On Limits I

What on earth could one usefully say about limits, that hasn’t been printed for centuries on a thousand calculus textbooks?

I don’t know about usefulness, but one can certainly say something different. For all their numerosity, calculus texts are remarkably homogeneous in their treatment of this elementary topic, and I see at least one point that could stand clarification.

Limits — especially expressions involving “dot dot dot”, such as 0.999… — still seem to be a source of confusion for many beginning calculus students. Since all the popular treatises give thorough mathematical definitions of the concept, the problem must lie somewhere other than mathematics. I conjecture that what’s missing is not a more easily understandable definition, but instead a straightforward discussion of how definitions work in mathematics.

To get into definitions, I’m going to start by talking about symbols — in particular, symbols that represent numbers.

I will assume that numbers are something that can be coherently referred to, just as tables and chairs. This does not entail that numbers exist in the same sense as tables and chairs, since it is also possible to refer coherently to Santa Claus: everyone agrees that he wears red and owns a sled.

Nevertheless, the content of today’s post depends only on the weak assumption that people can talk about numbers — via decimal representations, or fingers raised up in one hand, for instance — and agree with each other. In fact, today’s post merely clarifies that limits are a way of referring to numbers. How they refer to numbers is a bit subtler than, say, how decimal representations refer to numbers; and in a way their mechanism is, I daresay, almost unique to mathematics. Which may explain the difficulty encountered by many new students: very few people, in their daily life, refer to objects in the subtle way that limits refer to numbers.

Let’s start with an example: what number does the following symbol stand for?

Yes, the number one. People use it to count how many heads the average human has, or the number of stars that the Earth orbits around. In general, if an object is of a certain kind, and there are no other objects of the same kind, we say: “there is one object of that kind”.

Compare how ‘1’ refers to the number one, and how ‘ॐ’ refers to the sound ‘om’. (I know it refers to a lot more, but focus on the sound right now, for argument’s sake.) In a certain sense, each symbol just sort of “stands for” something else, and provides a way for people to talk about that something else.

Now, what number do the following symbols stand for?

Yes, the number two. Notice how this is qualitatively different from the simple ‘1’ above. It is a composite symbol, so to speak: it’s made up of smaller symbols, namely a couple ‘1’s and the plus sign. I will call such composite symbols expressions.

One crucial point is that it is not at all obvious when a few pencil strokes count as a symbol, or as an expression. There is nothing in the expression itself to tell you how to read it; it’s a convention which you must be taught. Someone uninformed, like a small child or an alien, might view ‘1+1’ as a single symbol with a few separated pieces, just like ‘i’ is a single letter, even though the dot is separated from the rest. If you can’t read japanese, you might make the same mistake! The japanese alphabet, though staggeringly large at first sight, actually contains many symbols which are combinations of other symbols, often drawn a bit smaller and on top of each other. For instance, here is the saying “onna sannin yoreba kashimashii”, meaning “when three women gather, it is noisy”:

女 三人 寄れば 姦しい

The symbol for ‘noisy’ contains three miniatures of the symbol for ‘woman’. Can you spot them?

A peripheral remark: we might find it undesirable to leave any part of mathematics to pure convention, let alone a part as important as the meaning of formulas. We might wish to write formulas in such a way that it would be clear how to read them. However, some level of arbitrariness is unavoidable: even if expressions came with instructions for proper reading, you’d need further instructions on how to read those; and so on, ad infinitum. If we’re going to communicate, it has to stop somewhere, and at that point we’ll have to assume that everyone just gets it. (This is a point forcefully made by Wittgenstein in his Philosophical Investigations.)

All right. How do we find out what number ‘1+1’ stands for? We were taught in school that ‘1+1’ is made up of a ‘1’, a ‘+’, and another ‘1’; that each ‘1’ stands for a number, and ‘+’ stands for addition, which is a way to combine numbers into new numbers; and that ‘+’ combines two ‘1’s into a number that everyone calls ‘two’.

Compare how ‘1+1’ refers to the number two, and how ‘pineapple juice, coconut milk and rum, shaken with ice’ refers to the piña colada. They both sort of give us some starting ingredients and a way to mix them. The same goes on with more complicated expressions like

.

The distinguishing feature of this method of referring to numbers is the following: we start with an expression containing symbols for numbers and rules; follow the rules for a while; and end up with the number that the expression refers to. Each step of following the rules is basically replacing a rule symbol, and its associated numbers, with the “result”. For example:

(1 + 3 x 5) /2

(1 + 15) / 2

(1+15) / 2

16 / 2

16 / 2

8

There is a subtlety implicit already in this very simple way of referring to numbers. Suppose I want to tell you a number that I’m thinking of, and the way I do it is by giving you a sequence of steps to follow. I say, “the end result of these steps, that’s the number I’ve got in mind”. For this promise to be held, that is, for the sequence of steps to actually represent a number, it is absolutely essential that the steps can be carried out.

For instance, suppose I tell you “1/0”. This is shorthand for “take the number 1, which you know, and the number 0, which you also know, and divide the former by the latter; the result of that division is the number I want to talk about”. It’s obvious that I haven’t told you any number at all, since the division rule doesn’t handle a 0 in the denominator. There’s nothing mysterious about this: I promised you a number at the end of some calculations, but they can’t be done. That just means I’m a flake. It doesn’t say anything deep about mathematics.

OK. So far we’ve agreed to communicate numbers to each other in various ways, one of which is giving a sequence of steps to “build” a number. We’ve seen that sometimes one of the steps is meaningless, and therefore the expression stands for no number at all. Another way that communication can go wrong is if I give you a list of steps, each of which is fine on its own, but the list is infinitely long. It is also obvious that such a list can’t denote a number, since the very essence of naming numbers by sequences of steps is each party’s unspoken meaning: “the end result of these steps, that’s the number I’ve got in mind”. If there’s no end result, there’s no number being communicated.

Enter the bane of online math forums everywhere:

How is one to make sense of this? Worse yet, how can it ever equal 1?

One natural way to go, and many people do go this way, is to establish a connection with something already familiar. For example, the simpler expression ‘0.99’. Here is a perfectly well understood sequence of symbols: it stands for the number

.

We also understand ‘0.999’, ‘0.9999’ and ‘0.99999’; each means a sum of fractions like the above, only with more terms.

Great! We are rarely in such good shape when solving a problem. We usually have to scratch our heads to come up with just one helpful analogy, but here is endless supply of them. This gives us confidence to assert that the expression ‘0.999…’ stands for that number which is obtained by the infinite sum

And boom, just like that, everything has gone to hell. We have committed the cardinal sin of trying to express a number by a sequence of steps which does not end; which, therefore, does not express any number at all.

Yet mathematicians still seem able to talk about ‘0.999…’ and agree with each other, just as they do with simpler number representations like ‘1’. To understand how this is possible — ultimately, to understand what ‘0.999…’ actually means — we must come to grips with a different kind of representation, a different way of talking about things. It is more abstract than the two fairly direct ways described above (the straightforward symbol, like ‘1’, and the symbol-with-rules, like ‘1+1’). Nevertheless, it is well within the grasp of any human being, once it is brought to their attention. For some reason, however, mathematics textbooks never come out and say it, which I suspect to be the main source of difficulty.

(Continued)