Re:Is 0 Odd or Even?
by nrr on Tuesday October 20, 2009 @02:25AM
(#891902)
Be careful with your language here. Zero isn't defined to be even; rather, it has the property of being even by way of the definition of an even number. Usually, this process involves making a conjecture and then proving or disproving it.
On the other hand, definitions are, well, whatever we want them to be. We can define a sheep to be woolly, and we can define a collection of crows to be a murder, but we have to justify conjectures that all sheep are black or that all corvids are crows.
Think about it this way. In Scheme/Lisp, consider the notions of functions and macros. The mathematical analogue to a definition is a function; they're defined in very much the same manner, and the idea is effectively the same. Likewise, the analogue to a theorem/corollary/lemma is a macro. A theorem allows one to traverse a logical chain from P to Q very easily, and a macro allows the Schemer/Lisp hacker to abstract dull, repetitive tasks into short, succinct statements.
The humdinger: the proof to a theorem is analogous to the macro expansion. :)
Now, I'm hoping I'm not being too terribly nitpicky, but 1/inf does not equal zero. It's still undefined. (See also: 1/0 does not equal infinity.) This mess hinges on the idea that infinity is not a number, and in general, it holds true for the kind of real analysis one would typically learn as an undergrad.
On the other hand, definitions are, well, whatever we want them to be. We can define a sheep to be woolly, and we can define a collection of crows to be a murder, but we have to justify conjectures that all sheep are black or that all corvids are crows.
Think about it this way. In Scheme/Lisp, consider the notions of functions and macros. The mathematical analogue to a definition is a function; they're defined in very much the same manner, and the idea is effectively the same. Likewise, the analogue to a theorem/corollary/lemma is a macro. A theorem allows one to traverse a logical chain from P to Q very easily, and a macro allows the Schemer/Lisp hacker to abstract dull, repetitive tasks into short, succinct statements.
The humdinger: the proof to a theorem is analogous to the macro expansion. :)
Now, I'm hoping I'm not being too terribly nitpicky, but 1/inf does not equal zero. It's still undefined. (See also: 1/0 does not equal infinity.) This mess hinges on the idea that infinity is not a number, and in general, it holds true for the kind of real analysis one would typically learn as an undergrad.