Deciphering the weirdness of mathematics
Looking more closely at what maths might really be about
Across the history of ideas, one of the most unexpected, and productive, intellectual revolutions was the adoption of mathematical methods in our scientific theories. While this approach seems almost self-evident today, if we hopped in a time machine to explain how we use mathematics in modern science to an Ancient Greek or Roman philosopher, they would likely be shocked, even offended.
These thinkers, following Plato and Pythagoras, saw mathematics as a perfect body of knowledge that described eternal truths. Our physical world, on the other hand, was imperfect, messy and inferior. Applying the perfect knowledge found in mathematics to the fallible physical world was, therefore, a waste of time (outside a few particular cases).
It is to all of our benefit that early scientists like Galileo and Newton, typically guided by theological assumptions, tried applying mathematics to a wide range of physical phenomena and discovered how well it worked. But, as often happens, success breeds familiarity. For if we look carefully, it isn’t clear why mathematics is so effective, nor even exactly what mathematics is, in any deeper philosophical sense.
These questions may not matter in the day to day but, as
reminds us, it is often when we look oddly at things and ask weird questions that we make progress. So bear with me while I take a closer, and different, look at the field of mathematics.Maths is weird
One of the first things to notice is that mathematical words and concepts work differently to other words and concepts. Let's take the number 'eight' as an example. It looks like a noun - the name of something. But it isn't like a normal noun that names or refers to something in the world: I cannot ever point to some thing in the world that is an 'eight' like I can point to, say, a sheep.
The fact I can’t point to some thing isn’t that unusual as humans refer to lots of things that are outside our immediate senses. But, while I could, in theory, point to a proton if I could see things at that small a scale, there is no way that my senses could change so that an 'eight' would become visible (or audible or touchable). Nor is it the type of thing that we refer based on direct experience. An 'eight' isn't like pain or hope or sadness, which all refer to psychological or internal experiences. We just don't have a recognisable experience of an 'eight'.
Given that 'eight' doesn't seem to function like a normal noun, then the next obvious option is that it is some kind of adjective, or refers to a property of objects. Colour and size, for example, work like this. There is no such thing as 'blue' that I can point to, but there are lots of blue things in the world. However, for each of these, say the shirt I am wearing, there is something about the object that makes them blue.
However, numbers don't work that way. There is nothing intrinsic to my shirt that determines it was one of eight blue shirts in my wardrobe this morning. The correct number of blue shirts in my wardrobe changes regularly, without anything about any of the items of clothing changing. And, at the same time, it might be one of eight blue shirts in my wardrobe, 17 blue shirts in the house, 85 items of clothing in the house and so on. The correct number that applies to an object depends on context (or the question we are asking). It is not something inherent to the object, or perhaps even in the physical world.
Given the inherent dependence on context, perhaps then we would say that mathematical words, like 'eight', are some kind of social construct. They are something we humans have constructed and apply to the world, but don't exist as a property of the world in any way. Our judgements about what is fashionable or legal are examples of constructed concepts - there is nothing objective about the universe that determines that denim is cool, or that we should drive on the right (or left) of the road.
However, socially constructed concepts change over time and space, and our judgements about them change. Whether something is legal in a remote African tribe is very different to what is legal in a modern liberal democracy. On the other hand, it doesn’t quite make sense to say that mathematical concepts like the number 'eight' vary across different social or cultural settings.1 That there are eight people eating a meal together (and not 6 or 12) is a testable fact that holds regardless of the cultural context.
So what is mathematics?
So mathematical concepts don’t work like other concepts we are familiar with. And yet, we commonly take mathematics as the archetype of undoubtable knowledge. It is logically certain that 2+2=4.2 How then do we make sense of mathematics? There are a few standard philosophical approaches.
One famous account derives from Plato. The Platonic account of mathematics maintains that all mathematical concepts, including numbers, do in fact refer to objects in the way that 'sheep' does. It is just that those objects are not physical, psychological, or spiritual, but are some kind of purely intellectual objects. These objectively exist but we can only access them intellectually.3 While this account seems a bit spooky and odd, it maintains the objective truth of mathematics and generally describes how mathematicians think about what they are doing.
In my mind, however, there are two main problems with a Platonic account. The first is the more obvious one: it is hard to know how to understand what it means for numbers to exist as objective abstract entities. The second, however, seems more decisive. Over the past couple of centuries, there has been a proliferation of different, often competing, mathematical systems. For example, there are now multiple different types of geometry (including various non-Euclidean systems) that are all mutually incompatible.
The same situation exists around numbers. To pick up our previous example, what objective, Platonic form could ‘eight’ refer to, when it has different meanings and functions depending on whether we are considering the natural numbers, integers, rational numbers, various systems of modulo arithmetic, p-adic numbers and so forth. Are there an infinite number of different objectively existing 'eights' corresponding to all the different systems, or is there one object that changes properties? Neither option is satisfactory if we are trying to establish the truth of mathematics by tying it to an objective reality.
A second way of explaining mathematics is to argue that it is really a subset of a different field of knowledge. In the early twentieth century, it was often thought, by people like Bertrand Russell, that mathematics was really just part of formal logic built by basic principles of reasoning. An alternative approach, advocated by WVO Quine, relied on the foundational role of mathematics in many scientific theories that we take as true. If a scientific theory that relies on mathematics is clearly true, then the mathematics also has to be true.
