezra Resnick in Politics, Science.
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In his first op-ed column for The New York Times, Bret Stephens blames a supposed lack of concern among the American public about climate change on science advocates who (according to Stephens) claim to have “100 percent of the truth” on their side:

Claiming total certainty about the science traduces the spirit of science and creates openings for doubt whenever a climate claim proves wrong. Demanding abrupt and expensive changes in public policy raises fair questions about ideological intentions. Censoriously asserting one’s moral superiority and treating skeptics as imbeciles and deplorables wins few converts.

Who are these censorious, arrogant ideologues, derailing our climate policy conversations with their claims of total scientific certainty? Curiously, Stephens never offers any examples. In any case, he’s just muddying the waters by focusing on tone rather than substance. The scientific method is the best tool we have for understanding reality and predicting the future (as close to certainty as is allowed by the available evidence), and it has been amazingly successful; so we ought to be basing our public policy on the scientific consensus. And unfortunately, abrupt and expensive changes are sometimes called for — especially when earlier scientific recommendations have been delayed and avoided for years. Being skeptical is certainly a good default position when considering claims not sufficiently supported by evidence, but willfully ignoring the judgement of the overwhelming majority of experts doesn’t make you a skeptic; it does, in fact, make you an imbecile.

Stephens’s anti-science nonsense only gets worse as the column concludes:

None of this is to deny climate change or the possible severity of its consequences. But ordinary citizens also have a right to be skeptical of an overweening scientism. They know — as all environmentalists should — that history is littered with the human wreckage of scientific errors married to political power.

… Perhaps if there were less certitude about our climate future, more Americans would be interested in having a reasoned conversation about it.

If you don’t remember learning about “scientism” in school, that’s because it’s a bullshit term — used to preemptively dismiss a scientific thesis you don’t like without actually confronting it. (Despite all his insinuation, Stephens doesn’t directly challenge the accepted climate science.) Here’s a trick you can try at home: the next time someone presents you with a well-reasoned argument whose consequences you wish to avoid, just say “I have a right to be skeptical of your overweening reasonablism.” If they persist, tell them their assertions of certitude and moral superiority raise fair questions about their ideological intentions, then blame them for the failure of conversation. You might lose a friend, but maybe you’ll get a column in the Times!

This would all be laughable if it weren’t so dangerous: let us not forget the political climate we are living in. We have a U.S. President who claimed global warming is a hoax perpetrated by the Chinese, appointed a climate-change denier to head the Environmental Protection Agency, proposed deep budget cuts to many scientific programs and agencies (including the EPA) — the list goes on. Sophistry and misdirection may deceive voters (and readers), but nature will not be cheated; we ignore reality at our peril. Stephens is right about one thing: we need a politically engaged public who care about climate change. But more fundamentally, what we need is a scientifically literate public — who are committed to a rational conversation based on objective facts; who demand policies based on the best scientific evidence; who can tell the difference between a skeptic and an imbecile.

The trade-off triangle (or: Culture wars explained) December 11, 2016

Posted by Ezra Resnick in Democracy, Freedom, Politics.

I think many of our worst political and social conflicts are caused by failures to recognize the trade-off between two values that are both important.

Let’s start with an easy example: the relationship between the efficiency and safety of our transportation systems. We naturally want our transportation systems to be both efficient and safe; and while it’s obviously possible to have a system that is both inefficient and unsafe, it’s less obvious that it’s not possible to have a system that is both maximally efficient and maximally safe. There’s a tension between the two: once we reach a certain level of efficiency and safety, further improvements to one will necessarily come at the expense of the other. Consider speed limits, for instance. We could doubtless save many lives by reducing all speed limits to 10 mph, but we don’t do so because of the intolerable loss of efficiency; conversely, we could save many lost hours by abolishing all speed limits, but we don’t do so because of the intolerable risk to safety.

