The subtitle of this book, How Big Data Increases Inequality and Threatens Democracy really says it all. Big data has come into our lives in numerousThe subtitle of this book, How Big Data Increases Inequality and Threatens Democracy really says it all. Big data has come into our lives in numerous ways, and many of them are a scourge on our lives. Big data, in and of itself, is not to blame, but the uses to which it is put are often outrageous. Take the case of automated teacher evaluations. These are often based on the improvement of students' scores. It seems like a no-brainer, and since the scores take into account the improvement rather than the absolute scores, they seem to be very fair. However, one New York teacher received an abysmal score of 6 (out of 100) one year, and the following year received a wonderful score of 96. Obviously the teacher did not suddenly improve his teaching methods.
If a mathematician were to analyze the scores from such evaluations, their random scatter would be instantly recognized as meaningless. Yet, these automated evaluations are used for hiring/firing decisions, as well as compensation decisions. The worst aspect, is that there is no attempt to improve the algorithms. The algorithms used in these evaluations are opaque, and no attempt is made to apply feedback to tweak the scores to make them more accurate and fair.
Such algorithms are used in many avenues of life. Credit scores are used by loan companies--and even by auto insurance companies! Police departments use algorithms to plan targeting of police activity, while courts use algorithms to predict recidivism. Colleges, especially for-profit colleges, use algorithms to target potential students.
While this book comes across as very preachy, and very liberal, it does make some very good points. Big data is often used--intentionally or not--to punish the down-trodden and to increase inequality. Profitability is usually the goal, and while fairness is often the purported goal, these algorithms rarely turn out to have the effect of being fair.
I recommend this book to people who are curious about the effects of big data on their lives and on democracy....more
This is a fantastic book about scaling laws and how to understand them. Geoffrey West is a theoretical physicist, who has spent a lot of time at the SThis is a fantastic book about scaling laws and how to understand them. Geoffrey West is a theoretical physicist, who has spent a lot of time at the Santa Fe Institute, deriving theoretical scaling laws, and applying them successfully to biology, cities, and companies. He derives the theories from the structure of networks; arteries, capillaries in organisms, social networks and city infrastructure, and companies.
The scaling laws themselves are fascinating. In the very first chapter, Geoffrey West hits the reader with an astounding set of scaling laws that certainly surprised me. As to biology, there are about 50 different metrics that have interesting scaling laws--and West touches on a few of them. The scaling of metabolism, heart rates, brain matter, growth rates, life spans, aorta lengths, tree heights, and on and on; you get the picture. These scaling laws pertain across organisms, from the tiniest microbe to the blue whale; over 20 orders of magnitude.
But the really surprising aspect is that almost all of the scaling laws are factors of 1/4! For example, metabolic rate scales as Mass to the 3/4 power. Doubling the mass of a mammal increases its time to maturity by 1/4, its lifetime increases by 1/4, and its heart rate decreases by 1/4. And, these laws apply over the entire range of mammals, despite their diversity. Mitochondrial mass, relative to the total mass of an organism, goes as Mass to the -1/4 power. And, Geoffrey West describes how he and colleagues have derived theoretical scaling laws and growth curves from first principles. He shows how remarkably well the data fit his theoretical predictions.
As a physicist, West felt that this universal 1/4-power scaling tells us something fundamental about the dynamics, structure, and organization of life. These laws suggest dynamical processes that constrain evolution. And there are some surprising constants among all mammals. Blood pressure is approximately the same, and the number of heart beats in a lifetime is about the same, among all mammals!
But the discussion of scaling laws don't stop with biology. West finds fascinating scaling laws that apply to cities and to companies. The most perplexing question he addresses is, why do most cities live forever, while companies have short lifetimes? Cities are the prime drivers of economic development, not the nation state. And, most of the scaling laws associated with cities are either to the 0.85 or 1.15 power. That is to say, comparing two average cities, one twice as big as the other in population, the larger city will not have double the number of gas stations, but only 85% more than the smaller one. The larger city will have 115% higher wages, more doctors and lawyers, patents, GDP, number of cases of AIDS, crime and pollution. This scaling applies within all countries, but not across from one country to another.
The average half-life of companies is 10.5 years!And in any given year, the risk of a company disappearing (through bankruptcy, or merger, or acquisition) is the same, regardless of a company's size!
While cities become more diverse as they age and grow, companies do the opposite; they lose diversity, as they become more supportive of tried-and-true products in order to guarantee short-term returns. As companies grow, so too does their bureaucratic control. And, this is at the expense of innovation and R&D (Research and Development).
At times, the narrative deviates from scaling, and goes into various qualitative aspects of cities and companies. The author is rather opinionated in these areas, but his conjectures are interesting, though open to controversy. My only complaint about this book is that, while theoretical scaling laws in biology are developed and tested successfully against data, the book does not offer theoretical scaling laws for cities and companies. To some extent, these are more difficult to develop, because they depend on socio-economic structures and social networks. Data for these, especially for companies, are more difficult or expensive to obtain. Nevertheless, this book offers a wealth of information, and is endlessly fascinating. Highly recommended!...more
I enjoy thinking about algorithms as they are applied to technical problems. So, when I saw this book, I thought, "This is a book written just for me.I enjoy thinking about algorithms as they are applied to technical problems. So, when I saw this book, I thought, "This is a book written just for me." And, that assessment was absolutely correct. It is a fascinating book, all about how sophisticated algorithms are applicable to everyday problems.
The book starts out describing the "optimal stopping problem." It is also sometimes called the "secretary hiring problem", and I have seen it applied to dating to find a romantic partner, and this book points out that it can also be applied to looking for an apartment in a seller's market, or searching for a parking space. It can be summed up by the phrase, "Quit while you're ahead." Basically, you have a finite amount of time to complete a search for the "best" item or person. What do you do? It can be shown, mathematically, that you should conduct a search for 37% of your total available time, without making any decisions. Then, at the very next time you find an item or person that is better than all of the previous ones, you should jump on it! I read about this, beforehand, but I didn't realize that 37% comes from 1/e, where "e" is the natural exponent.
