Functions in programming

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

Here's a question for those who know about translating between category theory
for mathematicians and category theory for computer programmers.

In class today I was discussing functions with domain the empty set.  The
students don't have much background in formal set theory (and none in category
theory though I'm doing my best to sneak it in where I can) so they were
trying to get to grips with the idea that the _are_ functions from the empty
set, but just not very many of them.

Afterwards, one student asked about how this related to functions as used in
computer programming.  It seemed from what he said that he had some
understanding of the formal relationship between functions in mathematics and
functions in computer programs - beyond them having the same name.  He said
that a function that takes no input is known as a "constant function" and so
wasn't sure how to fit the two notions together.

I, on the other hand, am at the level of "Ooo, look!  Mathematicians and
computer programmers both use the word 'function'.  So do biologists and event
organisers.  Maybe we should organise a function whose function would be to
investigate all these different uses.' so I didn't know what answer to give.

The best that I could think of was that program functions have a 'hidden'
input: the fact that they have been called.  So a function defined on the
empty set corresponds to a function that can never be called.

Can anyone help me straighten this out?

Extra kudos for answers that I can just pass on to the student!

Thanks,

Andrew Stacey