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JoshuaZ comments on Artificial Addition - Less Wrong

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

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Comment author: JoshuaZ 30 December 2010 10:30:16PM *  0 points [-]

We want it to denote a number for simple consistency. .11111... is a number. It is a limit. 3.14159... should denote a number. Why should 1.99999?... Be any different? If we are going to be at all consistent in our notation they should all represent the same sort of series. Otherwise this is extremely irregular notation to no end.

Comment author: taryneast 30 December 2010 11:47:44PM -1 points [-]

Yes, I totally agree with you: consistency and convenience are why we have chosen to use 1.9999... notation to represent the limit, rather than the sequence.

consistency and convenience tends to drive most mathematical notational choices (with occasional other influences), for reasons that should be extremely obvious.

It just so happened that, o this occasion, I was not aware enough of either the actual convention, or of other "things that this notation would be consistent with" before I guessed at the meaning of this particular item of notation.

And so my guessed meaning was one of the two things that I thought would be "likely meanings" for the notation.

In this case, my guess was for the wrong one of the two.

I seem to be getting a lot of comments that are implying that I should have somehow naturally realised which of the two meanings was "correct"... and have tried very hard to explain why it is not obvious, and not somehow inevitable.

Both of my possible interpretations were potentially valid, and I'd like to insist that the sequence-one is wrong only by convention (ie maths has to pick one or the other meaning... it happens to be the most convenient for mathematicians, which happens in this case to be the limit-interpretation)... but as is clearly evidenced by the fact that there is so much confusion around the subject (ref the wikipedia page) - it is not obvious intuitively that one is "correct" and one is "not correct".

I maintain that without knowledge of the convention, you cannot know which is the "correct" interpretation. Any assumption otherwise is simply hindsight bias.