Reasonable Objections

REASONABLE OBJECTIONS

Competitions decided by an audience vote are often somewhat derided, and “experts” might poke fun at non-expert audience members who, the experts say, don’t know what they’re talking about. This happens when audiences love crossover singers who sing a famous and heart-rending excerpt from an opera (say, Nessun Dorma), but without “proper opera singing technique”. However, according to the rules of an audience vote, that stirring singer can win fairly and squarely. Unlike peer review it’s just a vote, and nobody has to justify their vote. In the process of peer review, objections need to be justified too, not just stated.

In normal life, outside some specialised situations, there isn’t a clearly defined panel acting as peer review on our logical arguments. In a court of law it’s the jury, and their decisions might be based less in the logic of the arguments and more in their emotional response to witness testimony. Either way, a jury is not generally required to justify its decision. For politicians, the “peer review” is the election – it doesn’t matter if they’re right or not, and it doesn’t matter if their arguments are sound or not, it just matters if people vote for them or not. Voters do not have to justify their vote either. You might hope that the votes will depend on the soundness of the politicians’ arguments, but probably not if you’ve ever experienced an election. For corporations, the “peer review” is the money: they just need to persuade people to buy their product, and it doesn’t necessarily matter if their methods are sound or logical (although if they’re outright fraudulent they might get into legal trouble).

Politicians, corporations, and anyone seeking to sway people’s opinions can be seen to be using non-logical methods to influence and manipulate them via their emotions. It is easy to be swept along by this, but if you don’t want to be so easily manipulated it is important to remain sceptical. This doesn’t mean outright denying everything everyone says, but requiring at least some level of justification, and having a framework so that you are prepared to believe someone if they achieve that level, and not if they don’t. This is the difference between reasonable and unreasonable scepticism, which Cheng also discusses in the source book’s Chapter 16.

Reasonable scepticism about a mathematical proof can arise in two ways:

1. Someone might think there’s a gap or error in your logic.
2. Your conclusion might contradict someone’s intuition.

The first is a simple logical objection, and is dealt with in a simple logical way, by filling in more of the logic to make clear that the gap can be filled in or that the supposed error isn’t one.

The second type of objection is more slippery. It has happened to me several times in my research and happens all the time in politics. It happens every time someone doesn’t believe some scientific research because it contradicts their own experience or strongly held belief. It’s why some people still believe that vaccinations cause autism although there is no scientific evidence for it. It’s why some people still believe the universe is only a few thousand years old or that the earth is flat or that human life did not originate in Africa or that Barack Obama was not born in Hawaii, despite evidence.

An intuitive objection is much harder to deal with than a logical objection, because you have to change someone’s intuition in order to convince them, and there is not a clearly defined way of doing that. Restating the evidence may not be enough, and telling people they are stupid definitely doesn’t help. The source book explores this further in Chapter 15. In theory, this type of objection has no place in rigorous mathematics, because if someone can’t find a fault in your proof then they do not have a valid objection. However, mathematical research is still about convincing humans of things, and so in practice this very human type of objection is a problem, not least because if humans aren’t convinced of your result then they won’t use it, build on it, or value it.

When this sort of objection happens to me I do console myself with the fact that logic contradicting intuition is one of the reasons we use logic at all – if logic always matched intuition it would be somewhat redundant to use it. Crucially, I don’t just dismiss the intuitive objection, but rather, I try to find the root of it so that I can resolve it. Often something seems intuitive from one point of view but counter-intuitive from another, and so resolving the conflict involves persuading someone to acknowledge the other point of view. But it is also important to acknowledge the validity of their point of view. To do this, we have to start by understanding what their point of view is, to see where their intuition is coming from.

Whether we’re writing mathematical papers and talks, or developing arguments to back up our own views, being able to imagine a sceptical person is an important discipline, so that we can pre-empt these situations and perhaps remove people’s reasonable scepticism in advance.

I often imagine a sceptical person arguing with me. We are allowed to imagine that they are as intelligent as us, which is why it is called peer review and not idiot review, but I imagine that they are highly sceptical of everything I’m saying, or that they are actively trying to find a mistake in my proof, so that I can find any possible mistakes myself.

Imagining someone very sceptical arguing with you is a good way to test your logic in life as well. It does require you to be able to think like someone else, but this in itself is an important skill and a crucial aspect of having discussions that build bridges with other people rather than widening divides. It can also genuinely open your eyes to new ways of thinking about something. I find that I understand things much more by teaching them, because I have to think about how to explain them to sceptical students. Even while writing this book I keep having new revelations about the interaction between logic and the world.

If you only imagine talking to people who already agree with you then you never have to test your arguments. Worse, many people only talk to people who agree with them, both in reality and in their imagination. The online version of this is the much-discussed “echo chamber” that we are apparently being kept inside by search engines and social media algorithms. To avoid this, I think it’s important to seek out different viewpoints to try and understand where they are coming from. I sometimes test out my powers by reading what I think is a reasonable article and then trying to guess what the main objections will be in the online comments section. Often the disagreements come from differences in fundamental beliefs, a topic addressed in the source book’s Chapter 11. But sometimes the comments are knee-jerk reactions that may well be true but have nothing much to do with the argument in the article. Whenever there is an article about dogs there is bound to be a thread of comments declaring, apropos of nothing in particular, that “they eat dogs in China”.