Math

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Read: Foundations and Fundamental Concepts of Mathematics

While Eves’ Foundations and Fundamental Concepts of Mathematics is certainly a bit outdated by now – it is 1997 reprint of a textbook originally published in 1990 – it was still fun and interesting to read.

The books offers a nice historical overview of fundamental concepts of mathematics (hence the title) that includes not just the historical background but a solid introduction of each concept itself. Of course, solid means here that the introduction just provides as much depth as is needed to understand what it is all about. As such the different chapters may whet one’s appetite for more on the respective topic. Just when it gets interesting the text stops. It has to. Otherwise, Eves would not be able to cover as much as he does.

Sometimes maybe, the little detail that is given can also already seem a bit too much. Getting to a theorem 48 in just one chapter shows that Eves is certainly not just skipping over details when he feels the reader may benefit from a rigorous presentation of the material.

Read: How not to be wrong

With “How not to be wrong” being about mathematical thinking I was a bit surprised about how much of it was about statistics. And even though it (may) lack(s) the depth of critique of the (ab)use of statistics that can be found in the works of Ziliak and McCloskey or Gigerenzer it is a very good popular treatment of the topic. Worth the read.

A particular additional added value is – in my opinion – the reminder that most things in the real world are not linear. Linearity is just an approximation, valid for only (very) small ranges. I agree with Ellenberg, we – I – forget this too often.

The only thing that I did not like was the sports references (I can condone idiosyncratic tastes in music). The book includes lots of footnotes and endnotes with references. So many, and so many recent ones that I, indeed, found a few new sources that I added to my to-read list. That is rare.

Skimmed: Counterexamples in Analysis

The problem with ordering your books online instead of browsing the shelves of a bricks and mortar bookstore is that you may end up with a disappointment. And, if you are ordering in bulk you may well exceed the time allowed for sending the book back for a refund before you have your first look at it. This is was happened to me here. Though it may have been unlikely to find the book on a shelf of any traditional book store.

Counterexamples in Analysis is an enumeration of a long list of curious mathematical terms that have one property but not another that is usually implied by the first. Apart from introducing some notation this is pretty much it: a list of mathematical terms without any embellishment. Certainly not a book that anyone would or should read cover to cover. Hence, I put it down and on my shelf after about the first 10% or so. There is a lot to learn in this little book. Yet, I do not think I can appreciate this rather bare list.

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Read: A Brief History of Infinity - The Quest to Think the Unthinkable

Infinity or infinitesimals really are something that can boggle your mind. Similar as zero, infinity was not always there in our (mathematical / philosophical) toolbox. Even though we nowadays use the concept of infinity and its reciprocal the infinitesimals almost nonchalantly, we do so without really considering the philosophy and history behind it.

Brian Clegg provides such a history. And if his book was not part of a larger series the books title would be the first pun: A brief history of infinity. There are others. The book even closes with one. Correspondingly the whole book has a rather light tone; Clegg’s rhetoric is almost colloquial. This makes the book rather enjoyable, the topic could have certainly also presented in a much duller way.

For anyone more generally interested in mathematics the book is, however, a disappointment. The focus is clearly on the history of infinity and not the mathematics or the deeper philosophical questions that are only commented upon en passant. And even the history part is certainly – as the classifier “brief” in the title indicates – not a complete and authoritative treatise. The author may also have padded the text with some material that seems to belong more to his own personal interests. There is surprisingly much space dedicated to Roger Bacon. Or, it is rather not so surprsingly; earlier Clegg wrote a whole book about this medieval scholar. In Clegg’s defense, Bacon really did contribute to the discussion on infinity.

Read: Rational Decisions

Ken Binmore’s Rational Decisions is not quite what I expected when I first read just the title. It is, however, something you should very much expect from Ken Binmore. He is an excellent Game Theorist. And even though he eyes current experimental economics rather critically, i. e. he is just more skeptical about some of the claims your standard experimental economists (or are these rather behavioral economists?) are willing to make, his research is very much based on empirical findings, experiments and the boundedness of human capabilities and facilities.

Rational Decisions is about Bayesian Decision Theory. It is about where it can apply and where not. Bayesian Decision Theory It is useful in Games, in situation in which everything is known up to an exactly quantified level of risk. It is utterly “ridiculous” if the decision maker instead is confronted with uncertainty as is the case in the real world where most events do not have a unique objective probability.

Rational Decisions is, therefore, about filling the gap between Decision Making under risk and Decision Making under uncertainty. In my opinion, it is ultimately about an instance of Bounded Rationality.

Binmore gives a comprehensive overview of the standard economic model of decision making and its limitations. He explores the foundations of this model, reviews the historical context and clarifies how Economic Theory has to be interpreted, that current Economic Theory (i. e. the concept of revealed preferences and utility) is not concerned with psychological motives at all.

His writing is witty and opinionated, yet, succinct and clear. Though, sometimes the math made me think twice… All in all, Rational Decisions is a very enlightening experience.

Read: Unknown Quantity - A Real And Imaginary History of Algebra

Unknown Quantity was part of my last year’s advent calender. A surprise since it was not on my wish list. Yet, it could have been on the list as I am – at least once in while – interested in math, or the history of math.

Unknown Quantity traces the history of algebra from its roots to modern times. While it seems quite comprehensive concerning the earlier phases of discoveries it becomes rather patchy in describing the development during the, say, last hundred years. Of course, a lot was happening during these years and most can only be understood with a proper degree in mathematics. A fact that led the author John Derbyshire (apparently quite a character) to add some more technical notes (on the easier topics) to the historic accounts and biographical notes.

The book is rather interesting and informative; though certainly it is not the authoritative source on the history of math, or even just algebra.

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