LOGIC BY ITSELF HAS no starting point. It consists of a way of making deductions from things we already know. So we have to start somewhere in order to deduce anything logically. Limits are often thought of as being at the end, but there are also limits at the beginning.
We have already touched on this limit to logic that comes from wondering what the root of all truth is. Just like thinking up words for a new language, we have to decide to start with some truths before we can apply logic to find more truths. In mathematics, the things we decide to start with are called axioms, and in life these are our core beliefs.
Axioms are the basic rules in the system. We do not try to prove axioms, we just accept or choose them as basic truths that generate other truths. There are broadly two different ways to approach the use of axioms in math, which I think of as being externally motivated and internally motivated.
The internal approach to thinking about axioms is where we pick some axioms and see what system this generates, logically. In this case, any axioms are valid because we are assuming them to be true in the system, to see what will result. The only problem is that if the axioms cause a contradiction then the whole system will collapse and become the null system, in which everything is both true and false. This isn’t mathematically incorrect, it just means that there is no sensible notion of truth in that system, so it’s not a very illuminating place in which to understand or model anything.
In the real world this gives us a way of conducting thought experiments. We might, for example, imagine a fantasy world in which there is no gender pay gap. Or a fantasy world in which perpetrators of sexual harassment are not tolerated, particularly not in positions of power and influence. It is informative to imagine a world in which everyone reporting sexual harassment were automatically believed. What would happen? First it would mean that huge quantities of women would report the sexual harassment that men inflict on them every day. (Yes, some men are victims too and some women are harassers.) Huge numbers of men would be removed from positions of power. They would either be replaced by women, or by men who would in turn be at great risk of being accused. Perhaps men would start to become very frightened of false accusations. Perhaps employers would start worrying about hiring men in case they were then accused of sexual harassment. If you think that this is an indefensible state to arrive at, it is worth turning it round and remembering that most if not all women fear sexual harassment all the time; we would be replacing this with men fearing accusation of harassment. It is also worth remembering that some people are loath to employ women “in case they get pregnant” (even though such discrimination is against the law in many countries); we would be replacing this with companies being loath to employ men in case they commit sexual harassment. We will later discuss using analogies to pivot between different points of view in this way. Imagining a world with these new basic axioms doesn’t mean we think it should happen, but it helps us understand the complex web of issues involved. We can understand some things about how far we are from that world and what might need to change in order to get there, together with what some unintended consequences might be.
The external approach is to start with a world that you are trying to understand, say numbers or shapes or relationships or surfaces, or the world we actually do live in. Axiomatizing it is the process of looking for basic truths that logically generate everything else that is true. One famous axiomatization is Euclid’s axiomatization of geometry, where he came up with five rules from which all other rules of geometry should be deducible.
The difference between the external and internal approaches is a bit like the difference between moving to another existing country and trying to understand the laws there, as opposed to setting up a new country from scratch and deciding what basic rules you would start with.
There are usually different ways of axiomatizing the same system, so we should consider what a good set of axioms is. First of all, the axioms should definitely be true. They should also somehow be basic, so they should be things you can’t really break down into smaller parts. It is often desirable to have as few axioms as possible, but it is also desirable for the axioms to be illuminating, highlighting some important aspect of the structure. Sometimes it is possible to narrow things down to a smaller set of axioms, but at the cost of some clarity.
Usually we don’t want axioms that are redundant–if we can deduce one of the axioms from the other ones, then we probably don’t need it as an axiom. For some years mathematicians suspected that Euclid’s fifth axiom (about parallel lines) was redundant, and tried to prove it from the other four. But they turned out to be wrong–dropping the fifth axiom leaves you with a perfectly good mathematical system, just a somewhat different type of geometry.
Axioms in mathematics are analogous to our personal core beliefs.
Axiomatizing our belief system in our own lives is somewhat like axiomatizing a mathematical system from an external point of view. We can start by thinking about all the things we believe to be true, and then we can try to boil them down to some basic beliefs from which everything else follows. Anything is valid really, as long as it doesn’t cause a logical contradiction, in which case your system of beliefs will collapse. Of course, this is only the case if you are trying to be a logical person. If you are not trying to be a logical person you might be perfectly happy believing a contradiction. But even two people who are logical might disagree about things just because they have different core beliefs–they are using different axioms. It doesn’t necessarily mean that one of them is being illogical.
There are then two slightly separate questions: how can we work out what our personal axioms are, and where did we get those axioms from?
We can work out what our fundamental beliefs are by starting with anything we believe and asking why we believe it. The process of repeatedly asking “Why?” is a way of uncovering the deep logic behind something. It is one way of understanding what mathematics is: if we ask why aspects of the physical world work the way they do, the questions may be answered by science. If we ask why science works the way it does, the questions are answered by mathematics. If on the other hand we ask why aspects of the human world work the way they do, we are likely to end up in psychology and ultimately philosophy.
Being able to answer our own “Why?” questions about beliefs requires us to have a certain amount of logical proficiency, so that we can uncover long chains of logical implications, as well as self-awareness. On the other hand, finding out what someone else’s axioms are requires us to have logical proficiency coupled with empathy. So we see that an interplay between logic and something more emotional comes in.
My personal axioms fall into three main groups:
1. Kindness: I believe in being kind to people. From this I deduce other beliefs about helping others, contributing to society, education, equality and fairness.
2. Knowledge: I believe in the frameworks that we have set up to access knowledge in various different disciplines. So I believe in scientific research and historical research, for example, within the confidence levels that those disciplines have established.
