The Riemann Hypothesis; the most famous unsolved problem in mathematics

I’ve read 3 books on this topic: The Music of the Primes by Marcus du Sautoy, Prime Obsession by John Derbyshire and Stalking the Riemann Hypothesis by Dan Rockmore (and I originally read them in that order). They are all worthy of recommendation, but only John Derbyshire makes a truly valiant attempt to explain the mathematics behind the ‘Hypothesis’ (for laypeople) so it’s his book that I studied most closely.

Now it’s impossible for me to provide an explanation for 2 reasons: one, I’m not mathematically equipped to do it; and two, this is a blog and not a book. So my intention is to try and instill some of the wonder that Riemann’s extraordinary gravity-defying intuitive leap passes onto those who can faintly grasp its mathematical ramifications (like myself).

In 1859 (the same year that Darwin published The Origin of the Species), a young Bernhard Riemann (aged 32) presented a paper to the Berlin Academy as part of his acceptance as a ‘corresponding member’, titled “On the Number of Prime Numbers Less Than a Given Quantity”. The paper contains a formula that provides a definitive number called π (not to be confused with pi, the well-known transcendental number). In fact, I noticed that Derbyshire uses π(x) as a function in an attempt to make a distinction. As Derbyshire points out, it’s a demonstration of the limitations arising from the use of the Greek alphabet to provide mathematical symbols – they double-up. So π(x) is the number of primes to be found below any positive Real number. Real numbers include rational numbers, irrationals and transcendental numbers, as well as integers. The formula is complex and its explication requires a convoluted journey into the realm of complex algebra, logarithms and calculus.

Eratosthenes was one of the librarians at the famous Alexandria Library, around 230 BC and roughly 70 years after Euclid. He famously measured the circumference of the Earth to within 2% of its current figure (see Wikipedia) using the sun and some basic geometry. But he also came up with the first recorded method for finding primes known as Eratosthenes’ Sieve. It’s so simple that it’s obvious once explained: leaving the number 1, take the first natural number (or integer) which is 2, then delete all numbers that are multiples of 2, which are all the even numbers. Then take the next number, 3, and delete all its multiples. The next number left standing is 5, and one just repeats the process over whatever group of numbers one is examining (like 100, for example) until you are left with all the primes less than 100. With truly gigantic numbers there are other methods, especially now we have computers that can grind out algorithms, but Eratosthenes demonstrates that scholars were fascinated by primes even in antiquity.

Euclid famously came up with a simple proof to show that there are an infinite number of primes, which, on the surface, seems a remarkable feat, considering it’s impossible to count to infinity. But it’s so simple that Stephen Fry was even able to explain it on his TV programme, QI. Assume you have found the biggest prime, then take all the primes up to and including that prime and multiply them all together. Then add 1. Obviously none of the primes you know can be factors of this number as they would all give a remainder of 1. Therefore the number is either a prime or can be factored by a prime that is higher than the ones you already know. Either way, there will always be a higher prime, no matter which one you select, so there must be an infinite number of primes.

The thing about primes, that has fascinated mathematicians for eons, is that there appears to be no rhyme or reason to their distribution, except they get thinner - further apart as one goes to higher numbers. But even this is not strictly correct because there appears to be an infinite number of twin primes, 2 primes separated by a non-prime (which must be even for obvious reasons).

Back to Riemann’s paper and its 150 year old legacy. Entailed in his formula is a formulation of the Zeta function. Richard Elwes provides a relatively succinct exposition in his encyclopaedic MATHS 1001, and I’m not even going to attempt to write it down here.  The point is that the Zeta function gives complex roots to infinity. Most people know what a quadratic root is from high school maths. If you take the graph of a parabola like y = ax² + bx + c, then it crosses the x axis where y = 0. It can cross the x axis in 2 places, or just touch it in 1 place or not cross it at all. The values of x that gives us a 0 value of y are called the roots of the equation. As a polynomial goes up in degree so does its number of roots. So a quadratic equation gives us 2 roots maximum but a polynomial with degree 3 (includes x³) will give us 3 roots and so on. Going back to the parabola, in the case where we don’t get any roots at all, it’s because we are trying to find square roots of negative numbers. However, if we use i (√-1), we get complex roots in the form of a + ib. (For a basic explanation see my Apr.12 post on imaginary numbers.) A trigonometric equation like sinθ can give us an infinite number of zeros and so can the Zeta function.

If you didn’t follow that, don’t worry, the important point is that Riemann’s Hypothesis says that all the complex zeros of the Zeta function (to infinity) have Real part ½. So they are all of the form ½ + ib. Riemann wasn’t able to provide a proof for this and neither have the best mathematical minds since. The critical point is that if his Hypothesis is correct then so is his formula for finding an exact number of primes to any given number.

In the 150 years since, Riemann’s Hypothesis has found its way into many fields of mathematics, including Hermitian matrices, which has implications for quantum mechanics. The Zeta function is a formidable mathematical beast to the uninitiated, and its relationship to the distribution of the primes was first intimated by Euler. Riemann’s genius was to introduce complex numbers, then make the convoluted mental journey to demonstrate their pivotal role in providing an exact result. Even then, his fundamental conjecture was effectively based on a hunch. At the time he presented his paper, he had only calculated the first 3 non-trivial zeros (non-trivial means complex in this context) and computers have calculated them in the trillions since, yet we still have no proof. It’s known that they become chaotic at extremely high numbers (beyond the number of atoms in the universe) so it’s by no means certain that Reimann’s hypothesis is correct.

It would be a huge disappointment to most mathematicians if either a proof was found to falsify it or an exception was found through brute computation. Riemann gave us a formula that gives us an accurate count of the primes (Derbyshire gives a worked example up to 1 million) that’s dependent on the Hypothesis being correct to specified values. It’s hard to imagine that this formula suddenly fails at some extremely high number that’s currently beyond our ken, yet it can’t be ruled out.

Marcus du Sautoy, in The Music of the Primes, contemplates the Riemann hypothesis in the context of Godel’s Incompleteness Theorem, which is germane to the entire edifice of mathematics. The primes have a history of providing hard-to-prove conjectures. Along with Riemann’s hypothesis, there is the twin prime conjecture I mentioned earlier and Goldbach’s conjecture, which states that every even number greater than 2 is the sum of 2 primes. These conjectures are also practical demonstrations of Turing’s halting problem concerning computers. If they are correct, a computer algorithm set to finding them may never stop, yet we can’t determine in advance whether it will or not, otherwise we’d know in advance if it was true or not.

As du Sautoy points out, a corollary to Godel’s theorem is that there are limits to the proofs from any axioms we know at any time. In essence, there may be mathematical truths that the axioms cannot cover. The solution is to expand the axioms. In other words, we need to expand the foundations of our mathematics to extend our knowledge at its stratospheric limits. Du Sautoy speculates that the Riemann Hypothesis, along with these other examples, may be Godel’s Incompleteness Theorem in action.

Exploring the Reimann Hypothesis, even at the rudimentary level that I can manage, reinforces my philosophical Platonist view of mathematics. These truths exist independently of our investigations. There are an infinity of these Zeta zeros (we know that much) the same as there are an infinity of primes, which means there will always exist mathematical entities that we can’t possibly know. But aside from that obvious fact, the relationship that exists between apparently obscure objects like Zeta zeros and the distribution of prime numbers is a wonder. Godel’s Theorem implies that no matter how much we learn, there will always be mathematical wonders beyond our ken.

Addendum: This is a reasonably easy-to-follow description of Riemann's famous Zeta function, plus lots more.

Sex, Lies and Julian Assange, according to the ABC

With Assange’s status again in the spotlight, and the British government threatening to revoke Ecuador’s diplomatic asylum status, using force if necessary, which would be unprecedented in the modern world, it is worth looking at what all the fuss is about.

Almost a month ago, ABC’s 4 Corners aired its own investigations of the allegations against Assange initiated in Sweden. What the programme demonstrates is just how farcical this entire episode is.

