Limits (essay)

essay prompt: Is the idea of 'limit' indispensable? Neutral? A hindrance?

I contend that some concept of 'limit' is indispensable in mathematics, physics, and beyond. I will make this case inductively, beginning with an exposition of the facts of reality that gave rise to the need for such a concept, and concluding with a proper definition of 'limit'.

[TODO maybe also talk about invalid versions of the concept?]

Fact of reality 1:

The Earth is a sphere, but its radius is so large, and the circumstances of everyday life take place on a scale so small, that in most contexts, it is valid to think of the Earth as a plane.

We may say, informally, that a plane is the limit of a sphere as its radius "tends towards infinity". [TODO one has to be careful about which limit one is taking, TODO harry doesn't like the word infinity, but I don't know how else to say it]

We may also say that in the context set by many everyday situations, the difference between a plane, and a sphere with 4000mi radius, is nill.

One might ask: why would I ever treat the Earth as a plane when it is in fact a sphere? What's the point of passing to the limit? The point is that it helps the crow. Here are some hypothetical examples:

• You might find it useful to know that the angles of a triangle add up to 180˚. However, this fact is only valid for triangles on a plane; "triangles" on a sphere can have angles that add up to more than 180˚. If you are thinking of your back yard as a sphere, then you have to go through an extra step, where you first think: the size of this triangle is very small relative to the sphere, and therefore its angles add up to something very close to 180˚. This is true, but it is extra cognitive work, which can be avoided by thinking of your back yard as a plane.
• You might find it useful to know that the two parallel lines you drew in the ground will not intersect. However, this fact is only valid for parallel lines on a plane; "parallel lines" on a sphere will eventually intersect. If you are thinking of the ground as a sphere, then you will have to go through an extra step, where you first think: the point at which the curvature of the earth will cause these line to intersect lies many miles out into the Pacific Ocean, and so it's completely irrelevant for my goal of making a pretty design (or whatever the goal is). Again, that is true, but it is extra cognitive work, which can be avoided by thinking of the ground as a plane.

Fact of reality 2:

[TODO write neatly] The sum from j = 0 to N of 2^{-j} is tabulated below for various values of N

  N   sum

----|-----

  0   1

  1   1.5

  2   1.75

  3   1.875

  4   1.9375

  5   1.96875

  6   1.984375

    ...

  30  1.999999999068677425384521484375

It appears that the larger N gets, the closer this sum gets to 2.

One might ask: How do we really know that if we continue for long enough, we'll get increasingly closer to 2? All we've seen is that we get very close to 2 after we sum the first 30 numbers. But maybe something unexpected will happen if we sum up to 3000, 300, or even 31.

Such a question can be answered as follows. First, note that

  (1 + x + x^2 + ... + x^N)(1 - x) = 1 + x + x^2 + ... + x^N

                                       -(x + x^2 + ... + x^N) - x^{N+1}

                                   = 1 - x^{N+1}.

Therefore, rearranging, we find

  1 + x + x^2 + ... + x^N = (1 - x^{N+1})/(1 - x)

or for x = 1/2,

  1 + 1/2 + 1/2^2 + ... + 1/2^N = 2 - 2^{-N}.

Thus we have precisely quantified the difference between (the sum up to N) and 2; it is 2^{-N}. The larger our value of N gets, the smaller this difference will become.

In fact, for any nill value epsilon, there exists some N such that the difference between (the sum up to N) and 2 is nill.

--

One might ask: Who CARES about passing to the limit? What do we gain from talking about 2 instead of 1.984375 ? The answer is that the former is simpler than the latter.

I will introduce the principle of Occam's razor,[TODO]

Real example?

Why might you actually have to do this sum? Well let's say you're a contractor for a new airport, and they picked a design for the marble tiles that looks like a spiral: [TODO show picture]. Each tile has an area of 1 square meter. How much marble do you need per tile?

One way of thinking about it is that each tile is made up of a sequence of subtitles. The first subtile will be made of 1/2 of a square meter of marble, the second 1/4th of a square meter marble, the third 1/8th of a square meter of marble, and so on. You could add those quantities up for each subtile, and find that you need

$$1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256=0.99609375$$

square meters of marble, then round that up to 1 square meter of marble. Another, easier way of thinking about it, is to note that the pattern basically fills up the entire square meter (which is perceptually obvious in this case), so therefore you need 1 square meter of marble per tile.

Fact of reality 3: [TODO]

This one should be something having to do with a practical application of a derivative or an integral.

one idea is find the average height of a sine wave. This is contrived (when would I ever do it IRL?), and it's something you could figure out without calculus or limits.

starting over

I contend that some concept of limit is indispensable in mathematics, but that one must be very cautious in using standard mathematics' concept of limit.

Outline:

1. Explain what a limit is (on my view)
2. Explain why limits are useful, e.g. by giving examples of limits
3. Explain standard mathematics' definition of limit, and the problems that I think it has

What is a limit?

In this section, I will explain what a limit is. A limit can only be taken with respect to a sequence, so first, I must explain what a sequence is.

A sequence is a function from the natural numbers to a group $C$ of existents, where there is some notion of closeness or distance between the units of $C$. A sequence is like a set of instructions: you give me any natural number $n$, and I'll show you how to produce or identify some thing $a_{n}$, where $a_{n}$ is a $C$. Some examples:

1. numbers of a sequence converging to 2 [TODO]
2. circles [TODO]
3. secant lines [TODO]
4. solutions of a differential equation? [TODO]

Sometimes, but not always, sequences have the property that the elements of the sequence get closer and closer to a fixed point $L$ as the sequence progresses. In such cases, the sequence is said to converge to $L$, and $L$ is said to be the limit of the sequence. Examples

• TODO same as above

I will conclude with a concept-definition of a limit. The limit of a sequence is the place it goes to once we are well beyond the level of precision that is relevant in our given context.

Why are limits useful?

In this section, I will give some fictional but realistic scenarios in which limits could be used to great effect. [TODO]

1. converging to 2

example with the square

2. circles

earth is like a plane locally

3. tangent lines

how fast is the particle moving when it hits the ground?

What are the problems with limits?

[TODO] in a sense, the problem comes in when you have to make a formal definition. I don't know if it's actually a problem with the formal definition, or if it's just that formal definitions encourage you to drop context.