Tag: maths

Jul 10 2022

Book review: The Art of More by Michael Brooks

the_art_of_moreThe Art of More by Michael Brooks is a history of mathematics written by someone whose mathematical ability is quite close to mine – that’s to say we did pretty well with maths at school but when we went to university we reached a level where we stopped understanding what we were doing and started just manipulating symbols according to a recipe.

The book proceeds chronologically starting with origins of counting some 20,000 years ago and finishing with information theory in the mid-20th century with chapters covering arithmetic, geometry, algebra, calculus, logarithms, imaginary numbers, statistics and information theory.

It is probably chastening to modern mathematicians and scientists that much of the early work in maths on developing the number system, including zero and negative numbers, was driven by accounting and banking. Furthermore, much of the early innovation came from China, India and the Middle East with Western Europe only picking up the ideas of zero and negative numbers in around the 13th century.

Alongside the development of the number system, the ancient Greeks and others were developing geometry, the ancient Greeks seemed to go off numbers when they discovered irrational numbers – those which cannot be expressed exactly as a ratio of integers! Geometry is essential for construction, surveying, navigation and mapmaking – sailors have often been competent mathematicians – through necessity. Geometry also plays a part in the introduction of accurate perspective in drawings and paintings.

Complementing geometry is algebra, developed in the Arabic world. Our modern algebraic notation did not come into being until the 16th century with the introduction of the equals sign and what we would understand as equations. Prior to this problems were expressed either geometrically or rather verbosely.

Leading on from algebra was calculus – the maths of change. It started sometime around the beginning of the 17th century with Kepler calculating the volumes of wine barrels whilst he was preparing for his wedding. There was further work on the infinitesimals through the century before the work by Newton and Leibniz who are seen as the inventors of calculus. I was struck here by how all the key characters in the development of calculus Newton, Leibniz, Fermat, Descartes and the Bernoullis all sounded like deeply unpleasant men. Is this the result of the distance of history and the activities of various proponents for and against in the intervening centuries? Or were they really just deeply unpleasant men?

Doing a lot of calculation started to become a regular occurrence for sailors, as well as people such as Kepler and Newton working on the orbits of various celestial bodies. John Napier’s invention of logarithms and his tables of logarithms, published in 1614 greatly simplified calculations. It converted multiplication and division into addition and subtraction of values looked up in his tables of logarithms. The effort to create the tables was massive, it took 20 years for Napier to prepare his first set of tables, containing millions of values. Following Napier’s publication in 1614 logarithms reached their modern form (including natural logarithms) by 1630. In addition mechanical calculating devices like the slide rule were quickly invented. I grew up in a house with slides rules, although by the time I was old enough to appreciate them electronic calculators had taken over. Napier was also an early promoter of the modern decimal system. Logarithms also link to exponential growth, highly relevant as we still wait for the COVID pandemic to subside.

Historically the next area of maths is the invention of imaginary numbers, if you don’t know what these are then I’m not going to be explain in the space of a paragraph! There is a link here with natural logarithms through Euler’s identity which somewhat ridiculously manages to link e, pi and i in one really short equation. I was not previously familiar with Charles Steinmetz who introduced complex numbers into the analysis of electrical circuits responding to alternating currents – although it is a very elegant way of handling the problem and a method I used a lot at university. Largely when we talk about complex numbers we are discussing the addition of i, the square root of -1, to our calculations. But there are additionally quaternions, invented by William Hamilton, which add three complex numbers: i,j and k to the real numbers but the limit is octonions – a system of seven complex numbers and the real numbers. I am curious as to why we cannot have more than 7 flavours of complex numbers.

Statistics is my area of mathematics, I’m a member of the Royal Statistical Society. I think the thing I learned from this chapter was that the word "statistics" has its origins in German and "facts about the state". I quite liked Brooks’ description of p-values which seemed particularly clear to me. Brooks highlights some of the sordid eugenicist history of statistics, as well as the more enlightening work of Florence Nightingale and others.

The book finishes with a chapter on information theory, largely based on the work of Claude Shannon but with roots in the work of Leibniz and George Boole. George Boole invented his Boolean logic in an attempt to understand the mind in the mid-19th century but his work on "binary" logic was neglected for 70 or so years until it was revived by Shannon and other pioneers of early computing.

This is a fairly informal history of mathematics, I found it very readable but it includes a number of equations which might put off the completely non-mathematical.

Oct 28 2017

Book review: The Art of Strategy by Avinash K. Dixit and Barry J. Nalebuff

art_of_strategyNext up, some work related reading. The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life by Avinash K. Dixit and Barry J. Nalebuff.

The Art of Strategy is about game theory, a branch of economics / mathematics which considers such things as the “ultimate game” where one player choses how to split $100 (i.e. keeping $60 and giving away $40) and a second player decides to accept or reject the split, in the latter case neither of them gets any money. In the former case they get the offered split.

In the “prisoners dilemma” two prisoners are each offered the opportunity to give evidence against the other. If one of them does this, and the other doesn’t, then they will be set free, whilst their fellow prisoner services a sentence. If both betray the other then they will both serve a longer sentence than if they had both kept quiet.

