The Art of Logic in an Illogical World - Eugenia Cheng 2018
Emotions
Beyond Logic
WHEN LOGIC NEEDS HELP
EMOTIONS DO NOT LIE. They are never false. If you feel something you are definitely feeling it. If someone tells you that you are not justified in feeling it, that doesn’t help. Nor does it help if they tell you that they feel completely differently. There is still always a reason you are feeling it, and in this sense there is always some sort of logic to emotions. Instead of denying or suppressing emotions we should understand and explain them. I will go one step further and say we should even use them: it is important to remember that emotions can, and probably should, play a role even when we’re being logical. We use them when doing rigorous mathematics, as we have already seen, and so we should use them when making logical arguments in life as well. Our access to emotions is an important difference between us and computers. Emotions can help us in all of our logical endeavours, and I would even go as far as to say they are crucial.
First of all, emotions can help us work out what we really believe in, just like they help us guess what is logically correct in mathematics before we start trying to prove it. Then when we do start trying to justify things, emotions help us to arrive at logical justifications, if we closely analyse where our gut feelings are coming from.
The next stage of a (useful) logical process involves persuading other people of things. We are going to discuss the importance of using emotions for this. But the emotions shouldn’t supersede the logic—they should reinforce it.
Sometimes people try to argue that we should only use logic and scientific evidence to reach conclusions. However, if we then meet someone who isn’t convinced by logic and evidence, how are we going to persuade them to be convinced by it? We can’t use logic and evidence because that doesn’t convince them. We are going to have to use emotions.
In a way this means that emotions are much more powerful than logic, and are much more convincing than any other possible method of justification. If you feel something, there is absolutely no way to contradict it. This power should be harnessed in a good way, to back up logic rather than contradict it.
LOGIC AND EMOTIONS
Being emotional does not necessarily equate to being irrational: I think that is a false equivalence. This takes the form of a false dichotomy between
A: Using emotions.
B: Using logic.
I think this is the type of false dichotomy where it is possible to do both at the same time.
Using emotions is not inherently illogical, and using logic is not inherently emotionless either. We can be pushed to a fabricated disagreement in which one person says they believe in using emotions and another says they believe in using logic. But it is possible to do both.
At root this has turned into some needless antagonism about intelligence and sympathy.
It is possible to be neither emotional nor logical, and it is also possible to be both emotional and logical. That is to say I think it is possible to be in all parts of this Venn diagram:
For the left-hand side, there are aspects of mathematics that require one to be very strictly logical. This is not the beginning part, where we are generating new ideas, inventing new language, taking vague leaps in different directions to see what will happen. It is not the end part, where we are writing up our proof for the benefit of other people’s understanding. It is in the crucial middle part where we are meticulously proving all the logic steps of our theory to make sure that it is utterly watertight. This part, I think, demands that we be utterly dispassionate, not distracted by what we feel or what we want to be true, so that we can see if logic by itself will hold it up.
For the right-hand side, there are times when it is enjoyable and even beneficial to allow emotions to guide everything. This might be when enjoying sensory experiences, allowing ourselves to be open to art, or just when supporting another person through a difficult or a particularly joyful time. If someone is very hurt it is often no use applying any sort of logic, but the most helpful thing can be to simply sit with them in their emotions and feel things with them.
The only part I can see no good use for is the outside, where one is neither emotional nor logical (although perhaps we should count subconscious actions like reflex actions or automatic behavior like walking or brushing your teeth). By contrast I am going to argue that the most powerful place to be is the place in the middle, where logic and emotions coexist. Not only do they not need to compete with one another, but they can even strengthen each other.
Living too much in the logical world can make it difficult to deal with other people, as they do not usually, or ever, behave entirely logically. On the other hand, those who live too much in the emotional world may have trouble dealing with the world in as much as it does behave logically, and does have components and systems that interact with each other. But living very predominantly in the emotional world doesn’t mean being actively irrational, it might just mean being guided more by emotions than by logic. And it might mean being unable to follow complex reasoning about the complex world.
