12 - one of only two known "sublime numbers"

The Silk Road Lodge 1s 12 years old today. That is, we opened our doors to the public on 7th August, 2000.

We’ll have a cake in the office, as usual, to commemorate the occasion … but as it is not exactly a kroogliy data - a round figure – I don’t think we’ll be doing anything else special to mark the anniversary.

As it happens, however, the twelfth anniversary is a Silk Anniversary – which, I suppose, is quite appropriate for the Silk Road Lodge.

For the record: Here’s a list of traditional anniversaries:

• 1 year        - Paper
• 2 years      - Cotton
• 3 years      - Leather
• 4 years      - Linen
• 5 years      - Wood
• 6 years      - Iron
• 7 years      - Wool
• 8 years      - Bronze
• 9 years      - Copper
• 10 years   – Tin/Aluminium
• 11 years    - Steel
• 12 years    - Silk
• 13 years    - Lace
• 14 years    - Ivory
• 15 years    - Crystal
• 20 years   – China/Porcelain
• 25 years    - Silver
• 50 years   – Golden
• 60 years   – Diamond (when applied to the reign of Monarchs like Queen Elizabeth II)
• 70 years   – Platinum
• 75 years   – Diamond
• 100 years – Century

 The number 12, of course is a very special number, encrusted with mysticism and symbolism in almost every culture and apart from being a sublime number, it is also associated with : …

• the concept of a dozen;
• a jury which comprises “twelve good men and true”;
• there are twelve months of the year;
• the twelve tribes of Israel;
• the twelve disciples of Jesus; and
• the twelve Imams in Islam;
• the twelve Olympian Gods of Greek Mythology; and
• the twelve labours of Hercules;
• in Nordic mythology there were the twelve sons of Odin;
• the twelve signs of the zodiac, not forgetting
• the twelve days of Christmas;
• there are twelve stars on the flag of the European Union;
• King Arthur is supposed to have had twelve knights of the Round Table;
• there were twelve pence in a shilling; and
• twelve inches in a foot;
• there have been several songs, albums, novels and films called Twelve

… and so on.

My favourite association with the number 12, however, if the fact that it is a “sublime number“.

I often wonder who thinks up the various special designations for days of the calendar like International Beer Day, (which was last Sunday), or Fresh Breath Day, (yesterday), but I can see some value in them as far as Greeting Card Manufacturers are concerned, (hence they are often called Hallmark Days – after the Hallmark company).

When it comes to number theory, however, some of the designations seem to be so esoteric that they defy any practical use … or justification … other than the fact that they exist to define some obscure characteristic or quality.  Sublime numbers, for example, are numbers with an “perfect number” of factors, which add up to another perfect number.

For the record: a perfect number is defined as having factors which, (apart from itself),  add up to that number … for example 6, which has factors 1, 2, 3 and 6 … (1 x 6 = 6 and 2 x 3 = 6) … and 1+2+3 = 6.   There aren’t that many perfect numbers, the next one is 28, (Factors, 1, 2, 4, 7 14 and 28).  After that comes 496 and 8128 …  Those four were known to the ancient Greeks, but it took until the sixteenth for the next three to be discovered.  We still don’t know if there any perfect numbers that are odd … but any mathematician who finds one is likely to have his name made … even if they don’t make a fortune.

The number 12 is a sublime number because it has 6 factors, (1, 2, 3, 4, 6 and 12 – 6 being a perfect number), which add up to 28, (which is the second perfect number).

For the record, the other one is:
6086555670238378989670371734243169622657830773351885970528324860512791691264
which has 76 digits.

I am sure that there are some mathematicians out there who will be able to tell me the reasons why “sublime numbers” are so special … but I have to admit to a feeling, a temptation, to think it is a case of “from the sublime to the ridiculous”.

A seminal Glam Rock album

There are several songs about the “Perfect Day“, but the one that sticks in my mind is the one by Lou Reed from his 1972 album Transformer, (and which was the flip side of his, Walk on the Wild Side, released the same year and regraded by many people as his signature piece).

I was never that keen on Glam Rock, but there is something haunting about this song: the music with the repetition of the simple phrase “Perfect Day” together with the description of simple pleasures which go to make up such a “perfect day” … and … the promise in the closing refrain: “You’re going to reap just what you sow” (… or is it an implied threat?).

Anyway, Lou Reed’s song comes to mind because today, (28th June … 28/06), is a Perfect Day.  Both the day and the month are Perfect Numbers.

There is only one other date each year which might qualify, June 6th – 06/06, but today is even more special because it does not involve the repetition of the day and the month – they are different.

Perfect numbers are an obscure feature of mathematics, (number theory), which was known to the Ancient Greeks, and has proved the inspiration for a surprising amount of investigation by mathematicians down the years.   A Perfect Number is defined as one that is the sum of all its “proper divisors” – that is all its factors except itself.

OK – some examples:

• The number 6 has four factors, or divisors:  1, 2, 3, and 6 … 1×6=6 and 2×3=6.  Of these 1,2 and 3 are the proper divisors – that is, not including 6 itself.  If we calculate the sum of these proper factors we get: 1+2+3=6 … so 6 IS a Perfect number.
• The number 10 also has four factors, or divisors:  1, 2, 5 and 10 … 1×10=10 and 2×5=10.  Of these 1,2 and 5 are the proper divisors – that is, not including 10 itself.  If we calculate the sum of these proper factors we get: 1+2+5=8 … so 10 is NOT a Perfect number.
• The number 25 has just 3 factors, or divisors:  1, 5 and 25 … 1×25=25 and 5×5=5.  Of these 1 and 5 are the proper divisors – that is, not including 25 itself.  If we calculate the sum of these proper factors we get: 1+5=6 … so 25 is NOT a Perfect number.
• The number 28 has six factors, or divisors:  1, 2, 4, 7, 14 and 28 … 1×28=28, 2×14=28 and 4×7=28.  Of these 1,2, 4, 7 and14 are the proper divisors – that is, not including 28 itself.  If we calculate the sum of these proper factors we get: 1+2+4+7+14=28 … so 28 IS a Perfect number.

In fact there are very few perfect numbers … the ancient Greeks knew of just four of them, the firs four: 6, 28, 496 and 8128.  Apparently we know a few more these days … but the gaps between them seems to get larger and larger…

I suppose it would be a really perfect day if the day, month and year were all perfect numbers … but the last time that happened was on 28th June, 496, and the next time will be some time in the future … 28th June, 8128.

OK – I admit it … this particular postcard has nothing to do with Kyrgyzstan …

I did think about  trying to define what might make for a perfect day here in Bishkek: warm sunshine, (but not too hot, perhaps with a light breeze), a trip into the mountains, in the company of good friends, a picnic, a relaxing evening curled up with a good book or a an interesting movie … or, as he seems to have all the bases covered, trying to find a link with Lou Reed’s song such as drinking Sangia at the Mexican Cantina, taking a walk in the park, (of which we have many), and watching movies …  (feeding animals in the zoo would be particularly difficult as we don’t have one).

