Before the second world war the voting age in most countries was 21 but in the second half of the twentieth century the age of 18 gradually became very common. For example, the USA passed the 26th amendment to its constitution in 1971 largely because it was felt that to be conscripted at age 18 and not allowed to vote was anomalous. There are a few countries where the voting age is 16 (such as Austria) and a very few where it is higher than 18 (such as Kuwait and Bahrain). In many countries there have been movements to bring the age down to 16 (for example, the UK, Australia, and New Zealand) but these attempts were abandoned (although, for the 2014 Scottish Independence referendum, the voting age was 16).

The reason that 18 has emerged as the most common voting age seems to be that 18 is seen as the age when children become adults. That this transition should occur instantly on an eighteenth birthday is obviously a fiction and therefore it is legitimate to ask whether we have got the age right, and to ask whether adulthood is the criterion we should be using.

The assumption that adulthood is a prerequisite to vote is very ingrained. Surely, it is said, a certain maturity is essential before a considered vote can be cast. And yet the right candidate to vote for and the right set of policies to support are generally not decisions that are arrived at by the tools of knowledge and logic; because, if they were, there would be no dispute at the ballot box. Once one accepts that opinions are driven by prejudice and emotion who is to say that a 16 year old is not as qualified as an 18 year old? Let's be honest and admit that most people vote out of self interest even if they are able to rationalise their decisions.

So why should the voting age not be 16? But isn't that as arbitrary as 18? If you are thinking along those lines consider the very bold proposal by Cambridge Professor David Runciman. He suggests the voting age should be lowered to 6! He has admitted that his proposal is made with a certain tongue in cheek but is ready to defend it anyway. For example, what of the objection that children will just vote as their parents tell them? Why should they? Women did not vote the way their husbands said they should when they got the franchise?

For me, the most compelling reason for taking Runciman seriously is that our electorate is already top heavy with old people. The most compelling issues of our age - climate change, whether Scotland should have independence, how should institutionalised racialism be addressed in the USA and other countries - are long term issues. Why should old people have any say in these? Leave it to the citizens who have the most skin in the game. After all if, like me, you are appalled at how the majority of the UK voters voted in the EU referendum, consider what might have happened if the franchise had consisted also of 16 and 17 year olds (a proposal that was seriously floated in 2015). Don't you think their youthful good sense would have prevailed over the grey vote which, on the day, carried the Leave side over the line?

# The wonder of it all

## Monday, 31 December 2018

## Sunday, 29 April 2018

### The trauma of the United States

Recently I read "Loaded" by Roxanne Dunbar-Ortiz. It is a history about the US Constitution's Second Amendment and the mythology surrounding it. The Second Amendment is once again in the news for a reason that occurs with depressing frequency. This time it is the shooting that occurred at the Douglas High School in Florida in February and which has resulted in a mass movement Never Again led by some of the survivors at the school.

I must confess that I began Dunbar-Ortiz's book with pessimism. I expected to read the usual arguments for gun control and the usual counter-arguments and I thought cynically that believing anything would really change was just pie in the sky. That is not exactly how it worked out.

The first thing to say is that the book has a deep and well-researched thesis that the history of the USA from even before its foundation contains a number of deeply disturbing causes for the national gun culture that go way beyond the usual excuses. This is not about personal freedoms endowed to the citizenry by the Founding Fathers. It is not about the national hunting culture. Nor is it about the distrust fostered by the NRA that the government wants to disarm the people to enable their suppression ("from my cold dead hands").

No, it is about slavery and genocide. The US Constitution is not about all men being created equal; it is about the privileges of white male landowners and the second amendment is why white male landowners needed guns. They needed guns to contain their slaves through patrols looking for escapees. And they needed guns to carry out the systemic slaughter of Native Americans so that their lands could be seized. It is interesting to reflect that a key reason for the Declaration of Independence is that King George III tried to restrict the colonists' "right" to seize territory west of the Appalachian mountains whose inhabitants were described as "merciless Indian Savages".

This is not an interpretation of their history that most Americans will like. They are more used to a narrative in which heroic settlers fought for their freedom from an oppressive colonial tyrant and then, through the nineteenth century, expanded ever westwards claiming land for themselves. A history in which brave cowboys and fearless rangers protected farmers from violent attacks by Indians and they came into their manifest destiny much like the Old Testament Jews settling Canaan (just after God gave them their marching orders to kill every one of the original inhabitants). These myths protect them from their murderous pasts and are part of the defence of their wide gun ownership. They revere the part that guns played in taming their land, they laud the bravery of the family man whose gun is to protect his nearest and dearest as part of a long tradition, and they hold their constitution in almost superstitious awe.

