CLOSER

Pooh was walking round and round in a circle, thinking of
something else, and when Piglet called to him he just went on walking.
"Hallo!" said Piglet, "what are you doing?"
"Hunting," said Pooh.
"Hunting what?"
"Tracking something." said Winnie-the-Pooh mysteriously.
"Tracking what?" asked Piglet, coming closer.
"That just what I ask myself. I ask myself, What?"
"What do you think you'll answer?"
"I shall have to wait until I catch up with it," said Pooh.

    I had discovered why so many youngsters fail in learning mathematics. Being realistic: this is not the money shot. Important to parents and teachers who worry when their children or pupils fail, the truth is: not many care. It is so normal to be a maths dunce that many people will just declare it cheerfully: sometimes adding, as if to emphasize its complete social unimportance: 'that's why I have an accountant!'
    Even the opposite is true. Being good at maths so alarms some people, especially most women, that it is perfectly fatal to one's chances to confess it. How many times have I seen the light of interest die as soon as I have uttered: I teach maths. Whatever do they imagine this means: that for foreplay we substitute Fourier functions?
    The majority of children fail because they are taught to fail. Fortunately, they rarely all fail at once. Perhaps a little more notice might be taken if they did, although I wouldn't bet too much on that either. Fortunately or unfortunately what children generally do is fail by fractions: beginning with small fractions, getting bigger, every year, every year - until finally: ahah! here we are in their final years, and usually only a very small fraction now remain, and the majority will never want to do maths again!
    To have discovered why this happens must be of some importance. A few specialists might applaud - but it still did not seem of any great importance. It was not cosmic. Yet here was still this sensation of someone breathing on the back of my neck, and there was still the pricking of my thumbs.
    If there was more to it that I had understood - this could only mean that it would involve not just mathematics, but the people who make mathematics: the mathematicians. But what made this so very hard to believe - from all that I knew about them - is that professional mathematicians are without question the most unlikely people to know anything of general importance to the rest of the world - or, indeed, to care if they do. Most are content to exploit their talents quietly, to enjoy their salaries - even if these are rarely very grand - and never to interest themselves in anything outside their secure, secluded and very private world. Some will even boast that what makes them important is that their work has no relevance to ordinary lives.
    Godfrey Harold Hardy, for example, (1877-1947), a highly influential professor of pure mathematics, first at Oxford and then at Cambridge, estimated in his prime that he was one of the ten best pure mathematicians - at Cambridge he would be called a 'wrangler' - in the world. His own version of Plato's parable is famous:

    "I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations."

    He was wonderfully dismissive of his own importance to the rest of society - and, by implication, just as derisive of the importance of the rest of society to himself:

    "The 'real' mathematics of the 'real' mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless'(and this is as true of 'applied' as of 'pure' mathematics. It is not possible to justify the life of any genuine professional mathematician on the ground of the 'utility' of his work."

    Not much social conscience there. But we might be grateful that there is no meretricious moralizing either. Mathematicians are a very unusual breed. It is usually obvious that they are not working to improve the survival of the world. Hardy was only being honest. They do not think about morality because they believe they do not need to, although this can lead them into very dark corners indeed. To those who heard him, the anguish of Richard Feynman, one of the greatest mathematical physicists of the 20th century, on being asked to explain - perhaps to justify - his own vital contribution to the development of the atomic bomb will be for ever memorable: "We just didn't think. Okay? WE JUST DIDN'T THINK!"
    Somewhere in the world, mathematicians are almost certainly still working on problems whose solution promises its even swifter termination. They will think only of the interest of their mathematics: just as Enrico Fermi, the Italian-American nuclear physicist who was another of Feynman's colleagues, saw his own contribution as just 'interesting physics'.
