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.

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.