So, on reaching the end of the book, I had a sense of hopelessness. How on earth could one penetrate these myths so that a rational discussion about how to go forward could be held? After a few days I began to read some other historical material in the same vein and this left me feeling a little more optimistic. To begin with the USA is not the only country burdened by a very shameful history. The Doctrine of Discovery promulgated by Pope Alexander VI in 1493 initially gave the Spanish permission to take possession of colonise any lands they discovered which were not under the control of a Christian ruler and it became the justification for later European powers to colonise at will (and the American colonists inherited this idea from the British). The world is still suffering from the aftermath of colonisation by force but other colonial powers have at least begun to recognise their catastrophic agency with apologies, reparations, and deliberate reconciliation. The American haven't really started down this road but they did come late to the game - maybe their eyes will be opened in due time.

But also we should not overlook that there are movements in the USA that are trying to confront their racist misogynistic culture: the Woman's Movement, Metoo, the LGBT movement, Black Lives Matter and more. Some of these movements have extraordinary charismatic leaders who recognise the extreme difficulty rank and file Americans have in facing up to their past. I encourage you to view the lecture by Mark Charles; he is a native American who, for all his criticism of the oppression of his people, ends his lecture with some prescriptions that might change the discourse. We have a long way to go (and other colonial powers are still travelling that road) but it is important to try to keep reason, tolerance and understanding alive - and maybe in a couple of hundred years we can emerge in an enlightened sunshine.

I must confess that I began Dunbar-Ortiz's book with pessimism. I expected to read the usual arguments for gun control and the usual counter-arguments and I thought cynically that believing anything would really change was just pie in the sky. That is not exactly how it worked out.

The first thing to say is that the book has a deep and well-researched thesis that the history of the USA from even before its foundation contains a number of deeply disturbing causes for the national gun culture that go way beyond the usual excuses. This is not about personal freedoms endowed to the citizenry by the Founding Fathers. It is not about the national hunting culture. Nor is it about the distrust fostered by the NRA that the government wants to disarm the people to enable their suppression ("from my cold dead hands").

No, it is about slavery and genocide. The US Constitution is not about all men being created equal; it is about the privileges of white male landowners and the second amendment is why white male landowners needed guns. They needed guns to contain their slaves through patrols looking for escapees. And they needed guns to carry out the systemic slaughter of Native Americans so that their lands could be seized. It is interesting to reflect that a key reason for the Declaration of Independence is that King George III tried to restrict the colonists' "right" to seize territory west of the Appalachian mountains whose inhabitants were described as "merciless Indian Savages".

This is not an interpretation of their history that most Americans will like. They are more used to a narrative in which heroic settlers fought for their freedom from an oppressive colonial tyrant and then, through the nineteenth century, expanded ever westwards claiming land for themselves. A history in which brave cowboys and fearless rangers protected farmers from violent attacks by Indians and they came into their manifest destiny much like the Old Testament Jews settling Canaan (just after God gave them their marching orders to kill every one of the original inhabitants). These myths protect them from their murderous pasts and are part of the defence of their wide gun ownership. They revere the part that guns played in taming their land, they laud the bravery of the family man whose gun is to protect his nearest and dearest as part of a long tradition, and they hold their constitution in almost superstitious awe.

So, on reaching the end of the book, I had a sense of hopelessness. How on earth could one penetrate these myths so that a rational discussion about how to go forward could be held? After a few days I began to read some other historical material in the same vein and this left me feeling a little more optimistic. To begin with the USA is not the only country burdened by a very shameful history. The Doctrine of Discovery promulgated by Pope Alexander VI in 1493 initially gave the Spanish permission to take possession of colonise any lands they discovered which were not under the control of a Christian ruler and it became the justification for later European powers to colonise at will (and the American colonists inherited this idea from the British). The world is still suffering from the aftermath of colonisation by force but other colonial powers have at least begun to recognise their catastrophic agency with apologies, reparations, and deliberate reconciliation. The American haven't really started down this road but they did come late to the game - maybe their eyes will be opened in due time.

But also we should not overlook that there are movements in the USA that are trying to confront their racist misogynistic culture: the Woman's Movement, Metoo, the LGBT movement, Black Lives Matter and more. Some of these movements have extraordinary charismatic leaders who recognise the extreme difficulty rank and file Americans have in facing up to their past. I encourage you to view the lecture by Mark Charles; he is a native American who, for all his criticism of the oppression of his people, ends his lecture with some prescriptions that might change the discourse. We have a long way to go (and other colonial powers are still travelling that road) but it is important to try to keep reason, tolerance and understanding alive - and maybe in a couple of hundred years we can emerge in an enlightened sunshine.

## Friday, 20 April 2018

### A Higher Loyalty

Former FBI Director James Comey published his keenly anticipated book "A Higher Loyalty" this week. I listened to the Audible edition read by Comey himself. This turned out to be a good method of digesting the book because Comey's earnestness comes through very clearly in his own voice.

The book is autobiographical but not a full autobiography. Those who read it just for the events surrounding Donald Trump should not ignore the larger part of the book which deals with episodes from Comey's childhood and career in the law. They are dramatic, thoughtful, and interesting. Comey uses them to develop his views on ethical leadership, self-knowledge and humility describing several people he has known whose lives and personalities have been exemplars for him through the years.