    Remarkable people are never necessarily astute moral scientists. This is not what they are selected for. They may become alarmed by their own science. They may become engaged - as Einstein, Bethe, Oppenheimer became engaged - and attempt to control it; or they may be as unconcerned as the grotesque Edward Teller who pressed on with the development of the real Doomsday Machine, the perfectly unnecessary vastly more powerful hydrogen bomb, whilst accusing Oppenheimer of treachery for opposing it.
    Mathematicians are not all morally blind. Those who are, are not blind alone. But knowing some of their histories made it very difficult for me to think any general moral connection possible.
    Some mathematicians appear to inherit their skill. Some families have maintained remarkable intellectual eminence over generations. Money seems to help here: a lot. The Fugger family, for example, beginning as bankers in Augsburg in the early 15th century, became princes of the Holy Roman Empire under the Emperor Charles V who defeated Süleymann the Magnificent at the siege of Vienna in 1552. There is still a Fugger bank - more or less a showpiece - in Augsburg, in Southern Germany today. The Rothschilds are even better known. The five sons of Meyer Rothschild (1743-1812), all were made barons of the Austrian empire in 1822 - and their descendants continue their success today. It could be claimed that no-one can make and keep vast amounts of money without some purely practical genius in mathematics.
    Amongst the pure mathematical families, the most famous must surely be the Bernoullis - originally from Antwerp, settled in Basel. Over three generations between 1654 to 1863 they produced twelve university professors of mathematics or physics. Although they could be use to explain why most of us never learnt to understand long division - because our parents never did - the theory cannot hold. The majority of great mathematicians are nearly all solitary. Only rarely is their genius carried on.
    Leonhard Euler, for example, born in Basel (1707-1783), studied with the young Bernouillis and became so prolific in mathematics - he could calculate, it was said: "as men breathe, as eagles sustain themselves in the air" - to average 800 pages of mathematics every year, whilst also becoming the father to thirteen children. But one of them appears to have continued his genius. The compilation of Euler's lifetime of work still continues. It has so far filled seventy-five volumes and has seen three distinguished editors into their graves.
    Mathematics is indeed a vast world. Most of us hardly cross its borders.
    Born to another unremarkable family, Isaac Newton was another solitary. He is famous for explaining the gravity and light, but in 1696 he solved a problem offered by Jean Bernouilli as a challenge to all the mathematicians of Europe. In 1716 he did the same to one proposed by Leibniz (1646-1716), the German mathematician whose discoveries in calculus equalled his own. Newton never married.
    Whereas he is often called the greatest scientist, Karl Friedrich Gauss (1777-1855) is just as often called the greatest pure mathematicians. Karl was the son of a Brunswick canal bank attendant who naturally wanted his son to tend canal banks too: a valuable task. He was saved from this first by an attentive schoolmaster, later by the patronage of his Duke of Brunswick. He also had no heirs.
    Still more exceptional was the Hindu Srinivasa Iyengar Ramanuja Iyengar (1887-1920), usually known as Ramanujan. Born of a poor Brahmin family, he was rescued by Hardy (who called this 'the only romantic incident in my life' ), who brought by him to Trinity and arranged a stipend for him. Ramanujan baffled even Hardy. He was virtually self-taught and he told Hardy that his discoveries were given to him in his dreams. To his Hindu friends he explained that he found them written on his tongue every morning by the goddess Namakaal. Whatever one may make of this - many of his discoveries would cover several tongues - he also told them: "An equation has no meaning for me unless it expresses a thought of God."
    Hardy, who was himself an atheist, remained strangely convinced that "religion, except in a strictly material sense, played no important part in his life'. One can only wonders what he meant by 'except in a strictly material sense'.
    Before these men lived, mathematics might have been fairly described as the study of number, magnitudes, and ratios. In their lifetimes this ceased to be adequate at all. Through the constant collaboration of their minds, their almost continual exchange of their discoveries, proofs and ideas, mathematics became capable of explaining the behaviour of the planets, the tides, ships, artillery, machines - but also of describing entirely abstract mental structures, structures seemingly of pure thought possessing, for mathematicians, an unearthly beauty, exquisite elegance, total perfection, self-sufficiency. And existence.
