Numbers Versus Symbols
Are you a numbers person or a symbols person?
Look, if you don’t know what I mean then it’s simple, you are a numbers person.
It’s a question I often ask of my audience. It helps me judge how best to explain some mathematical concept. Do I do it with 5s and 7s or with Xs and Ys? The hardest audience is one where’s there’s a 50-50 split between numbers and symbols people.
In my experience all audiences seem to be that 50-50 split. Hey ho.
What difference does this make? A lot.
A non-mathematical audience wants to see everything in terms of numbers. Ok, they might not remember what 7 time 8 is (who does?) but they’ll at least understand the concept. Seeing numerical examples is reassuring.
Numbers are great for illustrating how things work. Add, subtract, multiply, divide, raise to a power, etc., you can’t fool anyone with numbers. If it can be done with numbers then it must be easy.
But there’s a downside to using numbers. If I try to explain, I don’t know, double entry bookkeeping using numbers then how do I know whether the $19.99 that appears twice is the same $19.99 each time? You can’t see structure with just numbers. For that you need symbols.
If that $19.99 represents the same quantity then both times it would be an X. If they are different $19.99s, because one represents one sale and the other a different sale that just happens to have the same value, then one would be an X and the other a Y.
But that’s a rather trivial example. You can’t exactly do much in quantum physics if symbols make you uncomfortable.
Symbols are great for showing structure, abstraction is always necessary if you are to go beyond mere arithmetic. The problem with symbols is that some people are frightened by them. And if you and I are used to using different types of symbols it may take some time before we fully understand each other. One could even be accidentally or deliberately confusing, throw in a symbol without a proper explanation and before you know it everyone is lost.
This isn’t important just for me and my lectures, it also matters in the education of our children. There comes a point where if we want to educate a new generation of physicists, engineers, quants, and mathematicians generally, then we have to teach them to think in symbols. And in the abstract generally. Going too far down the mathematics-is-counting-apples route is counterproductive.
People can become terrified of the subject at an early age if taught badly, with the result that they are probably forever lost. How often at dinner parties have we mathematicians heard the ever-so-original response to what we do for a living “I was terrible at maths at school, me!”? Said almost boastfully. I read recently that the part of the brain that does maths is right next to the part that registers fear. I don’t know whether it’s true but it certainly makes sense.
I am forever hearing politicians wittering on about how maths education in schools needs to be made more fun, and more, what’s the word? Practical! Misguided fools! Not a single GCSE maths above grade D among them. The point of mathematics is that it is supposed to be abstract. If all your maths comes from counting apples then you are going to be stymied by the real thing. Mathematics is abstract, that is the beauty of it. And that’s what actually makes it fun. Teach mathematics properly, don’t terrify children by asking them how long it takes ten politicians to dig themselves into ten holes, explain to the young the beauty of the abstract.
“In mathematics you don’t understand things. You just get used to them.” John von Neumann
Now if only someone would explain double-entry bookkeeping to me using symbols.