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.
I want to begin with fractions because, ultimately, it is by far the easier of the two — being only a semi-abstraction — and will provide an example of the process as background for stepping up to elementary algebra.
As was noted in the last entry, the complexity of mathematics begins to open up once we pass from considering numbers as referring to collections of objects and begin to think of them as objects in their own right. Once we have grasped that abstraction we can count numbers themselves, and operations on numbers, giving us the higher order construction of multiplication. Division operates in a similar way, providing an inverse to multiplication in the same way that subtraction provides an inverse to addition. That is, while addition asks “if I add a collection of size 3 to a collection of size 2, what size is the resulting collection?”, subtraction asks the inverse question “if I got a resulting collection of size 5 by adding some collection to a collection of size 3, how much must I have added?”; parallelling that we have multiplication asking “if I add together 5 collections of size 3, what size is the resulting collection”, and division reversing the question: “if I have a resulting collection of size 15, how many collections of size 3 must I have added together?”. Everything seems fine so far, but there is some subtlety here that complicates the issue.
If we are still thinking in terms of collections then dividing a collection of 2 objects into 4 parts doesn’t make sense. If we are viewing numbers and operations on them as entities in their own right then we can at least form the construction 2/4, and ask if it might have a practical use. It turns out that it does, since it allows a change of units. What do I mean by this? We can say a given collection has the property of having 2 objects in it, but to do so is make a decision about what constitutes a discrete object. Deciding what counts as an object, however, is not always clear — there are often several possible ways to do it, depending on what you wish to consider a whole object (that is, the base unit which you use to count objects in the collection). A simple example: in the World Cup soccer finals, do you count the number of teams, or the number of individual players? Both make sense depending on the kind of result you want to obtain, so considering a team, or each individual player, as a discrete object is a choice. The problem is even more common when dealing with measurement: a distance is measured as a certain number of basic lengths, but what you use as your basic length (the unit of measure) is quite arbitrary. We tend to measure highway distances in miles or kilometres and people’s heights in feet or metres, but we could just as easily switch to different units and measure highway distances in feet or metres and still be talking about the same distance. What matters is knowing what your base units are: we can count money in terms of dollars, or in terms of cents, but knowing which you are using makes big a difference.
Most importantly we can change our minds, or re-interpret, what constitutes a distinct object after the fact. Using this re-interpretation of what constitutes a discrete object, we can make sense of 2/4. If we re-interpret a distinct object such that what we had previously considered a single object is now considered two objects then we will have 4 objects in the collection, and we need 4 of these new objects to arrive at a collection that would be regarded as having 2 old objects. That is, 2/4 is expressible in terms of the re-interpreted objects, and in fact defines the relationship between old objects and new. But here’s the rub: we arrived at the new objects by considering each old object as 2 new objects, and so 1/2 expresses the same relationship between old and new objects: 1 old object reinterpreted as 2 results in the same new object as 2 old objects re-interpreted as 4.
Indeed, we can go on like this, with 3/6, 4/8, 5/10, and so on, all expressing the same relationship of new object to old – all different ways to arrive at the same “size” of new object. And so we have a catch – what are on inspection quite different expressions will, in practice, behave the same. Re-interpreting 1 object as 2, or 2 objects as 4 results in the “same” new objects, so counting, and hence addition, subtraction, and multiplication of these new objects will give the same result, whichever re-interpretation we use. Perhaps it doesn’t seem like much of revelation that 2/4 is the same as 1/2, but that is simply because we have learned, through practice, to automatically associate them. The reality is that 2/4 and 1/2 are quite distinct, and it is only because they behave identically with regard to arithmetic that we regard them as the same. In identifying them as the same we are abstracting over such expressions, forgetting the particularities of what size of initial collection we were dividing, caring only about the common behaviour with regard to arithmetic. Making sense of fractions involves abstracting over numbers – they are another level of abstraction, and this, I suspect, is why people find them difficult when they first encounter them.
There is an important idea in this particular abstraction that is worth paying attention to – it leads the way to algebra. We have an infinite number of different objects: 1/2, 2/4, 3/6, 4/8,… but because they all behave identically with respect to a given set of rules (in this case basic arithmetic) we pick a single symbol to denote the entire class of possible objects. Algebra can be thought of as extending that idea to its logical conclusion. The insight we need to make the step to algebra is that there is a subset of the rules of arithmetic for which all numbers behave identically. For example reversing the order of addition makes no difference to the result, no matter what numbers you are adding: 1+2=2+1, and 371+27=27+371. If you can identify which rules have the property that the specific numbers don’t matter, then you can pick a single symbol to denote the entire class of numbers for any manipulations within that set of rules. This is algebra.
