Tag Archives: Mathematics

BOOK: “Mathematical Finance: A Very Short Introduction” by Mark H. A. Davis

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Mathematical Finance: A Very Short Introduction (Very Short Introductions)Mathematical Finance: A Very Short Introduction by Mark H.A. Davis
My rating: 4 of 5 stars

Publisher Site – OUP

This is Oxford University Press’s Very Short Introduction to the field of the “Quants,” individuals who apply mathematics to questions of how to value financial assets and assess risks. The book begins by laying out how banking and financial markets work, then discusses how interest rates are determined, and then explores the quantification of various risks faced by lenders. The book finishes by discussing how the 2008 financial crisis impacted the field and how it operates in the wake of that event. (The 2008 crisis was described in an intriguing fashion in the book and movie The Big Short. It basically resulted from deceptive grading of mortgage-backed securities such that investors who thought they had the ultimate default-proof asset in fact had assets that not only could collapse, but — in fact — were bound to.)

Even though this book is a concise introduction, it shouldn’t be confused for a simple guide. It is not only mathematically intense but also jargon dense. It’s not a complete waste for someone without any advanced mathematics and / or economics / finance background to read, but there will be large patches that will likely be lost on one. (And if you’re not at all used to reading scholarly writing, it may be excessively daunting.)

If you want a quick guide to the field of quantitative finance, and you have an understanding of the mathematical notation used in calculus and statistics, I’d recommend this book. If you are interested in the topic but aren’t at all mathematical, you might start elsewhere (the aforementioned, The Big Short, might be a good place.)

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PROMPT: Pet Tricks

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If you could make your pet understand one thing, what would it be?

Partial differential equations. First of all, then it could explain them to me. Secondly, I could completely demoralize all the Westminster types who think they have “smart dogs.”

Statistician Limerick

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The police questioned an old statistician
 Whose department had suffered attrition.
   "My memo was wrecked
   by auto-correct:
 Distribution of 'Poisson' became 'Poison.'"

PROMPT: Don’t Understand

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Daily writing prompt
What’s something most people don’t understand?
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I’ve often been surprised how little intuitive grasp people have of basic mathematical or statistical ideas or relationships, even when they have had the education to understand with a little effort.

One example of this is what I call “unilateral mathematics” where people fixate on one term or side of an equation while ignoring that changing a term changes the equation’s other side (or to keep the other side static, something else has to give.) For example, I hear people getting so excited by the new salary they will earn when they move to a new place. Then they get to the new locale only to find that the cost of living is so much higher that even their hefty pay boost supports only a diminished quality of life. One sees this tendency a great deal in people’s policy discussions when someone will say, “just set a maximum (or minimum) price” without understanding that shortages or surpluses will come along for the ride. [The Law of Unintended Consequences is another good answer to this prompt.]

We all saw flaws in statistical thinking during the pandemic when people said things like, “See, she got the vaccine and then she got COVID, so obviously the vaccine doesn’t work!” I’m convinced this is because people don’t have good intuition for statistical thinking and — instead — they want to treat a low probability as an impossibility and a high probability as a certainty.

By the way, you see this from people of all persuasions, including those who are highly educated, conservatives, progressives, believers, atheists, etc. One can see the universality of the flaw most commonly in climate change comments. You’ll hear one person say, “See, it’s the hottest day on record, that’s evidence global warming is real!” Another person will say, “See, it’s the coldest day on record, global warming is obviously hokum!” Somehow, even with diametrically opposed viewpoints, these two manage to both be wrong because one day’s WEATHER is not instructive of what is happening to the CLIMATE. In other words, a sample of one provides no insight into state changes in the population. [Maybe it’s more appropriate to use Wolfgang Pauli’s terms and say the two are “not even wrong.”]

BOOK REVIEW: Introducing Fractals: A Graphic Guide by Nigel Lesmoir-Gordon & Will Rood

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Introducing Fractals: A Graphic GuideIntroducing Fractals: A Graphic Guide by Nigel Lesmoir-Gordon
My rating: 5 of 5 stars

Amazon.in Page

Fractal Geometry is a school of mathematics that contends with the kinds of shapes seen in nature, shapes which often appear irregular (at least on some scale,) but which are also frequently self-similar (i.e. the twig looks like the branch looks like the whole tree.) One problem that led to the discipline’s development was determining the distance of a coastline. The distance between measurements vastly alters the final measurement one gets. From the discipline’s origins in observation of the natural world and the problems found in nature, fractal geometry was put to use for problems in ecology, finance, technology, and art and music. The book touches upon this sprawl of the subject, as well as relating fractal geometry to Euclidian Geometry, Calculus, and theories of Chaos and Complexity.

