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.

The Quantum Universe by Brian Cox and Jeff Forshaw

I’ve recently read this tome, subtitled Everything that can happen does happen, which is a phrase they reiterate throughout the book. Cox is best known as a TV science presenter for BBC. His series on the universe can be highly recommended. His youthful and conversational delivery, combined with an erudite knowledge of physics, makes him ideal for television. The same style comes across in the book despite the inherent difficulty of the topic.

In the last chapter, an epilogue, he mentions writing in September 2011, so this book really is hot off the press. Whilst the book is meant to cater for people with a non-scientific background, I’m unsure if it succeeds at that level and I’m not in a position to judge it on that basis. I’m fairly well read in this area, and I mainly bought it to see if they could add anything new to my knowledge and to compare their approach to other physics writers I’ve read.

They reference Richard Feynman (along with many other contributors to quantum theory) quite a lot, and, in particular, they borrow the same method of exposition that Feynman used in his book, QED. In fact, I’d recommend that this book be read in conjunction with Feynman’s book even though they overlap. Feynman introduced the notion of a one handed clock to represent the phase, amplitude and frequency of the wave function that lies at the heart of quantum mechanics (refer my post on Schrodinger’s equation, May 2011).

Cox and Forshaw use this same analogous method very effectively throughout the book, but they never tell the reader specifically that the clock represents the wave function as I assume it does. In fact, in one part of the book they refer to clocks and wave functions independently in the same passage, which could lead the reader to believe they are different things. If they are different things then I’ve misconstrued their meaning.

Early in their description of clocks they mention that the number of turns is dependent on the particle’s mass, thus energy. This is a direct consequence of Planck’s equation that relates energy to frequency, yet they don’t explain this. Later in the book, when they introduce Planck’s equation, they write it in terms of wavelength, not frequency, as it is normally expressed. These are minor quibbles, some might say petty, yet I believe they would help to relate the use of Feynman’s clocks to what the reader might already know of the subject.

One of the significant facts I learnt from their book was how Feynman exploited the ‘least action principle’ in quantum mechanics. (For a brief exposition of the least action principle refer my post on The Laws of Nature, Mar. 2008). Feynman also describes its significance in gravity in Six-Not-So-Easy Pieces: the principle dictates the path of a body in a gravitational field. In effect, the ‘least action’ is the difference between the kinetic and potential energy of the body. Nature contrives that it will always be a minimum, hence the description, ‘principle of least action’.

Now, I already knew that Feynman had applied it to quantum mechanics, but Cox and Forshaw provide us with the story behind it. Dirac had written a paper in 1933 entitled ‘The Lagrangian in Quantum Mechanics’ (the Lagrangian is the mathematical formulation of least action). In 1941, Herbet Jehle, a European physicist visiting Princeton, told Feynman about Dirac’s paper. The next day, Feynman found the paper in the Princeton library, and with Jehle looking on, derived Schrodinger’s equation in one afternoon using the least action principle. Feynman later told Dirac about his discovery, and was surprised to learn that Dirac had not made the connection himself.

But the other interesting point is that the units for ‘action’ in physics are mx²/t which are the same units as Planck’s constant, h. In other words, the fundamental unit of quantum mechanics is an ‘action’ unit. Now, units are important concepts in physics because only entities with the same type of units can be added and subtracted in an equation. Physicists talk about dimensions, because units must have the same dimensions to be able to be combined or deducted. The dimensions for ‘action’, for instance, are 1 of mass, 2 of length and -1 of time. To give a more common example, the dimensions for velocity are 0 of mass, 1 of length and -1 of time. You can add and subtract areas, for example, (2 dimensions of length) but you can’t add a length to an area or deduct an area from a volume (3 dimensions of length). Obviously, multiplication and calculus allow one to transform dimensions.

One of the concepts that Cox and Forshaw emphasise throughout the book is the universality of quantum mechanics and how literally everything is interconnected. They point out that no 2 electrons can have exactly the same energy, not only in the same atom but in the same universe (the Pauli Exclusion Principle). Also individual photons can never be tracked. In fact, they point out a little-known fact that Planck’s law is incompatible with the notion of tracking individual photons; a discovery made by Ladislas Natanson as far back as 1911. No, I’d never heard of him either, or his remarkable insight.

Cox and Forshaw do a brilliant job of explaining Wolfgang Pauli’s famous principle that makes individual atoms, and therefore matter, stable. They also expound on Freeman Dyson’s and Andrew Leonard’s 1967 paper demonstrating that it’s the Pauli Exclusion Principle that stops you from falling through the floor. Dyson described ‘the proof as extraordinarily complicated, difficult and opaque’, which may help to explain why it took so long for someone to derive it.

They also do an excellent job of explaining how quantum mechanics allows transistors to work, which is arguably the most significant invention of the 20th Century. In fact, it’s probably the best exposition I’ve come across outside a text book.

But what comes across throughout their book, is that the quantum world obeys specific ‘rules’ and once you understand those rules, no matter how bizarre they may seem to our common sense view of the world, you can make accurate and consistent predictions. The catch is that probability plays a key role and deterministic interpretations are not compatible with the quantum universe. In fact, Cox and Forshaw point out that quantum mechanics exhibits true ‘randomness’ unlike the ‘chaotic’ randomness that is dependent on ultra-sensitive initial conditions. In a recent issue of New Scientist, I came across someone discussing free will or the lack of it (in a book review on the topic) and espousing the view that everything is deterministic from the Big Bang onwards. Personally, I find it very difficult to hold such a philosophical position when the bedrock of the entire physical universe insists on chance.

Cox and Forshaw don’t have much to say about the philosophical implications of quantum mechanics except in one brief passage where they reveal a preference for the 'many worlds' interpretation because it does away with the so-called ‘collapse’ or ‘decoherence’ of the wave function. In fact, they make no reference to ‘collapse’ or ‘decoherence’ at all. They prefer the idea that there is an uninterrupted history of the quantum wave function, even if it implies that its future lies in another universe or a multitude of universes. But they also give tacit acknowledgement to Feynman’s dictum: ‘…the position taken by the “shut up and calculate” school of physics, which deftly dismisses any attempt to talk about the reality of things.’

In the epilogue, Cox and Forshaw get into some serious physics where they explain how quantum mechanics gives us the famous Chandrasekhar limit, developed by Subrahmanyan Chandresekhar in 1930, which determines how big a star can be before it becomes a neutron star or a black hole. The answer is 1.4 solar masses (1.4 times the mass of our sun). Mind you, it has to go through a whole series of phases in between and that’s what Cox and Forshaw explain, using some fundamental algebra along with some generous assumptions to make the exposition digestible for laypeople. But the purpose of the exercise is to demonstrate that quantum phenomena can determine limits on a stellar scale that have been verified by observation. It also gives a good demonstration of the scientific method in practice, as they point out.

This is a good book for introducing people to the mysteries of quantum mechanics with no attempt to side-step the inherent weirdness and no attempt to provide simplistic answers. They do their best to follow the Feynman tradition of telling it exactly as it is and eschew the magic that mysteries tend to induce. Nature doesn’t provide loop holes for specious reasoning. Quantum mechanics is the latest in a long line of nature’s secret workings, mathematically cogent and reliable, but deeply counter-intuitive.