This is important because it is a layer of abstraction over and above the abstraction of numbers. With numbers we considered many different collections and abstracted away everything about them except a certain property — the number of objects they contain. This proved to be useful because with regard to a certain set of rules, the rules of arithmetic, that was the only aspect of the collection that made a difference. Now we are regarding numbers as objects in their own right and, having identified a set of rules under which the particular number is unimportant, we are abstracting away what particular number we are dealing with. With numbers we could perform calculations and have the result be true regardless of the particular nature of the collections beyond the number of objects. Now, with algebra, we can perform calculations and have the result be true regardless of the particular numbers involved. This is an exceptionally powerful abstraction: it essentially does for numbers what numbers do for collections. This is why the rules of algebra, that subset of arithmetic rules under which all numbers behave identically, are so important.
In particular we can say that, no matter what numbers x, y and z are, the following are always true:
- x+y=y+x and x×y=y×x. These are referred to as commutative properties.
- x+(y+z)=(x+y)+z and x×(y×z)=(x×y)×z. These are referred to as associative properties.
- x×(y+z)=x×y+x×z. This is referred to as a distributive property.
- x+0=x and x×1=x. This property of 0 and 1 is referred to as being an identity element for addition and multiplication (respectively).
- There is a number, denoted –x such that –x+x=0. This refers to the existence of inverses for addition.
We also have one odd one out — the existence of inverses for multiplication. The catch here is that it does matter what number x is; inverses exist for almost every number, but if x=0 there is no multiplicative inverse of x. Thus we have:
- If x is any number other than zero then there is a number, denoted 1/x, such that (1/x)×x=1.
If you have any curiousity you will be wondering why this special case occurred, breaking the pattern. Remember that we are talking about abstract properties common to all numbers, so the fact that this is a special case says something quite deep about both multiplication, fractions, and the number zero. Indeed, because we are two layers of abstraction up, referring to all numbers, which in turn each refer to all collections with a given property, the fact that this is a special case has significance with regard to almost everything in the physical world. It is worth spending some time thinking about what it truly means.
We have some further properties with regard to how numbers can be ordered. I haven’t touched on this topic yet — we’ve only referred to numbers as a property of collections, and not as an ordering — but it is sufficiently intuitive (that is, most people have a firm enough grasp on numbers) that I won’t get into details here; just be forewarned that numbers as order and numbers as size are actually distinct concepts that, at some point, we will have to carefully tease apart.
- Either x<y, y<x, or x=y.
- If x<y and y<z then x<z.
- If x<y then x+z<y+z.
- If x<y and 0<z then x×z<y×z.
Note that, again, 0 and multiplication have a significant interaction and provide another special case.Note that I gave names to properties 1 through 5 because these properties will keep cropping up again and again later; some will prove to be important, others less so. Which ones are important and which are not may be somewhat of a surprise, but I’ll leave that surprise till later.
At this point it is worth taking stock of how far we’ve come. Not only have we built up two layers of abstraction, each of which can be used to great practical effect (just witness how much of modern technology and engineering is built upon arithmetic and elementary algebra!), in doing so we’ve begun to uncover an even deeper principle — the principle that will form the foundation for much of the modern mathematics that is to follow. What do I mean? There is a common thread to how these successive abstractions have been built: we discerned a set of rules for which an entire class of objects (potentially even completely abstract objects) behave identically, and this allowed us to abstract over the entire class. The broader the class the broader the results we can draw; the higher the abstraction (in terms of successive layers) the deeper the results we can draw. The approach now will be to seek out rules, and classes that they allow us to abstract over; the broader and more layered, the better. In so doing we will part ways with numbers entirely. Fractions, ordering, and the difficulties of 0, will lead us towards a kind of generalised geometry, while consideration of properties 1-6 will lead us to a language of symmetry.
We have come to the first truly significant incline on our road. Behind us lies a vast plain of numbers, fractions, and algebra. There is much more to explore there — we haven’t even touched on popular topics such as trigonometry — but in following the path we have, we have stumbled across a road that leads deep into the mountains. We have identified a common property to the abstractions we are making, and will now seek to generalise it. The importance of this cannot be overstated! We are abstracting over the process of abstraction itself! This is the path to high places from which, when we finally arrive, we can look out, over all the plains we now leave behind, with fresh eyes, and deeper understanding.