Tag Archives: maths

Book Review: Enlightening Symbols by Joseph Mazur

Subtitled ‘A Short History of Mathematical Notation and Its Hidden Powers’ I was first made aware of this book on GrrlScientist’s blog on the Guardian website. I added it to a wishlist and was given it as a present for my birthday earlier in the autumn. Having been educated in maths to a degree further than most, I have used rather a library of symbols in my time and had cursory historical overviews of their development but I have not previously read a thorough history.

The book is split into three parts, the first focusing on the development of numerals, the second on algebra and the third on the power of symbols. All three parts are quite distinct and ought to be looked at one by one.

I must confess, I didn’t find the first part particularly coherent. That is partly a feature of the fact that the history of the representation of numbers is itself quite muddled. In reading this, I got the impression that Mazur, who I don’t ever recall coming across before, was more of a mathematician than a historian. As it turns out, this seems to be a fair characterisation, though, like me, he takes a very keen interest in history and (we’ll come to it below) into other areas as well.

The history of numerals is summed up on page 78 as follows: “There have been many scrupulous studies on the origins of our system, but even after a hundred years of scholarly wide-ranging research, we are left with only sketchy guesses of its beginning and evolution.” Perhaps this should have been an executive summary at the start of the section rather than a conclusion reached after having gone round the houses a few times. It’s not that the history is uninteresting, it is really quite captivating. It’s just that Mazur’s take on it didn’t allow this reader to get a grasp on it, so it was quite bewildering. So I must confess that I wasn’t overly enamoured with Mazur’s writing and as I finished the first part, I feared that the last 2/3rds of the book would be a bit of an unenjoyable trudge.

How glad I was to be proved wrong. For in moving from numeral to algebra, fresh life was breathed into the text and I was treated to the book that I had hoped to read.

As with the first section, the story is not straightforward, but we do get to see some of the significant historical developments in fresh light rather than the fairly dim gloom we had beforehand. The first major figure we encounter is Diophantus. His name should be familiar to most maths students, though if you haven’t come across him then this would be a good place to gain an introduction. The basic story is that problems that we think of as algebraic did not begin with symbolic representations.

If you had a good maths teacher (and I’ve been blessed by having a few) then you will have been presented with “word problems” where some question or other is asked which involves numbers and where the answer is required in the form of a number. The student is then asked to convert the word problem into a symbolic form and then manipulate that symbolic form using the methods taught to arrive at an answer. What Mazur gives us is an unpacking of this, showing that most early algebra consisted of such word problems.

We get to meet al-Khwarizmi and see some of the problems he posed in his seminal work Al-Kitab al-mukhtasar fi hi sab al-gabr wa’l-muqabala (yes, I did have to copy that carefully). We see the development of symbolic representation such as those for multiplication, powers and division. Without trying to summarise it here (I confess, this part of the review was written a couple of weeks after I finished reading the book), I would heartily recommend it to you. For those who dropped maths after their GCSEs, I will say that it might not be particularly applicable. For those who are university educated or who can still recall their A-levels then the final step will be very familiar, but it’s a fascinating story as to how we got here.

The final third of the book carried on in the same vein as the second part had, with less of a major change in tone that there was between the first and the second. As I read through the first two parts, I was struck by a quite sobering (or maybe dispiriting might be a better term) thought that in spite of having studied maths to a greater level than most people in the world, was my understanding of it merely the understanding of manipulation of symbols?

There is reassurance at the end, though. Mazur’s view is that our ability to shorthand things in symbolic frees up the mind to truly understand what is going on. This seems to coincide with how I view the abstraction in maths in general, as well as some specific aspects like Fourier transforms; here we phrase a question in a specific way, abstractify to the general case, solve the general case and then you have a template for answering the specific case. By working with symbols we may temporarily lose sight of exactly what it is we are calculating, but that lack of sight allows us to avoid getting bogged down in unnecessary detail. By all means, if we wish to come to back to an intermediate stage in the calculation and convert into word problems, we can – that is the power of symbolic maths.

The final section also deals with some other matters peripheral to our understanding of mathematics, such as the psychology and philosophy of maths. So it was little surprise to see Wittgenstein referenced at this point. Though Mazur was readily more accessible than Wittgenstein was. The breadth of this final view reveals the author to be more than just a mathematician, he is a bit of a polymath. So while the book was not hugely coherent to begin with, the last two-thirds are very creditable and I would recommend it to anyone interested in maths and the history thereof.

Book Review: A Mathematician Reads The Newspaper by John Allen Paulos

This has been my latest coffee table book that I dip into a few pages at a time. The premise is that maths is not well understood, but that it’s all around us. Paulos’ plan then is to educate us through a number of examples, which run to just a few pages each.