He has much to say about what makes an inspiring leader and it is clear that he wishes he will have been seen as such. Readers will naturally be on their guard for self-serving stories when he talks about the organisations he himself has led but Comey's frequent admissions of his own mistakes and weaknesses lead one to think that we are reading a broadly honest account.

To many people his investigation into Hilary Clinton's unclassified email server will be the black mark forever held against him. These investigations are described in greater detail than anything else in the book. Comey admits that another FBI Director might have handled things somewhat differently but, reading his account, it is difficult not to sympathise with the way he made some incredibly difficult decisions. If (like I had been) you are slightly unclear on what this investigation was about this part of the book is a very complete account. The bottom line is that Clinton was very careless in using an unclassified email server. That conclusion was easily reached but it took the FBI much longer to trawl through the emails before being able to declare that there was nothing further to the whole issue. Comey reported the closure of the investigation to Congress in July 2016. But two weeks before the 2016 election, the investigation had to be re-opened because further emails were found on another laptop and Comey had to decide whether to inform Congress the investigation was being re-opened. He chose to do so and it is this that Clinton and others believe cost the Democrats the election. Should he have waited until the reopened investigation was complete? That might have resulted in Clinton being elected and then the public finding out she was about to be indicted.

I feel there is an unanswered question lurking over this reopening of the investigation: Comey says he had to make this public because the ongoing examination was projected to take weeks yet, in the event, it was completed before the election; had that been known why could the announcement not have been delayed until the investigation was complete?

Donald Trump enters only in the final stages of the book but it is this part of the book that has attracted the most attention. In some ways though, one can read the previous part of the book as setting the scene for Comey's withering criticism of the US President. Comey has constructed a number of ethical principles based on leaders who have influenced his own life and we know, as the moment approaches, that Trump is going to fall very far short of those standards. And, for once, Trump does not disappoint us.

I write this as one who thinks that Donald Trump is a peculiarly awful individual and president. But, of course, some Republicans will not agree. The attacks on Comey have already begun and are being spearheaded by the web site lyincomey.com. I've just spent some time perusing this site and found that its rebuttals are weak and ludicrous. So I think A Higher Loyalty is a very credible attack on the Trump presidency and will only add to the maelstrom of difficulties now swirling around it.

The book is autobiographical but not a full autobiography. Those who read it just for the events surrounding Donald Trump should not ignore the larger part of the book which deals with episodes from Comey's childhood and career in the law. They are dramatic, thoughtful, and interesting. Comey uses them to develop his views on ethical leadership, self-knowledge and humility describing several people he has known whose lives and personalities have been exemplars for him through the years.

He has much to say about what makes an inspiring leader and it is clear that he wishes he will have been seen as such. Readers will naturally be on their guard for self-serving stories when he talks about the organisations he himself has led but Comey's frequent admissions of his own mistakes and weaknesses lead one to think that we are reading a broadly honest account.

To many people his investigation into Hilary Clinton's unclassified email server will be the black mark forever held against him. These investigations are described in greater detail than anything else in the book. Comey admits that another FBI Director might have handled things somewhat differently but, reading his account, it is difficult not to sympathise with the way he made some incredibly difficult decisions. If (like I had been) you are slightly unclear on what this investigation was about this part of the book is a very complete account. The bottom line is that Clinton was very careless in using an unclassified email server. That conclusion was easily reached but it took the FBI much longer to trawl through the emails before being able to declare that there was nothing further to the whole issue. Comey reported the closure of the investigation to Congress in July 2016. But two weeks before the 2016 election, the investigation had to be re-opened because further emails were found on another laptop and Comey had to decide whether to inform Congress the investigation was being re-opened. He chose to do so and it is this that Clinton and others believe cost the Democrats the election. Should he have waited until the reopened investigation was complete? That might have resulted in Clinton being elected and then the public finding out she was about to be indicted.

I feel there is an unanswered question lurking over this reopening of the investigation: Comey says he had to make this public because the ongoing examination was projected to take weeks yet, in the event, it was completed before the election; had that been known why could the announcement not have been delayed until the investigation was complete?

Donald Trump enters only in the final stages of the book but it is this part of the book that has attracted the most attention. In some ways though, one can read the previous part of the book as setting the scene for Comey's withering criticism of the US President. Comey has constructed a number of ethical principles based on leaders who have influenced his own life and we know, as the moment approaches, that Trump is going to fall very far short of those standards. And, for once, Trump does not disappoint us.

I write this as one who thinks that Donald Trump is a peculiarly awful individual and president. But, of course, some Republicans will not agree. The attacks on Comey have already begun and are being spearheaded by the web site lyincomey.com. I've just spent some time perusing this site and found that its rebuttals are weak and ludicrous. So I think A Higher Loyalty is a very credible attack on the Trump presidency and will only add to the maelstrom of difficulties now swirling around it.

## Friday, 6 April 2018

### Prime conjunctions, and ruling families

This is positively my last post on prime conjunctions and I expect it to be read by nerds only. My purpose is to give two concrete examples of the more technical discussions in my two previous posts on prime conjunctions.

