Libertarianism and the Leap of Faith – The Origins of a Political Cult

By Philip Pilkington, a journalist and writer living in Dublin, Ireland

You wanted God’s ideas about what was best for you to coincide with your ideas, but you also wanted him to be the almighty Creator of heaven and earth so that he could properly fulfil your wish. And yet, if he were to share your ideas, he would cease to be the almighty Father.

– Søren Kierkegaard

Political cults often have the strangest and most obscure origins. Take Marxism, for example. Today it is well-known that Marxist doctrine essentially sprang out of the obscure 19th century economic debates over the source of ‘value’. By ‘proving’ – that is, lifting the assumption from classical political economy – that all ‘value’ came from labour, Karl Marx went on to show that it was therefore only logical to assume the existence of something called ‘surplus value’ that was sucked out of labourers by a parasitic capitalist class. From out of this obscure debate flowed an awesome political movement – and a tyranny to match.

What is less well-known is that today’s most popular political cult – that is, libertarianism – was born in very similar circumstances; it too, arrived into the world out of the obscure 19th century debates over economic ‘value’. But before we explore this in any detail it might be appropriate to speculate a little on what characterises a political cult and why so many of these find their sustenance in economic theories of value.

What is a Political Cult and Why Do they Often Love Economic Value Theory?

A political cult is characterised by a political or economic doctrine that answers all the ‘big questions’ about life, the world and everything else. The doctrine that is handed down is then to be conceived of as a way to live one’s life – a project, handed down from Mount Sinai, that one is under the moral obligation to spread far and wide. This is why we refer to these movements as cults. And it is this that gives them such an awesome status in the glazed eyes of their devotees.

Under such circumstances, politics becomes a sort of religious calling. In these doctrines there is usually an ‘Evil Being’ who is opposing the spread of the ‘Good’ on earth and it is these that are to blame for all the bad things in the world. In Marxism this Evil Being is the capitalist; in libertarianism it is the figure who is at different times referred to as the ‘collectivist’, the ‘liberal’ or the ‘socialist’. Needless to say that, since these figures are usually ones of Extreme Evil they must be ‘liquidated’ or ‘eliminated’ at the first possible opportunity lest they spread their Demonic Gospel to the masses.

Political cults thus provide their devotees with a firm identity in an otherwise changeable and, let us be frank, confusing world. Like all cults they provide an anchor for their devotees with which they can fasten themselves to a rigid doctrine. They also typically lend their devotees a Holier-Than-Thou attitude as they provide them with ‘secrets’ that those outside of the cult cannot grasp. Not only does this allow the devotees to feel ‘special’, in modern political cults it also gives them practical, albeit ‘secret’ advice about what they should do in their day-to-day lives. (Think of the advice to buy gold or foreign stocks coming out of certain libertarian front men, for example).

Finally, the political cult will usually offer their followers the possibility of a Heaven on Earth. If the follower behaves well and spreads their beliefs to others they will eventually arrive at some sort of Utopia. This is their reward for believing in the doctrines, despite these doctrines being ridiculed by others.

So, why do these cults spring out of economic doctrines based on value? Well, this is a very complex question but there is one key aspect that is absolutely fundamental. In order to understand it a little better we must think for a moment about what economic ‘value’ supposedly is. It is, in fact, when we boil it right down, a moral entity. If we can tell what people ‘value’ and why, then we can make prognostications on what is Good for society as a whole.

In times past organised religions handed down fixed value systems to their adherents. Today people have become disillusioned with religious systems – ostensibly because they conflict with these peoples’ supposedly ‘scientific’ worldview. But the impulse among some for the self-assurance provided by a religion is so strong that they seek out ‘scientific’ systems that operate in an identical manner to religious or cult systems.

This is why the economic doctrine of ‘value’ is such a good foundational stone for such a cult. It provides a pseudo-scientific account about how people attribute value to things and in doing so tells the cult member a ‘Truth’ that they can use to make turn the world into a Utopia in which the optimal amount of ‘value’ is realised by the optimal amount of people.

Karl Marx claimed that ‘value’ was embodied labour and hence his followers concluded with him that all that was Good sprang from labour and that society should thus be based on free labour. The libertarians – together with the neoclassicals that they otherwise scorn – believe that all ‘value’ springs from utility maximisation. While the neoclassicals simply tinker with toy-models of ‘value’ to bolster their pseudo-scientific prestige, the libertarians undertake a leap of faith into the unknown and claim that in the theory of marginal utility they have found a ‘Truth’ that must be brought down from Heaven to Earth.

The Birthing of a Cult

Libertarianism was born out of the late 19th century doctrine of marginalism; a doctrine that went on to gain popularity with those opposed to Marxism. We will not dwell too much on the doctrine of marginalism when applied to the analysis of ‘value’ – having done so elsewhere. Here we will merely note that marginalism provides a moral defence for the supposedly ‘free market’ system that we live under today.

Marginalism, when applied to ‘value’ analysis, holds that it is in Man’s nature to follow a certain path in his consumption habits. These habits are determined by his maximising his utility. Most modern marginalists claim that they can use this concept to show that a ‘free market’ system is the fairest social system possible, since it responds automatically to Man’s marginal utility preference it delivers ‘value’ in a perfect and harmonious manner.

Markets deliver this ‘value’ through the mechanism of price. Prices, which reflect peoples’ desires to maximise their marginal utility, ensure that the most equitable distribution of ‘value-in-the-abstract’ is accommodated for by the ‘free market system’. And this is the point at which marginal value analysis becomes a value judgment in a very real sense.

The neoclassicals held, and continue to hold, that their models could capture this complex dynamic. Such an assertion was and is, of course, absolute rubbish. But the Austrians took a different tack. “Yes,” they said, “marginal utility theory is the correct way to go, but we cannot formulate models that adequately capture the inner workings of this great mechanism.”

In their book Modern Political Economics: Making Sense of the Post-2008 World, the authors provide a good summary of this approach. In the book they discuss what effect the discovery of marginalism’s inherent uselessness had on the Austrians:

Faced with the impossibility of mathematically deriving prices and quantities on the one hand and a metric of social welfare on the other, some Marginalists understood the limitations of their utility calculus. Mainly of an Austrian persuasion (most notably Ludwig von Mises, Friedrich von Hayek and Joseph Schumpeter), they even gallantly tried to use this failure to the advantage of their claims on behalf of untrammelled markets and against the encroachments of collective agencies, trade unions, governments etc.

This was a clever move. While the neoclassicals tinkered with their silly toy-models, trying to show how prices are determined through a sort of grand marginal calculus, the Austrians shrugged their shoulders as to how such a Divine Event could occur. Instead they began to think of price as a sort of Miracle that proved the divinity of the Market mechanism. They then went on deploy this argument to show that anything that encroached upon this Divine Being’s presence was inherently Evil:

If no degree of mathematical sophistication can pin down the ‘right’ prices and quantities, how can a government or any other form of collective agency work them out? How could a socialist economy, or even a national health service, ever price things? Thus, the market mechanism is indispensible because of the radical indeterminacy of prices.

Note what is happening here. The Austrians, like their marginalist brothers and sisters, thought that in marginal utility theory they had found the source from which ‘value’ truly flowed. They never for one moment questioned that. Even when they came to conclude that marginalist analysis could never definitively show anything useful about price determination, they remained confident – indeed, they became even more confident – that such an analysis was Truth.

In short, they postulated a theory and then when confronted with the inconsistencies of the theory when it was applied to any practical ventures they simply threw up their arms and claimed that such inconsistency showed just how true theory was and how much we should respect it. The knowledge that the theory imparted then became, in a very real sense, Divine, in that we meagre humans would never be able to grasp it and instead should simply bow down in front of the Great Being that possessed this knowledge – that is: the Market.

This is what gives the libertarians their religious zeal. In their quest for the Grand Truth they find this Truth to be inaccessible to Man. But in this inaccessibility they find a Higher Truth again; namely, that there is some other entity out there – a benevolent entity called ‘the Market’ – that possesses this Truth and all we have to do is follow the Laws which it has handed down to us and we will eventually reach Utopia. This is, of course, a leap of faith – a truly Kierkegaardian Leap of Faith.

