Who is this aimed at? Recreating Euclid? Then, yes, I think there are too many implicitly defined terms. Post-Euclid, who is helped by rigor? What advances are driven by rigor? The only one that springs to my mind is Gauss's correct statement of the fundamental theorem of algebra.
I'm also not convinced that Euclid's rigor mattered, but I don't know how to assess this.
Post-Euclid, who is helped by rigor? What advances are driven by rigor? T
The primary advantages of rigor are that it helps prevents one from going down blind alleys or getting hopelessly confused. Thus for example in the 17th century much ink was spilled over what 1 -1 +1 -1 +1 -1... summed to. They didn't have any rigorous notion of what the sum of an infinite series meant. And once this was answered the question essentially disappeared. Similarly, there were all sorts of apparent paradoxes and issues in calculus that were cleared up with delta-epsilon limit notions in the mid 19th century. And one can tell similar tales about early topology.
So apparently Richard Feynman once said:
I could be missing something, but this strikes me as a terrible answer.
When was the atomic hypothesis confirmed? If I recall correctly, it was only when chemists started noticing that the outputs of chemical reactions tended to factorize a certain way, which is to say that it took millennia after Democritus to get the point where the atomic hypothesis started making clearly relevant experimental predictions.
How about, "Stop trying to sound wise and come up with theories that make precise predictions about things you can measure in numbers."
I noticed this on Marginal Revolution, so I shall also state my candidate for the one most important sentence about macroeconomics: "You can't eat gold, so figure out how the heck money is relevant to making countries actually produce more or less food." This is a pretty large advance on how kings used to think before economics. I mean, Scott Sumner is usually pretty savvy (so is Richard Feynman btw) but his instruction to try to understand money is likely to fall on deaf ears, if it's just that one sentence. Think about money? Everyone wants more money! Yay, money! Let's build more gold mines! And "In the short run, governments are not households"? Really, Prof. Cowen, that's what you'd pass on to the next generation as they climb up from the radioactive soil?
*Cough.* Okay, I'm done. Does anyone want to take their own shot at doing better than Feynman did for their own discipline?