Clearly we need some reasonable balance between efficiency and safety. One problem, however, is that as humans we tend to be biased by our own personal experiences (among other things), which can cause us to focus too much on one side of the equation at the expense of the other. If you have not had any firsthand exposure to traffic deaths and injuries, for instance, but you have experienced many annoying traffic regulations and delays, you might consider safety concerns overblown and advocate against onerous safety precautions in the name of increased efficiency. If that path is followed too far, however, traffic casualties will inevitably rise; and someone who loses a loved one as a result may come to demonize the pursuit of efficiency and advocate for increasing road safety at any cost. Finding (and maintaining) an optimal compromise is not easy, but if we want to make progress while avoiding the harmful extremes we must recognize that efficiency and safety are both important, and also understand that there is a trade-off between them, so that whenever we increase one we are mindful of the impact on the other. There is never going to be a silver bullet that maximizes both.

Transportation safety versus efficiency is really just a special case of a more general dilemma: how to build a civil society while maintaining individual freedom. In this context, “freedom” refers to an individual’s ability to express themselves, make their own choices, and pursue their own goals without interference; while “civility” refers to a society’s cohesiveness and inclusiveness, the extent to which it affords equal opportunities to all its members and treats them all with justice and compassion, leaving no one behind. Unfortunately, there is a tension between freedom and civility, as illustrated by this graph:

Freedom vs Civility

At the bottom left, we have no freedom or civility, which is the lowest humanity can sink: think Nazism. (This, I believe, is the explanation for Godwin’s law: all discussions that go on long enough end up with a comparison to Hitler.) Notice, however, that it’s not possible to achieve maximal freedom and maximal civility simultaneously: we cannot escape the trade-off triangle. A world with complete freedom is a world with no civil society at all: law of the jungle, dog eat dog, every man for himself, might makes right. On the other hand, attaining perfect civility would allow no freedom: it would require enforced orthodoxy, suppression of individuality, intolerance of criticism and dissent, groupthink — a benign dictatorship, a “Brave New World” dystopia.

Most everyone agrees in principle that both freedom and civility are important and that both extremes are bad, but I think our personal experiences and other biases can lead us to focus on one axis exclusively while ignoring the trade-off implications for the other. For example, someone whose current position in society is more secure, who belongs to a more privileged class, or who has had personal experience with authoritarianism, might tend to consider freedom all-important, and might too easily dismiss calls for more civility as whining and weakness. Whereas someone whose current position in society is less secure, who belongs to an underprivileged class, or who has had personal experience with a failed state, might tend to consider civility all-important, and might too easily dismiss calls for more freedom as greed and callousness.

It’s not my intention to suggest that there are no right or wrong answers to specific policy questions, or that the correct point of balance always lies at the exact center between the two extremes: as I’ve said, there are no silver bullets. And sometimes the person disagreeing with you actually does have bad intentions. But I think there are some principles we can adopt to help keep the conversation productive, and hopefully move us closer to the optimal part of the trade-off triangle:

Recognize that both freedom and civility are important; that more of one can sometimes mean less of the other; that we need voices on both sides to stop us from going too far in either direction. Don’t live in a bubble or an echo chamber, don’t automatically take the side of your preferred axis on every issue, don’t demonize the other side or dismiss their concerns out of hand. Try to understand the point-of-view of those who have had different life experiences than you.

And always remember that despite our differences, there’s one thing we can all proudly agree on: At least we’re not Nazis.

Nobody has a better right to speak in the name of the people November 6, 2016

Posted by Ezra Resnick in Democracy, Politics.
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On January 30, 1937, Adolf Hitler gave a speech before the Reichstag, commemorating the four-year anniversary of his coming to power.

Surely nobody will doubt the fact that during the last four years a revolution of the most momentous character has passed like a storm over Germany. Who could compare this new Germany with that which existed on the 30th of January four years ago, when I took my oath of loyalty before the venerable President of the Reich? …

May we not speak of a revolution when the chaotic conditions brought about by parliamentary-democracy disappear in less than three months and a regime of order and discipline takes their place, and a new energy springs forth from a firmly welded unity and a comprehensive authoritative power such as Germany never before had? …

I myself, to whom the people have given their trust and who have been called to be their leader, come from the people. All the millions of German workers know that it is not a foreign dilettante or an international revolutionary apostle who is at the head of the Reich, but a German who has come from their own ranks…

The National Socialist Movement … refuses to allow the members of a foreign race to wield an influence over our political, intellectual, or cultural life. And we refuse to accord to the members of a foreign race any predominant position in our national economic system…

