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'.

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.

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.

The focus of this essay is mainly negative. I criticize two common attitudes towards limits: the "empiricist" attitude, which says that we don't need limits at all, and the "rationalist" attitude (the attitude of modern mathematics towards limits), which says that we can use limits without any thought as to how they connect to reality. What is needed is a proper, objective attitude towards limits. However, offering a positive theory of limits is a very difficult task, which I will mostly not attempt to do in this essay.

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 produce or identify some thing $a_{n}$, where $a_{n}$ is a $C$. If the sequence consists of fractions with an inverse power of 10 in the denominator, then another way of thinking of the sequence is as instructions for continuing the "..." in a decimal expansion; the expression "3.14159..." can be thought of as shorthand for a sequence. Some examples of sequences:

1. "$a_{1},a_{2},a_{3},\cdots$", where $a_{n}:=\sum _{i=1}^{n}2^{-i}$.
2. "$b_{1},b_{2},b_{3},\cdots$", where $b_{n}$ is a circle of radius $n$, centered at $(x,y)=(n,0)$.
3. "$c_{1},c_{2},c_{3},\cdots$", where $c_{n}$ is a secant line to the graph of $f(t)=10-10t^{2}$ near $t=1$. More formally, $c_{n}$ is the unique line passing through the points $\left(1+1/n,f(1+1/n)\right)$ and $\left(1-1/n,f(1-1/n)\right)$.
4. "$d_{1},d_{2},d_{3},\cdots$", where $d_{n}:=\log n$.
5. "$e_{1},e_{2},e_{3},\cdots$", where $e_{n}:=\sum _{k=0}^{n}{\frac {(2k+1)!}{2^{3k+1}(k!)^{2}}}$.

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:

1. The numbers "$a_{1},a_{2},a_{3},\cdots$" get closer and closer to 1, as $n$ gets larger and larger.
2. The circles "$b_{1},b_{2},b_{3},\cdots$" gets closer and closer to being the y-axis (from the perspective of someone sitting at the origin), as $n$ gets larger and larger.
3. The lines "$c_{1},c_{2},c_{3},\cdots$" get closer and closer to being the tangent line to the graph of $f(t)$ at $t=1$, as $n$ gets larger and larger.
4. The numbers "$d_{1},d_{2},d_{3},\cdots$" get larger and larger as $n$ gets larger and larger. The sequence does not converge; it does not have a limit.
5. The sequence "$e_{1},e_{2},e_{3},\cdots$" converges to ${\sqrt {2}}$. That is, as $n$ gets larger and larger, $e_{n}^{2}$ gets closer and closer to 2.

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.[TODO delete?]

Although the reader should keep in mind that the concept of limit extends far beyond the realm of numbers, for simplicity I will henceforth only consider sequences of numbers.[TODO orly?]

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 1

Suppose that the following design is proposed for some of the granite floor tiles at a new airport terminal: [TODO]

That is, each tile consists of many sub-tiles, forming the pattern above. Each big tile is 1 square meter, so the sub-tiles have sizes 0.5 square meters, 0.25 square meters, etc. How much granite is needed to make each big tile?

The answer to that question is perceptually obvious, given the picture of the design: about 1 square meter of granite is needed for each tile. But if what if we had to justify this to a blind person, or a person who hasn't seen the picture?

The empiricist way of proceeding would be to say: The big tile contains sub-tiles of sizes 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, and 1/256 square meters, and if you perform a long arithmetic computation, you will find that

$${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}+{\frac {1}{256}}=0.99609375,$$

which is close enough to 1 in the context at hand. Who cares about limits?

The rationalist way of proceeding would be to say: The big tile contains sub-tiles of sizes 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, and 1/256 square meters. The sequence of partial sums of these numbers is approximately the sequence $a_{n}$ (example 1 above). The limit of that sequence is 1. Therefore,

$${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}+{\frac {1}{256}}\approx 1.$$

One advantage of the rationalist approach is that it saves a lot of time and effort (especially in the days where people didn't have calculators). Another advantage of the rationalist approach is that it is more conceptual, and seems to give us a better idea about what's really going on: the sub-tiles form a pattern which, if continued in the natural way, would make their total area arbitrarily close to 1. In other words, the rationalist approach gives us an explanation for why the area of the sub-tiles is so close to 1.

The problem with the rationalist approach is that, unlike the empiricist approach, its connection to reality is not obvious. What do we actually mean by saying that "[t]he sequence of partial sums of these numbers is approximately the sequence $a_{n}$ "? And in our conclusion that 1/2 + ... + 1/256 ≈ 1, what do we mean by "≈", i.e. exactly how close can we conclude that 1/2 + ... + 1/256 is to 1?

