Proof, Progress, and Königsberg

Why Leonhard Euler will remain a more important contributor to mathematics than Open AI

This post presents a discussion of how the mathematical community develops understanding using Euler’s solution of the seven bridges of Königsberg as a guide. We also contrast that with the effect of Open AI’s recent research dump.

Introduction

According to the Guinness Book of World Records, Leonhard Euler remains the most prolific mathematician of all time. His 886 books and papers have been surpassed by Paul Erdös but the number of pages he wrote still ranks first.

More recently, Open AI placed 772 research looking papers in a Github repo, each one purportedly proving some well known mathematical conjecture of interest. I gotta say that I’ve looked through the papers and there are certainly some interesting items in there. Perhaps, one or two of those might make for another post, after I’ve got some time to digest them.

But that’s just the problem - digesting these things takes time and doing so is the important part. If AI produces the output of Leonhard Euler by volume every few weeks, we’ll never be able to understand it all. So, what’s the point?

Bill Thurston put it this way in his 1994 paper “On proof and progress in mathematics”:

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

Much more recently, Giordano Scarciotti posted a great video about this. In that video, he used Leonhard Euler’s solution of the seven bridges of Königsberg problem to explain why the concepts introduced in a mathematical paper can be much more important than the primary results of that paper.

I thought it might be fun to write about that problem and how this all fits with the AI assault on knowledge here in a way that my students can follow. That’s the point of this post.

The problem of the seven bridges

The mathematical history in this post draws from Euler and Königsberg’s Bridges: A historical view by Gerald Alexanderson and many of the basic facts are taken from Wikipedia. The map of Königsberg shown below, for example, can be found here and is used courtesy of a Creative Commons License.

Königsberg was a German city from the 13th century until the 19th. It became part of the Soviet Union after WWII and is now called Kaliningrad in Russia. The Pregolya River separates the city into four parts that were connected by seven bridges. Three of those bridges remain, two of which are from Euler’s time.

Even prior to Euler, a puzzle well known to the residents asked:

Can one walk through the city traversing each of the bridges exactly once?

Euler was asked this question by a prominent resident and amateur mathematician named Carl Ehler in 1735. Though initially uninterested, Euler ultimately answered the question in correspondence with Ehler in 1736. He published the result in the Proceedings of the St. Petersburg Academy later in 1741.

You can scroll to see a dynamically illustrated look at the problem:

import {make_map} from './components/make_map.js';
RasterMap = FileAttachment("./images/Plan_Der_Stadt_Koenigsberg-reduced.jpg").url();
svgDoc = FileAttachment("./images/bridges_and_paths.svg").xml("image/svg+xml");
koenigsbergSvgText = FileAttachment('./images/Konigsberg_graph.svg').text()

This map of Königsberg was made in 1763 during Euler’s lifetime.

The seven bridges appear mostly near the center of the city, though one is a bit further south than the others.

It’s not hard to cross five of the bridges exactly once, as shown here.

Once you do so, though, you’re left with an impossible choice to make on where to go next. Either way you go, you’re stuck at the next step.

A key step in Euler’s reasoning was to abstract the problem to its key elements using what we now call a network graph.

pic = make_map(RasterMap, svgDoc, koenigsbergSvgText);

supress = {
  pic.showStep(crTriggerIndex, crDirection)
}

Once he abstracted away the distracting details in the figure, Euler could identify the key feature in his network graph. He then asked the more general question:

What properties must a network graph have so that we can walk through the graph traversing each edge exactly once?

He realized that every time you traverse an edge into a point, you must also travel out from that point, unless it’s the first or last vertex. Thus, an even number of edges must meet at every vertex except possibly the first and last. A glance at the Königsberg graph shows that an odd number of edges meet at each of its four vertices, so the walk is impossible.

Generalization and understanding

Euler’s work did so much more than simply answer this one question.

• He explained it in a clear and simple way,
• He shared it with many others, and
• He generalized it to broaden its applicability.

More, this one paper laid the foundations for topology and graph theory - both of which are active research areas still today. This one paper is largely viewed as the seminal paper on both subjects. This is a perfect illustration of Thurston’s view that what we do should enable people to understand and think more clearly and effectively about mathematics.

Why not just ask a chatbot?

Imagine, if you will, that AI chatbots existed in the 1730s. Perhaps the Königsberg problem could’ve been answered something like so:

Is it possible to walk through Königsberg and cross each of the seven bridges exactly once?

No.

Uh, okay - thanks.

No problem! Would you like me to book a weekend stay at Zum Palmbaum and perhaps a meal at Blutgericht?

The point is that knowing the answer and understanding the solution are very different things. In mathematics, understanding is the real objective.

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Now, to be clear, the papers that were recently produced by Open AI do contain purported proofs. At this point almost a week after the drop though, nobody that I know of (not even an author) has claimed to have fully vetted one of the proofs. It remains to be seen what level of understanding these papers promote.

Just the sheer volume and the speed at which mathematical results are being produced is a major challenge. New mathematics has never been just pumped into a void; rather, there’s a community who reads it, vets it, and (potentially) uses it. Good mathematics is typically used by others. As Euler’s story shows, it’s not just the result that’s used but the ideas that produced that result.

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