The next topic in the book is the dichotomy between explore and exploit. That is to say, when you are looking for something to do, will you try something new (explore), or repeat something that has pleased you in the past (exploit). The answer has to do with how much more time you have in this situation. Movie sequels are all about exploiting, because movie revenues are decreasing. If you are new to an area, you will explore restaurants, but if you are moving away soon, you will return to restaurants you have previously visited. I enjoyed how this issue is related to the "multi-armed bandit" problem. This is a mathematical problem that has lots of applications; choosing the best slot machine in a casino, for example, or designing a web site that will maximize profits for a company.
Sorting is a big topic in the book. It turns out that top poker players sort themselves out, and don't want to play with players they think are better. Libraries use an inefficient caching system, putting recent acquisitions in front. Instead, they should put the most popular books in front. The Naguchi filing system involves returning all files to the extreme left. This goes against the recommendations of efficiency experts, who recommend sorting by topic. Leaving a pile of papers on top of a desk is an example of the MOST efficient filing system! I enjoyed reading an anecdote about Barack Obama visiting Google, before he became president. At Google, he was treated to a pretend job interview. He was asked, "if you had to design an algorithm to sort a million 32-bit integers, what would you do?" He replied that he would not use a bubble-sort algorithm, and all the people in the audience applauded his correct answer!
I enjoyed reading about human memory from an algorithmic point of view. Memory recall is a problem of organization. It was fascinating to read that a graph of the probability of remembering something as a function of elapsed time exactly mimics the probability that a word appears in a newspaper headline twice in a row!
Knowing more makes it harder to remember things; what we call memory decline is actually just learning. A brain fart is actually just a cache miss!
Related to sorting are various problems of scheduling. Examples from the book include multiple laundry loads to minimize total washing plus drying time, and minimizing the rotting of food from a CSA (Community-Supported Agriculture).
No book on everyday applications of algorithms should bypass Bayesian reasoning, and this is an important subject in this book. Our memories of images of plane crashes and car crashes are roughly the same; as a result, we are surprised that there are orders of magnitude more deaths in car crashes, than in plane crashes.
The dangers of over-fitting are discussed at length. As an example, police and FBI agents sometimes over-train; they sometimes use good gun-training etiquette during a shoot-out, with fatal consequences. I really appreciated the discussion of how over-fitting is avoided in biological evolution. It is dangerous for organisms to evolve to over-fit an ecological niche, because the species might not be able to rapidly adapt to an uncertain future environment.
I also thought that the description of "exponential backoff" was very well treated in the book. This esoteric-sounding algorithm is used in all computer networks, but is also used by many of us in treating flaky friends, and in punishment for probation violations.
The last chapter in the book was about game theory. It is helpful, in reading this book, to have some previous background, an understanding of a "Nash equilibrium". The following quote sums it up nicely:
Love is like organized crime. It changes the structure of the marriage game so that the equilibrium becomes the outcome that works best for everybody.
I highly recommend this book. Regardless of your mathematics background, it will intrigue you and amaze you, to see how math permeates our everyday lives....more
This is a fascinating book about the theoretical limits of our knowledge, our ability to make predictions, and to understand the universe. Marcus du SThis is a fascinating book about the theoretical limits of our knowledge, our ability to make predictions, and to understand the universe. Marcus du Sautoy is a professor of mathematics at Oxford University. A central theme of the book, is to show how mathematics works as a powerful tool in helping us understand nature, and the limits to our understanding of how it works.
The book begins with a discussion of throwing a 6-sided die. With perfect knowledge of the position, velocity, and rotation of the die, is the outcome predictable? Actually, the answer depends on the friction of the table onto which the die is thrown. For certain ranges of friction, chaos prevents predictability.c
So the book relates the story of King Oscar II of Norway and Sweden, who celebrated his 60th birthday by offering a gold medal prize to whomever solved a math puzzle; "Is the solar system stable?" Poincare worked on this puzzle, and submitted a paper analyzing the orbits of two planets and a speck of dust. He showed that they followed periodic paths. The editor of the journal showed a gap in the proof, so Poincare tried to stop publication, but it was too late. Poincare had discovered that small perturbations to a stable orbit could make the system fly apart. This led to the discovery of chaos. In 2009, French astronomers modeled the solar system's stability many times, adding slight perturbations. The orbits of Jupiter and Saturn were very stable, but in 1% of the simulations, Mercury's orbit had a resonance with Jupiter, allowing Mercury to collide with Venus, which then collides with the Earth.
It is well known that the Earth's atmosphere is a chaotic system. A small perturbation leads to unpredictable outcomes. This idea feeds the claims of climate-change deniers, that we cannot predict a change in climate. Sautoy brings up the quote,
Not believing in climate change because you can't trust weather reports is a bit like saying you can't tell when the next wave is going to break on Bondi Beach because you don't believe in tides.
Sautoy gives the clearest explanation of wave-particle duality and some of the paradoxes of quantum mechanics, that I have ever read. He discusses elementary particle physics. He brings up the paradox, that electrons behave as if they have no volume, yet they have mass. But then, what is their density? Is it infinite? Is each electron a tiny black hole?
Sautoy brings up a very interesting aspect of Heisenberg's uncertainty principle. Heisenberg's original paper on the subject is often misunderstood. It is not that the act of observation affects a system. Heisenberg was forced to put that example into his paper in order to get it approved by skeptical editors. Even without a direct interaction with a particle, the product of uncertainties in its position and momentum is limited. It's not that we cannot simultaneously precisely measure position and momentum. It is that they do not exist until we measure them.
Many scientists equate God with "things we cannot know." Heisenberg's uncertainty principle also holds for time and energy. When you look at empty space for a short period of time, there is uncertainty in the energy content; space can never be truly empty. Since energy can change into mass, particles can spontaneously appear in a vacuum. So, there is no need for a creator. Some people give the definition of God as the solution to a question, such as "why is there something rather than nothing?" The trouble with most religions, is that God is then given so many properties that god ends up having nothing to do with the definition. It is like working backwards, conjuring up properties without understanding the original definition. One could define God as the existence of things we cannot know. To declare oneself as an atheist is to mean there is nothing we cannot know. But to say that there are things we cannot know, then, is a proof that God exists.
