# Mathematics and the nature of reality

How many universes are there? What has made us into who we are? Is there absolute truth?

These are difficult questions, but mathematics has something to say about each of them. It can probe the physical reality that surrounds us, shed light on human interaction and psychology, and it answers, as well as raises, many of the philosophical questions our minds have allowed us to dream up.

On this page we bring together articles and podcasts that examine what mathematics can say about the nature of the reality we live in. They look at physical reality, the mind, consciousness, the emergence of life, philosophy and mathematics itself. The page will be continually updated with new relevant articles, so keep looking and get reading!

We've grouped our articles into three categories:

### Mathematics and physical reality

Hooray for Higgs — "It's a great day for particle physics," says Ben Allanach, a theoretical physicist at the University of Cambridge. "It's very exciting, I think we're on the verge of the Higgs discovery." And indeed, it seems like the Large Hadron Collider at CERN has given particle physics an early Christmas present — compelling evidence that the famous Higgs boson exists.

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What is time? — Newton thought that time was absolute. Einstein thought it was relative. Today some people think it doesn't exist at all. Or is it an emergent phenomenon? Paul Davies explores.

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This interview is also available as a podcast.

Is time travel allowed? — We're all on a journey towards the future, but can we travel into the past? Find out with Kip Thorne.

What happened before the Big Bang? — Did the Big Bang mark the beginning of time? Not if we live in a bubble multiverse, says John D. Barrow.

Shattering crystal symmetries —In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?

Exploding stars clinch Nobel Prize —This year's Nobel Prize in Physics was awarded for a discovery that proved Einstein wrong and right at the same time.

Countdown to the Higgs? — Are we close to finding the Higgs? Ben Allanach explains it is not about catching a glimpse of the beast itself, but instead keeping a careful count of the evidence it leaves behind.

Does quantum physics really describe reality? — Quantum physics is a funny thing. With counterintuitive ideas such as superposition and entanglement, it doesn't seem to resemble reality as we know it, yet it is an incredibly successful theory of how the physical world operates. In this podcast we speak to experts including John Polkinghorne and Roger Penrose about how we can resolve the mysteries of quantum physics with our experience of reality. And we find out why quantum physics is just like riding a bike...

Flying home with quantum physics — Quantum mechanics is usually associated with weird and counterintuitve phenomena we can't observe in real life. But it turns out that quantum processes can occur in living organisms, too, and with very concrete consequences. Some species of birds use quantum mechanics to navigate. Studying these little creatures' quantum compass may help us achieve the holy grail of computer science: building a quantum computer. You can also listen to the interview this article is based on as a podcast.

Biology's next microscope, mathematics' next physics — It is thought that the next great advances in biology and medicine will be discovered with mathematics. As biology stands on the brink of becoming a theoretical science, Thomas Fink asks if there is more to this collaboration than maths acting as biology's newest microscope. Will theoretical biology lead to new and exciting maths, just as theoretical physics did in the last two centuries? And is there a mathematically elegant story behind life?

And the Nobel Prize in Mathematics goes to... — Well, it goes to no one because there isn't a Nobel Prize for maths. Some have speculated that Alfred Nobel neglected maths because his wife ran off with a mathematician, but the rumour seems to be unfounded. But whatever the reason for its non-appearance in the Nobel list, it's maths that makes the science-based Nobel subjects possible and it usually plays a fundamental role in the some of the laureates' work. Here we'll have a look at two of the prizes awarded this year, in physics and economics.

What's happening at the LHC? — This article gives an update on what's happening at the Large Hadron Collider, as of February 2011. It is based on an interview with John Ellis, outgoing leader of the theory division at CERN, who describes some of the mysteries the LHC may solve.

Hidden dimensions — That geometry should be relevant to physics is no surprise. After all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be. *Plus* met up with mathematician Shing-Tung Yau to find out more. You can also read our review of a book co-written by Yau, called *The shape of inner space*.

Exotic spheres, or why 4-dimensional space is a crazy place — The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?

The Able Prize 2011 goes to John Milnor — This news story examines the work of John Milnor, who discovered (invented?) the exotic spheres described in the above article.

Is space like a chessboard? — Physicists at the University of California, Los Angeles set out to design a better transistor and ended up with a discovery that may lead to a new explanation of electron spin and possibly even the nature of space.

Benoît Mandelbrot has died — Benoît Mandelbrot, the father of fractal geometry, died in October 2010 at the age of 85. His work shed important light on how complexity can arise from simple rules.