From the Leap of Faith to the Knight of Faith

The Austrians were never quite content with the chicanery and political posturing that they had passed off as scientific debate. As alluded to above, their theories about market prices were forged in the debates with those who advocated a socilialistic planned economy. Being ideological to the core, the Austrians were, for a while at least, perfectly content with saying that while no economist could say anything worthwhile about price determination – and thus, any attempt at a socialist planned economy would be doomed to fail because there could be no perfectly informed coven of evil socialist economists who could administer it – they were still happy with the airy theory of market prices that they had just poked such a large hole in. Yes, they had undertaken a Leap of Faith by admitting that their logical constructions would never be whole but, as Kierkegaard well knew, every Leap of Faith needs a hero, a Knight of Faith – and the Austrians soon found theirs.

The Austrians had, although one suspects that they never fully realised this, essentially proved that their theories were inconsistent. There was always, lurking somewhere, that element that disturbed the calculation of prices in the market models.

Let us emphasise here that this element of disturbance was found, not in reality, but only in their models and in their minds. The fact is that the Austrians, even in out-stepping their neoclassical brethren, were still only exploring their own fantasies. This fact must always be kept in the front of one’s mind when considering their doctrines.

We highlight this because it was precisely at this point that the Austrians could have conceded that they were building castles in the sky – ideologically and emotionally motivated castles in the sky, no less – and that it might be time to grow up and give up on the whole sordid venture of trying to establish a ‘logical’ ‘economic’ basis for ‘value’ that would temper them with the moral certainty they needed to carry on their political crusade. But not so. Instead they found a Kierkegaardian Knight of Faith to fill the gap in their logic. And that Knight of Faith was the entrepreneur.

The Austrian economist Israel Kirzner put it as such in his fine paper ‘The Economic Calculation Debate: Lessons for Austrians’ (which is also an excellent historical overview of much of what we have here been discussing):

[T]he truth is that Hayek opened the door to an entirely new perspective on the “goodness” of economic policies and institutional arrangements. Instead of judging policies or institutional arrangements in terms of the resource-allocation pattern they are expected to produce (in comparison with the hypothetically optimal allocation pattern), we can now understand the possibility of judging them in terms of their ability to promote discovery.

And this ‘discovery’, of course, comes from the entrepreneur who was hereafter identified by the libertarian as the social hero who broke through all barriers in the pursuit of the creation of new ‘values’ – and by that, we mean economic ‘values’ – for the community as a whole. Kizner again:

For Austrians, prices emerge in an open-ended context in which entrepreneurs must grapple with true Knightian uncertainty. This context generates precisely the kind of choice that stimulates the competitive discovery process. In this context, the entrepreneur does not treat prices as parameters out of his control but, on the contrary, represents the very causal force that moves prices in coordinating directions.

In Kierkegaard’s writings which, like the writings of the Austrians sought to establish a theological metaphysic from which an individual could derive principles of moral certitude, it was the Knight of Faith – the true believer with complete faith and certainty in both himself and God – that filled in the logical gaps inherent in even the greatest philosophical systems. For the Austrians the entrepreneur filled the same role – except that this was a great hero that had both full faith in the Market and the ability to find opportunities to inject disequilibrium into the price system through innovation.

By now we are far outside the realm of anything even remotely resembling a science of ‘value’. What we have instead is a vast metaphysical and moral system that is built around a very specific – not to mention very narrow – conception of value, together with a sort of existential appendage in the form of the hero-entrepreneur. The hero veneers over the logical flaws in the metaphysical system, while that system remains in place as a faith-based explanatory schema which can be applied to the world around the libertarian.

Note how fantasy blends into reality almost completely at this point. No longer do we separate our supposedly ‘factual’ ideas about ‘value’ from the mythological figure of the entrepreneur. Fact and fantasy merge to form a sort of continuum the purpose of which is to insulate the devotee from any empirical evidence that might arise to prove them wrong – or, at least, misled – regarding, for example, more fundamental and more pressing macroeconomic questions. They simply know what is what because they have it all worked out – and no silly facts are going to tell them otherwise.

From the fertile source of marginal utility value calculus the Austrians thus constructed a pristine moral and metaphysical system. But in doing so – like all metaphysicians – they allowed their imaginations to run away with them. They never noticed the point at which they crossed that fateful line; that line that separates our attempts to represent the world accurately and dispassionately to ourselves from our attempts to create a fantasy world in which we can live. The Austrians had, at first, attempted to use their imaginations to explain the world around them and, in doing so, had fallen into a dream world of their own creation.

And so the foundations of the political cult we call libertarianism were firmly in place. It is an ingenious creation which even came to include what CG Jung and other mythologists might call a central ‘archetypal’ or mythic figure. Even more specifically, what the Austrians have done is insert into their narrative what the great American mythologist Joseph Campbell called the ‘monomyth’. The monomyth is a recurrent theme in mythologies from all over the world. It is essentially a ‘hero myth’ and, as Campbell argues, can be located in most major religious narratives (Christ, Buddha etc.). In this the Austrians provided the libertarian religion with their very own version of the monomyth.

That most libertarians are ignorant of the source of their beliefs – just as most of them are not very conversant with economic theory generally, their protestations to the contrary notwithstanding – only adds a sociological dimension to their cult. Their cult forms a hierarchy where those who are closer to the Grand Truth are supposed to know more than those who are less conversant. Those who are less conversant then scrutinise the Great Texts – which are largely taken to be Holy Writ – until they can advance up the priestly ranks.

The Malign Consequences of Political Cults

After experiencing what used to be called ‘Bolshevism’ we are well aware of the dangers of political cults if they should ever ascend to power. Indeed, we already had forewarnings of this danger in the cult of Reason that Robespierre erected in revolutionary France upon the intellectual architecture that Jean-Jacques Rousseau had constructed for him. All of these cults espouse liberty and freedom and end up creating regimes of pure tyranny. Why? Because in their violent desire to turn reality into a Utopia, they stamp all over reality as it fails to conform to the images in their minds.

Some have objected to fellow Naked Capitalism writer Andrew Dittmer’s ‘interview’ series as an attempt to misrepresent the libertarian movement by espousing the ideas of an extremist. This is unfair. The views of people like Hoppe may be fringe among libertarians – then again, they may not be – but the zealousness is the same across the whole movement.

Libertarians think that they have unearthed a Truth that no one else can grasp (because, of course, this Truth being so pure, anyone who could possibly grasp it must then by default recognise it as Truth). And they think that if they can get adequate social and political power to enforce this Truth we will all be better off for it. Hoppe’s vision of a totalitarian, corporatist future is thus realistic in that if libertarians were ever truly to get into power they would have to enact an immense violence upon the world to try to get it to conform to their vision of Utopia. In this, they are like every other political cult that has ever existed. And they are just as dangerous.

In fact, the libertarians are the direct heirs to the Marxist-Leninist throne. Even though their motives differ substantially, their Faith is based on very similar principles – which is not surprising given that both movements grew out of the same 19th century debate over economic value. In this regard it is useful to recall John Maynard Keynes’ characterisation of Marxism-Leninism:

[It] is the combination of two things which Europeans have kept for some centuries in different compartments of the soul – religion and business.

Keynes also highlighted an important point about how such cults become influenetial:

[They derive their] power not from the multitude but from a small minority of enthusiastic converts whose zeal and intolerance make each one equal in strength to a hundred indifferentists.

The goal may have changed, but the unswerving faith in pseudo-scientific – or, to be very precise, in the Austrians case, because they tend to eschew ‘scientificity’: pseudo-rational – economic doctrines has not. Let us just hope that such a cult does not deliver to us another era of primitive tyranny and medieval inquisition. It is our democracies that are at stake.

The mathematical equation that caused the banks to crash

 Ian Stewart in The Observer 21-02-12

It was the holy grail of investors. The Black-Scholes equation, brainchild of economists Fischer Black and Myron Scholes, provided a rational way to price a financial contract when it still had time to run. It was like buying or selling a bet on a horse, halfway through the race. It opened up a new world of ever more complex investments, blossoming into a gigantic global industry. But when the sub-prime mortgage market turned sour, the darling of the financial markets became the Black Hole equation, sucking money out of the universe in an unending stream.

