Let’s begin with a short practical experiment. Pick up a pen, or whatever similar sized object is handy, hold it a short distance above the ground, and drop it. The result — that the pen falls to the ground — is not a surprising one. The point of the experiment was not to note the result, however, but rather to note our lack of surprise at it. We expect the pen to fall to the ground; our expectation is based not on knowledge of the future however, but on abstraction from past experience. Chambers Dictionary defines “abstract”, the verb, to mean “to generalize about something from particular instances”, and it is precisely via this action that we come to expect the pen to fall to the ground. By synthesis of many previous instances of objects falling when we drop them, we have generalized the rule that things will always fall when we drop them*. We make this abstraction so instinctively, and take it so completely for granted, that it is worth dwelling on it for a moment so we can see how remarkable it actually is.
The circumstances surrounding each and every instance of you observing an object falling to the ground are quite unique. Were it not for our brains’ natural tendency to try to link together our experiences into some kind of narrative we would be left contemplating each dropped object as an entirely distinct instance, and be in no position to have any expectation as to what will happen; each time it would be an entirely new case. In our minds we have, rather than a vast array of disjoint and distinct instances, a single principle that knits together the common elements and generalises to new circumstances that share the same common elements of those past experiences. This is something we do unconsciously and automatically: our brains hunt for patterns in the world, generalises those patterns, and expects them to continue. Indeed, almost all our expectations are a result of such inductive knowledge and abstraction. Abstraction is fundamental to our experience.
Of course not all abstractions are correct. Our minds are constantly on the hunt for possible patterns, and don’t always pick out valid ones. The classic example is the Christmas goose, who, every day of the year has found the arrival of the farmer results in the goose getting fed — until Christmas day when the abstracted rule that “Farmer implies food” meets a painful end. Even if we find abstractions that are ostensibly correct, that doesn’t mean they’re ideal. A perfectly valid abstraction of dropping objects, for instance, would be the rule that a dropped object always accelerates away from your hand. Certainly that is true, but it leaves out what might be considered an important common property of things being dropped: the direction in which the dropped object accelerates. Alternatively we could note that different objects, say a feather compared to a stone, behave very differently when dropped and arrive at a vast array of rules, one for each different kind of object.
This leads to the dilemma of finding the most effective or efficient abstraction — the abstraction that most consistently produces the results you want with the least effort. That is where science comes in: it is a systematic effort to refine our abstracted rules and principles, and continually check them for consistency. We may, if we like, think of the different sciences as arising from the “results we want” clause of our definition of effective abstraction. A biologist tends to work with different abstractions (generally on very different scales) than a physicist because the sorts of results they are interested in determining are rather different.
This is all very interesting, but you are probably starting to wonder what any of it has to do with mathematics. The answer is that mathematics relies on precisely this sort of abstraction that is so integral to our experience of the world. Mathematics simply attempts to take the abstraction as far as it possibly can. Part of what makes mathematics “difficult” is that it tends to pile abstraction upon abstraction. That is to say, after developing a particular abstraction it is common for mathematics to then study that abstraction and, upon finding common properties when dealing with that abstraction, generalise that commonality into a new abstraction; mathematics develops abstractions not only from a synthesis of experiences of the external world, but also via synthesis of properties of existing abstractions. This layering means that unless you’ve gotten a good grasp of the preceding level of abstraction, the current one can be extremely hard to follow. Once you’ve wandered off the path, so to speak, it can be difficult to find your way back.
The other problem that people tend to face, when learning mathematics, is that as you pile up abstractions and climb higher, and hence more distant from everyday experience, intuition becomes less and less helpful, and an increasing degree of pedantry is required. To give an example of what I mean by this, lets take a detailed look at a mathematical abstraction that almost everyone takes for granted: numbers.
Natural numbers (also known as counting numbers) are one of those remarkable abstractions, like objects falling to the ground, that we take for granted. Natural numbers are, however, a very abstract concept – they simply don’t exist in the external world outside your own head. If you think otherwise, I challenge you to show me where the number 3 exists. You can point to some collection of 3 things, but that is only ever a particular instance that possesses the property common to all the particular instances from which the number 3 is generalised. Because we take natural numbers for granted we tend to make assumptions about how they work without thinking through the details. For instance we all know that 1 + 1 = 2 – but does it? Consider a raindrop running down a windowpane. Another raindrop can run down to meet it and the separate raindrops will merge into a single raindrop. One raindrop, plus another raindrop, results in one raindrop: 1 + 1 = 1. The correct response to this challenge to common sense is to say “but that’s not what I mean by 1 + 1 = 2” and going on to explain why this particular example doesn’t qualify. This, however, raises the question of what exactly we do mean when we say that 1 + 1 = 2. To properly specify what we mean, and rule out examples like putting 1 rabbit plus another rabbit in a box and (eventually) ending up with more than 2 rabbits, is rather harder than you might think, and requires a lot of pedantry. I’ll quote the always lucid Bertrand Russell** to explain exactly what we mean when we say that 1 + 1 = 2:
Omitting some niceties, the proposition ‘1 + 1 = 2’ can be interpreted as follows.