Cosmology is a science where not all questions are, in principle, answerable. This is a "Copernican" aspect of things. The universe is expanding and in the past seven billion years, the expansion rate has been accelerating. Dark energy is hypothesized as the cause. In the distant future, we won't be able to see other galaxies, as space will have disappeared from view, beyond the sphere of our visible universe. But we will still see the stars in our Milky Way galaxy. So, we are lucky to know that other galaxies even exist. Some day in the far future, this knowledge may be lost.
Sautoy describes an interesting set of experiments designed to fool the brain into thinking that your seat of consciousness is located elsewhere--in another person or even in a Barbie doll! Then there is a fascinating discussion of experiments dealing with consciousness. A group of philosophers believes that science can never solve the problem of consciousness. But, we are reminded of the prediction of the philosopher Comte, who foretold that we would never know what is at the heart of a star.
This book is written in a very accessible style. Sautoy gives crystal-clear explanations. I enjoyed every page of this book, as it gave me different ways to think about the universe, and the role of mathematics in unraveling its mysteries. ...more
This is an amazingly ambitious book. It covers such a wide range of topics--I have never seen such a comprehensive non-fiction book. It starts out witThis is an amazingly ambitious book. It covers such a wide range of topics--I have never seen such a comprehensive non-fiction book. It starts out with a detailed description of theories of the origins of the universe. Here, John Hands is at his best, as he sorts out the various theories. He reasons why some of the theories are still in the running, while others are not borne out by the available evidence.
John Hands continues to discuss the origins of life. He describes the prevalent theories, and gives his opinions about which theories are most realistic. He describes evolution from the earliest microbes to the present-day complex organisms. He considers the evolution of humans and the origins of consciousness. He points out the ways in which humans differ from all other animals.
Then, the book goes into a history of the human race. The development of science, technology, and philosophy are covered in some detail. He points out that many so-called "world-wide" philosophy books skip almost completely the philosophies of the East, and concentrate almost entirely on those of the West. While John Hands does devote attention to Eastern philosophies, he also spends much more time on Western ones.
This is not an entertaining book. There is not a trace of humor, and there is no effort taken to make it easy on the reader. However, the sheer scope of the book, and the intelligent unbiased descriptions of the science, history, and philosophy, make this a book deserving of one's attention. I recommend the book for people who are truly curious, and want an unbiased view of our understanding of human evolution....more
I wanted so much to love this book, but it was difficult. About half of the book is about Frenkel's life; and it was fascinating. The other half, inteI wanted so much to love this book, but it was difficult. About half of the book is about Frenkel's life; and it was fascinating. The other half, interleaved with his memoirs, are descriptions of Frenkel's mathematical work and discoveries. I had a great deal of trouble following the descriptions of the math. I am superficially familiar with many of the concepts, but it just gets more and more complex. Toward the end, especially, I became quite confused.
Frenkel grew up in a small town in the Soviet Union, a two-hour train ride from Moscow. Initially interested in physics, Frenkel's father introduced him to a mentor who showed him how modern physics, especially quantum mechanics, relies on some very modern concepts in mathematics. In that way, Frenkel got hooked on math.
When he was in high school, starting to think about college, Frenkel applied to the math department at Moscow State University. It was explained to him that he had no chance of being admitted. He had a Jewish last name, and anti-Semitism would prevent his admission. Frenkel applied for admission anyway, and he was grilled mercilessly during an oral entrance exam. The examiners found excuses to refuse him admission.
So, Frenkel went to undergraduate school at a different college, one that did not discriminate so much. Nevertheless, he attended lectures and seminars at Moscow State University. He didn't have a college ID, so he scaled the fence to get in! The Soviets put tight controls on photocopiers. While in undergraduate school, his research papers were secretly copied and smuggled out, and reached mathematicians around the world. One day, Frenkel received an invitation from the president of Harvard University to come to Harvard on a fellowship grant, and become an assistant professor. At that point in time, he didn't even have a PhD!
What I did get out of the math descriptions, is the inner beauty of math. Vastly different areas of math can be connected, through hidden connections, as if by magic. This attracted Frenkel to the Langlands Program, a grand unified theory of mathematics. Now a vast subject, the program tries to connect number theory, harmonic analysis, geometry, representation theory, and mathematical physics. Riemann geometry is the cornerstone of Einstein's Theory of General Relativity. It contains hidden connections with number theory. Frenkel's career goal is to establish connections between the dualities in physics and the dualities in mathematics.
In the last chapter, Frenkel spent a sabbatical in France, writing and producing a short film. He also co-stars in the film, titled Rites of Love and Math. The trailer is on a site on YouTube.com. In this allegorical film, a mathematician discovers a mathematical formula for love. He realizes the formula's importance, and that it could be used for good as well as for evil. He tattoos the formula on his lover's body. The film was screened at many film festivals to wide acclaim, but also received a lot of controversy.
You can read the book and skim or skip through the math. Frenkel does a good job of describing the hidden connections and beauty of math. But he leaves the average reader lost in the details....more
This is a fun little book by a young woman who found some degree of fame in the field of cryptography. Growing up in Ireland, Sarah Flannery was constThis is a fun little book by a young woman who found some degree of fame in the field of cryptography. Growing up in Ireland, Sarah Flannery was constantly exposed to mathematics; her father, a mathematician, gave math and logic puzzles to her and her brothers. As a high school student, Sarah took a "transition" year off during high school, and worked for a cryptography company. She learned how to code algorithms in the Mathematica computer language. At the company, she was supervised by some gifted mathematicians, who gave her some difficult, challenging projects. She did well, and presented a very good science fair project at her high school. Her project was good enough to make it to the national fair, and then to an international fair.
Then Sarah decided she wanted to continue on, and do some original research. With the help of a mathematician mentor, she invented a new "trap door" algorithm that was much faster than the state-of-the-art algorithms. She presented her improved science fair project, and won at the national and international fairs. She had quite a big exposure, as the media caught on to her discovery.
