I’ve been trying to understand the invisibility of mathematics. The fact that mathematics is so central to our technology, society and culture – yet goes unnoticed most of the time. I had problems deciding on a title for the post as it touches on so many things. It may seem a little bit far out.
It is obvious that our high tech society couldn’t exist without advanced mathematics. It’s not just electronics – which relies on physics, understood in terms of mathematical physics – it is also the logistics of administrating energy, materials and information that require mathematics & computation. I won’t bore the reader with more examples. A very thorough and still very up-to-date discussion can be found in Lynn Steen´s 1985 article: “Mathematics: Our invisible Culture” (I have no link – Lynn sent a copy to me as a response to a letter I wrote him in an early stage of this project).
Here’s a my take on the paradox. It’s by analogy with natural language (that is: any spoken and/or written human language). When you speak your own native language, then you are not (most of the time at least) conscious of that. It’s just something you do. You’re probably more conscious when writing since that is more difficult – it’s not so instantaneous – and its more reflective (that’s indeed one point of writing).
However, when I speak English – which is not my native language even though I’ve lived in England and the US – I’m much more conscious of speaking – curiously much more so in everyday situations than in academic situations. And I make a lot of errors.
Now, could it be that when mathematics teachers use the mathematical language (I’m thinking of both the formalism itself and the natural meta-language needed to communicate mathematics) they are not aware of the fact that they are using it? They speak (and write) as if the students were as fluent as themselves? It’s like in X-land long time ago when people couldn’t understand that not everyone spoke X-ish. But the typical student is not fluent. Mathematics is not a native language for most people.
Formal mathematics as taught in school is therefore highly visible to most people. But all the mathematics that is built into our society and technology is almost entirely invisible! Of course it is not formulas that are built into technology. There are no formulas in a cell phone. But mathematics was needed when designing it and the network infrastructure that makes it work. And all that is based on our knowledge of physics described in mathematical language. All this is well-known.
But can I really understand how mathematics is built into a cell-phone? I mean in some detail – could I give a plausible explanation to a student asking me?
So we all use mathematics all the time without being aware of it – but not formal mathematics – which is used by almost no-one except mathematicians and a subset (yes) of scientists, engineers, economists and logisticians, …
This is all very strange. It is as if mathematics is built into the very fabric of reality. All languages (as far as I know) have nouns, adjectives and verbs. This precisely corresponds to the fact that the world consists of “things” that have “properties” and can “perform actions”. This also, by the way, corresponds closely to the classes of Object-Oriented-Programming (OOP): classes are blueprints of things (abstract or concrete) that have properties and actions. It is a close step to think of classes as Plato’s ideas and the objects (the instantiations of classes, still speaking OOP) as concrete physical things.
And this leads over to philosophy of mathematics, but I think I shall leave that to another post. It is time for some breakfast.