Anyone who has followed the crisis will understand that the real economy of businesses and commodities is being upstaged by complicated financial instruments known as derivatives. These are not money or goods. They are investments in investments, bets about bets. Derivatives created a booming global economy, but they also led to turbulent markets, the credit crunch, the near collapse of the banking system and the economic slump. And it was the Black-Scholes equation that opened up the world of derivatives.

The equation itself wasn't the real problem. It was useful, it was precise, and its limitations were clearly stated. It provided an industry-standard method to assess the likely value of a financial derivative. So derivatives could be traded before they matured. The formula was fine if you used it sensibly and abandoned it when market conditions weren't appropriate. The trouble was its potential for abuse. It allowed derivatives to become commodities that could be traded in their own right. The financial sector called it the Midas Formula and saw it as a recipe for making everything turn to gold. But the markets forgot how the story of King Midas ended.

Black-Scholes underpinned massive economic growth. By 2007, the international financial system was trading derivatives valued at one quadrillion dollars per year. This is 10 times the total worth, adjusted for inflation, of all products made by the world's manufacturing industries over the last century. The downside was the invention of ever-more complex financial instruments whose value and risk were increasingly opaque. So companies hired mathematically talented analysts to develop similar formulas, telling them how much those new instruments were worth and how risky they were. Then, disastrously, they forgot to ask how reliable the answers would be if market conditions changed.

Black and Scholes invented their equation in 1973; Robert Merton supplied extra justification soon after. It applies to the simplest and oldest derivatives: options. There are two main kinds. A put option gives its buyer the right to sell a commodity at a specified time for an agreed price. A call option is similar, but it confers the right to buy instead of sell. The equation provides a systematic way to calculate the value of an option before it matures. Then the option can be sold at any time. The equation was so effective that it won Merton and Scholes the 1997 Nobel prize in economics. (Black had died by then, so he was ineligible.)

If everyone knows the correct value of a derivative and they all agree, how can anyone make money? The formula requires the user to estimate several numerical quantities. But the main way to make money on derivatives is to win your bet – to buy a derivative that can later be sold at a higher price, or matures with a higher value than predicted. The winners get their profit from the losers. In any given year, between 75% and 90% of all options traders lose money. The world's banks lost hundreds of billions when the sub-prime mortgage bubble burst. In the ensuing panic, taxpayers were forced to pick up the bill, but that was politics, not mathematical economics.

The Black-Scholes equation relates the recommended price of the option to four other quantities. Three can be measured directly: time, the price of the asset upon which the option is secured and the risk-free interest rate. This is the theoretical interest that could be earned by an investment with zero risk, such as government bonds. The fourth quantity is the volatility of the asset. This is a measure of how erratically its market value changes. The equation assumes that the asset's volatility remains the same for the lifetime of the option, which need not be correct. Volatility can be estimated by statistical analysis of price movements but it can't be measured in a precise, foolproof way, and estimates may not match reality.

The idea behind many financial models goes back to Louis Bachelier in 1900, who suggested that fluctuations of the stock market can be modelled by a random process known as Brownian motion. At each instant, the price of a stock either increases or decreases, and the model assumes fixed probabilities for these events. They may be equally likely, or one may be more probable than the other. It's like someone standing on a street and repeatedly tossing a coin to decide whether to move a small step forwards or backwards, so they zigzag back and forth erratically. Their position corresponds to the price of the stock, moving up or down at random. The most important statistical features of Brownian motion are its mean and its standard deviation. The mean is the short-term average price, which typically drifts in a specific direction, up or down depending on where the market thinks the stock is going. The standard deviation can be thought of as the average amount by which the price differs from the mean, calculated using a standard statistical formula. For stock prices this is called volatility, and it measures how erratically the price fluctuates. On a graph of price against time, volatility corresponds to how jagged the zigzag movements look.

Black-Scholes implements Bachelier's vision. It does not give the value of the option (the price at which it should be sold or bought) directly. It is what mathematicians call a partial differential equation, expressing the rate of change of the price in terms of the rates at which various other quantities are changing. Fortunately, the equation can be solved to provide a specific formula for the value of a put option, with a similar formula for call options.

The early success of Black-Scholes encouraged the financial sector to develop a host of related equations aimed at different financial instruments. Conventional banks could use these equations to justify loans and trades and assess the likely profits, always keeping an eye open for potential trouble. But less conventional businesses weren't so cautious. Soon, the banks followed them into increasingly speculative ventures.

Any mathematical model of reality relies on simplifications and assumptions. The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets. The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different.
When these assumptions are valid, risk is usually low, because large stock market fluctuations should be extremely rare. But on 19 October 1987, Black Monday, the world's stock markets lost more than 20% of their value within a few hours. An event this extreme is virtually impossible under the model's assumptions. In his bestseller The Black Swan, Nassim Nicholas Taleb, an expert in mathematical finance, calls extreme events of this kind black swans. In ancient times, all known swans were white and "black swan" was widely used in the same way we now refer to a flying pig. But in 1697, the Dutch explorer Willem de Vlamingh found masses of black swans on what became known as the Swan River in Australia. So the phrase now refers to an assumption that appears to be grounded in fact, but might at any moment turn out to be wildly mistaken.

Large fluctuations in the stock market are far more common than Brownian motion predicts. The reason is unrealistic assumptions – ignoring potential black swans. But usually the model performed very well, so as time passed and confidence grew, many bankers and traders forgot the model had limitations. They used the equation as a kind of talisman, a bit of mathematical magic to protect them against criticism if anything went wrong.

Banks, hedge funds, and other speculators were soon trading complicated derivatives such as credit default swaps – likened to insuring your neighbour's house against fire – in eye-watering quantities. They were priced and considered to be assets in their own right. That meant they could be used as security for other purchases. As everything got more complicated, the models used to assess value and risk deviated ever further from reality. Somewhere underneath it all was real property, and the markets assumed that property values would keep rising for ever, making these investments risk-free.
The Black-Scholes equation has its roots in mathematical physics, where quantities are infinitely divisible, time flows continuously and variables change smoothly. Such models may not be appropriate to the world of finance. Traditional mathematical economics doesn't always match reality, either, and when it fails, it fails badly. Physicists, mathematicians and economists are therefore looking for better models.

At the forefront of these efforts is complexity science, a new branch of mathematics that models the market as a collection of individuals interacting according to specified rules. These models reveal the damaging effects of the herd instinct: market traders copy other market traders. Virtually every financial crisis in the last century has been pushed over the edge by the herd instinct. It makes everything go belly-up at the same time. If engineers took that attitude, and one bridge in the world fell down, so would all the others.

By studying ecological systems, it can be shown that instability is common in economic models, mainly because of the poor design of the financial system. The facility to transfer billions at the click of a mouse may allow ever-quicker profits, but it also makes shocks propagate faster.

Was an equation to blame for the financial crash, then? Yes and no. Black-Scholes may have contributed to the crash, but only because it was abused. In any case, the equation was just one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation.

Despite its supposed expertise, the financial sector performs no better than random guesswork. The stock market has spent 20 years going nowhere. The system is too complex to be run on error-strewn hunches and gut feelings, but current mathematical models don't represent reality adequately. The entire system is poorly understood and dangerously unstable. The world economy desperately needs a radical overhaul and that requires more mathematics, not less. It may be rocket science, but magic it's not.
Ian Stewart is emeritus professor of mathematics at the University of Warwick.

Ian Stewart's top 10 popular mathematics books

Ian Stewart is an Emeritus Professor of Mathematics at Warwick University and a Fellow of the Royal Society. He has written over 80 books, mainly popular mathematics, and has won three gold medals for his work on the public understanding of science. In collaboration with Terry Pratchett and Jack Cohen he wrote the Science of Discworld series. His new book, 17 Equations That Changed the World, is published by Profile.

