Sunday, 10 November 2013

Women should .... and men should ....

There was a very interesting report in Global Voices last week.  It reported on an unusual way to measure negative attitudes to women's rights in different countries across the world.  Type phrases such as "Women should", "Women must", "Women shouldn't" etc into Google's search box and see what auto-completions are suggested.  These auto-completions are generated on the basis of previous searches conducted by users and so they measure the commonality of searches beginning with these phrases.  Some examples (from Google New Zealand):

  • Women should not (speak in church, work, preach, vote)
  • Women must not (teach, preach, wear men's clothing)
I was interested to compare these searches with corresponding searches where the word "Men" replaced "Women".  For the cases above I found
  • Men should not (marry, wear shorts, wear sandals)
  • Men must not (walk too late, cut down trees)
The Global Voices article gave many examples of "Women" searches in other languages.  It would be interesting to see the results with "Men" replacing "Women" there too.

In any case, this small experiment adds weight to the main point: that women remain disadvantaged and oppressed in many societies.

Sunday, 6 October 2013

Dangers of benign faith

In a discussion between religious believers and atheists there are two fundamental questions: whether a particular faith is actually true or whether (irrespective of truth) faith can be an overall force for good in the world.  For the first question it is quite difficult for a believer to respond to the charge that not all faiths can be true, and the charge that one's particular faith is almost always determined by one's cultural background.  But the second question is more subtle: it depends on what you count in the balance sheet.  Which wars should you count as being religiously motivated?  If, on the surface they were religiously motivated (like the Thirty Years War), would they have occurred even if religion had mysteriously disappeared from the human condition?  Nevertheless, most theists will concede that, in the past, many things have been done in the name of religion that nowadays we abhor.

Often such a concession is accompanied by two mitigating comments: that nowadays religion is more benign, and that much community good work is carried out by churches.  I accept both of these comments, am happy to discount the past and to make a balance sheet purely on today's religious activities.

In this post I am not going to point to the many examples around the world where violence and intolerance are founded on religious grounds.  Nor am I going to criticise the many well-meaning, sincere church-goers of all faiths who care deeply about their fellow men and women and live their lives trying to improve our society.  Instead I wish to make a case that the very presence of faith-based behaviour in our society has negative consequences.

Is it harmless to "touch wood" (you know what I mean, you naughty people) just in case there might possibly be something in that old superstition?  Is it harmless to believe that "what goes around comes around" or that "everything happens for a reason"?  Is it harmless to believe that your life is overseen by some benevolent creator, or to believe that praying for a friend to recover from an illness is effective?  Aren't all those things (and many others like them) at the very worst just personal musings?

In my opinion these beliefs are damaging in a subtle way.  The fabric of religious belief contains many strands: the examples I have given are less damaging than some others (such as killing people because of a conviction you think is implanted in you from your god, or refusing a woman birth control because a holy book prohibits it, or marginalising a gay person).  But I use the fabric metaphor because these beliefs are connected.  If you believe that "everything happens for a reason" you have taken a step along a superstitious strand and you will take another unthinking step a little more willingly.  In bad cases you may become so superstitious that you may one day come to believe that you are charged by a god to act in a certain way, and for some fervent believers that way may be bad for society (or bad for you).

In other words what I am claiming is that superstition, or believing things for which there is no evidence, is a pernicious thought habit.  Once you let a smidgeon of it pollute you, you are in danger of wading a little deeper into the pollution.  Maybe you will resist the next superstitious thought that you meet.  But isn't it likely that, having admitted to some unthinking superstition, your critical thinking faculties may let you down once more, or twice more.

If you remain unconvinced consider the prevalence of religious belief in the United States.  That rich culture allows many scales of superstition to flourish and some of them are surely harmful.  For example Justice Antonin Scala believes that Satan is a real person.  Doesn't it make you uneasy that a member of the Supreme Court of the US, someone who has to judge on human wickedness, believes in a superhuman incarnation of evil?  Or what about Congressman Paul Broun who is a Young Earth creationist.  He is a member of the House Science Committee!  Do you trust him to understand the science behind Global Warming (to take just one example).  And then of course there are the tele-evangelists who amass large fortunes while promoting hatred of gays (and in some cases like Bob Larson and Ted Haggard are caught in mind-boggling hypocrisy).  Of course, these examples are outliers but my point is that such irrational behaviour is only unremarkable in a culture where faith is exalted rather than pitied.

The problems of life are very complicated and superstitious thought patterns are sometimes useful as rough and ready rules of thumb in order for us to make quick decisions (don't walk under ladders so that a paint pot will miss you when it falls).  But don't credit your superstitions with mysterious properties or you will lead a less examined life.  Socrates thought that such lives were not worth living and I agree.

Monday, 26 August 2013

Syrian weapons of mass destruction

Suppose your husband is acting rather distantly.  You suspect he is having an affair.  He denies it but, in due course, you are proven right.  Husband is contrite, you are forgiving.  A few years later you notice the same symptoms of distance.  Is he having an affair?  You ask him.  He denies it.  Do you believe him?  And, in a nutshell, we have the problem of trust.  Why should you believe his declarations?  Would your girl-friends advise you to be very wary?  Quite frankly you need an unimpeachable source and your husband just doesn't cut it.

Now we hear the claim that the Syrian government has unleashed chemical weapons on the rebel insurgency.  The US is outraged, sabres rattle, and we hear the familiar prelude to a US-UK intervention.  Should we believe what we are told?  Doesn't our memory go back to 2002/2003 when we were told with cast-iron certainty that Iraq - another oil-rich country - had WMD and that armed invasion was therefore vital.

The US and the UK have a credibility problem of gigantic proportion.  Let's leave aside the question of whether they have a moral obligation to go to war (I think not, but that is not the point I want to address here).  Let's assume that war could be justified if the WMD claim is true.

But how can we the ordinary voters know what to think?  After all, it wasn't just the leaders of the US and the UK who sold us a pup 10 years ago.  Our media, including the "quality press" (such as the UK Observer - see the expose by Nick Davis in Flat Earth News), bought in to and passed on as truth, all their government's propaganda.  Seriously, folks - what could convince us that Syria is not another Iraq?  The resemblance is uncanny.

I don't know what could realistically be said or done to make us believe the claims of chemical weapons.  Obviously what the US and UK governments say is worthless. Obviously, what the US and UK press say is tainted by their past behaviour.  The chickens of their lies have come home to roost.

We need some credible sources and those sources have to be clearly not beholden to the US and UK.  So, how about inviting the Chinese to investigate?  How about flying in the Libyans?  Or the North Koreans?  Or some representatives of America's down-trodden black community who have suffered the most in the financial fallout from the two trillion dollar spend in the Iraq war?  I am not joking - I would trust these sources more than I trust our official overlords.

Monday, 19 August 2013

Despairing of the UK: Scottish opportunity

We are culturally disposed to feel respect and loyalty to our country of birth.  On the whole I believe this is a good thing.  It makes us more likely to sign up to the social contract: cooperate with law enforcement agencies, pay our taxes, behave responsibly about litter, and a host of other things to make us good citizens.  But what do you do if you think your country is taking measures against the populace, overt or covert, which are clearly a breach of their part in the social contract?  And, worse, if they ignore the wishes of large sections of the populace leading them to a genuine sense of disenfranchisement?

This is the situation I find myself in as a UK citizen largely as a result of the revelations given to us by Edward Snowden.  Of course I am not alone.  Other British people are similarly outraged.  Furthermore, residents of several other countries - the USA and New Zealand spring to mind - are also outraged by the behaviour of their own security agencies.  Instances of these agencies over-reaching their mandate are too many to enumerate but the present furore over the harassment of journalist Glenn Greenwald's partner David Miranda is typical.

To return to my question: what should I do?  The sad answer is that I have no immediate recourse.  All I can do is to continue to make my views known and to use the democratic process to express my strong disapproval of the slide towards totalitarianism.  We are still a long way from an East German state so peaceful protest has still to be order of the day.

However, in the UK, there is a constitutional issue on the horizon that could enable a large swathe of the UK population to express their disapproval of HM's government in a very significant manner.  Next year there will be a referendum on Scottish independence.  Scotland has felt a sense of disenfranchisement for many years.  Currently it has only one Conservative MP (out of over 50 seats) and it would not be an exaggeration to say that the Conservatives are hated in Scotland.  The arguments leading up to the referendum are already becoming heated and, in some cases, quite silly (just recently the question of whether Scotland would continue in the British Commonwealth has arisen, as if the Commonwealth meant anything other than just a consortium of sporting nations).  But the mood from Westminster is chillingly uncompromising and, if I were a Scot, I would not hesitate to vote "yes" for independence.

