Arithmetic

Title: Arithmetic

Author: Paul Lockhart

Publisher: Belknap Press of Harvard University Press, 2017 (First)

ISBN: 9780674972230

Pages: 223

Even though arithmetic forms only a small subset of mathematics (when you say something about math, it causes no harm using some of its parlance such as subsets), for many people it is the science of mathematics. And, it is the subject that scares away children from appreciating the art and experiencing the joy of doing math. It is forced learning by rote and mechanical procedures for performing arithmetical operations without stopping a moment to perceive what is going on that repels so many children away from it. Clearly, a new paradigm is needed to teach math in elementary schools, making it a participatory experience for the students. However, the sad fact is that most of the teachers were also churned out by the system of memorizing of multiplication tables and doing long division by carefully choreographed procedures which keeps you on tenterhooks until the final result appears. This book is a welcome change to the genre of popular math. It teaches a teacher of math how to teach it to young boys and girls and to make a generation quite confident of dealing with it. The methods and structures detailed in the book are highly oriented to practical learning, while clearly explaining the theoretical aspects in an easy to follow way. Paul Lockhart teaches math at St. Ann’s School in Brooklyn, New York. He is the quintessential teacher who renounced his prestigious career in the university to teach in a grade school. He is the author of two more books on learning math.

Arithmetic is the skillful arrangement of numerical information for ease of communication and comparison. Lockhart makes it categorically clear that being good at it does not make you particularly smart or mathematically inclined. It is like any other craft you can get good at if you wanted to, but it is no big deal either way, assures the author. The book gives an excellent introduction to the two main schemes of arranging numerals – the marked-value and place-value systems. It starts right from the activity of counting where you take stock of something by assigning a numeral to denote something in the real world. Actual examples from the Egyptian, Roman, Chinese and Indian systems are given. It is the place-value system developed in ancient India that revolutionized the way the world did its calculations. Invention of the symbol and concept of zero as a place holder to denote a null value was a novel concept, without which the explosion in information exchange wouldn’t have come into being. The Indian numerical system reached Europe in the thirteenth century through Arab traders. Leonardo of Pisa – better known as Fibonacci – showcased the new ideas in his book Liber Abaci (Book of the Abacus). Strangely, such a fool proof system was slow to catch on with the general public in Europe. Such is the aversion of people to change! Even as late as the eighteenth century, well-educated adults found it confusing and overly technical. Eventually, convenience and increasing availability of inexpensive paper won out over the traditional Roman numerals.

The book is an attempt right from the first page to the last to kindle the spirit of appreciation for mathematical beauty among the readers. In case anyone still ‘feared’ numbers, Lockhart encourages them with sage advice that mathematical operations are only good strategies for encoding and manipulating numerical information, and we can use them in any way we see fit. Instead of thinking in terms of systems and rules, we should think of it more as options and tools at our disposal (p.73). The entire gamut of basic mathematical operations like addition, subtraction, multiplication and division are covered in detail and provide new insight into the heart of the problem. Even those who are well-versed in math would learn one or two new ideas from this impressive book. Lockhart also attempts to detach numbers and its representations from its homologues in the real world. It is not always possible to find a related process happening around us. Math or arithmetic is abstract, making reification – the process of searching for counterparts in the real world – extremely difficult and sometimes impossible. Whether or not it is physically possible, arithmetic imagine or invent some sort of logically coherent structures and forms a part of mathematical reality. This disclaimer is issued in the context of negative numbers which can’t be compared readily to any real thing. But it provides an excellent tool for making calculations to arrive at the result that has real significance. Even though Lockhart doesn’t mention it in this book, complex numbers – which are square roots of negative numbers – are another realm of arithmetic which has no physical meaning, but is immensely useful in calculating and comparing real world data.

There is nothing to be said against the book, except its small type size and the absence of categorization into distinct chapters. The entire volume appears to be divided into several topics of varying lengths, more akin to an encyclopedia rather than a normal work of learning. Even though the focus is on aspiring teachers, the author has cleverly included several practice questions without making it appear as an exercise. The chapter on mechanical calculating machines could’ve been eliminated, as it proves a redundant chapter that doesn’t contribute anything to the general thread of argument. This book is an excellent choice for anyone who has set his heart on becoming a teacher of elementary schools, which will surely help them become an arithmetician of wit and intelligence.

