Paul P. Mealing

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Wednesday, 7 December 2016

How algebra turned mathematics into a language

A little while ago I wrote a post arguing that mathematics as language was just a metaphor. I’ve since taken the post down, though those who subscribe may still have a copy. In the almost 10 years I’ve been writing this blog it’s only the second time I’ve deleted a post. The other occasion was very early in its life when I posted an essay on existentialism (from memory) only to post something more relevant.

The reason I took the post down was because I thought I was being a bit petty in criticising some guy on YouTube who was probably actually doing some good in the world, even if I disagreed with him on a philosophical level. Instead, I wrote a comment on his video, challenging the premise of his talk that the reason mathematics is ‘difficult’ for many people is because it’s not taught as a language. I would still challenge the validity of that premise, but I would now change my own approach by acknowledging that there is a sense in which mathematics is a language, but not in a lingua franca sense.

In my last post – the review of Arrival – language and communication are major themes, and I make mention of a piece of expositional dialogue that I thought very insightful and stuck in my brain as a revelatory thought. To remind everyone: it was the realisation that language determines the limits of what we can think because we all think in a language. In other words, if a language doesn’t define the specific concepts we are trying to comprehend then we struggle to conjure up those concepts, and mathematics provides a good example.

The reason that mathematics is best not construed as a language is because mathematics, as it’s generally practiced, has its own language and that language is algebra. As I’ve said before: mathematics is not so much about numbers as the relationship between numbers, and the efficacy of algebra is that it allows one to see the relationships without the numbers.

And this is the thing, because some people find it easier to think in algebra than others. I will illustrate with examples.

A = k/B then B = k/A

If k is a constant (can’t change) and A and B are variables then there is an inverse relationship between A and B. In other words, if A gets larger then B must get smaller and vice versa. This can be written as A ∝ 1/B or B ∝ 1/A, where ∝ (in this context) means ‘is proportional to’. Note that if the number on the bottom gets smaller then the whole term must get larger and, of course, the converse is also true: if the number on the bottom gets larger then the whole term must get smaller.

People who are familiar with these concepts think this automatically. They also know that if you move a term from one side of an equation to the other, then you either invert it or take its negative. So if you have a language that captures these concepts, then you can think in these concepts with no great effort. It also means that you are not easily intimidated by equations.

To give another common example: the distributive rule, which is arguably the most commonly used rule in algebra.

A = B(C + D) is the same as A = BC + BD

And if A = -B(C - D) then A = BD – BC

(Note that multiplying by minus changes the sign: from + to - and - to +)

We could have done this differently because –(C – D) = D – C and B(D – C) = BD –BC   (So same answer)

This is all very simple stuff and it can be extended to include square roots (including square roots of -1), logarithms, trig functions and so on. Even calculus is just algebra with numbers disappearing into zero with the inverse of infinity.

One of the problems in learning mathematics is that we are trying to learn new concepts and simultaneously a new ‘language’ of symbols. But if the language of algebra allows one to think in new concepts, then a hurdle becomes a springboard to new knowledge.

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