Is graph theory a good model for Seven Bridges of Koenigsberg?

Question

I've asked this question on MathSE, but apparently people over there don't like philosophy.

Seven Bridges of Koenigsberg is the problem whose solution (by Euler) gave a rise to graph theory and (later) topology. The goal is to determine whether there exists a walk through the seven bridges in Koenigsberg crossing every bridge exactly once. Eveyrone knows it, as it is mentioned possibly in every univeristy mathematics course covering elements of graph theory.

Leonhard Euler came up with the following intuition: The physical map of the city, the exact location of the bridges, distances don't matter. All that matters mathematically is just the list of which regions are connected by bridges.

We could easily imagine a wrong mathematical model of this problem. Yet, it's commonly believed we can accurately describe the problem using graph theory and provide a solution within this theory, that correctly answers the original problem. How to ensure that a mathematical theory can describe a given problem correctly? What criterion to use here? Is it okay if nobody finds any flaw in the mathematical model within one week? One month, year, ten years, one hundred years, a thousand years? There might come a counterexample tomorrow, showing that this is NOT the right model of the problem. Intuition often deceives us.

What do mathematicians and philosophers could say about it?

Some may argue that it's unthinkable it may be wrong. Yes, it's a very easy, natural transition from informal problem to its mathematical description. But we can give examples of more difficult problems to model mathematically that aren't that obvious. I've taken a decision support systems course. We took problems expressed in natural language and tried to model them and find an algorithmic solution. How 'correct' were the models we've come up with? Up to which point in complexity of problems should we trust our thoughts, imagination and intuition?

Answer 1

Of course there are important things left out of the mathematical model. Maybe one of the islands is inhabited by bears who will certainly eat anyone who tries to enter. Maybe one of the bridges is incapable of supporting the weight of a human. Maybe there's an eighth bridge that somehow nobody's noticed before.

I have a great deal of trouble seeing what the issue is here. No matter what you choose to talk about, it's always possible you've overlooked something important. Once that thing is pointed out, you might or might not be perfectly happy to continue ignoring it. That's why we generally like to run our arguments past other people --- because sometimes they notice things we haven't noticed.

Your question seems to be: "How can we be 100% certain of never overlooking anything we might care about?". The answer is that we can't, and the counter-question is "Why ever would you have thought that we could?"

Edited to add: In the particular case of the Konigsberg bridges, I am sure that Euler would have been perfectly happy to go right on ignoring the bears or the collapsible bridge, because he was never terribly interested in traversing the islands in the first place --- instead he was interested in the mathematical problem the islands inspired, and that problem would have remained interesting with or without its "real-world" counterpart.

Answer 2

Let's make this simpler. Are numbers a good way to model amounts of discrete objects? This sentence has fifty four letters including spaces. But does it? Does it really? Isn't it possible we've made a mistake, and numbers are not a good model of anything?

Welllllll.

Technically, yes. Practically, no. Nihilism just isn't a very effective life strategy.

So, on to the bridges. The description of the problem, along with properties of the connectedness of space as experienced by us, prevents any divergence between the solutions given from graph theory and the solutions available in real life.

If you want to check, you ask questions like the following: what land can I reach without crossing a bridge? Does being in some spot in that land prevent me from reaching some bridges that touch that land? (No, barriers are not part of the problem.) Every point on the land is thus equivalent, so we can replace it with its equivalence set (a single point) for the purposes of this particular problem. Now, regarding bridges: if you start at one end of the bridge and go to the other, is there any possible path you can take such that you end up on a land unconnected to some other land you reached by crossing that same bridge? (No, each bridge end is fully contained within one bit of land.) So you can replace the bridge with a line between two bits of land (i.e. two points).

This sort of reasoning can be made more precise if one wishes, but really, this is not at all hard if done with care by people who know how to do such things. Thus, the chance of error is exceedingly, vanishingly small, to the point that you should no more worry about that than that you could count the number of 'o's in the body of this answer. (117, by the way.)

Answer 3

The loophole needed is that the original problem is phrased in a language. In our case, the language is English. English turns out to be a really poor language for discussing bridge crossings, so we turn to the language of math. We define a translation of the original problem (in English) into a new problem, defined purely in the language of mathematics. We then do the problem in mathematics, and translate the answer back into English.

Anyone is free to question whether the translation of the original problem fully captures the English version. Anyone is free to question the translation of the mathematical answer into English. In fact, this happens on a regular basis. Look up any "proofs" regarding omnipotence and omniscience and you'll see just how much people disagree on translations for these words.

As for the Koenigsberg bridge problem, I would argue that you could translate the problem into a topology problem (every point in the city has "neighbors"), then use math to perfectly translate that into the graph problem.