But it’s really a question taste … one person’s idea of a perfect day might be a torment for another … so I decided to leave it with a a tribute to a rock icon and a mathematical oddity.

For the record: the lyrics of Perfect Day, from the Album Transformer, by Lou Reed, (1972)

Just A Perfect Day,
Drink Sangria In The Park,
And Then Later, When It Gets Dark,
We Go Home.
Just A Perfect Day,
Feed Animals In The Zoo
Then Later, A Movie, Too,
And Then Home.

Oh It’s Such A Perfect Day,
I’m Glad I Spent It With You.
Oh Such A Perfect Day,
You Just Keep Me Hanging On,
You Just Keep Me Hanging On.

Just A Perfect Day,
Problems All Left Alone,
Weekenders On Our Own.
It’s Such Fun.
Just A Perfect Day,
You Made Me Forget Myself.
I Thought I Was Someone Else,
Someone Good.

Oh It’s Such A Perfect Day,
I’m Glad I Spent It With You.
Oh Such A Perfect Day,
You Just Keep Me Hanging On,
You Just Keep Me Hanging On.

You’re Going To Reap Just What You Sow,
You’re Going To Reap Just What You Sow,
You’re Going To Reap Just What You Sow,
You’re Going To Reap Just What You Sow…

The infographic

Recently, the country’s most popular newspaper, Vecherniye Bishkek, (Evening Bishkek), published an interesting infographic under the headline “What do businessmen complain about?”.

Although it was called an Infographic, I am not sure to what extent it really qualified as such.

Let me explain:  An infographic presents information in a visual representation, rather than in words or symbols. The idea is to encapsulate and communicate complex ideas in a simple format which can be easily grasped and understood.  This particular example, however,  was really just a list of statistics presented with a graphic to illustrate each of the individual eight points.

Having said that, it was a very interesting article … and I think it is worth drawing attention to it.

The value of images should not be underestimated.  The Nuffield Maths Project used to have as its motto an old Chinese proverb: “I hear and I forget, I see and I remember, I do and I understand”.  There seems to be some intuitive truth behind this maxim.  Although some people have very good verbal skills and can remember intricate details of conversations in which they have been involved, or even simply overheard, for most people the visual image tends to stay with us.  The image also goes a long way towards convincing us, after all there is another adage that “Seeing is believing”.

As an aside: back in the 1960’s there was a research study done where people were presented with three versions of scientific concept and asked to indicate with which they felt more comfortable and more convincing.  Take for example, Pythagoras’ Theorem.  Presented with the three versions:

The theorem in words …

... and as symbols

... as a graphic ...

Most people felt at home with the graphic, some people preferred the comfort of the phrase they had learned parrot fashion and only a few people – Mathematicians felt at ease with the symbols.

Anyway, as I said, this particular example was really just a list of statistics presented with a graphic to illustrate each of the individual eight points.  Under the title, “What inhibits business in Kyrgyzstan?”, and a heading “The Top 8”, was the list of concerns which according to the credit at the bottom, came from information supplied by the World Bank.  Unfortunately, there was no other information, for example about the survey which revealed these concerns, (and apparently covered several countries in the region), or what other concerns might have been raised.

Also, although each of the eight issues was numbered implying that they had been placed in some form of rank order,  it was not clear to me the criterion on which this was based.  Bottom of the list, for example, in eighth place was Transport with 31% of firms in the survey considering it to be a big problem whilst in seventh place was Licensing and different types of permits, but only 26% of respondents reported that they gave “gifts” in order to receive a permission, especially when it came to imports.

It would have been nice to see a bit more of the justification for the inclusion of each item on the list, although there weren’t many surprises.  The issues are all fairly general and crop up in other surveys, such as the Investment Survey conducted by the International Business Council, (IBC).

No. 1: Electricity supply

At the top of the list was concern over electricity supply with 58% of respondents considering that unreliability of supply was a major obstacle that they faced in trying to conduct their business.   This surprised me a little, but I can see the point.  We have been fairly lucky in our office in that although we have experienced power cuts, there haven’t been that many of them, and only a relatively few have occurred during the working day or lasted long enough to require us to close the office.

When I first came here seventeen years ago, the power supply was a major problem, so nothing much has changed in that respect, although there has been a difference in the nature of  the problem.  Back then, the problem was usually one of variations in voltage with the occasional failure of some part of the system leading to a power cut.   In recent years the variations in voltage levels, although they still exist, seem to be less marked but other forms of disruption to the regularity of supply have become almost institutionalized, with planned rolling power cuts, and system failures.

If we have suffered relatively little disruption, there have been considerable problems faced in some areas, and the survey reported that on average firms experience about 3 power cuts each month losing almost seven and a half hours.  Those are average figures, so if we have been fortunate in having fewer power outages, there are firms which have suffered far more.  It’s been estimated that over 10% of production has been lost as a result of the unreliability of power supply.  I remember how one manufacturer, (a well known soft drinks manufacturer), complained that the rolling power cuts was doing irreparable harm to the country’s economy and I don’t think it was a just case of “special pleading”.

Interestingly, and ironically, one of the considerations that is often quoted as being attractive for  potential investors is the availability of low cost energy.

No.2 : Access to Finance

In second place was access to finance.  This was cited by almost 28% of the firms surveyed.  Apparently just over one percent of all the financial capital in the economy involves credit finance, and only about 14% of the capital investment in the country is provided by the banks.

Although one might want to praise businesses for keeping within their capabilities and resources, that’s pretty low, (amongst the lowest in the region), and hinders development of new business and the introduction of improvements and modernization.

It’s not that businessmen don’t want to take credit, almost 20% of the firms in the survey said that they consulted banks, asking for information or applying for credit,  but that’s actually quite a low proportion when compared to other countries.

There are all sorts of reasons why this situation might exist.  Until recently the banking system was mainly used for transactions and there was very little work in the field of credit management.  Interest rates tend to be high, for all sorts of reasons – not just profit.  The term of a loan was often quite short, partly because the tax system seemed to encourage this.  Many of the businesses in Kyrgyzstan didn’t have the collateral to offer to guarantee a loan.  As it happens, many of the credits implemented in Kyrgyzstan have actually been  quite small – too small for some of the banks, and this has led to the rise in Microcredit organizations, (although these are not always able to meet the demand of the marketplace).

Alternative  methods of financing have been explored: in particular leasing and the Islamic system of banking; but with different levels of success when if comes to implementation.

No. 3 : Corruption (For the record: Vzyatka = Bribe)

I had expected the next issue to feature higher than the previous two; in third place, rated by almost 60% of the respondents as an obstacle to doing business, was the issue of Corruption.