    As Hardy explained in his Apology - which is no apology at all in the ordinary sense, for he persists in quite indecently thumbing his nose at all of the barbarians outside his college gates - these structures appear to be more solid, more enduring, and even more necessary, than anything experienced directly by our senses. Theirs is the realm of supersensory reality: of the perfectly enduring unchanging Ideas that Plato described, the level of existence that is the source of our confused experience of all imperfect, continually changing, decaying Things making up the busy, buzzing, blooming confusion: all that we believe is reality. Why the first should give rise to the second - rather than the other way around - is of course only an aesthetic prejudice. It is not logically necessary at all. It has its parallel in the belief that Heaven must be a nicer place to be.
    I am not a mathematician. Even if I wished to be one, I know I have not a hundredth part of the talent needed. But that is not the real obstacle. The real obstacle is that I have no interest in their work. What modern mathematicians do, wrote a modern expert historian is to "carry on a highly sophisticated intellectual activity which is not easily defined"
    An intellectual activity which is not easily defined is not very likely to be of very great general value to the world. That was certainly Hardy's claim, and we should accept it. My own interest was always to try to understand something far easier to define, and yet - and now I will not modestly claim - much more difficult to do. The evidence for this is right there in Feynman's anguish and in Teller's insanity. Whilst such wonderfully clever minds as theirs were exerting themselves to the utmost to make possible the killing of hundreds of millions, no-one could think of a better way to make peace: except to offer the penalty of mutually assured annihilation. MAD is not very smart.
    The qualities that mathematicians have must include a remarkable degree of tenacity, of stubbornness, self-confidence, autonomy, determination, courage - all of which may be certainly rendered, rather less kindly, as obsessiveness and arrogance.
    More: they need refinement in the faculty for noticing the significance of details - sometimes absurdly little details - that others may have missed, sometimes repeatedly, or which others may never have seen. And, finally, there must be a remarkable interest in what is not known - and with this, an absolutely fearlessness in looking for it.
    One begins to see some connections between this Something and that Bigger Problem.
    Many years ago I asked a young friend of mine - he was then the only really active mathematician whom I knew: "Well, how do you discover something new?"
    He replied, and here I have to paraphrase, for it is really only the end which I remember for sure: 'It's rather like being in a completely dark room in a house with many rooms. Most of these room - just like this one - you already think you know pretty well, and you think you know all their contents too. Then you find that in this room there is something which you have not noticed before. It is like a piece of furniture that you bump into. Apparently substantial - apparently complete - it resists being moved or changed in certain unusual ways. And as you explore it further, you begin to realize that it is actually new; that no-one has described it before. You may the first to have discovered it: first to learn of all its shape, as it were; then what kind of object it is - and even its nature: what it does.'
    "So," I commented, and perhaps I was too excited. He was a most amiable young man but it was usually very difficult to draw anything out of him: "You are discovering new mathematics: and not inventing it!"
    He flushed, just as if I had just found him possessing some faintly discreditable idea, and his retort was unusually brusque. "Well," he replied: "That's what it feels like!"
    Now, I had thought - if I had thought at all - that he would describe a much more systematic, dispassionate, even rather mechanistic, process: just the way, in fact, that we make and expect machines to work. Only many years later did I realize how very interesting it was that he insisted that the feeling is so important. I know now that a pretty good working definition of the soul is that it is just this function of mind which recognize truth - not necessarily infallibly, but always as a very distinctive feeling; and only after this feeling do we say: Ahah!
    But now we can finally return to my young Robert, still smouldering with resentment - and to the rest of his class - all unaware we have just solved a puzzle of at least two millennia.
