Alan Turing’s 100th Birthday today

Alan Turing is not as well known as Albert Einstein, yet he arguably had a greater impact on the 20^th Century and was no less a genius. Turing was not only one of the great minds of the 20^th Century but one of the great minds in Western philosophy. In fact, in January, Nature called him “one of the top scientific minds of all time”. He literally invented the modern computer in his head in the 1930s as a thought experiment, whilst simultaneously solving one of the great mathematical problems of his age: the so-called ‘halting problem’. I’ve described this in a previous post (Jan. 2008) whilst reviewing Gregory Chaitin’s book, Thinking about Godel and Turing, but the occasion warrants some repetition.

The 2 June 2012 edition of New Scientist had a feature on Turing by John Graham-Cumming, and it covers in greater detail and erudition anything I can write here. For the public at large, Turing is probably best known for his role at Bletchley Park, in the 2^nd World War, deciphering the Enigma code used by German U-boats. Turing’s contribution remained ‘classified’ until after his death, though, according to Wikipedia, he received an OBE ‘for his work at the Foreign Office’. Turing worked with Gordon Welchman on the Bombe, a machine they designed to run ‘cribs’ to decipher the enigma code. And, with mathematician Bill Tutte, he also developed a method to decode the Tunny cipher, which was used for high-level messages in Hitler’s command.

Turing also developed a ‘portable’ code called ‘Delilah’, which was unique in that it depended on clock-arithmetic, making it very difficult to decode compared to other ciphers. According to Graham-Cumming, the details of this have only recently been declassified.

Turing also became fascinated with mathematics in nature in his childhood, like the recurrence of Fibonacci sequences in spiral patterns in daisy petals and sunflower heads. In 1952 he published a paper on “The chemical basis of morphogenesis”, whereby ‘…specific chemical reactions were responsible for the irregular spots and patches on the skin of animals like leopards or cows, and the ridges inside the roof of the mouth.’ He provided a mathematical model (a computer simulation) of 2 chemicals interacting via diffusion and reaction in a chaotic yet repetitive fashion that would result in a variegated pattern. He speculated that this could become manifest as a literal pattern on animal skins if the 2 chemicals either turned on or off specific cells. Again, according to Graham-Cumming, as recently as January this year, researchers at King’s College London demonstrated Turing’s theory ‘…that 2 chemicals control the ridge patterns inside a mouse’s mouth.’

But, in scientific and mathematical circles, Turing is best known for his ‘proof’ of the ‘halting problem’, which is actually very simple to formulate but difficult to prove. Basically, Turing conjured a thought experiment of a machine that could compute an algorithm until it either found an answer or it didn’t, which meant it could run forever (the ‘halting problem’). Turing was able to prove that one could not determine in advance whether the algorithm would stop or not. An example is Goldbach’s conjecture, which can be easily formulated by an algorithm and run on a computer. At present there is no proof of the Goldbach conjecture but it has been derived by computers up to 100 trillion or 10¹⁴. Obviously, if we knew it could stop or not we could determine if it was true or not to infinity. The same is true for Riemann’s hypothesis, probably the most famous unsolved problem in mathematics. Chaitin (mentioned above) has invented a term, Ω (Omega) to provide a probability of Turing’s algorithm stopping. To quote from a previous post:

Chaitin claims that this is his major contribution to mathematics, arising from his invention of the term ‘Ω’ (Omega), though he calls it a discovery, to designate the probability of a programme ‘halting’, otherwise known as the ‘halting probability’.

But it was in conjuring his ‘thought experiment’ that Turing mentally invented what we now call a computer. I expect computers would have been invented without Turing in the same way relativity would have been discovered without Einstein, yet that is not to diminish either man’s genius or singular contribution. Turing’s insight was to imagine a ‘tape’ of infinite length with instructions that not only performed the algorithm but performed actions on the tape itself. It’s what we recognise today as software. Turing realised that this allowed a ‘universal’ machine to exist, now called a ‘universal Turing machine’, because the tape could instruct one machine to do what all possible machines could do. All modern computers are examples of Universal Turing Machines, including the one I’m using to write and post this blog.

One cannot discuss Turing without talking about the circumstances of his death, because it was a tragedy comparable to the deaths of Socrates and Lavoisier. Turing was persecuted for being a homosexual after he went to the Police to report a burglary. He was given a choice of imprisonment or ‘medical castration’ by hormone treatment, which he accepted. In 1954, at the relatively young age of 41, he committed suicide and the world lost a visionary, a genius and a truly great mind. John Graham-Cumming, the author of the 5 page feature in New Scientist, successfully campaigned for an official apology for Turing from the UK government in 2009. Given the current debate about gay marriage, it is apposite to remember the injustice that was done just over half a century ago to one of the greatest minds of all time. I’ve no doubt that there are many people who believe that Turing could have been ‘cured', such is the ignorance that still pervades many of the world’s societies, and is often promulgated by conservative religious groups, who have a peculiarly backward and anachronistic view of the world. Turing was ahead of his time in many ways, but in one way, tragically.

Addendum: For more detailed information, there is the Wiki site linked above, and Andrew Hodges dedicated Site. The Stanford Encyclopedia of Philosophy gives a good account of Turing’s seminal work in artificial intelligence. Andrew Hodges gives a good account of his untimely attitude to being openly homosexual and an insight into his modest character. There is a very strong sense of an extraordinary visionary intellect who was a victim of prejudice. 

Prometheus, the movie

Everyone is comparing Ridley Scott’s new film with his original Alien, and there are parallels, not just the fact that it’s meant to be a prequel. The crew include an android, a corporate nasty and a gutsy heroine, just like the first two movies. There are also encounters with unpleasant creatures. Alien was a seminal movie, which spawned its own sequels, albeit under different directors, yet it was more horror movie than Sci-Fi. But SF often combines genres and is invariably expected to be a thriller. Prometheus is not as graphically or viscerally scary as Alien, but it’s more a true Sci-Fi than a horror flick. In that respect I think it’s a better movie, though most reviewers I’ve read disagree with me.

Prometheus is a good title because it’s the Greek story about the Gods giving some of their abilities to humankind. Scott’s tale is a 21^st Century creation myth, whereby mankind goes in search of the ‘people’ who supposedly ‘engineered’ us. One of the characters in the film quips in response to this claim: ‘There goes 3 centuries of Darwinism.’ From a purely scientific perspective, it’s possible that DNA originally came from somewhere else, either as spores or in meteorites or an icy comet, but it would have been very simple life forms at the start of evolution not the end of it. The idea that someone engineered our DNA so it would be compatible with Earthbound DNA destroys the suspension of disbelief required for the story, so it’s best to ignore that point.

But lots of Sci-Fi stories overlook this fundamental point when aliens meet Earthlings and interbreed for example (Avatar). And I’ve done it myself (in my fiction) though only to the extent that humans could eat food found on another planet. I suspect we could only do that, in reality, if the food contained DNA with the same chirality as ours. The universal unidirectional chirality of DNA is one of the strongest evidential factors that all life on Earth had a common origin.

But I have to admit that Ridley has me intrigued and I’m looking forward to the sequel, as the final scenes effectively promise us one. One of the major differences with Alien and its spinoffs is that there is a mystery in this story and the heroine is bent on finding the answer to it. She wants to find out who made us and where they came from and why they did it. There is an obvious religious allusion here, but this is closer to the Greek gods, suggested by the title rather than the Biblical god. Having said that, our heroine wears a cross and this is emphasised. I expect Ridley wants us to make a religious connection.

Good Sci-Fi in my view should contain a bit of philosophy – make us think about stuff. In this case, stuff includes the possibility of life on other worlds and the possibility that there may exist civilizations greater than ours, to the extent that they could have created us. We find it hard to imagine that we are the end result of a process that started from stardust; that something as complex and intelligent as us could not have been created by a greater intelligence. Ridley brings that point home when the android asks someone how would they feel about meeting their maker, as he has had to. So I’m happy to see where Ridley is going with this – it’s a question that most people have asked and not been satisfied with the answer. I don’t think Ridley is going to give us a metaphysical answer. I expect he’s going to challenge what it means to be human and what responsibilities that entails in the universe’s creation. 

Philosophy in action - on gay marriage

Last night I went and saw a live stage production of 8 by Dustin Lance Black (whose screenwriting credits include Milk and J. Edgar), a one-off production at Her Majesty’s Theatre in Melbourne. It was a fund-raiser for the lobby group, Australian Marriage Equality, so tickets were not cheap yet the theatre was packed.

The play is based on a real-life trial held in California in 2010, when 2 same-sex couples (Kristin Perry and Sandy Stier, and Paul Katami and Jeff Zarillo) challenged the passing of Proposition 8 as unconstitutional. Effectively, Proposition 8, under Governor Arnold Schwarzenegger, prevented gays and lesbians from getting married.  There was a strong TV campaign supporting Proposition 8, which I’ll address later, and some of these were shown to the theatre audience as background.

It was also relayed to the audience, right at the beginning, how the play came about. Requests by the plaintiff’s team to have the trial broadcast were overturned by their opponents, but transcripts can’t be denied forever and most of the play is taken directly from the transcript. The play is actually read, with almost no props, yet real actors were used to give it authenticity.

There is an on-line version on YouTube including George Clooney, Martin Sheen and Brad Pitt as part of the cast. The Australian production I saw included its own well-known actors like Rachel Griffiths, Lisa McLune, Shane Jacobson and Magda Szubanski (from Babe for international readers). It also included Kate Whitbread as one of the plaintiffs and she was instrumental in getting the production performed. Incidentally, Kate has been producer to Aussie film-maker, Sandra Sciberras (Max’s Dreaming, Caterpillar Wish and Surviving Georgia).

This is not a play that will attract opponents of gay marriage – it was clear from the audience’s reaction that most, if not all, members were advocates. Being a fund-raiser you wouldn’t expect anything else. Opponents, no doubt would call it propaganda and biased, but the ‘opponents’ in the trial come off very badly indeed. In fact, this is the salient point because it demonstrated how weak their arguments were when subjected to the rigours of courtroom dissection and cross-examination. It’s no wonder they opposed it being broadcast.

And that’s why I call it ‘philosophy in action’ because it demonstrated the difference between a glib, emotive, made-for-TV advertising programme and critical, evidence-based argument. It was obvious from the pro-proposition 8 campaign and other rhetoric we hear in the production, that it was based on fear. Fear that same-sex marriage will infect children (yes, I mean infect not affect). Their whole campaign was based around the need to protect children from the ‘evils’ of gay parents and gays generally.

It was obvious that many conservatives actually believe that lesbianism and homosexuality are contagious – not biologically contagious, but socially contagious like cigarette smoking or alcohol consumption or drug-taking. They have a genuine fear, despite all the evidence to the contrary, that more children will become gay if gay marriage is legalized because it’s a choice that they didn’t have before. In other words, gay marriage is a lifestyle choice and has nothing to do with biology. Allowing gays and lesbians to be perceived as ‘normal’ is dangerous because kids will become ‘infected’, whereas at present they are still ‘protected’. That’s their argument in a nutshell.