In between her autobiographical narration, Sarah provides some fun mathematical puzzles. I loved the puzzle about the insurance salesman. Also, she guides the reader through elementary number theory, giving the reader enough understanding to grasp the essentials of modern-day cryptography. ...more
This book is an engaging, comprehensive guide to strategies, as applied to everyday life. The first part of the book focuses on standard game theory,This book is an engaging, comprehensive guide to strategies, as applied to everyday life. The first part of the book focuses on standard game theory, graphical notations for various problems, and applications of the prisoners' dilemma to everyday situations. The second part of the book concentrates more on everyday and business problems, and strategies to achieve optimal solutions. Game theory is not always applicable to all of these problems, but logic and rational problem-solving and a bit of mathematics are ever-present.
The book explores the voting issue in some detail. When two candidates are running against each other, the best strategy of course is to vote for your first choice. When three or more candidates are running, it is not always best to vote for your first choice, especially if you believe that your first choice has no chance of winning. For example, in the presidential election of 2000, there were three candidates, Bush, Gore, and Nader. If you preferred Nader to the others, you could vote for him, but your vote would be pretty much wasted, as he had little chance of winning. It would be best to vote for your second choice. But, what if the election was predicted to be much closer; what would the best strategy be then?
Furthermore, the book explores other voting systems that would allow you to list all of your voting preferences? For example, what if you could vote on all of the candidates, listing their names in preferential order. Various vote-tallying systems could take these preferences into account, and come up with a fairer assessment of the most-preferred candidate.
But here's the rub; there are numerous vote-tallying systems, each of them objective, but depending on which one is chosen, a different candidate could win. The book goes into some detail in considering the different outcomes of the 2002 presidential race, considering several of these systems.
The book also describes three different systems for auctions. Although the systems differ dramatically, the optimum strategy is the same for all of the systems.
The book describes various approaches for political negotiations. Examples include incentives and threats. But a threat is only good if it is credible. The book describes some historical approaches that have made threats credible. Another type of strategy is how a company can best compete with other companies, by setting prices that will maximize profitability.
The book has a set of exercises to try out your newly-gained understanding. One of the exercises is to consider how to make a good first impression on a first date. You are faced with two simultaneous problems; how to prove your sincerity and quality to your date, and how best to assess the sincerity and quality of your date. In other words, what is the best strategy for signaling and screening?
This book is best appreciated if you are not afraid of some simple algebra. However, many of the strategies are not at all mathematical, but simply rely on logic. I thoroughly enjoyed this book; some of the chapters were a bit repetitive, but not overly so. ...more
This is a wonderful book about mathematics and its application to everyday life. Jordan Ellenberg shows that the certainty that people associate withThis is a wonderful book about mathematics and its application to everyday life. Jordan Ellenberg shows that the certainty that people associate with math is often misplaced; some areas of math are devoted to uncertainty, and that's where things get very interesting.
Ellenberg starts the book with a beautiful example of application of mathematics, logic, and thinking out of the box. During World War II, a group of mathematicians working for the Statistical Research Group were given a problem by some Air Force officers. Fighter planes returning from missions were analyzed for bullet holes. The number of bullet holes per square foot were counted. For example, there were 1.11 bullet holes per square foot in the vicinity of the engine, 1.73 in the fuselage, 1.55 in the fuel system, and 1.8 in the rest of the plane. The officers wanted to add some armor to the planes; the question was where? The planes could only support so much weight, and where would additional armor be most advantageous? The officers thought that since the fuselage had the greatest density of bullets, that would be the logical location for more armor. A mathematician named Abraham Wald said exactly the opposite; more armor is needed where the bullet holes aren't, namely, around the engines. Planes with lots of bullet holes in the engine did not return at all!
The book discusses the issue of statistical significance. Scientific experiment often use a 95% confidence threshold as an indicator of statistical significance. This means that if a truly random outcome were expected, a positive correlation would be seen only 5% of the time. Ellenberg includes an xkcd cartoon that shows how easy it would be to perform a set of experiments that could come up with statistically significant results like "Green jelly beans linked to acne! at the 95% confidence level.
Some of the section and chapter titles are hilarious. For example, in the chapter titled "Are you there, God? It's me, Bayesian Inference", Ellenberg brings up a scary example of the use of "big data". Based on a teen-age girl's purchases of unscented lotion, mineral supplements, and cotton balls, the retail store "Target" began sending her coupons for baby gear, because of the (correct) inference that she was pregnant. Another great section title is "One more thing about God, then I promise we're done."
Another interesting title is "The Cat in the Hat, the Cleanest man in school, and the creation of the universe", in which Ellenberg reviews some of the probabilistic arguments for and against the existence of god. And I love the famous quote by Richard Feynman:
You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW375. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!
I also love the chapter title, "If Gambling is exciting, you're doing it wrong". Ellenberg describes how several groups capitalized on several state lotteries. Due to some strange lottery rules, it is (was?) possible to reliably make a profit, given enough investment of resources. No illegal shenanigans--the states make money no matter what you do. You could make a profit by taking advantage of the rules, and of the people who buy lottery tickets without a coherent strategy. And, I did not realize that Voltaire made his fortune by taking advantage of state lotteries!
Ellenberg brings up the phenomenon of Nate Silver predicting the outcome of the Obama-vs.-Romney election. Silver predicted the probability of both candidates winning state by state, along with the margin of error. By adding up the probable errors, he estimated that he would be wrong by 2.83 states. Critics seemed to have ignored the fact that he was not wrong by this many states--in fact he correctly predicted the outcome in all 50 states!
I highly recommend this book to all people who are even vaguely interested in math, probability, logic, and the application to everyday life. This is an excellent book!...more
This book is a wonderful introduction to history of predicting stock prices using mathematics and concepts from physics. It is basically a history ofThis book is a wonderful introduction to history of predicting stock prices using mathematics and concepts from physics. It is basically a history of pricing models; from the earliest mathematical models to the most modern ones. Of course, the best ones are maintained in secret by some super-secretive investment companies, for good reason. The only way a pricing model can be profitable is it to be better than most others being used.
The author, James Weatherall, has a PhD in physics, and is presently an assistant professor of logic and philosophy of science at the University of California, Irvine. He writes with clarity and an engaging style. His narrative follows a logical path, and does not take big diversions along the way.