1. Seventeen Equations that Changed the World
2. by Ian Stewart
3. 
4. Buy it from the Guardian bookshop

Search the Guardian bookshop

1. Tell us what you think: Star-rate and review this book

Buy 17 Equations That Changed the World from the Guardian bookshop
"'Popular mathematics' may sound like a contradiction in terms. That's what makes the genre so important: we have to change that perception. Mathematics is the Cinderella science: undervalued, underestimated, and misunderstood. Yet it has been one of the main driving forces behind human society for at least three millennia, it powers all of today's technology, and it underpins almost every aspect of our daily lives.
"It's not really surprising that few outside the subject appreciate it, though. School mathematics is so focused on getting the right answer and passing the exam that there is seldom an opportunity to find out what it's all for. The hard core of real mathematics is extremely difficult, and it takes six or seven years to train a research mathematician after they leave school. Popular mathematics provides an entry route for non-specialists. It allows them to appreciate where mathematics came from, who created it, what it's good for, and where it's going, without getting tangled up in the technicalities. It's like listening to music instead of composing it.
"There are many ways to make real mathematics accessible. Its history reveals the subject as a human activity and gives a feel for the broad flow of ideas over the centuries. Biographies of great mathematicians tell us what it's like to work at the frontiers of human knowledge. The great problems, the ones that hit the news media when they are finally solved after centuries of effort, are always fascinating. So are the unsolved ones and the latest hot research areas. The myriad applications of mathematics, from medicine to the iPad, are an almost inexhaustible source of inspiration."

1. The Man Who Knew Infinity by Robert Kanigel

The self-taught Indian genius Srinivasa Ramanujan had a flair for strange and beautiful formulas, so unusual that mathematicians are still coming to grips with their true meaning. He was born into a poor Brahmin family in 1887 and was pursuing original research in his teens. In 1912, he was brought to work at Cambridge. He died of malnutrition and other unknown causes in 1920, leaving a rich legacy that is still not fully understood. There has never been another mathematical life story like it: absolutely riveting.

2. Gödel, Escher, Bach by Douglas Hofstadter

One of the great cult books, a very original take on the logical paradoxes associated with self-reference, such as "this statement is false". Hofstadter combines the mathematical logic of Kurt Gödel, who proved that some questions in arithmetic can never be answered, with the etchings of Maurits Escher and the music of Bach. Frequent dramatic dialogues between Lewis Carroll's characters Achilles and the Tortoise motivate key topics in a highly original manner, along with their friend Crab who invents the tortoise-chomping record player. DNA and computers get extensive treatment too.

3. The Colossal Book of Mathematics by Martin Gardner

In his long-running Mathematical Games column in Scientific American, Gardner – a journalist with no mathematical training – created the field of recreational mathematics. On the surface his columns were about puzzles and games, but they all concealed mathematical principles, some simple, some surprisingly deep. He combined a playful and clear approach to his subject with a well-developed taste for what was mathematically significant. The book consists of numerous selections from his columns, classified according to the mathematical area involved. Learn how to make a hexaflexagon and why playing Brussels sprouts is a waste of time.

4. Euclid in the Rainforest by Joseph Mazur

A thoroughly readable account of the meaning of truth in mathematics, presented through a series of quirky adventures in the Greek Islands, the jungles around the Orinoco River, and elsewhere. Examines tricky concepts like infinity, topology, and probability through tall tales and anecdotes. Three different kinds of truth are examined: formal classical logic, the role of the infinite, and inference by plausible reasoning. The story of the student who believed nothing except his calculator is an object lesson for everyone who thinks mathematics is just 'sums'.

5. Four Colours Suffice by Robin Wilson

In 1852 Francis Guthrie, a young South African mathematician, was attempting to colour the counties in a map of England. Guthrie discovered that he needed only four different colours to ensure that any two adjacent counties had different colours. After some experimentation he convinced himself that the same goes for any map whatsoever. This is the remarkable story of how mathematicians eventually proved he was right, but only with the aid of computers, bringing into question the meaning of "proof". It contains enough detail to be satisfying, but remains accessible and informative throughout.

6. What is Mathematics Really? by Reuben Hersh

The classic text What is Mathematics? by Richard Courant and Herbert Robbins focused on the subject's nuts and bolts. It answered its title question by example. Hersh takes a more philosophical view, based on his experience as a professional mathematician. The common working philosophy of most mathematicians is a kind of vague Platonism: mathematical concepts have some sort of independent existence in some ideal world. Although this is what it feels like to insiders, Hersh argues that mathematics is a collective human construct – like money or the Supreme Court. However, it is a construct constrained by its own internal logic; it's not arbitrary. You choose the concepts that interest you, but you don't get to choose how they behave.

7. Magical Mathematics by Persi Diaconis and Ron Graham

Both authors are top-rank mathematicians with years of stage performances behind them, and their speciality is mathematical magic. They show how mathematics relates to juggling and reveal the secrets behind some amazing card tricks. Here's one. The magician mails a pack of cards to anyone, asking them to shuffle it and choose a card. Then he shuffles the cards again, and mails half of them to the magician—not saying whether the chosen card is included. By return mail, the magician names the selected card. No trickery: it all depends on the mathematics of shuffles.

8. Games of Life by Karl Sigmund

Biologists' understanding of many vital features of the living world, such as sex and survival, depends on the theory of evolution. One of the basic theoretical tools here is the mathematics of game theory, in which several players compete by choosing from a list of possible strategies. The children's game of rock-paper-scissors is a good example. The book illuminates such questions as how genes spread through a population and the evolution of cooperation, by finding the best strategies for games such as cat and mouse, the battle of the sexes, and the prisoner's dilemma. On the borderline between popular science and an academic text, but eminently readable without specialist knowledge.

9. Mathenauts: Tales of Mathematical Wonder edited by Rudy Rucker

A collection of 23 science fiction short stories, each of which centres on mathematics. Two are by Martin Gardner, and many of the great writers of SF are represented: Isaac Asimov, Gregory Benford, Larry Niven, Frederik Pohl. The high point is Norman Kagan's utterly hilarious "The Mathenauts", in which only mathematicians can travel through space, because space is mathematical – and, conversely, anything mathematical can be reality. An isomorphomechanism is essential equipment. Between them, these tales cover most of the undergraduate mathematics syllabus, though not in examinable form.

10. The Mathematical Principles of Natural Philosophy by Isaac Newton

There ought to be a great classic in this top 10, and there is none greater. I've put it last because it's not popularisation in the strict sense. However, it slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics. Using nothing more complicated than Euclid's geometry, Newton developed his laws of motion and gravity, applying them to the motion of the planets and strange wobbles in the position of the Moon. He famously said that he "stood on the shoulders of giants", and so he did, but this book set the scientific world alight. As John Maynard Keyes wrote, Newton was a transitional figure of immense stature: "the last of the magicians … the last wonderchild to whom the Magi could do sincere and appropriate homage." No mathematical book has had more impact.

• © 2012 Guardian News and Media Limited or its affiliated companies. All rights reserved.

Pick a Card, Any Card

• DECEMBER 10, 2011 The Wall Street Journal

By ALEX STONE

The standard way to mix a deck of playing cards—the one used everywhere from casinos to rec rooms—is what is known as a riffle (or "dovetail") shuffle. You begin by splitting the deck into two roughly equal stacks. Then you flick the cards with your thumbs off the bottoms of the piles in alternating fashion, interleaving the two stacks.

For games like blackjack or poker to be truly fair, the order of the cards must be completely random when the game begins. Otherwise a skilled cheat can exploit the lack of randomness to gain an advantage over other players.

How many riffle shuffles does it take to adequately mix a deck of 52 playing cards?

Francesco Abrignani/Alamy

As it turns out, you have to shuffle seven times before a deck becomes truly scrambled. Not only that, the cards become mixed in a highly unusual way: The amount of randomness in the deck does not increase smoothly. The first few shuffles do little to disturb the original order, and even after six shuffles, you can still pick out distinctly non-random patches.

But right around the seventh shuffle something remarkable happens. Shuffling hits its tipping point, and the cards rapidly decay into chaos.

