Mathematics

This one is mathematical, but seems to address philosophical issues regarding definition and the nature of mathematical truth. So: If, for any x, x^0 = 1, and, for any y, 0^y = 0, then what is the value of 0^0?

Comments

There are no right or wrong answers when it comes to making definitions in mathematics. We can define things however we please. However, mathematicians generally try to make definitions that will be useful, and one way to do this is to preserve general rules.

Now, you have identified a case in which this strategy leads to conflicting answers. One general rule, x0 = 1, suggests that we should define 00 to be 1. Another rule, 0y = 0, suggests that we should define 00 to be 0. There is no way to define 00 and preserve both rules, so we have to make a choice. I don't think there is universal agreement among mathematicians about how to define 00--sometimes it is regarded as being undefined. However, in many contexts mathematicians take 00 to be 1.

In choosing which definition of 00 is most useful or natural, it may help to think about why the rules x0 = 1 and 0y = 0 make sense:

x0 can be thought of as representing an empty product, but what does that mean? It may help to think first about empty sums. I think most people would find it natural to consider an empty sum to be 0: If you're going to add up a bunch of numbers, then you can imagine yourself starting with a sum of 0, and then adding your numbers to this sum. If you've got no numbers to add, then you end up with 0. Similarly, if you're going to multiply a bunch of numbers, you can think of yourself as starting with a product of 1 and then multiplying this product by your numbers. If there are no numbers to multiply, the product is 1.

0y = 0 works if y is a positive integer, since it represents 0 multiplied by itself some number of times. But notice that it doesn't work if y is negative: For example, 0-1 = 1/0, which is undefined, not 0. Should the case y = 0 work like positive y's or negative y's, or is it not like either one? This rule seems to give less clearcut guidance about 00 than the previous rule.

There are many other rules that seem to work better if we define 00 to be 1 rather than 0. There's a Wikipedia entry that has more to say about this.

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