So, if I were a Scot, I would be adding another injustice to an already long list suffered under the English jackboot.  This government has proclaimed loud and clear that it does not care about the civil liberties of its people.  The Scots have a chance to save themselves and who could blame them for seizing it?

Saturday, 10 August 2013

Obama's press conference: more lies

At his press conference at the weekend President Obama promised "appropriate reforms" to the domestic surveillance carried out by the NSA.  He also used the term "safeguards against abuse" but it is crystal clear that he does not intend any dismantling of the apparatus that has come under such criticism since Edward Snowden's revelations.  In fact Obama's track record on truthfulness is so patchy (along with almost all administration apologists) that we cannot have any trust in even the weak promises that he made.  The sad fact is that Obama is a liar, just one of many in the US government.

His hypocrisy is mind-blowing.  The extraordinary claim that his administration would eventually have gotten round to reining in the NSA is pathetic in the light of the over 10 year abuses that have gone unchecked until the public was eventually enlightened.  And let us never forget that this enlightenment came from a man who is being pursued by the authorities as a criminal.

Mr Obama: you should be ashamed of yourself.  Not only for such dissembling but also for thinking that you retain any credibility at all.

Monday, 5 August 2013

Is there life on other planets?

The title of this post is surely one of the biggest questions for which we have no answer.  Of course it is one of those questions - like "Does God exist" - for which we shall never know the answer if it is "No" but could only know the answer if it is "Yes".  Put another way, the hypothesis that there is no extra-terrestrial life can be falsified but its opposite can't.  Or to paraphrase yet again: the null hypothesis is that there is no extra-terrestrial life.  I labour this point because often in science one's position is to accept the null hypothesis until we know otherwise (are there fairies at the bottom of your garden) and yet the scientific and popular literature is full of assertions about the overwhelming likelihood that life elsewhere exists.

The basis of these assertions is almost always the famous Drake equation which is an estimate for the total number N of intelligent civilisations in our galaxy based on multiplying together several unknown quantities.  In 1961, when Frank Drake invented his formula, we had virtually no idea about planets beyond our solar system.  But over 50 years later we now know that they are very numerous and we therefore have a much better idea about some of the factors in the formula.  This knowledge has made some people very hopeful that not only is N>1 but that N might be very large.  To put it crudely the argument goes: "Since there are literally billions of planets it is exceedingly likely that some of them will contain life".

I don't accept this argument.  The Drake equation contains another factor that measures the probability of life beginning (and I mean beginning, not evolving) on a planet with the conditions to sustain life.  We know nothing about this probability.  It may be so small that, even though the number of planets is vast, our own existence may be just a one-off fluke.  I am not saying that no extra-terrestrial life exists; I am saying that we do not know enough to put an estimate on how many instances of it there are, and in particular we cannot say whether this number is likely to be greater than 1.

Why then is there such a lot of firmly-held opinion on the question?  I think the belief in alien life is partly the result of wishful thinking.  It's a cool idea that there may be beings elsewhere in the galaxy, one that has inspired very many science fiction stories.  Wouldn't it be sad to consign all that imaginative writing to the fantasy bucket?  At the end of this post I'll advance another possible explanation for this "belief without evidence".

By the way, we have a much better understanding of how life can evolve once cellular life forms exist so the "trick" might be to explain how readily chemistry can give rise to very complex molecules.  However, that is not the only hurdle we have to vault: some scientists believe that other rare factors enabled the development of life on Earth (such as the shielding effect of Jupiter's gravitational field as an asteroid deflector).

But, before that, let me just touch on some of the scientific estimates for how much extra-terrestrial life there is out there.  First we have Andrew Watson (reported here) who, in 2008,  devised a model that suggested that the probability of (intelligent) life evolving anywhere else is less than 0.01%.  Watson's model took into account the steps from the formation of single-celled bacteria through to intelligent life with an established language.  I haven't managed to access his original paper but I think it must be problematic to estimate the chance of replicating molecules arising.

Next we have Edwin Turner and David Spiegel writing in the Proceedings of the National Academy of Sciences in 2012 (and reported here).  They too are pessimistic that life exists elsewhere and they specifically addressed the objection "But it exists on Earth, so why not elsewhere" using a Bayesian analysis.

In the other corner we have an opinion poll: 85% of those who answered a recent poll voted for extra-terrestrial intelligent life.  Most of the believers do not advance serious arguments.  Some scientists, like Stephen Hawkins, merely make the claim that alien life is extremely likely (see here).  Or Harvard physicist and SETI leader Paul Horowitz.  He stated in a 1996 interview with TIME Magazine, "Intelligent life in the universe? Guaranteed. Intelligent life in our galaxy? So overwhelmingly likely that I'd give you almost any odds you'd like."  These types of statement are extremely common.  They stem from a knowledge of how many planets there seem to be but ignore the difficulty of life getting started.

I would like to advance another reason why some scientists and many non-scientists are prone to advance the unsubstantiated opinion that alien life is common.  We live in a culture that is very different from those in times past in the respect that today we see ourselves as occupants of the universe rather than occupants of Planet Earth.  Even 200 years ago most people's metaphysical concerns were with their destination after death: heaven or hell; and these domains figured in their thinking as the only domains other than earth itself (and, from a practical point of view, this was true even of those who pointed their telescopes skywards).  As knowledge of astronomy percolated into the realm of general knowledge and (for many people) the simplistic beliefs in heaven and hell retreated our view of our place in the cosmos was transformed.  We now know to think of ourselves not as the centre of the universe but as a microscopic agglomeration of carbon molecules in one arm of one galaxy among billions.  This has produced a feeling of loneliness that the tales of science fiction, both in literature and in films such as Star Trek, can tap into.  In my opinion, the void that religious world views used to occupy has been filled by another fantasy in which we are not alone; and this is the reason why we are prone to believe that there is indeed life beyond our own planet.

A final thought.  Suppose it is the case that we are alone in the universe.  What does that say for the tussle between theists and atheists?  I think it's fairly balanced.  The theist's world view is strengthened because humankind really would be as special as their churches tell them.  But, for the atheist there is an interesting counterpoint.  Your fine-tuning argument argument looks to be in fragments now, does it not?

Friday, 28 June 2013

Demonising Edward Snowden

Make no mistake: the US administration is very badly rattled by the revelations that they spy, at a massive scale, on all its citizens and many non-US citizens.  One common response I see repeatedly in the blog/news sphere is the comment "We already knew this was happening; Snowden has done little to advance our knowledge".  No.  We might reasonably have suspected it.  We might even have had evidence of some aspects of it.  But it is a far cry from that to the now absolute certainty that it is happening and on such a vast scale.  What Snowden did was heroic, informative, and very dangerous for him.

Another common response from the administration apologists is to declare that innocent people have nothing to hide.  Many people have said, but it is worth repeating, that this is unbelievably naive.  We have all got something to hide.  We all need privacy.  I do not want my government to use my private secrets against me if ever I do need to defend myself against an accusation.  I do not want to use my toilet in a glass-walled room.  I do not want the opinions I had thirty years ago to be used against me.  Isn't that absolutely obvious?  And isn't it absolutely obvious that I have a right to that protection?

I would love the consequence of the NSA's activities coming to light to be a rueful admission by the US administration that they have gone way too far in ignoring the rights of their people.  Obviously that would in turn also be unbelievably naive.  So what do I expect to actually happen?

Well,  I don't think they will try to seize Snowden and "render" him back to the US.  I believe they will be much more subtle and wage a campaign in the media to gradually discredit him.  Already we see that officialdom's main response is to ignore the actual content of the leaks and concentrate on his breaking of US law.  That is going to continue to be the issue that they hammer home.  Unfortunately they have a huge advantage in a war of opinion.  They have thousands of opinion-moulders and free access to every major news outlet.  Snowden and his supporters cannot match that propaganda machine.

To begin with, many US journalists who might well have known or suspected what was going on and turned an expedient (or cowardly) blind eye have clearly felt some irritation at missing out on the scoop of the decade.  Where in the US Press do we see the staunch defenders of their constitution?  Where is the outrage at what has been done to their society?  Instead we see nuanced discussions about how or whether Snowden has broken the law.  Where is the recognition that it is clearly impossible for Snowden to get a fair hearing from the US authorities?

Furthermore the US administration can channel opinion pieces in droves to the media.  Overwhelmingly we shall therefore see articles attacking Snowden personally and down-playing the surveillance issues.  Oh - you didn't think that some US officials might not tow the administration line, did you?  Why should they when they haven't done hitherto?

Only time will tell whether the administration can successfully distract the public attention from the real issue.  If that sounds pessimistic let me close with one optimistic comment.  I have read many article in the US Press about Snowden and, although most of them ignore or downplay the enormity of the NSA's trampling over civil rights, the comments of readers are much more supportive.  What I hope will happen is that this evidently large supportive constituency of folk, who understand that the true issue is civil liberties not breach of employment contract, will eventually have an effect on those powerful vested interests who wish to sweep the real issue under the carpet.

