On Tuesday, September 3, 2013 9:17:00 PM UTC-4, Lite Beta wrote: > Is there a reason why most theorems in Euclidean geometry are IFF theorems?
Interesting question, though I am not quite sure about the "most" part. How would you quantify that and (assuming you can and the "most" is justified) how different is Euclidean Geometry from e.g. number theory?
One possible answer is that Euclidean Geometry is a mature branch of mathematics and that (over time) results tend to move towards IFF theorems. A non-specific example of what I mean by this is, suppose that at one stage in the development of geometry it was known that parallelograms have a certain property but later it was discovered that, more generally, quadrilaterals have that property. The second of the two results would have been what made its way into Euclid's Elements (assuming that it was known by then) but the second result (with a weaker premise) would have greater likelihood of having a true converse.