From the start, though, one is struck by a very heavy American bias. I think he tries to name drop by using examples of people he thinks his readers will know, but outside of the USA, names of the justices of the US supreme court are not commonly known pieces of trivia. That left this UK-based reader a little nonplussed, as it could have been made far more inclusive.

It’s a real shame, particularly as I read through the first part, which was on the subject of politics, its relentless US-centricism detracted from some otherwise very good prose. Paulos doesn’t really go into much mathematics here. His focus is more about rational thinking and how that can apply to things of a mathematical nature. So do not expect a particularly pedagogical text or worked examples. Numbers are fairly thin on the ground. As such, some who, like me, picked up the book expecting a book primarily about mathematics might be left wondering if the title wasn’t a little misleading.

In truth, it’s much more about general rationality than it is about maths. Given the expectations generated from the title, this inevitably left me rather disappointed. I know it was a follow up to an earlier book of his, entitled Innumeracy, which may have been closer to a better title for this work than the one it has.

The way the book is supposed to be structured is meant to roughly mirror a newspaper. So the front part of the book has more politics, the middle is more ‘lifestyle’ and there is a bit about sports (almost invariably US-based sports) towards the end. At times, the link to the typical newspaper seems rather tenuous, even if the general thrust of the argument is sound. Yet for a science writer, Paulos just seems to lack any great level of enthusiasm. Many of the best science writers (I think here of the likes of Feynman, Dawkins, Penrose and Sagan) have an attitude of “[isn’t this brilliant? Come, let me show you]” whereas Paulos is more towards the brow-beating end of the spectrum. There is little joy to be found, with an air of despondency at other people’s lack of nous.

The other fact that cannot be avoided is that, though it was only written in the mid 1990s, it hasn’t aged well. Any talk he has of computers or the possible threat the internet would be to the newspaper industry seem rather dated. That cannot be a criticism against the author, though, as one cannot expect him to be a prophet. Rather, it is a word of caution intended for any potential reader. Though I cannot say I would be in a great rush to recommend this book to anyone. The material covered here may be found in many a popular level book on mathematics and are dealt with in more detail and with a greater level of engagement than may be found here.

Book Review: Professor Stewart’s Cabinet of Mathematical Curiosities by Ian Stewart

This has been my “coffee table” book for the last few months, following on from Julian Baggini’s The Pig That Wants to be Eaten. This is because it’s full of lots of little bits, with no overall narrative. It’s not quite a school exercise book, but it does have quite a lot of puzzles for you to think through, some of which require some scribbling with some pen & paper or plugging numbers into a calculator. As well as these, there are lots of little vignettes of mathematical thought which inform but require less input from the reader.

So my initial advice for any readers of this would be get a notepad and some pens and keep them nearby. Fans of recreational mathematics will find much that is familiar here, as some problems recur in just about every such ‘popular’ level book on maths, such as the problem of the bridges of Konigsberg or lots of factoids about pi.

That may sound like damning with faint praise, but there is a depth of mathematics on display here that is rather splendid. Many of the ideas are really quite profound, yet the way they are presented makes them quite accessible. A non mathematician might disagree with me, but it may be interesting to find out from others if there are areas where they get stuck.

There is a general trend for the puzzles to get a little bit more difficult later on in the book. So we are given some treats that will be unfamiliar even to those who did maths at A-level. We deal with topics ranging from geometry, number theory, topology and even some complexity is thrown in at the end.

I probably ought to add that for any sections that ask questions there are answers provided at the back of the book. Most are pretty good, though if the book does have any weaknesses, it is here, where some of the answers are given with not enough explanation. Though for recreational mathematics, one of the litmus tests has to be how well the solution to the Monty Hall problem is described and this one is very fair.

There is a follow-up book that Ian Stewart wrote, in the same vein but with a different set of problems. Given the quality of this work, I will be reading that as well, so you can look forward to seeing another review like this in a few months. For my next coffee table book, though, I will be turning to Plato and a Platypus; a book I searched for for some years but only got my hands on recently.

Book Review: The Simpsons and Their Mathematical Secrets by Simon Singh

First he studied quarks. Then he battled quacks. Now he looks at quirks.

From the outset, one gets the impression that this was great fun to write. Maybe it was even more fun to write than it was to read. The basic premise is that a lot of the scriptwriters of The Simpsons are highly scientifically literate and that throughout the many years that the show has been broadcast there have been a number of gags that rely on an understanding of maths.

Singh has spoken to a number of the creators of The Simpsons in order to ascertain whether the theories of some of the earliest discussion boards on the internet, like some UseNet groups were right when they analyzed various episodes, as well as to get an insight into the creative process that goes into creating one of the most successful television programmes of the last 25 years.

