Math

Infinity or infinitesimals really are something that can boggle your mind. Similar as zero, infiity 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.

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.

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 mondern 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.

Despite the luring title “Nerds - Who they are and why we need more of them” I would not have bought the book – it was a gift. The author David Anderegg, PhD has his academic degree printed on the book’s cover. I am always a bit suspicious of people insisting on the public use of their degree. At least as a means to promote their work, to lend credibility to their expertise. Nevertheless, I mostly enjoyed reading the book.

The author gives an assessment of the nerd and geek stereotype in American culture. He pays special attention to children and adolescents and how the nerd label may affect their personal attitude towards academic achievements. I think his observations are quite to the point. Even though the book is rather American-centric the same observation can be made in Germany. There is a general aura of anti-intellectualism in Germany as well. People are actually proud of their lack of math skills and little understanding of science at large.

This is a serious problem for society as people (children) who do not want to be labeled as nerds or geeks may choose not to reach their full potential. This is a loss for society. Technological and scientific advancement could be improved if only it was not hip to be dumb or just mediocre.