Mathematics in the wild

This is an essay draft that I wrote about a year ago. My views on mathematics and language have shifted a bit but I'm posting this here anyway. Comments welcome!

Most people today would probably agree that mathematics is useful. We live in a quantitative world, and math is becoming increasingly important in many professional domains, from business and finance to technology. Still, it is probably common to view mathematics mainly as a tool for solving practical problems. In fact, people who do not use math in their profession are rarely attracted to it, and are often intimidated by it. I hope to argue here that "mathematical thought" is, from a certain perspective, a very natural and fundamental process that is already hidden in the way we all view the world, but we need be identify and recognize it more widely in order to unleash its full potential.

The first natural question is: what is mathematics? Perhaps surprisingly, there is no universally agreed-upon definition. Modern mathematics is typically categorized into different research areas, such as "number-theory," "combinatorics," "algebraic geometry," "numerical analysis," and others. There is also a broader classification into "pure" and "applied" mathematics. But the classification into subfields is not clear-cut, and can often be attributed to historical reasons.

What many people probably do not realize is that virtually anything can be subject of mathematical study. Deep mathematical theories have been developed for classifying knots, for moving dirt, or for describing the shape of soap bubbles; math can also answer questions such as how to seat people at weddings or how to eat pizza. More broadly, mathematics can deal with very abstract categories and their relations. The only essential requirements for a mathematical theory are in fact:

1. A clear set of objects and rules,

2. A rigorous investigation of the consequences of these rules.

From this perspective, the domains of application of mathematics are almost limitless.

Usually, the objects and rules that are studied in a mathematical theory are abstractions of patterns that exist the physical world. The most basic examples of abstractions are numbers and geometric entities. More generally, we can break down mathematical thought into two types of processes:

1. Abstraction, for deciding what are the important objects and rules to study.
2. Logic, for carrying out a formal analysis of the consequences of the rules.

These two processes are actually essential for acquiring any type of knowledge. Abstraction is associated with inductive reasoning, that is, going from the "particular" to" general"; logic is associated with deductive reasoning, that is, going from "general" to "particular." Every day we use inductive and deductive forms of reasoning to make sense of the world: we look for patterns around us and use them to our advantage. This should illustrate an intimate relationship between mathematics and broader forms of "rational reasoning." Any rational thought is, at heart, a logical manipulation of abstract concepts, and the same is true for mathematics.

In practice, the most important difference between traditional rational and mathematical thought is simply in the amount of rigor. The more rigorous a reasoning is, the less room it leaves for inconsistencies, and the harder it is confute it. Contrary to everyday reasoning, mathematical arguments and proofs are considered the gold standard of rigor. It is however interesting to note that even within mathematics the amount of rigor can vary widely. Many mathematical arguments developed prior to the 19th century would not be considered rigorous by today's standards,¹ and even modern proofs are almost never completely rigorous in comparison to algorithmic proof systems. In particular, even though the difference in rigor between traditional forms of rational and mathematical thought is significant, the distinction between the two forms of reasoning is perhaps more quantitative than qualitative.

That said, I think that there are some important reasons for the high standard of rigor that is achieved in mathematical reasoning, so I'll try to discuss some of these.

First, mathematics is based on a very strict separation between the two fundamental processes of abstraction and logic. In other words, once a "model" consisting of objects and rules is in place, only these abstract entities are used for logical analysis.² For example, if we use game theory to model cooperation among "rational agents," then we can apply precise (mathematical/logical) rules to deduce optimal strategies of behavior, without having to resort to external concepts, such as "real-life" versions of the agents. Reasoning only about abstract entities forces models to be well-defined (objects have a clear and fixed meaning) and consistent (rules can be followed without reaching contradictions).

Importantly, this is not the way we usually think about things. When we try to make a rational argument about the real world, the contents of our thoughts are almost never entities with a clear and fixed meaning, but rather fuzzy concepts that are influenced by our personal experiences, intuitions, and emotions.

