Book review

Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications

by Nassim Nicholas Taleb


I’ve just started reading this, and wanted to say TYPOS before I forgot. Not that I could forget, there’s one every couple of pages. Which is annoying, as there is clearly good stuff in here, from which this detracts – those formulae that I can’t quite follow through, is that a typo, an error by the author, or just me being dim? Hmm… I shall report more as I get further in…

Right, we’ve finished it now. Taleb is certainly opinionated – very opinionated about other people and their views, at times, and personally I felt this rather detracted from the writing. Keep it for the Twitter wars, mate, really.

The book is a compilation of a number of papers, some of them quite technical, about the subject matter. Throughout, I ran into the issue mentioned in the first paragraph – typos are rife, and quite disturbing in the middle of formulae or proofs.

Much of what Taleb says, however, is important, even if he does rather aim at getting in his retribution first against his perceived opposition. He’s quite right, fundamentally, about the difference between probability and reality: in the real world, there are no probabilities – things happen, or they don’t. Probability expresses our uncertainty, not a real-world property of objects (perhaps quantum mechanics aside – although by some interpretations, this is just as true there, too). So we need to be very very careful when we apply simple to manipulate theorems of probability (such as the Gaussian distribution) to real-world problems, especially when we are making high-impact decisions based on the result of those manipulations. Examples abound, but the most dramatic is perhaps the Value at Risk (VaR) calculations that were supposed to keep our financial institutions “safe” during market turbulence, and which so spectacularly failed to do so in the 2008 financial crisis.

Tough for me to review this book usefully. I’m pretty mathematically smart, but I still couldn’t follow all of the arguments. I’m sure others with more of the appropriate background could do so, but in this case I’m not sure it went far enough. Replacing VaR by Extreme Value Theory is all very well to talk about in principle, but you’re going to need a lot more detail to make it practical, for example.

Three stars, in the end, for the sound argument, counterbalanced by TYPOS and unnecessary sarcasm from the author.

Book review

Our Mathematical Universe

My Quest for the Ultimate Nature of Reality

by Max Tegmark

I found this a thoroughly entertaining account by Max Tegmark of some of the “big questions” of modern physics, with a particular slant towards cosmology – what is our universe composed of, how did it come into being, what might be its ultimate fate, that kind of thing. Initially I wasn’t sure about the very personal style in which it was written, but after a few chapters I began to like it. Tegmark has been at the forefront of a number of these discoveries, and there are lots of semi-autobiographical anecdotes in here: ultimately, the cheerful and enthusiastic style did win me over, and I found it definitely added to my enjoyment of the book.

I realised as I was reading this that I haven’t really read a popular science account of these topics for perhaps 20 years, so I was definitely behind the curve with regard to things like dark matter and dark energy, and their abundance in the universe (making up something like 20% and 70% of all the stuff out there, respectively). This was a great account of all these theories, working through the evidence in particular from the cosmic microwave background, which has yielded two different Nobel physics prizes.

Tegmark moves the reader through all this at the same time as he elucidates his view of how it all comes together with quantum mechanics. I was pleased here to see the increasing acceptance by physicists of Everett’s “many worlds” interpretation of quantum mechanics, and how it’s taken over from the more historically popular Copenhagen interpretation, with its reliance on the horribly ill-defined “observer” causing a collapse of the quantum wave function. To the very limited degree that I’ve ever managed to get my head around quantum mechanics, it’s always seemed a complete fudge to say “these equations describe the universe as we know it to thirteen decimal places, but I don’t really like what they say about the nature of reality, so I’m just going to paper it over with this observer thing and then I’ll feel more comfortable”!

At this point, Tegmark starts to roll out the big guns of his theory. Not only are there multiple physically separate universes (because the initial expansion phase was so fast that light will never be able to travel from one to another), but also these universes will have different starting conditions. This provides an explanation as to why our universe is so “well tuned” for life – there are 30 or so physical constants whose values would only need to vary by a very small amount to have prevented life ever existing. Well that’s straightforward in this view- those other variations are just in other universes (inaccessible to our light cone). And quantum mechanics now says that physical reality also has other places where the alternate paths of quantum interactions are playing out – in some of these parallel worlds, Schrodinger’s famous cat is alive, in others it’s dead, and that’s all ok (unless perhaps you are the cat!)

