New complex systems of maths

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There are different approaches to the subject of mathematics, and it is possible to posit a number of wholly consistent alternative systems which produce some results which everyday life might consider to be more than a little strange. Back in my sixth form days, I was somewhat taken by the hyperbolic form of non-Euclidean geometry, in which the sum of the angles in a triangle varies according to the size of the triangle – with the extremes ranging from zero for an infinitely large triangle to 180 degrees for an infinitely small one. It’s of limited practical use in daily life (where we are, in cosmological terms, close to being infinitely small), but none the less fascinating for that.

It’s not uncommon for me to be more than a little harsh on the extent to which Sunak and his not-so-merry men are mathematically challenged, as a result of watching them struggle with some very basic arithmetic. Perhaps I’m being unfair; perhaps they’ve just developed a hitherto unknown version of mathematics, in which things which make little sense to most of us are actually part of an internally consistent system of logic which the rest of us are just too stupid to comprehend. This was reinforced today by the sight of some Tory spokesmen apparently trying to tell the world that yesterday’s round of elections was, in fact, a huge success for the party. The fringe elements have gone further: the falling number of votes ‘proves’ that the public at large has an appetite for even more extreme policies. It’s a strange equation they’ve developed in which people who refuse to vote for increasingly extreme policies can somehow be brought back into the fold by adopting even more extreme policies. Absolute success is thus equivalent to an absolute lack of votes; only when no-one at all votes for them will they feel fully vindicated. They might even have once possessed a mathematical proof of their theorem, which they’ve emulated Fermat by losing after scribbling something in the margin.

It's an interesting conjecture although, like non-Euclidean geometry, it’s of rather limited practical use to anyone else. On the other hand, my old favourite, Occam, might suggest that the simpler solution might be a more appropriate conclusion to draw. They really are just not very good at maths.

Proof of the theorem?

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In relaunching his mathematics strategy yesterday, having done little or nothing about implementing it since the last time he launched it (anyone else see a pattern emerging there?), the Prime Minister claimed that the UK economy is losing “tens of billions” of pounds a year due to functional innumeracy. The first question which sprung into my mind was that often voiced by my old maths teacher, “show me your workings”. That thought was closely followed by memories of the same teacher's strictures about applying the ‘reasonableness test’ – if an answer looks or feels wrong, it probably is.

If we take “tens of billions” to mean £20 billion – the lowest possible mathematical interpretation of the number – and compare it to the UK’s total GDP of £3.1 trillion, it implies that the UK is losing just over 0.6% of GDP every year, purely (according to Sunak) as a result of the fact that children are only taught mathematics until they are 16 rather than 18. According to the government’s own figures, that’s equivalent to between 7 and 8 trade deals with Australia. In ten years’ time. ‘Show me your workings’ is an apt response.

I don’t doubt that there is an economic cost to innumeracy (although a substantial part of that cost may well be the result of having a surfeit of innumerate politicians who pluck arbitrary figures out of the air), but attributing that to the fact that most pupils are not taught any maths after the age of 16 looks like a non-sequitur to me. As someone with three maths A levels to my name (I’ve always liked numbers and what they can tell me), I can honestly say that I don’t think that much of what I learned in the field after reaching 16 has been of much practical use to me in life, no matter how interesting I found it at the time. It seems far more likely to me that the problem of functional innumeracy lies not in the age to which maths is taught in schools, but in the extent to which what is learnt matches what is being taught. That is to say, at its simplest, that maths teaching up to the age of 16 is far from universally effective, and that putting that right would be a better use of any additional resources which can be made available. That isn’t a criticism of teachers as such – we know that there is a huge shortage of specialist teachers of the subject, and has been for a long time. It’s just that assuming that lessons delivered = lessons learned, especially when they are delivered by non-specialist teachers, is an invalid premise. And if they can’t deliver the resources and the processes to get the basics right, the chances of a successful outcome to Sunak’s latest half-baked proposal are close to zero, a statement which can be confidently made even without having enough data to calculate a precise probability.

There is an old adage which claims that 97.8% of all politicians’ statistics are made up on the spot (although estimates of the precise percentage vary), and in this case Sunak’s numbers appear to provide proof of that theorem. It's not exactly a first for him.