Let's begin with the children of Elizabeth II. They are (listed not by age but by the order in the year that their birthdays occur):

Let's begin with the children of Elizabeth II. They are (listed not by age but by the order in the year that their birthdays occur):

- Andrew b. 19/2/1960
- Edward b. 10/3/1964
- Anne b. 15/8/1950
- Charles b. 14/11/1948

Their ages at the beginning of 2018 were 57, 53, 67, 69. There's a stroke of luck - they are all odd. Indeed they are odd during the time interval 14/11/1999 to 18/2/2000. Such an odd conjunction period occurs every two years so maybe we shall find many prime conjunctions. Sadly, that set of ages has remainders 0, 2, 1, 0 on division by 3. In other words the 3-condition of my previous post fails - whenever the ages are odd, therefore, at least one age will be divisible by 3. So the only chance of a prime conjunction is when one of the is 3 itself and that would have to Edward's age. When he was 3 during the odd conjunction period the set of ages was 7, 3, 17, 19 and they are all prime. So the Royal siblings have had a prime conjunction (which they no doubt celebrated uproariously) but there will never be another.

What about 5 individuals who I shall call (in the order of their birthdays throughout the year)

- Barron b. 20/3/2006
- Melania b. 26/4/1970
- Donald b. 14/6/1946
- Tiffany b. 13/10/93
- Ivanka b. 30/10/1981

At the beginning of 2018 their ages are 11, 47, 71, 36, 24. Oh dear - not all prime and neither are they all odd. However they do have the property that they are bunch of odds followed by a bunch of evens and so we know that eventually they will have an odd conjunction. The nearest odd conjunctive period is 14/6/2017 to 29/10/2017 when their ages are 11, 47, 71, 35, 23. Nearly all prime but not quite - maybe this will also prove impossible.

Since there are 5 people in this case we have to check both the 3-condition and the 5-condition. The 3-residues are 2, 2, 2, 2, 2 while the 5-residues are 1, 2, 1, 0, 3. Both conditions hold! So Dickson's conjecture would predict that, not only will these 5 individuals have a prime conjunction, they will have infinitely many. How do we find them?

Their earliest prime conjunction occurred when the ages were 7, 43, 67, 31, 19. To find later prime conjunctions we must add a number d to all of them that has the properties

- d is even
- d has remainder 0 or 1 when divided by 3
- d is exactly divisible by 5

By a procedure called the Chinese Remainder Theorem these 3 conditions are equivalent to the condition d has remainder 0 or 10 when divided by 30. So the choices for d are 0, 10, 30, 40, 60, 70 etc. In fact d = 30 works (37, 73, 97, 61, 79). The next successful value of d is 40 and I'll leave you to find others.

## Wednesday, 14 March 2018

### Prime conjunctions and Dickson's conjecture

Way back in 2013 I blogged about "Prime Conjunctions". If you want to get the most out of this posting you will need to read that article first (use the search box) but I will supply some background here anyway.

This all started when one day I noticed that all my children's ages were prime numbers (and, of course, they remained like that until one of them had a birthday and spoilt everything). Not every set of people are destined to enjoy such a prime conjunction but how lucky do they have to be to have one? Since most prime numbers are odd it is unsurprising that a major thing to investigate is whether there will be a period when all their ages are odd and this was the main focus of my previous posting. But before recounting that story let's briefly consider the case that the prime 2 might occur as part of a prime conjunction. To determine whether that is possible all you need do is note the ages of your other children in the full year of one of your children being two; throughout the year these ages will change slightly as your other children have their birthdays and you see whether they ever become prime together. You have to do this for each child and the period in which they are two.

Having dispensed with the case that a prime conjunction might involve the prime number 2 we shall, from now on, just consider whether a prime conjunction with odd primes only can occur. So we shall begin by considering whether your children can have an odd conjunction: a period when all their ages are odd. It is not difficult to find the condition. If you list their ages at the beginning of the year in the order in which their birthdays occur you can look forward to them all being odd if that list is either a bunch of odds followed by a bunch of evens or a bunch of evens followed by a bunch of odds.

For example if on 1 January the ages in birthday order through the year are 10, 6, 9 then when the 6 year old turns 7 the ages will be 11, 7, 9 (because the 10 year old has already turned 11). On the other hand if their ages on 1 January had been 10, 5, 8 there is never a period when all their ages are odd.

It is also easy to see that even when an odd conjunction is possible there is only one period between birthdays when it occurs (in the example 10, 6, 9 that is the period in the year when the first two birthdays have occurred but the third not yet - and this is in every other year).

Essentially then if we are given a set of people we can tell if an odd conjunction is possible and, if it is possible, we can determine the unique interval in the year when it occurs. I shall mention in passing here that, obviously, if an odd conjunction occurs, then other odd conjunctions occur at two year intervals so once we have one odd conjunction we shall have infinitely many. That suggests a related question: if there is a prime conjunction are there infinitely many? That may seem like a much stronger requirement but we shall see that it really isn't. So, for the moment, let's consider whether there are an infinite number of prime conjunctions. At the end of this article I'll return to the original question.