Considering he was allowed to leave Sweden by Sweden’s public prosecutor, you would have to wonder, what changed? Is it a coincidence that Sweden’s change of mind - complete reversal in fact - came about when Assange elevated his whistle-blowing campaign against America?

Going by the rhetoric coming out of London, it’s fairly obvious, no matter what decision Ecuador comes to, Assange will be extradited to Sweden, and then we will find out if America will finally reveal its hand.

Why is there something rather than nothing?

Jim Holt has written an entire book on this subject, titled Why Does the World Exist? An Existential Detective Story. Holt is a philosopher and frequent contributor to The New Yorker, the New York Times and the London Review of Books, according to the blurb on the inner title page. He’s also very knowledgeable in mathematics and physics, and has the intellectual credentials to gain access to some of the world’s most eminent thinkers, like David Deutsch, Richard Swinburne, Steven Weinberg, Roger Penrose and the late John Updike, amongst others. I’m stating the obvious when I say that he is both cleverer and better read than me.

The above-referenced, often-quoted existential question is generally attributed to Gottfried Leibniz, in the early 18^th Century and towards the end of his life, in his treatise on the “Principle of Sufficient Reason”, which, according to Holt, ‘…says, in essence, that there is an explanation for every fact, an answer to every question.’ Given the time in which he lived, it’s not surprising that Leibniz’s answer was ‘God’.  Whilst Leibniz acknowledged the physical world is contingent, God, on the other hand, is a ‘necessary being’.

For some people (like Richard Swinburne), this is still the only relevant and pertinent answer, but considering Holt makes this point on page 21 of a 280 page book, it’s obviously an historical starting point and not a conclusion. He goes on to discuss Hume’s and Kant’s responses but I’ll digress. In Feb. 2011, I wrote a post on metaphysics, where I point out that there is no reason for God to exist if we didn’t exist, so I think the logic is back to front. As I’ve argued elsewhere (March 2012), the argument for a God existing independently of humanity is a non sequitur. This is not something I’ll dwell on – I’m just putting the argument for God into perspective and don’t intend to reference it again.

Sorry, I’ll take that back. In Nov 2011, I got into an argument with Emanuel Rutten on his blog, after he claimed that he had proven that God ‘necessarily exists’ using modal logic. Interestingly, Holt, who understands modal logic better than me, raises this same issue. Holt references Alvin Platinga’s argument, which he describes as ‘dauntingly technical’. In a nutshell: because of God’s ‘maximal greatness’, if one concedes he can exist in one possible world, he must necessarily exist in all possible worlds because ‘maximal greatness’ must exist in all possible worlds. Apparently, this was the basis of Godel’s argument (by logic) for the existence of God. But Holt contends that the argument can just as easily be reversed by claiming that there exists a possible world where ‘maximal greatness’ is absent’. And ‘if God is absent from any possible world, he is absent from all possible worlds…’ (italics in the original). Rutten, by the way, tried to have it both ways: a personal God necessarily exists, but a non-personal God must necessarily not exist. If you don’t believe me, check out the argument thread on his own blog which I link from my own post, Trying to define God (Nov. 2011).

Holt starts off with a brief history lesson, and just when you think: what else can he possibly say on the subject? he takes us on a globe-trotting journey, engaging some truly Olympian intellects. As the book progressed I found the topic more engaging and more thought-provoking. At the very least, Holt makes you think, as all good philosophy should. Holt acknowledges an influence and respect for Thomas Nagel, whom he didn’t speak with, but ‘…a philosopher I have always revered for his originality, depth and integrity.’

I found the most interesting person Holt interviewed to be David Deutsch, who is best known as an advocate for Hugh Everett’s ‘many worlds’ interpretation of quantum mechanics. Holt had expected a frosty response from Deutsch, based on a review he’d written on Deutsch’s book, The Fabric of Reality, for the Wall Street Journal where he’d used the famous description given to Lord Byron: “mad, bad and dangerous to know”. But he left Deutsch’s company with quite a different impression, where ‘…he had revealed a real sweetness of character and intellectual generosity.’