Magical Mathematics

By Persi Diaconis and Ron Graham
Princeton, 244 pages, $29.95

The seven-shuffles finding applies to messy, imperfect riffle shuffles. The deck might not be divided exactly in half, for instance, or the cards might be riffled together in a haphazard way. Far from undesirable, a little sloppiness is actually the key to a random shuffle.

A perfect (or "faro") shuffle, meanwhile, wherein the deck is split precisely in half and the two halves are zippered together in perfect alternation, isn't random at all. In fact, it's completely predictable. Eight perfect shuffles will return a 52-card deck to its original order, with every card cycling back to its starting position.
And this doesn't just work for 52 cards. A deck of any size will eventually return to its starting order after a finite sequence of faro shuffles, although the number of faros required isn't always eight—and doesn't increase linearly. If you have 104 cards, for instance, it takes 51 faros to restore the deck. For a thousand cards, it takes 36.

These findings are among the many fascinating results explored in "Magical Mathematics," a dazzling tour of math-based magic tricks. The authors, Persi Diaconis and Ron Graham, are distinguished mathematicians with high-powered academic pedigrees. Both are also accomplished magicians who have taught courses on mathematical magic at Harvard and Stanford.

Mr. Diaconis has an especially unusual résumé for a mathematician. In 1959, at age 14, he ran away from home to study with the great 20th-century sleight-of-hand master Dai Vernon—a man who once fooled Harry Houdini with a card trick. After spending 10 years under Vernon's tutelage, Mr. Diaconis returned home to New York and enrolled in night school, eventually earning a full ride to a Ph.D. program in mathematics at Harvard.

The book's title may strike some people as odd in its pairing of magic and math, but the two subjects share a common lineage that goes back centuries. In fact, some of the earliest recorded magic tricks were based in math. Fibonacci's 1202 manuscript "Liber Abaci," the foundation of modern arithmetic, contains a number of magic tricks, including several versions of the famous three-object divination, wherein a spectator mentally selects one of three objects and the magician correctly identifies the spectator's choice.

The earliest recorded card tricks, meanwhile, appear in a math text written around 1500 by a Tuscan friar who was close friends with Leonardo da Vinci. And one of the first magic manuals was compiled in the 17th century by Claude Gaspard Bachet de Méziriac, an early number theorist.

Player/Alamy

But mathematical magic truly came of age in the 20th century, with the growth of magic as a mainstream hobby. "In the past hundred years, a revolution has taken place," the authors write, citing the thousands of math-based magic tricks now in circulation.

In their breezy yet authoritative book, Messrs. Diaconis and Graham showcase some of the genre's best creations as well as many new ones of their own devising. Included are tricks with coins and cards (the reader will want to have a deck handy), a divination routine that employs the I Ching—the 5,000-year-old Chinese fortune-telling book—and, my personal favorite, a gambling demonstration in which the spectator shuffles a deck of cards but somehow still manages to deal himself a royal flush in spades.

This last effect exploits something known as the Gilbreath Principle, a beautiful property discovered in the 1950s by a mathematician who worked for many years at the Rand Corp. Take a deck of cards and arrange it in alternating red-black order. Now deal half of the deck facedown into a pile—thus reversing its order—and riffle shuffle the two piles together. Finally, deal the cards face up in pairs.

Each pair will contain one red and one black card (though not necessarily in alternating order). This is the Gilbreath Principle. This same idea applies to any repeating pattern of cards. If, for instance, the deck is arranged so that the cards cycle through the four suits—clubs, hearts, spades, diamonds, clubs, hearts, spades, diamonds, and so on throughout the deck—and the same procedure is executed, then every four cards dealt off the top will contain a complete set of suits. This result, combined with a few clever subtleties, is the basis of the royal-flush effect.

All the tricks in "Magical Mathematics" are of the "self-working" variety—meaning they require little or no physical skill—and while a grasp of the underlying mathematics is helpful, it is by no means a necessity. Even math-phobes will be able to astound audiences by simply following the directions and consulting the many full-color illustrations provided throughout the text.

The mixing of magic and math is more than just a means to new tricks. It has also spawned a host of major mathematical breakthroughs. "Some magic tricks use 'real mathematics' and lead to questions beyond the limits of modern mathematics," the authors write. "Sometimes, we have been able to solve the math problems."

The seven-shuffles result is one such solution. Mr. Diaconis became interested in the math of shuffling after he encountered a card trick published in the early part of the 20th century by Charles Jordan, a chicken farmer and champion puzzle solver who invented several groundbreaking card tricks. In this particular effect—called "Long Distance Mind Reading," because it could be performed through the mail—the spectator shuffles before and after picking a card, but the magician still finds his selection.

Mr. Diaconis realized that for the trick to work shuffling had to be less effective than people generally assumed. While at Harvard, he teamed up with a mathematician named David Bayer and the two undertook a theoretical analysis, building on work done at Bell Labs in the 1950s. Their landmark 1992 paper—"Trailing the Dovetail Shuffle to its Lair"—rigorously proved that anything less than seven shuffles is inadequate. Not only that, their results had implications for a wide class of "mixing" phenomena—from stirring cake batter to compounding chemicals.

Similarly, the remarkable "looping" property of perfect shuffles is a facet of group theory—a branch of abstract mathematics that deals with, among other things, symmetric structures. Group theory has applications to chemistry, biology and, most notably, physics, where it provides the mathematical framework for the Standard Model—the overarching theory of subatomic particles and forces.

There's also a deep link between the perfect shuffle and the binary number system—the universal language of modern computing. To appreciate the connection, you first have to understand that there are two ways to do a faro. You can either weave the cards together so that the top and bottom cards stay in place—this is called an "out-faro"—or you can do what is known as an in-faro, in which the top and bottom cards each move inward by one card.

Now let's say that the ace of spades is on top, and you want to move 25 cards above it, so that the ace will be 26th from the top. The sequence of faros required to bring about this arrangement can be found by writing the number 25 in binary notation, like this: 11001. For each 1, you do an in-faro, and for each 0 you perform an out-faro. In this case, you would do two in-faros (11), followed by two outs (00) and, lastly, one more in (1).

Shuffling is one example of something seemingly ordinary that subtends an elegant mathematical structure. Juggling is another. "Mathematics is often described as the science of patterns," Messrs. Diaconis and Graham (a former president of the International Jugglers' Association) write. "Juggling can be thought of as the art of controlling patterns in time and space. Both activities offer unbounded challenges."

The central challenge in the mathematical study of juggling is to figure out which sequences of throws are possible and to categorize them according to the number of balls they require and their length—or period. Toward that end, mathematicians have developed a notation, called "siteswap," that uniquely describes all possible throwing sequences.

A siteswap pattern consists of a string of numbers, each of which specifies how much time one ball—or club, or chainsaw, or banana—spends in the air. The classic three-ball cascade, for instance, is denoted 333, because each ball is aloft for the same amount of time (three beats), and the sequence repeats after every third throw.

The remarkable thing about siteswap is that it allows jugglers to devise new patterns on paper and determine whether they're juggleable with a few simple calculations, all without tossing a single ball. What's more, the average of the digits in a pattern tells you the number of objects needed to juggle it—3 in the case of 441, for example, since the average of 4, 4 and 1 is 3.

Siteswap has led to the discovery of hundreds of unknown throwing sequences, many with just three or four balls. "Once the connection has been made between juggling (sequences) and mathematics, all kinds of doors, both mathematical as well as juggling, are thrown wide open," the authors note. "Many jugglers have been working hard to master the almost unlimited number of new patterns suggested by siteswaps."

Throughout the book, Messrs. Diaconis and Graham shuttle back and forth between magic and math, probing each trick for hidden mathematical insights and developing new magic based on what they find. In the process, they encounter a number of unsolved problems, some of which have prize money attached to them. It's a fun ride, even if you don't follow the nuances of every theorem and proof, and a refreshing change from the bombastic sort of magic one typically encounters on television.

