Dear Luca, thanks for reading my essay and commenting on it in such a positively high note. Meantime I've read and rated your essay and I'll leave some comments too as to what I found interesting or problematic in it. As to your question posed in this paragraph that I quote
"A mystery in mathematics, that is puzzling me recently, is the richness of structure inherent in simple axioms. For instance from counting, we can get, addition, then multiplication, prime factor decomposition and cryptography. I really don't understand, where this richness is coming from. Was it already in the natural numbers?"
I can only say that if you read the classic monograph of Dedekind 'What are the numbers and what are they for?' it should certainly through a brighter light and enrich on your current concepts and ideas about the concept of number and set. Then you'll realise that even though he iteratively started from N to Z, Q and then defining R with his famous irrational cuts and then onto C he was not happy with the fundamental role taken for granted to N and tried to logically ground N on more abstract concepts of sets and mappings and which ultimately and jointly with Cantor and Cantor's more Platonistic approach led them to modern set theory that was further axiomatised successfully by Zermelo and Fraenkel but not only them. Obviously clarity and precision in the ZF axiom system and as you say richness of axioms and set theory concepts was something that was discovered after much work was done by Skolem, Russell, Von Neumann, Weyl, Hilbert, Ackerman, Tarski and not lastly by Godel and Turing to which our theme refers, and this is only before the 40s.
As to the role that intuition and imagination play in maths or physics I think that these is such a rich subject that it definitely deserves and essay contest of its own in the future, not to mention intuitionistic logic and its connection with multi-valed logic or Brower's famous ban on the principle of excluded middle and reductio ad absurdum method of proof in mathematics. Best regards!