A surprisingly common sentiment among some programmers is that “computer science is math.” Certainly, computer science as a rigorous discipline emerged from mathematics. Now, we consider such foundational work to be theoretical computer science. For example, Alonzo Church’s lambda calculus and Alan Turing’s Turing machine provided a theoretical foundation for computation. At the time, the two self-identified as mathematicians, and were clearly doing mathematics. So if the foundations of computer science are math, how is it that computer science as a whole is not math?

Simply, computer science has grown well beyond its purely theoretical roots. We invented real computers, which are not theoretical devices. In doing so, we had to deal with the complicated and messy reality of designing, implementing, using and programming computers. Those areas of study are also computer science. My operating definition of computer science is: everything to do with computation, both in the abstract and in the implementation.

The relationship I am claiming:

Camping Buddies

Much like physics, we have two camps: theory and experimentation. However, the relationship between the two camps is not the same as it is in physics. In physics, experimentalists often have the job of testing the theories produced by the theoreticians. If the experimentalists are ahead of the theoreticians, then theoreticians must develop new theories to explain results discovered by the experimentalists that are inconsistent with current understanding.

The “I explain your results” and “I test your theories” relationship does not exist in computer science. Our version of experimentalists are generally called systems researchers. When a theoretical computer scientist proves that matrix multiplication is O(n^2.3727), a systems researcher is never going to produce any results that disagree with that theoretical result. The theoreticians have discovered a mathematical fact—and yes, I use that word deliberately.

What systems people may do is provide evidence that while such a result is theoretically interesting, real systems may never take advantage of it. We (yes, I include myself in this group) do so by designing and implementing novel systems, from which we learn what is feasible and useful.

Computer science theoreticians and systems researchers do not always work in isolation. I am good enough in math and theory to know when I am not good enough in math and theory. I have worked on a project where in order to solve an interesting systems problem, I needed a sophisticated model that was beyond my ability to discover. In response, the theoreticians I worked with had to quiz me to understand what kind of information we could reliably measure from our system. In order for them to build a model, they had to know what kind of reliable information our system could provide. All of us were doing “computer science,” despite performing very different tasks.

Naming Names

I am in the systems camp. I have (at least) an intuition for the whole system stack, from knowing what kind of code a compiler is likely to emit for particular language semantics, to how the operating system will behave under that workload, and what the processor itself must do to execute it. My research almost always has messy empirical results. Broadly, I am interested in improving the performance of software, which means lots of experiments, lots of results and lots of interpretation. That process is not math.

But there are people who are not only theoreticians or only systems researchers. I used a broad brush when painting the divide between theory and systems. It does not capture the entirety of the field; many people work in both theory and systems, and there are probably people who feel that the two categories don’t capture what they do. Which, of course, is my point: computer science is a large discipline that goes far beyond the parts that we all agree is math.

There are plenty of computer scientists who straddle the divide. I think this is particularly common in programming languages. Results in programming language research may be theoretical. The same researchers who are able to prove something about, say, a type system, are often the same people to design a language and implement a compiler that embodies the theoretical result. In short, the divide between theory and systems research is not as clean as it is in physics.

Example the second: consider networking. The algorithms that govern how individual TCP connections avoid congestion is certainly computer science. There is also a large amount of mathematical reasoning that goes into designing and understanding how individual connections governed by these algorithms will behave. But, in the end, what matters is how they work in practice. These algorithms are the result of design, experimentation, interpretation of results and iterating. (And iterating.)

When someone simply says “computer science is math,” they are doing a disservice to all of these other fields in the discipline that are clearly not just math. Of course, we use mathematical reasoning whenever we can, but so does all of science and engineering. Math is the common language across all empirical disciplines, but they do not all tell the same story.

Aside from programming languages and networking, the field of computer science also includes operating systems, databases, artificial intelligence, file and storage systems, processor design, graphics, scheduling, distributed and parallel systems—more than I can exhaustively list, but luckily, someone else has. All of these areas use math to a varying degree, and some even have highly theoretical sub-fields. To the point, even, that I would agree that the theoretical basis for some of those areas is arguably math. For example, relational algebra is math, but it’s also the theoretical foundations of relational databases. But if we make the blanket statement “databases is math,” we miss all of the implementation and design on the systems side that allows actual databases to exist in our world.

SCIENCE!

It’s impossible to discuss the nature of computer science without recognizing the elephant in the room: is it science? I won’t discuss that—not out of lack of interest, but because others have done a better job than I could. Cristina Videira Lopes covered the topic in an excellent essay, where I also learned about Stefan Hanenberg’s paper on a similar topic. Everything I have to say on the subject is derivative of their points.

Best Intentions

Those who claim that “computer science is math” generally have good intentions. They are usually responding to the notion that computer science is just programming, which is, of course, false. Anyone who has taught beginning programmers knows how difficult it is to convey to them that underneath all of the accidental complexities lies something fundamental.

But it is still a gross simplification to call the entire discipline of computer science “math.” Related to math, foundations in math—sure. But after a while, it makes sense to group the theoretical foundations of computation along with the design and implementation itself. That grouping is computer science.