How an equation contributed to the GFC

Ian Stewart is well known to anyone interested in mathematics - alongside Marcus du Sautoy, he is one of the great popularisers of the subject. His book, 17 Equations that Changed the World, lives up to its brief. Stewart not only gives insights into the mathematics of 17 disparate topics, but explains how they’ve affected our lives in ways we don’t see. I’ve read a number of books along similar lines, all commendable, but Stewart succeeds better than most in demonstrating how so-called pure mathematics has shaped the modern world that we all take for granted. (By all, I mean anyone who can read this via a computer and the internet.)

The book includes the usual suspects like Pythagoras, Newton, Maxwell, Einstein, Schrodinger and lesser known ones like Boltzman, Shannon, the Bell curve, chaos theory and the Fourier transform. In all cases he explains how they have affected what we loosely call civilization. But it is the last chapter in the book that covers the Black-Scholes equation, which is most relevant to the present state of the world, and what Stewart aptly coins ‘the Midas formula’. This is the Nobel-prize-winning formula that effectively created the GFC (global financial crisis).

I was lucky enough to see the movie, The Inside Job, which had a limited release in this country, but ran for well over a month in one art-house cinema in Melbourne, such was its morbid appeal. It’s a depressing yet illuminating film because, not only do you get a recent history lesson, but you realise that no one has learnt anything and it will happen all over again.

Stewart is a mathematician yet he explains the machinations that created the current economic catastrophe with remarkable clarity and erudition, and provides antecedents that teach us how we never learn from history.

Some quotable quotes:

The banks behave like one of those cartoon characters who wanders off the edge of a cliff, hovers in space until he looks down, and then plunges to the ground.

How did the biggest financial train wreck in human history come about? Arguably, one contributor was a mathematical equation.

He then goes on to explain what derivatives are and how they became monopoly money in the hands of the biggest financial institutions in the world.

As Stewart expounds:

In 1998 the international financial system traded roughly $100 trillion in derivatives. By 2007 this had grown to one quadrillion US dollars… To put this figure into context, the total value of all the products made by the world’s manufacturing industries, for the last thousand years, is about 100 trillion US dollars, adjusted for inflation. That’s one tenth of one year’s derivative trading.

Curiously, it was a mathematician, Mary Poovey, professor of humanities and director of the Institute for the Production of Knowledge at New York University, who rang alarm bells in August 2002, when she gave a lecture at the International Congress of Mathematicians in Beijing, titled ‘Can numbers ensure honesty?’ The lecture was subtitled ‘Unrealistic expectations and the US accounting scandal’.  She pointed out, amongst other things, that ‘by 1995 [the] economy of virtual money had overtaken the real economy of manufacturing.’ She argued that  ‘[this] deliberately confusing virtual and real money… was leading to a culture in which the values of both goods and financial instruments were… liable to explode or collapse at the click of a mouse.’ This, of course, was the year after the collapse of Enron, the biggest bankruptcy in American history (at the time) to the tune of $11 billion to shareholders.

Stewart’s major point is that people used the Black-Scholes equation routinely, with no appreciation of its dependence on key assumptions. Change the assumptions and the consequences could be dire as we have since witnessed world-wide. Its proliferation was guaranteed by its Nobel-prize-winning status and the simple fact that everyone else was using it. What’s more, it could be converted into a computer algorithm, ensuring its ubiquity.

Economics doesn’t follow natural laws like gravity, nevertheless I expect chaos theory could provide some insights. It’s the human factor that appears to be the element that people leave out or ignore. I’m not an economist – it’s the area I least understand – yet a mathematician can explain to me what went wrong in the past decade in a way that makes sense. If I can understand it, why can’t the people who run economies and financial institutions?

Stewart’s final comment:

The financial system is too complex to be run on human hunches and vague reasoning. It desperately needs more mathematics, not less. But it also needs to learn how to use mathematics intelligently, rather than as some kind of magical talisman.

Addendum 1: Stewart also explains how mathematics gives credibility to human-generated climate-change, although that’s another issue. In particular, he claims: Global warming was predicted in the 1950s, and the predicted temperature increase is in line with what has been observed.

