A friend of mine says that the act of "doing math" is the process of inventing a language (and its associated notation) in which it's easy to talk about the problem you're working on. Once you have the notation and language, the problem becomes easy. Well easier.

Certainly the notation for juggling has made a whole slew of juggling tricks easier to think about, analyse, and learn. A good notation is like a good programming language, it enables, and then gets out of the way.

Not only does the problem become easier, using the right notation enables us to do things that we just couldn't do before. Humans (or at least me) can only hold a few simple things in our head at once. If the problem is too complicated, we can't deal with it, and the only way to approach it is to split it down into simpler abstractions that we can hold in our head all at once. I think this is a large part of what mathematics is - taking concrete concepts that are hard to hold in our head, and turning them into an abstraction that describes their important and general properties in a way that is simple enough that we can keep them in our head. So much of mathematics is the pursuit of different abstractions and general properties of things.

I can juggle with 3 balls, and I'm able to do a passable Mills-Mess, but I'm not sure I have the experience to judge the utility of your juggling notation. However, it certainly seems to be doing a similar thing. Your notation contains exactly the same information as the entire pattern but it can actually be kept in ones head and worked with. Even being able to juggle Mills-mess, I'm still not really sure what's going on, and certainly struggle to imagine the whole pattern. I'm not surprised it took you 2 and half pages to describe!

You might like my page about it: http://www.solipsys.co.uk/new/MillsMess.html?HN0

And the frame-by-frame analysis: http://www.solipsys.co.uk/new/images/MillsAnalysis/Analysis....

Though Gauss famously remarked, "What we need are notions, not notations." :)

That's true, but without notation at one level, it's hard to get the notions at the next level. Familiarity, comfort even, with the notation for the things you understand and the things others have done releases your mind to have the notions at the next level.

After that, you develop the notation for these new ideas, and the cycle repeats.

And in addition, very few are as good as Gauss. It's possible, even plausible, that he didn't need the notations at all. I, for one, certainly do.

For what it's worth, I agree. Ramanujan would also be in that category of people who don't need notation.