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.
Friday, 29 March 2013
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.
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.
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.
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.
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'!
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.
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.
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?
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.
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.
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