Now, many of the subjects of this book are not physics at all, but applied mathematics. For example, the so-called "black box" model does not use any physics, but use purely statistical associations that are discovered algorithmically. Their name stems from the fact that they are opaque; they may make accurate predictions, but they offer no insight into the reasons for their predictions. Hence, it is difficult to judge how much confidence should be given to their results. But, for example, a model that predicts an investment strategy that returns a hundred times the S&P 500 over a fifteen-year period is nothing to be sneezed at.
Some of the models are definitely an application of physics concepts, such as the gauge theory model. I find if fascinating that this arcane physics concept has some practical applications in economics and predictions.
Some people blame these computer models for the disastrous economic downturns and stock market volatility that occur from time to time. While Weatherall sympathizes with this attitude, he wholeheartedly endorses the models, as they are simply tools. Sometimes, the assumptions and limitations of these tools are ignored, with dire consequences....more
The concept behind this book is wonderful. Scientists and science fiction authors discuss the relationships between science and stories.
In practice, hThe concept behind this book is wonderful. Scientists and science fiction authors discuss the relationships between science and stories.
In practice, however, much of the book was boring. Some of the essays--the more philosophical ones--simply didn't make any sense to me. Many pages are devoted to a telling of the story of the "Three Little Pigs"; a side-by-side telling via cartoons and mathematical symbology, as if it were some sort of mathematical proof. I didn't really see the point of this.
The most interesting essay is about Einstein and how his theories of relativity were accepted--or not. At one point, Einstein suddenly became world famous, and he went on a world tour. Many critics sprang up; they didn't understand the concepts, and they thought that his theories were "pure math" or simply a new theoretical philosophy that didn't have any practical application or corroboration with experiment.
This book had been on my wishlist for years. Some of the authors are among my favorite science fiction authors. But the fact is, I'm not happy I read it....more
The first half of this book is a review of modern physics on the macro and micro scales. The second half of this book is a discussion of the author'sThe first half of this book is a review of modern physics on the macro and micro scales. The second half of this book is a discussion of the author's speculation, that the universe is a mathematical structure. Max Tegmark is quite clear--he is not saying that the universe is described by mathematics, but that the universe is mathematics. He calls this the "Mathematical Universe Hypothesis", or "MUH" for short.
Tegmark asserts that this idea is a testable, falsifiable hypothesis. I did not find the experimental test, but perhaps I simply missed it. To be perfectly frank, I don't even understand his reasoning.
The last chapter of the book switches gears entirely, and discusses the existential threats to human survival. Things like asteroid collisions, eventual expansion of the sun, and so on. Then the book describes the two most immediate threats to human existence. The first is a nuclear war. And the second is--are you sitting down for this?--the singularity, where artificial intelligence takes over the world.
I didn't read this book--I listened to the audiobook, read by Rob Shapiro. He did an excellent job, giving the narration an aura of authenticity. It's just too bad that the content of the book is not as good as the narration....more
Simon Singh has the ability to present a story about a mathematics problem, and tell it like a detective story. He makes the subject exciting, even thSimon Singh has the ability to present a story about a mathematics problem, and tell it like a detective story. He makes the subject exciting, even though the outcome is well known. Singh intersperses history with discussions about the mathematics, and makes it quite understandable.
Singh starts with the roots of the famous Fermat's Last Theorem, by recounting the stories and mathematics of Pythagoras, Euclid, and Euler. Other, less well-known mathematicians are also given credit, for example Sophie Germain, Daniel Bernoulli, Augustin Cauchy, and Evariste Galois.
Three hundred fifty years ago, Fermat wrote the following theorem in the margin of a mathematics book: And, Fermat wrote that he had a marvelous proof, but no room in the margin for it. For centuries, mathematicians have attempted to prove the theorem, without success. It had been sort of a "holy grail" of mathematicians to prove the theorem, and many brilliant minds spent years on it. Perhaps in was unprovable, and worst of all, Kurt Godel showed that some theorems are actually undecidable--that is to say, it is impossible even to decide whether or not a theorem is true.
Singh recounts a fascinating story of the gifted mathematician, Paul Wolfskehl. He was very depressed, and decided to commit suicide on a particular night, at midnight. While waiting for that time to arrive, he started to read about the failed attempts to prove Fermat's Last Theorem. He became so engrossed in the subject, that he worked well past midnight. He found a gap in the logic of a predecessor, and was so proud of himself that he gained a new desire for life. And, in his will he established a fund of 100,000 marks to be given to the mathematician who first completes the proof of the theorem!
Much of the book describes how Andrew Wiles developed a growing interest in the theorem. He worked in almost total isolation for seven years, in order not to be distracted. He occasionally published little tidbits unrelated to his real endeavor, in order to dispel suspicions about what his real work entailed.
The central piece of the proof entailed proving the Taniyama-Shimura conjecture, that linked modular forms with elliptic equations. This was a linkage between two far-flung branches of mathematics that seemed to be totally unrelated. To prove the conjecture would allow incredible advances to be made. And then, Ken Ribet showed that a proof of the Taniyama-Shimura conjecture would, in effect, be a direct proof of Fermat's Last Theorem. But many people tried and failed to develop the proof. But that is exactly what Andrew Wiles worked on for so many years.
I had previously read that during Andrew Wiles' famous lecture, he just casually let the unsuspecting audience know, "and that is a proof of Fermat's Last Theorem." Well, this book tells a somewhat different story. Most of the audience had heard rumors that the third of Wiles' lectures would be of historical significance. They came prepared with cameras, and took photographs during the lecture. So, it was a surprise, but not a total surprise.
After Wiles' manuscript of the proof was sent to a publisher, six mathematicians reviewed it, and a crucial gap was found in it. Wiles worked furiously for a nightmarish year, and with the help of Richard Taylor, finally closed the gap. Today, Wiles is recognized as the one who developed the proof. But it is clear, that Wiles "stood on the shoulders of giants"; he used--and developed--mathematical techniques that had not existed just a few decades previously.
Simon Singh writes with a wonderful style. It is clear, not too jargon-heavy but contains enough mathematical "meat" to seem satisfying. The book is followed by ten appendixes that contain more details about some of the mathematics; they are not overly technical, and give the reader a better understanding of some of the issues. I highly recommend this book to everyone interested in math. ...more
This is a very good book about methods for computing probability puzzles. There are two types of methods discussed here: analytic solutions and computThis is a very good book about methods for computing probability puzzles. There are two types of methods discussed here: analytic solutions and computer simulations. The book is intended for readers with a strong mathematics background. Good algebra skills are essential, and calculus is also required to solve some of the puzzles analytically.