Lovers of recreational mathematics, and especially fans of the late Martin Gardner, who contributed the foreword, will find many pleasures in "Magical Mathematics." And while exposing magic secrets in a book intended for the general public may raise hackles among some old-guard magicians, exploring the math behind these tricks will, in truth, only deepen the mystery. For, as the authors remind us, sometimes the methods are as magical as the tricks themselves.
 

—Mr. Stone is the author of the forthcoming "Fooling Houdini: Magicians, Mentalists, Math Geeks, and the Hidden Powers of the Mind."

Is modern science Biblical or Greek?

By Spengler

The "founders of modern science", writes David Curzon in Jewish Ideas Daily [1] of October 18, "were all believers in the truths of the opening chapter in the Hebrew Bible. The belief implicit in Genesis, that nature was created by a law-giving God and so must be governed by "laws of nature," played a necessary role in the emergence of modern science in 17th-century Europe. Equally necessary was the belief that human beings are made in the image of God and, as a consequence, can understand these "laws of nature."

Curzon argues that the modern idea of "laws of nature" stems from the Bible rather than classical Greece, for "ancient Greeks certainly believed that nature was intelligible and that its regularities could be made explicit. But Greek gods such as Zeus were not understood to have created the processes of nature; therefore, they could not have given the laws governing these processes."

Is this just a matter of semantics? Is there a difference between the "Greek" concept of intelligibility, and what Curzon calls the biblical concept of laws of nature? After all, the achievements of Greek science remain a monument to the human spirit. The Greek geometer Eratosthenesin the third century BCE calculated the tilt of the earth's axis, the circumference of the earth, and (possibly) the earth's distance from the sun. Archimedes used converging infinite series to calculate the area of conic sections, approximating the calculus that Newton and Leibniz discovered in the 17th century.

An enormous leap of mind, though, separates Archimedes' approximations from the new mathematics of the 17th century, which opened a path to achievements undreamed of by the Greeks. Something changed in the way that the moderns thought about nature. But does the rubric "laws of nature" explain that change? Curzon is on to something, but the biblical roots of modern science go much deeper.

Before turning to the scientific issues as such, it is helpful to think about the differences in the way Greeks and Hebrews saw the world. The literary theorist Erich Auerbach famously contrasted Greek and Hebrew modes of thought [2] by comparing two stories: the binding of Isaac in Genesis 22, and the story of Odysseus' scar told in flashback (Odyssey, Book 19).

Homer's hero has returned incognito to his home on the island of Ithaca, fearful that prospective usurpers will murder him. An elderly serving woman washes his feet and sees a scar he had received on a boar hunt two decades earlier, before leaving for the Trojan War, and recognizes him. Homer then provides a detailed account of the boar hunt before returning to his narrative.

Homer seeks to bring all to the surface, Auerbach explained in his classic essay. "The separate elements of a phenomenon are most clearly placed in relation to one another; a large number of conjunctions, adverbs, particles, and other syntactical tools, all clearly circumscribed and delicately differentiated in meaning, delimit persons, things, and portions of incidents in respect to one another, and at the same time bring them together in a continuous and ever flexible connection; like the separate phenomena themselves, their relationships - their temporal, local, causal, final, consecutive, comparative, concessive, antithetical, and conditional limitations - are brought to light in perfect fullness; so that a continuous rhythmic procession of phenomena passes by, and never is there a form left fragmentary or half-illuminated, never a lacuna, never a gap, never a glimpse of unplumbed depths."

Auerbach adds, "And this procession of phenomena takes place in the foreground - that is, in a local and temporal present which is absolute. One might think that the many interpolations, the frequent moving back and forth, would create a sort of perspective in time and place; but the Homeric style never gives any such impression."

Stark and spare, by contrast, is the story of God's summons to Abraham to sacrifice his beloved son Isaac. Where Homer tells us everything, the Bible tells us very little. God speaks to Abraham, and Abraham says, "Here I am." Auerbach observes, "Where are the two speakers? We are not told. The reader, however, knows that they are not normally to be found together in one place on earth, that one of them, God, in order to speak to Abraham, must come from somewhere, must enter the earthly realm from some unknown heights or depths. Whence does he come, whence does he call to Abraham? We are not told."

Abraham and Isaac travel together. Auerbach writes, "Thus the journey is like a silent progress through the indeterminate and the contingent, a holding of the breath, a process which has no present, which is inserted, like a blank duration, between what has passed and what lies ahead, and which yet is measured: three days!" Auerbach concludes:

On the one hand, externalized, uniformly illuminated phenomena, at a definite time and in a definite place, connected together without lacunae in a perpetual foreground; thoughts and feeling completely expressed; events taking place in leisurely fashion and with very little of suspense. On the other hand, the externalization of only so much of the phenomena as is necessary for the purpose of the narrative, all else left in obscurity; the decisive points of the narrative alone are emphasized, what lies between is nonexistent; time and place are undefined and call for interpretation; thoughts and feeling remain unexpressed, are only suggested by the silence and the fragmentary speeches; the whole, permeated with the most unrelieved suspense and directed toward a single goal (and to that extent far more of a unity), remains mysterious and "fraught with background".

Literary analysis may seem an unlikely starting-point for a discussion of science. But the Hebrew Bible's embodiment of what Auerbach called "the indeterminate and the contingent" has everything to do with the spirit of modern science. This emerges most vividly in the difference between the Greek and Hebrew understanding of time, the medium through which we consider infinity and eternity.

What separates Archimedes' approximation from Leibniz' calculus? The answer lies in the concept of infinity itself. Infinity was a stumbling-block for the Greeks, for the concept was alien to what Auerbach called their "perpetual foreground." Aristotle taught that whatever was in the mind was first in the senses. But by definition infinity is impossible to perceive. In the very large, we can never finish counting it; in the very small (for example infinitely diminishing quantities), we cannot perceive it. Infinity and eternity are inseparable concepts, for we think of infinity as a count that never ends.

For the Greeks, time is merely the demarcation of events. Plato understands time as an effect of celestial mechanics in Timaeus, while Aristotle in the Physics thinks of time as nothing more than the faucet-drip of events. That is Homer's time, in Auerbach's account. Biblical time is an enigma. That is implicit in Genesis, as Auerbach notes, but explicit in the Book of Ecclesiastes. Greek time is an "absolute temporal present."

In Hebrew time, it is the moment itself that remains imperceptible. Here is Ecclesiastes 3:15 in the Koren translation (by the 19th-century rabbi Michael Friedländer): "That which is, already has been; and that which is to be has already been; and only God can find the fleeting moment." As I wrote in another context, [3] Rabbi Friedländer's translation probably drew upon the celebrated wager that Faust offered the Devil in Goethe's drama. Faust would lose his soul will if he attempted to hold on to the passing moment, that is, to try to grasp what only God can find. The impulse to grab the moment and hold onto it is idolatrous; it is an attempt to cheat eternity, to make ourselves into gods.

A red thread connects the biblical notion of time to modern science, and it is spun by St Augustine of Hippo, the 4th-century Church father and polymath. His reflection on time as relative rather than absolute appears in Book 11 of his Confessions. And his speculation on the nature of number in time takes us eventually to the modern conceptual world of Leibniz and the calculus Aristotle's description of time as a sequence of moments, in Augustine's view, leads to absurdities.

To consider durations in time, we must measure what is past, for the moment as such has no duration. Events that have passed no longer exist, which means that measuring past time is an attempt to measure something that is not there at all. Augustine argues instead that we measure the memory of past events rather than the past itself: ''It is in you, my mind, that I measure times,'' he writes. Our perception of past events thus depends on memory, and our thoughts about future events depend on expectation. Memory and expectation are linked by ''consideration.'' For ''the mind expects, it considers, it remembers; so that which it expects, through that which it considers, passes into that which it remembers.''

Time is not independent of the intellect in Augustine's reading. Expectation and memory, Augustine adds, determine our perception of distant past and future: ''It is not then future time that is long, for as yet it is not: But a long future, is 'a long expectation of the future,' nor is it time past, which now is not, that is long; but a long past is 'a long memory of the past.''' This is the insight that allows Augustine to link perception of time to the remembrance of revelation and the expectation of redemption.

