Simpson's paradox

But one of the examples in there of how their methodology ends up successful is when he's discussing the possible reasons lighting and thunder occur at different times.

One possibility thrown out is that they are actually occurring at different times. But another is that they occur at the same time but one takes longer to reach the viewer than the other.

On its own, these two ideas don't indicate the correct answer.

But then Lucretius ties the latter to another observation - that this seems similar to how a drummer in the distance can be seen to beat the drums before you would hear the drums.

Essentially in an age without the methodology of testable predictions, they circumvented that shortcoming by considering multiple hypotheses for multiple naturally occurring observations and looking for overlaps between them.

This seems to have pointed them in the correct direction on a number of major topics, especially relative to their contemporaries who were generally arguing for a particular hypothesis with various appeals to rhetoric or principle (like Aristotle claiming the leader of a bee hive couldn't be female because it had a stinger and "the gods don't give women weapons").

The times the Epicureans completely miss the mark is generally when they disregarded their principle of avoiding false negatives and discounted things with insufficient observational evidence (for example, they had pretty bad cosmology and they rejected the Stoic pre-gravity due to their incorrect base assumption of infinite amounts of matter). The times they kept an open mind and considered how concepts overlapped, even when they were wrong about the 'why' of an initial assumption they were often correct in secondary assumptions when tying it into multiple other systems and observations.

Your comment made it all fit together. May not be the most accurate (I dont know) but it helped me. Thank you.

If you have any written material I can access, from your courses, I'd be interested to read.

Thanks for this.

Since you didn't specify under what system we need to prove that 0=1 doesn't exist, I vaguely remembered or figured there was a simpler version of arithmetic under which that concept makes sense, but which wouldn't be strong enough to fall into incompleteness territory (so it would have to be weaker than Peano arithmetic, like you said).

So I just looked it up (thanks ChatGPT), and there's something called Presburger Arithmetic, to quote Wikipedia: (https://en.wikipedia.org/wiki/Presburger_arithmetic)

> The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The theory is computably axiomatizable; the axioms include a schema of induction.

So a very dumbed-down version of arithmetic, but which does contain a notion like 0=1, and which is complete and consistent, so it can't contain a proof of 0=1.

Obviously, this is probably not the kind of thing you meant, hence my cheekily bringing it up :)