In a promotional review of the play in last weekend’s Age, both Kate Whitbread and Bruce Myles (director of the Aussie version) give their more parochial reasons for putting it on. Bruce said he was ‘disgusted’ by Bob Katter’s political advertisement in the recent Queensland state election, whereby Katter used lewd images of homosexual couples juxtaposed with Campbell Newman’s (Queensland’s Liberal party contender and shoe-in to win) statement that he supported gay marriage. It was an obvious ploy on Katter’s part to exploit homophobia to undermine Newman’s commanding lead in the polls.

Both Bruce and Kate expressed outrage at six Catholic bishops in Victoria sending out 80,000 letters exhorting parishioners to lobby against gay marriage. Apparently, few parishioners were as alarmed as the bishops, going by the response. In fact, both in Australia and the US, it’s conservative religious groups who are the most vocal opponents to gay marriage. Arguments based on arcane religious texts are arguably the least relevant to the debate. It’s effectively an argument to maintain a longstanding prejudice because the Bible tells us so.

Spencer McLaren, who plays the courtroom advocate defending proposition 8, said: “What it is really about is putting prejudice and fear on trial and showing the inhumanity of the discrimination that is occurring.”

For those interested, here is the online version (90 mins).

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.

Stephen Hawking

Someone offered me this after seeing my blog. So my thanks to Peter Kim.




Created by: Online PhD

How chaos drives the evolution of the universe and life

The Cosmic Blueprint is the very first book of Paul Davies I ever read nearly a quarter of a century ago, and I’ve read many others since. I heard him being interviewed about it on a car trip from Melbourne to Mulwala (on the Victorian, New South Wales border) and that was the first time I’d heard of him. The book was published in 1987, so it was probably 1988.

Davies received the Templeton Foundation Prize in 1995, though not the wrath of Dawkins for accepting it. He’s also received the 2002 Michael Faraday Prize from the Royal Society and the 2001 Kelvin Medal and Prize from the UK Institute of Physics. He was resident in Australia for a couple of decades but now resides in the US where he’s an astro-biologist at the University of Arizona.

In America, Davies has been accused of being a ‘creationist in disguise’ by people whose ignorance is only out-weighed by their narrow-mindedness (they think there are atheists and there are creationists with nothing in between). The 2004 edition of this book is published by the Templeton Foundation and the first word in the opening chapter is ‘God’ as part of a quote by Ilya Prigogine, who features prominently in the book. But anyone who thinks this is a thesis for Intelligent Design will be disappointed; it’s anything but. In fact, one of the book’s great virtues is its attempt to explain complexity in the universe and evolution as a natural occurrence and not a Divine one.

I’ve long believed that Davies writes about science and philosophy better than anyone else, not least because he seems to be equally erudite in the disciplines of physics, cosmology, biology and philosophy. He’s not a member of the ‘strong atheist’ brigade, which puts him offside with many philosophers and commentators, but his argument against ID in The Goldilocks Enigma (2006) was so compelling that Stephen Law borrowed it for himself.

I remember The Cosmic Blueprint primarily as introducing me to chaos theory; it was the new kid on the block in popular consciousness with fractals and Mandelbroit’s set just becoming conspicuous in pop culture. Reading it now, I’m surprised at how much better it is than I remember it, but that’s partly due to what I’ve learnt in between. A lot of it would have gone over my head, which is not to say it still doesn’t, but less so than before.

More than any other writer on science, Davies demonstrates how much we don’t know and he doesn’t shy away from awkward questions. In particular, he is critical of reductionism as the only method of explanation, especially when it explains things away rather than explicating them; consciousness and life’s emergence being good examples.

I like Davies because his ideas reflect some of my own ruminations, for example that natural selection and mutations can’t possibly explain the whole story of evolution. We think we are on the edge of knowing everything, yet future generations will look back and marvel at our ignorance just as we do with our forebears.

There is an overriding thesis in The Cosmic Blueprint that is obvious once it’s formulated yet is largely ignored in popular writing. It’s fundamentally that there are two arrows of time: one being the well known 2^nd law of thermodynamics or entropy; and the other being equally obvious but less understood as the increase in complexity at all levels in the universe from the formation of galaxies, stars and planets to the evolution of life on Earth, and possibly elsewhere. Both of which demonstrate irreversibility as a key attribute.  And whilst many see them as contradictory and therefore evidence of Divine intervention, Davies sees them as complementary and part of the universe’s overall evolvement.

Davies explains how complexity and self-organisation can occur when dynamic systems are pushed beyond equilibrium with an open source of energy. Entropy, on the other hand, is a natural consequence of systems in equilibrium.

In the early pages, Davies explains chaotic behaviour with a simple-to-follow example that’s purely mathematical. In particular, he demonstrates how the system is completely deterministic yet totally unpredictable because the initial conditions are mathematically impossible to define. This occurs in nature all the time, like coin tosses, so that the outcome is totally random but only because the initial conditions are impossible to determine, not because the coin follows non-deterministic laws. This is a subtle but significant distinction.

A commonly cited example is cellular automata that can be generated by a computer programme. Stephen Wolfram of the Institute for Advanced Study, Princeton, has done a detailed study of one-dimensional automata that could give an insight into evolution. Davies quotes Wolfram:

“…the cellular automaton evolution concentrates the probabilities for particular configurations, thereby reducing entropy. This phenomenon allows for the possibility of self-organization by enhancing the probabilities of organized configurations and suppressing disorganized configurations.”

Wolfram is cited by Gregory Chaitin, in Thinking about Godel and Turing, as speculating that the universe may be pseudo-random and chaos theory provides an innate mechanism: deterministic laws that can’t be predicted. However, it seems that the universe’s innate chaotic laws provide opportunities for a diverse range of evolutionary possibilities, and the sheer magnitude of the universe in space and time, along with a propensity for self-organisation, in direct opposition to entropy, may be enough to ensure intelligent life as an outcome. The truth is that we don’t know. (Btw, Davies wrote the forward to Chaitin’s book.)