Tuesday, 25 June 2013

A limerick for the NSA

Their actions are dirty and base
But now they've been put in their place
They cannot have known
When the whistle was blown
How much egg they would find on their face

Friday, 14 June 2013

American lies

The US is claiming that the Syrian government has used poison gas against their rebel opponents, and they are using this as a reason to engage in the conflict on the rebel side.  Should we believe or discount their claim?  I have no idea but the fact that the claim was made adds nothing to whether I think it might be true.  The problem is that, in providing excuses for war, the US has absolutely no credibility.  I think about the Iraq war and the lies they told then about weapons of mass destruction.

The FBI is claiming that Edward Snowden's revelations have jeopardized American lives.  Should we believe or discount this claim?  Again, I have no idea.  Again the claim adds nothing to whether I think it might be true.  This time I think about the Bradley Manning leaks about the conduct of the wars in Iraq and Afghanistan where the claims that they jeopardized American lives have not been confirmed (and you would certainly expect the State Department to be trumpeting them abroad if there was confirmation).

The problem about lies is that they destroy credibility.  In the case of individual liars we don't have to be all that worldly-wise to withhold our judgement or immediately discount their statements.  Which of us, when Bill Clinton proclaimed "I did not have sex with that woman", immediately accepted the statement?  Or which of us, when former US Senator Todd Akin made the claim that victims of  "legitimate rape" cannot become pregnant immediately accepted the statement?  We are free to accept or reject the personal statements of individual politicians usually without any significant harm done.  If Clinton or Akin tried to sell me a second-hand car I wouldn't buy it, nor do I given any credence to any other personal protestations they might make.

But when a state lies, especially one as powerful as the US, we all have a problem.  Of course, states have always lied but nowadays the stakes are higher.  

To begin with, if we live in a democracy, we are part of a social contract.  We pay our taxes and we observe the laws of our country.  In return we are entitled to be honestly governed by the people we have chosen in a plebiscite where we have weighed the pros and cons of competing candidates.  If governments violate their side of this bargain they endanger this social contract.  If they betray our trust they lose our cooperation and our goodwill.  This is a downside for a government who, presumably, wish to be re-elected.  The problem is that damage is being done to the political fabric in the longer term.  When politicians see that the only downside is to be kicked out in favour of the next lot they may, when their personal economic advantage is at stake, simply accept this downside.  Worse still is that the lying precedent that has been set can all the more easily followed by the next lot.  In this way, the social contract continues to be degraded.

In my opinion much has gone wrong with American governance over the last generation and especially since the 9/11 attacks.  The implosion of the former USSR removed a check on the behaviour of the US - and their greater power led to greater corruption.  Then, in the wake of 9/11, when (understandably) many Americans looked for very strong leadership, that leadership abused its power in the most shocking ways.  Internationally, they went to war for the flimsiest of reasons.  Domestically, they instigated a regime of surveillance and harassment against their internal enemies in which many innocent citizens became victims.  All of this is possible only because liars are not held to account.  They can lie with impunity and now a climate of scepticism shrouds all governmental announcements.  Unfortunately this is bound to lead to a rise in those who believe in conspiracy theories.  How ironic that the lies over the Benghazi attack might encourage people to believe that the US government brought down the Twin Towers.

So what should we as individuals do?  Obviously, it is a rare person who can make a great difference.  But we shouldn't just leave it to others and we mustn't be cowed.  At every opportunity we must denounce the lies - and not care that this will lead to much repetition.  Remember: governments should fear the people, not people fear the government.

Don't be afraid to say: Clinton was a liar over Monica Lewinsky, Bush was a liar over Iraq's weapons of mass destruction,  Obama lied when he said that Mitt Romney planned to raise taxes by $2000 on middle-income tax-payers.

I am not saying disbelieve anything the government says.  No, let's not encourage conspiracy-mongering.  But I am saying don't believe anything until you have some corroborative evidence.  Politicians need to win back their credibility - it's in all our interests.

Wednesday, 22 May 2013

A surprise in the world of prime numbers

The world of mathematics is buzzing with the news of a breakthrough in the theory of prime numbers by a virtually unknown mathematician named Yitang Zhang.  Prime numbers are numbers like 3, 5, 23, 97 which cannot be expressed as a product of two smaller numbers.  Prime numbers have been studied since the time of the Ancient Greeks and about 23 centuries ago the famous geometer Euclid discovered that there are an infinite number of them.  Since that time the study of prime numbers has produced some of the deepest results in the whole of mathematics yet these results are often very easy to understand (but the reasons the results are true are another matter).  

It is easy to see that 2 and 3 are the only consecutive prime numbers (because if you have two consecutive numbers one of them must be even but 2 is the only even prime number).  What about prime numbers differing by 2?  Here there are more: (3, 5), (5, 7), (11, 13) for example and there are many other so-called twin primes.  But are there are an infinite supply of twin primes?  Despite hundreds of years of research we don't know the answer to this innocuous question.

What about pairs of prime numbers that differ by 4 (like (3,7) or (19, 23))?  We don't know if there are infinitely many such pairs either.  And if you replace 4 by any other even number at all we still don't know.  For example, we don't know whether there are infinitely many prime pairs that differ by 30 say (like 31 and 61).

Enter Yitang Zhang.  In his 50's he is definitely beyond the age where most mathematicians make their mark and until now he has been practically unknown.  After he got his PhD in 1992 he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.  But he never gave up doing mathematics and eventually was appointed at the University of New Hampshire.  There he pursued an unremarkable research career with no publication since 2001 but he was loved by his students apparently because he set easy exams.  Now he has burst onto the world's mathematical stage with a result that has surprised all the experts.  He has proved that there is some number k that "works" for prime pairs.  We don't know that k=2 or k=4; all we know is that k is less than 70 million.  And for this k, whatever it is, Zhang has proved that there are infinitely many pairs of prime numbers that differ by k.

The reason that this has excited mathematicians is that no result like this has ever been proved before and it gives some hope that the original prime twin problem might eventually be solved.  It is also a surprise result in that it hardly ever happens that a giant step like this is taken by such an obscure mathematician.  When it has happened before (such as for the Indian genius Ramanajuan) the newcomer is usually much younger since mathematics at its most creative is usually a young person's metier.

Zhang's theorem probably won't have much practical use.  If you like problems about ages and birthdays here's a consequence that might be appealing.  Somewhere, sometime, there were two mammals of different ages and there will be an infinite number of years when both their ages are prime numbers.  Might there be two human beings with this property?  We don't know because humans haven't been around for 70 million years whereas mammals have.

Sunday, 19 May 2013

Freedom of information: a window?

You may not agree that the glory days of newspaper journalism are long gone.  Perhaps you think that the Washington Post's Watergate investigation which led to the eventual resignation of US President Nixon was less than stellar reporting.  Maybe you think that the meek way almost the entire mainstream media accepted the US and UK lies about Iraqi weapons of mass destruction as an excuse to go to war was an isolated collective error of judgement.  If so, you and I have quite different opinions on the way that newspaper reporting has changed over the years.

But perhaps we agree on something else: that the Wikileaks revelations told us things about how the US conducted its wars in Afghanistan and Iraq, and how its emissaries around the world conducted business with foreign powers gave us an unprecedented grandstand view of events that most of us never hear about.  Perhaps you think, along with Senator Joe Lieberman, that the leaks were "outrageous, reckless and despicable" but I hope you would fall short of Sarah Palin's call to pursue Julian Assange with the same urgency that Al-Qaida leaders were pursued, or Congressman Mike Rogers' threat to have foreign national Assange executed for treason against the US.  Nevertheless the scale of what we learnt via Bradley Manning and Julian Assange cannot be denied.

The point of this post is not to persuade you to my view (that Manning and Assange are among the great heros of our time).  It is more to alert you to the fact that, if you care about knowing what is going on in the world, you have just lived through a short period where we the people had information about how great powers operate of a magnitude that we might not see again for a very long time.

The intense efforts that the US and UK governments went to in order to suppress events in the Afghan and Iraqi wars, and the publication of US diplomatic cables tell us how much they were embarrassed by the Wikileaks collaboration with the Guardian and other newspapers.  I have no doubt that they have ramped up their security to prevent a repeat.  In any case the US retaliation against Manning has been so severe that other potential leakers of his persuasion might well think again (certainly Julian Assange himself is in no hurry to be extradited to Sweden in case he is handed over to US authorities).  Furthermore the US authorities are now pursuing the Associated Press organisation by subpoena-ing phone records that might bear on the CIA successfully thwarting a plot by al-Qaeda in Yemen to blow up a U.S. jetliner (this is not the first such aggressive subpoena act).