Addendum 2 (6 Sep 2012): I've just seen the movie, Margin Call, a well-drawn fictional account of this issue, with some big-name actors: Kevin Spacey, Jeremy Irons, Demi Moore, Paul Bettany and Simon Baker, amongst others. There is a reference to this equation early in the film in a dialogue between the Demi Moore character and Simon Baker character, though its significance is not explained nor its title given. Demi's character says: 'I told you not to use that equation...' (or words to that effect) and Simon's character says: 'Everyone else is using it...' (or words to that effect). An intriguing piece of dialogue that only 'people-in-the-know' would pick up on.

Addendum 3 (3 Nov 2012): This interview with Greg Smith (formerly with Goldman Sachs) reveals the real story behind Wall Street, its culture, its hypocrisy (how it wants zero government interference in the good times and government bale-out in the bad times) and, most importantly, how nothing has changed since the GFC.

Addendum 4 (30 Jun 2013): I changed the title from 'Mathematics and the Real World'. I think it was misleading and the new title is more relevant to the discussion.

This is so COOL

This is a brilliant piece of simple, yet profound, scientific understanding, that anyone with a high school education should be able to follow. I can't imbed it so I provide this link.

John D. Barrow, in his book, The Constants of Nature provides a very neat graphic (p. 222 of 2003 Vintage paperback edition) that demonstrates why 3 dimensions of space and 1 of time provide the most 'livable' universe (my term, not his). Barrow has written extensively on the 'Anthropic Principle' in all its manifestations, and I keep promising myself that I'll write a post on it one day.

Why the argument for the existence of God (as an independent entity) is a non sequitur

This has been a point of discussion on Stephen Law’s blog recently, following Law’s debate with William Lane Craig last year. My contention is that people argue as if God is something objective, when, clearly it isn’t: God is totally subjective.

God is a feeling, not an entity or a being. God is something that people find within themselves, which is neither good nor bad; it’s completely dependent on the individual. Religiosity is a totally subjective phenomenon, but it has cultural references, which determine to a lesser or greater extent what one ‘believes’. Arguing over the objective validity of such subjective perspectives is epistemologically a non sequitur.

Craig’s argument takes two predominant strands. One is that atheists can’t explain the where-with-all from whence the universe arose and theists can. It’s like playing a trump card: what’s your explanation? Nil. Well, here’s mine, God: game over. If Craig wants to argue for an abstract, Platonic, non-personal God that represents the laws of the universe prior to its physical existence, then he may have an argument. But to equate a Platonic set of mathematical laws with the Biblical God is a stretch, to say the least, especially since the Bible has nothing to say on the matter.

The other strand to his argument is the Holy Spirit that apparently is available to us all. As I said earlier, God is a feeling that some people experience, but I think it’s more a projection based on one’s core beliefs. I don’t dismiss this out of hand, partly because it’s so common, and partly because I see it as a personal aspiration. It represents the ideal that an individual aspires to, and that can be good or bad, depending on the individual, as I said above, but it’s also entirely subjective.

Craig loves the so-called ‘cosmological’ argument based on ‘first cause’, but it should be pointed out that there are numerous speculative scientific theories about the origin of the universe (refer John D. Barrow’s The Book of Universes, which I discussed May 2011). Also Paul Davies’ The Goldilocks Enigma gives a synopsis on all the current ‘flavours’ of the universe, from the ridiculous to the more scientifically acceptable. Wherever science meets philosophy or where there are scientific ‘gaps’ in our knowledge, especially concerning cosmology or life, evangelists like Craig try to get a foothold, reinterpreting an ancient text of mythologies to explain what science can’t.

In other posts on his blog, Stephen Law discusses the issue, ‘Why is there something instead of nothing?’ Quite frankly, I don’t think this question can ever be answered. Science has no problem with the universe coming from nothing – Alan Guth, who gave us inflationary theory also claimed that ‘the universe is the ultimate free lunch’ (Davies, God and the New Physics, 1983). The laws of quantum mechanics appear to be the substrate for the entire universe, and it’s feasible that a purely quantum mechanical universe existed prior to ours and possibly without time. In fact, this is the Hartle-Hawking model of the universe (one of many) where the time dimension was once a fourth dimension of space. Highly speculative, but not impossible based on what we currently know.