A glimpse of what Augustine's theory of time implies for mathematics appears in his later book, Six Books on Music. I argued in a 2009 essay for First Things: [4]

In De Musica, Augustine seeks to portray ''consideration'' as a form of musical number, that is, numeri judiciales, ''numbers of judgment.'' These ''numbers of judgment'' bridge eternity and mortal time; they are eternal in character and lie outside of rhythm itself, but act as an ordering principle for all other rhythms. They stand at the head of a hierarchy of numbers that begins with ''sounding rhythms'' - the sounds as such - which are in turn inferior to ''memorized rhythms.''

Only the ''numbers of judgment'' are immortal, for the others pass away instantly as they sound, or fade gradually from memory over time. They are, moreover, a gift from God, for ''from where should we believe that the soul is given what is eternal and unchangeable, if not from the one, eternal, and unchangeable God?'' For that reason the ''numbers of judgment,'' by which the lower-order rhythms are ordered, do not exist in time but order time itself and are superior in beauty; without them there could be no perception of time. Memory and expectation are linked by the ''numbers of judgment,'' which themselves stand outside of time, are eternal, and come from God.

That is an intimation of a higher order of number. Because it is buried in a treatise on musical time, Augustine's idea about "numbers of judgment" has elicited scant scholarly interest. But it is clear that his "numbers of judgment" are consistent with his much-discussed theory of "divine illumination." He wrote in Confessions, "The mind needs to be enlightened by light from outside itself, so that it can participate in truth, because it is not itself the nature of truth. You will light my lamp, Lord."

Descartes' "innate ideas" and Kant's "synthetic reason" descend from Augustine, although Kant recast the concept in terms of hard-wiring of the brain rather than divine assistance. The founder of neo-Kantian philosophy, Hermann Cohen (1842-1918) built his career out of the insight that the fact that infinitesimals in the calculus add up to a definite sum proves the existence of something like synthetic reason. That is why Kant triumphed in philosophy and the Aristotelians were reduced to a grumpy band of exiled irredentists.

Augustine's idea finds its way into modern science through Cardinal Nicholas of Cusa (1401-1464). Theologian and mathematician, Cusa noticed that musicians were tuning their instruments to ratios that corresponded to irrational numbers. The "natural" intervals of music tuning clashed with the new counterpoint of the Renaissance, so the musicians adjusted (or "tempered") the intervals to fit their requirements.

The Greeks abhorred the notion of irrational number because they abhorred infinity. Aristotle understood that infinity lurked in the irrational numbers, for we can come infinitely close to an irrational number through an infinite series of approximations, but never quite get there. And the notion of an "actual infinity" offended the Greek notion of intelligibility. To medieval mathematicians, the irrationals were surds, or ''deaf'' numbers, that is, numbers that could not be heard in audible harmonic ratios. The association of rational numbers with musical tones was embedded so firmly in medieval thinking that the existence of an irrational harmonic number was unthinkable.

The practice of musicians, Cusa argued, overthrew Aristotle's objections. The human mind, Cusa argued, could not perceive such numbers through reason (ratio), ie the measuring and categorizing faculty of the mind, but only through the intellect (intellectus), which depended on participation (participatio) in the Mind of God.

Cusa's use of Augustinian terminology to describe the irrationals - numbers ''too simple for our mind to understand'' - heralded a problem that took four centuries to solve (and, according to the few remaining "Aristotelian realists," remains unsolved to this day).

Not until the 19th century did mathematicians arrive at a rigorous definition of irrational number, as the limit of an infinite converging sequence of rational numbers. That is simple, but our mind cannot understand it directly. Sense-perception fails us; instead, we require an intellectual leap to the seemingly paradoxical concept of a convergent infinite series of rational numbers whose limit is an irrational number.

The irrational numbers thus lead us out of the mathematics of sense-perception, the world of Euclid and Aristotle, into the higher mathematics foreshadowed by Augustine (see my article, ''Nicholas of Cusa's Contribution to Music Theory,'' in RivistaInternazionale di Musica Sacra, Vol 10, July-December 1989).

Once irrational numbers had forced their way into Western thinking, the agenda had changed. Professor Peter Pesic [5] recently published an excellent account of the impact of irrational numbers in musical tuning on mathematics and philosophy. [6]

Another two centuries passed before Leibniz averred, ''I am so in favor of the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that nature makes frequent use of it everywhere, in order to show more effectively the perfections of its author.'' Theological concerns, one might add, also motivated Leibniz' work, as I sought to show in ''The God of the Mathematicians'' (First Things, August-September 2010).

Unlike Archimedes, who still thought in terms of approximations using rational numbers, Leibniz believed that he had discovered a new kind of calculation that embodied the infinite. Leibniz' infinitesimals (as I reported in ''God and the Mathematicians'') lead us eventually to George Cantor's discovery of different orders of infinity and the transfinite numbers that designate them; Cantor cited Cusa as well as Leibniz as his antecedents, explaining ''Transfinite integers themselves are, in a certain sense, new irrationalities. Indeed, in my opinion, the method for the definition of finite irrational numbers is quite analogous, I can say, is the same one as my method for introducing transfinite integers. It can be certainly said: transfinite integers stand and fall together with finite irrational numbers.''

Gilles DeLeuze (in Leibniz and the Baroque) reports that Leibniz ''took up in detail'' Cusa's idea of ''the most simple'' number: ''The question of harmonic unity becomes that of the 'most simple' number, as Nicolas of Cusa states, for whom the number is irrational. But, although Leibniz also happens to relate the irrational to the existent, or to consider the irrational as a number of the existent, he feels he can discover an infinite series of rationals enveloped or hidden in the incommensurable.'' Leibniz thus stands between Cusa in the fifteenth century and the flowering of the mathematics of infinite series in the nineteenth century. That is a triumph of the biblical viewpoint in modern science.

We can thus draw a red line from the Hebrew Bible (most clearly from Ecclesiastes) to Augustine, and through Nicholas of Cusa to G W Leibniz and the higher mathematics and physics of the modern world. The Hebrew Bible remains a force in modern science, despite the best efforts of rationalists and materialists to send it into exile.

Kurt Goedel, perhaps the greatest mathematician of the 20th century, approached all his work with the conviction that no adequate account of nature was possible without the presence of God. Inspired by Leibniz, Goedel destroyed all hope of a mechanistic ontology through his two Incompleteness Theorems as well as his work (with Paul Cohen) on the undecidability of the Continuum Hypothesis, as I reported in a recent First Things essay. [7]

There is always a temptation to offer simple homilies in honor of the Bible, for example, "intelligent design" theory, which in my view tells us nothing of real importance. An atheist like Spinoza also would contend that God designed the world, because in his philosophy God is the same thing as nature. Design contains no information about the unique and personal God of the Bible.

Curzon's discussion of the laws of nature is by no means wrong, but it would be wrong to leave the matter there. "The fear of God is the beginning of wisdom." As Ecclesiastes (3:11) said, "I have observed the task which God has given the sons of man to be concerned with: He made everything beautiful in its time; He also put an enigma [sometimes "eternity"] into their minds so that man cannot comprehend what God has done from beginning to end" (Ecclesiastes 3:11, Artscroll translation). Eternity is in our minds but the whole of creation is hidden from us. Steven Hawking has gone so far as to conjecture that something like Goedel's Incompleteness Principle might apply to physics as well as mathematics.

What divides Hebrews from Greeks, above all, is a sense of wonder at the infinitude of creation and human limitation. The Odyssey is intended to be heard and enjoyed; Genesis 22 is to be searched and searched again for layers of meaning that are withheld from the surface. The Greek gods were like men, only stronger, better-looking and longer lived, immortal but not eternal, and the Greeks emulated them by seeking become masters of a nature infested by gods. The Hebrews sought to be a junior partner in the unending work of creation. With due honor to the great achievements of the Greeks, modernity began at Mount Sinai.

Graduates in Science, Engineering and Maths are more versatile than others

The versatility of science graduates should be celebrated not criticised. What's the problem if science graduates end up in alternative careers? If anything, we need more of it.