So don't expect anything as informative as the Wikileak bonanza to strike again for a very long time.
Therefore my take home message is this.  Have a good luck at what we already have.  You can go to the Wikileaks web site itself.  However you might find this is more than you can handle!  It is vast.  Instead you might try the Guardian site which is very well-organised and will tell you also a lot about the politics associated with the reaction of the US and the UK.

Saturday, 20 April 2013

A serious case of poor economic analysis

In this post I want to give some more publicity to a story that has, so far, only been run in specialised economic news columns.  It is a dramatic one that suggests much of the world has taken a wrong financial turn partly because of one influential academic paper, around which there are now some serious questions.

Two economists,  Carmen Reinhart and Kenneth Rogoff at Harvard University, released a paper in 2008“Growth in a Time of Debt.” whose main conclusion was that countries whose public debt was over 90% of their Gross Domestic Product have below average growth rates, slightly negative in fact.  Their paper has been widely cited (454 Google Scholar citations as of 21 April, 2013).  In addition it has been used by government exchequers to justify severe austerity measures.  The most significant such example is when Paul Ryan, the Republican chair of the US House of Representatives budget committee, a notable economic hawk who has pushed for rapid fiscal tightening, cited the Reinhart-Rogoff paper as “conclusive empirical evidence that total debt exceeding 90 per cent of the economy has a significant negative effect on economic growth”.  In addition, as reported in the Guardian, British Chancellor George Osborne has repeatedly said that his monetary policy was highly influenced by the paper.

Last week there was a bombshell.  Three researchers from the University of Massachusetts, Thomas Herndon, Michael Ash, and Robert Pollin, published a paper "Does high public debt consistently stifle economic growth" which seriously undermined (some have said "tore to shreds") the Reinhart-Rogoff conclusions.  The new paper challenges the Reinhart-Rogoff methodology in two ways: by criticising the ways in R&R excluded some data that did not accord with their conclusion, and by criticising how they weighted their summations.  You can read the actual details in the source articles.  More sensationally, Herndon et al found an error in the spreadsheet that R&R had used to construct their main summative table.

In their response R&R who were obviously deeply embarrassed by the spreadsheet error nevertheless argued that their main conclusions were still sound and a debate is now beginning between the two sides.  So far Herndon et al robustly maintain that the R&R study is not only flawed in the way that they originally claimed but that the results are almost the opposite of those claimed by R&R.

I hope that the debate does not simply recede into exchanges between academics guarding their reputations.  This issue is of fundamental importance to the way in which we regulate our economies.  I am so far dismayed by the entrenched attitudes of those who originally espoused the R&R conclusions.  In particular the response from George Osborne's office has been abysmal - just a claim that their economic policy does not rest on one academic paper and that "the majority of economists" back Osborne's strategy.

It seems to me that economists, and the entire discipline of Economics, faces a challenge.  They need to sort out for policy-makers just what the situation is vis a vis debt versus growth.  I know this might be a big ask but they must do their level best.  If we see two camps emerging divided by political affiliations we would be tempted to kiss goodbye to any pretence that Economics is a science.

Wednesday, 17 April 2013

Prime conjunctions

I have 4 children and around 10 years ago their ages were 17, 23, 29 and 31.  This period lasted only for a few weeks until one of them had a birthday but I made much of it at the time for these ages are all prime numbers.  One day they will be 47, 53, 59 and 61 - another prime conjunction and I hope I shall still be around to celebrate it.  I will definitely miss the next (and so will they): 167, 173, 179, 181.

Of course I am extraordinarily fortunate in having so wonderful a quartet of children.  Am I fortunate in another way:- that their ages allow such prime conjunctions?  We shall soon see.

Let's consider a set of n children (think of n as being 2, 3 or 4 if you like).  How likely is it that one day they will all have prime ages?  I'm going to simplify matters in three ways.  The first simplification is really unnecessary in practice: it is that we'll assume no two children were born at exactly the same time of day on the same day of the year.  The second simplification is that we'll only worry about odd primes.  Ignoring the prime 2 is not such a big deal.  This special case is actually quite significant if you are just interested in whether the children have a prime conjunction at all but not if you are more concerned with whether they have lots of them.  The third simplification is apparently quite bizarre but at the end of the post I will explain that it is not as bizarre as all that - so bear with me.  This simplification is that rather than thinking about prime conjunctions we think about occasions (odd conjunctions) when all the children's ages are odd.

We want to think about their ages and we have to write down these ages in some order.  Rather than that order being highest to lowest or lowest to highest I shall write them down in order of birthday throughout the year.  For example three children born on 2 March 1991, 6 November 1994, 4 May 1997 will be ordered  with the March birthday coming first, then the May birthday, then the November birthday.  So, at the beginning of this year, their ages were 22, 15, 18.  These remain their ages until 2 March when the 22 becomes 23.  Another change (from 15 to 16 occurs on 4 May) and another change occurs on 6 November.

In terms of evenness and oddness it was Even, Odd, Even at the beginning of the year and then it went
Odd, Odd, Even
Odd, Even, Even
Odd, Even, Odd
and, continuing to track the changes in 2014,
Even, Even, Odd
Even, Even, Odd
Even, Odd, Odd
and then we come back to
Even, Odd, Even

Put more concisely we begin with a list of E's and O's.  We change the first one, then we change the second, then we change the third, and then we go back to the start of the list and change the first (then the second, then the third).
and now the cycle repeats.

For this particular set of birthdays we never find OOO so these children are never all of an odd age (and therefore never all odd prime ages (as it happens, it doesn't matter in this case whether we allow the prime 2)).

OK.  Let's jump to the general case.  Now we have n children.  As before we list their evenness or oddness of age by order of their birthday.  Suppose at the beginning of the year that gives us a bunch of evens followed by a bunch of odds.  At some stage in the year those initial evens will all have changed to odds and all of them will be an odd age.  Instead suppose at the beginning of the year we have a bunch of odds followed by a bunch of evens.  Part way through the year everyone will now be even and so, a year later, everyone will be odd.

But what if we don't have one of these types of initial odd/even lists.  If that happens then either we begin with an even, later in the list we have some odd, and still later we have another even (or we might have a similar case with the roles of evens and odds exchanged):
E ... O ... E ...
We start our process of changing the evens and odds one by one from the beginning.  It is easy to see that, no matter where we are in the process, the three symbols at these places cannot all be equal; so we can never get all odds.

It is not too difficult to work out that 2n even/odd lists have the property that they consist of a bunch of one type (even or odd) followed by a bunch of the other type (odd or even).  But there are 2n sequences in all.  So what that means is that, with n children, the chance of their having an odd conjunction is n/2n-1.  And notice that if they do have an odd conjunction then they will have others at two-yearly intervals - so many of them.

Therefore, given that I have 4 children, the chance of them having an odd conjunction is exactly 50%.

But, wait a minute, I began by asking about prime conjunctions!  All I have done is analyse the chance of getting an odd conjunction.  The passage from odd to prime leads us to some deep unsolved questions in number theory.

Suppose I have two children with the younger born between 1 and 3 years after the older.  Then some of the time their ages will differ by 2 and every two years there will be a period when their ages are both odd.  These children will have many prime conjunctions, the first few being (3, 5), (5, 7), (11, 13), (17, 19).  The Twin Prime Conjecture states that there are infinitely prime conjunctions in this case (prime numbers differing by 2).  It is a very long-standing unsolved problem and most mathematicians believe it is true.

What if I have two children born more than a year apart?  Then every two years their ages will differ by an even number k.  Now we are in the realm of Polignac's Conjecture: whether there are infinitely many primes that differ by some specific even integer k.  With more than two children we come to an even more general conjecture by Dickson.

What all this means is that if your children do not have an odd conjunction (and we have seen above how this may be tested, and how likely it is) then you will never see a prime conjunction. But, more usefully, if you are looking for a prime conjunction, you should first find an odd conjunction (if any) and then starting adding two to it repeatedly hoping to find a set of odd numbers that are indeed all prime. This is not guaranteed (e.g. 7, 9, 11 - all increments include a multiple of 3). However Dickson's theorem (which I will write about in a future blog) provides some refined help if needed. Finally, unless a large number of mathematicians are going to be very surprised indeed, if your children have "several" prime conjunctions, they will have infinitely many prime conjunctions.

Update 22 May 2013

The mathematical world is buzzing with the news that a virtually unknown mathematician Yitang Zhang at the University of New Hampshire has proved that there is a number k<70,000,000 for which Polignac's conjecture is true.  So we know now that somewhere sometime there must have been two mammals (because mammals go back more than 70 million years) who enjoy an infinite number of prime conjunctions.

