Going on the marketing, what we have here is a counter to fake news and dodgy argumentation in the form of mathematical logic. The back cover tells us 'Newspaper headlines and social media use emotions to warp the facts. Politicians and companies master rhetoric to mislead us. What one book could help us make sense of it all?' Admittedly they don't answer their rhetorical question, but I assume the answer is intended to be The Art of Logic. (Did the company behind this book realise it was using rhetoric, though presumably not to mislead us?)
What we actually have is a combination of a lucid and interesting explanation of the basics of logic with the mathematical equivalent of those books such as Algorithms to Live By that were so popular a couple of years ago. They used the logic of algorithms (differently worded, and, to me, easier to understand), the heart of computer programs, to look at everyday problems. Here, Cheng is using the purer mathematical form of logic to the same end.
Cheng does a good job at explaining logic from a mathematical viewpoint and gives a useful brief dip into her own field of category theory. Her illustrations of concepts like the lost middle are effective, and though it sometimes feels points are being laboured, this can be an alien area to many, and a slow and steady approach is undoubtedly best.
There were a few small content issues. We are told that scientists pick their confidence limits based on the seriousness of the situation - but this seems at odds with the way that physicists use vastly higher confidence limits when dealing with the fascinating, but hardly life-changing Higgs boson than psychologists do when trying to understand and improve human behaviour. There's quite a lot in the book about blame, some of which doesn't sit well with the meaning of the word. We are told that both dropping a glass and a hard floor are 'to blame' for a glass breaking. But where both are causal, blame can't be ascribed to a passive object. And there's a total misunderstanding of the origins of airline overbooking. However, these are small points - overall, the book is engaging and effective in putting across its message.
So far, so good. The problem - and the reason I think this is an important book - comes in two ways when Cheng attempts to apply logic to everyday life. Mathematics works by starting with axioms and building up a logical structure piece by piece. As Cheng says, this is part of its wonderful appeal if you can get past the fear of maths. But what is not emphasised enough is how axioms can cause difficulties. Mathematical axioms seem extremely straightforward statements such as 'A straight line segment can be drawn joining any two points,' or 'two sets are equal if and only if they have the same elements.' But Cheng's axioms are all about what she feels is right. I'm not saying I disagree with her ethics, but rather that value judgements are a poor basis for logical axioms.
The other aspect of the problem is that, as Cheng examines, in applying logic you can select different levels of abstraction from, say, the experience of an individual person up to all people. In her examples, she makes the choice of which level to render that abstraction: yet that choice itself has a major influence on the outcome that isn't recognised in her logical structure. As a mathematician, she should know from the history of set theory that when choice enters the game, even mathematics has problems.
By ignoring these two issues, Cheng gets to a position where, for example, she is prepared to argue that justice should not be blind, but rather the scales of justice should be weighted in favour of those she decides are disadvantaged (as opposed to privileged). Unfortunately, history shows that when society decides to weight justice to favour a particular viewpoint - however apparently worthy - that society is on the road to totalitarianism.
Of course, there is no suggestion that this is Cheng's intention. But this brings me back to back to why I think this book is important. Unlike the algorithms books, which generally concentrate on trying to use logic to deal with everyday practical tasks, Cheng applies logic to societal structures and relationships. In doing so, she demonstrates why taking a mathematical logic approach to life is not only impractical, but quite possibly dangerous.
Review by Brian Clegg