So let's assume you have carried out the calculation above and found that your children do have an odd conjunction. If there are going to be prime conjunctions then they must occur in the same period of the year when the odd conjunctions occur. The mathematical expression of the question therefore is: given k odd numbers a

As an easy test case consider divisibility by 3. Is it possible that no matter what value n we add to the set of ages we shall always find one that is a multiple of 3? Yes, indeed! The condition is simply that among the k remainders when the a

A similar condition must hold for every prime p: if the remainders when a

are divided by p do not include every possible remainder 0, 1, ..., p-1 then there are infinitely many values of n for which every sum a

So here's the $64000 question. Well, actually, it's more like the $64000000000 question. If the p-condition always holds will we necessarily have a prime conjunction? This is where Dickson's conjecture enters the scene. It states: Let a

Let's restate that conjecture in the special case that all the b

Thus, assuming Dickson's conjecture, the bottom line on how to find a prime conjunctions (if they exist) given a set of ages a

Of course the big downside of all of this is that we don't know whether Dickson's conjecture is indeed true! A proof of it is not likely to be found any time soon. Notice that the case k=2 and with n and n+2 is the twin primes conjecture: whether there are infinitely many primes differing by 2. The twin primes conjecture is a famous unproven conjecture and widely thought to be true. Clearly Dickson's conjecture is an even tougher nut.

Finally, let's return to the original question: does there exist at least one prime conjunction? One case that may arise is when one of the primes occurring in the conjunction is 2: it's easy to check such cases. So let's consider just odd primes again. We shall need the condition that odd conjunctions exist just as before. Also, we can rely on Dickson's conjecture if the p-condition holds for all odd primes. So the only cases where we might have a prime conjunction not covered by the analysis above is when the p-condition fails for one or more primes. But such a prime p is necessarily no more than k so there are only a finite number of primes to consider. Here, in any prime conjunction, one of the terms must be divisible by p and so must be p itself. So the prime conjunctions in question must have some prime no more than k and these can be inspected on a case by case basis.

For example, 2, 5, 7 is a prime conjunction which never reoccurs. And 3,5,7 is another because the 3-condition fails.

This all started when one day I noticed that all my children's ages were prime numbers (and, of course, they remained like that until one of them had a birthday and spoilt everything). Not every set of people are destined to enjoy such a prime conjunction but how lucky do they have to be to have one? Since most prime numbers are odd it is unsurprising that a major thing to investigate is whether there will be a period when all their ages are odd and this was the main focus of my previous posting. But before recounting that story let's briefly consider the case that the prime 2 might occur as part of a prime conjunction. To determine whether that is possible all you need do is note the ages of your other children in the full year of one of your children being two; throughout the year these ages will change slightly as your other children have their birthdays and you see whether they ever become prime together. You have to do this for each child and the period in which they are two.

Having dispensed with the case that a prime conjunction might involve the prime number 2 we shall, from now on, just consider whether a prime conjunction with odd primes only can occur. So we shall begin by considering whether your children can have an odd conjunction: a period when all their ages are odd. It is not difficult to find the condition. If you list their ages at the beginning of the year in the order in which their birthdays occur you can look forward to them all being odd if that list is either a bunch of odds followed by a bunch of evens or a bunch of evens followed by a bunch of odds.

For example if on 1 January the ages in birthday order through the year are 10, 6, 9 then when the 6 year old turns 7 the ages will be 11, 7, 9 (because the 10 year old has already turned 11). On the other hand if their ages on 1 January had been 10, 5, 8 there is never a period when all their ages are odd.

It is also easy to see that even when an odd conjunction is possible there is only one period between birthdays when it occurs (in the example 10, 6, 9 that is the period in the year when the first two birthdays have occurred but the third not yet - and this is in every other year).

Essentially then if we are given a set of people we can tell if an odd conjunction is possible and, if it is possible, we can determine the unique interval in the year when it occurs. I shall mention in passing here that, obviously, if an odd conjunction occurs, then other odd conjunctions occur at two year intervals so once we have one odd conjunction we shall have infinitely many. That suggests a related question: if there is a prime conjunction are there infinitely many? That may seem like a much stronger requirement but we shall see that it really isn't. So, for the moment, let's consider whether there are an infinite number of prime conjunctions. At the end of this article I'll return to the original question.