Children often live very predominantly in the emotional world. All their emotions are valid and strongly felt, but they are unable to see more complex long-term arguments such as: if you only ever eat ice cream then eventually this will probably not be very good for you. Or even: if you roll around in the snow it might well be fun, but your clothes will get wet and then you’ll be miserable.
One aspect of growing up is developing the ability to comprehend longer chains of causation and logic. One concrete way this manifests itself is in the ability to make long-term plans, or make short-term sacrifices for long-term gains, rather than just living for instant gratification in the moment. At least, this is one of my personal axioms; at the other extreme there are some people who strongly believe in only living in the moment, or living entirely emotionally. When adults live strongly in the emotional world it doesn’t necessarily mean they are neglecting the logical world. I believe I live very strongly in both. I respect and trust my emotions, but always look for logical explanations of them so that they’re not “just” emotions. The two are not mutually exclusive.
EMOTIONS OVERRIDING LOGIC
There are plenty of ways to use emotional responses that do not back up logic, and even to contradict it. One powerful tactic is shock and fear. Once someone is afraid, it doesn’t matter whether the thing they’re afraid of is real or not. Horror movies are still scary, although they’re not real. Clickbait headlines and catchy slogans are also appealing to emotions instead of logic, and too often the logic doesn’t hold up. Clickbait headlines often do not accurately reflect what is in the actual article, even if the article itself holds up logically. I read one recently that said “Letter invoking Article 50 declared illegal by judicial review”, which was astonishing to me. But when I read the article what it reported was actually that a retired doctor had decided that the letter invoking Article 50 was illegal and was trying to instigate a judicial review: very different. It doesn’t even matter if you’re not British and don’t know what the furor over Article 50 is about, it should be clear that “X is declared illegal by judicial review” is not the same as “A retired doctor thinks X is illegal and is trying to instigate a judicial review”.
Catchy slogans often sound good but either don’t make sense or have no content. “Weight is just a number”, people like to say, or “age is just a number”. But numbers can be very informative if they are treated in the right way. My weight happens to correlate very well with how much fat I’m carrying around my stomach, and hence with what clothes fit me. Certain medical risks go up with weight and with age. You might as well say “Medical risk is just a number”. Or even, when you’re running a dangerous fever, “temperature is just a number”.
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. Fear also gets in the way in interpersonal situations even if it’s not being used as a deliberate means of manipulation. If someone feels under attack it can cause them to override logic, or it causes them to be unable to use logic. Or it causes them to cling on much too strongly to a position they don’t really hold. What we need, for productive arguments, is to find ways to build bridges between positions so that people can move, rather than cling on to a position because we’ve backed them into a corner. And, incidentally, for many people, even if they are open to changing their mind, it might be something they would rather do in private when nobody is looking, like changing clothes.
We are now going to discuss some important ways in which we should make use of emotions, beginning with the language we use.
PERSUASIVE EMOTIONS
In Chapter 10 we discussed the starting points of language and the fact that some basic words have to come from somewhere before a language can develop in even a vaguely logical way. Those words can have strong emotional effects on people. When mathematicians choose words for new mathematical concepts they often think very hard about what sort of emotional response they want to elicit. There is something in a name after all, even if “that which we call a rose by any name other would smell as sweet” according to Shakespeare’s Juliet. She was, of course, being touchingly naive. Would you be able to sniff a rose seriously if it was suddenly renamed “diarrhoea”? It might take some mental effort.
I had cause to think about the logic and emotions of words when boarding planes recently because American Airlines changed its boarding procedure. Instead of having different “priority groups” boarding before Group 1, it renamed those priority groups with the numbers 1—4, shifting the first normal boarding group to become Group 5. This caused all sorts of confusion because people were not able to listen to instructions, read their boarding pass, or understand the logic of it (or all three). Personally I think, logically, it makes no structural difference as the boarding groups still board in the same order, and surely it doesn’t matter whether the groups are called Platinum and Gold or red and blue or banana and frog, or 1 and 2. But apparently some people are very upset about not being in a group called “priority” any more. To them, that word really matters. It is causing an emotional effect and not a logical effect. The people who were upset about not being in a priority group were upset about the word, not about the actual boarding process.