The book is highly recommended.

Rating: 5 Star

A Strange Wilderness

 

Title: A Strange Wilderness – The Lives of the Great Mathematicians

Author: Amir D Aczel

Publisher: Sterling New York, 2011 (First)

ISBN: 9781402785849

Pages: 284

Some people among us don’t relish the prospect of studying mathematics. The probable reason for this aversion is mostly improper assimilation of fundamentals caused due to lapses in the method of teachers who taught them in primary schools. Such people opt for the inexact sciences like biology or humanities like history when the time comes to make a choice. However, reading about the development of mathematics and the lives of its pioneers is as exciting and satisfying as any. So, this book will be interesting for both math-philes and math-phobes equally. Man innately possesses the ability to compute with simple numbers. Research states that even birds do retain a basic sense of number! The origins of mathematics was surely associated with counting, as those early settlers on the fertile river valleys of Nile and Euphrates-Tigris used them to keep account of their livestock. Gradually, other applications developed, like keeping track of the seasons by counting elapsed days. Early astronomers used it extensively to predict the sowing time. As time went on, mathematics became more complex and began to be applied to all aspects of life. An amusing example of a peculiar rule of marriage among the aborigines of New Guinea presented in the book shows that mathematics can be extended to human relations as well. Amir D Aczel has produced nearly a dozen books on science and mathematics. He lives in the United States and contributes to newspapers and television also. In this nice book, he tells the story of mathematics developing from humble origins to what it is today – touching the everyday lives of all civilized societies in numerous ways. Some books on the mechanism of human brain state that the faculty of language and mathematics will not be developed simultaneously in people. However, this book presents several mathematicians who were adept at both. This pleasantly readable work is a must-have for students of mathematics.

The first two parts of the book neatly sums up the work done by ancient scholars in Egypt, Greece, India, China and the Arab world. Contrary to our expectation, intellectuals in the ancient period also traveled far and wide in search of knowledge. We read about Greek scholars visiting Babylon and Egypt to partake of the knowledge amassed in these cradles of civilization. Thales of Miletus was inspired to formulate the first theorem of mathematics on a visit to the Great Pyramid of Cheops in the 6^th century BCE. Anxious to find the height of the pyramid, he devised an ingenious way by measuring the length of the shadow cast by the structure, which is still intriguing. Restriction of knowledge to the initiates alone had begun in those times in the case of Pythagoras and his disciples, who were very particular in keeping the word to themselves and even going as far as to kill some of their brethren who wanted to spread the message on the existence of irrational numbers which challenged their own intellectual foundations. Aczel gives a fitting representation of Indian thought guided by Aryabhata and Vishnugupta. Though he remarks that the contributions of these masters may have been guided by assimilation of Greek thought diffused through increased trade between the two countries, he has been straightforward in assigning the invention of algebraic and trigonometric ideas to India. Greece excelled in geometry. When the classical age ended in Greece and Alexandria, the beacon of learning passed to the Arabs who kept it lit till Renaissance, when it was handed over to Europe. Combining elements from Greece and India and producing original thought of their own, Arab mathematicians founded the roots of some of the branches of modern mathematics. The term algebra derives its etymology from a treatise called ‘Al Gabr Wa’l Muqabala’ by Muhammad ibn Musa al-Khwarizmi who lived in the court of caliph al-Mamun. Signs of influence of Brahmagupta’s work ‘Brahmasphuta Siddhanta’ are said to be unmistakable in al-Khwarizmi’s work (p.46). With Jamshid al-Kashi (1380 – 1429), Arab scholarship faded into oblivion. Arabs translated ancient Greek manuscripts and Indian numerical notation to Arabic, which was translated to Latin in the Middle Ages, which helped Renaissance science to flourish. The book also sets aside a chapter on Chinese origins of mathematical concepts.