So let's assume you have carried out the calculation above and found that your children do have an odd conjunction. If there are going to be prime conjunctions then they must occur in the same period of the year when the odd conjunctions occur. The mathematical expression of the question therefore is: given k odd numbers a

_{1}, a_{2}, ..., a_{k}can we find some number n (indeed, an infinite collection of numbers n) to add to all of them making them prime? These numbers n will, of course, all be even.As an easy test case consider divisibility by 3. Is it possible that no matter what value n we add to the set of ages we shall always find one that is a multiple of 3? Yes, indeed! The condition is simply that among the k remainders when the a

_{k}are divided by 3 all the possible remainders 0, 1 and 2 occur; if so this condition must persist when a number n is added. For example the ages 7, 9, 11 never become prime if we a positive number n to all of them as their remainders on division by 3 are 1, 0 and 2. But notice also that if only two (or one) possible remainders occurred (as, for example, 7, 9, 13) then there will be an infinite set of values n (1, 4, 7, 10, ... ) where none of a_{1}+ n, a_{2}+ n, ..., a_{k}+ n are divisible by 3.A similar condition must hold for every prime p: if the remainders when a

_{1}, a_{2}, ..., a_{k}are divided by p do not include every possible remainder 0, 1, ..., p-1 then there are infinitely many values of n for which every sum a

_{1}+ n, a_{2}+ n, ..., a_{k}+ n is not divisible by p. It's convenient to give a name to this phenomenon.We shall say that a_{1}, a_{2}, ..., a_{k}satisfies the p-condition if, among the remainders on dividing a_{1}, a_{2}, ..., a_{k }by p, at least one remainder does not occur. The point is that, when the p-condition holds, then there will be infinitely many values of n for which the set a_{l}+ n, a_{2 }+ n,...,a_{k}+ n contains a number not divisible by p. So if we are looking for an infinite number of prime conjunctions we want the p-condition to hold for all values of p. This seems like a strong condition to require but it isn't: it's easy to see that the p-condition almost always holds. Indeed it holds for all p>k (because then k remainders cannot exhaust the set of all possible remainders).So here's the $64000 question. Well, actually, it's more like the $64000000000 question. If the p-condition always holds will we necessarily have a prime conjunction? This is where Dickson's conjecture enters the scene. It states: Let a

_{1}, a_{2}, ..., a_{k}and b_{1}, b_{2}, ..., b_{k}be positive integers. Suppose that there is no prime p which divides the product (a_{l}+ nb_{1})(a_{2 }+ nb_{2})...(a_{k}+ nb_{k}) for all values of n. Then there are infinitely many values of n for which all of a_{l}+ nb_{1}, a_{2 }+ nb_{2},...,a_{k}+ nb_{k}are prime.Let's restate that conjecture in the special case that all the b

_{i}are equal to 1. It then states: suppose there is no prime p that divides at least one of a_{l}+ n, a_{2 }+ n, ... , a_{k}+ n for all values of n. Then, for infinitely many values of n, all of a_{l}+ n, a_{2 }+ n, ...,, a_{k}+ n are prime. To put this in the language we have been using the conjecture says: if the p-condition holds for all primes p then a_{l}, a_{2 },..., a_{k}have infinitely many prime conjunctions.Thus, assuming Dickson's conjecture, the bottom line on how to find a prime conjunctions (if they exist) given a set of ages a

_{1}, ..., a_{k}on 1 January is the following:- Determine whether there is an interval in the next two year period when all the ages are odd. If so replace the ages by this new set.
- For the primes p≤k see if the p-condition holds

Of course the big downside of all of this is that we don't know whether Dickson's conjecture is indeed true! A proof of it is not likely to be found any time soon. Notice that the case k=2 and with n and n+2 is the twin primes conjecture: whether there are infinitely many primes differing by 2. The twin primes conjecture is a famous unproven conjecture and widely thought to be true. Clearly Dickson's conjecture is an even tougher nut.

Finally, let's return to the original question: does there exist at least one prime conjunction? One case that may arise is when one of the primes occurring in the conjunction is 2: it's easy to check such cases. So let's consider just odd primes again. We shall need the condition that odd conjunctions exist just as before. Also, we can rely on Dickson's conjecture if the p-condition holds for all odd primes. So the only cases where we might have a prime conjunction not covered by the analysis above is when the p-condition fails for one or more primes. But such a prime p is necessarily no more than k so there are only a finite number of primes to consider. Here, in any prime conjunction, one of the terms must be divisible by p and so must be p itself. So the prime conjunctions in question must have some prime no more than k and these can be inspected on a case by case basis.

For example, 2, 5, 7 is a prime conjunction which never reoccurs. And 3,5,7 is another because the 3-condition fails.

## Friday, 6 October 2017

### Our Moral Obligations

This post is about an issue in moral philosophy that I don't understand and its main purpose is to help me wrestle with my confusion. Let me begin with some invented scenarios.

Suppose my adult brother who lives far away robs someone and then disappears. Do I have a moral obligation to make any restitution to the victim for the harm they have sustained? So long as I didn't encourage or aid the crime I imagine the answer is No because I myself took no part in the act of harm. If you disagree I'd like to hear from you via a blog comment!