A much more serious version of this mistake over language is a study1 of young male university students finding that they seem to think forced intercourse might not be rape. The study found that many more men admitted that they had coerced somebody to intercourse than admitted they had raped anybody. One might call it a “false inequivalence” situation where some men think that non-consensual intercourse is not the same thing as rape. It tragically demonstrates how far we still have to go when it comes to education around consent.
The emotional connotations of language can be exploited deliberately, as in the use of the nickname “Obamacare” for the Affordable Care Act (ACA) in the US, as we discussed in Chapter 3. If someone supports ACA but not Obamacare it is painfully clear that they are not evaluating things by their merits, but by their names. This shows the power of language, again in a “false inequivalence” situation. This is all an example of how something emotional can be used to guide or even manipulate people’s thought processes in a way that has nothing to do with logic.
The idea of manipulation has negative associations. If someone is called manipulative that is not a good thing. The world is trying to manipulate us all the time, especially corporations, politicians and the media. One reason to be able to think more clearly is to be able to withstand the onslaught of manipulation that is flooding our world.
However, as with other emotional tools, there are reasons to embrace the power of emotional manipulation if it can help us overcome things in our own lives. Emotional manipulation helped me overcome my fear of flying, when statistics alone wouldn’t do it: the logic was no match for my emotions. Emotional engagement was also crucial to me losing weight, as statistics about health risks were not enough—it was not until I became viscerally afraid of becoming morbidly obese that I was able to do it. (It is not acceptable to use this as a technique on other people as it consists of “fat-shaming”.)
Sometimes people use gray areas to manipulate us, but we can also use gray areas to manipulate ourselves. I use emotional tricks on myself to motivate myself, overcome mind blocks, stop procrastinating. Perhaps if I were a more logical person I would never lack motivation, get mind blocks, or procrastinate. My mother, the most logical person I know, never procrastinates. But that is a level of logic that is beyond me. Berating myself does not enable me to become logical, and berating others will not make them become more logical either. Thus we need to deal with others and ourselves in an emotional way when we are dealing with the emotional part of ourselves.
In the “post-truth” world, feelings are often presented as facts. Some people accuse others of conflating feelings with facts, especially if they seem to be convinced by something other than logic and evidence. And indeed many people are convinced by personal experience, someone charismatic telling them something, peer pressure, tribe mentality, fear or love. Advertising and marketing drive much of our experience of the world. Marketing is not about demonstrating that a product is better; it is about making people feel that a product is better, even if it’s essentially the same as all the other ones (or worse).
But these methods of convincing people of things aren’t all bad. We generally seem to accept that we become wiser through personal experience, and that there is something valid to be learned from it. Charismatic teachers are an important part of what can make a very good education. Peer pressure and tribe mentality can contribute to change for good as well as bad, for example in the civil rights movement. If people give up smoking because of peer pressure rather than because the evidence says smoking is bad for you, they have at least still given up smoking. But making decisions based on fear is often said to be a negative way of approaching life, and elections based on fear are unpleasant and divisive.
Some fields have become particularly good at invoking emotions to convince people of their message. This might include religious leaders, public speakers, some teachers, advertisers and artists. Science sometimes suffers from the belief that only evidence and logic should be used to convey a conclusion. But this is unrealistic. If we are to do that, then we’ll have to start by convincing everyone to be persuaded only by evidence and logic. And how do we do that using only evidence and logic, if people are not already convinced by evidence and logic? We end up in something like Russell’s paradox.
Moreover, it is unrealistic to think that people can be persuaded by logic alone. In fact, it is in too many cases hypocritical, as scientists themselves are not immune from veering into drawing conclusions based on personal experience or emotions on controversial topics such as women in science.
Instead of denigrating emotions in a quest for more rigorous discourse, we should acknowledge their truth and seek to find the sense in which there is logic to them.
THE LOGIC IN EMOTIONS
Feelings are not facts. Or are they? It depends what we mean. If I “feel” that 1 + 1 = 3 that doesn’t make it true in normal life. (There is a mathematical world in which it is true, but that’s a different matter.) Similarly if I “feel” that someone arrested for a crime is guilty that doesn’t make it true.