Davies calls this position ‘predestiny’ but he’s quick to qualify it thus: ‘Predestiny is a way of thinking about the world. It is not a scientific theory. It receives support, however, from those experiments that show how complexity and organization arise spontaneously and naturally under a wide range of conditions.’

This view is mirrored in the anthropic principle, which Davies also briefly discusses, but there are two version, as expounded by Frank Tipler and John Barrow in The Anthropic Cosmological Principle: the weak anthropic principle and the strong anthropic principle; and ‘predestiny’ is effectively the strong anthropic principle.

Roughly twenty years later, in The Goldilocks Enigma, Davies elaborates on this philosophical viewpoint when he argues for the ‘self-explaining universe’ amongst a critique of all the current ‘flavours’ of universe explanations: ‘I have suggested that only self-consistent loops capable of understanding themselves can create themselves, so that only universes with (at least the potential for) life and mind really exist.’ This is effectively a description of John Wheeler’s speculative cosmic quantum loop explanation of the universe’s existence – it exists because we’re in it. Davies argues that such a universe is ‘self-activating’ to avoid religious connotations: ‘…perhaps existence isn’t something that gets bestowed from outside…’

Teleological is a word that most scientists avoid, but Davies points out that the development of every organism is teleological because it follows a ‘blueprint’ or ‘plan’ entailed in its DNA. How this occurs is not entirely understood, but Davies makes an analogy with software which is apposite, as DNA provides coded instructions that ultimately result in fully developed organisms like us. He explores a concept called ‘downward causation’ whereby information can actually ‘cause’ materialistic events and software in computers provide the best example. In fact, as Davies hypothesises, one could imagine a software programme that makes physical changes to the computer that it’s operating on. Perhaps this is how the ‘mind’ works, which is similar to Douglas Hofstadter’s idea of a ‘strange loop’ that he introduced in Godel Escher Bach (which I reviewed in Feb. 2009) and later explored in another tome called I am a Strange Loop (which I haven’t read).

Davies introduces the concept of ‘downward causation’ in his discussion on quantum mechanics because it’s the measurement or observation that crystallises the quantum phenomenon into the real world. According to Davies, Wheeler speculated that ‘downward causation’ in quantum mechanics is ‘backwards in time’ and suggested a ‘delayed-choice’ thought experiment. To quote Davies: ‘The experiment has recently been conducted, and accords entirely with Wheeler’s expectations. It must be understood, however, that no actual communication with the past is involved.’

It’s impossible to discuss every aspect of this book, covering as it does: chaos theory, fractals, cosmological evolution, biological evolution, quantum mechanics and mind and matter.

Towards the end, Davies reveals some of his own philosophical prejudices, which, unsurprisingly, are mirrored in The Goldilocks Enigma twenty years on.

The very fact that the universe is creative, and that the laws have permitted complex structures to emerge and develop to the point of consciousness – in other words, that the universe has organized its own self-awareness – is for me powerful evidence that there is ‘something going on’ behind it all.

This last phrase elicits the ‘design’ word, many years before Intelligent Design was introduced as a ‘wedge’ tactic for creationists, but Davies has been an outspoken critic of both creationism and ID, as I explained above. Davies strongly believes the universe has a purpose and the evidence supports that point of view. But it’s a philosophical point of view, not a scientific one.

This leads to the logical question: is the universe teleological? I think chaos theory provides an answer. In the same way that chaotic phenomena, which includes all complex dynamics in the universe (like evolution) are deterministic yet unpredictable, the universe could be purposeful yet not teleological. In other words, the purpose is not predetermined but the universe’s dynamics allow purpose to evolve.

How does language work?

This topic became a source of disagreement on Rust Belt Philosophy a couple of weeks ago, so I would like to point out that this essay was written prior to that discourse.

In fact, the title is the ‘Question of the Month’ in the last issue of Philosophy Now (Issue 88, Jan/Feb 2012). That issue contained selected entries of the previous Question of the Month, which was ‘How can I be happy?’ I (amongst 7 others) won a book for my entry (On Evil by Adam Morton). The editors invited me to submit for the next question of the month, hence this post.

I know of at least one professor of linguistics who reads this blog, so he may wish to challenge my thesis or theses.

Human language is unique to humanity in many respects. For a start, we think in a language and secondly it’s a cultural attribute that is effectively downloaded, independently of our genes, from generation to generation. Language in other species is ‘hardwired’ or genetically determined, like nest-building is in birds, and it’s hard to imagine that any other species thinks in a language the way we do. So what do they think in? I suggest that dreams provide the answer because we dream in imagery and emotion, and I suspect most animals think emotionally. There are animals that use logic, which we witness when they use ‘tools’, including other primates and some birds like crows, but they can only express that logic through demonstration rather than through language.

For each and every one of us there is an external and internal world and the most familiar bridge between those worlds is language. Herein lies the key because language reflects the modality of the world in form as well as function. The smallest ‘atomic’ component of language is individual words, but it’s only in the context of a sentence that they gain leverage in meaning, because the entire sentence provides a meaning that the individual words cannot. Sentences are combined to provide arguments, stories, explanations, just like I’m doing now. But the external world follows this same model because it is made up of ‘atoms’ at various levels that combine into entities, like, for example, individual cells forming a fully developed human being. The human brain can ‘nest’ concepts within concepts and language is the most familiar manifestation of this unique ability. Furthermore, language allows us to not only express concepts within concepts, but to actually think them, and these concepts within concepts are analogous to the worlds within worlds that we investigate and explicate.

