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Mathematics is the Best Way of Thinking

I believe in the power of logic. Without logic, there will be no peace, and if there is no peace, there is no love, and if there is no love, there is no hope for humanity. We need a world where logic is in charge, and irrationality has surrendered. So our thoughts just need to be acceptable by logic. Otherwise, the world stays mired in conflict, divisive arguments, fake news, yelling, bigotry, prejudice, hate, exploitation, and victimization.

For a long time, people and the things that they do in this world do not make sense. Logical and well-intentioned people have been looking for solutions to make sense and help us. One of the most powerful tools that those well-intentioned people use is mathematics. The application of mathematics is helpful to science. For instance, differential equations and solving equations are the ways in which mathematics is useful in science.

Science is useful to engineering and medicine, and engineering and medicine are helping our world. Thus, mathematics is constructive. But mathematics is more than just that. Although most parts of the world use applied mathematics, everything starts with pure mathematics, which taught as the most abstract subject in education. There is a beautiful connection between pure mathematics and applied mathematics. Doctor Eugenia Cheng has a good definition of pure mathematics: pure math is a framework for agreeing on things.

I am an incoming graduate student in pure mathematics, and what I have learned so far about pure mathematics is that it an essential tool to learn how to think rationally better. I firmly believe that there is a strong connection between how to think and mathematics. That makes me feel that mathematics is about our lives, or if we think profoundly, mathematics is about being human; because only humans have the ability to think and act rationally. Unfortunately, there are so many people around us who ignore mathematics and wait for other people to think, make some discoveries, and solve the problems in our world. On the other hand, those people always hope the best. But still, some people spend most of their time to think or want to learn mathematics.

Analogies find something in different things or situations, and that takes us into the abstract world. In the abstract world, we remove ambiguities, and different things become the same by analogy, which is not possible in our daily lives. For instance, if we think about two horses, two tigers, and two cars, basically they have something in common, which is 2. They are two things, and this analogy gives us number 2. So, we find a common thing that is abstract from different things by using an analogy. This analogy encompasses a lot of stuff.

On the other hand, we can find another analogy that encompasses less number of things. For instance, when we say two animals, the analogy will not contain cars or other objects but animals only. In this example, this analogy encompasses fewer things.

If we want to encompass almost everything, then we can just say two things.

These simple arguments of analogy are used to teach little kids how to count, but the point of doing that is not to scare children. There is more behind it. To introduce the abstract concept to little kids, we show those simple examples over and over again to leap abstraction. Furthermore, we can go to a more profound level in analogy when we teach simple operations in mathematics, such as addition and multiplication. For instance, when we say 1+2, 3+5, 2x5, or 6x9, all of them are examples of “a+b” or “a x b”.

The idea of abstraction is to encompass more examples and study a lot of different things at the same time to save brainpower and be more efficient. It is a very good thing because we don’t want to do the same thing over and over again. It is a perfect solution for lazy people. After you do the same thing a few times, and then you say, “I don’t want to do that anymore, I’m going to make a theory that will do it for me.” It is like getting a dishwasher after a while. Those examples of “a something b” are different ways of combining two things.

That is called a binary operation in mathematics. Math students use group theory to look at things with binary operations. Some binary operations are at different levels. If you try to study different operations at the same time, some processes will not work. For example, commutative property works for both addition and multiplication. 1+2 = 2+1 and 2x3 = 3x2. And if you go one level deeper, you see that even the undoing process works for addition, but it does not work for multiplication. If you multiply a number by 0, you cannot undo that because you cannot divide by any number apart from 0. So, we can say that at this level, addition and multiplication are not analogous.

So let me give you a real-life example. Some people think that marriage should be between a man and a woman, and those people do not believe that gay people should be allowed to marry each other. It is not an illogical argument. However, this also is not a profound argument because they say the same thing twice. It is saying this: I like orange juice because I like orange juice. The problem here is we do not want to say the same thing over and over again with different words. We need to make progress and gain understanding. So this is not an example of analogy. Thus, precision in abstract mathematics is critical.

Everything is interconnected. And we are all trying to make sense of a complicated world. And that requires somehow simplifying the world. Unfortunately, many people often like to blame precisely one thing, or precisely one person, or one group of people. And that is a way to simplify situations, but it is not a very good way. A better way of doing it is to become more intelligent because then the world becomes simpler relative to your intelligence. And then, you simplify the world. Here is one of my favorite diagrams of interconnections.

This picture is not geographically accurate, and it doesn’t show a geographical location. What it shows is the interconnections of stations by different tube lines. That is all we need to know if we’re trying to get from one tube station to another. Once we’re on the train, we don’t need to know exactly where the tunnel goes, necessarily. We just need to know where it is going to pop out. It is an excellent example of an abstract picture because it’s just showing the interconnections of things.

It is much harder to understand if you’re trying to use it to get from A to B, but it’s quite interesting to see where the tube lines go and to know where the tube stations are in New York. So, these different types of maps are not like one of them is right; the other one is wrong. It is about what you’re trying to achieve. You look at the things that are relevant to the thing you’re trying to achieve or understand, and then you can switch thinking about pertinent things of situations. That’s one of the things that abstract mathematics does a lot. Abstract mathematics gives a full explanation of a situation. For instance, one day, you can find a description of an aspect of a situation, and then on a different day when you need something else, you can think about another aspect.

