If you look for tools in a Sears store, you’ll find large numbers of different kinds of screw drivers. And yet, we tend to think of screw drivers are general purpose tools. In contrast, a tooth brush is a very special purpose tool. I tend to use mine for only one thing, and if I ever were to clean the fire place using my tooth brush, I would no longer want to use it for its original purpose. My tooth brush is such a special-purpose device that I wouldn’t even want to lend it out to anybody else.
It is useful to think of ‘general purpose’ and ‘special purpose’ as directions on a scale rather than precise points on that scale. Compared to me using my teeth, a Swiss Army knife is a special purpose tool. If it has a cork screw, it is considerably better suited to pull a cork than my teeth are. And yet no wine dude in a restaurant would want to be caught dead using a Swiss Army knife to open a wine bottle. The wine dude uses a very specialized tool that allow the cork to be pulled swiftly, gracefully, and with a very satisfying pop. Special purpose tools can be extraordinarily important for certain professions, cost a great deal, and yet be useless for any other task. The specialized expensive cork screw, so good for pulling corks, is useless as a screw driver, whereas a lowly nickle may work much better.
For a water main, you can buy a special shut-off wrench that looks like this:
To shut off the main, you can also use a general tool, like a crescent wrench:
but you’d discover that the tool fits awkwardly, and that it is hard to get the leverage you need. The trade-off between general purpose tools and special purpose tools makes for interesting practices, like what tools you keep in your workshop, what tools you keep in your van, and what tools you bring on a backpack trip.
In mathematics, certain tools are extremely general purpose, such as addition of numbers. In comparison, calculating square roots is much more special purpose. Middle schoolers often learn a very general purpose tool for solving equations, usually stated as something like: “do the same thing on both sides of the equal sign”. They also learn a very special purpose tool for solving equations of a special type where and are known but arbitrary numbers, and is the number you’re trying to find out. This tool is often stated something like: “multiply the two numbers across and divide by the number opposite the unknown.”
In addition to operations and methods, an other aspect of mathematics we could rate on the scale of general purpose to special purpose is representations. The decimal number system is a general purpose system for representing numbers, whereas making tally marks is a special purpose representation system that is well-adapted for counting.
A four-function calculator is a general-purpose tool, which I could use in a special situation, such as converting temperatures in Centigrade to temperatures in Fahrenheit. If I have a big thermometer with scales indicating in Centigrade on the left and Fahrenheit on the right, I can use that as a special purpose tool for converting values in Centigrade to Fahrenheit – much more direct.
Professionals appreciate a wide variety of tools. They don’t use their teeth for everything, nor will they be paralyzed without their water main shutoff wrench. It is the same with mathematics.