What then proceeds is an exploration of some of the more fun aspects of maths. Those who have read a lot of recreational maths may struggle to find much that is new here, but what is presented is done so clearly, with great humour and evident enthusiasm. One will not be surprised to find discussions here on prime numbers, pi and Fermat’s last theorem. To those familiar with the concepts, this is like meeting an old friend in a bar. You may have heard their stories before, but they are fun to be around nonetheless.

For me, one of the most interesting chapters was on sports statistics. I’ve been aware for some time that some sports teams have used statisticians to try to bolster their results, but have had little more insight than that. Singh gives us a nice overview of the subject here, under the heading of ‘sabermetrics‘.

Some of the links between the maths and The Simpsons can be a bit tenuous at times. Some other reviewers have commented on this, but I wouldn’t be too critical on this point. The book is more about maths than The Simpsons, with the latter being a springboard from which Singh dives; and rather elegantly at that.

Yet The Simpsons isn’t the only springboard used. Approximately 2/3rds of the way through, we start to look at another of Matt Groening’s creations, Futurama. The book, though, keeps the same format and uses this to look at a greater range of issues, which was possible because Futurama was ostensibly more sci-fi than The Simpsons.

In this part of the book, we get treated to one of my favourite tales from the history of modern maths: the meeting of G. H. Hardy and Srinivasa Ramanujan, talking about 1,729 (though Singh fails to note that if you get hold of a mathematician’s debit or credit card, 1729 would be a good guess for the PIN!). If you haven’t heard of those two people or the number 1,729 means nothing to you, then please do buy this book.

Although the book is excellent, it does have a few flaws. It triggers one of my pet hates of using what is euphemistically known as ‘American spelling’ or in more plain terms, it is littered with typos. Though the subject is an American show, it is written by an English author with an English publisher, so one would have hoped for correct spellings. Though that might be levelled at the book’s editor, there is one critique for the author, and that is a bizarre dichotomy he tries to draw between science and maths, as though the two were somehow different disciplines. One might demonstrate with this quote from page 45: “…scientists have to cope with reality and all its imperfections and demands, whereas mathematicians practice their craft in an ideal abstract world.” As a mathematician, this is a view I profoundly disagree with, not least given the very simple fact that mathematics departments are typically located with the faculty of science at any university. It was certainly was at my alma mater, where I recall Simon Singh once gave a guest lecture while I was an undergraduate.

I’ve followed the fortunes of Simon Singh for a few years now. His books on Fermat’s Last Theorem (which crops up here) and on code breaking have both proved popular and critical successes. In recent years though he has been the subject of libel proceedings and has become something of a world-weary figure after his long legal battle with the British Chiropractic Association and his subsequent work in libel reform. Early on in the book I got the impression that this was a book to signal that he had moved on and was now back to enjoying work and that this was a breath of fresh air. This impression was deepened after a sly reference at the start of the book notably using the word ‘bogus’ that had gotten him into so much trouble in the first place. If there was any doubt, though, it would be thoroughly dispelled in the acknowledgements at the end of the book where he takes time and space to thank those who have supported him and his campaigns.

It’s a coda which indicates the more serious aspects of science to which the main content of the book is the joyous, trivial flipside. I look forward to seeing what he writes next. If it is as good as The Simpsons and their Mathematical Secrets, it will be well worth a read.

It’s just a theory

Last week I was a watching a repeat of an edition of Horizon which was looking at the idea of what was around before the Big Bang. For UK readers, it may still be available on iPlayer and would highly recommend watching it. It features a few of my favourite scientists: the physicists Neil Turok and Lee Smolin and the mathematician Roger Penrose. Those who know me well may recall that for my master’s degree I studied Roger’s twistor theory under the supervision of one of Roger’s former students.

The programme featured a variety of views from these and other scientists about a controversial notion. I’m familiar with Roger’s view, conformal cyclic cosmology, from reading his work, Cycles of Time. The other views I had heard before (not least from when the programme was first aired) but was less acquainted with.

One post I saw on Facebook read as follows: “So watching Horizon about the Big Bang, and loving the fact that scientists clearly have not got a clue about the universe started. Mmm…

One of the comments below stated: “Most of science is all theory and guess work!

These two combined rather got my hackles up. I chose not to enter into a debate then and there but rather think about it for a few days and write something up the following weekend (which, at 9:31pm on a Saturday night, I am now doing).

The first comment

The phrase that I object to here “have not got a clue”. Cosmology may be a relatively young science with much of the universe still to explore, but the notion that there isn’t a clue simply flies in the face of the evidence.

After the discovery of galaxies and the measures of their redshifts allowed for us to realise that we live in an expanding universe. This has been backed up by numerous astronomical observations and, to the best of knowledge, no observations have been made that falsify the idea of the expanding universe.