Confronted with this new and vigorous ideal, all idols and relics of the past which had been upheld by dynastic interests, tribal affiliations and even party interests, now began to lose their glamour. That is why the whole party system of former times completely collapsed in a few weeks, without giving rise to the feeling that something had been lost. They were superseded by a better ideal. A new movement took their place. A re-organization of our people into a national unit that includes all those whose labour is productive simply pushed aside the old organizations…

There could be no more eloquent proof of how profoundly the German people have understood the significance of this change and new development than the manner in which the nation sanctioned our regime at the polls on so many occasions during the years that followed. So, of all those who like to point again and again to the democratic form of government as the institution which is based on the universal will of the people, in contrast to dictatorships, nobody has a better right to speak in the name of the people than I have…

But the meaning and purpose of human organizations and of all human activities can be measured by asking what value they are for the maintenance of the race or people, which is the one existing element that must abide. The people — the race — is the primary thing. Party, State, Army, the national economic structure, Justice etc, all these are only secondary and accidental. They are only the means to the end and the end is the preservation of this nation. These public institutions are right and useful according to the measure in which their energies are directed towards this task. If they are incapable of fulfilling it, then their existence is harmful and they must either be reformed or removed and replaced by something better.


One true dialogue September 27, 2016

Posted by Ezra Resnick in Religion, Science.
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For some reason, Nature has published an essay by Kathryn Pritchard entitled “Religion and science can have a true dialogue”:

I work for the Archbishops’ Council in the Church of England, and this summer I did something that many people would think is impossible. I sat in a dark lecture theatre engrossed in a computationally generated 3D journey through the Universe… I listened to cosmologists speak on research into dark matter, particle physics, the rate at which the growth of the Universe is accelerating and the possibi­lity of multi­verses. I asked questions and they responded.

According to the popular narrative on the relationship between science and religion, this event should not have happened. The entire audience was made up of bishops and church leaders. Science and faith, we are constantly told, are in conflict and have little in common. Yet in this enjoyable, high-energy context, there was much to tease out together in terms of big questions about human origins, purpose and destiny. What would it mean for belief in God and the story and themes of Christian faith if there were multiverses? Where is the Universe heading, and what does that tell us about human purpose and destiny?

Pritchard apparently has no idea why people talk of science and faith as being in conflict. Unfortunately, it has nothing to do with religious people’s ability to enjoy 3D planetarium shows without falling asleep or to converse amiably about physics without burning anyone at the stake. Clues to the real conflict can actually be found right on the Church of England’s own website:

The Church of England is part of the One, Holy, Catholic and Apostolic Church. It worships the one true God, who is Father, Son and Holy Spirit.

It professes the faith that is uniquely revealed in the Bible and set forth in the Catholic Creeds…

The Church makes very specific assertions about the nature of the world we live in: there is one true God, worthy of worship, consisting of three persons, uniquely revealed in the Christian Bible — along with everything that entails. Anglicans are presumably aware that members of other faiths profess different, contradictory creeds, so why are they convinced theirs is true? Another clue:

The Church of England encourages people to use scripture, tradition and reason to come to a considered view on many subjects.

And therein lies the conflict. Religion makes grandiose claims about how the world works, citing uniquely revealed truths, and tells us to accept them “on faith” while relying on scripture and tradition — ahead of reason and evidence — as a basis for belief; while science is pretty much the exact opposite of that. Of course, we all know which method actually progresses reliably towards a better and better understanding of reality. Pritchard talks about “the conviction that science and theology … can illuminate one another to the benefit of all” and promises to “report on the results”, but, as enjoyable as it might be to reconcile interpretations of quantum mechanics with the stories of Christianity (or Scientology, or Harry Potter), I highly doubt the cosmologists are awaiting the outcome with bated breath. The dialogue between religion and science is entirely one-sided — it usually goes something like this:

Rᴇʟɪɢɪᴏɴ: Here’s what our magic book says about the universe. We know it’s true, because it says so in the book!

Sᴄɪᴇɴᴄᴇ: Sorry, that’s wrong: the evidence says otherwise.

Rᴇʟɪɢɪᴏɴ: How arrogant! Did we mention that our book is a unique revelation by the One True God?

Sᴄɪᴇɴᴄᴇ: Be that as it may, we’re going to see how much progress we can make by being skeptical and following the evidence wherever it leads.