Friday, 29 March 2013

From Daniel Ellsberg to Bradley Manning

When I post a blog entry I like to have something more to say than just a rehash of known facts.  But today I don't.  I write out of rage for the treatment of Private Bradley Manning, a hero who has put his freedom in jeopardy for principles that all of us who value honesty and decency should share.  Manning is currently on trial for sending to Wikileaks material about the war in Iraq and Afghanistan.  The most infamous disclosure was footage of a US helicopter firing on civilians in Baghdad with a voice-over from the operators whose vicious delight in killing their victims is evil and sickening.  The material he released seems not to be compromising to US military strategy, nor to endanger the lives of serving personnel; but it gives a perspective on US thinking that must be deeply embarrassing for the High Command.

In my opinion Manning's actions should be applauded.  But, not only has he been put on trial, he has been treated with cruelty out of all proportion to the nature of the offences - including over a year of solitary confinement under brutal conditions.

Plus ça change, plus c'est la même chose.  Over forty years ago another whistle-blower, Daniel Ellsberg, was arraigned by the US courts.  His crime was to have leaked the so-called Pentagon Papers that contained the damning revelation that the US government had known years earlier that the Vietnam war almost certainly could not be won, had lied to the public and had continued to wage the war causing tens of thousands of US deaths.  At Ellsberg's trial it came to light that, in an attempt to discredit him, the US administration had broken into the office of his psychiatrist and had installed illegal wiretaps.  As a result of this incredible persecution the judge threw out all charges.

Both Ellsberg and Manning broke the law.  Yet with the perspective of history the vast majority of us on all points of the political spectrum applaud Ellsberg.  Without his courage we would not have known about the crimes of the Johnson and Nixon administrations, and there would have been no Watergate.  But at the time Ellsberg was excoriated in the same terms that Manning is now suffering.  I am absolutely sure that history will judge Manning just as favourably as we now judge Ellsberg.

Finally, an interesting and uplifting postscript.  Daniel Ellsberg has had a distinguished career as a proponent of open government and in March 2011, two weeks before he turned 80, he showed that his passion for social justice burned just as brightly: he was arrested in a protest demonstration against Manning's incarceration.

Sunday, 17 March 2013

The Swerve

I've just finished reading "The Swerve" by Stephen Greenblatt.  It's a fascinating account of how in the early 15th century a copy of "On the Nature of Things" by the Roman author Lucretius was discovered in a remote monastery.  The book is a powerful and passionate poem inspired by the Greek philosopher Epicurus and had long been thought to be lost forever.

Greenblatt brilliantly conjures up the atmosphere of late medieval Europe by writing about the life and times of the man, Poggio Bracciolini, who made the discovery.  He goes on to describe how the prevailing theology of the day was challenged by exposure to the ideas of Epicurus and makes a good case that "On the Nature of Things" was one of the drivers towards the more enlightened ideas we have today.  I encourage you to read "The Swerve" as Lucretius' original "De Rerum Natura" is possibly a little inaccessible unless you happen to be a Latin scholar.

For me the deepest impression was about the contrast between the philosophy of Epicurus and the teachings of the Christian churches.

We tend to think that "Epicurean" means unthinking abandonment to licentiousness.  But that completely distorts the Epicurean message.  Indeed that message does advocate that one should pursue pleasure.  However, the point of the pursuit is to live life to the full because this life is all there is.  So not only the bodily passions are important but also the passions of the mind and the satisfactions of creating and living out one's own thoughtful purposes.  Everything is made of atoms and when our body dies the atoms are reformed.  The soul is also made of atoms and it too does not survive our deaths.  Isn't that incredibly modern?

Contrast that with the central message of Christianity: bear your privations in this life so that you may enjoy the eternal one that follows (because if you don't you'll be enduring an infinite torment afterwards).  I literally shudder to think of how many lives have been blighted by this message.  How much effort has gone into refining the Christian dogma and imposing it on its adherents.  We could have begun the enlightenment 1700 years earlier if the Christian religion had not had the supreme good luck to be adopted as Ancient Rome's official religion.

But don't read my ranting: go out and buy Stephen Greenblatt's masterpiece.

Wednesday, 6 March 2013

Deductive and inductive inferences 2

In my last post I extolled the virtues of inductive inference and I verged on claiming that it was a more useful tool than deductive inference.  In this post I want to say something about deductive inference, where its strengths lie, and comment on why its use is sometimes more difficult than we expect.

In its purest form deductive inference starts with various statement known to be true, carries out a logical sequence of steps using these statements, and arrives at one or more further statements.  If the initial statements really are true and the reasoning steps are logically valid then the new statements will also be true.  Further deductive inferences can then be made starting from these new statements and, repeating this many times, very long chains of deductive inference can be created.

Mathematics is just a collection of these inferential chains.  Deep Mathematics is when the chains are long and 'good' Mathematics is when the resulting statements are deemed to be interesting (which is often a subjective judgement).  This pure form of deductive inference does occur outside Mathematics but usually the inferential chains are very short; however, when it can be used, it has the edge over inductive inference in that the statements it arrives at will be believable beyond doubt.

However, in my previous post I promoted both deductive and inductive inferences as ways of making good judgements, or making good decisions as often as possible.  The pure form of deductive inference described above seems to be an 'all or nothing' process (and, in that case, will usually be a 'nothing' process because if even a single one of the links in the inferential chain cannot be made then the final result will be worthless).

What rescues deductive inference from being largely useless outside Mathematics is the notion of probability which gives us a way of ascribing a likelihood of truth to the conclusion we have reached.  This is not the place in which to write about probability but the sort of thing I mean can be appreciated by a simple example.  If I know it is 90% likely (probability 0.9) that my roses have black spot fungus and my fungicide has a 60% success rate (probability 0.6) at treating black spot then, by applying the fungicide I have a 0.9 * 0.4 = 0.36 chance of  my roses continuing to have black spot after applying the fungicide; therefore no black spot is a 1 - 0.36 = 0.64 chance.

If you didn't follow that, don't worry.  The main message of this post is that such reasoning is not easy because our brains don't seem to have a good intuitive grasp of probability.  Formal training in probabilistic assessment is almost essential in order to reliably calculate the odds that any particular event will occur.  It alerts you to the common logical fallacies but is definitely not a guarantee you will not fall into one.  The remainder of this post is a description of some of the counter-intuitive conclusions that you might come to in assessing the likelihood of an event.

The Birthday paradox.  

At a party or 24 or more people it is more likely than not that there will be two people with the same birthday.  Most people find this surprising.  The justification is no more than a few lines of probabilistic calculation but you have to know what you are doing.

The Monty Hall problem.  

A game show contestant is asked to pick one of three doors.  One of the doors conceals a valuable prize and there is nothing behind the other two doors.  The contestant makes their choice and then the game show host (who knows which door conceals the prize) opens a door that was not picked to reveal no prize behind that door.  The contestant is asked whether they wish to change their mind and go for the other unopened door.  Should they change their mind or not?

It is very tempting to believe that the contestant neither increases or decreases their chance of winning by changing and that the prize is behind either remaining door with a 50-50 chance.  In fact they double their chance of winning by changing their choice.  This so counter-intuitive that (see this Wikipedia article) almost 1000 PhD graduates, including the famous Paul Erdös, were fooled.

Boy girl combinations

If someone says to you "I have two children and (at least) one is a boy" what is the probability that they have two boys?  The logical trap is to reason that the remaining child is as likely to be a boy as a girl and therefore the answer is 50%.  In fact the actual probability of two boys is 33% (probability 1/3).  The reason that this problem fools people is that the 'sample space' (the set of equally likely different possibilities from which one has to choose) is not what, at first, you think it might be (indeed, this is often the pitfall that probability presents).  The sample space is BB, BG, GB (these combinations being the order by age of the children).  Only one out of three represents two boys.

See how subtle this is?  Had you been told "I have two children and the elder is a boy" then the 50% answer would have been correct.

But if you thought that was bad enough enough consider being told this.  "I have two children.  One is a boy born on a Tuesday".  Now what is the probability of there being two boys?  The answer is 13/27.  This astonishing answer is discussed in a  Science News article along with various caveats and is worth reading for reinforcing my warnings about the danger of trusting your probabilistic intuition.

If you are still with me but are beginning to feel that these apparent paradoxes are not important in the real world please think again.  We have to make decisions every day and most of the time we act with imperfect knowledge.  But that doesn't mean that we cannot improve the choices we make.  We have to realise that some choices are more likely to be successful than others, and that there are ways to find these successful choices more often than not.  The science of Probability is the key to such informed decision-making.  Should we eat organic foods?  Should we get a 'flu shot?  How much Life Insurance should we buy?  The list is long.  And we should remember that our gut feel about probability is very likely to be unreliable.