The idea of the Big Bang (a term originally used as a derisory attempt to discredit it by the steady state advocate, Fred Hoyle) was then developed by ‘turning back the clock’ and asking: if the universe is expanding, then in the past it was smaller, but how far back does that go.

The serendipitous discovery of the cosmic microwave background radiation by Penzias and Wilson was the nail in the coffin for Hoyle’s steady state hypothesis. It also tied in neatly with the developing theory of the Big Bang.

To date, it is the best theory of how the universe came into being. It accords with the best available evidence and has been studied in great detail in papers published in peer-reviewed journals – the gold standard of scientific epistemology. (Though it should be added that it’s not foolproof, as recently attested)

That doesn’t mean our knowledge and understanding is perfect and complete. If it were, there would be no need for further analysis. A lack of understanding is not a cause for abandonment or of making presumptions, it is a cause for further study. The Big Bang theory has its limits and the Horizon programme probed at some of these limits, asking important questions. If it turns out that another theory will be needed in order to give more details into the universe’s origins then it must take into account the evidence to date.

To take an historical example, Einstein’s general relativity recast Newton’s earlier work. The formalisms look quite different, but if one starts from Einstein and make some simplifications, one can rederive Newton. But Einstein had to start with Newton. Newton’s work was necessary to begin with. It seems unlikely that the more ‘obvious’ formalism would be overlooked and that someone could have come up with the more sophisticated theory. Yet Einstein didn’t overthrow Newton. The former built upon and refined the latter.

The case may be the same with the Big Bang. If a new theory is needed it will need to incorporate the evidence gathered to date. It would likely have an element which, if simplified, would result in something that looks like the formalism of the Big Bang. It’s doubtful many  people would understand it at first. But the lack of understanding is not a good reason for rejection. It should be quite exciting.

This is why I reject the idea that “scientists clearly have not got a clue about the universe started”. We have many clues. We have a coherent and consistent model of how it came about. It’s not complete, with some aspects as yet not understood. The work of the cosmologist is to try to bridge that gap in understanding.

The second comment 

So then, with that in mind, what about the second point?

“Most of science is all theory and guess work!”

First, we have the appalling grammar to deal with. Is it all or is it most? It can be difficult to judge tone in a written text (goodness knows what you make of my tone on this blog sometimes!), especially in one so short, but I read this is a statement not made by someone well-versed in science but rather as a condemnatory statement.

One of the key words here is “theory”. It is an example of where words which have fairly precise, technical meanings are also used in colloquial English. For another example in a different context, see my take here on the use of ‘complex’.

The Oxford English Dictionary defines theory as “1. an idea or set of ideas that is intended to explain something. 2. a set of principles on which an activity is based.” A footnote adds that it is derived from the Greek theoria which is translated as ‘speculation’.

This is the more common use of the term, and represents an unscientific viewpoint. That is not to say it is an inferior viewpoint in any way. It’s just a use of the term that is not context-specific. Contrast it with this, from Encyclopaedia Britannica: “a systematic ideational structure of broad scope, conceived by the imagination of man, that encompasses a family of empirical (experiential) laws regarding regularities existing in objects and events, both observed and posited – a structure suggested by these laws and devised to explain them in a scientifically rational manner.”

I would paraphrase this as: a framework of understanding, based on observation, by which reality is modelled.

However you prefer it, the scientific theory is a lot more than speculation. For that we would generally use the term hypothesis or conjecture.

But this brings us back to the comment. We test what we want to find out, in other words, what we do not know. Science tends to come in two parts: theoretical and experimental. I’m much more of a theorist. The work of the theorist is to develop the models, usually mathematical, and develop them into a coherent system which first of all agrees with all known observed data and secondly makes new, testable predictions. Depending on what area of science one works in, one will have different standards of evidence. Mathematics is the most precise of the sciences, where the only evidence accepted is that of proof. It’s a watertight logic. In physics, such proof is hard to come by. But physics is extremely rigorous, with 5-sigma being the level of certainty required before announcing a discovery . The use of this was well documented at the Large Hadron Collider in two recent instances: 1) the discovery of the Higgs boson and 2) the possibility (later explained) that a particle had travelled faster than the speed of light.

The latter point makes for an interesting case study in the scientific method. The apparent result would, if true, have contradicted a century’s worth of physics. That was not a reason to either throw Einstein out of the window or to reject the experiment as a hoax. It demanded to be taken seriously. It was the fact that Einstein’s relativity has been tested numerous times and that the framework which is established by that theory is one that we use everyday (e.g. for satellite navigation) that made a possible falsification a prospect that was at once thrilling and threatening. Had it been wrong, how would the body of evidence that supports it be accounted for?

I could go on even more. But I hope I’ve made my point.

