How Abstract Mathematical Logic Can Help Us in Real Life

The internet is a rich and endless source of ﬂawed arguments. There has been an alarming gradual increase in non-experts dismissing expert consensus as elite conspiracy, as with climate science and vaccinations. Just because a lot of people agree about something doesn’t mean there is a conspiracy. Many people agree that Roger Federer won Wimbledon in 2017. In fact, probably everyone who is aware of it agrees. This doesn’t mean it’s a conspiracy: it means there are very clear rules for how to win Wimbledon, and many, many people could all watch him do it and verify that he did in fact win, according to the rules.

The trouble with science and mathematics in this regard is that the rules are harder to understand, so it is more difﬁcult for non-experts to verify that the rules have been followed. But this lack of understanding goes back to a much more basic level: different uses of the word “theory”. In some uses, a “theory” is just a proposed explanation for something. In science, a “theory” is an explanation that is rigorously tested according to a clear framework, and deemed to be statistically highly likely to be correct. (More accurately, it is deemed statistically unlikely that the outcome would occur without the explanation being correct.)

In mathematics, though, a “theory” is a set of results that has been proved to be true according to logic. There is no probability involved, no evidence required, and no doubt. The doubt and questions come in when we ask how this theory models the world around us, but the results that are true inside this theory must logically be true, and mathematicians can all agree on it. If they doubt it, they have to ﬁnd an error in the proof; it is not acceptable just to shout about it.

It is a noticeable feature of mathematics that mathematicians are surprisingly good at agreeing about what is and isn’t true. We have open questions, where we don’t know the answer yet, but mathematics from 2,000 years ago is still considered true and indeed is still taught. This is different from science, which is continually being reﬁned and updated. I’m not sure that much science from 2,000 years ago is still taught, except in a history of science class. The basic reason is that the framework for showing that something is true in mathematics is logical proof, and the framework is clear enough for mathematicians to agree on it. It doesn’t mean a conspiracy is afoot.

“Some of the disagreement around arguments in real life is unavoidable, as it stems from genuine uncertainty about the world. But some of the disagreement is avoidable, and we can avoid it by using logic.”

Mathematics is, of course, not life, and logical proofs don’t quite work in real life. This is because real life has much more nuance and uncertainty than the mathematical world. The mathematical world has been set up speciﬁcally to eliminate that uncertainty, but we can’t just ignore that aspect of real life. Or rather, it’s there whether we ignore it or not.

Thus arguments to back something up in real life aren’t as clean as mathematical proofs, and that is one obvious source of disagreements. However, logical arguments should have a lot in common with proofs, even if they’re not quite as clear cut. Some of the disagreement around arguments in real life is unavoidable, as it stems from genuine uncertainty about the world. But some of the disagreement is avoidable, and we can avoid it by using logic. That is the part we are going to focus on.

Mathematical proofs are usually much longer and more complex than typical arguments in normal life. One of the problems with arguments in normal life is that they often happen rather quickly and there is no time to build up a complex argument. Even if there were time, attention spans have become notoriously short. If you don’t get to the point in one momentous revelation, it is likely that many people won’t follow.

By contrast a single proof in math might take 10 pages to write out, and a year to construct. In fact, the one I’m working on now has been 11 years in the planning, and has surpassed 200 pages in my notes. As a mathematician I am very well practiced at planning long and complex proofs.

A 200-page argument is almost certainly too long for arguments in daily life (although it’s probably not that unusual for legal rulings). However, 280 characters is rather too short. Solving problems in daily life is not simple, and we shouldn’t expect to be able to do so in arguments of one or two sentences, or by straightforward use of intuition. I will argue that the ability to build up, communicate and follow complex logical arguments is an important skill of an intelligently rational human. Doing mathematical proofs is like when athletes train at very high altitude, so that when they come back to normal air pressure things feel much easier. But instead of training our bodies physically, we are training our minds logically, and that happens in the abstract world.

Most real objects do not behave according to logic. I don’t. You don’t. My computer certainly doesn’t. If you give a child a cookie and another cookie, how many cookies will they have? Possibly none, as they will have eaten them.