In real life, interconnections may cause relationships breakdown. And when relationships break down, everyone starts blaming each other, and people take sides. People and things go in both directions. Here’s an example of it.
Suppose Hazel is someone who does not like feeling disrespected, and her boyfriend Dillard is someone who likes to feel loved. If Hazel feels disrespected, then she is unable to show love. If Dillard feels unloved, then Dillard is unable to show respect, in which case Hazel feels disrespected. We get in this cycle. It can get worse, and worse, and worse!

We can classify this situation by using arrows. We can have arrows for people’s actions that are taken in response to something, and we can have arrows for the feelings that people have in response to something. So we can get a very bad vicious cycle. Once we see the interconnections between these arrows, we can think about how to break the cycle. We can think about which of these arrows is the weakest one, the one which we can cut. We can think about whether it’s more challenging to change feelings or actions. For some people, it’s harder to change feelings because feelings are just responses to things, but we do not have to act on those feelings. But some other people cannot stop themselves acting on those feelings, so they have to change their feelings. However, at least we can then have a more transparent discussion about what’s going on in a situation.

There’s an analogous situation that makes the same picture, and possibly it is more urgent; the police violence against African American people. It is a particular issue in America, where we have another vicious circle. Police may feel threatened by African American people. And that’s why they aggressively defend themselves against African American people. But in turn, African American people feel very intimidated by the police, which means that they aggressively try to protect themselves against the police, which means that police feel threatened by them.

Then people start taking sides. Half the people on the internet yell, and support the police and believe that if people do what the police tell them they will not get shot. The other half of the people believe that the police never shoot them because it is unfair. Again, this separates into feeling responses and action responses. Once we’ve observed that there is an interconnection between causes, then we can have a more productive argument.

So, whose responsibility is it to change something? I think in these situations, whoever has more power should take responsibility for changing it. Let me give you another example of interconnections to understand difficult situations.

When one of the United flights was overbooked, the cabin crew picked someone to kick off the plane, but the person that they chose did not want to leave the plane. Then they called security, and the security hauled him off violently, and he sustained quite severe injuries on the way out.

Then people started retaking sides on the internet. Half of the people blamed the passenger and believed that if the passenger left the plane, he would not get injured. And the other half of the internet said that the security guys should not beat people up because they do not want o leave a plane. And then a journalist blamed all the people and said “It’s all your fault because you’re sometimes late for your flight. And that means that they overbook flights in case people are late. And it’s because of that overbooking that they had to get someone off the plane. So it’s your fault for being late for a flight.”

Anyway, the result was that an injury was inflicted because a passenger refused to leave the plane because he needed to get to work, and the security used force. But why did the airline even choose him in the first place? Why did they also need to kick people off in the first place? Whose fault was that? Why did nobody volunteer to leave the plane even the airline offered them money? Were they stingy, or why did they want to get home? That happened because the flight was too full, and they needed to get some crew to somewhere because there was a scheduling issue. The flight was too full. It was overbooked because the crew was hoping that there would be enough no-shows.

Now, whose fault is that? Is it the airline’s fault for overbooking, or is it the people’s fault for not showing up? I think that changing any one of those factors would have changed the outcome, but it doesn’t mean that anyone of those factors by themselves is individually to blame. It’s the interconnections of the whole system.

Let’s look at another example. Almost everyone talks about losing weight, and people periodically say, “It’s not rocket science, you just have to eat less and exercise more.” If it is so simple, why do we put on weight? Technically it is because we take in more energy than we burn. But it’s also because of our metabolism. What affects our metabolism? Our metabolism slows down if we eat too little, and if we exercise too much. It’s also some genetics, the amount of sleep we get.

Why do we overeat? We may like food, or it happens because of our emotions. Yes, we emotionally eat. Why do we eat too much because of feelings? We have an emotional connection to food when we are growing up. And there is also social pressure to eat more. The social norms also cause emotions that cause us to eat more. Then there is also time. If we are very busy, we get stressed, and we even sleepless. There’s also the food industry. So there’s a multibillion-pound food industry that is trying to get us to eat too more because that’s how they make money. So advertising causes us to eat too much, and the food industry makes money out of us. Maybe it is money that’s at the root of everything. Some people say, “You cannot blame everyone else all the time.”

So there’s the fact that if we gain weight, then we try to eat less, and exercise more, and we get stressed about it. And we get all of these vicious cycles, and interconnections, and a conglomeration of factors. It seems not simple, but it is simple. And the point is if you get used to understanding interconnected networks of elements, then everything is simple. Your brain can comprehend these situations quickly. That’s what I mean by becoming more intelligent to make things simpler.

So this is the logic inside these politics and sociological situations. Understanding the logic of the interconnections right at the outset can help us start to think about all of the other issues. It is definitely not the full understanding, but it makes it so much more evident to us, and it means that we can see what’s going on with other people’s arguments about it.

