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From reading the title I thought that 'juggling' and 'tricks' where metaphors for how choosing the right notation in mathematics can make a problem trivial, whi
by jpallen 14y ago
From reading the title I thought that 'juggling' and 'tricks' where metaphors for how choosing the right notation in mathematics can make a problem trivial, while choosing the wrong notation can make it nearly impossible. Even though the article is talking about a concrete example of juggling, I think it can still be taken as an appropriate metaphor.
Perhaps the oldest and most obvious example of the power of notation is the invention of algebra itself. Before that, people used to write maths in words. Fermat's Last Theorem for example was stated as:
It is impossible to separate a cube into two cubes, or
a fourth power into two fourth powers, or in general, any
power higher than the second, into two like powers.
This isn't very easy to understand! With algebra we can just write
There are no integers, a, b and c such
that a^n + b^n = c^n when n is an integer n > 2.
This is much easier to read once you're familiar with algebra, but much more importantly, the notation is very conducive to being able to manipulate the expression to derive new results from it.
Perhaps that's the criteria for good notation - it makes additional manipulations and insights easy to the point of trivial.
- ColinWright 14y agoA 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.
- jpallen 14y agoNot 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!
- ColinWright 14y agoYou might like my page about it: http://www.solipsys.co.uk/new/MillsMess.html?HN0 http://www.solipsys.co.uk/new/MillsMess.html?HN0 And the frame-by-frame analysis: http://www.solipsys.co.uk/new/images/MillsAnalysis/Analysis.html?HN0 http://www.solipsys.co.uk/new/images/MillsAnalysis/Analysis....
- dalke 14y agoThough Gauss famously remarked, "What we need are notions, not notations." :)
- ColinWright 14y agoThat'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.
- dalke 14y agoFor what it's worth, I agree. Ramanujan would also be in that category of people who don't need notation.