### Programming, maths and the other things

Sarah Mei argues that programming is not math, arguing instead that programming is language. I don’t think it’s hard to see the truth in the first part, though due to geopolitical influences on my personality I’d make the incrementally longer statement that programming is not maths.

## But there’s maths in programming

Let’s agree to leave aside the situations in which we use programming to solve mathematics problems, such as geometry or financial modelling. These are situations in which the maths is intrinsic to the *problem domain*, and worrying about the amount of maths involved could potentially confuse two different sources of maths.

Nonetheless, one may argue that the computer is simulating a mathematical structure, and that therefore you need to understand the mathematical model of the structure in order to get the computer to do the correct thing. I can model the computer’s behaviour using the lambda calculus, and I’ve got a mathematically-rich model. I can model the computer’s behaviour as a sequence of operations applied to an infinite paper tape, and I’ve got a different model. These two models can be interchanged, even if they surface different aspects of the real situation being modelled.

It’s the second of the two models that leads to the conclusion that capability at advanced maths is not *intrinsic* to success at programming. If what the computer’s doing can be understood in terms of relatively simple operations like changing the position of a tape head and tallying numbers, then you could in principle not only *understand* a program but even *replicate it yourself* without a deep knowledge of mathematics. Indeed that principle provides the foundation to one argument on the nature of software as intellectual property: a computer program is nothing more than a sequence of instructions that could be followed by someone with a pencil and paper, and therefore cannot represent a patentable invention.

## Maths, while not intrinsic, may be important to programming

It may, but it probably isn’t. Indeed it’s likely that many thousands of programmers every day ignore key results in the mathematical investigation of programming, and still manage to produce software that (at least sort-of) works.

Take the assertion as an example. Here’s a feature of many programming environments that has its root in the predicate systems that can be used to reason about computer programs. The maths is simple enough: if predicate P is true before executing statement S, and consequent Q will hold after its execution, then the result of executing S will be P AND Q.

From this, and knowledge of what your program *should* be doing, then you can *prove* the correctness of your program. Because of the additive nature, if you can *prove* the outcome of every statement in a subroutine, then you can make an overall statement about the outcome of that subroutine. Similarly, you can compose the statements you can make about subroutines to prove the behaviour of a program composed out of those subroutines. Such is the basis of techniques like design by contract, and even more formal techniques like proof-carrying code and model-carrying code.

…which are, by and large, not used. You can write an assertion without having the mathematical knowledge described above, and you can write a program without any assertions.

Here’s what Tony Hoare said about what programmers know of their programs, back in 1969 (pronoun choice is original):

At present, the method which a programmer uses to convince himself of the correctness of his program is to try it out in particular cases and to modify it if the results produced do not correspond to his intentions. After he has found a reasonably wide variety of example cases on which the program seems to work, he believes that it will always work.

Since then, we’ve definitely (though not consistently) adopted tools to automate the trying out of programs (or routines in programs) on particular cases, and largely not adopted tools for automatic proof construction and checking. Such blazing progress in only 45 years!

## There are more things in heaven and earth, Horatio

Here’s a potted, incomplete list of things required of someone or a group of people making software. The list is presented in order of remembering them, and doesn’t reflect any intrinsic dependencies.

- know of a problem that needs solving
- think of a solution to the problem
- presume or demonstrate that the solution can be made out of software
- understand whether the solution will be useful and valuable
- understand the constraints within which the solution will operate
- understand and evaluate the changes to the system that arise from the solution existing
- design an implementation of the solution to fit the constraints (maths is optional here, but index cards and arrows on whiteboards will often work)
- build that solution (maths is optional here)
- demonstrate that the implementation does indeed solve the problem (maths is optional here)
- convince people that they in fact need this problem solved in this particular way
- explain to people how to use this implementation
- react to changes that occur in the implementation’s environment
- pay for all of the above (some maths helps here)

The point is that *actual programming* is a very small part of everything that’s going on. Even if maths *were* intrinsic to parts of that activity, it would still be a tiny contribution to the overall situation. Programming *is* maths in the same way that cooking *is* thermodynamics.