Message #37 Posted by Crawl on 29 Sept 2010, 8:55 p.m.,
in response to message #1 by John Mosand
I'm gonna go way off base here:
I'm not convinced that irrational numbers exist.
Someone who wanted to dispute that might say, "Oh, well, you must not understand the proof that the squareroot of 2 is irrational. Because it is very well established that that number is irrational, and therefore that irrational numbers exist."
But that's not true. There is a proof that there is no rational number that, when squared, gives 2. That does NOT prove that the squareroot of 2 is irrational. A third possibility is that there is NO number that equals the squareroot of 2 (if only rational numbers exist).
Of course, there are numbers that, when squared, are approximately 2. And they can get better and better as they get more decimal digits. As far as that is concerned, we might as well say the squareroot of 2 exists. Also, we can treat it formally -- eg., (x-sqrt(2))(x+sqrt(2))= x^2-2, whether or not sqrt(2) is a "number" or a symbol or whatever.
So, okay, we can ACT like irrational numbers exist. But do they really?
I saw a mathematician who worked on some sort of computer math (using a computer to find proofs, I think) who argued that irrational numbers don't exist from a completely different point of view: It could be possible to make some "ultimate computer" for this universe, by using every single particle in the universe to make it. This computer can only work with so many finite digits. It might be enormous, but there is a limit. Maybe you can calculate 5 trillion digits of pi, maybe you can calculate a trillion trillion digits of pi, but you might not be able to calculate a trillion trillion trillion trillion digits.
(Yes, there are formulas to calculate digits of pi without calculating preceding digits. But this ultimate computer presumably has a memory limit, so to calculate more digits would require dropping others. So even if you could calculate the trillion trillion trillion trillion trillionth + 1 digit, you might not be able to hold all those digits *simultaneously*)
If you can't know all those digits at once, is it meaningful that they "exist" in some philosophical sense?
You can actually use those infinite cardinal numbers in proofs. It's possible to show the cardinality of rational numbers is less than the cardinality of real numbers, hence irrational numbers exist. Or that the cardinality of algebraic numbers is less than the cardinality of real numbers, hence transcendental numbers exist.
Well, this isn't a way out of the above paradoxes. It just links the concept of infinite cardinal numbers to the concept of irrational numbers. We can ACT like infinite cardinal numbers exist, just like we can ACT like irrational numbers can exist. And if you think that one exists, you probably should think that the other exists, too. But infinite cardinal numbers are probably even less intuitive than irrational numbers, so I don't think looking at this issue from the former perspective have any persuasive power for the latter perspective.