The Power Of Mathematical Thinking

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How Not to Be Wrong explains the mathematics behind some of simplest day-to-day thinking.[4] It then goes into more complex decisions people make.[5][6] For example, Ellenberg explains many misconceptions about lotteries and whether or not they can be mathematically beaten.[7][8]

Chapter 1, Less Like Sweden: Ellenberg encourages his readers to think nonlinearly, and know that "where you should go depends on where you are". To develop his thought, he relates this to Voodoo economics and the Laffer curve of taxation. Although there are few numbers in this chapter, the point is that the overall concept still ties back to mathematical thinking.[10]

Chapter 2, Straight Locally, Curved Globally: This chapter puts an emphasis on recognizing that "not every curve is a straight line", and makes reference to multiple mathematical concepts including the Pythagorean theorem, the derivation of Pi, Zeno's paradox, and non-standard analysis.[10]

Chapter 7, Dead Fish Don't Read Minds: This chapter touches on a lot of things. The basis for this chapter are stories about a dead salmon's MRI, trial and error in algebra, and birth control statistics as well as basketball statistics (the "hot hand"). He also notes that poetry can be compared to mathematics in that it's "trained by exposure to stimuli, and manipulable in the lab". Additionally, he writes of a few other mathematical concepts, including the Null hypothesis and the Quartic function.[11]

Chapter 10, Are You There, God? It's Me, Bayesian Inference: This chapter relates algorithms to things ranging from God, to Netflix movie recommendations, and to terrorism on Facebook. Ellenberg goes through quite a few mathematical concepts in this chapter, which include conditional probabilities relating back to "P value", posterior possibilities, Bayesian inference, and Bayes theorem as they correlate to radio psychics and probability. Additionally, he uses Punnett squares and other methods to explore the probability of God's existence.[11]

Chapter 12, Miss More Planes: The mathematical concepts in this chapter include utility and utils, and the Laffer curve again. This chapter discusses the amount of time spent in the airport as it relates to flights being missed, Daniel Ellsberg, Blaise Pascal's Pense's, the probability of God once more, and the St. Petersburg paradox.[12]

Chapter 13, Where the Train Tracks Meet: This chapter includes discussions about the lottery again, and geometry in renaissance paintings. It introduces some things about coding, including error correcting code, Hamming code, and code words. It also mentions Hamming distance as it relates to language. The mathematical concepts included in this chapter are variance, the projective plane, the Fano plane, and the face-centered cubic lattice.[12]

Chapter 17, There Is No Such Thing As Public Opinion: This chapter delves into the workings of a majority rules system, and points out the contradictions and confusion of it all, ultimately stating that public opinion doesn't exist. It uses many examples to make its point, including different election statistics, the death sentence of a mentally retarded person, and a case with Justice Antonin Scalia. It also includes mathematical terms/concepts such as independence of irrelevant alternatives, asymmetric domination effect, Australia's single transferable vote, and Condorcet paradoxes.[14]

Business insider said it's "A collection of fascinating examples of math and its surprising applications...How Not To Be Wrong is full of interesting and weird mathematical tools and observations".[22]

Times Higher Education notes "How Not To Be Wrong is beautifully written, holding the reader's attention throughout with well-chosen material, illuminating exposition, wit, and helpful examples...Ellenberg shares Gardner's remarkable ability to write clearly and entertainingly, bringing in deep mathematical ideas without the reader registering their difficulty".[24]

If your acquaintance with mathematics comes entirely from school, you have been told a story that is very limited, and in some important ways false. School mathematics is largely made up of a sequence of facts and rules, facts which are certain, rules which come from a higher authority and cannot be questioned. It treats mathematical matters as completely settled.

Toward the end of each chapter, Ellenberg broadens from these specific examples to a series of questions about how else some of the ideas in the chapter might be used, what kinds of mathematical questions are left to answer, and what kinds of real-life problems they might eventually solve.

Knowing math provides you with a tool to gain a deeper understanding of the world around you. You can use mathematical thinking to get problems of commerce, politics and even theology right. When students ask:

The mathematical concepts explored here are relatively basic, though a reader should have a basic understanding of geometry, algebra, probability and data visualizing to gain the most utility from this book.

When considering the various types of mathematical facts, Ellenberg segments them into four categories based on a 2 x 2 matrix (left). On one axis, facts are categorized by the complexity of the computations involved: some are simple, while others are more complicated.

In this section, Ellenberg describes numerous cases in which challenging the assumption of linearity can lead to interesting findings. Sometimes common phenomena are assumed to be linear but are, in fact, non-linear. In other cases, phenomena are assumed to be non-linear but can, in fact, be considered linear when viewed on a different scale. Understanding when to apply linear and non-linear thinking leads to useful insights.

Inference is the basis of many statistical methods. Simply stated, it is the assertion that conclusions about full sets of data (or populations) can be drawn by observing a randomly selected subset of that data (samples). Inference is a very powerful statistical tool that has led to many discoveries in traditional fields such as science, engineering, finance and medicine, but also in surprising places such as theology.

However, like other mathematical tools, when used incorrectly, statistical tests rooted in inference will reject true conclusions and fail to reject untrue hypotheses. While using inference as a basis for decision making (whether formally through statistics or informally through crude observation) it is important to understand that improbable things happen all the time.

For this reason (and his mathematical inclination toward understanding uncertainty), Pascal viewed the decision to believe in the God of Christianity as a wager. In order to decide if to believe or not, he carried out a straightforward expected value computation: expected value of believing God is real = chance that God is real × infinite joy + chance that God is not real × some finite cost of a life of piety.

Given the title, it is only fitting that Ellenberg concludes with a summary of how to use the mathematical principals described in the text to always be right. However, the answer is not what many readers want to hear.

The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one. And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life is another, and we should keep track of the differences if we can.

I think one of them is to not only be able to compute, but to look at the answer and see whether your answer makes sense, which is sort of a species of skepticism, right? You should always have skepticism about your own computations whether they're mathematical computations or moral computations or whatever they are. The example I give in the book is that if you're working a word problem that asks you to compute how much mass of water is left in a jug and you get negative four, part of mathematics is to be able to look and say, "No, there [are] not -4 grams of water in the jug. That is not actually an amount of water. I don't know what I did wrong, I don't know where I switched a plus for a minus or switched a 2 for a 9, but I know I did something wrong." ...

One thing that is very special is that mathematical knowledge is something the student can create and experience directly. It doesn't rely on authority. If you study history, in the end you're not experiencing those things directly. You learn that a certain thing happened because your teacher said it happened ... but you're not experiencing that yourself. And the truths of mathematics are different. I think this is where, in school, we teach kids that knowledge can be created and that it can be experienced directly, and that if your teacher says something that doesn't make sense, you can show them that what they said doesn't make sense.

It's kind of that same reason that if we're like, "Who has the best batting average in the American league?" we don't count every single batter because there's going to be one guy who only came up once, and he got a hit and he has a perfect batting average. And we don't look at that guy and be like, "That guy is the best hitter in the entire league." We don't do that because on some level we know that small sample sizes don't give reliable measurements. And what the mathematics does is it's there to formalize that. It's there to say exactly how much more unreliable is a smaller sample size that's this much smaller. So it's taking an insight that we already have available to us, that's part of our cognition, and just making it more precise, more strong, more powerful, more generalizable, more correct.

Statistics This book may contain the single greatest explanation of the p value, its uses, and its faults that I have ever read. Ellenberg adds further value with insights on sample size (n), representative samples, Bayesian thinking, the need for (and difficulty with) replication in science, and numerous examples of reputable medical journals that miss the boat on both conceptual and mathematical grounds. 781b155fdc