From Gödel -> Faith

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Gӧdel’s incompleteness theorem says any sufficiently complicated mathematical system is either incomplete or inconsistent. There’s a lot of complicated math here that I don’t totally understand, this basically means that in any logical system, there are logical statements that you can make that are true, that cannot be proven to be true from within the system.

“1 + 1 = 2” is a statement you can make using basic arithmetic. And it’s true according to the rules of basic arithmetic. Because we assume that it’s meaningful to talk about things like numbers, and addition, and equality when we are doing arithmetic. And we define 2 as being equal to 1 + 1. But you can never really prove that 2 = 1 + 1 using only arithmetic.

If we were arithmetical beings that could only communicate using mathematical statements, I could prove to you that 3 + 2 = 5 by saying:

“2 = 1 + 1

4 = 1 + 1 + 1 + 1

4 = 1 + 1 + 1 + 1 = 2 + 1 + 1 = 2 + 2

4 = 2 + 2”

If you were suspicious of my claim that 2 + 2 = 4, but agreed with me that 2 = 1 + 1 and that 4 = 1 + 1 + 1 + 1, then you would be forced by my argument to accept that 4 does, in fact, = 2 + 2.

But suppose you objected to my claim that 2 = 1 + 1. How could I answer you? I can’t, since I can only speak to you using arithmetic, and part of what it means to do arithmetic is to assume that 2 = 1 + 1. If we were non-arithmetical beings (imagine we were humans instead) I could try to convince you using words. I could say “Listen up punk, the fact that 2 = 1 + 1 is the foundation of arithmetic, so it’s something you just have to accept if you’re doing arithmetic.” But this is not an argument you can make to an arithmetical being, because words aren’t available to beings living in the arithmetical world. The fact that 2 = 1 + 1 forms the basis of the arithmetical world, and so cannot be proved from within it.

So it is with the world we live in. There are facts about the universe that are true, that cannot be rationally proven by beings like us that exist within that universe. Some of the biggest ones I can think of are:

  • Morality exists; i.e., there are certain things that people should do, and certain things people should not do

  • Cause and effect exist

  • The people around me are conscious beings just like I am

  • The natural world is governed by mathematical principles

You might disagree with these statements. And as much as I love to argue and debate about them, there is no way I can rationally prove to you that my views are the correct ones. Because the tools I’d need to prove them aren’t available to rational beings living in the natural world. I believe these facts form the basis of the world we live in, and cannot be proved.

So how can someone reasonably come to believe in facts that cannot be proven rationally? This is a really important question, because these “unprovable truths” form the basis of the world we live in. I would say you should believe what seems the most reasonable to you. For some people that will mean religious belief, for some it will mean a strict commitment to scientific inquiry, and for some it will mean skepticism. And even though some of these people are right and some of them are wrong, it’s not possible to make rational arguments proving that this is so.

And for some people maybe none of this matters. Maybe not all our beliefs don’t need to be rationally built up from reasonable assumptions about what the basic facts of the universe are. But that’s a topic for another time.