But when philosophers and scientists suggest that the ‘why something’ question is an epistemological dead end, evangelists like Craig see this is as a capitulation to their theistic point of view. I’ve said in a previous post (on Chaos theory, Mar. 2012) that the universe has purpose but is not teleological, which is not the oxymoron it appears to be when one appreciates that ‘chaos’, which drives the universe’s creations, including life, is deterministic but not predictable. In other words, the universe’s purpose is not predetermined but has evolved.

Some people, many in fact, see the universe’s purposefulness as evidence that there is something behind it all. This probably lies at the heart of the religious-science debate, but, as I expounded in a post on metaphysics (Feb. 2011): between chaos theory, the second law of thermodynamics and quantum mechanics, a teleological universe is difficult to defend. I tend to agree with Stephen Jay Gould that if the universe was re-run it would be completely different.

Addendum 1: Just one small point that I’ve raised before: without consciousness, there might as well be nothing. It’s only consciousness that allows meaning to even arise. This has been addressed in a later post.

Addendum 2: I've added a caveat to the title, which is explained in the opening of the post. If humans are the only link between the Universe and a 'creator' God (as all monotheistic religions believe) then God has no purpose without humanity.

This is meant to be Australia

Ranjini was found to be a genuine refugee before ASIO decided last week she is a security risk for Australia. But the government won't tell her why, and now she's facing a life in detention. (The Age, 18 May 2012, front page)

It’s unbelievable that you can be detained indefinitely in this country without being given a reason, so that there is no defence procedure by law and no appeal process. The defendant in this case, Ranjini, can’t even confess because she’s a ‘risk’, not a criminal, apparently. As far as we can tell, she’s being detained in case she plans to execute a terrorist act; the truth is we don’t know because no one is allowed to tell us. What is unimaginably cruel is to give someone hope and then take it away with a phone call and a brief, closed interview. She’s been living in Australia since 2004.

To quote The Age:

Because she does not know what she is accused of doing, or saying, she cannot defend herself. Because there is no mechanism for an independent review of ASIO's finding, she, like the other 46, faces indefinite detention, along with two boys who were beginning to show signs of recovering from the traumas of their past.

Under the guise of ‘security reasons’, an apparent law-abiding housewife (who is also pregnant) can be incarcerated with her 2 school-age boys without even her husband knowing why. Australia is not meant to be a totalitarian government so why do we behave like one. The Minister for the Attorney General’s Department, Nicola Roxon, has so far dodged any questions on the issue. This is a law that is clearly unworkable (if it can’t be appealed or defended) born out of the post-9/11 paranoia that has seized all Western democratic countries and compromised our principles.

As is evident in the Haneef case in 2007, police and investigators tread a thin line in prosecuting possible terrorist suspects and protecting their civil liberties. In Haneef’s case, who was eventually not convicted, and other cases that have been successfully prosecuted, there have been specific accusations, involvement of the DPP and Federal Police, as well as ASIO. In the case of Ranjini, from what has been revealed thus far, there is only a risk assessment from ASIO and no specific accusations. One suspects that, because she’s a refugee, no one would care or kick up a fuss, or that the story would become front-page news in The Age.

This is not a law suited to a 21^st Century, Western democratic country; it’s a law suited to a paranoid totalitarian government.

Addendum 1: Here is a TV presentation of the story.

Addendum 2: This whole issue has a history going back 6 months at least and revealed here. We actually treat criminals better than this. The reason that the government gets away with this is because refugees are demonised in our society. Refugees don't vote and lots of people who do vote think that all refugees should be locked up indefinitely or sent back to where they come from. It's a sad indictment on our society.

