TITLE: Are there are any inherent mathematical reasons some proofs are difficult?
QUESTION [26 upvotes]: This is not a complaint about my proofs course being difficult, or how I can learn to prove things better, as all other questions of this flavour on Google seem to be. I am asking in a purely technical sense (but still only with regards to mathematics; that's why I deemed this question most appropriate to this Stack Exchange).
To elaborate: it seems to me that if you have a few (mathematical) assumptions and there is a logical conclusion which can be made from those assumptions, that conclusion shouldn't be too hard to draw. It literally follows from the assumptions! However, this clearly isn't the case (for a lot of proofs, at least). The Poincaré Conjecture took just short of a century to prove. I haven't read the proof itself, but it being ~320 pages long doesn't really suggest easiness. And there are countless better examples of difficulty. In 1993, Wiles announced the final proof of Fermat's Last Theorem, which was originally stated by Fermat in 1637 and was "considered inaccessible to prove by contemporaneous mathematicians" (Wikipedia article on the proof).
So clearly, in many cases, mathematicians have to bend over backwards to prove certain logical conclusions. What is the reason for this? Is it humans' lack of intelligence? Lack of creativity? There is the field of automated theorem proving which I tried to seek some insight from, but it looks like the algorithms produced from this field are subpar when compared to humans, and even these algorithms are obscenely difficult to implement. So seemingly certain proofs are actually inherently difficult. So I plead again - why is this?
(EDIT) To rephrase my question: are there any inherent mathematical reasons that contribute to explaining why proofs can be incredibly difficult?

REPLY [3 votes]: I will try to complement the discussion. I think the question is also linked to this other: why is so hard to choose good axioms?
Of course every theorem, even Poincaré Conjecture, is trivial if we insert it as an axiom in a system (we can always do that for a true formula). We could be less extreme and reduce to axioms only the difficult lemmas used to proof something. Then the theorem would follow easily. Take for example the classical triad of equivalent theorems in set theory: the Well-ordering Theorem that states that every set can be ordered; the Zorn lemma and the axiom of choice. There is a joke that says:

The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

It is complex (at least for me) to proof the Zorn lemma from the axiom of choice. But the well-ordering principle follows easily from the Zorn lemma. See the discussion on the above link.
Reverse mathematics is the fields that do the following: give me theorems and I will find which axiom is needed. What seems interesting to me is that this approach allows someone more philosophically inclined to be more precise about things he want to avoid.
But let say our philosophy is something like a proof pragmatism: given a set of theorems, is there a optimal axiom system for it? With optimal we mean not only the size of axiom, but perhaps also time of computation consumed (or both of them). Well, if we can assume that the set of axiom and the set of rules operating it can be implemented in a computer, this question turn to be linked to computational complexity. In special, there is an analogous of the Gödel Incompleteness theorem in this context, called Chaitin Theorem.
There is also one thing worth mentioning I think: given the truth value of some propositions, to verify if some formula of proposition logic is true is a piece of cake, just substitute the truth values in it. Given a formula, to find the truth values that turn the formula true: that is more difficult, in some cases the computers we have today can do it, and in the worst case will take a lot of time (not viable for our computers). This is the SAT-problem. But this problem will be still decidable by any abstract computer (a computer that don't have problems of time): do the truth table. Now the problem seems to start when we have a logic with predicates and quantifiers.  It is not decidable by any abstract computer to know whether a formula in predicate logic is satisfatible.