What do mathematicians want from their #notations
? Why are they so hard to learn? And why is it cool if you do? Mathematical writing is a couple of millenia(!)
old. Almost all of that time it relied on hand drawn formula.
In this post I will be arguing for hand drawn notations. It has been inspired by +Xah Lee
's critique of TeX for mathematical publishing, a
very much different opinion:https://plus.google.com/+XahLee/posts/6QmrKzL8KE6
I'm linking the above 2-year old post first, because it has some discussion, the latest is this one:https://plus.google.com/+XahLee/posts/Cg181xxeutF(1)
- First and before all, the notation should reflect what is going on. To the point that everything that can be said about a structure, can also be written down
. Uhm, that sounds weird... But it's not, let my try again:
What i'm demanding here is that the language has to be powerful enough to capture what you intend to say with it.
Say, you have an idea, followed by cognition, a little enlightenment so to speak, and you now want to share it with your friends. So you make up a story to convice them. Better yet, start writing it down so you can be sure it doesn't change after you fiddled it to your satisfaction.
Your story might contain nouns for all objects of interests, verbs that allow to relate them in each of the important ways, or be structured in a very different way. If your language is one of the better ones, it'll allow others to reason in interesting directions you haven't thought of before.(2)
- All notation should be concise. People working with results
(to apply them, do research, or teach them) are expected to fill loads of blank space with scribbles in that notation.
I find it interesting that that often makes reading math books hard. Sure, finding answers often took one or many researchers's life-long dedication, but learning notation is no piece of cake either! Some areas easily require you to invest a year of studying to get a decent insight from scratch.
Still, it is much more economic to read those pesky books than to find out all by yourself! You know, there are lots of beginner's introductions for almost any topic, but they are fashionable, ephemeral after a while, whereas the original writings often stay popular for decades, centuries, or even longer
Maybe mathematicians like TeX because it feels more like drawing pictures. And that one guy, Donald, felt that he should be holding the pen, because he couldn't trust people who'd typeset for him to make the best decisions. And that that'd be worth the effort!(3)
- Make it look similar to everyone else's. You do want to lean on the plenty of natural language or existing math notation to communicate with your readers. Ideally one would use the same symbols, but if there is something to describe that could be really new, that might not be the best choice:(4)
- Make it subtely differ
from everyone else's. Yes, I know, I just stated the opposite opinion. It is just impossible to tell beforehand which other notation will appear next to our favorite kind of hieroglyphs:
For one, you don't know about relations to other areas that haven't been discovered yet, and in mathematical reasoning that tends to be the case more often than not!
And secondly, still, people will want to try and look if they can find such connections, even if that turns out not to be the case.
All of these reasons only make sense if we understand written mathematics as a form of communication between human beings. To make these ideas do something
in the physical world
, for example, tell a computer to perform a computation, is the job of engineers. Like typesetters they are better at their job than mathematicians... Oh wait, I'm contradicting myself again!(5)
- Nowadays we have another fascinating development: Proof assistants
. Simple programming languages, like
Sage, that do computations (even symbolic ones, reading and printing equations) don't count here!
Well, mathematicians do compute, but their top notch results are theorems
. Sometimes that means presenting the outcome of large computations, but not too many researchers end up publishing tables with numbers (or symbols) in mathematics.
So to them, a calculator isn't the right tool to write up their kind of results, and to communicate those to others. Or even to make significant progress, because calculators can only yield the easy fruits. The ones hard to get require a creative thinker to find a new way to get where no one has gone before
But a proof assistant just might provide a significant edge over doing mathematics in the traditional way. And that would get many people writing maths in a format a computer can handle semantically.
What mathematics is, and how it is practiced are subject to change. I bet there'll always be a place for people doing geometry with a stick on a beach. But there will also be new fields born out of traditional ones.
And sometimes the medium changes, too. I'd guess that TeX's availability as open source has also been an important step in the ongoing struggle to free math papers from publishers, who lock them up, and let you have a look only if you pay them.
For a piece about change in mathematics, have a look at John Von Neumann
's short "The Mathematician"
Thanks to +Stefan Huber
for posting about it here:https://plus.google.com/105612404698873272691/posts/guyS3e689sn
The picture I found because +Richard Green
used it here:"Continued fractions"https://plus.google.com/101584889282878921052/posts/2mbZvUrewmC
It's by Lucas Vieira Barbosa
, read Richard's post to find out more about him.
Still reading? Are you ready for more on complex numbers? Look out for my upcoming post!#mathematics