I didn’t know this, but Deutsch had extended Turing’s proof of a universal computer to a quantum version, whereby  ‘…in principle, it could simulate any physically possible environment. It was the ultimate “virtual reality” machine.’ In fact, Deutsch had presented his proof to Richard Feynman just before his death in 1988, who got up, as Deutsch was writing it on a blackboard, took the chalk off him and finished it off. Holt found out, from his conversation with Deutsch, that he didn’t believe we live in a ‘quantum computer simulation’.

Deutsch outlined his philosophy in The Fabric of Reality, according to Holt (I haven’t read it):

Life and thought, [Deutsch] declared, determine the very warp and woof of the quantum multiverse… knowledge-bearing structures – embodied in physical minds – arise from evolutionary processes that ensure they are nearly identical across different universes. From the perspective of the quantum multiverse as a whole, mind is a pervasive ordering principle, like a giant crystal.

When Holt asked Deutsch ‘Why is there a “fabric of reality” at all?’ he said “[it] could only be answered by finding a more encompassing fabric of which the physical multiverse was a part. But there is no ultimate answer.” He said “I would start with the principle of comprehensibility.”

He gave the example of a quasar in the universe and a model of the quasar in someone’s brain “…yet they embody the same mathematical relationships.” For Deutsch, it’s the comprehensibility of the universe (in particular, its mathematical comprehensibility) that provides a basis for the ‘fabric of reality’. I’ll return to this point later.

The most insightful aspect of Holt’s discourse with Deutsch was his differentiation between explanation by laws and explanation of specifics. For example, Newton’s theory of gravitation gave laws to explain what Kepler could only explain by specifics: the orbits of planets in the solar system. Likewise, Darwin and Wallace’s theory of natural selection gave a law for evolutionary speciation rather than an explanation for every individual species. Despite his affinity for ‘comprehensibility’, Deutsch also claimed: “No, none of the laws of physics can possibly answer the question of why the multiverse is there.”

It needs to be pointed out that Deutsch’s quantum multiverse is not the same as the multiverse propagated by an ‘eternally-inflating universe’. Apparently, Leonard Susskind has argued that “the two may really be the same thing”, but Steven Weinberg, in conversation with Holt, thinks they’re “completely perpendicular”.

Holt’s conversation with Penrose held few surprises for me. In particular, Penrose described his 3 worlds philosophy: the Platonic (mathematical) world, the physical world and the mental world. I’ve expounded on this in previous posts, including the one on metaphysics I mentioned earlier but also when I reviewed Mario Livio’s book, Is God a Mathematician? (March 2009).

Penrose argues that mathematics is part of our mental world (in fact, the most complex and advanced part) whilst our mental world is produced by the most advanced and complex part of the physical world (our brains). But Penrose is a mathematical Platonist, and conjectures that the universe is effectively a product of the Platonic world, which creates an existential circle when you contemplate all three. Holt found Penrose’s ideas too ‘mystical’ and suggests that he was perhaps more Pythagorean than Platonist. However, I couldn’t help but see a connection with Deutsch’s ‘comprehensibility’ philosophy. The mathematical model in the brain (of a quasar, for example) having the same ‘mathematical relationships’ as the quasar itself. Epistemologically, mathematics is the bridge between our comprehensibility and the machinations of the universe.

One thing that struck me right from the start of Holt’s book, yet he doesn’t address till the very end, is the fact that without consciousness there might as well be nothing. Nothingness is what happens when we die, and what existed before we were born. It’s consciousness that determines the difference between ‘something’ and ‘nothing’. Schrodinger, in What is Life? made the observation that consciousness exists in a continuous present. Possibly, it’s the only thing that does. After all, we know that photons don’t. As Raymond Tallis keeps reminding us, without consciousness, there is no past, present or future. It also means that without memory we would not experience consciousness. So some states of unconsciousness could simply mean that we are not creating any memories.

