I have a reading recommendation for you. It’s not about math, but you'll probably see the implicit analogy.

​Utopian for Beginners, a 2012 article from The New Yorker, is one of my favorite pieces of long form journalism/narrative nonfiction. It’s been rattling around in the back of my mind ever since I first read it many years ago.

The premise is this:

A man named John Quijada develops an invented language called Ithkuil during his off hours from his job at the California DMV.

The language is designed to be maximally precise and maximally efficient: it can convey complex information and subtle gradations of meaning in as few words as possible.

In 2004 he publishes the language to a web site, which soon receives a disproportionate amount of traffic from Russia. One thing leads to another and he's invited to speak at a conference in Kalmykia, a semi-autonomous province in the Russian Federation known mostly for its chess prowess.

The reveal comes when Quijada and the author of the article travel to the conference and discover that the group who invited him are linked to terrorism and have some nefarious ideas about using his language in their quest to develop superhuman intelligence.

One theme here is of course the question about what happens when an invention is used for purposes that its inventor abhors. We've seen this story elsewhere in the history of science.

But the article also has quite a lot to say about the relationship between language and how people think.

Arguably, the ideas and thoughts that are available to us are dictated by the boundaries of what our language can convey.

So a more precise and efficient language can lead to more precise and efficient thinking.

We See This in Math History

As I read stories from the history of math, I'm constantly struck by the similarity between this idea and the development of mathematical notation.

While perhaps math has always existed and humans have merely discovered it, the notation that we've invented to organize and communicate this math is like the "decoder ring" that makes complex ideas comprehensible.

Take exponentiation.

The little superscript numeral that we are all so familiar with is an invention thousands of years in the making.

Archimedes was the first to lay out the familiar exponent rule:

But without suitable notation or a concept of exponents as we know them today, his explanation is long and tedious:

Over the centuries, different ad hoc notations for exponents emerged both in Europe and India, but they failed to convey the concept in a way that is as simple as we understand it today.

Each power often had a different notation, obscuring the simple relationship between the powers.

Also, exponents were only applied to unknowns, not to constants. (After all, why would you need them for constants? Just do the repeated multiplication, I guess!)

Here is an illustration of how the Ancient Greek mathematician Diophantus notated his exponents, from A Short History of Greek Mathematics by James Gow:

Subsequent notation methods made incremental progress, and in fact John Napier (not to mention Jost Bürgi around the same time) invented the system of logarithms BEFORE exponents as we know them today came into common usage.

When Descartes finally introduced superscript numerals denoting an operation that's applied generally to both variables and constants, this concept that once required complicated explanations involving geometric series became simple enough for any pre-teen to learn.

The notation opens the door for new patterns to be revealed, and for mathematical progress to take off.

What I'm working on

I've been thinking a lot about this idea of things that are now taken for granted but were once obscure in the context of the video I'm working on about al-Khwarizmi.

I'm working to articulate an answer to the question: "what exactly was his unique contribution?"

Even though he's often referred to as the "father of algebra," there was quite a bit of precedent for most of what he wrote in his book on al-jabr.

So, was he just a stenographer? A mere popularizer?

It seems that he did indeed make unique and important contributions to the field, but it's not as simple as saying that he invented "x" or something like that (he definitely didn't invent "x," by the way. Al-Khwarizmi's algebra was completely rhetorical, with no equations or symbolic variables as we know them today).

The actual answer has to do with his method of turning quantities into algebraic objects -- generalized abstractions which can always be manipulated according to the same rules.

An unknown value no longer needs to refer specifically to, for example, an unknown side length (meaning that x^2 becomes an area, etc). Instead the unknown can just be any number or magnitude ... any quantity at all, and his rules of "balancing" and "completing" an equation can be applied to it.

In the process, the equation itself becomes the object of study.

I am still working out the best way to articulate this.

If it sounds a bit confusing, it's also because these concepts are so obvious to us today that we can forget how remarkable they are. So talking about it feels a little bit like trying to tell a fish about the discovery of water.

My challenge is to use the video format to somehow make something that a modern viewer would consider to be obvious feel non-obvious again...

Until next time,

Ben