But human language has another unique feature that has allowed us to leave all other species in our cognitive wake. Language allows us to carry memories across generations - even before scripts were invented - and this has led to the development of cultures and civilizations that grow with accumulated knowledge. Ultimately, language allows us to think and conceptualise as well as record, and that is what makes humanity unique.


Addendum: Speaking of Philosophy Now, here is someone who claims that chimpanzees can be taught language.

Gay marriage

Three posts in 2 days is unheard of for me, and all politically motivated. But I couldn’t resist this, which is a post by Sally Whitwell, which she’s borrowed from You-Tube.

Gay marriage is inevitable because all the arguments against it crash on the rock of equality. This is between 2 people, not between them and governments or them and the church. When gay marriage is finally allowed, it will have an enormous effect on those who support it and absolutely no effect on those who oppose it.

Addendum: The above link is no longer available, but this POST is more informative about the debate.

Technology changes but human nature doesn’t

For science fiction writers and want-to-be science fiction writers, like myself, technology is overtaking our imaginations. Last Wednesday, the issue of drones and robotic warfare was raised and discussed, on ABC’s Lateline programme. I’ve posted on this issue twice before, over a year ago, in Jan. 2011 and Nov. 2010, but it’s more advanced than I thought.

 Unmanned ‘predator’ aircraft are becoming the weapon of choice for war strategists in the US and we can expect other countries to follow. The ability to target and kill your enemy remotely (from the other side of the world) is becoming too seductive to resist. People are already talking about giving robots decision-making abilities to engage the enemy.

In the short term it will lead to a bigger gulf between techno-savvy (therefore wealthy) countries and poorer nations – absolutely guaranteed to boost anti-Western paranoia. In the long term it may lead to warfare between drones or attempts to conduct war in space to eliminate satellites that unmanned aircraft depend on for navigation.

Ballistic and cruise missiles were developed in the cold war because they allowed one to attack a country without setting foot in it. Drone aircraft allow the exact same scenario, which is why they are so popular with politicians and military strategists. The psychological and ethical consequences are being glossed over, but is bombing by stealth with no visible or targetable combatant any less a terrorist act than suicide bombing? I guess it depends which side you’re on.

History reveals that when one opponent has a technological advantage over their adversary, then the adversary adopts strategies that are considered unprincipled by their superior opponent.

Why Finland is the best (in education)

This is an interview with Pasi Sahlberg, Finnish Director of Education, who is currently visiting Australia and gives an insight into Finland’s unique and enviable position in education. For a start, teachers have the same status as doctors and are remunerated accordingly; secondly, there is no private school system; and thirdly, there is a culture of trust between teachers, administrators, students and community.

What Sahlberg reveals is that the obsession with competition between schools and with standardised testing, that drives education in other OEDC countries, are the antithesis of Finland’s education policies. The message is clear and obvious, yet I don’t expect anyone outside of Finland to heed it.

There is another world

I’ve been a contributor to Plan for decades now, though my contributions are modest. They send me a magazine from time to time, which I usually ignore, but this time they had a cover story titled: Bringing an end to child marriage.

When I look at all the squabbles we have in domestic politics, not just here, in Australia, but in other Western countries, this issue helps to put things in perspective. In the past week, the Australian government, despite having arguably the most resilient economy in the Western world, did it’s best to self-destruct by publicly brawling over a leadership challenge that had obviously been festering for years. In America, politicians argue over the fundamentals of health care as if it distinguishes a free economy from a State-run monopoly, even though much of the rest of the so-called Free World moved on from that debate decades ago.

There is another world that most of us don’t see or hear about or care about, but it comprises the bulk of the Earth’s population. In this world, our political debates seem downright petty, considering that most of us have a fridge with food in it, running water, electricity and heating, as well as a roof over our head.

The education of women is something we take for granted in the West, yet, in many cultures, young girls are still treated as bargaining chips in a household economy. If we weren’t so egocentric and culturally insulated from the rest of the world we might see how important this issue is and that we are in a position to help.

 I strongly believe that women are the key to the world’s future. I would like to see more aid given to women in developing countries directly because I think they are more likely to use it for their children’s benefit, whether it be in schooling or nutrition. The all-pervading patriarchal society is past its use-by date, not just in the West, but globally. Until it is universally recognised that women deserve exactly the same rights as men, then the disparity in wealth, prosperity and health will continue between the West and the rest.

This report depicts the clash between Western feminist values and traditional culture, where being born a woman is perceived as a liability by both sexes. This attitude is pervasive in much of the world – the Western perspective is not only recent but the exception.

Economics of the future

In March 2010 I wrote a post titled, The world badly needs a radical idea.  Well, last Thursday I heard an interview with Guy Standing, Professor of Economic Security at the University of Bath, UK, who does have at least one radical idea as well as a perspective that coincides with mine.

In particular, he challenges the pervasive definition we give to ‘work’. Essentially, that ‘work’ must contribute to the economy. In other words, in the West, we have a distorted view that work only counts if we earn money from it. He gives the example: if a man hires a housekeeper, whom he pays, she is part of the economy, but if he marries her she effectively disappears, economically. I’ve long argued that the most important job you will ever do, you will never be paid for, which is raising children.

To give another very personal example, I make no money from writing fiction, therefore any time I spend writing fiction is a self-indulgence. On the other hand, if I did make money from writing fiction, then any time I didn’t spend writing fiction would be considered a waste of time. By the way, I don’t consider writing as work, because, if I did, I probably wouldn’t do it or I wouldn’t be motivated to do it. Writing fiction is the hardest thing I’ve ever done and treating it as work would only make it harder.