Our brain is geared towards thinking about relationships between things. And it can apply to many essential things in the world around us. For instance, the factors of 30 are 1, and 2, 3, 5, 6, 10, 15, and 30. This bunch of numbers in a straight line does not look very interesting because I wrote them into a one-dimensional straight line. So to make it more interesting, I will find some numbers which are also factors of each other. And to do this, I have something similar to a family tree to show the relationships.

So we have 30 at the top like a kind of great-grandparent. And then, we have 6, 10, and 15 that go into 30. We have 5 goes into 10 and 15, 2 that goes into 6 and 10, and 3 goes into 6 and 15. And 1 goes into 2, 3, and 5. So we have the corners of a cube. We can also see a hierarchy going on here, because you might notice that 1 is at the bottom because it’s the smallest, and then 2, 3, and 5 are directly above it because nothing else goes into them apart from 1, which is a reason they are prime numbers.

Then at the level above that, we have products of two prime numbers. And then at the top, we have a product of three prime numbers. So we have a hierarchy according to the number of prime factors that each number has. It is also called subset interaction. The interesting thing here is it does not matter what these numbers are at all. They could have been a, b, and c. So we’ve turned our numbers into letters.

It’s very powerful because we can try it with something else. So we could switch to doing the factors of 42. And we can draw a diagram similarly where we start with 1 and then prime factors 2, 3, and 7, then the products of two prime factors, and the product of prime factors. So we get the same picture. It is the point of abstraction that many things become the same. That is going to be crucial in abstraction.

Those a, b, c do not have to be numbered, and they can be anything. For example, they can be three types of privilege — rich, white, and male. Then at the next level down, we’ll have people with two types of privilege, rich and white, rich and male, and white male, and then at the next level down, just rich, white, or male, and then at the bottom, people with none of those types of privilege.

If I put back in other words for emphasis, we have rich, white, non-men, we have rich non-white men, we have non-rich white men. We have rich non-white non-men, non-rich white non-men, and non-rich non-white men. At the bottom, non-rich non-white non-men. So we have gone from a diagram of factors of 30 to a diagram of the interaction of different types of privilege.

There are many things I think we can learn from this diagram. The first is that each arrow represents a direct loss of one type of privilege. It doesn’t mean that all the people who are white are better off than all the people who are not white because you can see along with the second level that there are some people with no arrows in between them. So there is no direct loss of privilege along with that level.

So what is intelligence? First, intelligence involves being reasonable, being not just logical, but powerfully logical, and also being helpful. So if you just sit in a cave and use logic, I don’t think that’s particularly intelligent, because you’re not helping anything. You can disagree with me about this, but I don’t think that’s very intelligent.

So what do these things mean? I think that being reasonable means is not just that you use logic, but you have some kind of a framework for telling when you should change your mind. I think that’s important. You can tell when someone isn’t reasonable if there’s no possible situation in which they would change their mind if there is no possible evidence that would convince them that the moon landings actually happened, then that isn’t very reasonable.
So what about being powerfully logical? Being powerfully logical involves not just using logic, but also using techniques to build up logical arguments and unravel logical arguments so that you can get somewhere with your logic.
Finally, being helpful crucially involves some techniques for being helpful, but it consists of using our emotions because people are emotional beings. If we don’t use emotions to communicate with other people, then we’ll just be throwing cold logic at them all the time instead of understanding what it is that they are feeling and why they are feeling it.

So I like Carlo Cipolla’s theory of stupidity. It’s quite a serious theory. He starts by saying he thinks that in any given group of people, there’s the same proportion of stupid people, whether those are scientists, people who work at Google, maybe more in politicians, professors, Nobel Prize winners, prisoners, criminals or street sweepers. He thinks that there’s the same proportion of stupid people everywhere and that it’s more than you are ever expecting, even after you’ve taken that into account.

So what does he mean by stupid? Well, he has a definition that involves a graph with two axes. And one axis is how much you benefit yourself, and the other axis is how much you help other people. So if you benefit yourself and hurt other people, then you are a bandit. If you benefit other people but hurt yourself, then you are what he calls unfortunate. I sometimes think of this as being a martyr, where you sacrifice yourself all the time for the good of others. And I used to do this a lot because that it’s noble, and we’re supposed to efface ourselves for the good of our people.

I’ve eventually realized that that’s not very productive in the long run. If you hurt others and hurt yourself at the same time, you are stupid. That’s how he defines stupid. Whereas, if you benefit other people and you benefit yourself at the same time, that’s intelligent. And I think this is what intelligence is. I don’t believe it has anything to do with qualifications, degrees, education, money, status, job, achievements, prizes, accolades, or any of those things.
We should use logic and emotions together so that we can benefit ourselves and other people at the same time. In that way, create a virtuous circle, because logic and abstract mathematical thinking can help us to understand how other people think from their point of view rather than from our point of view, which allows us to empathize with other people. If we can understand with other people, then that will help us act reasonably with them and with all of us together so that we can create this virtuous circle.

I think it might be a surprising idea that abstract mathematics can do that for us, but I genuinely believe that it can, and I hope that I can persuade many people to join in with this lofty aim.

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