From debates about same-sex marriage to white privilege … A book of pure maths applied to the real world makes the case for thinking more clearly and logically in politics
What connects Russell’s paradox, intolerance and battenberg cake? Or Euclid’s axiomatisation of geometry and sexual harassment? The definition of marriage and lasagne? They are all sets of concepts that readers will find completely sensible and illuminating by the end of this mind-expanding book about “how people construct misleading arguments, and how we can argue back”.
First, a spoiler. The Art of Logic will not teach you how to win arguments. Do not buy it as a step-by-step guide to beating Twitter using maths. Its aim is “to help achieve better mutual understanding”, as Cheng explains in a disarming final chapter: “What I want to see in the world is more good arguments.”
The book is divided into three sections, The Power of Logic, The Limits of Logic and Beyond Logic, and it is about as difficult to summarise as a complex mathematical formula. There is a sense in which it is a complex mathematical formula – a set of logical proofs in which A implies B, and B implies C, and so on, until Z is inevitable assuming that we accept A. This makes for a very persuasive argument. Cheng also includes a chapter called How to Be Right in which she recommends prefacing certain statements with the clarifying logical phrase, “there is a sense in which … ”
She begins by describing what logic does and doesn’t mean to a mathematician – “Maths isn’t about right and wrong, and nor are most arguments” – and goes on to demonstrate the concept of logical implication, “that ‘if’ one thing is true, ‘then’ another thing must be true, using logic”. Working backwards through a person’s logical implications, much like a child repeatedly asking “why?”, can reveal their fundamental beliefs, or axioms. Two people can both be logical but still disagree, she points out, if their axioms differ.
A person is illogical if: “1. Your beliefs cause contradictions, or 2. there are things you believe that you cannot deduce from your fundamental beliefs, or 3. there are logical implications of things you believe that you do not believe.” A person is “unreasonable” if they are unwilling to consider changing their beliefs.
Along the way, Cheng explores converse errors, grey areas, oversimplifying and levels of abstraction. She highlights the difference between false equivalence and false dichotomy. This is pure maths applied to the real world, in a surprisingly practical way. It also happily accepts that there is a limit to logic – and that logical people must use empathy and emotion if we are to stand any chance of understanding each other and getting on.
Cheng makes her own axioms clear from the outset, and someone arguing with her on Twitter might describe them as particularly lefty and woke. (Those are not terms used in mathematical logic.) Writing about false equivalences, she complains: “The most pernicious straw man argument I see regularly is ‘All lives matter’ as a response to ‘Black lives matter’.” A cynical reader might think it stretches coincidence that logic, in almost every example given, seems to support her modern liberal point of view. But that reader, if she were being logical, could trace Cheng’s conclusions back to her axioms. In a chapter about how to axiomatise a system of beliefs, she makes the important point that not every deeply held belief is an axiom; saying that “I oppose same-sex marriage because I believe marriage should be between a man and a woman” is like saying that “the ingredients for lasagne are: lasagne”. But by interrogating that belief we can discover what the axiomatic belief is behind such a statement, or that the statement is not logical after all.
It’s a shame that not everyone can read this book, but Cheng claims it is incumbent on those of us who can to use compassion and logic to argue productively with those who can’t. In this way, advanced mathematics could make a meaningful contribution to creating a better society as well as happier conversations and relationships. There is a sense in which this book is proof it can.A series of simple sketches shows why some white men are so upset and angry to hear that they have “privilege”, and it is strangely moving to see someone else’s point of view so clearly in a diagram. Comparing “the prisoner’s dilemma” – in which two criminals face a worse outcome if they act in self-interest than if they remain loyal to each other – to climate change explains how it can be logical to betray our fellow humans. In general, arguments “can be exploited by politicians, the media, or just people who have more to gain from discord than from unity. Fear causes people to override logic, and that is a good thing in emergency situations. But fabricating fear in order to get people to override logic is not a good thing … One reason to be able to think more clearly is to be able to withstand the onslaught of manipulation that is flooding our world.”
- The Art of Logic by Eugenia Cheng (Profile, £14.99). To order a copy for £10.49, saving 30%, go to guardianbookshop.com or call 0330 333 6846. Free UK p&p over £10, online orders only. Phone orders min. p&p of £1.99.
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