This book is divided into three parts. The first part is a description of 21 puzzles. The descriptions go into some detail, and provide hints. The second part contains the analytic solutions to the puzzles. The third part contains computer programs for simulating the solutions; the programs are written in Matlab, which for me is just perfect.
The puzzles have interesting names, for example, "How to Ask an Embarrassing Question", "When Idiots Duels", "Who Pays for the Coffee", "The Unsinkable Tub is Sinking!", "The Blind Spider and the Fly." The puzzles are all different; solving any one of them requires an approach different from any other. This is a well thought-out book, highly recommended for anybody with a need to try challenging puzzles that require some thought and mathematical skills....more
The Universe in Zero Words is a beautiful book, in all senses of the word. Dana Mackenzie devotes several pages to each of many equations that made aThe Universe in Zero Words is a beautiful book, in all senses of the word. Dana Mackenzie devotes several pages to each of many equations that made a difference in the spheres of mathematics, science, and economics. The history of each equation is told superbly, as well as the meaning of each equation and its applications in the real world. The explanations are geared toward the layman--you don't have to be a mathematician to understand most of the explanations.
Dana Mackenzie is a mathematician, and his enthusiasm for the subject shines through his writing. He chose equations that have been important in the development of mathematics and science. I found his choice of equations to be quite good, and I learned a lot, even though I was already very familiar with the majority of the equations. A few, however, were quite new to me. The equation of quarternions is brand new for me, and Mackenzie tells why; the concept is rarely mentioned in modern-day physics textbooks. Also, the Black-Scholes equation is entirely new to me, as it is used by financial analysts and "quants".
The book is filled with beautiful illustrations and diagrams. In fact--I strongly urge interested readers to obtain a printed copy of the book, rather than an e-book version. The beauty of the printing, the layout, typography and illustrations add a considerable amount of charm to the reading experience. ...more
This is a fun book of puzzles of all types; mathematical, logical, algorithms, estimation, mind games and creativity. There is also lots of interviewThis is a fun book of puzzles of all types; mathematical, logical, algorithms, estimation, mind games and creativity. There is also lots of interview advice, when applying for a job. The advice is good not just for Google, but for many other companies as well.
It was fun trying to solve the puzzles. Lots of them are quite tricky. I was able to solve some, others I flubbed.
The book correctly points out that, even though lots of companies rely on such puzzles during interviews, they are not reliable predictors of eventual performance on the job. In fact, sometimes there is zero correlation between interview performance and job performance.
The book mentions the "20% project" at Google, where employees are allowed to work on any idea they may have, for one day a week. The book cites a list of highly-regarded products that came out of these 20% projects. I recently read that, unfortunately, this perk has been rescinded at Google.
The last half of the book supplies not just the answers to all the puzzles, but detailed explanations as well. The style of the writing has a light touch, and is often subtly humorous. I recommend the book for all those who like a diverse range of challenging puzzles....more
This is a fantastic book about predictions. I enjoyed every page. The book is filled to the brim with diagrams and charts that help get the points acrThis is a fantastic book about predictions. I enjoyed every page. The book is filled to the brim with diagrams and charts that help get the points across. The book is divided into two parts. The first part is an examination of all the ways that predictions go wrong. The second part is about how applying Bayes Theorem can make predictions go right.
The book focuses on predictions in a wide variety of topics; economics, the stock market, politics, baseball, basketball, weather, climate, earthquakes, chess, epidemics, poker, and terrorism! Each topic is covered lucidly, in sufficient detail, so that the reader gets a good grasp of the problems and issues for predictions.
There are so many fascinating insights, I can only try to convey a few. At the present time, it is impossible to predict earthquakes, that is, to state ahead of time when and where a certain magnitude earthquake will occur. But it is possible to forecast earthquakes in a probabilistic sense, using a power law. Likewise, it may be possible to forecast terrorism, because that too, follows a power law! (Well, it follows a power law in NATO countries, probably because of the efforts to combat terrorists. But in Israel, the tail of the curve falls below the power law, likely because of the stronger anti-terror emphasis there.)
The accuracy of weather predictions increases slowly but steadily, year by year. Ensembles of computer model runs are part of the story, but human judgment add value, and increases the accuracy. Weather forecasts issued by the National Weather Service are unbiased in a probabilistic sense. But weather forecasts by the TV weatherman are very strongly biased--the weatherman over-predicts precipitation by a significant amount.
Nate Silver shows that the people who are most confident are the ones that make the worst predictions. The best predictions are those that are couched in quantitative uncertainties. Silver shows how Bayes Theorem can be applied to improve predictions; it is all about probabilities. And I just love this footnote,
A conspiracy theory might be thought of as the laziest form of signal analysis. As the Harvard professor H.L. "Skip" Gates says, "Conspiracy theories are an irresistible labor-saving device in the face of complexity."
Paul Erdos was a prolific, well-known mathematician. He wrote over 1400 journal articles in various mathematical publications, many of them collaboratPaul Erdos was a prolific, well-known mathematician. He wrote over 1400 journal articles in various mathematical publications, many of them collaborations. Those people who collaborated with him earned an Erdos "number 1". Those who collaborated with someone who collaborated with him earned a "number 2", and so on.
To say that Erdos was "eccentric" would be an understatement. He had no home--he carried a suitcase with a single change of clothes in it, and traveled the world, visiting one mathematician after another. He would stay at a mathematician's home until he became unwelcome--and that was not long at all. Erdos only slept a few hours at night, so he kept his hosts pretty busy! He was physically inept, so help left trails and messes in his wake. The collection of anecdotes about his life are amusing, and usually center on his single-mindedness about mathematics.
Erdos' main area of expertise was number theory. Paul Hoffman has written a very readable book, expertly interleaving chapters about number theory with Erdos' biography. This gives a layman some understanding about the sorts of problems that Erdos solved. I learned some interesting things about mathematics, and also about the psychology of mathematicians. This was a fun book to read, and I can recommend it to anybody.
