Ok, say I'm an ancient greek and want to develop a new field of mathematics: irrational numbers. How would I apply the scientific method to 'start off' this new field ?
From what I recall, the Greek fella who identified the existence of irrational numbers (Hippasus of Metapontum, I had to look it up on Wikipedia) was killed by the other Pythagorean mathematicians for it.
Wikpedia: Irrational Numbers
They go into some detail about it including giving Hipassus' proof.