Just under half of those in the survey confirmed that the gave “gifts” to government employees in order to resolve problems.

Everyone seems to take it for granted that corruption is endemic in Kyrgyzstan, (and that’s not just a question of Vzyatki, or bribes, but all forms of abuse of power such as nepotism or taking decisions that favour one party at the expense of another).

It’s interesting to note that the major global survey about corruption by Transparency International is called the Corruption Perception Index.  By its very nature, it is almost impossible to garner concrete and reliable information about corruption and so the measurable statistic is the perceived level of corruption.  As I said, it is almost taken for granted that corruption is endemic and a way of life here.

It is possible, however, to survive and do business without submitting to corruption.  It may not be easy, but it is possible.  In fact, I think that there is a growing impatience with corruption with the authorities making “lots of noise” about it and introducing measures to combat it.  Perhaps more importantly, ordinary people seem to be making a stand and showing that they feel offended at “being used” in this way.

No. 4: Tax Policy and Administration

The next issue was that of the tax system, (policy and administration).  About a third of the forms considered that the administration of the tax system caused problems for the conduct of business and almost a half considered that tax rates were a problem.

This is something that I find confusing as.  Let’s face it, nobody likes paying taxes so any tax rate, no matter how low, is going to be perceived as a problem.  The alternative, however, of no taxes, and therefore no government services, is a recipe for anarchy which is not something that businesses would welcome.  People often say that “the government should do something”, but in order to do so it must receive some revenue in the form of taxes.

How the government raises that revenue, however, is another issue.  It seems to me that the system is complex – almost too complex to be efficient and appears as if it is designed to create areas of ambiguity and difficulty – thus creating the potential for corruption.

A former Prime Minister offered businessmen the advice: “Pay your taxes and sleep soundly!”  If only!  We try to be scrupulous about our tax affairs, but this doesn’t stop us facing and fending off inspectors intent on finding fault and applying an ingenuity which has to be admired for its creativity – even if, at the same time, it has to be deplored for it erroneous logic.

Nowhere, it seems to me, is this more evident than in VAT, (Value Added Tax).  Now, in theory VAT is an excellent tax; fair and efficient.  The consumer pays on the consumption and it can be targeted towards particular products and services,  so that essential expenses can be exempt or taxed at a lower rate whilst luxuries or less desirable products and services, (tobacco and alcohol, for example), could be taxed at a higher rate.  Through a system of rebates the tax is collected at various stages of the economic cycle, and the full burden falls on on the final consumer.  However, it is a complex tax, fraught with regulations and conditionalities, which add complexity and confusion.  Although there are many difficulties associated with them, I have come to prefer simplified taxes such as a straight forward sales tax.

No 5: The Shadow Economy

To me, there is a question of striking the balance between funding the essential services that government should provide and yet keeping taxes low enough and simple enough to administer, so as not to be a disincentive to innovation and work – or an incentive to hide in the shadow economy, which, incidentally, was the next concern on the list.

It was estimated that something like 67% of firms face competition  from firms that operate in the shadow economy – aren’t registered or work unofficially, (effectively, unfair competition as, for example, they do not pay taxes they are able to charge lower prices).    It was also estimated that firms work are able to continue working “in the shadow” for over four years, which is much longer than elsewhere in the region.  Over 35% of the firms taking part in the survey felt that this unfair competition was a handicap which prevented them from working normally.

This “not paying tax” is an insipid feature of society, depriving the state of the wherewithal to carry out its functions.

There is yet another factor to bear in mind.  Just as “corruption” is considered to be a fact of life, so is the fact that many businesses work in this “informal economy”.  People want lower prices and are always looking “for a deal”, which helps to perpetuate the system with all the iniquities that it involves, preventing the development of  a a caring and efficient society.

No 6: Availability of Qualified Staff

In the sixth place ion this list of concerns was the low level of education and training of the workforce.    The issue of the the availability of suitably qualified staff is always one of the top ten concerns highlighted in the IBC Investment Survey, so I wasn’t too surprised to see it in this list as well.

I know that this is a perennial complaint in tourism, for example.  Of the seventeen or so institutes of Higher Education where students follow “Tourism Courses”, tour firms always complain that graduates’ knowledge is patchy and it falls to the new employer to retrain their new recruit afresh.

Oddly enough, I seem to remember exactly the same complaints from employers back in the United Kingdom, twenty years ago.

As it happens I think that the issue is a bit more fundamental. Perhaps it is because I grew up in Britain, but I see Education and Training as being a process and the final result is not a graduate who simply knows things, but is able to do things; is adaptable, resourceful and able to use what they have learned.

As and aside: The graphic quoted statistics to show that institutes of Higher Education in Kyrgyzstan apparently occupy poor positions in the various world rankings.  In itself, that doesn’t worry me.  Someone has to be at the bottom of the table supporting all the others above them – and that doesn’t necessarily mean that the quality of education is bad.  It might be excellent, because it is excellent everywhere.  (Ok, I do understand that is unlikely to be the case … but my point is that the statistic doesn’t necessarily support the contention)

There were other contentious statistics, each of which may be telling and important in their own right, but seemed to be quoted in an inappropriate manner.  It was pointed out, for example, that there is limited expenditure by firms in Kyrgyzstan on the attraction of,  search for and engagement of new employees.  As it happens, I think that’s more symptomatic of a poor approach to staff recruitment than a comment on the avail;ability of suitably qualified staff.

(Incidentally, I am not sure why a low level of investment in internet websites and use of email was reported in this particular item.   No doubt the authors had their reasons, but it wasn’t clear to me.  What’s more, although I can’t argue with their figures – especially when it comes to making comparisons with other countries – as I don’t have access to the data – but the claims seem counter intuitive to my experience.)

No. 7: Licenses and Permissions

Which brings us to the seventh issue identified in the infographic: Licenses and Permissions.

Theoretically, at least, this strikes me as a fairly major issue.  I am not opposed to controls – where they are necessary, and  some are indeed necessary – for reasons of, for example,  Health and safety, national security or to prevent abuse of power and market position by holders of a monopoly.  The problem, it seems to me, that there is a tendency to impose a regulatory requirement when there is no such need.  Not only does this impose a burden on business, it also leads to a mentality of “going through the motions” and “meeting the minimum standard”, (LCD, or the lowest common denominator), rather than a commitment to quality.  In other words, in terms of consumer protection  it can be counter productive.

Most importantly, however, it opens the door to corruption … it provides an invitation to corruption.  In fact, 26% of respondents reported that they gave “gifts” in order to receive a permission, especially when it came to imports.

In such an environment, to what extent can we rely on the the veracity, validity and reliability of the system.  To what extent can we trust it?

No.8: Transport

The final concern cited in the infographic was that of  Transport, with 31% of firms surveyed highlighting it.