The seventeenth century CE may be credited with the honour of the origin and development of modern mathematics. Descartes, Newton and Leibniz shone with meridian splendour in this period, among an impressive array of scholars. The sharp disparity between England and continental countries like Germany are seen here. While in England it was possible for a talented man to find avenues for further study and research such as Cambridge and Oxford, without worrying too much about the financial circumstances of leading their daily lives, in Germany and other countries, the scholar had had to apply for patronage to a feudal lord or leading members of the clergy. Naturally, such a system was vulnerable to the fortunes of the patron in a battle or to the loss of favour of the patron himself with the king. Wherever there was a stable government, scholarship flourished. France led the field till the beginning of the nineteenth century on account of this, while Germany was splintered among a plethora of weak city states. After the downfall of Napoleon and amid the unsettled political turmoil which followed it, France lost its position it had enjoyed with the work of Laplace, Legendre, Galois, d’Alembert and Lagrange. Germany, consolidated in this century on the political front, and its repercussions were seen in mathematics as well, with the advent of notable personalities like Cantor, Dedekind, Weierstrass and others. We note another noteworthy fact in this regard. Many mathematicians in the Renaissance era were devout Christians, Newton being the most prominent. Mathematicians’ personal beliefs inevitably seeped into their work too. Newton studied the solar system in light of gravitational forces exerted by the bodies in orbit and reached the conclusion that it is stable in the long term due to God’s intervention. Laplace, an atheist who studied the same problem in a Europe conditioned by Enlightenment, declared boastfully that the stability of the solar system is not in need of the god hypothesis. As can be expected, he also reached the conclusion that the solar system is stable.

When we reach the modern period, mathematics has grown complex and out of reach of common people. No fundamental advance has taken place in the last 150 years, except perhaps the impetus made in non-Euclidean geometry by the development of Einstein’s theory of relativity. Researchers studied some of the highly specialized attributes of a theory, aloof from the buzzle of the street and away from any concern to find an application for the theory. Practitioners of pure mathematics take pride in the fact that the extreme abstractness of their field precludes the necessity to look for a practical way to employ the theory. When no path breaking advances were forthcoming, mediocrity set in. Even though Aczel praises the effort of Nicolaus Bourbaki, a group of maverick mathematicians posing as an individual, and Alexander Grothendieck, readers get a feel that instead of pioneering new ways, they have gone in search of cheap popularity tricks and pranks. Grothendieck was a researcher who suddenly turned to politics and environmentalism and effaced himself from public view by hiding somewhere in the Pyrenees. In an act of sheer irresponsibility, he burnt all his contributions to mathematics in addition to taking all electronic content off the Internet. Aczel revers this man, but readers believe that he is an impostor.

The text is very easy to read through, presented in a concise but effective way. All the usual anecdotes and events are included, but the book doesn’t advance any original ideas except the flawed one on the greatness of Grothendieck. There are no exclusive information available in this book, which is unattainable from others. Lot of photographs and paintings are included, along with a good index. The bibliography is extensive. However, the narration abruptly ends, without a proper epilogue or musing about the future course of mathematics. In this vein, it may be thought of as a description without insight or any contribution from the author apart from compiling data about various mathematicians. However, the author gives a respectable mention of Indian masters of old and new, and wholeheartedly acknowledges their pioneering roles. A number of sidebars are provided, but they blend confusingly with the text as the layout doesn’t neatly separate them from the main text.

The book is recommended.

Rating: 3 Star

Is God A Mathematician?

Title: Is God A Mathematician?