We shall say that φ is a unit property if it has the two following properties:
- there is an object a having the property φ;
- whatever property f may be, and whatever object x may be, if a has property f and x does not, then x does not have the property φ.
We shall say that χ is a dual property if there is an object c such that there is an object d such that:
- there is a property F belonging to c but not to d;
- c has the property χ and d has the property χ;
- whatever properties f and g may be, and whatever object x may be, if c has the property f and d has the property g and x has neither, then x does not have property χ.
We can now enunciate ‘1 + 1 = 2’ as follows: If φ and ψ are unit properties, and there is an object which has property φ but not the property ψ, then “φ or ψ” is a dual property.
It is a tribute to the giant intellects of school children that they grasp this great truth so readily.
We define ‘1’ as being the property of being a unit property and ‘2’ as being the property of being a dual property.
The point of this rigmarole is to show that ‘1 + 1 = 2’ can be enunciated without mention of either ‘1’ or ‘2’. The point may become clearer if we take an illustration. Suppose Mr A has one son and one daughter. It is required to prove that he has two children. We intend to state the premise and the conclusion in a way not involving the words ‘one’ or ‘two’.
We translate the above general statement by putting:
φx. = .x is a son of Mr A,
ψx. = .x is a daughter of Mr A.
Then there is an object having the property φ, namely Mr A junior; whatever x may be, if it has some property that Mr A junior does not have, it is not Mr A junior, and therefore not a son of Mr A senior. This is what we mean by saying that ‘being a son of Mr A’ is a unit property. Similarly ‘being a daughter of Mr A’ is a unit property. Now consider the property ‘being a son or daughter of Mr A’, which we will call χ. There are objects, the son and daughter, of which (1) the son has the property of being male, which the daughter has not; (2) the son has the property χ and the daughter has the property χ; (3) if x is an object which lacks some property possessed by the son and also some property possessed by the daughter, then x is not a son or a daughter of Mr A. It follows that χ is a dual property. In short … a man who has one son and one daughter has two children.
As you can see, once we try to be specific about exactly what we mean, even simple facts that we assume to be self-evident become mired in technical detail. For the most part people have a sufficiently solid intuitive grasp of concepts like “number” and “addition” that they can see that 1 + 1 = 2 without having to worry about the technical pedantry. The more abstractions we pile atop one another, however, the less intuition people have about the concepts involved and it becomes increasingly important to spell things out explicitly. In fact much of modern mathematics has reached the point where it is sufficiently far divorced from everyday experience that common intuition is counterproductive, and leads to false conclusions. If you think that sounds silly then consider that modern physics has also passed this threshold — few people can claim quantum mechanics to be intuitive. Our experience of the world is actually remarkably narrow, and thus so is our intuition.
At about this point I imagine some readers are wondering, if it is so easy to get lost and mired in technicality, why bother with all of these abstractions? I like to think of this process of increasing abstraction as akin to a road deep into the mountains. At times the road can look awfully steep. At other times, to avoid a sheer climb, the road is forced to take a tortuous winding path. As you progress deeper into the mountains, however, you will find places where the road opens out to present you with a glorious vista looking out over where you have come from. Each new view allows an ever broader view of the landscape, allowing you to see further and more clearly, while also seeing all the other different roads that all lead to this same peak. It is these unexpected moments, upon rounding a corner, of beauty, and clarity and insight, that, to me, make the study of mathematics worthwhile. I hope that, in this journey into the interior of mathematics, I can impart to you some of those moments of beauty and wonder.
* Note that in practice some expectations have been, to some extent, pre-wired into our brains, and this particular discussion is simply an illustrative example of the general concept of abstraction.
** From Essays in Analysis, Section V, Chapter 15: Is Mathematics Purely Linguistic, pages 301 and 302.