    "What is Truth?" asked Pilatus. A far better question is: 'What is knowledge?' Feynman was once asked to lecture on a particular kind of problem, which he had helped to solve. After he looking at it again, he had to ask for more time: because he said, although it was true that he had indeed been instrumental in achieving its solution, he had discovered that he didn't understand it well enough to explain it.
    Ahah!
For years I had been sorely perplexed. No matter how much I varied my teaching to suit every different class, the same proportion failed every year. The reason had suddenly emerged. A lesser being than Robert - a lesser mind - would have gone away and dutifully scribbled nonsense to satisfy my demand. And I would have learnt nothing from this at all.
    Knowledge is not what you can satisfactorily do. Knowledge is what you can satisfactorily explain to another.
    My pupils were failing because I was teaching them as if they were all little machines. I was programming them to act instead of helping them to have any comprehension at all why their actions should succeed. Without this vital understanding, the majority were as doomed as little Elle's brightly caparisoned snail creeping towards the road. Most must fail eventually, for all that most were doing was to commit long sequences of actions to memory without any correspondence whatever with anything. They had no understanding of their necessity, nor of their alternatives; and least of all of their meaning. I never explained any of this. They never asked. I was not imparting knowledge.
    Sometimes a few would begin naturally to detect some reasons - some purpose - in what they were doing. Finding this exciting or intriguing, they might keep this to themselves or they might try to share with their friends. In either case they would continue to succeed in mathematics - and enjoy it. They form a permanent upper social class. They may come from an upper social class. They may return to it.
    For the rest, however, the trap is already beginning to close. Everything depended for them on their memory. In this case, do that; in that case, do this - and so on. If you listen unnoticed to children showing each other what they understand, this is what you will hear. Usually what they are sharing are instructions of how to do things. And if you - as the adult - then ask them: 'But why did you do that: and not this?' They will falter: "But that's always what we're supposed to do!" they will usually reply. And then, alarmed that this may still be wrong, they will ask: "Isn't it?"
    They know no reasons. Being asked for reasons frightens them.
    The next shock to men - no less surprising, more painful - came within the year. I was leaving a meeting with a primary colleague. She was a great friend of mine, we had known each other for years. The meeting had been to discuss the future of a boy she had taught just a few years before. His future now was bleak. Once bright, eager, confident and trusting, he had turned sullen, dishonest, disruptive. Once popular and trusted, he was now shunned and distrusted. He was about to crash and burn.
    My dear friend had been silent for some time. Suddenly she stopped. I stopped too. "Just what is it" she asked me angrily, "that you do to our children in your school?" - and then she left me standing.
    Good children - in the main - come to many, many schools just like mine. They are all aged around eleven or so, and are eager to learn. By the age of twelve or more - like rats in a maze: and I mean no disrespect: this is just what our kind of 'teaching' creates - they are having to memorize literally hundreds of responses to stimuli. Neither the stimuli nor their responses have any particular meaning to them. They only know that pressing this lever like this produces rewards; or following routes like this from that and then to this avoids a shock. Children have fantastic memories. If they are told explanations, they can even remember the explanations too, perfectly; and repeat them, without a fault - but all the time without actually knowing what the explanation means!
    Why? Because nobody asks them.
    Eventually the memories of these good eager children are just not sufficient to contain - and manage: which is the harder part - this enormous volume of disconnected facts. The levers they are required to push and pull now demand unknown, untested sequences. The maze they are still trying to follow has now too many turnings. The shocks for failing are too frequent and too painful. There are no more easy rewards. They must now choose between two different stratagems.
    They can rebel. A minority always do. Or they can begin to conceal that they really do not understand what they are doing. They must become dishonest. The open rebellion of the minority brings some satisfaction to them all as almost everyone enjoys seeing that their teachers can also be humiliated or frightened. But for the first time there now appears the division between 'them' and 'us'. Democracy begins to die right here.
