The power of notation in problem solving

It’s trivial when you think about it:

good mathematical notation is one way of making a problem easier to solve.

In my introduction to advanced probability theory I put emphasis on how probability theory has developed a clever and natural way to describe the processes we deal with. If you think about it some more, the design of the notation is an important part of slicing the subject into manageable blocks.

And now that I’m thinking about it, I see an analogy with programming languages and with spreadsheets.

In my experience, a good spreadsheet is easy to use because its formatting, layout and design suggest to the user how they should use it.

Similarly, a programming language that allows you to solve a problem in the way you expected, without too many tricky bits in the middle (or before the start!), is good.

All of these have the same underlying principle: the design of the syntax (or language, etc.) naturally describes or maps onto the problems the system is designed to tackle.

And this, incidentally, is why Leibniz’s notation for calculus was so powerful. The rules of calculus can be mapped onto the same rules as fractions, so the dy/dx notation works without the user having to think too much — that’s why Leibniz thought it was a good idea, no doubt.

Which means it is a pity that courses in pure mathematics (and analysis in particular) tend to treat the ‘high-school’ approach to calculus as just a bundle of nonsense without any rigorous foundations. Of course, you can fall into traps if you use the simplistic notation without too much care, but don’t you think it’s amazing how you can solve a huge amount of physics problems just with a few rules for calculus and the benefits of Leibniz’s notation?