But here's a second scenario. Suppose my father and his brother (my uncle) were in a quarrel that resulted in my father being able to steal very significant assets from his brother. Maybe this happened years before I was even born and both protagonists are dead. Do I have a moral obligation to my uncle's children (my cousins) to make restitution. This seems like the same as the first scenario. But I think it is actually more complicated if my father was able to pass on to me wealth that resulted indirectly from the fraud against my uncle and my cousins are therefore poorer than they would be otherwise. In that case I am profiting inadvertently from an act in the past and my cousins are suffering. So, should I compensate my cousins for their loss?

Here is an issue on which people might disagree. On the one hand I committed no crime so I should suffer no penalty. On the other hand I profited (albeit inadvertently) from a crime and should make redress. Is there some sort of statute of moral limitations at work here? Would the onus to make redress lapse after a generation? A century? A millenium? It seems clear that for personal moral sins the obligation to make amends should remain until redress has occurred. But what about moral sins that you did not commit but just profited by?

So, to restate now in general terms the dilemma I am wrestling with: if you profit from a crime that you did not commit, possibly long in the past, do you have an obligation to make restitution? I'm aware that the answer might be very much more complicated than a simple Yes or No. Indeed possibly neither of these extreme answers would satisfy most people. If you think you know criteria that might resolve particular instances of the question I'd like to hear from you.

I have a feeling that many people will answer with some sort of qualified Yes, hedged with remarks about circumstances to be taken into account, the practical matter of verifying the facts of the crime if committed long ago, and the moral complexity of being accountable for very many actions beyond ones control. Nevertheless a reluctant "Yes, in general" is how most people will answer the question.

And this brings me to some of the greatest conundrums of our time. How should the countries that used to be colonial powers compensate their former colonies? This applies to some very wealthy countries such as Britain and Spain who stripped their colonies of immense material and cultural wealth. Another question: How should countries that, historically, have repressed groups within their borders compensate these groups? Here an obvious example is the USA's behaviour towards its black citizens.

These are questions so large that we run away from them most of the time because we cannot bear the enormous guilt that honest answers would cause. Do wealthy Englishmen whose estates have been purchased or maintained with the plundered wealth of African countries think too much about the suffering of present day Africans whose ancestors were robbed? Do rich white Americans think about how their wealth has been accumulated by enslaving black people? No, because the guilt would be crushing.

So what do the beneficiaries of their plundering ancestors do instead? They either ignore the moral question or they construct complex narratives that absolve their guilt.

What can we do about this? I accept that facing up to their moral guilt is too much for most people. But we have to begin somewhere. We have to change the narrative of entitlement that the rich countries cling to. We have to talk about these moral questions, make people aware that their present comfortable circumstances have been won by actions taken in the past that cause suffering in the present. This will hardly begin to redress the injustices we ignore every day but recognising how we became so rich and fortunate is surely the first step.

Suppose my adult brother who lives far away robs someone and then disappears. Do I have a moral obligation to make any restitution to the victim for the harm they have sustained? So long as I didn't encourage or aid the crime I imagine the answer is No because I myself took no part in the act of harm. If you disagree I'd like to hear from you via a blog comment!

But here's a second scenario. Suppose my father and his brother (my uncle) were in a quarrel that resulted in my father being able to steal very significant assets from his brother. Maybe this happened years before I was even born and both protagonists are dead. Do I have a moral obligation to my uncle's children (my cousins) to make restitution. This seems like the same as the first scenario. But I think it is actually more complicated if my father was able to pass on to me wealth that resulted indirectly from the fraud against my uncle and my cousins are therefore poorer than they would be otherwise. In that case I am profiting inadvertently from an act in the past and my cousins are suffering. So, should I compensate my cousins for their loss?

Here is an issue on which people might disagree. On the one hand I committed no crime so I should suffer no penalty. On the other hand I profited (albeit inadvertently) from a crime and should make redress. Is there some sort of statute of moral limitations at work here? Would the onus to make redress lapse after a generation? A century? A millenium? It seems clear that for personal moral sins the obligation to make amends should remain until redress has occurred. But what about moral sins that you did not commit but just profited by?

So, to restate now in general terms the dilemma I am wrestling with: if you profit from a crime that you did not commit, possibly long in the past, do you have an obligation to make restitution? I'm aware that the answer might be very much more complicated than a simple Yes or No. Indeed possibly neither of these extreme answers would satisfy most people. If you think you know criteria that might resolve particular instances of the question I'd like to hear from you.

I have a feeling that many people will answer with some sort of qualified Yes, hedged with remarks about circumstances to be taken into account, the practical matter of verifying the facts of the crime if committed long ago, and the moral complexity of being accountable for very many actions beyond ones control. Nevertheless a reluctant "Yes, in general" is how most people will answer the question.

And this brings me to some of the greatest conundrums of our time. How should the countries that used to be colonial powers compensate their former colonies? This applies to some very wealthy countries such as Britain and Spain who stripped their colonies of immense material and cultural wealth. Another question: How should countries that, historically, have repressed groups within their borders compensate these groups? Here an obvious example is the USA's behaviour towards its black citizens.