Rᴇʟɪɢɪᴏɴ (a century later): OK, so we figured out a way to reinterpret our magic book, and what it really means is what you said before. So it turns out we were right all along!

Sᴄɪᴇɴᴄᴇ: Don’t you have anything better to do with your time?

Rᴇʟɪɢɪᴏɴ: Glad we could help. Don’t forget to let us know when you want to do another awesome dialogue! Maybe we could publish it in Nature.


(via Why Evolution is True)

High stakes trolley September 17, 2016

Posted by Ezra Resnick in Ethics, Philosophy.
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“A runaway trolley is barreling down the railway tracks. Ahead, in the trolley’s path, five people are trapped and unable to get out of the way: they will surely be killed if the trolley continues on its present course. You are standing next to a lever that can switch the trolley to a different set of tracks in time to save the five; but there is one person trapped on the side track who will surely be killed if you pull the lever. What do you do?”

“I’d pull the lever: better one person dies than five.”

“What if the five trapped people are all registered organ donors, and you knew for a fact that after being killed by the trolley their organs would be used to save the lives of ten people (who would die otherwise)?”

“In that case I should probably just do nothing: five will die, but more lives will be saved.”

“What if the five trapped people are also the only ones who know the location of a ticking bomb that will kill a hundred people unless the five are saved?”

“I guess pulling the lever would be the lesser evil, after all.”

“Well, what if the person trapped on the side track is a uniquely brilliant computer scientist, who has just figured out how to build an Artificial Intelligence that would tell us how to instantly eradicate malaria — which currently kills a thousand people every day?”

“Then it seems like leaving the trolley alone is actually the greater good.”

“But what if you knew that unleashing such a powerful AI at this point in time, before we’re fully prepared to contain it, would set off a chain of events leading inexorably to the extinction of humanity?”

“OK, I would definitely pull the lever to save humanity.”

“But what if you also knew that allowing this AI to develop uninhibited is the only way to ensure it becomes conscious, which will result in its evolving into a being far more rational, compassionate and ethical than humans, eventually filling the universe with levels of happiness and beauty unimaginable to (and unachievable by) us?”

“Fine: I’d bite the bullet and get out of the way. Are you happy now? Can I go?”

“Almost… You may take your blindfold off.”

“What the… Hey! You down there, get off the tracks! Don’t you see the trolley coming? I can’t save all of you! Damn it…”


Proud to be the party that protects human life July 24, 2016

Posted by Ezra Resnick in Belief, Freedom, Politics, Religion.
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The Republicans have published their 2016 Party Platform, in which they claim to stand for noble ideals:

We denounce bigotry, racism, anti-Semitism, ethnic prejudice, and religious intolerance. Therefore, we oppose discrimination based on race, sex, religion, creed, disability, or national origin and support statutes to end such discrimination… Our ranks include Americans from every faith and tradition, and we respect the right of each American to follow his or her deeply held beliefs.

Not bad, but they’re missing a crucial bit at the end there: it should read, “we respect the right of each American to follow his or her deeply held beliefs — so long as they do not infringe upon the rights of others.” We don’t condone polygamy, for instance, or human sacrifice, no matter how deeply someone might believe in them. But the omission was not accidental: before you know it, the authors’ own “deeply held beliefs” are trumping their supposed opposition to bigotry and discrimination.

We … condemn the Supreme Court’s lawless ruling in Obergefell v. Hodges… [in which] five unelected lawyers robbed 320 million Americans of their legitimate constitutional authority to define marriage as the union of one man and one woman…

We endorse the First Amendment Defense Act… which will bar government discrimination against individuals and businesses for acting on the belief that marriage is the union of one man and one woman.

This obstinate resistance to same-sex marriage (in spite of the fact that a majority of Americans support it) might seem puzzling, since the Republican platform is all about promoting “married family life as the basis of a stable and prosperous society”, and it praises adoption as well (“Families formed or enlarged by adoption strengthen our communities and ennoble our nation”). Why, then, are they hellbent on preventing same-sex couples from marrying and adopting? The only justification they offer is a depressingly familiar one: “traditional religious beliefs that have been held across the world for thousands of years”. Do those who use that argument really not realize that it applies equally well to innumerable atrocities we have worked hard to leave behind us, from witch hunting to slavery?

