yume no ato
The summer grasses:
The high bravery of men-at-arms,
The vestiges of dream.
— Matsuo Basho, on visiting Hiraizumi, once home to the great Fujiwara clan whose splendid castles had been reduced to overgrown grass mounds.*
A good haiku not only arrests our attention, it also demands reflection and contemplation of deeper themes. In Basho’s Oku no Hosomichi, The Narrow Road to the Interior, the haiku often serve as a point of pause amidst the travelogue, asking the reader to slow down and take in all that is being said. The slow road to understanding is often the easiest way to get there. At the same time the travelogue itself provides context for the haiku. Without that context, both from the travelogue, and from our own experiences of the world upon which the haiku asks us to reflect, the poem becomes shallow: you can appreciate the sounds and the structure, but the deeper meaning — the real essence of the haiku — is lost.
Mathematics bears surprising similarities. A well crafted theorem or proof demands reflection and contemplation of its deep and wide ranging implications. As with the haiku, however, this depth is something that can only be provided by context. A traditional approach to advanced mathematics, and indeed the approach you will find in most textbooks, is the axiomatic approach: you lay down the rules you wish to play by, assuming the bare minimum of required knowledge, and rapidly build a path straight up the mountainside. This is certainly an efficient way to get to great heights, but the view from the top is often not rewarding unless you have spent time wandering through the landscape you now look out upon. Simply put, you lack the context to truly appreciate the elegant and deep insights that the theorems have to offer; like the haiku it becomes shallow.
My task, then, is to provide you with the necessary experiences in the mathematical landscape; to provide context for the insights that are to follow. In the previous entry, On Abstraction, I discussed the process of abstraction, and how mathematics builds up layer upon layer of abstraction. The road we must take, the slow road, is the path that winds its way through these layers. Each layer is, in a sense, a small plateau amidst the mountains; to be explored before the next rise begins.
The place to start, therefore, is with the area of mathematics that most people already have a fairly strong intuitive sense for: numbers. Many people tend to assume that mathematics is all about numbers, something that simply isn’t the case. Numbers are just one of the more extreme abstractions from the external world, amongst many different abstractions that make up modern mathematics. Even in antiquity mathematics was divided into arithmetic, which abstracted quantity, and geometry, which abstracted shape and form. Numbers are, however, something that almost everyone has (or thinks they have) a solid intuitive grasp of — and studying the nature of that abstraction, and how it is made, will provide some context for other similar abstractions, as well as providing a solid base from which to build further layers of abstraction.
The concept of number is both a greedy abstraction, and a remarkable one. It is greedy in that it tries to abstract away as much detail as possible. Given a collection of objects (for now we’ll take “collection” as intuitive, most people’s everyday experience is sufficient for elementary numbers and doesn’t run afoul of the pathological cases that require strict definitions to avoid) we forget absolutely everything about the collection, and about the objects themselves, except for a single particular property. The abstraction is remarkable because, by being so very greedy, it is applicable to everything — there is simply nothing in our experience of the world that doesn’t fall under the umbrella of this particular generalisation: everything can be quantified in some sense, albeit trivially (as “one”) in many cases.
This is the power of mathematics: by seeking greedy abstractions, by generalising as much as possible, it finds properties or concepts that have near universal applicability. Mathematics allows you to speak about everything at once. The catch, of course, is that by abstracting too far you leave yourself unable to say anything useful (you can say nothing about everything). The trick is to forget as much as possible about particular instances, while still leaving some property that can be worked with in a constructive manner toward some purpose or other. Whether you’ve forgotten too much depends on your particular purpose — that is, what structure or property you are interested in. Mathematics is the art of effective forgetting.
The effectiveness of numbers comes from arithmetic. The basic operation of addition allows us to describe the results of bringing together two collections and regarding them as one. By regarding this notion of combination in the very abstract terms of addition of numbers we gain two things: first, by removing the messy particularity of the world via abstraction we make the process simple to deal with; second, by using the greedy abstraction of numbers we produce universally applicable results; 2 + 3 = 5 is a statement about any collection of 2 things and any collection of 3 things. Performing an addition is generalising across incredibly broad classes of real world situations. We are saying an enormous amount incredibly simply.
The beauty and complexity starts to unfold when, due to the greediness of the abstraction, we find that numbers can reflect back on themselves. For example, additions can form collections, and, as noted, collections have quantitative properties. Thus we can talk about a particular number of additions; for example we might have 5 additions of 3, and arrive at multiplication. That is 3+3+3+3+3=5×3. We can talk about a particular number of multiplications and arrive at exponentiation, and so on. By folding the abstraction back on itself we can build layers of structure – structure that may be far more complicated than we might first imagine. In introducing multiplication we raise the question of its inverse, division. That is, if we can find the quantity that results from some number of additions, we can ask to go the other way and decompose a quantity into some number of additions. In doing this, however, we introduce prime numbers (those which cannot be decomposed into any integer number of additions of integers) and fractions. Whole new expanses of complexity and structure open up before you – there is, apparently, a whole world to explore.
Next time we’ll look into what kind of abstractions we can make from the world of numbers, and dip our toes into the beginnings of algebra. In the meantime, however, I’ll leave you with a question about numbers to ponder:
Can every even integer greater than 2 be written as the sum of 2 primes?
This is commonly known as Goldbach’s conjecture, and it remains an open problem to this day; no one knows the answer. It is worth taking a moment to think about the problem yourself, and wonder why it may, or may not, be true, and what it really means, and also how little we really know about the strange world of numbers.
* Translation by Earl Miner