Because computing science (see below), and even computer
programming, is applied mathematics.
Ed Dubinsky, a mathematics educator who was once a professional
programmer himself, has said:
A person's mathematical knowledge is her or his tendency to respond
to certain kinds of perceived problem situations by constructing,
reconstructing and organizing mental processes and objects to use in
dealing with the situations.
At a slightly less general level, consider what mathematics is. You choose or
create a language in which you can express certain ideas and then do
symbolic manipulation according to a set of rules you've also chosen
or created to come up with to create more valid statements in that
language according to those rules. If you're not careful to do this
correctly, you may come out with invalid statements. The results you
come up with may have some sort of application in the "real world"
(e.g., I can use the language and rules of "integers" to help keep
track of what people owe me and I owe them) or may be just work to
help you better understand how you can use the language and rules and
how they can be helpful to you in further use of them.
In many parts of mathematics we use particular symbols called
"numbers" and have a large library of oft-shared rules and languages
related to this, but there are other areas of mathematics that don't
use numbers at all (e.g., category theory), or, though they can be
applied to numbers, are not really about numbers per se (group
theory, algebraic structures, many more).
Even before you get into the study or use of particular algorithms and
the like, writing a computer program is basically what I described
above. Many of the "simplest" concepts in computer programming that we
use every day, such as the idea of a function, are purely mathematical
Now as you've seen, it's perfectly possible to attack real-world
problems with these mathematical tools in a non-rigorous way and get
useful results. Typically the results will not be truly correct (i.e.,
your programs will have bugs), but they will be "correct enough" to do
the job. (For a well written program in industry, you may never even
encounter the situations that would demonstrate that it's incorrect.)
That's what the discipline of engineering is: getting results that
work well enough in the real world at acceptable cost.
But even when you're doing engineering, much of what you do works well
only because someone has gone and done enough mathematical heavy
lifting to give you concepts and tools that you can use to do this.
You may not have a really good understanding of what a function or a
relation is, but your programming language or database system works
because somebody did figure those out.
And the people who did that work are the computing scientists.
All this has been known and seriously contemplated for a long time. I
think it's particularly well demonstrated by a comment in Peter
Landin's classic 1966 paper ["The Next 700 Programming
The most important contribution of LISP was not in list processing
or storage allocation or notation, but in the logical properties
lying behind the notation. here ISWIM makes little improvement
because, except for a few minor details, LISP left none to make.
There are two equivalant ways of stating these properties.
(a) LISP simplified the equivalence relations that determine the
extent to which pieces of a program can be interchanged without
affecting the outcome.
(b) LISP brought the class of entities that are denoted by
expressions a programmer can write nearer to those that arise in
models of physical systems and in mathematical and logical systems.
If you understand this (which probably requires some at least
intuitive understanding of the of the lambda calculus or similar), you
probably realize that a lot of the problems we deal with today are
still the same basically mathematical problems that were being
investigated back in the '60s when we were first seriously
investigating what a "programming language" really is and means.
On Working Programs
One can also look at this from the more narrow viewpoint of, "I just
want to write a program and make sure it works." Even here this
becomes math if you take as a constraint "I really do want to, as best
I can, make sure it works." Dijkstra's EWD303, "On the Reliability
of Programs," makes this argument in detail. His summary:
Reliability concerns force us to restrict ourselves to
intellectually manageable programs. This faces us with the questions
"But how do we manage complex structure intellectually? What mental
aids do we have, what patterns of thought are efficient? What are
the intrinsic limitations of the human mind that we had better
respect?" Without knowledge and experience, such questions would be
very hard to answer, but luckily enough, our culture harbours with a
tradition of centuries an intellectual discipline whose main purpose
it is to apply efficient structuring to otherwise intellectually
unmanageable complexity. This discipline is called "Mathematics". If
we take the existence of the impressive body of Mathematics as the
experimental evidence for the opinion that for the human mind the
mathematical method is, indeed, the most effective way to come to
grips with complexity, we have no choice any longer: we should
reshape our field of programming in such a way that their methods of
understanding become equally applicable, for there are no other
On "Computing Science" versus "Computer Science"
Some amongst us, including the University of Alberta, find the
more common name of the discipline slightly misleading and instead
prefer to call it Computing Science. As Keith Smillie said in
"Computing Science at the University of Alberta, 1957 -
The choice of the name "computing science" instead of the more
common "computer science" was deliberate in order to indicate that
computing rather than computers was to be the foundation of the
Thinking about what we are wrangling with as "computing" rather than
"computers" way may help you remember that all the software running
the world today is much more dependent on the mathematical tools we
use to be able to effectively and accurately model our problems and
the world than on the hardware on which it runs.