natsukusa ya
tsuwamonodomo ga
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

### 11 Responses to “The Slow Road”

1. Everything Blog » Blog Archive » The Slow Road Says:

[…] Original post by lmcinnes […]

2. Jesse Merriman Says:

3. William Lee Says:

Hi I really enjoyed your essays and find them very enlightening. I have always been deficient in math since elementary but I have found a renewed interest and passion in it recently as I have been studying finance. I found the overview of abstraction and your reference to Matsuo Basho to be very appropriate. Hope you can post more in the near future and would you be able to recommend any websites or books on General Introductions to Mathematical Theory? My major complaint on math textbooks are that they fail to deal with the philosophical underpinnings of the theory unlike what your writing

4. lmcinnes Says:

Hopefully I’ll have the next post, on fractions and algebra, up in a couple of weeks. Again we’ll be focussing on the philosophical underpinnings rather than the details.

As to suggested texts – that’s hard, since as I said, most books tat treat any interesting mathematics tend to assume experience and take an axiomatic bootstrapping approach which doesn’t always serve the mathematically inexperienced reader quite so well. I think you’ll fare better by hunting down books on mathematical philosophy. On that front I have to go with Bertrand Russell, whose “Principles of Mathematics” and “Introduction to Mathematical Philosophy” are both very readable, and quite deep.

5. Florian Says:

Hi,
I found your blog via google by accident and have to admit that youve a really interesting blog 🙂
Just saved your feed in my reader, have a nice day 🙂

6. The Narrow Road » Blog Archive » A Fraction of Algebra Says:

[…] As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing. In the previous entries we have dealt with subjects (abstraction in general, and the abstraction of numbers) that most people have a natural intuitive grasp of, even if the details, once exposed, prove to be more complex than most people give them credit for. It is time to start looking at subjects that often prove to be early stumbling blocks for some people: fractions and algebra.There is a reason that these subjects give people pause when they first encounter them, and that is, quite simply, that they are difficult. They are difficult in that they represent another order of abstraction. Both fractions and elementary algebra must be built from, or abstracted from, the basic concept of numbers. Because of the sheer prevalence of numbers and counting in our lives from practically the moment we are born, people quickly develop a feel for this first, albeit dramatic, abstraction. It is when people encounter the next step, the next layer of abstraction, in the form of fractions and/or algebra, that they have to actively stretch their minds to embrace a significant abstraction for the first time. Most of us, having won this battle long ago, struggle to see the problem in hindsight — we might recall that we had trouble with the subject when we were younger, but would have a hard time saying why. We have developed the same sort of intuitive feel for fractions and algebra as we have for numbers and have forgotten that this is hard won knowledge. […]

7. Amanda Kerik’s WordPress Blog If We Taught English the Way We Teach Mathematics... « Says:

[…] If you climb up those layers of abstraction you can use that broad brush to paint beautiful pictures — the vast scope of the language that mathematics gives you allows simple statements to draw together and connect the unruly diversity of the world. A good mathematical theorem can be like a succinct poem; but only if the reader has the context to see the rich connections that the theorem lays bare. Without the opportunity to step back and see the forest for the trees, to see the broad landscape that the abstract nature of mathematics allows us to address, it is rare for people to see the elegance of mathematical statements. By failing to address how mathematics works, how it speaks broadly about the world, and what it means, we hobble children’s ability to appreciate mathematics — how can they appreciate something when they never learn what it is? The formulas and manipulations children learn, while a necessary part of mathematics, are ultimately just the mechanics of the subject; equally important is why those mechanics are valuable, not just in terms of what they can do, but in terms of why they can do so much. […]

8. The Narrow Road » Blog Archive » Paradoxes of the Continuum, Part II Says:

[…] Now that we at least have some idea of what these sequences might look like, it is time to take a step back and consider what is actually going on here. Back in The Slow Road we constructed natural numbers as a property of collections of objects. Then, in A Fraction of Algebra, we created fractions to allow us to re-interpret an object within a collection. This was another layer of abstraction — fractions were not really numbers in the same way that natural numbers were — fractions were a way of re-interpreting collections, and we could describe those re-interpretations by pairs of natural numbers. Perhaps rather providentially it turned out that the rules of algebra, the rules of arithmetic that were true no matter what natural numbers we chose, also happened to be true no matter what fractions we chose. It is this stroke of good fortune, combined with the fact that certain fractions can take the role of the natural numbers, that allows us to treat what are really quite different things in principle (fractions and natural numbers) as the same thing in practice: for practical purposes we usually simply consider natural numbers and fractions as “numbers” and don’t notice that, at heart, they are fundamentally different concepts. Now we are about to add a new layer of abstraction, built atop fractions, to allow us to describe points in a continuum. While all that was required to describe the re-interpretation of objects that constituted a fraction was a pair of numbers, points in a continuum can only*** be described by an infinite Cauchy sequence of fractions. Thus, in the same way that natural numbers and fractions are actually very different object, so fractions and points in a continuum are quite different. Again, however, we find that when we define arithmetic on sequences (which occurs in the obvious natural way) they all behave appropriately under our algebraic rules. When we consider that it is easy enough to find sequences that behave as fractions (any constant sequence for instance) it is clear that, again, for practical purposes, we can call these things numbers and assume we’re talking about the same thing regardless of whether we are actually dealing with natural numbers, fractions, or points in a continuum. […]

9. Kimora Lee Simmons Says: