The real world is finite in many ways. As a result, the tools that we use to approximate the world as infinitely divisible—real numbers, continuous surfaces, integrals, and so on—are useful, but sometimes not applicable.
Of course there are problems with puncturing a piece of paper infinitely many times at mathematically perfectly correct locations. With a real piece of paper and real tools it cannot be done. However, we can make a mathematically ideal model of your paper, in which case our infinite-world approximation tools apply.
In such a model, we can say that your paper is essentially like the unit square $S = [0,1]\times [0,1]$ in $\mathbb{R}^2$.
$S$ represents an infinitely thin, infinitely divisible two dimensional surface.
We can cut out the irrational points, resulting in the perforated piece of paper $P\equiv S \cap (\mathbb{Q}\times \mathbb{Q})$. (Here, $\mathbb{Q}$ denotes the set of rational numbers, which we want to retain, getting rid of the remaining irrational points.)
Excitingly, the tools of real analysis are exactly the ones we use to model the properties you care about:
- The area of $P$, as determined by an integral, tells you how "much" of the paper is there. You can use this as a measure of the mass of the paper remaining.
- You can take the ratio of the area of $P$ and the original area of $S$ to compute the "density" of the remaining paper in the space it originally occupied. This can tell you something about the "opacity" of $P$— an idealized measure of what fraction of uniformly-bright light falling on the original area $S$ will pass through $P$, assuming for simplicity that the paper is made of a material that transmits no light itself (the paper itself is perfectly opaque, so light only gets through empty spaces).
- This density constitutes a kind of transparency, if you like— not strictly because the medium becomes more translucent to light the way greased paper does, but rather because it is full of holes.
And hence we can answer all of these questions in a similar way. One of our most tools for measuring area in this context is the Lebesgue measure: it extends the Riemann integral studied in elementary calculus in such a way that you can compute the area of functions like:
$f(x)=\begin{cases}1&\text{if }x\text{ is irrational}\\0&\text{if }x\text{ is rational}\end{cases}$
Now for some notation. If $A$ is any set in $\mathbb{R}^n$, let ${1}_A$ denote the characteristic function $$1_A(x) = \begin{cases}1 & x \in A\\ 0 & x \notin A\end{cases}$$
So if $U=[a,b]$ is an interval on the real line, then $\int_{-\infty}^\infty 1_U(x)\,dx = \int_a^b 1 dx = b-a.$
The general rule is the integral of a set's characteristic function represents the size of the set.
As another example, the original "size" (area) of your paper $S$ is:
$$\iint_{\mathbb{R}^2} 1_{S} = \int_{0}^1\int_0^1 1 \,dx = 1$$
as you'd expect
Now we can figure out the "size" of your paper $P$. We want:
$$\iint_{\mathbb{R}^2} 1_{P} = \iint_{[0,1]^2 \cap (\mathbb{Q}\times\mathbb{Q})} 1$$
Without explaining how to compute this Lebesgue integral (which takes some developed machinery to show), it turns out that this integral is 0. As a result, a reasonable mathematically-ideal answer to your questions is as follows:
- The perforated paper has exactly zero total area.
- The fraction of paper you've removed is essentially all of it.
- There is technically some paper remaining ($P$ is not an empty set), but—in a way that cannot happen in real life—so little paper remains that it has no mass. In real analysis, this is referred to as a set of measure zero.
- As such, exactly zero percent of incoming light will reflect off the paper and into your eyes. In any reasonable sense, you will not be able to see anything because there is essentially no paper left to see.
- Similarly, light shining from behind the paper will essentially not run into anything, so the light will not be obstructed or diminished in any way. The paper will be perfectly "transparent".
- And the paper will be perfectly massless.
These should be taken to be surprising results about our models involving real numbers, rational numbers, irrational numbers, and integration, rather than claims about what happens with genuine perforated pieces of paper. Taken as mathematical results, they offer neat insights into the way we've formalized our ideas of number, divisibility, and size, and the ways in which those formalisms sometimes defy our geometric intuitions. (For another similar surprising result, see for example the Banach-Tarski paradox.)
Good question, and good luck on your studies of real analysis!