3. Existence: I believe we exist, mainly as a pragmatic approach to getting on with life. I’m not so sure about this one and I suspect that if I believed the opposite it wouldn’t make much difference, so I’ve chosen to include it as it seems more helpful than the opposite.
The second point is important to me because it means that I will take some things on trust if I judge that the evidence makes that reasonable. This is not exactly logical overall, because I have not traced all of those conclusions back to their fully logical beginnings. But adding this second axiom means that I have traced those conclusions back to their fully logical beginnings inside my axiom system. For example, I believe in gravity although I don’t understand it logically. So I don’t know how to trace it back to first principles in mathematics, but I do know how to trace it back to the axioms of my personal belief system as it comes from my belief that scientists are probably right about it.
I think we have to accept some starting points in our logical system in order to get anywhere. This is true in mathematics and life. The important thing is to be clear what they are. As we’ll see, this can help us identify more complicated beliefs, and also pin down why someone else might disagree with us about some more complicated beliefs.
The question of where we came up with our own axioms is more philosophical, but important as it may help us understand other people who disagree with us fundamentally. Most of us get our personal beliefs from some combination of our upbringing, society, education, life experience and gut feeling. Some things are instilled in us by our parents, but most of us don’t have exactly the same opinions as our parents, which means something else must influence us. Education can expand people’s world view and lead them to see things differently from their parents. So can life experience, especially if the parents grew up in a very different era, culture, or economic environment. Some beliefs seem to come from nowhere in particular except personal conviction, but if we think about it very hard, we might be able to see where that personal conviction comes from.
For example, I believe that it is more important to be kind than to be right, and I simply feel this very strongly. But if I think about it hard, I see that it comes from my life experience, and incidents when I’ve been so hurt by other people’s behavior that I’ve come to believe more and more strongly in the importance of kindness.
I believe that education is the most important way in which I can contribute to the world, and again, I simply feel this very strongly. But if I examine the source of that feeling, it is a combination of values instilled in me by my parents and piano teacher, together with evidence that I am not really cut out to be a doctor (too much memorizing in medical school) or to brave war zones to rescue people (I am too afraid of physical danger).
Some people can trace all their fundamental beliefs back to religion. That still leaves the question of where their belief in religion comes from. For some people it might come from their parents and their upbringing, possibly reinforced by their education. For others it comes from a particular time in their life, unfortunately often after a tragedy or trauma. It might also come from being swayed by a very influential person. Understanding what people’s fundamental beliefs are helps us find the root of disagreements, and understanding where they get their beliefs can help us understand how we might be able to address those beliefs.
One way to axiomatize a system of beliefs is to take every single belief as an axiom. This certainly means all your beliefs can be derived from the axioms using logic, but it has not achieved anything. It would be a bit like a recipe for lasagne where the only ingredient is “lasagne”. Rather, the point of axiomatization is to understand the roots of a system and what holds it together.
Taking all beliefs as axioms would obliterate the need to follow any sort of logical deduction, and although it’s not exactly illogical (it doesn’t contradict logic) it’s hardly a well-developed point of view. It would be a rather extreme situation, but we do meet people who are unable to justify certain beliefs of theirs very much at all–they take quite complex beliefs as fundamental, without justification. Or in some cases there is some justification but it doesn’t get us very far. For example, a person might say, “I oppose same-sex marriage because I believe marriage should be between a man and a woman”. This might sound like a justification as it does have the word “because” in it, but it’s really just a restatement of the initial belief.
Our fundamental beliefs are rooted in something beyond logic. However, sometimes abstraction can help us find something more fundamental inside what we believe. As I discussed in Chapter 2, I have discovered that my belief in taxation and social services is rooted in a more fundamental belief that false negatives are worse than false positives. We will come back to this in Chapter 13.
As a related issue, I believe that everyone’s situation is a combination of their own input and their circumstances. I believe that humans are not isolated creatures, and that we exist inextricably connected to the communities around us, and thus there is some collective responsibility for both success and misfortune. This in turn comes down to my more fundamental belief that we should understand all things in terms of systems rather than individuals, as in Chapter 5, whether this is people, factors causing a situation, or mathematical objects–the latter being why I do mathematical research in category theory, a discipline that focuses on relationships between things, and the systems those form.
Having said that, I do believe that everyone should take responsibility for themselves, while at the same time believing that we should all look after each other. Where exactly personal responsibility ends and collective responsibility kicks in, I am not quite sure. This does however lead me to another more fundamental belief: that there are many gray areas in life and that it’s important we understand them for what they are rather than ignore them or force them to be black and white. This might mean that we can’t be entirely logical where gray areas are concerned. In the next chapter we’re going to look at the different ways that logic has of dealing with gray areas, and see that some of them are rather undesirable, pushing us to extremes in ways that don’t feel right. We will see that as we move through a gray area things may start out seeming “the same” and then gradually morph into something that seems different, although somehow the framework was the same. In Chapter 13 we will take this further and analyse how analogies work, and how we use them to pivot between things that aren’t the same, by means of some way in which they are the same. A big issue with analogies is deciding when they count as a good analogy or when they have been pushed too far. In Chapter 14 we will talk about when things should count as the same or not. False equivalence is a widespread logical fallacy, but in between true logical equivalence and false equivalence is a gray area of things that are the same in some sense, we just have to find it. In Chapter 15 we’ll look at how we can use all these techniques to engage our and other people’s emotions, to try and better see eye to eye with other humans, and finally in the last chapter we’ll paint a portrait of a good rational human–not a perfect computer–and what good rational arguments should look like.