Deductive and Inductive inferences 1

Suppose you observe someone tossing a coin and 'Heads' comes up every time in their first 99 tosses. What would you say about the 100th coin toss?  One answer is that there have already been way too many 'Heads' and therefore it is about time to call 'Tails'.  Another answer is that, as coin tosses are independent of one another, the next result is as likely to be 'Heads' as 'Tails'.  A third answer is that the coin is evidently biased and the next result will be 'Heads'.

If there any readers who favour the first alternative they had better stop reading now as the rest of this post won't make much sense.  The second and third alternatives characterise two different types of inference: deductive inference and inductive inference.  It seems that they demand somewhat different abilities. In the situation I have just given, either might be deployed but sometimes one is overwhelmingly more effective than another.

I want to discuss the making of inferences (or coming to a judgment, or coming to a decision) where one does not have complete information.  The goal is to try to make the right inference as often as possible.

In the sequel to this post I shall concentrate on deductive inference and how we humans seem to have an ingrained blind spot.  Deductive inference is what, as a mathematician, I have been trained in but in this first post I want to argue that a good inductive reasoner may very well outperform a good deductive reasoner more often than not.  My arguments begin with how we have managed to understand the natural world and the rate at which our understanding has advanced throughout history.

Of our present day sciences Mathematics is a huge outlier in that virtually nothing that it has discovered has been thrown away as invalid.  Our Physics, Chemistry, Astronomy, Biology and Geology have existed (although not with these names) for just as long as Mathematics but almost everything we thought we knew about them 1000 years ago is now known to be entirely wrong.  The reason for Mathematics being such an exception is that it proceeds by deductive inference (I oversimplify a little of course but, bear with me, it is clearly different to the other sciences).  The other subjects had no chance to make the same progress while they were being pursued by deductive inference because there was an axiom that completely undermined our thinking.  That false axiom was, of course, religion.  Deductions based on false premises are correct only by great good luck so it is no surprise that advances in the other sciences had to await our discarding (or ignoring) the religion premise.

It was axiomatic that God created Man in His own image.  How then could evolution even be contemplated?  It was axiomatic that the Earth was the centre of the Universe.  How then could astronomy develop?  It was axiomatic that the Earth had been created for Man.  How then could Geology say otherwise?  But the religion axiom only applied to the natural world, not to the abstractions of Mathematics so it was not so hamstrung.

But, with the Renaissance and then the Enlightenment, the Church's grip on the minds of the people loosened to the extent that the religion axiom could be ignored (if not dismissed).  Coupled with this occurred a revolution in thought: the rise of the experimental method.  Experiments are not deductions.  They are gatherers of information.  A single experiment rarely gathers complete information about a natural phenomenon but it may suggest a hypothesis.  Further experiments can then be conducted to test the hypothesis and sometimes a hypothesis survives all of these experiments and we can tentatively claim that we have discovered something.  Of course I am aware that the connection between experiments, evidence, knowledge, falsifiability etc. is the subject of much philosophical debate but the simple picture I have presented is not very controversial.  To put it another way our experiments allow us to infer knowledge about the natural world - and, quite obviously, this is inductive not deductive inference.

I will not deny that deductive inference has played a part in the successes of the non-mathematical sciences but I suggest that inductive inference through the experimental method has been the major player.  So there's a very good justification for inductive over deductive inference:  the triumph of most of science and its spin-offs for the way we live such comfortable lives.  But it doesn't stop there!

As we progress through our lives we acquire more and more experience in dealing with situations that call for judgement.  This experience can be very hard to describe.  How can we describe the advice from a seasoned fisherman on where to cast our line?  Does he have just a 'gut feel'?  More likely he has many memories of similar conditions that prevailed where fish usually gathered in the spot he recommends to us.  No guarantee - just the normal use of inductive inference even if carried out subconsciously.

Here is another example, personal for me and shared by many bridge players.  How do I decide what to do at a particular point during the play of a bridge hand?  As I age I get increasingly better at making the right choice (but I definitely know that my deductive abilities have waned over the years).  I can do this because I have seen many similar instances in the past and can rely on inductive inference much of the time (and, in any case, there is often not enough time to go through a full deductive analysis).

In other words, personal experience is often grounded in inductive inference.  We live and prosper by it.

Finally, if I still have not persuaded you of the merits of inductive inference, consider again the coin-tossing scenario I began with.  Further suppose that you had been told that the coin was unbiassed.  Would this make a difference to what you might think the 100th toss would be?  I submit that it may very well not make any difference.  Yes, you have been told something but all the evidence points to your having been told a lie.  Just as we have released ourselves slowly and painfully from divine revelation and rejected a false axiom, so here we should trust the evidence.  That coin is biassed - call 'Heads'!

Tuesday, 12 February 2013

On Richard Prosser

NZ First MP Richard Prosser has rightly come in for international condemnation for his Investigate Magazine column in which he suggested young Muslims shouldn't be allowed to travel on Western airlines because "most terrorists are Muslims".  It doesn't matter how he hedged this statement and his subsequent mealy-mouthed apology has not repaired the damage to New Zealand's reputation as a tolerant and inclusive society.

I became a NZ citizen three months ago.  During my application for this privilege I was often asked why I wanted to be a citizen rather than just a permanent resident.  My answer was always the same: I love the tolerant spirit of New Zealand, its relative absence of a class system, and its secular system of government.  That Richard Prosser should make such dishonest remarks, and still more that he serves as an MP, makes me extremely distressed for our international image.

Dishonest remarks?  Absolutely.  What's a terrorist?  Surely someone who terrorises innocent people whether the terrorist is a member of a guerilla group, an outright criminal, or a member of some country's official armed forces.  How about the terror inflicted by US drones in Pakistan and Afghanistan, or the brutality of the Israeli government to the Palestinians, or the genocide of the pygmies in the Congo Civil War; these are just a few examples.  Terror acts committed by Muslims are in a minority.

Even if Richard Prosser made a full and contrite apology which showed a genuine acceptance that he got things badly wrong he has nevertheless shown such poor judgement that he should be sacked immediately.  But to compound the aftermath his party leader Winston Peters has said there was an "element of truth" to Mr Prosser's comments and says he does not believe Prosser should apologise.

What a disgusting pair they are.

Sunday, 10 February 2013

On mathematical Platonism and faith

Mathematical Platonism is, roughly described, the belief that the entities considered in mathematics (numbers, functions, algebraic systems, etc) have a reality beyond their existence as neuronal configurations in our brains.  Non-mathematicians are often disposed to reject the belief ("Where do these entities exist?" seems to be a rather compelling objection) but many practising mathematicians are Platonists, and many more (maybe a majority, but I don't have any statistics to support this) behave as though they are Platonists.

Another way of expressing the belief is to say that mathematics is "discovered" rather than "invented".  In his short essay Barry Mazur calls this "The Question" that all mathematicians come to at some point in their metaphysical speculations.

Why does Platonism have such a strong foothold among mathematicians?  I am fascinated by this question because when I was a young pure mathematical researcher I would often maintain that the entities I thought about had an independent existence (and this was when I was only dimly aware of Platonism at all).  I had, for example, an absolute conviction that when humankind came across star-faring species they would have a mathematics in which simple groups (a particular interest of mine at the time) would play the same fundamental role that they play in terrestrial mathematics.  Why did I hold this view so strongly?  In this post I want to speculate about this psychological predisposition rather than whether it is defensible.

I suggest that Platonism has such a powerful grip on the mathematical mind because mathematical discourse is packed with the language of discovery rather than the language of invention.  This language biases us to subconsciously accept that we are finding out about things that already exist.  This tradition is very deeply rooted in our mathematical discourse and certainly appears in Euclid ("The sum of the three interior angles of a triangle equals two right angles": no doubt here that a triangle exists somewhere outside our minds).  In our modern mathematics how often do we say "There exists..."? - we even have a mathematical symbol for this phrase.  Mathematics is written in a certain style and this style abounds with phrases that suggest discovery of already existing objects.  For example here is paper published today that I selected at random: here is its abstract.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.
I suggest that mathematicians cannot read that abstract without having their Platonic tendencies confirmed.  This and the entire manner in which mathematics is written explains why Platonism is the default subconscious belief of so many working mathematicians.

What this tells us is that a Platonist ought to consider his/her position in the light of these strong linguistic pressures that produce cognitive bias.  Maybe they can mount a metaphysical defence (and I confess to a wistful hope that they can).  But if they cannot it is only honest to admit that they are holding similar opinions to a committed theist.  And that brings me to the final point I wish to make.

A few studies have shown that, among scientists, mathematicians tend to hold a theistic stance more so than others.  A few years ago the US National Academy of Sciences conducted a survey about god beliefs among their members.  Apparently 14.6% of mathematicians believe in god whereas the figure for biologists is 5.5%.  I am sure we should be cautious about such figures but other surveys have also shown that mathematicians tend to belief in god in higher proportions than other scientists.

