Can we trust logic

For some time I have been thinking about writing a series of articles on the problems with the scientific approach. I believe in the scientific approach, and I think it’s correct, but there are problems with it and only with a proper understanding of them, can we hope to not be bitten by those problems.

This is the first of those articles, and is about the problems with reasoning within a formal system. It’s a bit of an abstract topic, so I will attempt to explain it, and what is wrong with it.

Consider the case of one of the most famous scientific moments of all time; Newton, sitting under a tree. An apple is supposed to have fallen from the tree, and after hitting poor Newton on the head, inspired him with the theory of gravity.

Before long, Newton came up with a theory, that the force exerted on an object by another object is related to the mass of both, and the
distance between them. He proposed an equation to express what the force is. As soon as you introduce an equation, you have to deal with maths. At this point we have two vitally important questions. How can we be sure that the equation successfully predicts the behaviour of gravity, and how can we be sure that the maths itself is going to give the correct answer?

I want to make it clear the difference between the two. The first question is “Do we have the right idea as to how gravity works – are we using the right equation?”. The second question is “Does maths work?”, does a multiplying two numbers actually give the number that it should.

You may well be wondering at this point, am I asking if multiplying two numbers together gives the answer to multiplying two numbers, or if adding up 1+1=2. That is exactly what I am asking. Of course, there are proofs for this. If you ask any mathematician to prove 1+1=2 they can, (or at least can look it up, it’s rather messy apparently).

But proving 1+1=2 does not prove all of mathematics works, nor would showing that multiplying two numbers works. Before we can be sure of using maths to get the right answer, we need to know that every single part of maths is valid, that it works, and here is the crux – you cannot do this using mathematics. Maths alone is not enough to prove that maths works.

However, maths can be shown to work, using logic. Using the rules of logic, all the basic, atomic bits of maths can be proven, and have been. Problem solved.
Well no, not quite. How do we know that logic works? Logic is made up of statements like “if something is true, then something else is false” and so on, but is that really the case? It might seen obvious to us that it is, but that is because the English language has been built up assuming its true, and our brains and grown up thinking that way. It turns out, we cannot prove it. Can we use logic to demonstrate its true? No, for exactly the same reasons we cannot use maths to prove that maths is true. We need some greater system than logic to prove that logic is a valid way of thinking.

Can you spot the problem yet? Let’s assume we have a way of thinking called Super Logic and this new way of thinking clearly shows that Logic is OK to use. Then we have to ask how we can be sure about Super Logic, and so on, and on, and on. As one great man once said “its elephants all the way down”.

Now it turns out, we do not have Super Logic. We stop at logic. All of our reasoning is based on the assumption that it’s OK to think using logic, but we have no real way of backing that up.

We believe in logic, and we believe in it without any proof. We accept logic as an act of faith.

Now it’s not a complete leap of faith, we have built a hell of a lot of “stuff” based on this belief in logic. Ship, bridges, planes, cars, and all of them work in a way that can be explained using logic. So we have a hell of a lot of evidence that it’s OK to do this, but we still cannot be sure, we can never be sure.

Note: I came accross this article in the amazing book Gödel, Escher, Bach, and if you want to know more about this subject, or just enjoyed reading it, then this book is a must.

Technorati Tags: , , ,

By |September 21st, 2006|General|0 Comments