Yes, science is theory, but that’s how it works. In the development of our theories there are hypotheses made. These are not random guesses flung out, but are the result of disciplined work. Once those hypotheses are tested they might be rejected or incorporated into the current theory. We have a lot of clues as to how the universe began and what goes on within it. We don’t know everything, and that is why research continues.

Book Review: The History of Mathematics: A Very Short Introduction by Jacqueline Stedall

Having read a few of the science-based Very Short Introductions (VSIs), I have become more inclined to read those on subjects which I know little or nothing about, or where my knowledge is somewhat scatty. This volume falls squarely in the last of these. Having studied maths at university and read quite a few books about maths (for some reviews see here, here, here and here), I have picked up snippets here and there, but have never before read anything like a formal history.

Stedall opens with something of a case study: that of Fermat’s last theorem. To anyone who has a GCSE in maths, this is an understandable problem, even if the solution is still only understood by a handful of “elite” mathematicians in the world. It is through this problem that the author looks at various approaches one might take to mathematical history. At this point the unwary reader might start to get a little confused.

The reason for that is that Stedall takes her history seriously. Though the book is interspersed with the stories of some of the problems that have puzzled men and women for millennia, their friendships and rivalries, a history which is limited to dates and discoveries is one that would be superficial and misleading, as Stedall points out. An additional problem is highlighted, in that some of our sources are extremely sparse, with very little known about the likes of Pythagoras or Diophantus and even less about the development of the science outside Europe.

This begs the question: what is mathematics? It might sound like a stupid question, until you actually think about it. Much of what I did as part of my masters degree might well have been considered theoretical physics; and that was mostly studying scientific developments from the 18th up the 20th century. Most of this has been developed in western Europe. So what about the rest of history and the rest of the world? To answer that Stedall looks at a wider scope than many readers will be familiar with. Of particular note is the look at Arabic science, a topic I hope to follow up later this year with Jim Al-Khalili’s book, Pathfinders.

The next two chapters look at the spread of mathematical ideas. The first, from the perspective of the mathematician to the mathematician, the second, from the perspective of teacher to student. I must confess, I’d not really considered the dissemination of mathematical ideas from an historical perspective. The insight we get into an English Victorian education is something I probably ought to have known, but which was nonetheless a revelation to me. Less surprising, though sadly so, was the discussion on women’s mathematical education, which is a sad indictment on our education system until far too recently.

It’s only after all this that the reader who, like me, expected a more conventional story of some individual’s lives, get what they were expecting. Piecemeal, we have already had some insight but here we fill in some of the more obvious gaps, such as the lives of Euler and Lagrange. Though these are dealt with in a brief manner, that is kind of the point of this series of books.

Only towards the end do we actually see any maths. For some readers, this may come as a relief, for others, it may leave us wondering how good a history of ideas might be without the explicit statement of some of those ideas.

Stedall’s writing is perfect for the job. She is engaging, insightful and thought-provoking. As ever, we are provided with a great list for potential follow-up reading. I can’t say that I now have an encyclopaedic knowledge of the history of maths, but I have been able to glimpse some extra insights. Yet I don’t feel rushed to follow these up, but follow them up I may well do, but all in good time.

Book Review: Mathematics: A Very Short Introduction by Timothy Gowers

Carrying on a look at some of the books in the Very Short Introduction (VSI) series, I wanted to revisit some of the joy of university life by returning to mathematics, the subject in which I obtained a first class Masters degree. With the title as it is, one might wonder what sort of level as it’s pitched at. Here, one could be lulled into a false sense of security by mistaking it for “Arithmetic: A Very Short Introduction”. Do not expect this to be “a very simple introduction”. To anyone who has studied maths at university, this will be a very simple book. To anyone studying maths at A-level, they should find it a little challenging in places, but it should provide good food for thought, building on some familiar principles. The author says that it should be OK for anyone with a good GCSE grade, though I would express a little scepticism at that sentiment.

That said, I do think it’s a marvellous little book. One of the first things that Gowers discusses is the cumulative nature of maths. i.e. some things can be very simply stated, but only in terms of other things which need to be well-defined and understood. The danger then is how far back do you regress to be able to find ground which is widely understood?

Gowers deals with this brilliantly by having his opening chapter on mathematical modelling. In so doing, he grounds mathematics in the real, the physical, the tangible, instead of diving off into the realms of pure mathematics straight away. Though I must admit, the appeal to me comes about primarily because he enunciates the way I have thought of maths for most of the last 3 decades.

From here, he starts to ask some more fundamental questions about the nature of numbers, including complex numbers (but not quaternions) and some “proper” algebra though he cunningly avoids the use of terms such group, ring or field whilst ensuring the reader is familiar with their rules by means of definitions followed-up with examples. He also touches on some rules regarding logarithms which perplex some people, but are dealt with very well.