Same-sex marriage is not the only bogeyman the Republicans squander their energy on: abortion is mentioned in the platform thirty-four times. (Climate change is mentioned just nine times — the scientific consensus is rejected, in case you were wondering.) And this is where we really go through the looking glass. The platform says things like “We affirm our moral obligation to assist, rather than penalize, women who face an unplanned pregnancy,” and “We are proud to be the party that protects human life and offers real solutions for women,” but those words must not mean what I think they mean.

we assert the sanctity of human life and affirm that the unborn child has a fundamental right to life which cannot be infringed. We support a human life amendment to the Constitution and legislation to make clear that the Fourteenth Amendment’s protections apply to children before birth…

We will not fund or subsidize healthcare that includes abortion coverage…

We condemn the Supreme Court’s activist decision in Whole Woman’s Health v. Hellerstedt striking down commonsense Texas laws providing for basic health and safety standards in abortion clinics.

Needless to say, those laws were designed solely to reduce the availability of abortions: Texas could not provide the Court with a single example of a woman whose health would have benefited from the laws’ provisions. And in addition to making it as difficult as possible for women to get abortions, the Republicans also want to make it more likely that they’ll need them:

We renew our call for replacing “family planning” programs for teens with sexual risk avoidance education that sets abstinence until marriage as the responsible and respected standard of behavior. That approach — the only one always effective against premarital pregnancy and sexually-transmitted disease — empowers teens to achieve optimal health outcomes. We oppose school-based clinics that provide referral or counseling for abortion and contraception…

This despite the fact that abstinence-only sex education has been shown to be ineffective at preventing unwanted pregnancy or the spread of STDs — in the U.S., abstinence education was actually found to be positively correlated with teen pregnancy. And just for good measure, the platform also opposes embryonic stem cell research, which has the potential to produce therapies for many horrible injuries and diseases. But don’t forget, they’re “proud to be the party that protects human life”.

Why does the Republican Party care more about clumps of cells than the health and happiness of actual human beings? Why do they obsess over controlling people’s sex lives (despite claiming to be “the party of independent individuals”)? It all starts with their definition of “the fundamental precepts of American government”:

That God bestows certain inalienable rights on every individual… that man-made law must be consistent with God-given, natural rights; and that if God-given, natural, inalienable rights come in conflict with government, court, or human-granted rights, God-given, natural, inalienable rights always prevail…

And they have a specific God in mind, of course:

We support the public display of the Ten Commandments as a reflection of our history and our country’s Judeo-Christian heritage…

Yes, the same Ten Commandments that teach us to kill people for worshiping the wrong god. So much for respecting Americans from every faith and tradition. The Constitution, of course, never mentions God at all. Because, if you care about reality and about the wellbeing of those living in it, you need to base your policies on reason and critical thinking — instead of blindly maintaining ancient beliefs and traditions just because they’re ancient, in the face of all evidence against them.

Assisted suicide is yet another humane policy opposed by the Republican platform. But some things should really be allowed to die.


Who’s afraid of evidence? July 9, 2016

Posted by Ezra Resnick in Politics, Reason, Science.
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Astrophysicist Neil deGrasse Tyson recently tweeted:

Earth needs a virtual country: #Rationalia, with a one-line Constitution: All policy shall be based on the weight of evidence

Apart from theocrats and dictators, you’d think most everyone could agree with that sentiment in principle. But not UCLA sociologist Jeffrey Guhin: he says it’s “a very stupid idea”. Guhin, you see, has uncovered a fatal flaw in the scientific method (brace yourself): Scientists aren’t perfect!

… experts usually don’t know nearly as much as they think they do. Experts often get it wrong, thanks to their inherently irrational brains that, through overconfidence, bubbles of like-minded thinkers, or just wanting to believe their vision of the world can be true, mislead us and misinterpret information. Rationality is subjective. All humans experience such biases; the real problem is when we forget that scientists and experts are human too—that they approach evidence and reasoned deliberation with the same prior commitments and unspoken assumptions as anyone else. Scientists: They’re just like us.

