Functional programming has its roots in lambda calculus, a formal system developed in the 1930s to investigate computability, the Entscheidungs problem, function definition, function application and recursion. Many functional programming languages can be viewed as elaborations on the lambda calculus, where computation is treated as the evaluation of mathematical functions and avoids state and mutable data. In the other well known declarative programming paradigm, logic programming, relations are at the base of respective languages.

Functional design may seem like an odd constraint to work under. Why should you avoid objects and side effects? There are theoretical and practical advantages to the functional style:

• Formal provability
  For a long time researchers have been interested in finding ways to mathematically prove programs correct.
• Modularity
  A more practical benefit of functional programming is that it forces you to break apart your problem into small pieces. Programs are more modular as a result. It’s easier to specify and write a small function that does one thing than a large function that performs a complicated transformation. Small functions are also easier to read and to check for errors.
• Ease of debugging and testing
  Debugging is simplified because functions are generally small and clearly specified. When a program doesn’t work, each function is an interface point where you can check that the data are correct. You can look at the intermediate inputs and outputs to quickly isolate the function that’s responsible for a bug.
• Composability
  As you work on a functional-style program, you’ll write a number of functions with varying inputs and outputs. Some of these functions will be unavoidably specialized to a particular application, but others will be useful in a wide variety of programs.

One of the things that became clear in the responses was that many people who were debating whether a given language X was or was not a functional programming language didn’t have a good idea of what a functional programming language was. The situation was made worse, I think, by the fact that many of the key terms in the programming industry do not come with rigorous definitions and thus tend to shade into buzzwords at the edges. This is not the case with functional- there is a very rigorous definition of what is meant by "functional" and I’d like to introduce people to it.