Ultimately, Tegmark comes to argue that in fact these different worlds may not only have different starting conditions, but also entirely different laws. Not just (for example) different numbers of space or time dimensions, which is weird enough, but different kinds of physical existences. All our physical laws, he argues, are the result of mathematical symmetries – all the different kinds of particles (quarks, photons, etc) are predicted by symmetries in group theory, for example. Thus there will be other universes with other symmetries, based on other groups. An overwhelming number of these will be too simple to do anything very interesting, but some of them will be complex – perhaps complex enough to form other kinds of particles, and indeed other kinds of life, too strange to even imagine!

Since I recently read Group Theory and Physics, I was pleasantly surprised by how much I sympathised with Tegmark’s thesis here. It’s even reflected in science fiction in books like Distress, which again I read recently, and which is about how a “theory of everything” might decide which universe (which mathematical symmetries) we really live in. I’m obviously being swept up in a wave of theories and stories all about this – perhaps the cosmos really is trying to tell me something!?

Ultimately, these kinds of questions about the underlying nature of reality are profound, and perhaps even beyond our real knowing, but they’re great fun to think about, and Max Tegmark is exactly the sort of person who you’d want to be sat beside in the pub while you explored it all. This book is as close as I’m ever going to get to that, and I thoroughly enjoyed the opportunity.

Book review

Group Theory and Physics

by Shlomo Sternberg

Group Theory and Physics by Shlomo Sternberg

In this book Professor Sternberg of Harvard University takes a fairly detailed look at the deep links between mathematical group theory and the underpinnings of much of modern physics. It’s heavy going in many parts, requiring at least an undergraduate degree in mathematics (one containing lots of group theory) to even vaguely follow it. If you can at least follow along (and I was very much reduced to skimming the deeper technical material), then you’ll find beauty, and deep truth. Eugene Wigner talked about the “unreasonable effectiveness” of mathematics in explaining the physical world, and that’s what you will find herein.

In Chapter 1, Prof. Sternberg introduces key ideas like the action of a group on a set, and uses them to classify the finite subgroups of O(3), the orthogonal group of order 3 (i.e. the group of ways in which you can transform things in three dimensions such that distances between things are preserved). There is an immediate physical application in crystallography, and Prof. Sternberg takes us through this, introducing me at least to some marvelously-named crystals and compounds (iodosuccinimide, diaboleite, or wulfenite, anyone?)

Chapter 2 introduces representation theory for finite groups, which is a way of representing abstract objects as matrices, with the operations on those objects as equivalent to things like matrix multiplication. Using these techniques, you can produce character tables, which are a simple two-dimensional table that can capture much of the deep structure of a particular group. This material is set up for use in the following chapters.

Chapter 3 starts to build on this, first with a simple physical model of molecules as point masses connected by springs, and then adding in some quantum mechanics. This allows us to look at the ways in which such model molecules would “resonate” in vibration, which in turn leads to direct physical applications in spectrography. (Appendix F contains an interesting, detailed, and entirely non-technical review of how spectography evolved through the 19th century – one of the highlights of the book, for me.) This leads to a discussion of the symmetries of space-time, and out of this emerges a very natural representation of the fundamental properties of subatomic particles. Finally we come to Wigner’s truly remarkable 1951 paper in which he identifies these subatomic particles “as” the irreducible representations of the group. By doing this, Wigner was able to lay out a model that has directed particle physics for at least the last 60 years – he earned the Nobel Prize for Physics for this in 1963, and subsequent modifications of his work by Gell-Mann and others have lead to further Nobel Prize level discoveries.

It’s worth pausing on this point: here we have a use of the most abstract parts of mathematics, out of which directly flows deep and powerful results about the universe in which we live. It’s not that we can (say) simply model a molecule as a bunch of weights joined by springs: the types and properties of the most fundamental things we know about in the world spring directly from this abstract mathematics. It’s truly a remarkable, and very beautiful, thing.