Well, that’s a surprise to precisely no one. Apart from the “rationality is subjective” nonsense, scientists would certainly agree with the above, Tyson included. That’s why the scientific method has developed tools to help correct for error and minimize bias: randomized and blinded experiments, peer review, meta-analysis, etc. Which is how, despite the human flaws of individuals scientists, science has been so amazingly successful at expanding our knowledge and improving our lives: electromagnetism, evolution, genetics, cosmology — the list goes on and on. Advances in medical science have doubled our life expectancy over the last century. Guhin, however, is not impressed:

… science has no business telling people how to live. It’s striking how easily we forget the evil following “science” can do. So many times throughout history, humans have thought they were behaving in logical and rational ways only to realize that such acts have yielded morally heinous policies that were only enacted because reasonable people were swayed by “evidence.” Phrenology—the determination of someone’s character through the shape and size of their cranium—was cutting-edge science. (Unsurprisingly, the upper class had great head ratios.) Eugenics was science, as was social Darwinism and the worst justifications of the Soviet and Nazi regimes. Scientific racism was data-driven too, and incredibly well respected. Scientists in the 19th century felt quite justified in claiming “the weight of evidence” supported African slavery, white supremacy, and the concerted effort to limit the reproduction of the lesser races…

And yet, despite its abysmal track record, people continue to have extremely positive opinions of “science.”

You’ve got to be kidding me: “abysmal track record”!? Just last week, NASA’s spacecraft Juno entered Jupiter’s orbit after travelling 1.7 billion miles over five years — and it arrived within one second of the predicted time. Now, it’s true that following the scientific method does not guarantee immunity from mistakes: reality is complicated. But Guhin’s purported examples of “the evil following science can do” are actually not scientific at all: from pseudoscience (like phrenology) to fascistic propaganda (like Nazism), the great mistakes of history were caused by ideological dogmatism, and would have been prevented by more skepticism and more insistence on rational evaluation of the evidence — exactly the lesson Tyson wants us to learn. Sure, the bad guys tried to leech off science’s good reputation by claiming it was on their side, but saying something is scientific doesn’t make it so. Does Guhin think Scientology is a scientific organization? Does he consider North Korea a democracy just because it calls itself “the Democratic People’s Republic of Korea”?

Democracy is actually a good example of another system we stick with even though it’s imperfect — because it’s better than the alternatives. And that is the crux of the matter. Science is hard, and we should do our best to understand the ways it can fail so as to mitigate them; but come decision time, the only relevant question is whether there’s a superior alternative. If experts and evidence are stupid, what does Guhin think we should base our policy on instead? What method has a better track record than science? He does not get very specific about that.

Science may give us data, but that doesn’t mean that data points to truth—it just means that’s what we currently understand as truth. So how we act on that data requires nuance and judgment. It’s philosophical, maybe religious, and certainly political.

Oh, we just need to use “nuance and judgement”! Genius. What else? “Maybe religious,” he says — but which religion would that be? There are many, their prescriptions usually conflicting. And since religious beliefs aren’t evidence-based, religious differences cannot be resolved through rational discourse (witness the wonderful policies of ISIS, for instance). As for philosophy and politics, I would hope those are based on reason and evidence — otherwise we’re just back to religion again. Evidence is what grounds us to reality, and losing touch with reality inevitably turns out badly. Science is no more and no less than our best honest attempt to figure out what’s really true about the world we live in — and that’s exactly what you want to base your life decisions on.

Why, then, is Guhin so irrationally opposed to Rationalia? I don’t know. Perhaps he’s afraid that if the demand for reasonable arguments supported by evidence ever becomes widespread, he’ll have a hard time getting published.

Juno image by NASA

Turing’s library June 19, 2016

Posted by Ezra Resnick in Computer science, Logic.
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You are in an infinite library, containing the code of every possible computer program. In the library’s central space there is a computer that can be used for running the programs: you enter the code of the program you wish to run, along with the program’s input, then you push the Run button. A red light indicates that the program is running. Some output may be printed, and when the program’s execution is complete the light goes off. For some programs and inputs the run ends almost immediately, but for others the light stays on for a long time. If you get tired of waiting for the computer to halt, you can always press the Abort button to stop the current run.

Some programs in the library are short and simple, like this one:

Program M [input: two numbers, a and b]
Print the sum of a and b.

Unsurprisingly, running Program M with the numbers 2 and 3 as input prints “5”, denoted M[2, 3] = 5. Some programs are much more complex, of course. There is a program that translates English text to Japanese; a program that computes the shortest route between any two cities on a map; a program that, given the state of the board in a checkers game, finds the optimal next move.

