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 , and here we’ll start by bravely assuming that they exist. One result that follows from the existence of : Given a real number belonging to , we can find a natural number (e.g. 1, 2, 3 …) such that . This is known as the Archimedean property. To visualize this relationship, we draw an arrow from the existence of 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 to Theorem if the proof of relied on 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 , 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 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.