These examples represent the simplest two main types of game, the ultimate game is an example of a sequential game (where one player makes a decision followed by the other) whilst the prisoners dilemma is an example of a simultaneous game (where players make their decisions simultaneously). In real life, chess is an example of a sequential game and a sealed bid auction is a simultaneous game. Games are rarely played as a single instance, simultaneous games may be repeated (“the best out of 3”), and sequentially games may involve many moves. This repetition enables the development of strategies such as “tit for tat” and punishment. 

The ultimate game and the prisoners dilemma provide a test bed for game theory, normally illustrating that real humans don’t act as the rational agents that economics intended! For example, in the ultimate game players really should accept any non-zero offer since the alternative is getting nothing, in practice players will reject offers even as high as $10 or $20 as unjust. 

Sequential games are modelled using “game trees”, which are like “decision trees”. Simultaneously games are modelled with payoff tables. The complexity of real sequential games, such as chess, means we cannot inspect all possible paths in the game tree, even with high power computing.

The first part of the The Art finishes with some strategies for simultaneous games. These are to look for dominant strategies where they are available, i.e they are the best strategy regardless of what the other players do. If this isn’t possible eliminate dominated strategies, i.e. those which are always beaten by your opponent. Nash equilibria are those moves which could not be improved, even given knowledge of an opponents moves. There can be multiple Nash equilibria in a game, which means if strategies are not explicitly stated the the players must guess which strategy the other player is using and act accordingly. This section also covers how social context influences play, and ideas of “punishment”.

The second part of the book looks at how the strategies described in the first part are used in action, although these examples are sometimes somewhat hypothetical. This part also introduces randomness (called “mixed strategies”) as a component of strategies.

The final part of the book covers applications of game theory in the real world, including auctions, bargaining and voting. I was interested to learn of the several sorts of auction, the English, Dutch, Japanese and Vickrey. The English auction is perhaps the one we are the most familiar with, participants signal when they wish to make a bid, and the bid rises with time. The Japanese auction is similar in that the bid is always rising but in this case all bidders start in the auction with their hands raised (indicating they are bidding) and put their hands down when the price is too high. A Dutch auction is one in which the price starts high, and drops, the winner is the one who first makes a bid. Finally, a Vickrey auction is a sealed-bid auction where the winner is the one the makes the highest bid, but they pay the second highest value.

Auctions are big money, the UK 3G spectrum auction in 2000 raised £22.5 billion from the participants. It’s worth spending some money to get the very best game theorists to help if you are participating in such an auction. The section on bargaining is relevant in the UK at the moment given the Brexit negotiations, particularly the idea of the Best Alternative to a Negotiated Agreement (BATNA). Players must determine their pay off relative to the BATNA, and must convince their opponents that the BATNA is as good as possible.  

I found the brief descriptions of  concrete applications of game theory such as in the various “spectrum” auctions for mobile phone systems, and the formation of price fixing cartels the most compelling part of the book.

Game theory is a central topic in at least parts of economics, as witnessed by the award of the pseudo-Nobel Prize for Economics in this area – there is a handy list here (http://lcm.csa.iisc.ernet.in/gametheory/nobel.html), if you are interested.

The Art of Strategy has some overlap with books I have read previously, the decision tree/game trees have some relevance to Risk Assessment and Decision Analysis with Bayesian Networks by Fenton and Neil (which uses the Monty Hall problem as an illustration). The Undercover Economist by Tim Harford discusses game theory and its relevance to the mobile frequency auctions in the UK, as well as the example of information in buying second hand cars. The Signal and the Noise by Nate Silver has some discussion of gaming statistics.

Apr 08 2017

Book Review: Hidden Figures by Margot Lee Shetterly

hidden_figuresHidden Figures by Margot Lee Shetterly tells the story of Africa American women who worked as “computers” at NASA and its predecessor NACA during and after the Second World War.

In a first, this means I am currently reading both fiction and non-fiction by African-American women. (I’m also reading The Parable of the Sower by Octavia E. Butler)

The Hidden Figures worked initially in the West Area Computing Group at the NACA Langley Research Centre in Hampton, Virginia, which did reseach on aircraft and then rocket design. The Computing groups carried out calculations at the behest of engineers from around the Centre, this was at a time when calculation was manual or semi-manual compared to today. Over time they were co-opted directly into research groups, some of them to ultimately become engineers. The West Computing group was mirrored by the East Area Computing Group – comprised of white women.

There is some history for women acting as “computers”, and the necessity of World War II led to the government taking on Africa American women for the job, in face of historic segregation. For African American women this was a rare opportunity, until then the only recourse for African American woman with advanced training in maths was teaching. For a very few the Computing group ultimately acted as a stepping stone to working as an engineer.

Shetterly sees these women as a vanguard to the African Americans in the modern US who have every opportunity open to them. This jars a little to me when I see constant news from the US of, for example black people being more likely to be killed by the police, or a senior African American being brought together by the President with the policeman that aggressively interviewed him on his doorstep because the house looked too nice to belong to a black man. Or African Americans being purposefully disenfranchised.