There’s probably several aspects to this issue – given that Kyrgyzstan is a small landlocked country,  a long way from both suppliers and markets fort its goods and services.   What’s more, the country is divided into quite distinct small regions by the mountains that crisscross the landscape, making transport between communities difficult.  If such physical barriers did not present a big enough problem, there are man made administrative ones, such as borders that have to be navigated.  Then there is the question of the state of the transport infrastructure, much of it old and “well worn”, if not actually “worn out”.

We notice this quite a bit in tourism – the lack of air connections with the outside world;the limited rail network – effectively just one line to Bishkek and Balykchy;

One of the questions that immediately jumped into my mind when I saw this infographic was: “What’s missing?“.  It seems to me that there are several issues which could have been raised …

1. Regulations, whether it be environmental or labour legislation, or ownership (foreigners, for example, cannot own land), not just the question of licenses and permits.
2. With the regulatory framework, of course, comes Bureaucracy, with resulting  inspections, and the openings that they provide for stress and corruption; just part of the wider issue of relationships and the ease of dealing with the different levels of authority.
3. The predictability of the legislation which seems to be constantly changing.  Although there’s nothing particularly unusual in that, (all jurisdictions bring in amendments to their legislation – laws, rules and regulations – to take account of changing circumstances), but the scale and the frequency of such amendments in Kyrgyzstan can make it difficult to maintain an overall view and to keep track of the current requirements.
4. Relationships with trading partners … whether that is on the international stage or in the form of enforcing contracts, for example in the court system.  I have heard it said, for example, that businessmen try to avoid going to court to protect their rights in a commercial dispute because they don’t see court decisions as being impartial or consistent.
5. A crucial issue of course is Safety and Security … whether that be in terms of petty crime, organized crime, terrorism or corporate raids.  This not only adds to operating costs but also has to kept in mind and taken into account at every step of every corporate decision.  Once again, there is no specific Kyrgyz element to this, in that Safety and Security is an issue in all communities around the world … but whist there is uncertainty and unpredictability in society, planning for the future and for development is that much harder.

Pi

If you are an American, or happen to be in America, then you might come across references to today being Pi Day.

Pi, of course, is the ratio between the circumference of a circle and it’s radius, and most schoolchildren, (I won’t tempt fate and say any school child), will probably be able to tell you is approximately equal to 3.1415926 … I say approximately because it is what mathematicians call an “irrational number” which has nothing to do with the insanity that it seems to send people into, but that it cannot be expressed precisely in a rational format – that is, as a fraction: a/b. The stream of digits behind the decimal point in the number goes on, … and on, … and on … without ever repeating itself or forming a predictable pattern – or so the mathematicians inform us.

To most people, pi is equal to 3.14.  That’s not exact … but for most purposes it works perfectly adequately as an approximation.

And that’s why today, March 14th, is known as Pi day – especially in America.  Back in Britain we write the date in the order Day, Month, Year, (DD/MM/YY – so today’s date is 14th March, 2012 – or 14/03/12), but in America, for some unaccountable reason, they write the date in the order Month, Day Year, (that is, today’s date is 14th March, 2012 – or, in figures, 03/14/12 … or more simply 3.14).

Of course, March 14th, 2015, will be a very special day, (03/14/15 or 3.1415), but in the meantime, modern Mathematicians are happy to commemorate the occasion annually whilst planning special events for the centenary celebrations, (there’s already a special Facebook page to commemorate the occasion).

The history of Pi is quite interesting – but you would probably expect me to say that, having been a Maths teacher, and one who is interested in the History of Maths; in my opinion it was one of the best courses I ever took.

However, there is one incident in the history of Pi which is well worth recounting and I would like to share with you … the attempt by Dr. Edwin J. Goodwin to get the Indiana State Legislature to pass a law decreeing that the value of Pi in the state would not be the incredibly difficult to use value of 3.1415962… but, instead, to use the much simpler value of 3.2, (or maybe 3.2325 or 4 or even 9.2376 – Dr. Goodwin is not very clear in his various on the subject and the Bill doesn’t actually mention Pi … it’s actually about Squaring the Circle – something that Mathematicians have suspected for centuries to be an impossibility – which was finally proved in 1882 ).

As an aside: popular folklore says that he actually tried to change the value of Pi to 3, and the state is variously recorded as Kansas or Alabama …

Goodwin even managed to get his theories published in an early edition of what was to become the prestigious journal, the American Mathematical Monthly, (although they published a disclaimer … that it was “published by request of the author”).

Apparently there was some confusion amongst the elected members of the legislature when it came to debate the bill, (No 246 of 1897), possibly because the members didn’t understand the gobbledygook that was written in the substantiation of the measure, (at least one openly admitted as such), and were “blinded by science”.  Anyway, it mustered some support and looked like it might actually pass into law until one of the senators pointed out that no matter how august a body, it didn’t have the authority to “define mathematical truth”.

I often wonder what would have happened if Indiana had indeed passed the Bill.

This is one of the two stories I often tell people when discussing law making … politicians have a knack of thinking that they know the best course of action, but they are simply making decisions and guided by various principles and advice.  Sometimes they allow their fervor to get the better of them and make bad decisions, (albeit with the best of all possible motives), … but decisions that the rest of us then have to live with and which are often far more far reaching than was originally intended.

Oh, and the other story? … it was told to me by a lawyer and is about a politician who wanted to pass a law raising the age of puberty.

Sudoku

The kids spend an awful lot of time on the computer.  In a way, I am happy about this, after all I used to teach the subject and I know how powerful computers are, the capabilities they offer to children and the vast amount of information they can have at their fingertips thanks to sites like Wikipedia.

On the other hand, I am not so keen on what they spend most of their time doing when they are sitting at the computer … playing games.

I am not really very keen on computer games.  That may be because I am not very good at them … but it is also a feature of the sort of games that are available.  Most of them seem to be “Shoot ‘em up”, “Zap it” type games, and as someone who enjoys puzzles, I find most of these games extremely frustrating and a waste of time.

I do know that the development of computer games have been an important stimulus that has led to important developments in computing, but that doesn’t help to dispel my innate antipathy to them.

There’s one computer game, however, which sits on my desktop and there is hardly a day goes by when I don’t click on the icon and while away some time, unwinding and trying to beat the machine as it sets me a task: Sudoku.

I was introduced to Sudoku a few years ago when a tourist left a copy of a newspaper that they had brought with them from home as a “present” for me whilst they left Bishkek to go galavanting around the country.

Having scanned the headlines and read some of the articles, I was about to put the paper down when I turned the page and came across a strange grid with numbers in it.  It looked like a crossword, but as I am not particularly keen on crosswords I was about to put the paper down, but the numbers had caught the attention of the mathematician in me, and so I took a closer look.  I read the rules, and looked at the answer to the previous days puzzle, trying to work out what to do.