Chapter 4 continues the journey, moving on to compact groups. From this we get models which apply directly to the hydrogen spectrum, models of atomic nuclei, and the group underlying the periodic table. Again, we see the powerful linkages back to Wigner’s work on the Poincare group.

Chapter 5 moves on to quarks, and the predictions of and subsequent discoveries of the elements of quantum chromodynamics. Because this book was written in the early 1990s, it doesn’t include any discussion of grand unified models or sypersymmetric models like string theory or quantum gravity, which is a shame, as once again we see the mathematical and group theoretical underpinnings.

There is much to like about this book, and I enjoyed the journey. However, I spotted a number of annoying typographical errors, and I very much fear that if I’d been able to follow some of the denser passages, my journey would have been made harder by further typos. The index is also weak, and I was often reduced to paging back and fro to find what a particular notation or definition was. I also felt that since this book set out to link group theory and physics, I was often not at all certain what the point of a particular mathematical section was – a brief up-front discussion of the motivation would have been very helpful to me. The fact that it ends in the mid-1990s also means it doesn’t cover a lot of recent, equally interesting, work in theoretical physics.

Finally, to be honest, a lot of the book was somewhat beyond me: the author says “a course in multivariate calculus and linear algebra, together with an elementary physics course, should suffice”. It’s been a few years since I took my courses in these subjects, but I have very good retention of them, and they in no way adequately prepared me for some of the long technical passages!

Three stars for me – quite possibly more for you, if you’re further along the maths curve than I started.

Book review

The Indisputable Existence of Santa Claus

The Mathematics of Christmas

by Hannah Fry

The Indisputable Existence of Santa Claus by Hannah Fry

Very nice little book of mathematics, as applied to various Christmas problems.

Although this is in the long tradition of seasonal stocking-fillers, the execution was great: the maths was non-trivial, but not so hard as to require a degree in the subject (although there were extra-hard sums in the footnotes, where appropriate), and the whole subject was treated with the appropriate degree of tongue in cheek sillyness:

Maybe that’s how we should see Santa. An undecidable being. You can’t prove his existence one way or the other, you just have to close your eyes and decide for yourself.

In the meantime, everything from simple geometry to Markov chains was used to solve various seasonal sillynesses. Excellent!

Book review

How Not to Be Wrong: The Hidden Maths of Everyday Life

How Not to Be Wrong: The Hidden Maths of Everyday Life
Read date: Feb 2019

How Not to Be Wrong: The Hidden Maths of Everyday Life by Jordan Ellenberg
My rating: 4 of 5 stars

Four and a half stars, plus or minus half a star (to stay with the spirit of the book itself).

A delightful survey of a diverse array of subjects, showing how mathematics is important in our discussions in everyday life, from politics to podiatry. At the bottom, a lot of it comes down to logic and to numbers, and our ability to understand what’s going on is greatly aided by the wise words and examples that Jordan Ellenberg uses. Great discussions about p-values, about what use geometry is in our real lives, about how the Supreme Court makes its decisions, and more.

Loved it. Half a star off for the election rules discussion at the end – too much detail for even me to care about, and I really actually do care about it quite a lot.

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Book review

Four Colours Suffice: How The Map Problem Was Solved

Four Colours Suffice: How The Map Problem Was Solved
Read date: July 2018

Four Colours Suffice: How The Map Problem Was Solved by Robin J. Wilson
My rating: 4 of 5 stars

This book is a clear and entertaining account of the long history of the attempts to provr four colour theorem – that any map on can be coloured with at most four colour, such that no countries with a common border have the same colour. Although there are lots of interesting characters and asides, this is not a book for the mathematically faint of heart: in order to understand the approach that finally proved the conjecture, Robin Wilson takes you pretty far into the dense woods. If you enjoy following along, and are prepared (as I was) to skim over the more complex parts, you will still come away with a good appreciation of how it happened. You will also perhaps understand why many mathematicians at the time were skeptical about the proof: it needed over 1000 hours of computer time to complete the proof, and the approach is too complex to check by hand. Does that count as a “proof”? Nowadays, when computers are routinely used by mathematicians to check their own work, people would have fewer doubts, but in the 1970s, many felt this to be a real issue.

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