The shocking thing to me, as a Brit, was the degree to which US society was absolutely, formally segregated on racial grounds. In Virginia, where this story is set, segregation was preserved by the Democratic Party (perhaps some explanation as to why African Americans are not necessarily whole-heartedly Democrats). In Prince Edward County, Virginia they went as far as shutting down all the public schools for 5 years in order that black and white children would not be educated together – white children were given grants to study at private schools. Britain may have been racist in the past, it may still be racist today but it never enshrined it so deeply and widely into law.

In response to this Africa Americans ran a parallel community, segregation didn’t end because the segregation laws were repealed. It ended because African Americans saw the end of those laws as a door ajar which needed a serious push to pass through. Thus when Rosa Parks sat on the bus, Katherine Goble (from this book) went to university and Ruby Bridges went to school they didn’t do so entirely alone. They had the support of their community and the organisation of the NAACP to help them. They had to be twice as good as a white person to get the same job. At the same time they also saw themselves as representatives of their race, and examples to their children.

When you look at a man the age of Donald Trump, 70, it’s worth bearing in mind that his teenage years were spent during the end of segregation by law and his parents were the white generation which fought so hard to keep it.

The focus of the book is mainly the personal lives, and ambitions of the women. There is some description of the work they, and the Research Centre did, but not in any great depth. The book highlights again the transformative effect of, particularly, the Second World War on society in the US. The seeds of theses changes could be seen after the First World War. This mirrors similar changes in society in the UK.

Once “computing” became the realm of high capital machinery the importance of women as computers waned, high capital machinery being the preserve of men. We see the consequences of this even now.

The book finishes with the part Katherine Johnson, in particular, played in John Glenn’s first trip into orbit and her subsequent work on the Apollo moon landing and Apollo 13 recovery. Shetterly emphasises the legacy of this group of women that normalised the idea that Africa American women could ultimately become engineers, scientists or any other sort of professional.

Interestingly my wife and I disagreed on the prominence of the men on the cover of the book (see above). She thought they were central and thus important, I thought they were small and thus unimportant. In the text the men are bit-part players, they are husbands and sons, or drift in and out of the narrative having spoken their line.

Dec 13 2009

Drowning by numbers

I am not a mathematician, but physics and maths are intimately entwined. I suspect I stumble on a deep philosophical question when I ponder whether maths exists that has no physical meaning.

On a global scale I am moderately good at maths, I have two A levels* in the subject (maths and further maths), long years of training in physics have introduced me to a bit more. However, beyond this point I realised I was manipulating symbols to achieve correct results rather than really knowing what was going on. A lot of my work involves carry out calculations, but that’s not maths.

I did intend decorating this post with equations, I didn’t in the end, wary of a couple of things: firstly the statement by Stephen Hawking that every equation would half sales; secondly I discovered that putting equations into Blogger is non-trivial. Equations, statements in mathematical notation are the core of maths and much of my journey in maths has been in translating equations into an internal language I understand.

So here’s a pretty bit of maths, the Mandelbrot set, the amazing thing about the Mandelbrot set is how easy it is to generate such a complex structure. We can zoom into any part of the structure below and see more and more detail. Mathematics is the study of why such a thing is as it is, rather than just how to make such a thing.

Image by Wolfgang Beyer

I remember playing with Mandelbrot sets as a child, before I understood complex numbers, to me they were a problem in programming and a source of wonder as I plunged ever deeper into a pattern that just kept developing. Have a look yourself with this applet… *time passes as I re-acquaint myself with an old friend*. There is something of this towards the end of Carl Sagan’s novel Contact, where the protagonists discover a message hidden deep within the digits of π.

I fiddle with numbers when I see them, and I suspect mathematicians do too. So my Girovend card showed 17.29 recently which, without the decimal place is 1729 = 13+123 = 103+93, the smallest number that has the property of being the sum of two different pairs of positive cubes, it’s also a very common piece of numerology. The numbering of the chapters of “The Curious Incident of the Dog in the Night-time” by Mark Haddon, with consecutive prime numbers also appeals to me.

It’s become a tradition that I find ways to annoy the people I visit, and there’s no escape for mathematicians here. It seems like the best way is to annoy a mathematician is to assume they can do useful arithmetic, like a calculating a shared restaurant bill. Interestingly though, this may be a poor example, since fair division methods for important things, like cake, are an area of mathematical research.

It’s true that some mathematicians are a bit odd, but then so are some physicists and to be honest if reality TV has taught us anything, it’s that the world is full of very odd people in every walk of life. So if you meet a mathematician, don’t be afraid!

*A levels are the qualification for 18 year olds in the UK, when I was a student you would study for 3 or 4 A levels for 2 years.

Update: Since writing this I’ve discovered a couple more interesting sites for fractals, and for want of a better place to put them I record them: here you can find a pretty rendering of the quaternion Julia set, and here is an in depth exploration of the Julia and Mandelbrot sets (1/1/10).