Infinity: New Research Frontiers (Hardcover) by Michael Heller and W. Hugh Woodin (7 Feb 2011)
I was looking forward to reading this book and it was the first one I started in the last batch that I bought. However, I'm disappointed. It isn't obvious what level the book is pitched at. The book is composed of 14 different articles in 4 sections from 13 different researchers. The topics are all based on Infinity but each section considers the concept from different schools of thought: mathematics, physics, and theology. (NB: this is my full review, the Amazon version omits the bulk of the details).
Some articles, or parts of articles, are great while others are merely good. Unfortunately there are a couple of articles that really don't belong. The book is published by Cambridge Academic which should suggest that the articles are at a research level, that isn't quite the case. I'd say about 75% is fairly readable from a lay perspective and definitely readable by a keen/willing reader. The remaining 25% is either too mathematically abstruse or too frustrating to read; one article seems to use made up words that are closer to random unpronounceable strings than actual words (phonologically illegal). If you Google these words then it turns out some of them are 'real' but almost no explanation is given of them and I'd wager that a very narrow subset of professional researchers worldwide will know them.
As a concise volume of information for reference to topics on Infinity then this might be the book to have. That said, I still think it could have been better formed. Ultimately, there is a lot of good information in there but it could have been presented in a more coherent way. Part of the problem is that it is composed of several research articles that hang together by a common theme but don't really inter-related in a coherent enough way that you'd ideally want from a popular science/maths book. No doubt it was the intention to aim at the former rather than the latter but I think there is a lot of value in releasing a modern book about infinity that engages with cutting edge research, and that's what this book almost is.
For me the highlights were articles 2, 3, 10, 11 & 13.
Section I - History
Article 1: Provides a historical account of the evolution of the concept of infinity. Most of this is covered in other books and articles on Infinity, so it won't be anything new (as it wasn't for me) if you keenly read about Infinity (even from Wikipedia). The article is well referenced but slightly too wordy, I don't think the article flows well enough to be easily digestible. As a whole it is easily understandable from a lay perspective. The real annoyance is the inclusion of words spelt using Greek letters. While I can translate the sounds in my head I'm not that fast, so trying to pronounce some of the new words used in Greek antiquity is not easy. The words are not presented once in Greek but recur several times as if though they are normal terms that are easy to read. This isn't case, infact it is very frustrating to try to read them.
Section II - Mathematics
Article 2: Short and probably not new for anyone that's read a lot about infinity, but the information is well presented and reads well. If you lack of certain mathematical terms or expressions then you might struggle, there are a few equations but it should be accessible to most with an advanced high school level of knowledge (keen lay people). One the things I liked about this article is the mention of Turing Machines and Busy Beavers, I haven't come across those in many books but are worthwhile inclusions for considering large numbers and infinite processes.
Article 3: The first pages are quite acerbic and includes a reference to Shakespeare. I was close to giving up with the article after the end of second page but it gets better. The author actually has a valid point of view and by the end I had a deep respect for him. The main concern of this article is the foundations of mathematics, have we accepted some things into the canon of mathematics that is logical but perhaps not wholly consistent? I think these are valid concerns but the consensus seems to be that the concerns are too pessimistic. That's my naive observation at least.
Section III - Advanced Mathetmatics.
Article 4: I think I have to admit defeat here. This one was beyond me. The first 6 - 7 pages were ok, but beyond that I started to struggle. I could follow some of the logic and see that the arguments appeared consistent and logical but the actual meaning of the words was beyond my knowledge. This is some of the most abstract mathematics I've come across and I struggle with some of the fine details of transfinite sets.
Article 5: An interesting suggestion about how infinity plays a role in understanding if the universe is deterministic or non-deterministic. The idea is fascinating but I found that only the first 3 or 4 pages were readable. This one definitely isn't for a layman, which is a shame as I can see a lot of mileage in this idea.
Article 6: Concept Calculus. Great sounding concept but far too abstract for even someone with considerable mathematical training. The specifics are too deep into set theory for me to follow. Interestingly, I think there is some common ground between this idea and what I've written in other articles on this website.
Section IV - Cosmology
Article 7: This article by Rovelli is too simple. I have a deep respect for author (and would recommend his other writing) but this article is not focussed enough on Infinity.
Article 8: A fair review of cosmological inflation and how it relates to concepts of infinity. Most of it wasn't new to me but the part about eternal inflation is not something I had studied before, so I learnt something at least. I found this article to be readable and is a nice review of inflation for those that don't know it.
Article 9: This article covers some basic ideas regarding the cosmic microwave background and the geometry (shape) of the universe. It also expounds the notion that physical infinities are not the same as mathematical or metaphysical infinities. One is practical while the other is only theoretical, and eventually this ties in to whether shape of the universe is knowable if the universe is flat. It is best to read this one to get the full meaning. Probably one of the better articles in this book. I wasn't so enthused with the comments on art at the end though, could have done with that.
Article 10: This Michael Heller's article (chief editor) and is one of the highlights of the book. The writing style is highly readable and the content is good, I feel like I've actually learnt something new. The comments (criticisms) of multiverse theories is worth a read, and provides a fair cautionary on accepting such ideas. The ending part on theology is somewhat interesting but not the best perspectives on theology and infinity in this book.
Section V - Theology
Article 11: For me this article is another highlight. He recovers old ground with a definition of infinity as many of the others seem to do: either starting from a historical perspective or some definition of infinity and then working from there. It does become a bit tedious. In this article the author talks about the relation between the concept of infinity and of God. It is probably the best article in the theology section and I would definitely say I learnt something new. The article considers what it meant what a property of God is said to be infinite, such as infinitely good (or similar), and how that might conflict with saying that God is infinite in all regards. If we are cynical then we might say that theologians have an interesting perspective on infinity because they have to try to justify their ''crazy'' beliefs. I offer no criticism or praise on personal beliefs but rather would like to point out that this article contains many interesting thoughts regardless of the motivation. The philosophical curiosity should be enough for even the most ardent atheist (although it probably isn't).
Article 12: The most confusing article of the entire book. Did not enjoy; won't try to read again. This is the article with the phonologically illegal words.
Article 13: Another strong article, but it starts with historical references (too common in this book -> incoherent / too repetitive). It also retells the ideas of countability which are also expounded in quite a few previous article. However, the author goes on to mention ordinary infinity and what happens what you add/multiply ordinary infinities together. That is quite rare for articles on article, and isn't present elsewhere in this book. Finally, the author looks at Cantor's Absolute Infinity and asks about how to define it. He then compares Absolute Infinity to God and concludes that there is *something* that can be said about it but there is also so much that is entirely unknowable.
Article 14: A fair rebuttal of 12 and 13. He criticizes the comparison of God and Absolute Infinity and asks for a clearer definition. This is similar criticism levelled at article 12 but that's the least of the worries for article 12.