These are questions so large that we run away from them most of the time because we cannot bear the enormous guilt that honest answers would cause. Do wealthy Englishmen whose estates have been purchased or maintained with the plundered wealth of African countries think too much about the suffering of present day Africans whose ancestors were robbed? Do rich white Americans think about how their wealth has been accumulated by enslaving black people? No, because the guilt would be crushing.

So what do the beneficiaries of their plundering ancestors do instead? They either ignore the moral question or they construct complex narratives that absolve their guilt.

What can we do about this? I accept that facing up to their moral guilt is too much for most people. But we have to begin somewhere. We have to change the narrative of entitlement that the rich countries cling to. We have to talk about these moral questions, make people aware that their present comfortable circumstances have been won by actions taken in the past that cause suffering in the present. This will hardly begin to redress the injustices we ignore every day but recognising how we became so rich and fortunate is surely the first step.

## Tuesday, 22 August 2017

### From Theories to Narratives

When we face a complex issue that requires an action or a strategy on our part we can sometimes feel overwhelmed by the sheer number of components (facts or opinions or possibilities) that comprise it. We often try to make a start by trying to get all this sometimes conflicting data into some sort of more understandable assembly. But this organisational step may need to be carried out in a way that highlights some of the issues at the expense of others, or even disregards some issues entirely. Our hope is that what emerges will be a more manageable version of the original issue.

This process may be a number of things depending on the context but what I’d like to suggest is that while the name of the process may vary there is a commonality that runs through them all.

At the most technical level we can think of fitting a simple function to best approximate a seemingly unrelated set of data points. Least squares fitting of a straight line or another simple curve would be an example. Here we try to display a trend suggested by the data and it is obvious that some points do not obey the trend yet we shrug our shoulders and trade simplicity for accuracy.

A less technical, but still in the domain of science, is the formulation of theories to explain phenomena in the physical world. Newton’s theory of gravitation is such a theory. Actually, when Newton devised the theory he found an amazing fit of theory to data and it wasn’t until much later that inconsistencies were found. Would we have discarded the Newton theory if we had known of these inconsistencies? Almost certainly not but we might have regarded them as anomalies that could be explained away without discarding the theory. There are many examples from science and we are now used to being somewhat humble and being prepared to use a theory until we have a better one.

These scientific theories of the physical world have been amazingly successful in reducing a multitude of observed facts to a small number of basic principles. We like this very much! Unfortunately, when we go beyond phenomena of the physical world we don’t meet with the same success.

In the social, political and personal spheres there seems to be much greater complexity. No Newtonian theory of, say, Economics. Instead we have theories that hold sway for a while, are then discarded, may reappear slightly modified and then fall away again. These theories do not have the same explaining power as Gravitation but they are not useless. They impose a structural framework on an otherwise inchoate dataset, allowing us to make policy with some uncertainty but not entirely as though we relied on magic. They provide us with a way of looking at the economies of the world without being entirely overwhelmed.

And in the same sort of way social and political theories have some limited explanatory power helping us to govern our large complex societies at an international and national level. Their structure, whether it is real and true or not, enables us to act consistently and we make progress forwards or backwards rather than be stationary (which acting randomly might cause).

In these less precise domains the term “ideology” might perhaps be used instead of “theory”. For in these domains it is easy to lose sight of the fact we often really don’t know what is going on - but we pretend we do, and we elevate the ideology to a status that we call “belief”. Obviously this can be very dangerous and some of the cataclysmic events of human history have come about when an ideology has resisted challenge because its adherents believe in its utility for far longer than the evidence warrants.

Theories, ideologies, and beliefs are not the same but they do have much in common. All of them allow us to have an explanatory picture of a complex issue. We arrive at this picture by discarding many details (some of which perhaps should not be discarded). We adhere to our position by sometimes forgetting that we only have a partial picture. And we are often reluctant to abandon our position because we invest it with greater value than it should have.

There is another domain that almost everyone is familiar with where these ideas apply: the personal world of our interactions with family, friends and acquaintances. Why do people fall out with one another at a personal level? Sometimes an observer will remark that is all over nothing but not being party to the feelings of the protagonists means that they really cannot understand what is going on. Or why do we sometimes have very firm feelings about an issue of child-rearing? Most importantly, why do we find it so hard to relinquish a deeply-held personal position?

The answer to that question has very little to do with evidence and logic. Most often our sincerely held position has been strengthened because we have built narratives in our minds that enable our opinions to be consistent with those cherry-picked issues that support our own ideas. We believe we are right and we can argue our case passionately. The narrative is our “theory” but we do not subject it to the same scrutiny that Newton’s theory of gravitation suffered. On the contrary, we seek out more and more supporting tidbits that validate our narrative.

Obviously there’s a lesson here. We should question our personal narratives with the same zeal that Bishop Berkeley attacked the differential calculus. But that’s hard to do and it’s a habit best learnt young. If we succeed we may find that our own personal relations are less fraught. We will certainly be less opinionated, more compassionate and more pragmatic. To put it another way. we will not be dangerous men and women of principle.

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