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Houshalter comments on Artificial Addition - Less Wrong

36 Post author: Eliezer_Yudkowsky 20 November 2007 07:58AM

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Comment author: Houshalter 12 June 2013 07:48:26AM -1 points [-]

For what it's worth (and why do I have to pay karma to reply to this comment, I don't get it) there is an infinitesimal difference between the two. An infinitesimal is just like infinity in that it's not a real number. For all practical purposes it is equal to zero, but just like infinity, it has useful mathematical purposes in that it isn't exactly equal to zero. You could plug an infinitesimal into an equation to show how close you can get to zero without actually getting there. If you just replaced it with zero the equation could come out undefined or something.

Likewise using 1.999... because of the property that it isn't exactly equal to 2 but is practically equal to 2, could be useful.

Comment author: ialdabaoth 12 June 2013 07:57:57AM *  2 points [-]

er... I'm not sure if this is the right way to look at it.

1.999999... is 2. Exactly 2. The thing is, there is an infinitesimal difference between '2' and '2'. 1.999999.... isn't "Two minus epsilon", it's "The limit of two minus epsilon as epsilon approaches zero", which is two.

EDIT: And to explain the following objection:

Weird things happen when you apply infinity, but can it really change a rule that is true for all finite numbers?

Yes, absolutely. That's part of the point of infinity. One way of looking at certain kinds of infinity (note that there are several kinds of infinity) is that infinity is one of our placeholders for where rules break down.

Comment author: Houshalter 14 June 2013 05:55:56AM 2 points [-]

This is one of those things that isn't worth arguing over at all, but I will anyways because I'm interested. I'm probably wrong because people much smarter than me have thought about this before, but this still doesn't make any sense to me at all.

1.9 is just 2 minus 0.1, right? And 1.99 is just 2 minus 0.01. Each time you add another 9, you are dividing the number you are subtracting by 10. No matter how many times you divide 0.1 by ten, you will never exactly reach zero. And if it's not exactly zero, then two minus the number isn't exactly two.

Even if you do it 3^^^3 times, it will still be more than zero. Weird things happen when you apply infinity, but can it really change a rule that is true for all finite numbers? You can say it approaches 2 but that's not the same as it ever actually reaching it. Does this make any sense?