This book offers a simple and cursory overview of the subject. A reader expecting to learn how to employ Fractal Geometry will come away disappointed, but one who just wants to know the kind of problems its useful for and get a basic and intuitive explanation of why it’s useful can gain a great deal from the book. As the subtitle suggests, the book is illustrated and the graphics are far more useful in this volume than in most of the “Graphic Guide” series. That probably comes as no surprise as the subject is inherently more visual than average.

If you’re starting from ground zero, I’d highly recommend this book. Those with a mathematical background may yearn for more depth.


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BOOK REVIEW: The Information by James Gleick

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The Information: A History, a Theory, a FloodThe Information: A History, a Theory, a Flood by James Gleick
My rating: 5 of 5 stars

Amazon.in Page

Information is one of those topics that remains obscure not because it’s rare or hidden, but because it’s everywhere and the term is used for so many purposes it’s not thought of cohesively. It might seem like a book on this topic would be hopelessly boring by virtue of the fundamental meta-ness of the material. Instead, Gleick had a vast sea of topics and stories involving intense stakes for humanity from which to choose, e.g.: how did we learn to communicate and advance said capability until it was arguably the most important feature of our species, by what instructions are people “assembled,” might the most fundamental layer of reality be informational, and – in recent decades — will our species drown in flood of cheap information?

Given the vast sprawl of the subject matter, organization becomes a crucial question. In a sense the book is chronological, presenting humanity’s experience with information in more or less the order we came to think about the subject. I think this was a wise move as it starts from what most people think of when they think of information – i.e. language and its communication. That makes it easier to wrap one’s head around what comes later, and to see the conceptual commonalities. This approach might seem self-evident, but an argument could be made for starting with information as the word is used in Physics (as addressed in Ch. 7 – 9,) an argument that that approach is more fundamental and generically applicable, and while it might be both of those things, it wouldn’t be as easily intuitively grasped.

I found this book to be fascinating and easily followed — even though it covers some conceptually challenging topics, it does so in an approachable manner. It is over a decade old, but holds up well – though I think there is much more to say these days about the detrimental effects of information overload, a topic discussed at the end of the book. I recommend it for nonfiction readers.

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BOOK REVIEW: Nature’s Numbers by Ian Stewart

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Nature's Numbers. Discovering Order And Pattern In The UniverseNature’s Numbers. Discovering Order And Pattern In The Universe by Ian Stewart
My rating: 4 of 5 stars

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This popular mathematics book reflects upon the ways in which patterns appear in nature and how mathematics can shed light on said patterns. It explores why tides are predictable while weather patterns are anything but. It investigates why flowers disproportionately have a number of petals that is in the Fibonacci sequence (a list of numbers in which each is formed through the addition of the previous two numbers.) It shows one how an eyeball can evolve, and how long it would be expected to take. It describes where and how we see calculus, probability and statistic, chaos theory, and complexity in nature.

It’s unambiguously a pop math book, there’s not an equation in sight. It does use diagrams and various graphics to convey ideas, and these help to simplify and visualize the topic. If anything, I would say the book could have benefited from more graphics [and might even have benefited from a less strict rule about sticking to colloquial prose.] (Meaning, some of the analogies and attempts to relate clarified ideas better than others.)

I found the book highly readable, and believe that – overall – the author did a fine job of providing food for thought without getting too complicated for the general reader. There were points at which the author seemed to lose his train. For example, he off-ramped into criticisms of the division of mathematics into applied and theoretical branches and the tendency to more greatly value the applied side of this false dichotomy. I have no doubt this is a worthwhile subject of discussion, but not necessarily in this book.

If you’re looking for a readable discussion of how mathematics is used in the study of nature, this book is worth reading – especially if you are equation-phobic.