Author: Mario Livio

Publisher: Simon & Schuster 2010 (First published: 2009)

ISBN: 9780743294065

Pages: 308

Science is an attempt to read God’s mind which is evident in the physical reality as the rules and principles which hold the world together. Livio’s book is an elegant attempt to tell the epic story of man’s quest to peer into nature itself and to grasp its fundamental principles with the help of his greatest intellectual tool – mathematics. Its extraordinary ability to describe the world has been a source of wonder to philosophists ever. This feat comes in two varieties. In one category named active mode, scientists deduce mathematical laws applicable to an event after carefully observing it, while in the other, passive mode, mathematical functions which were formulated long ago in totally unrelated circumstances suddenly find application to explain new discoveries in science. Judging from the closeness with which mathematical predictions approach reality, we are tempted to think that God is a mathematician. So, the answer to the rhetorical question in the title is in the affirmative and the 250-odd pages explain why it is so. It may be mentioned in passing that another book titled ‘The Loom of God’ by Clifford Pickover (reviewed earlier in this blog) also follows a similar theme. Mario Livio is a noted author who is also an astrophysicist and the head of the Office of Public Outreach at the Hubble Telescope Science Institute. This is the 314th book review in this blog and it is a happy coincidence that a book related mathematics comes out as number 314 (remember pi is approximately 3.14?).

A noteworthy feature of mathematics is its strikingly effective provenance to explain natural features and phenomena. Why should it be so? Mathematics is anyway a product of human contemplation and analysis. If this fruit of human intelligence so faithfully displays an uncanny ability to explain and predict nature, it is no wonder that a group of philosophers – a large one indeed – postulated the existence of mathematics in an idealized Platonic world, whose reflections on the physical world constituted our everyday adventures. This raises the pertinent question whether mathematics is discovered or invented. The niceties of such philosophical speculation need not detain the readers, but Livio presents a deeply speculative question in an easy to digest way. The ideas of Platonic world and discovery are compatible, in the sense that the numbers and shapes already existed in a perfect, imaginary world until man stumbled upon them in a spark of intellectual brilliance. Just like America existed before it was ‘discovered’ by Columbus, or Vikings, or even by that Turkish guy – who provided some much needed comic relief in international discourse a few months ago – mathematics existed right from the universe’s moment of being. But quite a few philosophers, and such humble beings like myself, differs from this point of view. According to this theory, mathematics is an abstract concept developed by man with the help of his extraordinary ability to detect patterns in nature. The book provides ample room for general readers to get familiar with this dichotomy that surrounds mathematics’ existence.

History of science occupies a major portion of the book, but presented in an admirable way that commands attention from readers. Freely interspersed with witty anecdotes and informative quotes from authors present and past, the text stands tall as a testimony to the immense amount of research that had gone in to the publication of it. Livio identifies Archimedes, Newton and Gauss as the three greatest mathematicians of all time, but does not restrict his pen to these three. Would any discussion on the development of science through the Renaissance era be complete without a solid reference to that mathematics professor from Padua, Italy – Galileo Galilei, no less? Galileo’s trial and the stifling overlordship of blind faith over reason is a topic you would find described umpteen number of times in any book that deals with the history of science through the turbulent 17^th century. Livio’s description would feel to be delightfully elegant to new readers. Old readers also would find the narration to be very congenial. This book extends the story to other mathematicians, including Descartes, and the Bernoullis. The sibling rivalry between Jakob and his brother Johann Bernoulli is brought to light with a quote from a letter the younger Johann wrote to his friend in which he exults at defeating his elder brother in the solution to a vexing problem. Mathematicians are also human, after all!

Even though Livio considers Gauss to be one of the three greatest ever mathematicians, nothing much is said about him apart from casual references in the context of non-Euclidean geometry. But this shortfall is more than leveled by the extensive discussion on the new developments in mathematics that took place during the last two centuries. The new sprouts are so revolutionary as to merit the epithet that man had broken free from the shackles of classical learning and began to explore nature in the light of a new creative spirit. A mind boggling array of discoveries had taken place in this period, but ordinary readers find it difficult to comprehend the practical purpose of many of them. Non-Euclidean geometry is however very helpful in estimating the shortest possible distance between any two points on a spherical surface. Aircrafts usually follow these shortest routes. But such hyperbolic geometry is extended to such extreme lengths that no apparent use is evident – yet! At around this time, logic was also linked to mathematics so as to strengthen the mutual foundations. Boolean algebra originated as the systematic representation of logic as ordinary algebra was to scientific concepts. Enhancement of geometry to many more dimensions than three enabled it to stand as the structural framework of advanced theories on the origins of the cosmos in the form of string theory, which postulates ten dimensions. This also shows the effectiveness of the discipline as a faithful representative of nature. But the long chapters on logic and discussions on its consistency are hard to enjoy for average readers.

