How do I memorize axioms of a Hilbert system? I'm presented with a Hilbert system with just one inference rule (MP) and these axiom schemes:
$$A \supset (B \supset A)$$
$$(A \supset (B \supset C)) \supset ((A \supset B) \supset (A \supset C))$$
$$A \supset (B \supset A \wedge B)$$
$$A \wedge B \supset A$$
$$A \wedge B \supset B$$
$$(A \supset C) \supset ((B \supset C) \supset (A \vee B \supset C))$$
$$A \supset A \vee B$$
$$B \supset A \vee B$$
$$(A \supset B) \supset ((A \supset \neg B) \supset \neg A)$$
$$\neg \neg A \supset A$$
$$\textbf{F} \supset A$$
$$A \supset \textbf{T}$$
How am I supposed to memorize all of them? In other words, why these particular schemes? I think I've seen examples of every one of them to be necessary for various formal proofs, but I'm curious how the authors have come up with this.

To memorize them you might first rewrite all expressions in Polish notation.  This means that any expression (x@y) where @ indicates any binary operation becomes @xy.  ⊃ becomes a "C", ∧ a "K", ¬ a "N", and ∨ an "A".  Then denote all variables as lower case letters of your choosing (so long as each instance of a lower case letter gets replaced uniformly by one other lower case letter.  So, (A⊃(B⊃A)) becomes CaCba, but of course we need more than that.  Since you now have letters, you can make words out of the formulas, or silly/ridiculous/absurd sentences out of the formulas as a mnemonic.  For example, for CaCba you could use the mnemonic "Carrying Automobiles Causes Bad Arthritis."  Or you could write CeCae with the mnemonic "Crates Eat Clams And Eels".  The more ridiculous, absurd, outrageous, improbable, silly, etc. the words/sentences you come up with, the better.  Anything which invokes the senses will also help.  Or at least that's my experience and what the books on memory improvement I've read say.
Other than that I'd recommend seeing Carl Mummert's answer. 