Standing’s radical idea is that there should be a ‘minimum income’ as opposed to a minimum wage. Apparently, this has been introduced in some parts of Brasil and there is a programme to introduce it in India. In Brasil it was championed by a woman mayor who supported the programme if it was given to women. Standing claims that the most significant and measurable outcome is in the nutrition of babies and young children.

Now, many people will say that this is communism, but it’s not about overthrowing capitalism, it’s about redistribution of wealth, which has to be addressed if we are ever going to get through the 21^st Century without more devastating wars than we witnessed in the last century.

The core of the interview is about a new class, which he calls the ‘precariat’, who are the new disenfranchised in the modern world, partly a result of the concentration of wealth, created by those who still believe in the ‘trickledown fantasy’.

Is mathematics invented or discovered?

I've used this title before in Sep. 2007, even though it was really a discussion of George Lakoff's and Rafael E. Nunez's book, Where mathematics comes from. In fact, it was just my 6th post on this blog. This essay predates that post by 5 years (2002) and I found it by accident after someone returned a USB to me that I had lost. Though there is some repetition, this essay is written in the context of an overall epistemology, whilst the previous one is an argument against a specifically defined philosophical position. To avoid confusion, I will rename the Sep.07 post after the title of Lakoff's and Nunez's book.


I would argue that it is a mixture of both, in the same way that our scientific investigations are a combination of inventiveness and discovery. The difference is, that in science, the roles of creativity and discovery are more clearly delineated. We create theories, hypotheses and paradigms, and we perform experiments to observe results, and we also, sometimes, simply perform observations without a hypothesis and make discoveries, though this wouldn’t necessarily be considered scientific.

But there is a link between science and mathematics, because as our knowledge and investigations go deeper into uncovering nature’s secrets, we become more dependent on mathematics. In fact I would contend that the limit of our knowledge in science is determined by the limits of our mathematical abilities. It is only our ability to uncover complex and esoteric mathematical laws that has allowed us to uncover the most esoteric (some would say spooky) aspects of the natural universe. To the physicist there appears to be a link between mathematical laws and natural laws. Roger Penrose made the comment in a BBC programme, Lords of Time, to paraphrase him, that mathematics exists in nature. It is a sentiment that I would concur with. But to many philosophers, this link is an illusion of our own making.

Stanilas Debaene, in his book, The Number Sense, describes the cognitive aspect of our numeracy skills which can be found in pre-language infants as well as many animals. He argues a case, that numbers, the basic building blocks of all mathematics, are created in our minds, and that there is no such thing as natural numbers. The logical consequence of this argument is that if numbers are a product of the mind then so must be the whole edifice of mathematics. This is in agreement with both Russell and Wittgenstein, who are the most dominant figures in 20th Century philosophy. I have no problem with the notion that numbers exist only as a concept in the human mind, and that they even exist within the minds of some animals up to about 5 (if one reads Debaene’s book) though of course the animals aren’t aware that they have concepts – it’s just that they can count to a rudimentary level.

But mathematics, as we practice it, is not so much about numbers as the relationships that exist between numbers, which follow very precise rules and laws. In fact, the great beauty of algebra is that it strips mathematics of its numbers so that we can merely see the relationships. I have always maintained that mathematical rules are, by and large, not man made, and in fact are universal. From this perspective, Mathematics is a universal language, and it is the ideal tool for uncovering nature’s secrets because nature also obeys mathematical rules and laws. The modern philosopher argues that mathematics is merely logic, created by the human mind, albeit a very complex logic, from which we create models to approximate nature. This is a very persuasive argument, but do we bend mathematics to approximate nature, or is mathematics an inherent aspect of nature that allows an intelligence like ours to comprehend it?

I would argue that relationships like π and Pythagoras’s triangle, and the differential and integral calculus are discovered, not invented. We simply invent the symbols and the means to present them in a comprehensible form for our minds. If you have a problem and you cannot find the solution, does that mean the solution does not exist? Does the solution only exist when someone has unravelled it, like Fermat’s theorem? This is a bit like Schrodinger’s cat; it’s only dead or alive when someone has made an observation. So mathematical theorems and laws only exist when a cognitive mind somewhere reveals them. But do they also exist in nature like Bernoulli’s spiral found in the structure of a shell or a spider web, or Einstein’s equations describing the curvature of space? The modern philosopher would say Einstein’s equations are only an approximation, and he or she may be right, because nature has this habit of changing its laws depending on what scale we observe it at (see Addendum below), which leads paradoxically to the apparent incompatibility of Einstein’s equations with quantum mechanics. This is not unlike the mathematical conundrum of a circle, ellipse, parabola and hyperbola describing different aspects of a curve.

So what we have is this connection between the human mind and the natural world bridged by mathematics. Is mathematics an invention of the mind, a phenomenon of the natural world, or a confluence of both? I would argue that it is the last. Mathematics allows us to render nature’s laws in a coherent and accurate structure – it has the same infinite flexibility while maintaining a rigid consistency. This reads like a contradiction until you take into account two things. One is that nature is comprised of worlds within worlds, each one self-consistent but producing different entities at different levels. The best example is the biological cells that comprise the human body compared to the molecules that makes up the cells, and then in comparison with an individual human, the innumerable social entities that a number of humans can create. Secondly, that this level of complexity appears to be never ending so that our discoveries have infinite potential. This is despite the fact that in every age of technological discovery and invention, we have always believed that we almost know everything that there is to know. The current age is no different in this respect.