Another interesting personality in Holt’s engagements was Derek Parfit, who contemplated a hypothetical ‘selector’ to choose a universe. Both Holt and Parfit concluded, through pure logic, using ‘simplicity’ as the criterion, that there would be no selector and ‘lots of generic possibilities’ which would lead to a ‘thoroughly mediocre universe’. I’ve short-circuited the argument for brevity, but, contrary to Holt’s and Parfit’s conclusion, I would contend that it doesn’t fit the evidence. Our universe is far from mediocre if it’s produced life and consciousness. The ‘selector’, it should be pointed out, could be a condition like ‘goodness’ or ‘fullness’. But, after reading their discussion, I concluded that the logical ‘selector’ is the anthropic principle, because that’s what we’ve got: a universe that’s comprehensible containing conscious entities that comprehend it.

P.S. I wrote a post on The Anthropic Principle last month.

Addendum 1: In reference to the anthropic principle, the abovementioned post specifies a ‘weak’ version and a ‘strong’ version, but it’s perhaps best understood as a ‘passive’ version and an ‘active’ version. To combine both posts, I would argue that the fundamental ontological question in my title, raises an obvious, fundamental ontological fact that I expound upon in the second last paragraph: ‘without consciousness, there might as well be nothing.’ This leads me to be an advocate for the ‘strong’ version of the anthropic principle. I’m not saying that something can’t exist without consciousness, as it obviously can and has, but, without consciousness, it’s irrelevant.

Addendum 2 (18 Nov. 2012): Four months ago I wrote a comment in response to someone recommending Robert Amneus's book, The Origin of the Universe; Case Closed (only available as an e-book, apparently).

In particular, Amneus is correct in asserting that if you have an infinitely large universe with infinite time, then anything that could happen will happen an infinite number of times, which explains how the most improbable events can become, not only possible, but actual. So mathematically, given enough space and time, anything that can happen will happen. I would contend that this is as good an answer to the question in my heading as you are likely to get.

The real war in Afghanistan is set in hell for young girls

This is probably the most disturbing documentary I’ve seen on television, yet it elevates 4 Corners to the best current-affairs programme in Australia and, possibly, the world. I remember reading in USA Today, when American and coalition forces first went into Afghanistan after 9/11 (yes, I was in America at the time) a naïve journalist actually worrying that the change to democracy in Afghanistan might occur too quickly. I found it extraordinary that a journalist covering international affairs had such a limited view of the world outside their own country.

My understanding of Afghanistan is limited and obviously filtered through the eyes, ears and words of journalists, but there appears to be two worlds: one trying to break into the 21^st Century through youthful television programmes (amongst other means) and one dominated by tribal affiliations and centuries-old customs and laws. In the latter, it is the custom to settle disputes by the perpetrator’s family giving land or daughters to the victim’s family. In other words, daughters are treated as currency and as bargaining chips in negotiations. In recent times, this has had tragic consequences resulting from a NATO-backed policy to destroy opium crops, which is the only real way that Afghan farmers can make money. Opium is the source of income for the Taliban but the trade is run by drug smugglers, based in Pakistan. They are the Afghani equivalent of the mafia in that they are merciless. With the destruction of crops, that the drug smugglers finance, they are abducting the farmer’s daughters, from as young as 7 years (as evident in the 4 Corners programme) for payment of their debts. The government and NATO are simply ignoring the problem, and as far as the Taliban is concerned, it’s an issue between the drug smugglers and the farmers.

This is a world that most of us cannot construe. If you put yourself in their shoes and ask: What would I do? Unless you are delusional, the answer has to be that you would do the same as them: you’d have no choice. It’s hard for us to imagine that there exists a world where life is so cheap, yet poverty, perpetual conflict and no control over one’s destiny inevitably leads to such a world. I hope this programme opens people’s eyes and breaks through the cocoon skin that most of us inhabit.

More than anything else, it demonstrates the moral bankruptcy of the Taliban, the cultural ignorance of the coalition and the inadequacy of Pakistani law enforcement.

It’s time the Catholic Church came out of the Closet

This programme was aired a couple of weeks ago on ABC’s 4 Corners, but it demonstrates how out-of-touch the Catholic Church is, not only with reality, but with community expectations. More than anything else, the Church lets down its own followers, betrays them in fact.