This is why in mathematics we forget some details about the situation in order to get into a place where logic does work perfectly. So instead of thinking about one cookie and another cookie, we think about one plus one, forgetting the “cookie” aspect. The result of one plus one is then applicable to cookies, as long as we are careful about the ways in which cookies do and don’t behave according to logic.

Logic is a process of constructing arguments by careful deduction. We can try to do this in normal life with varying results, because things in normal life are logical to different extents. I would argue that nothing in normal life is truly entirely logical. Later we will explore how things fail to be logical: because of emotions, or because there is too much data for us to process, or because too much data is missing, or because there is an element of randomness.

So in order to study anything logically we have to forget the pesky details that prevent things from behaving logically. In the case of the child and the cookies, if they are allowed to eat the cookies, then the situation will not behave entirely logically. So we impose the condition that they are not allowed to eat the cookies, in which case those objects might as well not be cookies, but anything inedible as long as it is separated into discrete chunks. These are just “things”, with no distinguishable characteristics. This is what the number 1 is: it is the idea of a clearly distinguishable “thing”.

This move has taken us from the real world of objects to the abstract world of ideas. What does this gain us?

The advantage of making the move into the abstract world is that we are now in a place where everything behaves logically. If I add one and one under exactly the same conditions in the abstract world repeatedly, I will always get 2. (I can change the conditions and get the answer as something else instead, but then I’ll always get the same answer with those new conditions too.)

They say that insanity is doing the same thing over and over again and expecting something different to happen. I say that logic (or at least part of it) is doing the same thing over and over again and expecting the same thing to happen. Where my computer is concerned, it is this that causes me some insanity. I do the same thing every day and then periodically my computer refuses to connect to the wiﬁ. My computer is not logical.

A powerful aspect of abstraction is that many different situations become the same when you forget some details. I could consider one apple and another apple, or one bear and another bear, or one opera singer and another opera singer, and all of those situations would become “1 þ 1” in the abstract world. Once we discover that different things are somehow the same, we can study them at the same time, which is much more efﬁcient. That is, we can study the parts they have in common, and then look at the ways in which they’re different separately.

We get to ﬁnd many relationships between different situations, possibly unexpectedly. For example, I have found a relationship between a Bach prelude for the piano and the way we might braid our hair. Finding relationships between different situations helps us understand them from different points of view, but it is also fundamentally a unifying act. We can emphasize differences, or we can emphasize similarities. I am drawn to ﬁnding similarities between things, both in mathematics and in life. Mathematics is a framework for ﬁnding similarities between different parts of science, and my research ﬁeld, category theory, is a framework for ﬁnding similarities between different parts of math.

“They say that insanity is doing the same thing over and over again and expecting something different to happen. I say that logic (or at least part of it) is doing the same thing over and over again and expecting the same thing to happen.”

When we look for similarities between things we often have to discard more and more layers of outer details, until we get to the deep structures that are holding things together. This is just like the fact that we humans don’t look extremely alike on the surface, but if we strip ourselves all the way down to our skeletons we are all pretty much the same. Shedding outer layers, or boiling an argument down to its essence, can help us understand what we think and in particular can help us understand why we disagree with other people.

A particularly helpful feature of the abstract world is that everything exists as soon as you think of it. If you have an idea and you want to play with it, you can play with it immediately. You don’t have to go and buy it (or beg your parents to buy it for you, or beg your grant-awarding agency to give you the money to buy it). I wish my dinner would exist as soon as I think of it. But my dinner isn’t abstract, so it doesn’t. More seriously, this means that we can do thought experiments with our ideas about the world, following the logical implications through to see what will happen, without having to do real and possibly impractical experiments to get those ideas.

Getting to the abstract, logical world is the ﬁrst step towards thinking logically. Granted, in normal life we might not need to go there quite so explicitly in order to think logically about the world around us, but the process is still there when we are trying to ﬁnd the logic in a situation.

A new system was recently introduced on the London Underground, where green markings were painted onto the platforms indicating where the doors would open. Passengers waiting for the train were instructed to stand outside the green areas, so that those disembarking the arriving train would have space to do so, instead of being faced with a wall of people trying to get on. The aim was to try and improve the ﬂow of people and reduce the terrible congestion, especially during the rush hour.

