Can tactics be reduced to a finite number of simple patterns that can be learned? I realise that there are a finite number of tactical techniques such as forks and skewers and pins but I find them hard to identify in everyday play. I have started using chess.com's tactics trainer and am gradually making progress. However I think that this is ultimately a pattern recognition problem. 

So my question is whether it is possible to reduce all of the tactical tricks to a finite set of graphical images that I could stare at and memorise in the same way children learn their letters. I think that this may get my brain to identify these opportunities more quickly when they arise on a chess board.
I would say that, indeed, the vast majority of tactics falls into some kind of pattern and there are a number of them but not too many.  

In my opinion there are two issues that you need to improve in order to improve tactics in your games.  First, you need to know the basic patterns, that is you have to know how they look like and what is the main idea in exploiting them.  Often the basic ideas of the pattern cannot be applied out of the box and you will need to find the right moves to actually implement the combination.  The second issue is that, of course, you have to spot these patterns on the board and exploit them.

For picking up the patterns a tactics book that presents problems grouped by patterns, together with some explanations how to exploit them, is very useful.  E.g. "Tune Your Chess Tactics Antenna".  To facilitate the second step, such books often have mixed exercises at the end, where you are not told what the pattern is you should find, but then almost any tactics book with mixed problems will be okay.