The philosophical viewpoint that I prescribe to does not require a belief in the Platonic realm. From my point of view, I consider it to be more Pythagorean than Platonic, because my understanding is that Pythagoras saw mathematics in nature in much the same way that Penrose expresses it. I assume this view, even though we have little direct knowledge of Pythagoras’s teachings. Plato, on the other hand, prescribed an idealised world of forms. He believed that because we’ve had previous incarnations (an idea he picked up from Pythagoras, who was a religious teacher first, mathematician second), we come into this world with preconceived ideas, which are his ‘forms’. These ‘forms’ are an ideal perfect semblance from ‘heaven’, as opposed to the less perfect real objects in nature. This has led to the idea that anyone who prescribes to the notion that mathematical laws and relationships are discovered, must therefore believe in a Platonic realm where they already exist.

This aligns with the idea of God as mathematician. Herbet Westron Turnbull in his short tome, Great Mathematicians, rather poetically states it thus: ‘Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God.’ But mathematics does not have to be a religious connection for its laws to pre-exist. To me, they simply lie dormant awaiting an intelligence like ours to uncover them. The natural world already obeys them in ways that we are finding out, and no doubt, in ways that we are yet to comprehend.

Part of the whole philosophical mystery of our being and the whole extraordinary journey to our arrival on this planet at this time, is contained in this one idea. The universe, whether by accident or anthropic predestination, contains the ability to comprehend itself, and without mathematics that comprehension would be severely limited. Indeed, to return to my earliest point, which converges on Kant and Eco’s treatise in particular, Kant and the Platypus, our ability to comprehend the universe with any degree of certainty, is entirely dependent on our ability to uncover the secrets and details of mathematics. And consequently the limits of our knowledge of the natural world is largely dependent on the limits of our mathematical knowledge.


Addendum 1: This post has become popular, so I'm tempted to augment it, plus I've written a number of posts on the topic since. When studying physics, one is struck by the significance of scale in the emergence of nature's laws. In other words, scale determines what forces dominate and to what extent. This demonstrable fact, all by itself, signifies how mathematics is intrinsically bound into reality. Without a knowledge of mathematics (often at its most complex) we wouldn't know this, and without mathematics being bound into the Universe at a fundamental level, the significance of scale would not be a factor.

Addendum 2: Given the context of Addendum 1, this is a much later post that might be relevant: The Universe's natural units.

The anthropomorphism of computers

There are 2 commonly held myths associated with AI (Artificial Intelligence) that are being propagated through popular science, whether intentionally or not: that computers will inevitably become sentient and that brains work similarly to computers.

The first of these I dealt with indirectly in a post last month, when I reviewed Colin McGinn’s book, The Mysterious Flame. McGinn points out that there is no correlation between intelligence and sentience, as sentience evolved early. There is a strongly held belief, amongst many scientists and philosophers, that AI will eventually overtake human intelligence and at some point become sentient. Even if the first statement is true (depending on how one defines intelligence) the second part has no evidential basis. If computers were going to become sentient on the basis that they ‘think’ then they would already be sentient. Computers don’t really think, by the way, it’s just a metaphor. The important point (as McGinn points out) is that there is no evidence in the biological world that sentience increases with intelligence, so there is no reason to believe that it will even occur with computers if it hasn’t already.

This is not to say that AI or Von Neumann machines could not be Darwinianly successful, but it still wouldn’t make them necessarily sentient. After all, plants are hugely Darwinianly successful but are not sentient.

In the last issue of Philosophy Now (Issue 87, November/December 2011), the theme was ‘Brains & Minds’ and it’s probably the best one I’ve read since I subscribed to it. Namit Arora (based in San Francisco and creator of Shunya) wrote a very good article, titled The Minds of Machines, where he tackles this issue by referencing Heidegger, though I won’t dwell on that aspect of it. Most relevant to this topic, he quotes Hubert L. Dreyfuss and Stuart E. Dreyfus from Making a Mind vs Modelling the Brain:

“If [a simulated neural network] is to learn from its own ‘experiences’ to make associations that are human-like rather than be taught to make associations which have been specified by its trainer, it must also share our sense of appropriateness or outputs, and this means it must share our needs, and emotions, and have a human-like body with the same physical movements, abilities and possible injuries.”

In other words, we would need to build a comprehensive model of a human being complete with its emotional, cognitive and sensory abilities. In various ways this is what we attempt to do. We anthropomorphise its capabilities and then we interpret them anthropomorphically. No where is this more apparent than with computer-generated art.

Last week’s issue of New Scientist (14 January 2012) discusses in detail the success that computers have had with ‘creating’ art; in particular, The Painting Fool, the brain child of computer scientist, Simon Colton.

If we deliberately build computers and software systems to mimic human activities and abilities, we should not be surprised that they sometimes pass the Turing test with flying colours. According to Catherine de Lange, who wrote the article in New Scientist, the artistic Turing test has well and truly been passed both in visual art and music.

One must remember that visual art started by us copying nature (refer my post on The dawn of the human mind, Oct. 2011) so we now have robots copying us and quite successfully. The Painting Fool does create its own art, apparently, but it takes its ‘inspiration’ (i.e. cues) from social networks, like Facebook for example.

The most significant point of all this is that computers can create art but they are emotionally blind to its consequences. No one mentioned this point in the New Scientist article.

Below is a letter I wrote to New Scientist. It’s rather succinct as they have a 250 word limit.


As computers become better at simulating human cognition there is an increasing tendency to believe that brains and computers work in the same way, but they don’t.

Art is one of the things that separates us from other species because we can project our imaginations externally, be it visually, musically or in stories. Imagination is the ability to think about something that’s not in the here and now – what philosophers call intentionality – it can be in the past or the future, or another place, or it can be completely fictional. Computers can’t do this. Computers have something akin to semantic memory but nothing similar to episodic memory, which requires imagination.