Classroom activity: The game of life — One of the amazing things about life is its sheer complexity. How can a bunch of mindless cells combine to form something as complex as the human brain, or as delicate, beautiful and highly organised as the patterns on a butterfly's wing? Maths has some surprising answers you can explore yourself with this interactive activity.

The illusory Universe — With online socialising and alternative realities like Second Life it may seem as if reality has become a whole lot bigger over the last few years. In one branch of theoretical physics, though, things seem to be going the other way. String theorists have been developing the idea that the space and time we inhabit, including ourselves, might be nothing more than an illusion, a hologram conjured up by a reality which lacks a crucial feature of the world as we perceive it: the third dimension. *Plus* talks to Juan Maldacena to find out more.

Particle hunting at the LHC — It's hard to avoid CERN these days. Last year's successful switch-on of CERN's Large Hadron Collider, followed by a blow-out which is currently being fixed, sparked wide-spread media coverage, and currently CERN stars in the Tom Hanks movie Angels and Demons. So what goes on at CERN and why the hubbub about the Large Hadron Collider, known as the LHC?

Lambda marks the spot — the biggest problem in theoretical physics — The mathematical maps in theoretical physics have been highly successful in guiding our understanding of the universe at the largest and smallest scales. Linking these two scales together is one of the golden goals of theoretical physics. But, at the very edges of our understanding of these fields, one of the most controversial areas of physics lies where these maps merge: the cosmological constant problem.

Outer space: Are the constants of nature really constant? — Are the unchanging features of the Universe really unchanging?

### Mathematics and the mind

Gut logic — Human reasoning is biased and illogical. At least that's what a huge body of psychological research seems to show. But now a psychological scientist from the University of Toulouse in France has come up with a new theory: that logical and probabilistic thinking is an intuitive part of decision making, only its conclusions often lose out to heuristic reasoning.

Convex is complex — If computers are ever going to be as clever as humans they'll have to be able to solve really hard problems. But what do we mean by "hard" and how hard are the hardest problems? This news story explores the concept of NP completeness using a result on the convexity of mathematical functions as an example.

Born to count — People as well as animals are born with a sense for numbers. But is this inborn number sense related to mathematical ability? A new study suggests that it is.

Guilt counts — Guilt, so some people have suggested, is what makes us nice. When we do someone a favour or choose not to exploit someone vulnerable, we do it because we fear the guilt we'd feel otherwise. A team of neuroscientists, psychologists and economists has produced some new results in this area, using a model from psychological game theory.

Wiring up brains — The human brain faces a difficult trade-off. On the one hand it needs to be complex to ensure high performance, and on the other it needs to minimise "wiring cost" — the sum of the length of all the connections — because communication over distance takes a lot of energy. It's a problem well-known to computer scientists. And it seems that market driven human invention and natural selection have come up with similar solutions.

Neurotweets: #hashtagging the brain — We like to think of the human brain as special, but as we reported in the above article, it has quite a lot in common with worm brains and even with high-performance information processing systems. But how does it compare to online social networks? In a recent lecture the psychiatrist Ed Bullmore put this question to the test.

Struggling with your maths? — If you are, then you may be one of the 5 to 7% of the population suffering from dyscalculia, the mathematical equivalent of dyslexia. But unlike many dyslexia sufferers, you probably haven't received the help you need to cope with your condition. As a recent article published in the journal Science points out, dyscalculia is the "poor relation" of dyslexia.

Baby robots feel the love — Researchers have unveiled the first prototypes of robots that can develop emotions and express them too. If you treat these robots well, they'll form an attachment to you, looking for hugs when they feel sad and responding to reassuring strokes when they are distressed. But how do you get emotions into machines that only understand the language of maths?

Spaceships are doing it for themselves — It requires only a little processing power, but it's a giant leap for robotkind: engineers at the University of Southampton have developed a way of equipping spacecraft and satellites with human-like reasoning capabilities, which will enable them to make important decisions for themselves.

Finding your way home without knowing where you are — Foraging ants have a hard life, embarking on long and arduous trips several times a day, until they drop dead from exhaustion. The trips are not just long, they also follow complex zig-zag paths. So how do ants manage to find their way back home? And how do they manage to do so along a straight line? Their secret lies in a little geometry.

Uncoiling the spiral: Maths and visual hallucinations — Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. But it's not just hallucinogenic drugs that conjure up these geometric structures. People have reported seeing them in near-death experiences, following sensory deprivation, or even just after applying pressure to the eyeballs. So what can these patterns tell us about the structure of our brains?

Book review: Natural computing — Computing is at the heart of our modern world, but what are its frontiers? This book presents new trends in this fast growing field. Although the topics covered range from spacecraft control to embedding intelligence in bacteria, they all coincide in one fundamental point: the future of computing is a synthesis with nature.