By the way, baseball great Hank Aaron earned an "Erdos number 1"--read the book if you are curious to find out why!...more
Despite the title, this book is not primarily about Alan Turing. It is really about the group of people at the Institute of Advanced Studies at PrinceDespite the title, this book is not primarily about Alan Turing. It is really about the group of people at the Institute of Advanced Studies at Princeton. Much of the book focuses on John von Neumann, who spearheaded the effort to build some of the earliest electronic computers. These first computers were very unreliable--incorrect results were as likely due to faulty vacuum tubes as coding errors. In fact, circuits had to be designed to be robust to vacuum tubes that did not follow specs.
Quite a large chunk of the book--and the most fascinating--dealt with the types of mathematical and physical problems that the earliest computers could solve. In fact, that was the principal interest of von Neumann--learning what types of problems could be solved using computers. Here, Alan Turing and Kurt Godel played a large role in defining what sorts of problems might be solvable.
Among the problems that the earliest computers attacked was weather forecasting. In the late 1940's, there was much controversy about the feasibility of numerically computing forecasts in principle. Of course, weathermen wanted to continue to use their gut feelings to forecast the weather, while some scientists thought that, given sufficient spatial resolution, the weather could be forecast far in advance. It was not until later that people like Lorenz discovered that there are fundamental limits to how far in advance weather can be forecast.
The earliest computers were also used for developing the atomic bomb. Many aspects of the physics were not solvable using direct means. Simulations using a brand new numerical method called "Monte Carlo" were extremely significant for solving them. For this method to work, random numbers are required for initiating independent simulated trajectories. But random numbers were not easy to come by, so a special program helped to develop algorithms for computing them.
This book goes into considerable depth, in describing the people who developed and used the first computers at the institute. There are fascinating descriptions of the mathematical, physics, and biological puzzles that were attempted. I recommend the book highly, for those interested in the history of numerical computation.
This book develops the mathematical equations for modeling a variety of fascinating topics in evolution. If you are comfortable with equations, and haThis book develops the mathematical equations for modeling a variety of fascinating topics in evolution. If you are comfortable with equations, and have some background in linear algebra, then you are well-equipped for the formalism in this book.
What is best about this book, is how Martin Nowak develops the models starting from simple sets of equations. As the subject matter is developed, the models gain sophistication as additional parameters and feedback mechanisms are introduced. I especially like the game theory models, that are pretty much all based on the so-called "Prisoner's Dilemma". I also like the evolution of spatial structures through fractals.
I thought that I would enjoy the penultimate chapter, on language evolution. But then it suddenly hit me; none of the models are quantitatively compared against observations or data. The chapter on evolution of the HIV virus mentions that the model explains the long incubation period of AIDS. But, aside from such qualitative explanations, I was really disillusioned by the lack of model-data comparisons. The book also lacks good discussions about how the mathematical models can be interpreted and applied; they all contain parameters that seem to be totally arbitrary, and it is not at all clear what ranges are reasonable. The book would be much improved by additional interpretation of the models in terms of biology....more
This is quite a unique and enjoyable book. Encapsulated as a novel, the book covers elementary mathematics of infinity, set theory, and Euclidean andThis is quite a unique and enjoyable book. Encapsulated as a novel, the book covers elementary mathematics of infinity, set theory, and Euclidean and non-Euclidean geometry. It covers the history of these topics, as well. And it isn't dry--the subjects are covered in a easily-understood, "Socratic" approach. Nico, the professor of mathematics, knows how to motivate students, cultivate enthusiasm, and lure students into a deeper understanding of the topics of his course. Nico also shows how mathematicians think about their work, and about the world.
A major theme in the book is the hierarchy of provable theorems, built upon fundamental axioms that are unprovable, but "self-evidently true".
The main character is an Indian student, Ravi, who is enrolled as an economics major at Stanford University. He learns that his late grandfather, who he had adored as a child, had spent some time in a jail in New Jersey! It takes some time for Ravi to unfold the story of his grandfather, who had been a mathematician and an atheist. It seems that his grandfather had been arrested under an obscure law in New Jersey against blasphemy. Of course, this law is in contradiction with the freedom of speech clause in the Bill of Rights. A central question is whether mathematics is built on firmer foundations than religions.
While the story in the novel isn't really believable, it is a fun read if you enjoy mathematics, and highly educational at the same time. ...more
This is a very fun, entertaining book about the myriad ways in which random phenomena affect our lives. There is nothing really new here. As a physiciThis is a very fun, entertaining book about the myriad ways in which random phenomena affect our lives. There is nothing really new here. As a physicist, I am already well familiar will all of the concepts introduced, concerning probability and statistics. But oh--what a variety of fascinating applications!
I love the story about the "Ask Marilyn" column in Parade Magazine. Marilyn vos Savant holds the record for the world's highest IQ. She discussed the famous "Monty Hall" problem, and got aggravated letters from 10,000 readers, including 1,000 PhD's (many mathematicians!) who claimed her analysis was wrong. Nevertheless, she was absolutely correct--people just do not have a firm grasp of probability concepts.
The book explains lots of interesting puzzles and paradoxes. For me, the best part of the book is the discussion of how statistically random events conspire to make "outliers". This comes up again and again, in understanding "genius" mutual fund managers and fast-growing mega-companies.
My only disappointment, is the book's emphasis on the so-called "normal" (Gaussian) distribution, to the exclusion of other distributions. Many economic and natural environmental events are outliers that deviate from the normal distribution, as described so well in Benoit Mandelbrot's The (Mis)Behavior of Markets....more
I think this is the first book about Bayes' theorem and its applications, for the general reader. The book does not explicitly state the theorem as aI think this is the first book about Bayes' theorem and its applications, for the general reader. The book does not explicitly state the theorem as a mathematical formula, until the second appendix. However, the general idea is described, as well the general ideas behind it. The history of the theorem is described in some detail.
The ebb and flow in belief in the theorem over the course of 150 years is interesting. Applying Bayes theorem requires a prior probability, and this is often poorly known--it is often an educated, but subjective guess. Mathematicians and statisticians don't like guesses, they don't like subjectivity. As a result, the application of Bayes theorem was often in disrepute. But--despite the subjectivity, Bayes theorem usually works--and works very well! It was used most effectively during World War II, in the decryption of the Enigma code. It was also used effectively in anti-submarine warfare during the war, and in search and rescue operations. But its usage was classified, and as a result its power was hidden from statisticians.