While superficially attractive, all examples of this approach really just shuffle the core questions off to a different sphere of inquiry. Turning the question of what exactly ‘eight’ refers to into a question of logic, or scientific method, does not answer the question.
A third approach argues that mathematics isn't really true in an objective sense. It is socially constructed and/or fictional. We are comfortable saying that a statement about, say Othello, is true within the play but not of the real world. In the same way, mathematical statements are true in the context of mathematics but it is the wrong question to ask whether they are true about the world. This approach also shifts difficult questions - what exactly are fictional characters? But there is a deeper problem.
Fictional worlds are radically under-determined. Beyond what exists in a fictional text, there is no fact of the matter about what is true or false. There is no way of deciphering Othello's past military career, for example, beyond what is mentioned in the play. We can guess and speculate but not decide. Mathematics, on the other hand, is highly determined.4 There are things that are true or not about prime numbers, even if that is beyond what anyone has so far figured out. Mathematics has genuine discovery and surprise - we realise something is true that we didn't expect or finally answered a difficult question!
The genuine sense of discovery, that is shared by both children learning maths and professional mathematicians, is, I think, a key to better understanding what mathematics is.
So how do we make sense of mathematics?
Let’s look at a fairly typical journey of discovery that a child goes through when they learn maths. Importantly, as teachers often talk about, maths is a subject where learning needs to be ‘scaffolded’. That is, a child needs to learn and grasp certain skills or concepts and is able to move on only once they have learned those. There are no shortcuts as, until more basic concepts are clear, a child will always struggle to understand anything more advanced.
Given this, it makes sense to go back to the typical start of the journey of discovery - learning how to count. As parents and teachers know, teaching a child to count is not easy and can take a long time. A big part of the challenge is that there are separate things that we have to grasp together. One is to memorise the numbers in order. The harder step is to make the connection between a number and what it means. A child might know the number “three” and how to count to three, but will often not be able to correctly pick out three things.
However, once a child grasps the concept - that “three” means there is a collection of things and if I count them all, the highest number I get to is three - it all clicks and is much easier. In fact, in learning a lot of mathematics, learning to count is the hardest step. Once we have mastered counting, then addition and subtraction are relatively easy to grasp (if not do) as they are just extended steps of counting. In turn, multiplication and division are extended addition and subtraction, and we then get to square numbers, prime numbers and the whole of number theory. There are many non-trivial steps of discovery along the way, but all follow naturally from counting.
To put this differently, everything we know about numbers is built from a couple of basic, but not trivial, concepts. The first is conceptualising a single object, that is distinct from others, as ‘one’ thing. The second is the mechanics of counting, namely that putting another single, distinct object alongside an original one means we now have a distinct number of objects (i.e. 'two') and we can continue this process and produce further distinct numbers. We tend to take these for granted, but they are not easy to teach.
Once we take these two concepts together, a long list of inevitable consequences emerge, which we have codified into what we call arithmetic. There are also a number of natural, but also non-trivial, extensions of the concepts that expand into different types of mathematics (e.g. let's divide numbers into parts —> fractions, or can we continue subtraction below zero —> negative numbers, and so on). Importantly, in all these cases, the inevitable consequences form a particular structure and we know there are things that are definitely true, and definitely false, about this structure.
Other branches of mathematics function in a similar way, only they arise from thinking seriously with different foundational concepts. Geometry arises inevitably when we think systematically about lines and points. Category theory arises when we think about objects and relations between them (or functions). Probability theory emerges once we think about chance systematically. None of these foundational concepts are trivial or easy to grasp. But once we have got them, branches of mathematics inevitably, albeit often slowly, emerge.
In other words, mathematics is the systematic exploration and elucidation of the inevitable (or logical) consequences of different foundational concepts.
Usefully, this account takes us away from mysterious Platonic objects while still retaining definitive truth within mathematics. For example, there either is or isn’t a largest prime number based on the basic rules or concepts of counting. It also captures the internal coherence that fictionalist accounts point to, without sacrificing the highly determined nature of mathematics. And while it shifts the explanatory power onto the notion of accepting particular concepts, the basic process is something we are all familiar with. We know the ‘aha’ moment when we actually get our heads around a concept and it makes sense.
Even more usefully, this way of explaining mathematics can help explain the puzzles the Ancient Greeks would have over modern science. So why does mathematics work as an explanation of the physical world? Unfortunately, this post is already too long and I will have leave that to my next post. Stay tuned!
To be clear, different cultures have different ways of using and expressing mathematics, but the underlying concepts don’t vary. All cultures that have adopted counting end up in an equivalent place, even if they don’t always use a base 10 number system.
For the mathematically minded, I’m aware this doesn’t hold in certain systems of arithmetic, but these are exceptions.
For Plato, all concepts worked in this way and mathematics wasn’t an exception. He thought there are pure objects (or 'Forms') for all concepts and that the objects we see only fit a category to the extent that they adhere to the Form. So, for Plato, we can only rightly call an animal a dog if it conforms to the ideal Form of a dog.
It is an open question whether it is completely determined - there may be mathematical theorems that cannot be be proven either true or false.
Love it. My new favourite phrase is Fictional worlds are radically under-determined.