This deals specifically with a couple of cases in Australia, and it’s amazing that it takes investigative journalism to shine a light on them. Most damning for the Church, is evidence that protecting their pedophilic clergy and their own reputation was more important than protecting members of their congregation.

The most significant problem, highlighted by the programme, is the implicit belief, held by the Church and evidenced by their actions, that they are literally above the law that applies to everyone else.

This is an institution that claims to have the high moral ground on issues like abortion, therapeutic cloning, gay marriage, euthanasia, to nominate the most controversial ones, when it so clearly lacks any moral credibility. Most people in the West simply ignore the Catholic Church’s more inane teachings regarding contraception, but in developing countries, the Church has real clout. In countries where protection against AIDS and birth control are important issues, both for health and economic reasons, the Church’s attitude is morally irresponsible. The Catholic Church tries to pretend that it should be respected and taken seriously, and perhaps one day it will, when it enters the 21^st Century and actually commits to the same laws as the people it supposedly preaches to.

The Anthropic Principle

I’ve been procrastinating over this topic for some time, probably a whole year, such is the epistemological depth hidden behind its title; plus it has religious as well as scientific overtones. So I recently re-read John D. Barrow’s The Constants of Nature with this specific topic in mind. I’ve only read 3 of Barrow’s books, though his bibliography is extensive, and the anthropic principle is never far from the surface of his writing.

To put it into context, Barrow co-wrote a book titled, The Anthropic Cosmological Principle, with Frank J. Tipler in 1986, that covers the subject in enormous depth, both technically and historically. But it’s a dense read and The Constants of Nature, written in 2002, is not only more accessible but possibly more germane because it delineates the role of constants, dimensions and time in making the universe ultimately livable. I discussed Barrow’s The Book of Universes in May 2011, which, amongst other things, explains why the universe has to be so large and so old if life is to exist at all. In March this year, I also discussed the role of ‘chaos’ in the evolution of the universe and life, which leads me (at least) to contend that the universe is purpose-built for life to emerge (but I’m getting ahead of myself).

We have the unique ability (amongst species on this planet) to not only contemplate the origins of our existence, but to ruminate on the origins of the universe itself. Therefore it’s both humbling, and more than a little disconcerting, to learn that the universe is possibly even more unique than we are. This, in effect, is the subject of Barrow’s book.

Towards the end of the 19^th Century, an Irish physicist, George Johnstone, attempted to come up with a set of ‘units’ based on known physical constants like c (the speed of light), e (the charge on an electron) and G (Newton’s gravitational constant). At the start of the 20^th Century, Max Planck did the same, adding h (Planck’s quantum constant) to the mix. The problem was that these constants either produced very large numbers or very small ones, but they pointed the way to understanding the universe in terms of ‘Nature’s constants’.

Around the same time, Einstein developed his theory of relativity, which was effectively an extension of the Copernican principle that no observer has a special frame of reference compared to anyone else. Specifically, the constant, c, is constant irrespective of an observer’s position or velocity. In correspondence with Ilse Rosenthal-Schneider (1891-1990), Einstein expressed a wish that there would be dimensionless constants that arose from theory. In other words, Einstein wanted to believe that nature’s constants were not only absolute but absolutely no other value.  In his own words,  he wanted to know if “God had any choice in making the world”. In some respects this sums up Barrow’s book, because nature’s constants do, to a great extent, determine whether the universe could be life-producing.

On page 167 of the paperback edition (Vintage Books), Barrow produces a graph that shows the narrow region allowed by the electromagnetic coupling constant, α, and the mass ratio of an electron to a proton, β, for a habitable universe with stars and self-reproducible molecules. Not surprisingly, our universe is effectively in the middle of the region. On page 168, he produces another graph of α against the strong coupling constant, α_s, that allows the carbon atom to be stable. In this case, the region is extraordinarily small (in both graphs, the scales are logarithmic).