It looked simple enough – a nine by nine square, (nine rows and nine columns), divided into nine smaller, (three by three), squares.  Each row, each column and each of the smaller squares had to contain each of the numbers 1 to 9 … no number was to be repeated.  In the completed puzzle there should be nine 9′s, nine 8′s, nine 7′s … and so on.

OK – I thought to myself, that’s simple … and the Maths Teacher in me reminded me that, although the rules and constraints are different, it’s like a special kind of Latin Square – just like a Magic square.

Some of the numbers are given to you as a start, and you are supposed to then place the remaining digits in the remaining squares.  The digits that define the starting position are supposed to be selected in such as to make the solution to the puzzle unique.

At the time I didn’t know this, but there are a huge number of such squares where the digits 1 to 9 can be arranged according to these rules – apparently almost five and a half million of them.

Fortunately, it was a relatively easy example in that first first puzzle that I tried, although by no means trivial, and thought I would like to have another go.

So, rather than wait and hope that another tourist might bring me another newspaper with a puzzle in it, I decided to surf the net and see what I could discover.  I found a couple of sites on-line that published puzzles and so I printed them off and set about trying to solve them.  I soon realized that whereas examples were indeed trivial, others were monstrously difficult.

I worked out a few strategies for solving puzzles but at that point I was still very much a novice.

It turned out, however, that Sudoku was quite popular here as well and puzzles were published in some of the local newspapers.  Some of our staff were quite keen on them and we started swapping notes about strategies.  One of them showed me a computer program that they had downloaded from the net that generated puzzles and as well as allowing you to print out the puzzle and solve it on paper, also allowed you to fill in the grid on a computer.

I downloaded a copy and that is the version I play almost every day.

In a way, it’s a cheat because it won’t let you make a mistake an insert the wrong digit in a square.  Instead it flashes a message at you and keeps a count of how many mistakes you make.  Also, if you get really stuck, it offers the chance to ask for hints.

As a result I tend to print off the grid and try to solve it on paper, and only if I get really stuck on a “diabolical” puzzle do I use another feature of being able to enter a starting position and then solve it on the computer – asking for hints.

There are some people who say that Sudoku may involve numbers, but that it doesn’t involve Maths.  In a way they are right.  There is no arithmetic involved apart, perhaps, from basic counting; how many 9′s in that row, column or block?, or 8′s, etc.  On the other hand, they are totally wrong … it’s all about Mathematical thinking, logic and problem solving.

I wish I could get the kids to turn their hand to Sudoku.  I think it would be much more useful to them in the long run than Grand Theft Auto, Warcraft or any of the other violent “shoot ‘em up” variety.

Unfortunately it just doesn’t have the DADA, (the dynamism, action, drama or appeal), and neither do they, (the determination, application, discipline and ability to stick at it).

Still, I live in hope that one day … maybe … just maybe … one day … they will!

Analysing the election results

I came across a number of interesting articles about the election over the weekend, (both in Russian), comparing the number of votes won by candidates with the amount each spent as part of their campaign, which attracted the attention of the former Maths teacher in me.

The first compared the cost of each vote whilst a second addressed the question of the correlation between the two figures.

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Here’s a table giving a summary of the data:

*Approximate, I have used an assumed exchange rate of USD1 = 45 som

As I often say: “statistics raise more questions than they answer”, and these are no exception.

It looks, for example, as if the average cost of a vote was 457 som, although the articles suggest it was just 84 som.  This discrepancy arises because, for some reason, the author limited his analysis to just the top 11 candidates; those who attracted more than 3000 votes.  He presumably wanted to eliminate, (or at least minimize), any distorting effects that these lower figures might have on the figures, (such as inflating the average cost of a vote), but doesn’t explain why he chose this cut-off point apart from suggesting that this probably reflects the personal votes of family and friends – so votes that aren’t “bought” or obtained through the advertising.  Indeed, one of the candidates, (who was in the top 11), expressed surprise at how few votes he managed to gain from his home village, suggesting that this is indeed a factor that should be taken into account.  (Logic suggests to me, therefore, that perhaps we ought to deduct this figure from all the candidates before doing the calculations, but let’s let that pass.)  In fact, his choice of cut-off point hasn’t affected the range, (from 32 som to 2146 som per vote), which is in itself quite considerable.

Then there is the fact that the figures in the Campaign Fund column represent the official reported figures, but Atambaev, in particular, was the beneficiary of adverts placed by other parties in his support, and presumably paid for by them and so not included in these figures.

Also, there were reports on election day of someone apparently offering voters 500 som, (about USD11), to vote for a particular candidate although it’s not clear if anyone took him up on the offer, and, if they did, it probably wouldn’t be reflected here … which leads to a question: if there were other “unreported expenses”?

Still, it’s clear from this that Atambaev spent the most, and achieved the highest vote – winning the election.

The other issue that was considered was if there was a correlation between the amount spent by a candidate and the number of votes they managed to gain.

During the early days of the process, many of the original 83 presidential wannabes were complaining about two of the requirements in the process, the need to pay a 100,000 deposit, (returned in the event that they got over 5% of the vote), and having to collect some 30,000 signatures in support of their candidacy.  These, they argued, favoured the rich candidates and prevented poorer members of the community from exercising their “constitutional right” to contest the election, and so they “demanded” these provisions be revoked.

For the record: One of the features of this particular election was the number of candidates who described themselves on their nomination papers as “unemployed” and who, presumably, were without substantial funds to use in their pursuit of the highest office in the land.

In a way, these protests are right, but that doesn’t mean that these are a bad thing in themselves.  They are meant to help prevent “frivolous” candidacies – after all the election process costs the taxpayers money and aren’t really meant to be used for the purposes of self aggrandizement.

On the other hand, such nominations are not always for the purpose of self promotion.  There are times when ordinary individuals want to make “a stand against the system” and encourage protest votes.

For example: there was the case of Reg Keys, the father of a soldier killed in Iraq, who stood against Tony Blair in the 2005 General Election, on the single issue platform of opposition to the government’s policy on the Iraq and the war in Iraq. Although Tony Blair was comfortably re-elected, Keys got 4,252 votes (10.3% of the total), coming in fourth place not far behind the candidates from the main opposition parties.

Then there was the BBC correspondent Martin Bell who stood against the sitting conservative MP Neil Hamilton, (who was the subject of many allegations of “sleaze”) in the 1997 General Election.  Bell succeeded in overturning a substantial conservative majority, capturing the seat and taking his place in Parliament as a truly independent MP.