Imran Khan guardian.co.uk, Friday 9 September 2011 13.33 BST larger

'If you study engineering, physics or chemistry as your first degree, you're almost 90% likely to be in either full-time employment or further study three years later.' Photograph: Martin Shields/Alamy

The Guardian reported that "only about half of all science graduates find work that requires their scientific knowledge" – a fact that "casts doubt on the government's drive to encourage teenagers to study [science]". Yet year on year, the Confederation of British Industry (CBI) reports that its members are finding it difficult to get enough staff with science, technology, engineering and maths (Stem) skills. This year more than two in five employers had trouble. The Science Council has just released a report showing that a fifth our workforce is employed in a scientific role. So what's going on?



The concerns come from the paper, Is there a shortage of scientists? A re-analysis of supply for the UK. Its author suggests there is no shortage of scientists and engineers in the UK, despite what the CBI says and contrary to the messages of successive governments. However, both the paper and the Guardian's reporting are based on some pretty odd assumptions. While it's true that about half of Stem graduates end up in careers outside science, that's not an argument to say that too many young people are studying science.



For a start, a Stem degree is a fantastic preparation for a huge range of careers. We should celebrate that fact, not mourn it. Statistics show (table 7) that if you study engineering, physics or chemistry as your first degree, you're almost 90% likely to be in either full-time employment or further study three years later. Those figures compare with 73% for the creative arts, and 78% for languages and historical or philosophical studies. The average across all graduates is just above 80%. That's because a Stem degree gives you a huge range of skills that are in demand in wide variety of jobs, not just in science. Isn't that a good thing? We could "fix" it by training science graduates to be useless in the wider economy, but at the moment we have a higher education sector that is successfully producing young people equipped with highly transferable skills.



Moreover, what's the problem if Stem graduates end up in careers outside science and engineering? If anything, we need more of it. We're crying out for more scientists and engineers to teach in schools, get into politics and the civil service, and become involved in running companies. The scientific method should be more embedded in society, not less. In the UK, we have only two MPs with a PhD. China, the most populous country and fastest growing economy in the world, has been led for the past eight years by two men who are professional engineers. I'm not saying it's better – but wouldn't it be nice to have some diversity among all the lawyers and economists?



We don't worry when law graduates don't become lawyers, history graduates don't become historians, or English graduates don't become … er … So why be concerned about the versatile engineer or chemist? True, we do need more people going into research and development if the UK is to successfully rebalance its economy. To achieve that we must increase investment in research and skills so that employers have a reason to come here, and in turn attract our science and engineering graduates into science and engineering jobs. Yes, each company and lab leader will be looking for the very best staff, so with the best will in the world you're not going to get every single engineering graduate into their first-choice profession. But how is that different from any other type of graduate?



It's a shame that the Guardian's report focused on the misleading figures when there was much else of value in the study. We see that there is far too much social and gender stratification in the people who actually go into science and engineering. This is unacceptable, given the benefits that those subjects give to their students. It's 2011, and yet we still only have around one in 10 female graduate engineers. You're more likely to take science and maths A-levels if you attend an independent school, with pupils at state-maintained schools over-represented in arts and humanities subjects instead.



There is emphatically still a need for more scientists and engineers – and, far from retrenching support for science and engineering, we should be concentrating on making these subjects more accessible to everyone.

The man who proved that everyone is good at maths

By travelling all the way to Madagascar, the French academic Marc Chemillier has shown that humans have remarkable innate skills with numbers. Alex Duval Smith reports

Saturday, 16 July 2011 The Independent

Maths is simple. But to discover this requires travelling to the ends of the earth where an illiterate, chain-smoking fortune teller lives in a room with a double bed and a beehive.
As the sun rises over the hut belonging to Raoke, a 70-year-old witch doctor, a highly pitched din heralds bee rush hour. The insects he keeps shuttle madly in and out through the window.
This bizarre setting, near nowhere in the harsh cactus savannah of southern Madagascar, is where a leading French academic, Marc Chemillier, has achieved an extraordinary pairing of modern science and illiterate intuition.
In his book, Les Mathématiques Naturelles, the director of studies at EHESS (School for Advanced Studies in Social Sciences) argues that mathematics is not only simple, it is "rooted in human, sensorial intuition". And he believes that Madagascar's population, which remains relatively untouched by outside influences, can help him to prove this.
Mr Chemillier argues that children should be encouraged to do maths before they learn to read and write. "There is a strong link between counting and the number of fingers on our hands. Maths becomes complicated only when you abandon basic measures in nature, like the foot or the inch, or even the acre, which is the area that two bulls can plough in a day."
To make his point, Mr Chemillier chose to charge up his laptop computer, leave Paris and do the rounds of fortune tellers on the Indian Ocean island because its uninfluenced natural biodiversity also extends to its human population. Divinatory geomancy – reading random patterns, or sikidy to use the local word – is what Raoke does, when not smoking cigarettes rolled with paper from a school exercise book.
With a low table covered in pieces of wood – each of which has a particular medicinal virtue – Raoke sits on his straw mat and chants as he runs his fingers through a bag of shiny, dark brown tree seeds. "There were about 600 seeds in the bag to begin with but I have lost a few," he says. "They come from the fane tree and were selected for me many years ago. The fane from the valley of Tsivoanino produces some seeds that lie and others that tell the truth so it is very important to test each seed. I paid a specialist to do that," said the father of six.
Raoke pours a random number of seeds on to his mat, then picks them up singly or in twos and lays them in a grid from right to left. Each horizontal gridline has a name – son, livestock, woman or enemy – and each vertical one has a name, too: chief, zebu (cattle), brother and earth. Whether one or two seeds lie at the intersection of two gridlines determines the subject's fortune and informs Raoke as to the cure required, and its price. From the selection of wood pieces on his table, Raoke can mix concoctions to cure ailments, banish evil spirits and restore friendships.
A basic session with the seeds costs 10,000 ariary (£3), then a price is discussed for the cure. It seems there is nothing Raoke cannot achieve for the top price of one or two zebus – Malagasy beef cattle that cost about £300 each – though some remedies are available for the price of a sheep. "A white man came from Réunion with a stomach ailment that the hospitals in France could not cure. I gave him a powder to drink in a liquid. He vomited and then he was cured," said Raoke.
Given the thousands of plant species in Madagascar that are still undiscovered by mainstream medicine, it is entirely possible that Raoke holds the key to several miracle cures. But Mr Chemillier is not interested in the pharmacopaeic aspect of the fortune teller's work.
"Raoke is an expert in a reflexive view of maths of which we have lost sight in the West," says Mr Chemillier. "Even armed with my computer program, I do not fully comprehend Raoke's capacities for mental arithmetic. He can produce 65,536 grids with his seeds – I have them all in my computer now – but we still need to do more work to understand his mental capacity for obtaining the combinations of single seeds and pairs."
The way in which he poses questions over the seeds requires the same faculties for mental speculation as might be displayed by a winner of the Fields Medal, which is the top award any mathematician can aspire to, said Mr Chemillier.
Over the years, Mr Chemillier has earned respect from Raoke and other Malagasy fortune tellers. "Initially they thought France had sent me to steal their work in an attempt to become the world's most powerful fortune teller. But once I was able to share grids with them that had been through my computer program, we established a relationship of trust," says Mr Chemillier.
Raoke says God shows him how to position the seeds. He does not understand why "Monsieur Marc", and now this other visiting white person, keeps asking him why he lays the seeds in a certain way. Yet it is clear from a stack of grimy copybooks he keeps under his bed that if indeed God is a mathematician dictating to Raoke, then the Almighty keeps him busy. When not consulting clients, the diminutive fortune teller spends hours with his seeds, laying them in different formations and copying the dots down in pencil. Those grids have value and Raoke sells them to other fortune tellers.
Seeing that pages of the copybooks are being sacrificed to his roll-ups, I offer Raoke a packet of cigarette papers which he accepts with delight, having never seen them before. He buys his tobacco leaf in long plaits from the market. So I offer him a green plastic pouch of Golden Virginia. Raoke cannot read but he recognises the word "danger", written in red at the start of the government health warning. He drops the packet to the floor in shock and disgust.