But there is an important sense in which feelings are facts: feelings are always true. If you feel something then the fact that you feel it cannot be argued down by logic. There is rarely any point trying to persuade someone that they “should not” feel something, if they simply are feeling it. Emotions are very powerful at overriding logic.
A more productive process is to find the explanation behind the emotions, uncover the difference between that logic and one you are trying to convey, and use emotions to help bridge that gap. In this way instead of pitting logic against emotions, we separate out emotions and logic in the situation, and only pit logic against logic and emotions against emotions. In a way this is exactly what the process of teaching math well consists of. If a student gives a wrong answer, it rarely helps just to explain the right answer to them. First you have to uncover why they gave the wrong answer, understand the thought process behind it, and somehow convince them that your thought process is more sound.
The reason the emotions are so strong can often be traced back to some fundamental fear. But fear works in strange ways. Sometimes fear works to persuade people of something, and sometimes it doesn’t. People who believe the evidence about climate change are typically very afraid about the future of the planet, so feel that it is extremely urgent that we do something about it. Those who do not believe the evidence are typically not afraid of climate change and so don’t do anything about it. But why are we not able to make those people afraid enough to want to do something about it? Why by contrast are some people so easily whipped up into a frenzy of fear about refugees even though there is no evidence that refugees are more dangerous than, say, white Americans with guns? Why are some people not at all afraid of gun violence, or at least, not afraid enough to want to have tighter restrictions on guns?
Saying that political views are driven by fear is thus not really an explanation as it doesn’t work in all cases—it is an analogy but at the wrong level of abstraction. Perhaps a better analogy between those situations is to do with action. In the case of climate change, taking action against climate change requires some personal sacrifices (better use of resources, which might be expensive). Likewise taking action against guns requires some people’s personal sacrifices (giving up their guns). Whereas with refugees, it is being unafraid of refugees that requires personal sacrifices—giving up resources to look after refugees, accepting them in the community. Perhaps it is not so much fear that drives these arguments, but personal beliefs around sacrifices.
In this imagined argument we have used a key technique for engaging other people’s emotions: the use of analogies.
ANALOGIES FOR ENGAGING EMOTIONS
In Chapter 13 we talked about analogies as the result of performing an abstraction, explicitly or otherwise, and pivoting from one manifestation of it to another. At the simplest level, an analogy is a situation that has something in common with the one you’re really discussing. But I think that analogies are hiding something very powerful: a way to try and get people to feel things differently about a situation. Once they feel things differently, they might be able to see the logic differently. The power of the analogy is in doing this via emotions, without having to appeal to anyone’s understanding of the logic involved. Unless you are talking to someone already proficient in abstraction and logic, this might be the best you can do.
One way of doing this is to pivot from a situation that they don’t seem to feel anything about, to one that they feel a strong emotional reaction to. For example, trying to persuade white women to care more about racism, one might make an analogy with sexism and try to wake up their emotions that way. In a way, this entire technique is a pivot at a more abstract level, from the world of logic to the world of emotions:
One interesting thing about finding analogies is that when I’m thinking about a good one, I have to think abstractly to find the deep logical argument, and then apply it creatively to another situation to engage someone’s emotions. When I am doing this I do feel like I’m using my brain in a highly mathematical manner; it feels like I’m using the same part of my brain. But in arguments in real life the point of it is to enable others to feel my point without having to think in that highly mathematical manner. In a world where everyone has varying levels of proficiency in abstraction and logic, it is important to be able to find ways of explaining things that bypass that. This is why I make heavy use of analogies when explaining math to students or non-mathematicians, and much less when explaining it in research seminars.
Drawing together the subjects of the last three chapters, we will now show how to use analogies to engage someone emotionally around the issue of prejudice across power relationships. We have talked about the fact that men occupy a position of power over women in society in general. Some people argue that this means men making rude jokes about women is worse than women making rude jokes about men. Or to push it to a greater extreme, that men sexually harassing women is worse than women sexually harassing men. Others argue that this is the same.
There is no right or wrong answer to this, but there is a sense in which the two are the same and a sense in which they are different. The sense in which they are the same is that both consist of “people mistreating other people”. At this level of abstraction we have an equivalence.