A frequent source of controversy among mathematicians is the question whether its concepts should provide practical applications for human use. Such a notion itself is anathema to many practitioners who bask at the sheer glory of pure mathematics. Archimedes and G H Hardy were two mathematicians of this school. What would have been their impression when they saw their concepts eagerly accepted by the scholars and put to uses which provide immense value to their own societies? Archimedes is credited with the invention of a screw pump, levers of varying complexities, optical instruments and defensive apparatus, while much progress in cryptography is attributed to Hardy. There were mathematicians in the other camp as well, like Gerolamo Cardano, who wouldn’t conceptualize the definition of more dimensions than three because no practical utility was existent at that time, nor conceived to be feasible in the near future.

The book is splendidly written, having a good structure in presenting ideas. It is also graced with a good number of anecdotes, pictures and illustrations. There is an immense collection of notes mentioned in the main text and a sizable bibliography is listed. A nice and comprehensive index completes the attractive side of the book. On the negative part, about a quarter of the text starting from logic and its relations to mathematics is highly abstract, making life difficult for the readers. Fortunately, no harm is done even if you were to simply bypass those chapters and dive straight to the last one.

The book is highly recommended.

Rating: 3 Star

The Signal and the Noise

Title: The Signal and the Noise – The Art and Science of Prediction

Author: Nate Silver

Publisher: Allen Lane, 2012 (First)

ISBN: 978-1-846-14752-4

Pages: 454

Uncertainty is an inseparable feature of natural and social lives of man. We come across unpredictability at every corner, and encounter experts predicting the outcomes of various events based on painstaking research – at least that is what they say. Normally, this incertitude is so much a part of our way of life that we hardly pose to realize that there may be other ways, less uncertain, about them. This book is an excellent beginning to inspect those events in a rational way and to reach impressive conclusions. Even though I have used terms like uncertainty and unpredictability in a synonymous way, there are subtle differences between them which the author is at great pains to explain in the course of the narrative. And Nate Silver is just the right man for doing that, being a statistician and political forecaster at The New York Times. In 2012, he correctly predicted the outcome of all the states in the US presidential election. He has also been named one of Time’s 100 Most Influential People in the world. Being a forecaster himself, he explains the pitfalls many of them fall into, when analyzing complex fields such as electoral outcomes, stock markets, spread of contagious diseases, sports betting, weather, climate change and even some of the nuances in Chess tournaments. Every prediction is wrought with uncertainty, but the quantum of this factor is not always mentioned in some of the startling announcements. When skill is also a factor to account for, experts find it easy to outsmart the novices who are ignorant about the probabilities which determine the outcome to a great extent. Hence the importance of the book – it helps to assess the predictability of an event, the margin of error inherent in a prediction and how best to effectively use such advice in reaching conclusions that have impacts on the financial, political or climate fronts.

Silver starts his masterly discourse with a brief but inimitable introduction into the necessity of separating the information in the signal from the background noise. If only all authors used such lucid analysis to explain their concepts! The author asserts that mankind began facing the challenge of richness of data that originated with the invention of printing press, at which time the information revolution really began. The number of books skyrocketed in the years succeeding that momentous event and cost of books and printed information plummeted, making them affordable to a large class of common people. Along with this surge of information came noise, the signal which doesn’t carry any information at all. Man is evolutionarily well equipped to discern patterns in a forest of random shapes and the problem reared its ugly head when this supersensitive faculty was turned against the flood of data that suddenly became available. This ended up in a large number of predictions not matching up with the outcome. Silver describes about the art and science of prediction, the tools with which people go about predicting the results and the pitfalls that await them on the road

