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A Mathematical Theory of the Antilibrary<br>Garden statusPage typeEssay<br>Growth stageGrowing<br>Version0.1<br>Aug 06, 202610 min read<br>books<br>reading<br>mathematics<br>epistemology

A Mathematical Theory of the Antilibrary

I have too many books. This became very clear last year, when I moved.

During a long train ride, I also attempted to catalogue all the technical books I had read. I wanted to classify each one as “useful,” “evergreen,” or “garbage,” and keep track of how much I had actually benefited from it. Then, suddenly, the words of one of my favourite authors, Nassim Nicholas Taleb, came to mind: the antilibrary.

Taleb’s idea is that unread books are not useless possessions; they are a valuable reminder of everything we do not yet know. The story goes something like this, though I am unsure whether it is entirely true:

Umberto Eco famously kept a large personal library, and visitors tended to fall into two groups. Some admired how many books he had read. The more interesting visitors understood that the unread books mattered too: they mapped the edges of what he did not know.

I have always been fascinated by this concept, but none of my friends who gazed at my library seemed interested in listening to me explain it, let alone trying to formalize it mathematically.. multiple times.<br>So here I am. Hopefully, someone will find it interesting.

(I swear I am fun at parties. Sometimes.)

1) The binary model

Suppose a library contains N books. Give each book i a binary reading state

ri​={1,0,​if I have read it,if I have not.​<br>Then the usual split is simple:

L={i:ri​=1},A={i:ri​=0}.<br>The realized library is L and the antilibrary is A. Their shares are

ℓ=N∣L∣​,a=N∣A∣​=1−ℓ.<br>However, we all know that reading is not binary. I may know a reference book well without ever reading it from cover to cover. I may have technically finished a novel and remember nothing beyond the colour of its cover.<br>The binary model throws all of that into two buckets and calls the unread bucket profound.

2) The h-index temptation

What if we take inspiration from the h-index?

The usual h-index is the largest h for which a researcher has at least h papers with at least h citations each. It combines two quantities, the number of papers and the number of citations, by forcing them to meet on the diagonal.

We can do the same thing with books. Give each book a comprehension score ci​∈[0,1] and define the survival curve

F(p)=N1​i=1∑N​1{ci​≥p}.<br>F(p) is the fraction of books understood to at least level p. The Library Index is

HL​=sup{p∈[0,1]:F(p)≥p}.<br>If c(1)​≥c(2)​≥⋯≥c(N)​ are the scores in descending order, the same quantity is

HL​=1≤k≤Nmax​min(c(k)​,Nk​).<br>So HL​=0.72 means that at least 72% of the books are understood to at least the 72% level. This is almost a perfect analogue of the h-index.

It is also a brutal compression. A single number discards nearly everything interesting about the shape of the collection. The two-dimensional version makes that loss visible.

Can we use two variables?

A pair (b,p) naturally asks a yes-or-no question: are at least a fraction b of the books understood to at least level p?

We could encode the answer as the binary function

G(b,p)={1,0,​if F(p)≥b,otherwise.​<br>Equivalently,

G(b,p)=1{N1​#{i:ci​≥p}≥b}.<br>For example, G(0.7,0.8)=1 would mean that at least 70% of the books are understood to at least 80%. If G(0.9,0.8)=0, fewer than 90% reach that threshold.

This is mathematically clean, but still not very expressive: almost all the interesting information sits on the boundary between 0 and 1.

The feasible region

My preferred object is therefore not the binary output but the set of claims for which the output is true:

S={(b,p)∈[0,1]2:b≤F(p)}.<br>Every point (b,p)∈S represents the statement

At least 100b% of the library is understood to at least 100p%.

For the illustrative collection below:

Fraction bThreshold pFeasible?0.60.7✓0.80.8✗0.10.9✓

The shaded region contains every true claim. The staircase b=F(p) is its frontier: points to its left are feasible, while points to its right ask the library to guarantee more books than it can at that comprehension threshold. This looks a lot like a Pareto frontier. The interior is valid, but the boundary tells us the strongest claims the collection can support.

Reading the same boundary backwards

The inverse view starts with a desired fraction of the library and asks for the highest comprehension threshold it can guarantee:

T(b)=sup{p∈[0,1]:F(p)≥b}.<br>Then T(0.5)=0.82 means that half of the library is understood to at least 82%. It is the same boundary read in the other direction: F accepts a threshold and returns a fraction, while T accepts a fraction and returns a...