However, this has forgotten very many details about the situation. We could instead retain the information about power in society, and only abstract as far as “people with power mistreating people without power”. At this level, men mistreating women is equivalent to white people mistreating black people, but not equivalent to women mistreating men, or black people mistreating white people:
The question has become one of whether or not the intermediate level of abstraction is relevant: that is, whether or not power differential is a relevant factor. I believe it is relevant, but many people insist that it is not. We could try to push them to think about power differentials in general by invoking a more clear-cut example, about which they definitely feel something. For example, a teacher flirting with a pupil. There is a definite problem with power differences here, which is why even if an interaction appears to be consensual, it is illegal between a teacher and a pupil in some countries, just as between an adult and a minor. In the case of minors it is also to do with the absolute inability of a child to consent, but in the case of teachers and pupils the pupil might be over the age of consent but still considered to be unable to consent within that particular power relationship.
We have the following analogy (in fact, an analogous analogy):
I hope everyone would agree that a teacher making advances on a pupil is not equivalent to a pupil making advances on a teacher, because of the power differentials. Similarly a boss trying to seduce an employee is a different situation from an employee trying to seduce a boss, because of the power that the boss has over the employee.
If we can agree that power differentials at least sometimes make a difference in some clear-cut cases, then the argument becomes one about gray areas and where to draw the line, if anywhere—which power differentials do make a difference and which ones don’t? We can try to persuade people that because white people are so dominant in all levels of power, politics, management, entertainment, and all positions of influence, the position white people collectively hold over black people is analogous to the position a boss holds over an employee.
If they still disagree, it might come down to a question of transferring from the idea of an individual having direct power (boss over employee) and the idea of group power (white people over black people) and whether group power transfers at all to individuals. This is the question of structural racism.
In the process of writing this book and thinking through these arguments very carefully, stripping away layer upon layer to find further abstractions and logical points of view, I have realized how many of these arguments come down to tensions between the idea of individuals and the idea of groups. This applies to the idea of individual vs group responsibility, the extent to which everyone should look after themselves or whether there should be group care. It applies to whether people think a group’s treatment by society has any bearing on an individual’s. This may go back to a difference in basic personal axioms, in which case we will need to think about how to persuade someone to change their axioms.
We have discussed using analogies to uncover our own personal axioms. But we can also use analogies to uncover other people’s personal axioms, to understand why they are thinking the way they are. If we are disagreeing with them because of a different use of logic that is one thing, but if we are disagreeing because of different fundamental principles, it is hard to change those without invoking emotions.
For example, why are some people sometimes so unconvinced by scientific evidence? It might be because they are very invested in not believing it, in which case piling up more evidence won’t help: changing their sense of investment will help. Getting to the bottom of why they don’t believe it will help, rather than berating them for not believing it.
Once we have uncovered someone’s axioms that are at the root of a disagreement, we can start thinking about how to change them. If they are deeply rooted it can be hard, but it could be by experience, meeting people, education, empathy, but in all cases by engaging them emotionally. Our analysis does not tell us how to do that, but at least if we have reached an understanding of really why someone feels something, we are in a better position than if we just think they are stupid.
For many people, emotions and intuition are more convincing than logic. As I have discussed, this is true of mathematics as well, which is why I don’t think we should simply scorn the idea. In fact, my whole field of research, category theory, can be thought of as a field that makes precise our mathematical intuitions so that we can do calculations almost entirely by using our intuition, knowing that it will match rigorous logic.
Use of intuition has achieved a lot for mathematicians over the course of history, as long as it is backed up by rigorous justification. So I believe that intuition and emotions can achieve a lot in normal life too, if backed up by logic. Unfortunately while almost everyone feels feelings, not everyone can follow complicated logic. I believe it is therefore incumbent on more logical people to invoke emotional means to make sure logical thoughts are conveyed. This is the subject of the closing chapter of this book.
1 “Denying Rape but Endorsing Forceful Intercourse: Exploring Differences Among Responders”. Edwards Sarah R., Bradshaw Kathryn A., and Hinsz Verlin B. Violence and Gender. December 2014, Vol. 1, No. 4: 188—193