Mathematics

What are numbers? Are they unquestionably EVERYTHING? Let's take 17 and 18 for example: Isn't there an infinite amount of numbers that exist between 17 and 18? There is no such thing as the smallest number, and there is no such thing as the largest number. WHY?!

Comments

Well, there are a lot of questions there. I won't try to answer the first one: That's a topic for a book, not an internet posting. And I'm not sure I understand the second one. But regarding the next two, yes, of course there are infinitely many numbers between 17 and 18. Here are some of them: 17.1; 17.11; 17.111; 17.1111; etc. And between any two of those, there are infinitely many more. But if you want something that will really make your head spin: Consider all the rational numbers, that is, all the numbers that can be written as fractions m/n. There are no more of those numbers than there are "natural numbers", like 0, 1, 2, etc., and that is true even though there are infinitely many rational numbers in between any two natural numbers. The proof of this is not terribly difficult. The point is that we can order all the rational numbers like this: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, .... (What's the pattern?) Some rational numbers occur more than once, of course, but they can be weeded out. But once we have them in that order, we can "corrleate" them one-to-one with usual natural numbers. So there are just as many of the one as of the other.

Why is there no smallest number? Well, pretty much for the reason just given. (I'm assuming here that we mean no smallest positive number.) Between any two numbers, there is another. So if there were a smallest postive number, then there would be a number between zero and it, and that would be smaller. So there can't be a smallest number.

Why is there no largest number? (I'll assume here that we're talking about finite positive numbers.) Just because, given any number, there's always the number one greater than it. So if n were the greatest number, then we'd also have n+1, but then that would be greater. So there's no greatest number.

Now, of course, you might ask me why there's always a number that's one greater. And that's a perfectly reasonable question to ask. I've spent a good deal of time thinking about it myself. What is it about the numbers that makes this true? But that too is a topic for a book, not a posting.

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