This sounds like a good idea to me, but it was met with outcry from some regular commuters. Apparently some people were upset that these markings spoilt the “competitive edge” they had gained through years of commuting and studying train doors to learn where they would open. They were upset that random tourists who had never been to London before would now have just as much chance of boarding the train ﬁrst.

This complaint was met with ridicule in return, but I thought it gave an interesting insight into one of the thorny aspects of afﬁrmative action: if we give particular help to some previously disadvantaged people, then some of the people who don’t get this help are likely to feel hard done by. They think it’s unfair that only those other people get help. Like the absurdly outraged commuters, they might well feel miffed that they are losing their “competitive edge” that they feel they have earned, and they think that everyone else should have to earn it as well.

This is not an explicitly mathematical example but this way of making analogies is the essence of mathematical thinking, where we focus on important features of a situation to clarify it, and to make connections with other situations. In fact, mathematics as a whole can be thought of as the theory of analogies. Finding analogies involves stripping away some details that we deem irrelevant for present considerations, and ﬁnding the ideas that are at the very heart making it tick. This is a process of abstraction, and is how we get to the abstract world where we can more easily and effectively apply logic and examine the logic in a situation.

To perform this abstraction well, we need to separate out the things that are inherent from the things that are coincidental. Logical explanations come from the deep and unchanging meanings of things, rather than from sequences of events or personal decisions and tastes. The inherentness means that we should not have to rely on context to understand something.

We will see that our normal use of language depends on context all the time, as the same words can mean different things in different contexts, just as “quite” can mean “very” or “not much.” In normal language people judge things not only by context but also relative to their own experiences; logical explanations need to be independent of personal experiences.

Understanding what is inherent in a situation involves understanding why things are happening, in a very fundamental sense. It is very related to asking “why?”, repeatedly, like a small child, and not being satisﬁed with immediate and superﬁcial answers. We have to be very clear what we are talking about in the ﬁrst place. Logical arguments mostly come down to unpacking what things really mean, and in order to do that you have to understand what things mean very deeply. This can often seem like making an argument all about deﬁnitions. If you try having an argument about whether or not you exist, you’ll probably ﬁnd that the argument will quickly degenerate into an argument about what it means to “exist.” I usually ﬁnd that I might as well pick a deﬁnition that means I do exist, as that’s a more useful answer than saying “Nope, I don’t exist.”

I have already asserted the fact that nothing in the world actually behaves according to logic. So how can we use logic in the world around us? Mathematical arguments and justiﬁcations are unambiguous and robust, but we can’t use them to draw completely unambiguous conclusions about the world of humans. We can try to use logic to construct arguments about the real world, but no matter how unambiguously we build the argument, if we start with concepts that are ambiguous, there will be ambiguity in the result. We can use extremely secure building techniques, but if we use bricks made of polystyrene we’ll never get a very strong building.

However, understanding mathematical logic helps us understand ambiguity and disagreement. It helps us understand where the disagreement is coming from. It helps us understand whether it comes from different use of logic, or different building blocks. If two people are disagreeing about healthcare they might be disagreeing about whether or not everyone should have healthcare, or they might be disagreeing about the best way to provide everyone with healthcare. Those are two quite different types of disagreement.

“Understanding mathematical logic helps us understand ambiguity and disagreement. It helps us understand where the disagreement is coming from.”

If they are disagreeing about the latter, they could be using different criteria to evaluate the healthcare systems, for example cost to the government, cost to the individuals, coverage, or outcomes. Perhaps in one system average premiums have gone up but more people have access to insurance. Or it could be that they are using the same criteria but judging the systems differently against those same criteria: one way to evaluate cost to individuals is to look at premiums, but another way is to look at the amount they actually have to pay out of their own pockets for any treatment. And even focusing on premiums there are different ways to evaluate those: means, medians, or looking at the cost to the poorest portion of society.

If two people disagree about how to solve a problem, they might be disagreeing about what counts as a solution, or they might agree on what counts as a solution but disagree about how to reach it. I believe that understanding logic helps us understand how to clear up disagreements, by ﬁrst helping us understand where the root of the disagreement is.

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