Do you mean the example of 1 not being a prime, or the example in the article of Z[root -5] not having unique factorization? You could certainly discover the fo

Do you mean the example of 1 not being a prime, or the example in the article of Z[root -5] not having unique factorization? You could certainly discover the former, but not the latter as far as I can see. The idea is that the principle should guide you on what the 'right' set of definitions are, so that your theory detects interesting theorems and isn't riddled with edge cases. You might make an analogy to programming, finding good primitive data types and interfaces so that you can implement elegant and efficient algorithms and well structured modules. Perhaps the key insight of the 'too simple to be simple' principle is that we want the definition to impose both existence & uniqueness of something. In this case proper divisors (divisors strictly less than the number itself e.g. the proper divisors of 6 are 1,2, & 3). ------------------------------- Let me give a concrete example of how it might be used. - Uniqueness is captured algebraically here by the notion of a 'subsingleton' set: a subset where any pair of elements are equal. This can happen if (1) the subset has only a single element, or (2) the subset has NO elements (in which case the uniqueness requirement is vacuously true). - Existence is captured by the notion of a 'singleton' set: a subset with exactly 1 element: any two elements belonging to the set are unique, AND such an element exists. First lets apply this to the definition of the regular primes: the set of proper divisors of 1 is the empty set. The set of proper divisors of any other prime is the singleton set {1}. Thus the naive definition says 'proper divisors of primes form a subsingleton': it asserts they are unique, but not that they exist. The more sophisticated definition of the primes (which excludes 1) asserts the proper divisors of primes form a singleton: uniqueness and existence. Now lets try to apply it (somewhat informally) to the fictional example of even numbers and 'even primes'. Examples: 2 is an even prime, 4 = 2*2 not an even prime 6 is an even prime, 8 = 4*2 is not, 10 is even prime, 30 is an even prime is as well, and so on. Here I have uniqueness of proper divisors (they form a subsingleton), but only because existence of proper divisors fails for every even prime (not just 2)! Now what happens if we try and factor 60? 60 = 2*30 60 = 6*10 It not great surprise that uniqueness of prime factorization fails as well (one of several problems with this informal example). The principle didn't help me find this example, it suggests that if I use the above notion of an 'even prime', i'm not going to get a good set of theorems.