In the past 20 years, Bayes theorem has really taken hold. I personally use it daily in my work, where it is extremely useful. I vaguely remember reading about the controversies surrounding it. Now, at long last, you can learn about the true nature of the controversies in this enjoyable book....more
Benoit Mandelbrot is the inventor of the mathematical concept of fractals. His earlier book The Fractal Geometry of Nature was a truly groundbreakingBenoit Mandelbrot is the inventor of the mathematical concept of fractals. His earlier book The Fractal Geometry of Nature was a truly groundbreaking book about fractals and how they are seen in nature. In The Misbehavior of Markets he turns his attention to the application of fractal concepts to markets. Mandelbrot shows that price fluctuations: 1) are not independent from one time period to the next 2) appear to be the same, regardless of the time scale involved (hours/days/months/years) 3) do not obey a Gaussian (normal bell-like) distribution, but instead follow a power-law distribution. These characteristics are exactly the opposite of the assumptions that are normally used in financial circles. As a result, most financial models severely under-estimate financial risk. Most financial models use certain parameters (like the beta factor) that purport to measure price volatility. Mandelbrot shows that many of these parameters are worse than useless; they are so wrong, they are dangerous and can lead to world-wide financial ruin.
This book is also somewhat of a biography; Mandelbrot details some of the fascinating aspects of his life, and that of his parents. One of the reasons contributing to his move from France to the U.S. is the disdain of French mathematicians to applied research. The only problem with this book is that Mandelbrot writes in a tone that is too strident for my taste. Nevertheless, I strongly recommend this book to anyone with an interest in applied mathematics or finance.
Having dealt with mathematics as applied to physics all of my professional life, this book provides a welcome change. What amazed me is the sheer variHaving dealt with mathematics as applied to physics all of my professional life, this book provides a welcome change. What amazed me is the sheer variety of mathematical approaches that are being applied to biology, including Fibonacci sequences, networks, cellular automata, topology, game theory, multi-dimensional geometries. I had no idea that Alan Turing did work with reaction-diffusion equations, that can be used to model patterns in animal skin stripes versus spots.
The book is written very clearly, well organized, and is completely understandable to the layman. Stewart explains that while mathematical models are not completely realistic, simplifying approximations help to generate insights into the underlying biological mechanisms. And, this book is jam-packed with wonderful insights into an array of biology issues. The penultimate chapter deals with the question of life on other planets. While not dealing directly with mathematics, Stewart explains that the possibility of alien life does not necessarily require a planetary system "just like the Earth". A wide range of planetary conditions may be able to foster life, and we should not jump to hasty conclusions based on the conditions that seem "normal" to us on Earth....more
This is the type of book I like best; the author, Stuart Kauffman, describes his own research into a new field called "complexity theory". Kauffman buThis is the type of book I like best; the author, Stuart Kauffman, describes his own research into a new field called "complexity theory". Kauffman builds simulations of lattice networks, and explores their characteristics. He shows how the simulations are analogous to chemicals combining, and may shed light on the origin of life. He claims that the simulations show that the origin of life may not have been an improbable accident, but instead may have been almost inevitable. Auto-catalytic reactions may have driven chemicals to combine and "reproduce". In other simulations, Kauffman shows how evolution through natural selection may not have depended on improbable, random mutations. Genes may be "self-organized" in such a way as to make genetic improvements a very likely occurrence. Coevolution is also discussed in some detail.
Kauffman also shows the parallels between biological and technological evolution. These parallels are quite impressive. The concepts are amazingly thought-provoking.
This is not an easy-to-read book; it is filled with simple mathematics, though there are very few equations. Sometimes I had to read a page twice to really understand it, but was well worth the effort. Interestingly, the writing style alternates between straightforward technical writing, and lyrical. This alternation gives the book a nice change of pace....more
This wonderful collection of essays on mathematics is quite entertaining. I especially loved the essays on the Strasbourg Cathedral clock, the StatistThis wonderful collection of essays on mathematics is quite entertaining. I especially loved the essays on the Strasbourg Cathedral clock, the Statistics of Deadly Quarrels, and the Naming Names. The author does not simply "do scholarly research" to come up with his essays. For some of his essays, he actually performs a variety of mathematical calculations, computer simulations, and analyses to understand the topics....more
The first half of this book goes into some depth concerning Bertrand Russell's and Whitehead's Principia Mathematica, and then the work of Kurt Godel.The first half of this book goes into some depth concerning Bertrand Russell's and Whitehead's Principia Mathematica, and then the work of Kurt Godel. Hofstadter has an interesting description and point of view about this area. But the later portions of the book become steeped in philosophy, and quite frankly, became a bit boring. On the other hand, I had read his book Godel, Escher, Bach long ago, and found it to be excellent....more
This is a short book--a fast and easy read. The story describes how a good mathematician sank into an obsession that swallowed up his life. The storytThis is a short book--a fast and easy read. The story describes how a good mathematician sank into an obsession that swallowed up his life. The storyteller's mathematician friend, Sammy, mentions that the trail of a mathematical quest will be littered with intermediate, published results on a variety of topics. So, why didn't Uncle Petros publish his intermediate--but important--results?
Interestingly, I do not remember another novel with as many footnotes as this one! (Actually, I don't remember any novels with an author's footnotes.)...more
Truly a fascinating book! De Mesquita has used the game theory of John Nash to develop a series of models that help him to predict future events. TheTruly a fascinating book! De Mesquita has used the game theory of John Nash to develop a series of models that help him to predict future events. The predictions are not simply binary (yes/no) prognostications--they are in-depth analyses that describe what will happen, and why. The author claims a 90% success rate. The last few chapters include a set of detailed predictions made by HIS STUDENTS using his models. Some of the predictions (most notably, Pakistan) are starting to come to pass, now. Others will be proven--or disproven--within the near future. Of course, this book is sort of an advertisement for the author's consulting company. Nevertheless, I've been recommending this book to everybody. ...more