What’s the Most Important Theorem?

Mathematical truths are organized in an incredibly structured manner. We start with the basic properties of the natural numbers, called axioms, and slowly, painfully work our way up, reaching the real numbers, the joys of calculus, and far, far beyond. To prove new theorems, we make use of old theorems, creating a network of interconnected results—a mathematical house of cards.

So what’s the big picture view of this web of theorems? Here, we take a first look at a part of the `Theorem Network’, and uncover surprising facts about the ones that are important.  This is blatantly fun for us. Really.

Let’s go through an example starting with the real numbers.  Mathematicians like to write these numbers as $\mathbb{R}$, and here we’ll start by bravely assuming that they exist. One result that follows from the existence of $\mathbb{R}$: Given a real number $a$ belonging to $\mathbb{R}$, we can find a natural number $n$ (e.g. 1, 2, 3 …) such that $n>a$. This is known as the Archimedean property.  To visualize this relationship, we draw an arrow from the existence of $\mathbb{R}$ to the Archimedean property:

Now, the fact that real numbers satisfy the Archimedean property tells us something about sets that contain them. For example, more than a century ago, two guys named Heine and Borel used the Archimedean property to help prove their glorious, eponymous theorem.  We’ll now add an arrow leading from the Archimedean Property to the Heine Borel theorem, and we’ll include the one other component Heine and Borel needed:

All right: who is this De Morgan and what are his laws?  Back in the mid 1800’s, Augustus De Morgan dropped this bit of logical wizardry on the masses: “the negation of a conjunction is the disjunction of the negation.” We know, really exciting words.  If it’s not true that both A and B are true, then this is the same as saying either A or B or both are not true.  Better?

Before diving into a larger network, let’s think some more about these links.  One could prove the Fundamental Theorem of Calculus (which sounds important but could be just good branding) with nothing more than the axioms of ZFC set theory. But such a proof would be so long and tedious that any hope of conveying a clear understanding  would be lost.  Imagine taking all the atoms that make up a duck and trying to stick them together to create a duck; this would be the worst Lego kit ever.  And so in any mathematical analysis textbook, the theorems contain small stories of logic that are meaningful to mathematicians, and theorems that are connected are neither too close or too far apart.

For this post, what we’ve done is to take all of the theorems contained in the third edition of Walter Rudin’s Principles of Mathematical Analysis, and displayed them as nodes in a network. As for our simple networks above, directed edges are drawn from Theorem $A$ to Theorem $B$ if the proof of $B$ relied on $A$ explicitly. Here’s the full network:

Node size weighted by total incoming degree, colored by chapter, and laid out by Gephi’s Force Atlas.

We find that Lebesgue theory (capstoned by Lebesgue Dominated Convergence) lives on the fringe, not nearly as tied up with the properties of the real numbers as the Riemann-Stieltjes integral or the integration of differential forms. Visually, it appears that the integration of differential forms and functions of several variables rely the most on prior results. Over on the right, we’ve got things going on with sequences and series, where the well-known Cauchy Convergence criterion is labeled. By sizing the nodes proportional to their outgoing degree (i.e., the number of theorems they lead to), we observe that the basic properties of $\mathbb{R}$, of sets, and of topology (purple) lie at the core.

By considering the difference between outgoing and incoming degrees, we can find the most fundamental result (highest differential in outgoing and incoming degree, or net outgoing degree), and the most important or “end of the road” result (highest differential in incoming and outgoing degrees, or net incoming degree).  In Rudin’s text, the most fundamental result is De Morgan’s Laws, and the most important result is Multivariate Change of Variables in Integration Theorem (MCVIT, that’s a mouthful).

So the Fundamental Theorem of Calculus falls short of the mark with a net incoming degree 19, not even half of MCVIT’s net incoming degree of 45. And it is not the axioms of the real numbers that are the most fundamental, with the Existence of $\mathbb{R}$ having a net outgoing degree of 94, but instead the properties of sets shown by De Morgan with a whopping net outgoing degree of 122. Larry Page’s PageRank (the original algorithm behind Google) and Jon Kleinberg’s HITS algorithm also both rate the MCVIT as the most important result.

Would you agree that MCVIT is the most “important” result in Rudin’s text? It could just be the most technical.  We have only used a few lenses through which one might choose to evaluate the importance of theorems, so let us know what you think, or give it a try. Here’s a link to the Gephi files, containing all of the data used here.

Lastly, the network itself can be built differently by changing which theorems are included, or which are used in proofs. The resulting structures combine historical development with the author’s understanding. The goal of new textbooks is, in part, to organize the results in the most understandable fashion. With this view, we can start to think of the Theorem Network as the natural structuring of complex mathematical ideas for the human mind.

Now, one might idly think of extending this analysis to all of human knowledge. In that direction, Griff over at Griff’s Graphs has been making some very nice pictures leveraging the work of all those who edit Wikipedia.

Filed under mathematics, networks

10 responses to “What’s the Most Important Theorem?”

1. anton maier

Hi i’ve got an idea that won’t let me alone. As short as possible: Imagine you take all of “principles of mathematical analysis” put it into a latex-like format and rearrange the book structure, a tree, into a logical graph like you did here but still with code attached to it. Put this Graph-Text into git. Put all your research into this graph instead of papers. Put all your excercises for teaching into this graph. Think of it as another layer of the overall graph. (logically its just one hypergraph)
There is much to say about this idea, but it’s basically integrating linux-development method into science. Linux solved a huge problem in Software development, that made it the fastest and biggest software project of human kind.
I’d like to start this project, but i would start from excercise piece of the graph and adding more and more theory over time.

• Anton,
I think I get what you’re saying. Using github, you could allow research papers to be put themselves into a network like this. There was some work done on the git-linux data that’s worth a look too: http://arxiv.org/abs/0807.0014
Making this graph into an interactive teaching tool would be awesome, and maybe git could be the platform.

• Anton,
Is this what you’re thinking of?
https://svn.kwarc.info/repos/swim/doc/phd/phd.pdf

• Anton Maier

I’m definetly gonna read the dissertation. But on first sight his proposal is way too high level. low-level solutions work so much better and git is the center of a low-level solution….

2. Another approach to identifying important theorems is called Reverse Mathematics (http://en.wikipedia.org/wiki/Reverse_mathematics). The basic idea is that if you take (for example) the theorems in Hardy & Wright, certain subcollections can serve as axioms for proving all of the others. Consider now the set of all minimal collections that serve well as axioms. Certain theorems are in very many of these minimal collections; for example quadratic reciprocity. Quadratic reciprocity as an axiom? It makes everything else easier!

3. Andy

I think you are presenting the Archimedean property in a manner that is manifestly false: take a to be any negative real number, and b any positive real number, and you certainly can not find any natural number that makes n*a>b true. Why not just say that Archimedean property says, no matter which real number you choose, you can always find a bigger natural number.

• Very good Andy, thanks for the suggestion!

4. Proofs are not a feature of mathematics, they are human artifacts we use to explain mathematics to each other. One can turn any edifice of proofs upside down by swapping axioms for suitable theorems, Thus there are no “fundamental” or “importnat” results in mathematics per se, just as the words “top”, “bottom” or “central” have no meaning for the Earth’s surface. Those are features of our maps of reality, not of reality itself.

• john q

you’re a drag.