He then goes on to probably the most important idea in maths: proof. Though touching on a little philosophy, he tries to skirt around it and give a robust exposition of what a mathematician means when (s)he talks about proof and how it differs from the more lackadaisical use of the word in everyday (and even some other areas of scientific) usage.

Though any book on serious mathematics probably ought to contain a good amount on calculus, Gowers avoids this quite ostentatiously. Rather, he lays the groundwork for an understanding of it with a chapter on limits and infinity. In so doing, one might think he’s dodged a potential bullet of losing the interest of readers, though I think that anyone who hasn’t done calculus but who has understood this chapter will be well-placed to start studying calculus.

Moving on, we start to get a bit more geometrical. The first of these two chapters looks solely at the idea of dimensionality. One might instinctively have an idea of what a dimension is or how to count them. However, maths is rather more refined than such instinctive generalities and Gowers gives us some examples that any student would find in a 1st year linear algebra course. If anything, this is the acid test for those considering doing maths at university; if this is incomprehensible to you, then it’s best to turn away. But if, on the other hand, you can see there’s something there that can be grasped, if you don’t quite get it exactly at this stage, then you’re standing in good stead. The second of the geometrical chapters looks at much more fundamental geometry, looking at the classic issue of Euclid’s 5th postulate and the consequences of abandoning it.

The last main chapter is on one of my least favourite topics: estimates and approximations. This was a topic I did to death in my 2nd year at uni, not getting on with the lecturer (an angry Scot called Alan) and having a nightmare on the exam, barely scraping a 2:1 in it. Gowers doesn’t hide from the fact that this is not neat maths, as most of the rest of the book is, though he spares the reader from an insight into the gory detail of numerical analysis (or num anal, as we disparagingly called it). If there’s any downside to this wonderful introduction, it is this chapter. Not because it is badly written or poorly explained, it isn’t. But merely because it lacks the sexy panache that the rest of the topics have.

This is redeemed somewhat at the very end when he puts in some frequently asked questions, along with his answers. If anything, these give a far better insight into the working life of the mathematician than anything found in the rest of the book. He addresses some myths and common (mis)conceptions, giving an honest assessment on some issues where he needn’t have done so.

Overall, it’s a great book. If you hated maths at school, then this isn’t the book to get you back into it and loving it. But for anyone with a keen interest in science and how it’s done, then this is a gem. If you know any A-level students considering doing maths at uni, give them this book. Hey, they are students, so £8 for a little book is a lot of money. Go on!

Friday fun: Am I any good at playing Hearts?

As it’s Friday, I decided to take a bit of a break from the relatively serious blog posts I’ve been turning out of late and take a light look at probability and games. Games, after all, are a great arena for testing out ideas in combinatorics and statistics.

The question posed might seem to be an obvious ‘yes’ but just because something might seem obvious doesn’t always mean it’s true. I’ve had my current laptop now for a little over two years. In that time, I’ve played 345 games of Hearts on that computer. I will assume most of you have either played it, or at least know how to play. If not, here’s a quick guide.

Given that there are 4 players, if the game were entirely random, one would expect, over time, to win roughly a quarter of the games played. If your win percentage was higher, you might be justified in thinking that you’re better than the computer. If your win percentage was lower, you might think you were worse, though most people would quickly move on from that thought to trying to find an excuse.

So, how many games do you think I’ve won? Is it not 345/4 = 86.25? Obviously not, because I can only have won a discrete number. So should my expected number of wins be 86 or 87? Obvious rounding would say 86. Though since 87 is so close, that would also seem a reasonable number to have won. What about 90, though? That still seems to be within the realms of possibility. 100, perhaps? How about 120? That might seem less likely, as that would mean I’ve won over a third of the games, yet I only expect to win a quarter.

The answer is in fact 188, giving me a win percentage of just over 54%. Does that really prove that I’m good at Hearts? To try to give it some meaning, I’d like to know what the probability is of winning 188 games out 345 and compare this to the odds of winning 86 time out of 345. But there might be a snag. We would expect that the odds of winning 86 times would be the highest, but there are 345 different possible outcomes. So is one really much more likely to win 86 times than 87? Instinctively, it seems not, but we need to quantify to this to make much sense of this, and confirm or deny what “feels” right.

How do we work out the probability?

I’ll start off with a simple case, where we have only played 2 games. If it were random, we would expect to win 25% of the games. i.e. the probability of winning any given game is 0.25. Similarly, we would expect to lose 75% of the games i.e. a probability of 0.75.

