Kurt Gdel Proof Mathematical Rigor

“Alicia, does our relationship warrant long-term commitment? I need some kind of proof, some kind of verifiable, emprical data”

“I’m sorry, just give me a moment to redefine my girlish notions of romance.”

– from A Beautiful Mind

A paradox is something that appears to contradict itself.

Liar’s Paradox: This statement is false. So is it true?

Thomson’s lamp, Russell’s paradox, are more mathematical paradoxes among many others.

If the liar’s paradox is too vague for you, you may want to look into Russell’s paradox.

Of course people thought about this. And it bugged (pure) mathematicians especially, because mathematicians tend to be anal about ‘rigor’ and ‘proof.’

So a little more than a hundred years ago, a mathematician named David Hilbert decided it was about time people started answering questions like this. You know, what it meant for something to be ‘true’ and what statements can be true or false. It was a really big thing-in fact, some people consider Hilbert to be the most influential mathematician of the twentieth century (you’ve heard of Einstein but you don’t know Hilbert? Physics is just applied math you know.)

Unfortunately, answering whether something really is true or not is beyond the realm of mathematics. It’s become more of a philosopher thing. What Hilbert and his army of mathematicians decided to do is make ‘math games’ like typographical number theory (TNT), and other ‘formal systems.’ Formal systems may be a subject for another article, but you might just want to read Gödel, Escher, Bach if you don’t think you can wait. But you can think of a formal system as a black box that when given axioms and rules of inference, spits out ‘truth’ and proved things.

So in modern mathematics, when mathematicians say that he/she has ‘a proof’, we usually mean we have an outline of how we could build formal systems to spit out what we want. But this does seem nice right? We have stuff that can just spit out proofs? What if we could make a formal system that spit out all mathematical truths? We wouldn’t need mathematicians anymore! Well unfortunately (or maybe fortunately for mathematicians who want to keep their jobs) Kurt Gödel proved that this was impossible at least for many different types of maths including number theory circa 1931.

I’m not going to address Zeno’s ‘paradox’ and the like. These kinds of paradoxes point out aspects of mathematics that seem counterintuitive, but the more interesting aspects are really more philosophical than mathematical.