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John_Maxwell_IV comments on New Year's Predictions Thread - Less Wrong
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Comments (426)
Surely you mean "my estimate rounds to 0%"?
I meant 0%, but you probably have a point that I should present the chance as negligible rather than non-existent. Is there a limit, though? Does it make sense to say that there's a non-zero chance that a state will propose secession and be allowed to leave by tomorrow morning?
Yep. It even makes sense to say that there's a non-zero chance that a state seceded last month, and that we haven't heard about it yet. The word 'epsilon' is useful in such cases; it means 'nearly zero' or 'too close to zero to calculate'.
"Negligible" is a much better word, in my opinion, since epsilon is (conventionally) an arbitrarily small number, not a sufficiently small number. You could use "infinitesimal", but nothing in reality is actually infinitesimally small (including probabilities), so again you'd be inaccurate. I always get frustrated when people misuse precise mathematical words that have lots of syllables in them. The syllables are there to discourage colloquial use! I don't mind if you try to show off your knowledge, but for heaven's sake don't screw up and use that precise brainy term wrong!
You're straddling a strange line here. You're demanding a certain amount of strictness that is itself short of perfect strictness.
There's no such thing as an "arbitrarily small number". There are numbers chosen when any positive number might have been chosen. In particular, a given epsilon need not be "negligible". Really, to conform to the strict mathematical usage, one shouldn't say "epsilon" without first saying "For every". Once you're not demanding that, you're not using the "precise mathematical words" in the precise mathematical way.
I'm not saying that you're on some slippery slope where anything goes. But I wouldn't say that AdeleneDawner is either.
Actually, I'm fine with people speaking vaguely, I just don't want to see terminology misused.
"Through adding zeroes between the decimal point and the 7 in the string '.7', the number we are representing can be made arbitrarily small." Is this a misuse of the word "arbitrarily"?
The important think about an epsilon in a mathematical proof is, conventionally, that it can be made arbitrarily small. This is a human interpretation I am adding on to the proof itself. If the important thing about a variable in a proof was that the variable could become arbitrarily large, my guess is that a variable other than epsilon would not be used.
Your usage is fine, so long as it's clear that "arbitrarily small" is a feature of the set from which you are choosing numbers, or of the process by which you are constructing numbers, and not of any particular number in that set. This is clear with the context that you give above. It wasn't as clear to me when you wrote that "epsilon is (conventionally) an arbitrarily small number".
'Kay.
I'm not the only one you should be ranting at, though - I picked it up here, not in a math class, and I suggested it because it's in common use.
Yep, it is probably unrealistic to expect random folks to avoid picking up multisyllable terms in the way they pick up regular words.
Don't forget "modulo".
Suppose that Nancy meant 0% except for a few special cases that she didn't think should be relevant. Then she could say, '0% modulo some special cases'.
I often use epsilon in the same informal way AdeleneDawner does, though I'm perfectly aware of the formal use. Still, I think the informal use of "modulo" is more defensible - it maps more closely to the mathematical meaning of "ignoring this particular class of ways of being different"
Could you explain this in greater detail? This way of using "modulo" bothers me significantly, and I think it's because I either don't know about one of the ways "modulo" is used in math, or I have an insufficiently deep understanding of the one way I do know that it's used.
In modulo arithmetic, adding or subtracting the base does not change the value. Thus, 12 modulo 9 is the same as 3 modulo 9. Thus, for example, "my iPhone is working great modulo the Wifi connection" implies that if you can subtract the base ("the Wifi connection") you can transform a description of the current state of my iPhone into "working great".
(For your amusement: modulo in the Jargon File. Epsilon is there too.)
Edit: Actually, in this case, you would have to add the base, because my Wifi isn't working, but the statement remains the same.