Some programs take a long time to run, but you notice that some programs are guaranteed to never halt, at least for some inputs. For example:

Program N [input: a number, n]
As long as n equals 0, keep printing “0”.

For any nonzero input, Program N halts immediately; but N[0] gets stuck in an infinite loop, printing zeros forever (or until you hit the Abort button).

Unfortunately, you find that for many programs you cannot easily tell just by examining their code whether or not they will run forever. If a program is still running after a week, or a year, should you give up and abort? Perhaps it only needs a minute more, or an hour, or a day — or, it might be stuck in an infinite loop that will never end. It occurs to you that it would be nice to know in advance, before running a program, whether it will eventually halt (for a given input), or whether it is destined to run forever. Surely, you think, there must be a program somewhere in the library that can answer that question! After all, a computer program is just text, so there’s no reason why the code of one program cannot serve as the input for another. The special program you are seeking, call it Program H, would take as its input the code of the program to be evaluated (p), along with that program’s proposed input (i); Program H would then print “True” (i.e., H[p, i] = True) in cases where p[i] would halt, or “False” (i.e., H[p, i] = False) in cases where p[i] would run forever. You spend a long time searching the library for Program H.

One day you happen upon someone who introduces himself as the librarian. Excited, you describe Program H, and ask where in the library it can be found. The librarian smiles. The library is infinite, he says, containing every possible computer program — but none of them is Program H. You ask how he can be certain, since by his own admission the library is infinite. Let me prove it to you, he says. If Program H existed in the library, there must also exist Program G, defined as follows:

Program G [input: the code of a computer program, x]
Use the same method as Program H to determine whether program x will halt when running with the code of program x as its input. If so (i.e., H[x, x] = True), then loop forever; otherwise (i.e., H[x, x] = False), halt.

Now, says the librarian, ask yourself: What will happen if we run Program G with its own code as input? Will G[G] run forever or halt? We ought to be able to compute the answer using Program H. Let’s first assume that H[G, G] = True, which means that G[G] will halt. But, based on the definition of Program G, G[G] would only halt if Program H reports that G[G] will not halt, i.e., H[G, G] = False — contradicting our assumption. Let’s assume, then, that H[G, G] = False, which means that G[G] will not halt. But, based on the definition of Program G, G[G] would only fail to halt if Program H reports that G[G] will halt, i.e., H[G, G] = True — also a contradiction!

We therefore have no choice but to conclude that Program H is like a square circle: it cannot logically exist. The library is infinite, containing every possible computer program — but none of them is Program H: some problems are impossible to solve.

Look on the bright side, says the librarian, sensing your disappointment. What’s that, you ask. The librarian smiles: At least you won’t waste any more time searching for something that isn’t there.


Theirs is a repressive, hateful ideology June 14, 2016

Posted by Ezra Resnick in Equality, Politics.
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After Omar Mateen killed 49 people in a gay nightclub in Florida, Speaker of the House of Representatives Paul Ryan tweeted “We pray for those brutally attacked in Orlando”, followed by:

It is horrifying to see so many innocent lives cut short by such cowardice. Tonight, and in the long days ahead, we will grieve with the families…

As we heal, we need to be clear-eyed about who did this. We are a nation at war with Islamist terrorists. Theirs is a repressive, hateful ideology that respects no borders…

I’m certainly not going to minimize the problem of Islamist terrorists; and yet, they don’t hold a monopoly on repressive, hateful ideology. Take Paul Ryan, for instance. He has voted against same-sex couples adopting children in Washington D.C., opposed the repeal of the “don’t ask, don’t tell” policy (which barred openly gay persons from military service), and supported a constitutional ban on same-sex marriage. In other words, he has actively contributed to the denigration and demonization of gays in our society, and has helped perpetuate the discrimination and injustice they face. And then, when gays are targeted for violence, he condemns someone else’s hateful ideology.

Perhaps, instead of praying, the Speaker could try doing something actually useful: he could speak up in support of equal rights for all, and against the dogmas that divide us — domestic as well as foreign. Being clear-eyed about one’s own failings does take courage; but Paul Ryan is no coward, is he?