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BOOK REVIEW: Introducing Chaos: A Graphic Guide by Ziauddin Sardar

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Introducing Chaos: A Graphic Guide (Introducing...)Introducing Chaos: A Graphic Guide by Ziauddin Sardar
My rating: 5 of 5 stars

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This book provides a brief overview of the mathematical and scientific concept called “Chaos” (as opposed to the colloquial definition.) Chaos theory is most popularly associated with “the butterfly effect” in which small changes in initial conditions can result in large and / or unpredictable variations in outcome (e.g. the Houston butterfly that causes a typhoon in Hong Kong.) Chaos profoundly changed the landscape in many domains of science. Before Chaos, it was generally assumed that if one had a relatively simple model without random elements that one could make short work of developing predictions. Scientists working in Chaos discovered that this wasn’t necessarily the case, despite the intuitive appeal. In fact, one could have a relatively simple model without random elements that still resulted in irregular behaviors / outcomes.

Chaos overlaps with a number of subjects including the science of Complexity and Fractal Geometry. The book explores these connections, and gives the reader a basic understanding of how those subjects differ and what they share in common with Chaos. The book also draws examples from a number of different disciplines including meteorology, biology, city planning, etc. This is a beneficial way to broaden one’s understanding of this fundamentally interdisciplinary science.

I’ve read many titles in this series because they are available on Amazon Prime and provide readable overviews of subjects that are suitable for a neophyte reader. I found this to be one of the better titles in the series. I thought the author did a good job of explaining the concepts in clear, approachable language, aided by graphics. If you’re looking for a non-mathematical overview of Chaos theory, this is a fine book to consider.


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BOOK REVIEW: The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner

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The Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Paul Wigner
My rating: 4 of 5 stars

Available online here

This brief essay asks why math proves so effective for describing / codifying physical laws, and whether our physical theories — built on (phenomenally successful) mathematics — offer the truth, the whole truth, and nothing but the truth.

There’s a popular story in which a drunk man is found on his hands and knees under a lamppost at night when a police officer comes along. The cops says, “What-cha doin’?” To which the drunk replies, “I dropped my keys, and I’m looking for them?” So, the cop says, “Well, they’re clearly not where you’re looking, why not look elsewhere?” And the drunk says, “Cuz this is where the light is.” I think this story can help us understand what Wigner is getting on about, if only we replace the drunk’s “light” with the scientist’s “elegant mathematics.” Wigner reflects upon why it should be that so many laws of nature seem to be independent from all but a few variables (which is the only way scientists could have discovered them –historically, mathematically, and realistically speaking.) On the other hand, could it be that Physics has led itself into epistemological cul-de-sacs by chasing elegant mathematics?

There’s no doubt that (for whatever the reason turns out to be) mathematics has been tremendously successful in facilitating the construction of theories that make predictions that can be tested with high levels of accuracy. However, that doesn’t mean that some of those theories won’t prove to be mirages.

A few of the examples used in this paper are somewhat esoteric and won’t be readily understood by the average (non-expert) reader. That said, Wigner puts his basic arguments and questions in reasonably clear (if academic) language. The essay is definitely worth reading for its thought-provoking insights.


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BOOK REVIEW: The Joy of X by Steven H. Strogatz

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The Joy of X: A Guided Tour of Mathematics, from One to InfinityThe Joy of X: A Guided Tour of Mathematics, from One to Infinity by Steven H. Strogatz
My rating: 5 of 5 stars

Amazon.in Page

This is a mile-wide and inch-deep overview of mathematics. That is to say, it shines a light on a wide variety of subdisciplines, running from counting through subjects like topology, using rudimentary examples to give the reader insight into the kind of problems that can be solved. The book employs graphics, intuitive examples, and step-by-step explanation to clarify mathematics for individuals who didn’t get on so well with the subject the first time around.

The book’s thirty short chapters are divided into six sections: numbers (arithmetic,) relationships (roots, powers, etc.,) shapes (geometry,) change (calculus,) data (statistics,) and frontiers (group theory, topology, analysis, etc.) Like most popular mathematics books, formulas and equations are avoided to the extent possible. Even the notes that elaborate for curious readers use mathematical notation sparingly.

If you’re looking to give math a second try, this wouldn’t be a bad overview to get started. I don’t think it would be of much benefit to anyone who’s stayed in touch with mathematics, but it makes a fine light overview of the subject.


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