Art triggers a response from us because it has an emotional content that we respond to. With computer art we respond to an emotional content that the computer never feels. So any artistic merit is what we put into it, because we anthropomorphise the computer’s creation.

Artistic creation does occur largely in our subconscious, but there is one state where we all experience this directly and that is in dreams. Computers don’t dream so the analogy breaks down.

So computers produce art with no emotional input and we appraise it based on our own emotional response. Computers may be able to create art but they can’t appreciate it, which is why if feels so wrong.

Postscript: People forget that it requires imagination to appreciate art as well as to create it. Computers can do one without the other, which is anomalous, even absurd.

The Battle for ideals is the battle for the future

The opposition to gay marriage, especially as espoused by the Catholic Church, and Pope Benedict in particular, is a symptom of a deeper problem: ignorance over enlightenment; prejudice over reason.

There are people who would love to freeze our societies, freeze politics and freeze cultural norms. This is why they are called conservatives. Ironically, it’s conservatives, or their policies, that are creating more change than anything else. A belief in infinite economic growth, the limited role of women in society and the denial of human-affected climate change will create more change in the 21st Century than anyone wants to see, and none for the better. An overpopulated planet depleted of resources, with an increase in the global wealth gap, rising sea levels, increased frequency of droughts and floods and the depletion of species are all being driven by conservative political policies.

The one symptom of human nature that holds all these positions together is denial, including the Pope’s message. They also, in various ways, defy what scientific endeavours are trying to tell us.

In Australia, the debate over climate change has become one of public opinion versus science. There seems to be a belief that we can vote for or against climate change as if it’s a political position rather than a natural phenomenon. The arguments against climate change in this country are that the scientists are all involved in a conspiracy, so they can hold onto their jobs, and all we have to do is tell them to produce the data we want to see and climate change will go away.

Yes, a touch sarcastic, but that’s the prevailing attitude. At a rally held on Parliament House lawns last year, someone with a megaphone stood up and told the CSIRO (Australia’s most esteemed scientific organization) to “Stop writing crap” on climate change, as if the person making the exhortation would actually be able to tell the difference.

If science could be overturned by popular opinion, Einstein’s theories of relativity would be consigned to the rubbish bin, quantum mechanics would be pure fantasy and evolution would never have happened. It would also mean that there would be no transistors or computers or mobile phones (without quantum mechanics) or GPS (without relativity) and virus mutations would be inexplicable (without evolution).

Many of the things that modern society take for granted are dependent on science that most people don’t understand, even vaguely. Yet when scientists start making predictions that people don’t want to hear, they are suddenly ‘writing crap’. People think I’m being alarmist, yet in 2010 New Scientist listed 9 ecological criteria that affect the future of our planet, only one of which has been curtailed, the ozone hole (refer my post Mar. 2010).

Unfortunately, the only people who even know about this are nerds like me, and, as for politicians, they don’t want to know. Politicians in democratic societies can’t afford to tell anyone bad news because they get dumped at the next election. Consequently, as we’ve recently witnessed in Europe, politicians only deliver bad news after everyone has already been affected by it, and they can no longer pretend it isn’t happening. The same thing will happen with climate change. They’ve already put any action off till 2020: The Durban Agreement, reported in New Scientist (17 December 2011, pp.8-9); because they know no one will notice anything between now and then, even though the scientists are telling us we have to take action now.

What has climate change to do with the Pope’s anti-gay rhetoric? They are both examples of polarised politics, a symptom of our age: the political tension created by trying to hang onto the past and resist the future. There are those who can see the future and know we need to adapt to it and there are those who live in the past and think the future can be avoided by freezing our culture.

According to the Pope: "This is not a simple social convention, but rather the fundamental cell of every society. Consequently, policies which undermine the family threaten human dignity and the future of humanity itself,"

There is politics within the Catholic Church and not everyone who is part of the Church shares the Pope’s views, but it’s only conservative members who are promoted through its hierarchy, as the news item behind the link demonstrates.

According to the item: ‘The Roman Catholic Church, which has some 1.3 billion members worldwide, teaches that while homosexual tendencies are not sinful, homosexual acts are, and that children should grow up in a traditional family with a mother and a father.’

And herein lies the legerdemain: the Catholic Church is not against gays per se but only against gay marriage. However, this argument doesn’t stick. As Australian philosopher, Raymond Gaita, pointed out in a Q&A panel last year, the aversion to gay marriage is the direct consequence of an unstated aversion to homosexual acts. They can’t say they are against homosexuality but they can say they are against gay marriage. And science has played a major role in bringing gays and lesbians out of the closet. We no longer consider homosexuality to be a psychiatric illness, as people did 50 years ago, and it’s no longer considered a criminal offence. Sexual orientation is something you are born with – it’s not a lifestyle choice - but anti-gay advocates will tell you otherwise because they can’t understand why everyone else isn’t just like them.

The Catholic Church is more than a religious institution, it’s a global political entity. It still argues for the lack of birth control and thinks oral contraception was one of the worst inventions of all time. Not just because it undermines one of its more perverse inculcations, but because it’s what gave impetus to modern feminism and gave women the sexual independence and freedom that had previously been the sole providence of men.

And this too has an effect on our future, because it’s only through the emancipation and education of women, worldwide, that we will ever achieve zero population growth. This is arguably the most important issue of our century, and the most significant for our planet’s future.

There is an ideological battle going on in the West between conservative and liberal political forces, yet nature will dictate the outcome because nature has no political affiliations and nature has no preference for the human race. Science studies nature and is our best predictor of future events. But politicians, and the public at large, have little interest in science – it’s only our economic fate that concerns us. Such short-sightedness may well be our species’ undoing.