Predictions that mainly come our way in our normal course of life are about political events like the result of an election. The author submits the flurry of TV predictions to an exhaustive analysis to come out with the stunning observation that all of them don’t stand a chance better than flicking a coin. But the efforts to predict the future career of baseball players are not that random. Here, software as well inquisitive researchers have made proven track record in identifying talent from early stages. The author himself is immensely attracted to this field, who has made software for predicting this, and the readers gets the impression that Silver is not totally unbiased when he argues that the computer’s efforts in baseball is entirely worthwhile. Another common task is predicting the weather. Here, the meteorologist is solidly assisted with two things – persistence, which maintains that the weather tomorrow would be very similar to what it is today and climatology, which states the statistical probability of a day’s weather collected from data collated over many previous years. In order to classify a weather prediction as accurate, the person must exceed the utility provided by the two. However, the commercial analysis of weather is not unbiased. A wet bias is argued to exist, in which the predictor assigns a chance to rain when in fact the data claims the chance to be very small. This is because people tend to ignore non-occurrence of rain when it was predicted than the other case of rain occurring when it was predicted not to, which may ruin their picnic.

Climate change in the form of man-made global warming as the result of increased carbon dioxide emissions from industrial processes is a phenomenon seems to be occurring on a planetary scale. The UN-spawned IPCC (Inter-governmental Panel on Climate Change) monitors the temperatures regularly and comes down with predictions about long-term averages. The predictions of IPCC do not take into account the full measures of the complexity of the situation, as the author asserts. There is a full chapter on global warming in the book but uncharacteristically it does not delve deep into the details and don’t say conclusively whether the UN-body’s prediction would be right or wrong. Silver is contented with presenting a balanced picture, the arguments for and against the theory. There was indeed a rising trend from 1970 to 2000, but the first decade of the present century was relatively cool. But the author quickly picks up his Bayesian calculator and claims that the probability of the theory to be still true is a solid 85% even after accounting for the cool decade. A new argument is also presented to be behind the decline. This has to do with sulphur dioxide. The molecules of this gas spreads as aerosol in the upper layers of the atmosphere and reflect sunlight back to the space, thereby lessening the greenhouse effect. But the substance is highly polluting, being the source of acid rain. Sulphur emissions were cut down drastically as a sequel to the enactment of Clean Air Act in the mid-70s. The reduction might have contributed to the disappearance of the cooling effect of sulphur dioxide in the period leading up to 2000. Then how did the mercury go down in the next 10 years? According to Silver the impetus to industrial production in China, which doesn’t enforce any environmental regulations would have pumped more Sulphur into the atmosphere, ensuring a cooler decade. He ends with a premise that IPCC’s predictions of temperatures, revised in 1995, may well be true.

Silver’s examples and fields of application for his original thought and insightful ideas are very apt and fitting for the issue at hand. Unfortunately this fine discretion is unfortunately not applied in a few examples on prediction related to sports. The vile contraption going by the name of baseball dominates American thinking, even though nowhere else would you find sensible people pitching for this strange game. The author devotes a full chapter to the nitty-gritty of baseball prediction, which is really a pain-in-the-neck for the non-American readers who are not at all familiar with how the game is played. A similar argument holds for Poker, which is also one of the author’s favourite pastimes that have come to haunt the reader. This must surely be counted as a disadvantage to the book. At the same time, however, the author more than makes up for the shortcoming through several other chapters excellently structured with relevant concepts. We need not look further than the section in which he introduces Bayesian theorem which evaluates the probability of an event occurring due to a phenomenon which has a definite prior probability of occurring. Silver explains the concepts with an extremely hilarious instance of calculating the chances that your partner is cheating on you, if you happen to find a piece of underwear in the wardrobe which does not belong to you. If the prior probability of a cheating partner is 4% (collected from social data), Silver asserts humorously that, even after finding the suspicious object mentioned above, the probability that the person is cheating only rises to 29%. The reasoning is crystal clear, but the probability of a person being consoled by such figures is highly unlikely.

This book is highly recommended and is a must read. I would have given it a 4-star rating, if the author was not so particular about the lengthy chapters on baseball and poker.

Rating: 3 Star