I was surprised to learn that Immanuel Kant was possibly the first to appreciate the relationship between Newton’s theory of gravity being an inverse square law and the 3 dimensions of space. He concluded that the universe was 3D because of the inverse square law, whereas, in fact, we would conclude the converse. Paul Ehrenfest (1890 – 1933), who was a friend of Einstein, extended Kant’s insight when he theorised that stable planetary orbits were only possible in 3 dimensions (refer my post, This is so COOL, May 2012). But Ehrenfest made another revelation when he realised that 3 dimensional waves were special. In even dimensions, different parts of a ‘wavy disturbance’ travel at different speeds, and, whilst waves in odd dimensions have disturbances all travelling at the same speed, they become increasingly distorted in dimensions other than 3. On page 222, Barrow produces another graph demonstrating that only a universe with 3 dimensions of space and one of time, can produce a universe that is neither unpredictable, unstable nor too simple.

But the most intriguing and informative chapter in his book concerns research performed by himself, John Webb, Mike Murphy, Victor Flambaum, Vladimir Dzuba, Chris Churchill, Michael Drinkwater, Jason Prochaska and Art Wolfe that the fine structure constant (α) may have been a different value in the far distant past by the miniscule amount of 0.5 x 10^-5, which equates to 5 x 10^-16 per year. Barrow speculates that there are fundamentally 3 ages to the universe, which he calls the radiation age, the cold dark matter age and the vacuum energy age or curvature age (being negative curvature) and we are at the start of the third age. He simplifies this as the radiation era, the dust era and the curvature era. He contends that the fine structure constant increased in the dust era but is constant in the curvature era. Likewise, he believes that the gravitational constant, G, has decreased in the dust era but remains constant in the curvature era. He contends: ‘The vacuum energy and the curvature are the brake-pads of the Universe that turn off variations in the constants of Nature.’

Towards the end of the book, he contemplates the idea of the multiverse, and unlike other discussions on the topic, points out how many variations one can have. Do you just have different constants or do you have different dimensions, of both space and/or time? If you have every possible universe then you can have an infinite number, which means that there are an infinite number of every universe, including ours. He made this point in The Book of Universes as well.

I’ve barely scratched the surface of Barrow’s book, which, over 300 pages, provides ample discussion on all of the above topics plus more. But I can’t leave the subject without providing a definition of both the weak anthropic principle and the strong anthropic principle as given by Brandon Carter.

The weak principle: ‘that what we can expect to observe must be restricted by the condition necessary for our presence as observers.’

The strong principle: ‘that the universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers with it at some stage.’

The weak principle is effectively a tautology: only a universe that could produce observers could actually be observed. The strong principle is a stronger contention and is an existential one. Note that the ‘observers’ need not be human, and, given the sheer expanse of the universe, it is plausible that other ‘intelligent’ life-forms could exist that could also comprehend the universe. Having said that, Tipler and Barrow, in The Anthropic Cosmological Principle, contended that the consensus amongst evolutionary biologists was that the evolution of human-like intelligent beings elsewhere in the universe was unlikely.

Whilst this was written in 1986, Nick Lane (first Provost Venture Research Fellow at University College London) has done research on the origin of life, (funded by Leverhulme Trust) and reported in New Scientist (23 June 2012, pp.33-37) that complex life was a ‘once in four billion years of evolution… freak accident’.  Lane provides a compelling argument, based on evidence and the energy requirements for cellular life, that simple life is plausibly widespread in the universe but complex life (requiring mitochondria) ‘…seems to hinge on a single fluke event – the acquisition of one simple cell by another.’ As he points out: ‘All the complex life on Earth – animals, plants, fungi and so on – are eukaryotes, and they all evolved from the same ancestor.’

I’ve said before that the greatest mystery of the universe is that it created the means to understand itself. We just happen to be the means, and, yes, that makes us special, whether we like it or not. Another species could have evolved to the same degree and may do over many more billions of years and may have elsewhere in the universe, though Nick Lane’s research suggests that this is less likely than is widely believed.

The universe, and life on Earth, could have evolved differently as chaos theory tells us, so some other forms of intelligence could have evolved, and possibly have that we are unaware of. The Universe has provided a window for life, consciousness and intelligence to evolve, and we are the evidence. Everything else is speculation.