    The 'them' are the authority figures. The teachers are now understood to be both representatives and the servants of remote, unreachable, unchallengeable power. They set the endless tests of obedience, they monitor the degrees of their pupil's ability, but also their subservience. The 'us' are the pupils, for the majority must now choose - no: not choose, essentially there is no choice - who must now show that they have the ability to do what is required. They may still not understand why. This ceases to matter.
    The rebels - now a permanent social and intellectual underclass - must first be marginalized; then, if possible, dumped elsewhere. If neither of these is possible, they may soon terrorize an entire school. For the majority, the second strategy is more rewarding than revolt. It is also their only alternative. They must continue to hide that they do not understand. This will soon become automatic. It is then no longer even thought of as dishonesty. It is self-defence. Like karate.
    Other pupils will actively assist them; and actively support each other in frustrating any teacher's too intrusive inquiries. Sometimes, to their surprise they may find that some of their teachers actually want to assist their defence. It is no longer in the interest of anyone to admit the whole nasty truth: that Honesty, the Best Policy is dead.
    Massive failure has be avoided. Some failure - as 'natural failure' - is permitted every year. But other moral changes are taking place as well as these social divisions. As well dishonesty, the larger group begin to find that selfishness is an asset too: for this helps to make clearer the inferiority of the weakest in their class, the 'thickies' and the 'dummies' whom they will now neither help, nor treat kindly, nor any longer allow into their circle. These last poor souls, increasingly confused, now also isolated, are usually the ones of whom their teachers say: 'they did not try hard enough.
    I think I had just used this formula myself. I was ashamed. All of this had been under my nose for over ten years. I had not tried hard enough.
    To explain more precisely what it is that we do to children that destroys so many - and also to avoid to do it - I spent the next half of the year writing a book. It was called Speak Maths Better - not a very clever title, but it said what I wanted it to say - inside - in several thousand more words. Knowledge is not what you can do: it is what you can explain. If children are to learn anything better, they must be given opportunities to talk. It is basically as simple as that.
    When I had finished my first twenty drafts of my book, I took a copy to a friend, Philip Stewart, an Oxford don and polymath (and not all are both). We were sitting in his study and he had read only a part of my introduction when he stabbed at a line with a finger.
    "No-one has written that before" he told me. It was my turn to be surprised. I looked where he was pointing. It was a line in which I had written that "the reason why the early Greeks developed mathematical argument and democracy at the same time, is because they support each other: both use the same kind of argument."
    "But," I protested: "everyone must know that is true."
    Philip shook his head at me impatiently. "No." And then he poked the same bony finger in my direction. "You know it because you teach it: But -" here now was the scholar speaking: "no-one has ever written it before!"
    I would not like to confess that I gawped. It might similarly be thought an exaggeration - by the Witch especially, who is always sharply critical of this habit of mine - if I report the feeling of vast massive blocks, of enormous ponderous weights, suddenly sliding smoothly into place in my mind: and into the place they were always meant to be. But that is - unfortunately - precisely the sensation that I had. The Witch will never believe it.
    I don't remember thanking Philip. I do remember it was a lovely, cold autumn day, and that the leaves were turning to gold on the trees along the Banbury Road. I remember cycling home to the little flat I was about to buy from Mags, repeating to myself, over and over again. "They don't know! They don't know! Everywhere in the world they're teaching mathematics: and they don't know what it is they are doing!"
    The point is, you see, that whilst mathematicians are just as morally imperfect as the rest of us, in the science that they create every one of their proofs has to persuade others that it is true: with no compulsion whatsoever to help it.
    The mathematicians maybe absurdly elitist, like Hardy; joyfully demotic - like Feynman, with his bongo-drums; or they may be sociopaths like Teller.
    None of this matters.
    What matters is that to understand their science properly, to explain their science properly, one has to explain why it is democratic: it is so that anyone can take part in the argument: it is because this is the best way for any society to involve all its people: it is because this is how knowledge can be most generously shared: it is because essentially it is an expression of trust in others: it is because to trust is also to love.

11/05/05

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