Perhaps the most celebrated protest vote candidate in Britain was Screaming Lord Sutch.  A musician in the 1960’s he contested 41 elections before his death in 1999.  Only once did he receive more than 1000 votes, (which represented just 4.2% of the vote) and losing his deposit every single time.  His first election was contested as a candidate for the National Teenage Party, but going on in 1983 to found the Official Monster Raving Loony Party.  Although they are sometimes seen a “joke”, with a manifest containing things that seem to be impossible or absurd, they have had some success, with members being elected to local councils, (sometimes unopposed, but not always), and some of their proposals, (such as lowering the voting age to 18, passports for pets, and “all day opening for pubs” have made it into legislation and it was a “Loony” who created a scandal revealing that under European Community rules milk and butter surpluses were “being dumped down mineshafts” in order to maintain price levels.

No doubt some of the 83 candidates that submitted proposals represented the protest vote, (anyway, the ballot paper included a protest vote option – “Against all” which attracted a considerable 9483 votes … and there were a considerable amount of “spoilt papers”), or single issue candidates, (there was one candidate who seemed to have one issue foremost in his campaign; “I am for dual citizenship with Russia”).

In the West, this question of “funding democracy” has long been an issue.  Millions of pounds are spent by parties and individual candidates in a British General election and lions of dollars in a American election.  Even on these figures the candidates spent over three million dollars between them.

Using a statistical analysis, the article pointed out that there did appear to be a correlation between the amount spent and where a particular candidate appeared in the final results.  The correlation coefficient when considering all the candidates is 0.78, (and based on just the candidates who received more than three thousand votes it was 0.86).  That’s quite high; a perfect correlation would give a result of plus or minus 1.

Doing a regression analysis, however, does not produce such convincing results.  The author using just the top 11 candidates gets a line of regression which shows only Atambaev lying above the line; that is receiving more votes than the analysis might predict for him given his level of expenditure.    Including the bottom five candidates skews the line a little, giving a slightly better picture, but still not very convincing.  It seems clear that that statistical adage that “correlation does not mean causation” is true.  There may be a relationship between the amount spent and the number of votes gained, but there were other factors at play as well, which are less easy to quantify; such as the authority of the candidate of the people, the appeal of his views and beliefs to voters, value for money of the way the election fund was used, and so on.

Despite the lack of a causal connection between election expenses and the eventual results, this question of “funding democracy” is an important one.  It is a hot topic, for example, in both the United States and Britain.  A recent survey suggested that something like 60% of nations offer some form of public funding for political parties to help defray the costs of elections.  Sometimes this funding is given to the central party apparatus, but in other cases it is given direct to the candidate.

There is a good case to be made for state funding for political parties.  Although parties should be able to rely on their supporters for finance, dependency on fund raising activities and private donations opens a party to undue influence from individuals and pressure groups.   The overall aim of state funding both in the States and Britain has been to stop corruption in election finance, but even in those countries which do provide a measure of finance from the public purse, there are still occasional stories of “sleaze” and “graft”; Tony Blair was accused of setting the price of the Labour Party conscience at one million pounds after a donation of that size appeared to lead to a change in policy exempting Formula One racing from a ban on tobacco advertising; and the sale of posts such as Ministerial responsibility, or honours, or contracts.

As “part of the deal” schemes for providing state financial support often include caps on the size of donations, (sometimes with a requirement for public declarations of donations above a certain threshold), and election spending, (with an aim to “leveling the playing field” in order to help less well off campaigns as well as protecting political parties from an ever increasing spiraling in costs likely to damage if not bankrupt parties).

The Jogorku Kenesh have also been debating the topic and it looks as if they will  decide to provide state funds for political parties based on their performance in the previous election. That means that five parties that have seats in the current parliament will receive funding, but the others won’t.  That means that the Butun party which narrowly missed out on obtaining seats in the last election, due to late changes in the election register, won’t receive any state financing.

I understand that the UK is currently considering a similar system linking state contributions to the number of votes won by the party at the previous election – but without the limit which would hit the Butun Party and count against new parties trying to break into the system, such as UKIP in Britain.

There are other ways of financing parties, apart from basing it performance at previous elections, such as providing for  Tax relief on donations, (as is often the case with charitable donations – perhaps over a particular threshold and up to a capped limit), or matched funding of private donations, (up to a limit) as I believe is the case in Germany.

Whatever the source of funding, and no system strikes me as ideal, it is important that it is spent wisely.  As well as looking at the results in Sunday’s election, remember the experience of the UK Independence Party, (UKIP), which spent some twenty million pounds campaigning, yet achieved not a single seat.

These weren’t the only webpages that provided a mass of statistical data.  The Central Election Commission ran their own site, (http://cec.shailoo.gov.kg/) which provided official data as the results came in polling station by polling station.  The tables were then reposted on many of the news services.

Perhaps the most interesting article, however, was “Mapping Kyrgyzstan’s Regional Divisions, Vote by Vote” on the eurasianet.org site.  It didn’t have tables of figures, but did have one graphic – a map of the country showing Atambaev’s share of the vote in different electoral districts around the country:

The map was produced by Dr. Fredrik M. Sjoberg, a visiting scholar at New York University, and created with data from Kyrgyzstan’s Central Election Commission.

Here is clearly visible the one thing that has dominated political debate in the country: the North-South divide.  It seems obvious from the graphic that there is a political and social fault line, mirroring the geological fault line that cuts across the country from the South East to the North West, and implying that there is an awful lot of work ahead of the president elect if he is going to be able to resolve it and unify the country.

Call My Bluff

Back in the days of Black and White television, (yes, that long ago), there was a popular programme on the BBC2 channel called “Call my Bluff!”.   It first appeared in 1965, (in black and white) and lasted until 1988, when it was seen in colour.

Call My Bluff was a panel game chaired by Robert Robinson and featured two teams of three people captained, originally, by Frank Muir and Robert Morely, who took turns in providing definitions for strange words taken from the Oxford English Dictionary, announced by the chairman with great flourish with the phrase: “And the next word is …”, ringing his little bell, “ding”.  There in the set behind him the next word emerged.

Of course, unless you were the Cheshire Cat, (in which case means what you want it to mean – a recipe for anarchy and solipsism), most words have just one meaning.   That meant that, as each of the three members of one team would, in turn, offer the members of the opposing team a definition of the chosen word, only one of the suggestions could be the true definition whilst the other two were “bluffs”.   So, the opposing team had to listen to and weigh up the probability of each definition being the correct one. If they guessed correctly, they scored a point, but if they got it wrong, the other team got the point.

When I say it was “popular”, of course, everything is relative.  BBC2 was the “elite” channel with intellectual and arts programmes.  It tended to attract a smaller audience than the more mass orientated BBC1 or TV, (Independent Television), which relied on advertising for its funding.  Even so, it was a well known programme and many features of the game were to pass quickly into popular culture.

Of course, it was a game you could easily play along with, whilst sitting at home.  The words were obviously chosen to be obscure, words like: Queach, Strongle, Ablewhacket, Hickboo, Jargoon, Zurf, Morepork, and Jirble.  For example, “Queach” was apparently defined as “a malicious caricature”, “a cross between a quince and a peach”, and “a mini-jungle of mixed vegetation”.  (For the record, the true definition is the last one, the other two were “bluffs”.