Is it not plausible that someone who has committed unthinkingly to a position of faith in the real existence of mathematical entities will be more prone to commit to a faith in the real existence of supernatural beings?

Friday, 1 February 2013

The poverty of faith

I live in Otago in the South Island of New Zealand.  We have a reasonably good local newspaper, The Otago Daily Times, that informs us of local, national and world issues in decreasing levels of detail.  On 1 February it published an opinion piece from a local resident called Ivan Grindlay entitled
No room for fantasy if you understand dynamics of God's plan.  Mr Grindlay is an elder at the Caversham Community Church. His article was a summary of a bizarre theology that told of God's intention for humanity, its perversion by the "defection" of Adam, and an ultimate resolution as the Second Coming brings the nation of Israel back into the Christian fold.  It wasn't clear whether this was Mr Grindlay's personal theology or that of the Caversham Community Church. At any rate this world-view cuts off its adherents from the understanding that we now have about the natural world.  The explanation of our world and the huge cosmos in which it is situated is grander by far than the explanations in Mr Grindlay's theology. Many readers must have smiled rather sadly that such arbitrary fantasies still hold the minds of some people in thrall.

Perhaps among the smilers will be found some members of mainstream religious beliefs. Good Christian folk who regularly attend their church, recite the Lord's Prayer and the Apostle's Creed, engaged in good works to support their own community and possibly communities overseas. Ladies and Gentlemen of these benign religious persuasions you are complicit in Mr Grindlay's ignorance.  Your own supernatural beliefs are the ambient culture that nurtures Mr Grindlay's less mainstream opinons. While you, through a mental contortion in which you cherry-pick those pieces of scripture that are convenient to believe, have access to the understanding of our wonderful universe (biology and the unity of all life because of the DNA molecule, physics and the mathematical explanation of nature's laws, astronomy and the workings of a cosmos now unimaginably larger than ever envisioned in religious tracts) you are doing more credulous folk a grave disservice. Shame on you.

Mr Grindlay has chosen a certain portion of scripture on which to found his world-view. Anglican Christians choose another portion while Roman-Catholics choose yet another (and even this ignores the schisms within these broad churches). Still another portion is seized on by Sunni Moslems, and again another by Shi'a Moslems. And then there are Jews, Hindus and many other groups each elevating certain historical writings to the status of divine revelation.

I called Mr Grindlay's fantasy an arbitrary one in my first paragraph. Indeed it is arbitrary. But so are the fantasies of every other religion. What distinguishes them from the Mr Grindlay fantasy is that they have more adherents and with numbers come respectability. But all of them to one degree or another prevent their followers from truly appreciating the world we live in - the only one we will ever have.

Thursday, 31 January 2013

It's hard to kill your neighbour

There is a series of psychological thought experiments (the Trolley problem) in which participants are asked moral questions which produce increasing levels of discomfort.  These experiments - investigations of philosophical consequentialism -  are a litmus test to the commitment of a utilitarian's position but this post is not really about that issue so much as about how it is connected to our feelings about personal and impersonal violence.

Two of the trolley problem scenarios concern what you should do if you see a trolley on some rail tracks hurtling towards a group of 5 people who will die on impact.  In one scenario you have a lever that will divert the trolley onto another track but, on the other track, there is a single person who will then be killed by the trolley.  Should you pull the lever? In another scenario you are standing on a bridge alongside a fat man and you know that, by toppling him over the bridge, he will halt the trolley but die in the process.  Should you push the man?

Even if we cannot come to a conclusion about what actions are right for these scenarios most of us will feel that the second scenario presents a more challenging moral conundrum, and its resolution has been debated at least since the time of Thomas Aquinas.  But philosophy and morality aside the scenarios point to an aspect of our innate makeups that we need to acknowledge if we are to make sense of why individuals and societies act as they do under stress.  We might call this aspect of us our human empathy.

I offer the speculation that whatever it is in our psychology which makes us more leery of positive action in the second scenario is also what has contributed to the greater slaughter of people (military and civilians) in modern warfare.

Of course the huge mortality rates in modern warfare are definitely connected to another factor: the much greater killing power of weapons of mass destruction (WMD).  (By the way, the definition that WMD just means nuclear weapons is, I think, completely arbitrary.  The London Blitz: 40000 dead , the Dresden firebombing: 25000 dead, the Hiroshima bomb: 100000 dead but estimates wildly vary).

However I think that if WMD were not activated at a distance, if the immediate connection between opening a bomb hatch and the resulting killing could be witnessed, then WMD would not be deployed as they have been.  Imagine having to achieve the same mortality figures if each life had to be taken by slitting the throat of the victims.  It is not simply that it would take more time: it would be viscerally opposed by our human empathy.

Of course, war warps our human empathy.  To be successful in war, military commanders have to force their troops to carry out actions that in peace-time they would not consider.  That is one of the reasons why very strict disciplines are practised.  And it must be one the reasons why, when the conflict is over, so many returning war heros are deeply traumatised (for example, in 2012, more US serving troops committed suicide than died in war action).

So there is the thesis: we are more leery about pushing a fat man off the bridge because it is more "hands-on" than pulling a lever, and this is for same empathetic reason that we would be more leery of knifing a civilian rather than bombing them.

In my opinion this human empathy is embedded in our brains.  Examples of its functioning exist in all areas of our society.  For example, it is a common phenomenon that when two individuals of different nationalities, skin colour, or political persuasion get to know each other they often overcome antipathies they might have felt towards one another before their empathetic sense was triggered.  This all suggests that our human society should become more harmonious the more we trigger our empathetic response - and this means that, as far as possible, we should engage with those we find the most at odds with our own predispositions.

Tuesday, 22 January 2013

On historical causes

One of the most boring episodes of my high school career was the few weeks in fifth form history when we were taught the causes of the French Revolution.  To this day I really can't remember much of what we were taught but, looking up Wikipedia today, I read that one cause was the high level of taxation of the peasant class while the nobility led lives of conspicuous consumption.

But what exactly does "X causes Y" mean in the context of history?  Can we deduce that if the French peasantry had not been taxed highly then the revolution would not have occurred?  Of course not.  In fact we cannot know the answer to that question because we cannot rerun history.  Does it mean that there is a tendency for Y to occur when X is present in any historical situation?  Perhaps so, and this is indeed a hypothesis that could be tested by an examination of historical situations where X was present.  But the dataset might still be unconvincingly small.

I was led to these thoughts by two massive books that I've read over the last two years.  One was "Guns, germs and steel" by Jared Diamond, the other was "Ideas, a history from fire to Freud" by Peter Watson.  Both of them are staggering accomplishments that offer a comprehensive account of how hunter-gatherer societies developed into modern-day societies.  Inevitably the authors colour their accounts with opinions that can only be hypotheses, despite their persuasive arguments to support their opinions.

Diamond's main intent is to give an explanation of the factors that contributed to the greater success of European civilisation compared to its historical counterparts.  He completely rejects explanations based on racial differences and argues convincingly for geographical differences.  Among his explanations are, for example, the greater number of domesticable animals in the Old World compared to the new; and the larger number of plants that could be farmed for food production.  I strongly recommend the book for it is thoughtfully presented and hugely informative.

Watson's book offers a number of reasons why some ideas surfaced at a particular moment in history, and why they originated in a particular country.  Again it is all plausible coherent stuff and I greatly appreciated how he related the historical contexts to the development of key ideas.

But, just like the causes of the French revolution, what does it mean to say that a particular constellation of geographical factors made the Old World develop more quickly than the New World?  What does it mean to say that a particular constellation of social and historical conditions caused a certain set of ideas to be developed?  To make these claims is not to say that if X was not present then Y would not have occurred.  History is not a repeatable experiment.

Neither in the case of the Diamond and Watson claims can we even test the hypothesis "Y tends to happen in the presence of X".  We can't do this because they have already used up all the experimental data.  In other words they did too complete a job!

If we were to encounter a plentiful number of alien races not too unlike us we would have more scope to make historical claims of the form "X causes Y".  Since this is not likely to happen we appear to be stuck.  However there is a possible avenue that might enable us to test this type of causality.  We could simulate various worlds and societies in a computer, seed them with various initial conditions, and study how they evolved.  Such simulations already exist, some on a very large scale: some online computer games depend for their very fascination on watching how a virtual society develops under the constraints of the game.

So here's the ideal academic project: conduct history and sociology (and other inexact disciplines) through the medium of computer gaming.  Wouldn't that be great fun for the young folk of today?