As each game is independent, we need to multiply the probabilities together. So the odds of winning:

  • 0 games out of 2 would be 0.75*0.75 = 0.5625
  • 1 game out of 2 would be 0.25*0.75 = 0.1875
  • 2 games out of 2 would be 0.25*0.75 = 0.0625

But there’s a problem. Add them all up and you should achieve certainty, a probability of 1. But instead, our total is 0.8125. It falls short of 1 by 0.1875. It’s no coincidence that that matches the probability of winning 1 game out of 2. That’s because there are 2 ways we would that single game: we could win the first and lose the second, or lose the second and win the first. So we have to multiply that by the number of ways you can choose 1 ‘slot’ given 2 ‘slots’ to choose from.

How about we try it with 5 games now (to 4 decimal places):

  • 0 games out 5 would be (0.25^0)*(0.75^5) = 0.2373
  • 1 game out 5 would be (0.25^1)*(0.75^4) = 0.0791
  • 2 games out 5 would be (0.25^2)*(0.75^3) = 0.0264
  • 3 games out 5 would be (0.25^3)*(0.75^2) = 0.0088
  • 4 games out 5 would be (0.25^4)*(0.75^1) = 0.0029
  • 5 games out 5 would be (0.25^5)*(0.75^0) = 0.0010

Again, we have undercounted. With the 0 and 5 cases, there is only one arrangement each by which you can win or lose all 5 games. But to win 1 game (or to lose 1 game) there are 5 ways to do this. So we need to multiply the 1 & 4 cases by 5.

But what about 2? How many ways are there to choose 2 games to win (or to lose) from 5 opportunities? Without going into all the detail it is 5!/((5-2)!)*2!) = 120/(2*6) = 10

So to take into account the multipliers, we get the probabilities:

  • 0 out of 5 = 1*0.2373 = 0.2373
  • 1 out of 5 = 5*0.0791 = 0.3955
  • 2 out of 5 = 10*0.0264 = 0.2637
  • 3 out of 5 = 10*0.0088 = 0.0879
  • 4 out of 5 = 10*0.0029 = 0.0146
  • 5 out of 5 = 1*0.0010 = 0.0010

Adding up, we get back to our reassurance that the sum of all probabilities is 1.

What about 345 games then? Well, we just extend the pattern. The odds of winning 188 games are:

(345!/((188!)*((345-188)!)))*(0.25^188)*(0.75^(345-188)) = well, something very small indeed.

A standard calculator won’t be able to do the calculation, but with a little help from a more powerful computer (aka Excel on my laptop), the answer is roughly 0.00000000000000000000000000000012063

That seems pretty darn small. But there are a lot of options (345 to be precise) and most look pretty small. What about our expected figure of 86?

The odds for that (which I leave to you check) are 0.049574108. It is the highest probability for any number of wins, but it may surprise some that the odds of getting 25% of the wins given the odds of winning are 25% are in fact slightly worse than 1 in 20, or 19/1 against. Suddenly, the idea of a direct comparison doesn’t seem so sensible anymore.

Though our odds of winning 86 games are about 410,937,214,868,030,000,000,000,000,000 greater than winning 188 games, I’m still sceptical. What we need to do is look at some kind of spread around our two values of 86 and 188.

Let’s look at each plus or minus 20 (arbitrarily chosen, I admit, please feel free to suggest or try alternatives). So what are the odds of winning between 66 and 106 games? And what are the odds of winning between 168 and 208 games?

For that, I refer back to the working spreadsheet (which I can send you if you don’t believe me and can’t figure out how to design it yourself) and we get the former to be a quite reasonable 0.992543. In other words, if there was no skill in Hearts than you would have more than a 99% probability of winning between 66 and 108 games out of 345.

The latter turns out to be 0.00000000000000000000230188 or 1 in 434,427,345,316,048,000,000.

It is not impossible that my winning so many games is a fluke of the probabilities and that I have hit upon a streak that no one else in the history of the universe will have ever likely encountered before. It just seems very very very very very unlikely.

Maybe, then, it’s not just luck. Maybe I am more skilled at playing the game than the computer is. I’d certainly like to think so, though ‘liking to think so’ can be the downfall of anyone look at statistics…

Book Review: Thinking in Numbers by Daniel Tammet

This was by no means one of the books that I had ever intended to read. I’d never heard of the author nor had this particular book been recommended to me. I found it whilst perusing the science section of a local bookshop, having already determined that I would leave with at least once science book in my possession. I think I’ve said before, certainly in person, if not on this blog, that I am a sucker for a good cover. Not only was the title good enough to make me pick up to take a closer look, but the description of the book and the recommendations printed on both the back and the front were enough to make me think this was worth paying a little bit of money for.

The book is written as a series of short essays, seemingly distinct and with little to no overall narrative to it. So it’s a good book to have lying around that can be picked up, read for 15-20 minutes and put down again.