To see a concrete way in which the post-Newton approach can go awry, let's consider the sequence $a_{1}',a_{2}',a_{3}',\cdots$, where

$$a_{n}':={\begin{cases}\sum _{i=0}^{n}2^{-i},&{\text{if }}n\leq 1000\\42,&{\text{otherwise}}\end{cases}}.$$

One might accidentally subsume 1/2 + ... + 1/256 under the sequence $\{a_{n}'\}$ instead of the sequence $\{a_{n}\}$, because in real life we're only going up to $n=8$, and these two sequences agree as long as $n\leq 1000$. The limit of the sequence $\{a_{n}'\}$ is 42, so if one proceeded as before by taking the limit, he would erroneously conclude that

$${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}+{\frac {1}{256}}\approx 42.$$

[TODO move? delete?] In conclusion, the concept of

A more objective approach is to dig into the proof that the limit of $\{a_{n}\}$ is 1. By basic algebra, one may verify that the following identity holds

$$x+x^{2}+\cdots +x^{n}={\frac {1}{1-x}}-1-{\frac {x^{n+1}}{1-x}}$$

for any number $x$ such that $x\neq 1$. [TODO finish?] For $x=2$, this yields

$${\frac {1}{2}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{2^{n}}}=1-{\frac {1}{2^{n}}},$$

so as $n$ gets larger and larger, this quantity gets closer and closer to 1. [TODO yuck]

2. total distance travelled

A car is parked on on a parabola-shaped road. At time 0 it changes gear to neutral so that it can roll freely back and forth. [TODO picture] By looking at a watch and the car's velocimeter, it is observed that the relationship between the time and the velocity of the car is described by a function $v(t)=A\sin(Bt)$, for some constants $A$ and $B$. Where is the car after $\pi /B$ seconds have passed?

The answer is given by finding the area under the curve $v(t)$, between $t=0$ and $t=\pi /B$.

If the car were moving at a constant velocity $V$ for some time $T$, it would have travelled a distance $VT$ in that time.

Rationalist approach:

• extrapolate that the true velocity of the car at time t is given by some constant, times the cosine of t.
• find the area under the curve by taking the limit of successive approximations (you don't actually have to do it, you can just use general theorems)
• now we have a function for the change in position

Empiricist approach:

• draw boxes
• add up the area of the boxes

possible advantages of rationalist approach:

• it's more accurate
• it connects to this other concept, 2
• it's computationally easier

2. tangent plane

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.

3. tangent lines

A typewriter[?TODO] is dropped from a height of 10 meters, and we wish to know if it will break when it hits the ground. As an intermediate step, we must determine how fast it will be moving when it hits the ground.

The height of the typewriter[TODO], as a function of time, is $h(t)=10-10t^{2}$, and it hits the ground after $t=1$ second has passed.

To find the speed of the typewriter as it hits the ground, we could find the slope of the line $c_{n}$ for some large $n$. For example, the slope of $c_{12}$ is

5. square root of 2

blahblahblah

criticism

Well, why do we have to pass all the way to the limit? Can't we just pick a large n and say that that's good enough?

The reason why is that limits simplify things. If you're designing a fence for your back yard, it is inc

Rationalist approach

Mathematics gives no constraints on what sort of sequences we can make. We can consider any sequence we like, no matter where it comes from.

Empiricist approach

What I will dub the "empiricist approach" to limits is basically to say, we don't ever need to pass to the limit of a sequence, we can just work with some member of the sequence $a_{n}$, where $n$ is large enough.

For example, in the sequence $e_{1},e_{2},e_{3},\cdots$ above which converges to ${\sqrt {2}}$, an empiricist might say: for my purposes, $e_{4}=1.414$ [TODO] is good enough. That's what I mean by the square root of 2, and I don't need to think about all this extra nonsense ($e_{5},e_{6},e_{7},$ and so on) that tells me how to compute ${\sqrt {2}}$ up to arbitrary accuracy. In other words, in my present context, ${\sqrt {2}}$ is $1.414\pm 0.001$.

In many cases like the above, the empiricist is right. However, we will see that his approach misses something important.

---

OR

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For example, in the sequence $a_{1},a_{2},a_{3},\cdots$ above which converges to 1, an empiricist might say: for my purposes, $a_{4}=0.998$ [TODO] is good enough. I don't need to think of it as 1; 1 and 0.998 are indistinguishable at up to the level of accuracy to which I'm working. In cases like the above, the empiricist is not really right. The reason why is the following general principle (some version of Occam's Razor), which I will propose:

If something can be validly identified in multiple ways, then one should identify it in the way which requires the least cognitive work.

The reason for this principle is simply that it is bad to do unnecessary cognitive work. In our example, "1" is a concept that I'm very familiar with; there lots of things in my mind that fall under the file folder "1." On the other hand, "0.998," besides being less compact and more like a noun phrase than a concept, has much less content under its file folder; I am hard-pressed to think of any examples of 0.998 of something, there are fewer properties of 0.998 that I can think of, etc. Therefore, the principle implies that if I can validly identify something as either 1 or 0.998, then I should identify it as 1. [TODO ugh]