You can, yes. However "programming" covers a lot more than sticking a few lines of code together. A degree in computer science will stand you in far better stead than self-teaching PHP, for example.
or:
http://see.stanford.edu/
covers it better than the vast majority of UK computer science courses too...
(obviously a whole computer science degree contains more than just coding - but as far as coding is concerned the above is good - most UK comp sci departments don't even have the balls to use C++ in undergrad courses these days anyway.)