It covers a variety of topics from Tammet’s point of view. It must be noted that Tammet (not his real name, he changed it to ‘better fit’ his identity) is described as a high functioning autistic savant. In short, he’s a really clever chap. Now I’ve come across one or two in my time and have been able to hold my own against them in some intellectual challenges. However, they usually get the upper hand on me and I can’t quite emulate their speed or agility of thought, which I admit has been a cause of some chagrin from the age of 17 onwards; before that, I immodestly add, I was always the cleverest person I knew.

So it was with some relish, and a little touch of rivalry, that I wanted to get to see the world through such a savant’s eyes. In many respects, what I was reading seemed to be the account of a more articulate version of myself, with the only difference that Tammet views numbers as colours. I knew several in the maths department at university who did this, but I always think in terms of ‘complements’ – i.e. what number would you need to add to make a round number? So if someone says 7, I think 3. If they say 83, I think 17.

I probably ought to confess that I finished this review a few weeks after reading the book, so I am relying a little on memory. While reading it, I found it quite fascinating, but a few weeks later the only things that really stick in my mind are the fact that he came from a very large family and a compelling account of his recitation of digits of pi. This last bit was especially impressive as it ran on for thousands of digits and the recitation took several hours to complete.

The book will be of note to anyone interested in maths or to those who are keen in trying to understand how other people tick. Indeed, if anyone wants to understand me, then this is an account that gets as close to me as any I think I have read; although there are some marked differences.

Book Review: The Emperor’s New Mind by Roger Penrose

This was another of my ‘books of shame’ that I felt the need to re-read. I actually got quite a long way into it first time and I can’t recall why I put it down. The aim of the book is to explore the notion of artificial intelligence (AI), whether or not machines can truly “think”. In order to get to this question, Penrose first spends a lot of time (most of the book, in fact) looking at a wide variety of seemingly unrelated topics.

After an initial discussion of AI, Penrose launches straight into what is probably the hardest chapter to get your head round. It’s all about algorithms, Turing machines and the computability of mathematical problems. He doesn’t spare the detail with pages of binary digits and computer programming languages. It takes a long time to work through, but if you can brave it, there is much easier, and more enjoyable, science in later chapters.

Once you get over the initial hump, we ease back into some gentle maths with Penrose first outlining his neo-Platonic view of notions of reality (one I admit that I share with him). He does this via some very basic complex analysis, looking at the detail of the Mandelbrot set, though without going into too much depth for the casual reader. From here he looks at the world of classical physics and then quantum physics, giving the reader a general grounding in the basics of modern physics whilst every now and then alluding back to the premise of the book, essentially asking if a machine could ever be constructed that would be capable of making the intuitive leaps that humans have managed in coming to our present understanding of the cosmos.

For the most part, this should be readily understandable with a modicum of scientific education, though to someone who didn’t do maths or physics at A-level, much of it may be new and take significantly longer to get to grips with. But even the expert reader shouldn’t get complacent. Penrose’s approach takes much which we may be familiar with and turns it sideways, giving good reason to scratch our heads and think things through anew. The 2nd half of the chapter on quantum mechanics is, admittedly, a bit tougher to get through; the section on spin was where I found my bookmark from the first time I tried to read it and gave up.

After finishing with quantum mechanics he looks at the thermodynamics of the universe, a line of thinking which led, many years later, to Cycles of Time. He ponders over some ideas of quantum gravity but not to any depth that one might be satisfied with. For other takes on that, I’d recommend The Road To Reality (also by Penrose), Brian Greene’s The Elegant Universe or The Three Roads To Quantum Gravity by Lee Smolin.

Eventually, Penrose comes back to the question of AI. In order to do this though, he needs to look at the basic physiology of the brain. Now Penrose is a mathematician and a physicist; he’s not a neurologist. As such, this section of the book doesn’t come across anywhere nearly as strong as the rest of the book. It is clear that this is a written by an educated amateur in the field rather than an expert. For much more detail on how the brain works, I would recommend John Ratey’s A User’s Guide to the Brain.

One fascinating idea that Penrose puts forth is that what may distinguish human intelligence and consciousness is not our rationality, but our irrationality. If all people behaved in accordance with a strict rationality (though even most rationalists, myself included, exhibit some irrational behaviour from time to time) then the strong AI proponents might have more of a case. But the very evidence of irrational behaviour is what Penrose finds most interesting.

Ultimately, no firm propositions are put forward in this volume. The book ends with some musings and a tentative point of view. I intend to follow up, albeit not for a while, with Penrose’s later volume, Shadows of the Mind. In the meantime, what we have is a book which is very loosely about artificial intelligence, but which is really a book about the foundation of computing, along with a tour of some of the great ideas of maths and physics.