Thursday, 17 January 2013

Weapons of personal destruction

In the aftermath of the shootings at Sandy Hook in Connecticut a vigorous debate is taking place about gun control and President Obama has just announced measures to put before Congress.  Outside America there is overwhelming support for much stricter gun control but inside the country opinion is more divided.

As an outsider it seemed to be a no-brainer to me until very recently; surely the case for gun control was overwhelming.  But two arguments in the debate have given me pause for thought.  Both of them derive from the obvious thought that what we should be trying to do is not restricting Second Amendment rights just for the sake of it, but preventing innocent victims being killed by guns.

The first argument is that there are already over 300 million guns in the hands of private citizens.  How can restricting the purchase of further guns have much effect?

The second is that nearly all the shooters in the depressing list of US gun tragedies are people with mental health problems.  Since the 1970s it has been US health policy to allow such people to be managed and cared for in their communities rather than under supervision.  Sadly, very little funding has been given to such communities.  The result is that a large proportion of homeless people are victims of mental illness.  Would it not be better to address the root cause of shooting incidents?

I don't maintain that these two arguments should necessarily carry the day in the debate but I would like to see it focussing on the actual problem: of preventing shooting incidents rather than bashing the odious NRA and perfectly responsible gun-owners.  That new debate would have to come to grips with the gun culture in the US and the glorification of ever more powerful weapons.

So how should one address a culture where so many people own killing machines, have easy access to them, where accidents happen (fatal or otherwise) together with deliberate abuses; all of them regarded as the necessary price for the freedom to own those machines?  Every developed country has faced such a problem in trying to make its roads safe from the automobile killing machine.  Can we learn anything from that parallel?

A generation ago there was a much laxer approach to the misuse of automobiles but there has been a sea-change in attitude.  It is no longer a lark to get behind the wheel of a car when impaired, and designating a safe driver has become routine.  In the last 20 years the number of road fatalities in my own country (New Zealand) has dropped by more than 50%; much of that decline is because of the attitudinal change to drunk-driving.  We know why this change has occurred: it is because of the far more severe penalties for driving while impaired.

There are other lessons also from the automobile parallel.  Many countries have regular safety checks on vehicles.  A road license fee is levied and is hopefully used to improve road safety.  Drivers themselves can be checked periodically when their own license is renewed.  Motoring organisations are strongly behind automobile safety concerns rather than (as the NRA) act as lobbying groups for the rights of their members.

I cannot pretend that to solve the gun death problem we merely have to imitate the solutions we have found to the automobile death problem.  The analogy is not perfect (not least because the automobile death problem is far more severe). The gun death problem will have some peculiarities of its own to reckon with but we have to stay focussed on solving the "gun death problem" not the "gun control problem".

Monday, 14 January 2013

Word game for those sleepless nights

I'm an enthusiastic solver of cryptic crossword puzzles and also have a keen interest in words (their spelling, meaning and origin).  I don't know which interest came first but they seem to reinforce one another.  In particular I love anagrams. I was tickled pink recently listening to an episode of the BBC's News Quiz which touched on the story last year of the Duchess of Cambridge being photographed topless: I learnt that 'Kate Middleton' is an anagram of 'Naked Tit Model'.

Anyway that is a bit off-track for this post, which is about a solitaire word game based on anagrams.  You take any word you like and remove its letters one by one (in any order you like) and, at each stage, you have to make an anagram of the current set of letters.  For example, if you start with 'wonderer' you could proceed as drowner, wonder, drown, down, own, no, o.  Such a word is deemed to be a success, otherwise it is a fail.

With practice the game can be played in your head and the more experienced you get the longer the word you can start with.  I find words of 9 or more letters a bit of a problem but I've discovered many 8 letter successes.

Here are some more examples of successful words to give you the idea:
many, man, an, a
words, word, row, or, o
demand, named, name, man, an, a
diligent, tingled, tingle, glint, lint, tin, in, i
efforts, forest, store, rote, toe, to, o

Obviously, to be a success the word has to contain at least one of the letters a, i, o (the only one-letter words).  A stronger necessary condition for success (but not so easily checked) is that the set of letters you start with must have letter subsets of all sizes which have anagrams.  For example 'ice' is a failure because it has no two-letter anagrams.  On the other hand

It is astonishing how many successful words there are (especially if you restrict yourself to words satisfying these necessary conditions).  When trying to "solve a word" I find it easiest to begin with the word itself and reduce it to one letter as in the examples above.  This would be 'top-down' approach.  But I suspect that a bottom-up approach - building up longer and longer words in several ways - would be a more efficient approach for a computer program; a natural use for the approach called 'dynamic programming'.

There are several ways in which one can 'mathematize' the basic idea.  For example one could define a graph whose vertices are English words and where there was a directed edge from one word to another if the second word was an anagram of the first word without one of its letters.  Then one might ask about the connected components of this graph.

But you don't have to be a mathematician to enjoy the game.  Have a go yourself (ourself, flours, flour, four, for, of, o).

Saturday, 12 January 2013

Atheism - proselytise or not?

I live in a country where we don't really have to worry much about an encroaching theocracy unlike many Islamic countries and the United States.  So there does not seem to be a strong imperative to persuade religious believers that they are wasting their energies devoting themselves to a deity.  Yet, as an educator dedicated to spreading truth and reason, I can't help wanting to reduce the religiosity of my society.  So, in an idle moment I had the fantasy of preparing and distributing a leaflet that would challenge two basic Christian beliefs: that the bible was true and that it was moral.  This is what I wrote.

Is the bible true?

Some people believe every word as the literal word of God.  Most people believe that at least some of the bible should be read metaphorically.  How should we decide which parts to believe and which to take as fables?  Indeed, why should we believe any of it?  The bible has a huge credibility problem  because it is riddled with inconsistencies.

Let's take just one example: the events around the discovery of the empty tomb.  The gospels describe the women who came to the sepulchre where Jesus' body lay but each of the four gospels tells a different tale:  John 20:1 claims that Mary Magdalene was first on the scene but Matthew 28:1  adds the mother of Jesus to the discovering party.  Mark 16:1 claims that Salome was also there whereas Luke tells us that there was Mary Magdalene, Mary the mother of Jesus, Joanna and 'other women' present.

A little later on that same resurrection day John 20:11 speaks of there being two angels in the sepulchre clothed in white, whereas Mark has it that there was just one young man clothed in a long white garment.  In other words, the bible does not give a consistent account of the most important Christian belief of all: - the resurrection of Jesus.  The gospel writers are in disagreement over many other aspects of Jesus' ministry.

Inconsistencies also occur in the account of the same writer.  In the gospel of John we learn that Peter (John 13:36) and Thomas (John 14:5) both ask  Jesus where he is going.  Yet in John 16:5 Jesus claims no-one has asked him where he is going.

These examples are the tip of a large iceberg.  If you would like to know more about the contradictions in the bible you only need to have internet access and a copy of the bible.  Just google "bible contradictions" and you will be overwhelmed with other examples.

The bottom line is that these contradictions and inconsistencies teach us that there is nothing sacred about biblical "truth".  The bible is a book compiled by many men (and apparently a very few women), fallible human beings with imperfect memories.  But it cannot have been divinely inspired or it would have told a more consistent tale.

Is the bible moral?

There are many good moral teachings in the bible.  The famous Ten Commandments have six commands that none of us should disagree with (although the first four seem to pander to Jehovah's jealousy).  The New Testament abounds with advice that we would do well to heed.

Yet there is much to condemn in the "good book".  The Old Testament is full of counsel to commit appalling acts.  In Isaiah 13:16 Jehovah proposes that "Their children also shall be dashed to pieces before their eyes; their houses shall be spoiled, and their wives ravished."

But it is not only in the Old Testament that we find instructions to act immorally.  We hear the apostle Paul condoning slavery in Ephesians 6:5 "Slaves, obey your earthly masters with deep respect and fear. Serve them sincerely as you would serve Christ."

Even Jesus himself seems to cast away the cloak of forgiveness in Matthew 10:34; "I came not to send peace, but a sword. For I am come to set a man at variance against his father, and the daughter against her mother, and the daughter in law against her mother in law."  And he really loses it in Luke 19:27, "But those mine enemies, which would not that I should reign over them, bring hither, and slay them before me."

Most of us want to live an ethical life and to do this we need to consider our actions  thoughtfully.  It would be nice if there was an authority that would always guide us on the complex moral questions we face both personally and societally.  The examples above show us that the bible is not such an authority.  This is not surprising since it was written hundreds of years ago and today's society has become much more complex.

Some of the issues we face today (such as stem cell research and global warming) could not have been anticipated by the writers of the bible; others we grapple with (such as women's rights and how we cope with societal offenders) in the light of centuries of historical experience.  It would be folly to rely on the bible as our moral compass: it is too unreliable and blunt an instrument.