Adding up the numbers

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Answering a question from the BBC’s Chris Mason this morning on nurses’ pay, the Prime Minister was at pains to point out that, allowing for turnout, only a minority of the whole RCN membership voted to reject the deal. He’s right, of course, and it proves that he can at least do some simple maths (maths being his chosen subject of the day). He is clearly implying that a majority of 54% to 46% doesn’t really count, because it’s not a majority of the whole electorate, and in any event the RCN had advised members to vote for the deal. I seem to remember another vote in the fairly recent past where, allowing for turnout, only a minority voted for the outcome and those running the vote (the government of the day) advised people to vote the other way. In that particular case, I could have sworn that Sunak was amongst those arguing that it was an absolutely definitive vote which must be respected. It’s probably just a new branch of applied mathematics (Sunakian rather than Euclidean) under which the validity of any arithmetical result depends on the ideological perspective of the observer. It fits the post-truth world in which we seem to be living, but I’m not sure that learning this new type of mathematics, even up to 18 years of age, will actually serve to make the UK population more numerate. But then, that isn’t really what he wants. A truly numerate population would be challenging the government on a whole range of issues where their numbers simply don't add up.

The real problem of innumeracy

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Despite deploying his very best efforts, even Rishi Sunak can’t be completely wrong about everything all the time, and this week he managed to get one thing at least half right in identifying that innumeracy among school leavers is a problem. His ‘solution’ (compulsory maths until age 18 for all English pupils), however, tells us rather more about his own tenuous connection with the meaning of numbers than it does about addressing the problem. The curriculum has probably changed since my school days, when taking double maths at A level was a soft option for someone like me who was lucky enough to have a fairly natural grasp of numbers and numerical operations: learning a few principles and formulae and knowing how to apply them was much easier than having to cram a whole load of facts into my head for examination purposes. A level maths was not the same thing as O level maths, however – I remember one maths teacher describing the former as ‘hard sums’ whilst the latter was ‘easy sums’.

It's not entirely clear what Sunak has in mind. I can’t believe that he’s stupid enough to believe it a good idea to put people who haven’t got to grips with ‘easy sums’ through the trauma of trying to learn how to do ‘hard sums’; and Calculus with its differentiation and integration doesn’t immediately strike me as being relevant to the future lives and careers of most pupils. But the alternative is that he’s planning the even stupider approach of forcing those who have failed to master ‘easy sums’ after five years of secondary education through another two years of the same. The logic of adding an extra 40% of input to a process which is clearly failing many pupils is ‘interesting’ to say the least, and demonstrates a lack of understanding of what it is that the low levels of numeracy are actually telling us - as well as a lack of the 'analytical ability' which he claims to be so important.

It's far from being the only example. Yesterday, he and other ministers were keen to tell us that the NHS has all the money it needs to solve its capacity problems. There’s some doubt as to whether that’s actually correct even in purely numerical terms, but let’s give him the benefit of the doubt for a moment at least. If the NHS lacks 10,000 doctors, and employing a doctor costs £80,000 a year, then chucking £800 million into NHS coffers does indeed ‘solve’ the problem. It is, though, rather divorced from the real life question of what that 10,000 represents, and assumes (in the manner of basic economic theory) that there are 10,000 doctors sitting on a shelf somewhere, just waiting for someone to offer them a job. In reality, there is a lengthy lead time involved in recruiting doctors with the right specialisations in the right locations – and an even longer lead time if they need to be trained first. That involves understanding that the 10,000 isn’t just a factor in an equation, it’s a symbolic representation of a serious problem. Such a simplistic mathematical response to the needs of the NHS is made worse when another department of the same government is performing its own mathematical computations to conclude that recruiting trained doctors overseas is ‘cheaper’ than training our own, and yet another department performs its own sums and attempts to reduce the number of overseas migrants.