Sometimes, however, just occasionally, a word cropped up which you might already know.  I remember watching when the word “googol” was revealed as the chairman rang his little bell.

A googol is a very large number, a 1 followed by a googol of zeros.  A google – oh, that’s easy, it’s also a very large number, a 1 followed by 100 zeros.  so you have:

10 – ten
100 – hundred
1 000 – thousand
10 000 – ten thousand
100 000 – hundred thousand
1 000 000 – million
10 000 000 – ten million

… and so until

10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 - that’s a googol

 … and a googolplex, is a 1 with a googol of zeros after it.

The term was invented in 1938 by the 9 year old nephew of a mathematician who was trying to think of a way to talk about infinity in a book that he was writing.  He asked his nephew what he would call a very large number: a 1 followed by a hundred zeros, and he came up with “googol”.

I have to say that I think this was a piece of genius … Googol is so much friendlier a name than the alternatives: ten duotrigintillion,  ten thousand sexdecillion or ten sexdecilliard.  

Although googol is indeed a large number, so large that it is difficult to imagine, it is a definite quantity and is not the same as infinity … and neither is a googolplex – the name the same little boy gave when he was asked what he would call a 1 followed by a googol of zeros.

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Happy Birthday Google!

Incidentally: Google – yes, google as in “google it” when you enter a term in search engine on the internet – is actually a deliberate misspelling of “googol”.

And Google was founded thirteen years ago today … on September 4th, 1998.

Tomorrow, (today’s not a working day), we’ll have a cake in the office to mark the occasion.

Perhaps the best Statistical graphic ever produced: Minard's map showing the progress of Napolean's Grande Armee in the advance on, and retreat from Moscow.

The last Sunday in April is Statisticians Day.

It’s another one of those “professional holidays” that seem to be so common here in Kyrgyzstan.  This one seems to be particularly appropriate because it often seems that here in Kyrgyzstan, statistics are given the sort of reverence normally reserved for holy writ.

The word itself comes from the Latin, “statis“,  which means the “political state” and one of the main tasks it is used for is the analysis of facts and figures for and about governments, resources and populations.  Hence the official fascination with statistical results, even though many people may not be fully conversant with the underlying principles that produce them.   Although Statistics provide a valuable tool, they are also open to abuse and misunderstandings, both intentional and unintentional.  

One definition of Statistics is: “the science of the collection, organization, and interpretation of data”.   Another states that it is the study of “ways that lots of data can be represented using a few numbers and the study of how such numbers can be chosen and used to draw reasonable conclusions about the data.”  Those definitions raise a number of important issues, which I have mentioned before in other postcards.  For example, taking the first definition  I would add the “presentation” of data, and to the second I have concerns over what is meant by “reasonable”.

Liker most people, my involvement with, and exposure to, Statistics began at school.  Of course we learned standard techniques for finding an average and graphing data, but it was only as I started to undertake my A-Levels, (University entrance level examinations, taken at 18+ in the UK), that I really began to understand what a fascinating topic it was.

My Maths teacher looked at me and said: “So, you are not taking Physics.”  ”No, Sir,” I replied.  ”Well, in that case we had better not enter you for the Mechanics paper … there’s an overlap between the two and studying the one would help your understanding of the other.   More to the point, the other candidates would have an advantage over you, because they will almost certainly be studying Physics, so they will find the questions easier. ”   At the time I didn’t understand the significance of that comment, but having worked as an examiner and attending grading meetings, I now see the wisdom it encomposed.  Then he said, “You don’t want to do just Pure Maths, because that means you’ll have to delve deeper into the subject in more detail and will have no idea of the practical applications of what you are studying, so it will have to be the Statistics option.  It will also fit in better with your other subjects,”  (Geography and Religious Knowledge … OK, I’m not sure that it added much to my understanding of the Bible, but it did prove useful in some of the Geographical topics that we studied, and was immensely useful later when it came to studying Psychology and Sociology, not to mention my current work in trying to run a business.)

Despite the fact that Statistics was my Applied Maths Option, Statistics is considered by many to be a distinct mathematical science rather than a branch of mathematics because of its empirical roots and its focus on applications.  It can be subdivided into a number of different fields such as Descriptive Statistics and Inferential Statistics, or Parametric as opposed to Non-Parametric Statistics.

It’s a fairly new branch of mathematics and is usually thought to have it’s origins in the 17th Century, with crucial concepts later being developed later by Blaise Pascal and Pierre de Fermat.  In the nineteenth century it came into its own as a serious subject and the advent of computers to aid in the essential “number crunching” involved and the vast number of applications where its results can be applied have meant that it has become a ever increasingly important influence in our lives.

My study of Statistics, however, has led me two conclusions that I constantly refer to: that “I am not afraid of numbers, but I am afraid of the way that they are used” and “Statistics raise more questions than they answer“.

Disraeli is reputed to have said when describing the ever escalating costs of producing Babbage’s Analytical Engine, (a sort of prototype calculating device that is hailed as a forerunner of the computer – which, incidentally, could have been steam driven), that “there are lies, damn lies and statistics.”   Most mathematicians, and in this I would include myself, feel that he may have been going a bit far when he said that, but would have a lot more sympathy with Andrew Lang, when he said, “He uses statistics as a drunken man uses a lamppost – for support rather than illumination.”

With proper and sensible use, however, Statistics can be a vital tool in raising issues, focusing attention and helping the decision making process.  Without them, we would be reliant upon instinct and emotion and much more likely to find ourselves “lead up the garden path”.

Those working in the field of collection, organization, analysis and representing of data do a valuable service as “back room” boys,  and deserve the recognition given to them by this “professional holiday”.

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For someone interested in history, I find it interesting how some dates in the year seem to attract important anniversaries whilst others seem to be bereft of them.  Take April 15th, for example, which is not only the anniversary of the sinking of the Titanic, (which struck an iceberg the night before), and the death of Abraham Lincoln, (who was shot in the Ford Theater the night before), it’s also the birthday of two of my heroes: Leonardo da Vinci, (1452) and Leonhard Euler, (1707).

Most people will know the stories of the sinking of the Titanic and the assassination of Abraham Lincoln and have some idea of who Leonardo da Vinci was.  Leonhard Euler, however, will probably be a closed book to most and I suspect that few people will have any idea of how to pronounce his name.

I may be wrong, of course, because schoolchildren learn Euler’s formula (V − E + F = 2, connecting the number of vertices, edges, and faces of a convex polyhedron – for example a cube has eight corners, twelve edges and six faces: 8 – 12 + 6 = 2), in mathematics classes and his solution for the famous Konigsberg bridge problem, (where seven bridges connect two islands and the two banks of a river, and the problem being to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point – it isn’t and Euler developed the proof to show this is so).