Addendum 3: A lawyer is about to challenge the law in Australia's High Court. The last time it was challenged, the High Court rejected it 4 to 3, from memory, which only demonstrates that even the highest people in the land will follow political lines rather than the basic human rights of individuals.

i, the magic number that transformed mathematics and physics

You might wonder why I bother to beleaguer people with such esoteric topics like complex algebra and Schrodinger’s equation (May 2011, refer link below). The reason is that I’ve struggled with these mathematical milestones myself, but, having found some limited understanding, I attempt to pass on my revelations.

Firstly, I contend that calling i an imaginary number is a misnomer; it’s really an imaginary dimension. And if it was called such it would dispel much of the confusion that surrounds it. We define i as:

i = √-1

But it’s more intuitive to give the inverse relationship:

i² = -1

Because, when we square an imaginary number, we transfer it from the imaginary plane to the Real plane. Graphically, i rotates a complex number by 90⁰ in the anti-clockwise direction on the complex plane (or Argand diagram). Or, to be more precise, multiplying any complex number (which has both an imaginary and a Real component) by i will rotate its entire graphical representation through 90⁰. In fact, complex algebra is a lot easier to comprehend when it is demonstrated graphically via an Argand diagram. An Argand diagram is similar to a Cartesian diagram only the x axis represents the Real numbers and the y axis is replaced by the i axis, hence representing the i dimension, not the number i.

It’s not unusual to have mathematical dimensions that are not intuitively perceived. Any dimension above 3 is impossible for us to visualise. And we even have fractional dimensions that are called fractals (Davies, The Cosmic Blueprint, 1987). So an imaginary dimension is not such a leap of imagination (excuse the pun) in this context. Whereas calling i an imaginary number is nonsensical since it quantifies nothing.

In an equation, i appears to be a number, and to all intents and purposes is treated like one, but it’s more appropriate to treat it as an operator. It converts numbers from Real to imaginary and back to Real again.

In quantum mechanics, Schrodinger’s wave function is a differential complex equation, which of itself tells us nothing about the particle it’s describing in the physical world. It’s only by squaring the modulus of the wave function (actually multiplying it by its conjugate to be technically correct) that we get a Real number, which gives a probability of finding the particle in the physical world.

Without complex algebra (therefore i ) we would not have a mathematical representation of quantum mechanics at all, which is a sobering thought. We have long passed the point in our epistemology of the physical universe whereby our comprehension is limited by our mathematical abilities and knowledge.

There are 2 ways to represent a complex number, and we need to thank Leonhard Euler for pointing this out. In 1748 he discovered the mathematical relationship that bears his name, and it has arguably become the most famous equation in mathematics.

Exponential and trigonometric functions can be expressed as infinite power series. In fact, the exponential function is defined by the power series:

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ….

Where n! (called n factorial) is defined as: n! = n x (n-1) x (n-2) x …. 2 x 1

But the common trig functions, sin x and cos x, can also be expressed as infinite power series (Taylor’s theorem):

sin x = x – x³/3! + x⁵/5! – x⁷/7! + ….

cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + ….

Euler’s simple manipulation of these series by invoking i was a stroke of genius.

e^ix = I +ix – x²/2! – ix³/3! + x⁴/4! + ix⁵/5! – x⁶/6! – ix⁷/7! + …

i sin x = ix – ix³/3! + ix⁵/5! – ix⁷/7! + …

I’ll let the reader demonstrate for themselves that if they add the power series for cos x and isin x they’ll get the power series for e^ix .

Therefore:   e^ix = cos x + i sin x

But there is more: x in this equation is obviously an angle, and if you make x = π, which is the same as 180⁰, you get:

sin 180⁰ = sin 0 = 0

cos 180⁰ = - cos 0 = -1

Therefore:  e^iπ = -1

This is more commonly expressed thus:

e^iπ  + 1 = 0

And is known as Euler’s identity. Richard Feynman, who discovered it for himself just before his 15^th birthday, called it “The most remarkable formula in math”.

It brings together the 2 most fundamental integers, 1 and 0 (the only digits you need for binary arithmetic), the 2 most commonly known transcendental numbers, e and π, and the operator i.