It's an approach for which Sunak’s ‘financial’ background did much to prepare him. Bankers ‘know’ that if you can reduce the cost of production (X) and increase the productivity rate of staff (Y), the amount of profit flowing into their accounts (Z) will increase. But Z is the only factor which they have any ability to relate to the real world. The impact of job cuts for some and longer hours for others on people, lives and wellbeing – which is what X and Y actually represent – is beyond their comprehension. It’s all just mathematics. That really goes to the heart of Sunak’s category error in talking about levels of innumeracy among school leavers. The bigger problem is not that many do not have the ability to handle numbers well, it’s that some of those who can handle numbers well have lost all conception of what those numbers represent. And by some strange process, these are the people who end up in charge of the government and the economy.

Understanding numbers

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Following the opening salvos in the Great Sausage War of 2021, as it will surely come to be known, the PM told us yesterday, in relation to his belief that it should be possible to avoid barriers to trade either in the Irish Sea or on the island of Ireland, that he’s “very very optimistic about this” and that he thinks that it is “easily doable”. He’s right, of course. Conceptually, it’s extremely simple, just like crossbreeding a narwhal with a horse to produce a unicorn is conceptually very simple. The devil is in the detail of how to make it happen in practice. In fairness, he is at least being unusually consistent: what he’s calling for is what he’s always believed that the UK is entitled to – completely free trade with the EU whilst not following any of the EU’s rules. The problem, from the outset, has been that Europeans simply don’t understand how special and unique the UK is, and are making things unnecessarily difficult.

Achieving what he says he wants simply requires one of two things to happen. Either the UK agrees to align its controls on food quality with those in operation at the EU border, or the EU abandons its regulations and allows sausages to flow freely from anywhere in the world. Of the two, it is entirely obvious to everyone (well, everyone of any importance in the eyes of English exceptionalists) why it is the EU which should surrender. Not only are British Sausages inherently superior to all others (you can tell by the union flag on the packet), but more importantly the UK market of 66 million is bigger than the EU market of a mere 450 million, giving the UK the upper hand at all times.

Purist mathematicians may quibble slightly at that, but they need to get with the programme and understand the NewMath which is now the dominant strain within the government party. It’s far from the only example, and nor is it anything new. We were also told yesterday by a Welsh Conservative that 16 is bigger than 43, and the current Welsh Secretary is on record as believing that 1 is bigger than 7. And in relation to a comparison of votes received, they believe that winning 365 seats out of 650 in 2019 gives them an absolute right to rule, whereas the 71 seats out of 129 won by pro-Indy parties in Scotland in 2020 shows that independence has been resoundingly rejected.

One of the problems with NewMath is that, faced with two numbers, none of us can ever be certain which is the larger, or how to interpret them, until the Tories have explained it to us. Then the truth becomes obvious and inarguable. As obvious as the fact that there was never a time when the UK did not exist and that we have always been at war with the rest of Europe over sausages. What Orwell thought was a dystopian novel has become an instruction manual.

Tweedledum and Tweedledee

Tweet My Tory opponent tries to argue again this week in the letters column of the 'Carmarthen Journal' that the choice in this constituency is between Labour and Tory. I suppose I should at least acknowledge that he's no longer trying to base his claim on the mathematical absurdity that 1 is larger than 7.

Unsurprisingly, I still take issue with his claim.

Firstly, although I am not a great believer that either by-elections or mid-term opinion polls are accurate predictors of general election results, they can and do indicate a trend, particularly if they continue over a lengthy period. Volatility – or even collapse - in Labour support changes the nature of the coming campaign. And the Assembly result (the only other election fought on current boundaries) showed that this constituency can be seen as a three-way marginal.

Secondly, I really do challenge his assertion that Labour-Tory is any real sort of choice. Tories close rural schools; Labour close rural schools. Tories close 3,500 post offices; Labour add another 2,500. Labour launch illegal wars; Tories support them. Doesn't look like much of a choice to me. Given the Conservative reluctance to spell out what their policies are (on anything!) – and their refusal to commit to reversing any Labour policies – what is their basis for arguing that they will do anything different?

Finally, his suggestion that Plaid have only announced last week that we are in favour of Independence for Wales shows a lack of knowledge of Welsh politics which is truly staggering; although sadly not untypical of the English Conservative Party.

I doubt that anyone will be surprised if I say that my aim is to show people that there is a real alternative to either of the London-centric parties – and that is for us to take our future into our own hands by supporting Plaid.