QuoteOriginally posted by: Traden4Alpha Proofs aren't just mental exercises (like they were in class), they are the delicious meals latent in a carefully selected set of ingredient axioms. The point is that one can hand-pick any axioms one wants to fit a real world application and then derive some very powerful conclusions about all systems for all times that are described by those axioms.A proof is a means or way to a theorem. Perhaps you were referring to theorems instead of proofs in the first sentence above. A powerful or insightful theorem is the produced (or cooked) delicious meal (or dish) that interestingly can be used again and again in the production of many other delicious meals.In the history of mathematics, once mathematicians began to realize that they are not bound by a rigid set of axioms in doing mathematical activity, the floodgates of mathematical outpouring and progress were cast wide open.

QuoteOriginally posted by: quantystQuoteOriginally posted by: Traden4Alpha Proofs aren't just mental exercises (like they were in class), they are the delicious meals latent in a carefully selected set of ingredient axioms. The point is that one can hand-pick any axioms one wants to fit a real world application and then derive some very powerful conclusions about all systems for all times that are described by those axioms.A proof is a means or way to a theorem. Perhaps you were referring to theorems instead of proofs in the first sentence above. A powerful or insightful theorem is the produced (or cooked) delicious meal (or dish) that interestingly can be used again and again in the production of many other delicious meals.Yes, the first sentence is analogically confused -- the theorem is the delicious meal and the proof is the recipe and cooking required to convert raw axioms into that meal.QuoteOriginally posted by: quantystIn the history of mathematics, once mathematicians began to realize that they are not bound by a rigid set of axioms in doing mathematical activity, the floodgates of mathematical outpouring and progress were cast wide open.Exactly! For example, the attempt to find a proof for Euclid's 5th postulate led to the invention of non-Euclidean geometry. Non-Euclidean geometry seemed like a bizarre counterintuitive abstraction until Einstein's work suggested that the universe was non-Euclidean. And if an "obvious" axiom can be wrong, then why not play games with the other axioms to see what pops out.

QuoteOriginally posted by: Traden4Alpha-- the theorem is the delicious meal and the proof is the recipe and cooking required to convert raw axioms into that meal.Surprise! I take exactly this argument to propagate the mathematical-cuisine. One cannot create an outstanding meal by combinatoric approaches, reasoning is required. Only very few chefs do this (intuitively). They are usually also capable of replicating this in cookbooks. More recent, Alan Ducasse, Gordon Ramsay, Heston Blumenthal, Ferran Adria (to the extreme) ...