I’m a newbie that studies applied maths and I have been learning about measure theory for the past few weeks and I came across things like Banach–Tarski paradox, Gödel’s incompleteness theorems, axiom of choice.. Which made me feel super uncomfortable and confused so my question is why is maths seem to be working if the system within which maths is built is seemingly “incomplete”? Is it luck?How is maths even reliable if it’s not always reliable?
P.S: Not sure if I’m asking in the right forum so I’d appreciate it if suggest me one in which I could find an answer.
The key to your answer lies in model theory.
Mathematicians are interested in studying abstract mathematical objects, that we call models. These mathematical objects are interesting to us because they describe the world, and also because they are fascinating on their own right.
But how can you learn more about a concrete model?
The first step is to pin in down – by describing true facts about it. Hence, the first step in any mathematical theory is to write down definitions and axioms that describe the model of interest in as much detail as possible.
Then the mathematical apparatus can be used to prove new facts about these models. My topology teacher used to describe this part as “churning the wheel”. These is when new theorems and proofs are made – and if they follow from the axioms, then we conclude that every model satisfying the axioms must satisfy these results.
With these in mind, thee kind of paradoxes can arise:
- A contradiction: there is no model satisfying the axioms we have set. Then the exercise is pointless – we will be able to prove anything we want, since we are describing something which does not exist. When mathematicians find a contradiction, like Russell’s paradox, they are forced to abandon their theory and rethink their axioms.
- An unintuive consequence: the axioms we have chosen describe models that are unintuitive to us – as with Banach Tarski and the axiom of choice.
- A loose specification: there are many models satisfying the axioms, some with unintuitive properties. Gödel’s work showed us that this situation is not only possible but common – no matter how you try to axiomatize a simple model like the arithmetic of the natural numbers you will never be able to completely pin it down. Your description of arithmetic will always include bizarre models that are not the natural numbers. Since all the consequences of the axioms must be true on all models that satisfy the axioms, this means that there will be some propositions about the natural numbers you will never be able to prove; because they are false on the impostor models that still satisfy your axioms but are not the theory of arithmetic. This is what Gödel Incompleteness means.
Understanding better the consequences of different axiomatizations and the models they describe it is at the core of mathematics. Paradoxes are not something to be afraid of, but to celebrate – each one we discover improves our understanding of mathematics, and by extension, of reality.