What I find remarkable is that by adding 2 infinite power series we get one of the simplest and most profound relationships in mathematics.

But Euler’s equation (Euler’s identity is a special case): e^iθ = cos θ + i sin θ

gives us 2 ways of expressing a complex number, one in polar co-ordinates and one in Cartesian co-ordinates.

We use z by convention to express a complex number, as opposed to x or y.

So  z = x + iy (Cartesian co-ordinates)

And z = re^iθ  (polar co-ordinates)

Where r is called the modulus (radius) and θ is the argument (angle).

If one looks at an Argand diagram, one can see from Pythagoras’s theorem that:

r² = x² + y²

But the same can be derived by multiplying the complex number by its conjugate, x – iy

So  (x + iy)(x – iy) = x² + y² = r² 

(I’ll let the reader expand the equation for themselves to demonstrate the result)

But also from the Argand diagram, using basic trigonometry, we can see:

x = r cos θ  and y = r sin θ (from cos θ = x/r and sin θ = y/r)

So  x + iy  becomes  r cos θ + i r sin θ

There is an advantage in using the polar co-ordinate version of complex numbers when it comes to multiplication, because you multiply the moduli and add the arguments.

So, if:    z₁ = r₁e^iθ1   and   z₂ = r₂e^iθ2

Then:   z₁ x z₂ = r₁e^iθ1 x r₂e^iθ2 = r₁r₂e^{i(θ1 + θ2)}

And, obviously, you can do this graphically on an Argand diagram (complex plane), by multiplying the moduli (radii) and adding the arguments (angles).


Addendum 1: Given its role in quantum mechanics, I think i should be called the 'invisible dimension'.

Addendum 2: I've been re-reading Paul J. Nahin's very comprehensive book on this subject, An Imaginary Tale: The Story of √-1, and he reminds me of something pretty basic, even obvious once you've seen it.

tan θ = sin θ/cos θ or y/x (refer the Argand Diagram)

So θ = tan^-1(y/x) where this represents the inverse function of tan (you can calculate the angle from the ratio of y over x, or the imaginary component over the Real component).

You can find this function on any scientific calculator usually by pressing an 'inverse' button and then the 'tan' button.

The point is that you can go from Cartesian co-ordinates to polar co-ordinates without using e. According to Nahin, Caspar Wessel discovered this without knowing about Euler's earlier discovery. But Wessel, apparently, was the first to appreciate that you sum angles when multiplying complex numbers and invented the imaginary axis when he realised that multiplying by i rotated everything by 90⁰ anticlockwise.

A necessary law to protect women from an archaic, anachronistic, life-destroying practice

It’s extraordinary that in Australia, in the 21st Century, the Government is proposing an act of Parliament to make it illegal to marry a girl without her consent.

There were parts of this programme that had me shaking, but as teenage girls are becoming better educated their families are becoming more deceiving in arranging unwanted marriages. This programme tells the story of 4 women who dared to take control of their own lives so that they could have a future that was worth living.

What one finds unbelievable is that parents could force their daughters into a life of unhappiness and servitude against their will, obviously unaware of the opportunities they have for realising the potential of their educations.

As Ayaan Hirsi Ali wrote in her autobiography (I reviewed a year ago, March 2010), in some so-called ‘traditional’ cultures, women are never treated as mature adults, who are capable of intellectual and moral autonomy. And whilst, in the West, we find this culpable, it’s only in the last century that women have been given the benefit of the doubt, to put it kindly, that they can live and make decisions independent of men.

As Kerry O’Brien says in his summing up, the stories revealed here are both depressing and inspiring. I find it interesting that one of the girls featured (promised to a cousin in a foreign country at the age of 12, whom she first met on her supposed wedding day at age 17) had turned her father around after stubbornly refusing to recognise 2 marriages (one in Pakistan and one in Australia). He eventually realised (apparently, as he’s not interviewed) that his daughter’s happiness meant more to him than following a centuries-old tradition.

For many people, this is another arrow to fling at Islam, but there are Muslim feminists (I’ve met them) and it is they who can change this cultural relic, as it was changed in our society.