Proofs and Refutations

Forewarning: This is one of those posts where I ramble about something unrelated to religion and then make a tenuous theological point at the end. Today’s topic: math…. no don’t go, it won’t be that bad.

I was a counselor on a summer camp. A friend of mine knew I was into math and puzzles and set me this one.

There are three houses, each needs to be connected to three utilities. Can you connect them in 2d without any pipes crossing?

I sketched a couple of attempts first and it looked unlikely, so I resorted to a strategy that served me well in Math Olympiad type problems: try to prove it impossible. I remember my line of reasoning.

First I figured that the solution didn’t depend on where the utilities or houses were. Any solution would be essentially the same solution if you dragged the end points around, or stretched or bent the pipes. In math terms, this is topology – you can move stuff about as long as you do so with continuous deformations. Locations or pipes could be moved and stretched and reshaped, but could never jump.

Then I started from a simple problem – the 2 house 2 utility problem, and I played around until I saw there was only one solution to that. Any other routing would be stretchable back to that one solution. (I didn’t, if I remember rightly, prove this properly, I did it mostly by intuition, but I was right — from this point on however, my reasoning does form the outline of a valid proof)

The solution to the 2×2 problem divides space into two regions. It is impossible to get from one region to another without crossing a pipe. So when we go to the 3×2 problem, the next house has to be in one region or the other. And whichever region it is in, the pipes connecting it to the two utilities must be in the same region. So the result is the same in either case. All solutions to the 3×2 problem are topologically identical to this.

Which splits space into three regions. But notice that each region only borders two houses.

Whichever region the next utility is placed in, it will not have access to exactly one house. If it is in the red region, it won’t have access to the red house, the green house is inaccessible from the green region, and the blue house from the blue region.

Therefore, the 3×3 problem is impossible. As long as my intuition about the 2×2 problem was correct, I had a watertight proof that the problem couldn’t be solved.

And I said so. “It’s impossible, I can prove it.”

“No it isn’t”, said my friend, and drew the solution.

All utilities are connected to all houses, no pipes cross any other pipe. I was wrong.

I was wrong in a very important way. I had invented constraints that weren’t there. I had set myself a different problem to the real one.

I had proved my artificial problem was impossible, but who cares if the artificial problem you invented turns out to be impossible? Who cares if you have enough advanced math to construct a proof of it? (I’ve since figured out several other ways to prove the same result, it has become a favorite brain game of mine to find new ways to prove this irrelevancy).

Five years later, after describing this to my PhD supervisor, he pointed me at “Proofs and Refutations“, an excellent book on the philosophy of math by Imre Lakatos. Which discusses this phenomenon in detail. I enjoyed learning this lesson, and it has stayed with me ever since.

… and hence to the theological point …

I’ve been thinking about the “Argument between Science and Religion” recently, and reading around it. Everything I read seems to be on either the science side (which is usually but not exclusively anti-religion) or the religion side (which is almost always in favor of there being no conflict), and everything I read is unsatisfactory. And today I realized why. Both sides have manufactured their own (slightly different) imaginary problems. This usually happens in the prologue or first chapter. The authors then spend hundreds of pages arguing very powerfully and convincingly for why they are right. And pretty much, they persuade me. But even if they could be said to prove they were right, it doesn’t change the fact they’ve essentially invented an imaginary question. One that is, as far as I can see, utterly irrelevant. Nobody seems to want to step up to the really hard challenges on the other side.

As to what they are, in my opinion, I’ll leave for another post. But feel free to pre-empt me with your thoughts…

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3 Comments

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3 responses to “Proofs and Refutations

  1. I’ve not thought much about “Science vs. Religion” issue because I don’t think about either in general terms since both of the terms are so vague. When working with vague abstract terms, problems are inevitable and each sets up the words to match their agendas.

    But when someone makes an empirical claim, I expect them to be open to empirical criteria for criticism. When someone says there claim can not be tested, I get that — just as long as they don’t later add an empirical claim attached to it.

    Your story was fantastic. I find that many problems of theology and philosophy are set up incorrectly because of assumptions of language. I’m curious how you will use the above example to address apparent conflicts between science and religion.

  2. exrelayman

    I have a reaction to the ‘solution’. There is an assumption contrary to the normal situation in it. Since there are only 2 dimensions, the only way the deeper green utility reaches the second and third houses without crossing the other pipes is by going through the interior of houses. Now how many utilities connected to house A run through the inside of another house B? My take is, your initial assumptions were correct, and the ‘solution’ is a cheat. Maybe that’s just me.

    Now where you move to theological arguments and assumptions, boy that’s a whole ‘nother ball game. All those philosophical arguments with no real world referent get very hairy and abstract, and thus to me lose any power of convincing (even when I myself lack the mental power to combat the argument – it took a long time and better thinkers than me to refute Zeno’s false paradox).

    I will be enjoying what is coming next on this topic.

  3. Ian

    ex, I’d encourage you to keep thinking in those terms. Clearly you’re right. A flatlander wouldn’t stand for a utility running through their house, as it would permanently divide the house in two. But then flatlanders who had more than one utility delivered would never be able to go anywhere without passing through their neighbors houses. So maybe the underlying problem – the one that the abstract problem is merely an abstraction of, is not about flatland.

    It could be that there is only a tiny band of depths that the utilities can run in, because, say, they need to run between a sealant layer and the foundation layer, and the area they can run in is too thin to allow pies to cross in 3D. In that case, having a pipe run right under a property is no problem.

    And so on… there are justifications either way.

    But the only thing we have to go on is the question as asked. And that was deliberately asked in terms that allowed the answer the questioner had in mind. So although I initially reacted as you did, and said the ‘solution’ was a cheat. I’m now pretty certain I was wrong. The purpose of the question was to test my lateral thinking, and I failed.

    @sabio – yup, the question of empirical claims is the heart of my frustration…

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