### Mathematics, philosophy and truth

Freedom and physics — Most of us think that we have the capacity to act freely. Our sense of morality, our legal system, our whole culture is based on the idea that there is such a thing as free will. It's embarrassing then that classical physics seems to tell a different story. And what does quantum theory have to say about free will?

Free, from top to bottom? — A traditional view of science holds that every system — including ourselves — is no more than the sum of its parts. To understand it, all you have to do is take it apart and see what's happening to the smallest constituents. But the mathematician and cosmologist George Ellis disagrees. He believes that complexity can arise from simple components and physical effects can have non-physical causes, opening a door for our free will to make a difference in a physical world.

John Conway: discovering free will (part I) — On August 19, 2004, John Conway was standing with his friend Simon Kochen at the blackboard in Kochen’s office in Princeton. They had been trying to understand a thought experiment involving quantum physics and relativity. What they discovered, and how they described it, created one of the most controversial theorems of their careers: The Free Will Theorem.

John Conway: discovering free will (part II) — In the second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem.

John Conway: discovering free will (part III) — Inthe second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem.

This is not a carrot: Paraconsistent mathematics — Paraconsistent mathematics is a type of mathematics in which contradictions may be true.
In such a system it is perfectly possible for a statement *A* and its negation *not A* to both be true. How can this be, and be coherent? What does it all mean?

The philosophy of applied mathematics — We all take for granted that mathematics can be used to describe the world, but when you think about it this fact is rather stunning. This article explores what the applicability of maths says about the various branches of mathematical philosophy.

Searching for the missing truth — Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.

Picking holes in mathematics — In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?

The revelation game — Is it rational to believe in a god? The most famous rational argument in favour of belief was made by Blaise Pascal, but what happens if we apply modern game theory to the question? This article has been adapted from the book *Game theory and the humanities*, which has been reviewed in *Plus*.

Constructive mathematics — If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article Phil Wilson looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between.

Unreasonable effectiveness — When it comes to describing natural phenomena, mathematics is amazingly — even unreasonably — effective. In this article Mario Livio looks at an example of strings and knots, taking us from the mysteries of physical matter to the most esoteric outpost of pure mathematics, and back again.

Infinite investigators: Part I — What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? This article explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice.

Infinite investigators: Part II — We continue the investigation into Cantor and Cohen's work, looking at the continuum hypothesis, the question that caused Cantor so much grief.

Teacher package: Logic — In some sense, all of maths should come under the label "logic", but mathematical logic has shown that mathematics isn't entirely logical. Makes sense? If not, then this teacher package may help.

Book review: The big questions, mathematics — With twenty skillfully written essays Tony Crilly paints a broad-stroke picture of modern mathematics, focusing on some of the most exciting topics. This book is intended for people whose acquaintance with mathematics is limited to their high school years, but who want to know "what all this fuss is about". It is ideal for those who have heard that mathematicians talk about imaginary numbers and unbreakable codes, and want to know how much of it, if any, is true.

## Comments

## Model versus Reality

Very nice site with well-put-together packages.

One thing that all too frequently occurs in science and mathematics articles, especially in lay publications such as newspapers and magazines for the general public, but often in the technical articles themselves, is a confusion of the model with reality. Our scientific and mathematical models NEVER completely describe reality -- there is always error -- from simplifying assumptions, measurement error, model application limitations, data acquisition limitations, inadequate observation, induced error from calculation processes, and the inadvertent introduction of extraneous data.

Under ideal conditions, our models are adequate predictors, but under some circumstances, our models are woefully inadequate. Then there is the problem of apparently credentialed individuals unfamiliar with the applicable limits misapplying the models, sometimes with very misleading results.

The best we can do with our mathematical and scientific models is "good enough for now," with the admonition that better and more refined models might be needed in the future - consistent with more massive computing power to process the models, along with more extensive and and relevant data acquisition.

The most prominent violators of the "model is not reality" dictum appear to be astronomers, who cavalierly describe theories, such as the big bang, and the expanding universe, as established facts. Other gross violators are the "warming earth" crowd, whose political fervor for their cause and supposed cure, are unworthy to even be called "science."

## The importance of science communication

Thanks for your comment. We hope that our articles explain how mathematical models reflect reality and the kind of interpretations you can make from these results. From our experience working with astronomers and scientists using climate models, they are always very aware of the strengths and limitations of their models. But often during heated discussions in public forums and the press these subtleties are often overlooked. We hope that magazines like Plus play an important role in helping scientists to communicate the technical details of their research in an accessible and engaging way.