Irrationals Do … What? (Warning: Graphic!)

As I see it, this is a very graphic demonstration of how irrational numbers relate to rational numbers. It definitely concerns some inherent property of rational numbers. The problem is, I really don’t know what that property is or how to describe it.

We start with simple periodic functions like sine and cosine. Being periodic, they repeat with a certain period : ) But that period can be controlled. Below, Cos(x), Cos(3/2*x), and Cos(2*x) are all plotted on top of each othe, the first in blue, second in red, third in yellow-ish. As x increases, the functions oscillates and eventually repeat.

The Clearly, Cos(2*x) oscillates twice as fast as Cos(x) and has a period half as long.

Basic stuff, but then we start combining things.

Math as Language

Via slashdot, I found this article on an Amazonian tribe that does not seem words to describe numbers and counting. Rather, they have relative quantifiers that indicate a quantity from 1 to 4, 5 or 6, or ‘many’. In effect, small, medium, or large.

Of course, it’s probably a bit of a stretch, but as counting forms much of the basis of math (though I wouldn’t mind hearing discussions on that), this relates, as I see it, to whether mathematics is something inherent to the universe itself, or rather something more akin to an evolutionarily advantageous artifact of our brains. Some of our brains, anyway.

Evolution almost certainly plays a role here, as the language in question is spoken by about 300 people, and they likely haven’t spent a lot of time doing commerce with other cultures.

There were some interesting thoughts raised in the comments. For instance, one person noted that most people can easily recall about five separate objects at a glance, and the perception of quantity is likely linked to how easy it is to remember the objects. Another noted that this may be a different kind of language artifact – the researchers were having them count foreign objects, but should’ve asked them to count how many children they had, or some other familiar kind of object, the idea being that perhaps different words were used for different types of things. This is interesting, because linking quantity and type descriptors would greatly diversify what could be expressed.

There are far more questions I’d like to ask. For example, the researches only went up to 10 objects. What if they went straight from 10 to 1000? While both are ‘many’, there’s a pretty significant visual difference from the sheer amounts.

Also, I do wonder whether or not, even with the limited range of quantifiers, if the speakers here make use of any kind of computational-esque rules. For example, two ‘medium’ groups combined would always produce a ‘large’ group. Very fuzzy sort of addition. Of course, two ’small’ groups combined don’t necessarily produce anything definite.

It’s all very interesting.