This strict "decoupling" between the processes of abstraction and logical deduction is more or less standard practice in the hard sciences. However, this form of reasoning becomes less frequent as we move closer to the humanities (and it is especially rare in politics!). The reason for this is understandable: complex social dynamics and human behavior are extremely difficult to axiomatize in a satisfactory way, so simple models are unlikely to provide accurate abstractions of reality. Still, in any domain, if our goal is to reason and communicate effectively, we should always aim to develop clear conceptual frameworks that are both consistent and well-defined.³ I believe in general that abstract and "model-based" thinking is a powerful tool that we all should learn to use more often.

Another way that mathematics is rigorous is thanks to its specialized language. It might be obvious that a mathematical formula has a precise and unambiguous meaning. After all, algebraic symbols are just a set of conventions that mathematicians have agreed to use. But the added rigor might seem to come at the cost of obscure and difficult notation. In other words, by using mathematical language, we might be choosing to sacrifice intuitive and immediate communication in favor of rigor. There is certainly some truth to this, and mathematical jargon can often be a huge barrier for non-specialists. But technical language is often unavoidable when dealing complex concepts and, overall, I think that mathematical language is key to making abstract concepts more intuitive, rather than more obscure.

The reason for this is that mathematics is typically based on a hierarchical ontology of concepts of ideas. That is, terminology and notation is introduced through successive definitions and "constructions" that are based on prior ideas. This may seem trivial, but it's hard to overstate how useful and important it is. Our brains have limited processing power, and they're also much better at "fast" associative thinking rather than "slow" analytical reasoning. By leveraging hierarchical abstractions, we are effectively turning deep concepts into shallow ones. Once we become comfortable with a high-level abstraction, we can reason about deep concepts in an intuitive way.⁴

This hierarchical organization of ideas also makes it much easier to notice patterns and similarities among seemingly very different objects. When we think of something in abstract terms, we discard some of its properties and focus only on what we consider relevant. This often allows us to notice new connections that were previously hidden by the clutter of unnecessary details. And finding these similarities is more than just an intellectual exercise: the more connections idea has, the deeper is our understanding of it.⁵ By building abstract conceptual models, we are making things intuitive, not the other way around.

Once again, this is not how we usually talk about things. Our everyday language is organized mostly hierarchically when it deals with physical objects (e.g., a "Steinway" -> "piano" -> "musical instrument", etc.), but it is very "horizontal" when it deals with concepts and relations. For example, when talking about social or political issues, we rarely reason in terms of abstractions, but rather focus on what we consider to be plain "facts."⁶ Like we already said, this is in part because the world is complex and messy, so organizing concepts in a very structured way is difficult, considering that most concepts are not even rigorously defined. But I believe that another reason for this is that our natural language is limited and rigid. By this I mean that we do not have the possibility to easily "define" a term that might succinctly describe a new phenomenon that we encounter. This implies that complicated ideas will generally require complicated descriptions, and that they will remain unintuitive and difficult to process.

So what does all this mean in practice? Is there anything that we can learn from the way mathematical language is structured? I think there is. First, we should realize that our everyday language is in many ways similar to mathematical formalism: it is only a convention, and it is completely separate from the real world. I believe that if we all were more aware of this, many of our disagreements could be tackled in a more rational and objective way. Second, we should not be afraid to modify and improve our language, especially by introducing new concepts that could help us describe the world. A few centuries ago, the "%" symbol was not widely used, and the general concept "percentage" was probably not as natural and clear as it is to us to us today, even though fractions had been used in trade for a very long time. In the coming years, I think that we will see other technical concepts, ideas like "local minima," "entropy," "positive feedback loop,"⁷ become more common and part of people's general knowledge. While this process will happen naturally on its own, I think we should encourage it, and not be afraid to learn and use new technical concepts in our everyday life. In the end, having a better vocabulary will make it easier to reason about this complex world.

Emotions will always play an important role in our lives, and we should recognize that some parts of our experience will likely remain impossible to fully express in words, no matter how much we enrich our language. But this is not the case for all aspects of life, and in order to collaborate and understand each other we should always be trying to improve the way we think and talk about things. Taking inspiration from mathematics is one of the best ways to achieve this.