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Rider has set me straight, but here's my original beef: I said, quite clearly, "I understand that mathematics isn't simply about physical phenomenon. I underst

Rider has set me straight, but here's my original beef: I said, quite clearly, "I understand that mathematics isn't simply about physical phenomenon. I understand the beauty of i, for example." You mention 3 and the square root of 2. You plug both of those in to: (x == x+1) and you'll get an inequality. Not so with infinity. Again, Rider has set me straight that I need to learn more. I can believe that, but when I see how often in the physical world we creep up on infinity only to have it not exist, I'm definitely skeptical about plugging in infinity into equations like x == x + 1. It seems it doesn't make any sense at all.

cousin_it 17y ago
It's true that you can't extend the number system to include something called "infinity" that would behave like a number. Just like you can't extend the number system to make division by zero make sense while keeping all the usual laws of arithmetic. But this doesn't lead to any philosophical implications like "infinity doesn't exist". In every context where it makes sense to talk about infinities without leading to contradictions, we want to do that, because it enriches our vocabulary and allows us to solve more problems. Allow me to illustrate. Take the following problem stated in school geometry terms: A square is cut into triangles of equal area, now prove that their number is even. Interestingly, we don't yet know any elementary solution to this problem that can be stated in school math terms. The only solution known was discovered in the 1970s and relies on something called "p-adic numbers". What are those p-adic numbers you ask? They are weird number-like things that have infinitely many digits to the left of the decimal point. Freaky constructs without any counterpart in the real world, but you can still add them, multiply them and all that. Do such things "really exist"? Don't know, don't care. But they helped us solve a difficult problem and that's all that matters.
- david927 17y ago
  Cool. Thanks. Edit: but that still seems to me to reflect infinitely, the adjective, as opposed to infinity, the noun. Still, thanks.

cousin_it 17y ago

It's true that you can't extend the number system to include something called "infinity" that would behave like a number. Just like you can't extend the number system to make division by zero make sense while keeping all the usual laws of arithmetic. But this doesn't lead to any philosophical implications like "infinity doesn't exist". In every context where it makes sense to talk about infinities without leading to contradictions, we want to do that, because it enriches our vocabulary and allows us to solve more problems. Allow me to illustrate. Take the following problem stated in school geometry terms: A square is cut into triangles of equal area, now prove that their number is even. Interestingly, we don't yet know any elementary solution to this problem that can be stated in school math terms. The only solution known was discovered in the 1970s and relies on something called "p-adic numbers". What are those p-adic numbers you ask? They are weird number-like things that have infinitely many digits to the left of the decimal point. Freaky constructs without any counterpart in the real world, but you can still add them, multiply them and all that. Do such things "really exist"? Don't know, don't care. But they helped us solve a difficult problem and that's all that matters.

david927 17y ago
Cool. Thanks. Edit: but that still seems to me to reflect infinitely, the adjective, as opposed to infinity, the noun. Still, thanks.