He was born in Switzerland and was a great mathematician – one of the most prolific mathematicians of all time.   He made important discoveries in fields as diverse as acoustics, algebra, artillery, astronomy, calculus, finance, fluid dynamics, geometry, graph theory, mechanics, navigation, number theory, optics, and probability.  Amongst his greatest discovery was the classic identity :

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which unites the five great mathematical constants.  He was also responsible for the introduction of much of modern mathematical terminology and notation.

He spent a lot of his career in the Russian court of Peter the Great and Catherine the Great.  In fact, his biography is littered with names of famous people, especially mathematicians, philosophers and other literati, such as Bernoulli, Laplace, Mersenne, Voltaire, ….  My favourite story about him was the first one I ever came across and involves an encounter with the French philosopher Diderot, who was visiting St Petersburg and had been invited to the court of Catherine the Great.

Diderot was a self-proclaimed rationalist and atheist, whereas Euler was a devout Christian and believed in the efficacy of the Bible.  The Russian court, of course, was Orthodox and the Empress was alarmed that the philosopher’s arguments for atheism were influencing members of her court.  Euler was asked to confront the Frenchman and as a result Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced, “Sir,

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hence God exists—reply!“, and then sat down. According to the story, Diderot stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted.

I came across an account of the incident in a book on the history of mathematics.  The author was quoting it as an example of how important it was to have an enquiring mind which asked questions.  He pointed out that the notation, with which we are now only too familiar, was at that time new and unfamiliar.  Diderot, he said, was flummoxed by the new language and instead of asking for an explanation, allowed himself to be confounded.

As I read the story for the first time, I remembered a similar situation that had confronted me when I had just sat my O-levels, (the 16+ school leaving examination), and saw an A-level Maths paper, (the 18+, University entrance examination), for the first time.   It didn’t look at all like a Maths paper that I was used to.  There were all sorts of symbols that I had never encountered before, terms that meant nothing to me.  Like Diderot, I felt a sense of shock and disorientation.

Fortunately, I wasn’t under the same pressures of time as Diderot.  He had to respond immediately, and in the face of this incomprehensible statement he is alleged to have frozen and panicked. I, however, realized that I wasn’t supposed to know this yet … it would be two more years before I had to answer questions like these, and in the meantime I would learn the topics and, hopefully, be prepared to answer them.

If only Diderot had asked for an explanation, then Euler would have had to say something like:

“Take three quantities and call them a, b and n. multiply the second quantity by itself n  times, add the first quantity, a, and divide the result by the third quantity, n.  Call the result x.”

Diderot would have then been justified in asking what this had to do with God, and why Euler considered that it constituted a proof.

In the face of such questioning, it would have been Euler’s turn to be embarrassed, flummoxed and confounded … although, I daresay that he would have had a suitable strategy planned for meeting with such a challenge.

Unfortunately, this story is almost certainly untrue.  Diderot was by no means a mathematical ignoramus and had published mathematical treatises of his own.  It does, however, make for a good story, and one with a message: When confronted with something you don’t understand … Ask for an explanation!  You might be surprised by the result and discover that things are not what they seem to be.

It’s a lesson I have always tried to follow.

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The quotation, “Read Euler, read Euler, he is our master in all things,” comes from another great Mathematician, Laplace.

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“To bodly go … ” is probably the best known example of a split infinitive that there has ever been.  English teachers around the world have probably been cringing over it ever since, back in 1966, our television screens went black as a disembodied voice announced, “Space … the final frontier … ” and launched the first episode in the adventures of the crew of the Starship Enterprise.

Although it is now seen as a cult, with several films, four spin-off series, animated series, books, comics, computer games and other associated merchandising … it almost never happened.  The original pilot, (true, with a different cast), was rejected by Paramount.  I bet they were glad that they changed their minds … however, although the opening credits stated that the crew had a “five year mission … to explore strange new worlds, to seek out new civilizations, to boldly go where no man has gone before”, the series was cancelled after just three seasons.

They must have realized, however, that they may have been premature and re-runs of the series on numerous stations helped to create the cult following which led to the spawning of those numerous spin-offs.

When I was teaching, the successor series, “Star Trek, the Next Generation”, was being shown on satellite television.  Unfortunately it was screened at five o’clock, which was far too early for me to get home from school in order to watch.  However, it was also shown again at ten o’clock, and I usually got see each episode in this later time slot.

It’s funny, but it became a bit of an obsession, even though I felt that it wasn’t as good as the original series.  The plots and the acting were just as outrageous, the science and special effects were better, (even if the script simply called it “technobabble”), but there was something groundbreaking about the original series.

I have similar feelings towards the Star Wars series – and even towards Dr. Who.

The Next Generation was followed by Deep Space Nine.  I didn’t think that this series stood much of a chance of success, based as it was on a Space Station, orbiting a planet, rather than scouring the myriad of star systems in the galaxy.  To be fair, however, the writers were imaginative and employed a number of techniques to introduce twists and turns in the plot to keep the viewer’s attention.

One of these involved the discovery that near to the space station was a stable “wormhole”.

That’s got nothing to do with little creatures that burrow through the soil, and magically regenerate themselves if you accidentally cut them in half.  A wormhole is a topological concept, (a branch of Mathematics), which arises out of Einstein’s theory of Relativity.  Now, no matter how many times I read up on the subject, I never seem to be able to get my head round more than the main “headlines” of the subject.  A wormhole is basically a compact region of space whose interior has special characteristics which enable it to act as a sort of shortcut through the curved multi-dimensional space-time of the cosmos, rather like a bridge or tunnel, with the two ends in different places allows the traveler to pass over an obstacle without having to travel much further in order to circumnavigate it.

It’s a bit the magical wardrobe in C.S. Lewis’ “The Lion, the Witch and the Wardrobe” which acted as a portal between Earth and the land of Narnia.  What’s more, just like in the Narnia Chronicles, it could warp time itself so many months could be spent in Narnia whilst just moments just passed on the other end of the portal.

If all this sounds fanciful, and there is no actual proof that such a phenomenon exists, then there are mathematical equations which back up the idea, which I won’t even pretend to understand.

I mention this because over the weekend I came across a report on the UPI newswire about a team of scientists working in Bishkek who have been working on some of these equations.   Led by physicist Vladimir Folomeev, their results suggest that pairs of stars could be joined by such wormholes full of an exotic material known as “phantom matter.”  The existence of this “phantom matter” is purely hypothetical, it just happens to make some of the equations work providing a possible explanation for the accelerating expansion of the universe.  As a corollary, it seems that, (if it does actually exist), then its properties could also enable a wormhole to be kept open.

As with the whole subject of wormholes, the idea is not without controversy and sparks debate.  Even Folomeev admits that the team’s analysis is not complete and there is apparently a lot of calculations and other work to do in order to to either find support or to disprove the speculation that they have made.