To expand on Kalium’s comment. Sometimes it takes massive amounts of cognitive effort to estimate a small probability, and it just isn’t worth it for the purposes of some discussion. For example, it is extremely unlikely that the Illuminati have been secretly running the world governments for the last 300 years. It is extremely unlikely that ZPP is not contained in P^R where R is an oracle for the finite ring isomorphism problem. It is extremely unlikely that RIPD is going to win any Oscars. I don’t need to work out exact probabilities for any of these to recognize that they are very small, other than to note that the first one is probably less likely than the second, which is probably less likely than the third.
In that context, saying epsilon to mean a small but hard to precisely estimate probability is reasonable.
While I agree with the reasonability of such a shorthand, what bothers me about this choice of term is that in mathematical usage, ‘epsilon’ generally stands for a variable, i.e. is bound by a quantifier: “for every epsilon > 0...”, etc. Thus, although one informally thinks of such an “epsilon” as “a small number”, the real point is that it’s a number that “moves”, usually “hitting” every positive number in some open interval containing 0. This is quite different from standing for a fixed but unknown number.
That seems like a not very persuasive complaint since even professional mathematicians will “epsilon” to mean a very small difference in an informal setting. To use a recent example from discussing a calculus midterm that we were going to make two versions, one of the professors said something like “the midterms will be within epsilon of each other” and no one batted an eye.
Because professional mathematicians understand and depend on the technical usage, there’s little risk of the technical sense becoming diluted by such quasi-humorous, figurative allusions to the technical jargon, which can serve as a means of in-group bonding. When outsiders do it, hover, it’s no longer clearly an allusion to something else, and risks being mistaken for a distinct technical usage in its own right, in addition to losing the slight humor/bonding value.
Another mathematical in-term that has been subject to similar abuse by outsiders is the word “isomorphic”. When a mathematician speaks to a colleague of all the local cafeterias being isomorphic, this is clearly hyperbole—but it’s only clear if one understands the actual meaning and normal context of the word.
From what I’ve seen of cafeterias on large college campuses, it isn’t actually hyperbole to say that “all the local cafeterias are isomorphic”. They’re technically distinct, but under a transformation that preserves all relevant properties, they can all be mapped to each other; they are the same up to isomorphism.
To expand on Kalium’s comment. Sometimes it takes massive amounts of cognitive effort to estimate a small probability, and it just isn’t worth it for the purposes of some discussion. For example, it is extremely unlikely that the Illuminati have been secretly running the world governments for the last 300 years. It is extremely unlikely that ZPP is not contained in P^R where R is an oracle for the finite ring isomorphism problem. It is extremely unlikely that RIPD is going to win any Oscars. I don’t need to work out exact probabilities for any of these to recognize that they are very small, other than to note that the first one is probably less likely than the second, which is probably less likely than the third.
In that context, saying epsilon to mean a small but hard to precisely estimate probability is reasonable.
While I agree with the reasonability of such a shorthand, what bothers me about this choice of term is that in mathematical usage, ‘epsilon’ generally stands for a variable, i.e. is bound by a quantifier: “for every epsilon > 0...”, etc. Thus, although one informally thinks of such an “epsilon” as “a small number”, the real point is that it’s a number that “moves”, usually “hitting” every positive number in some open interval containing 0. This is quite different from standing for a fixed but unknown number.
That seems like a not very persuasive complaint since even professional mathematicians will “epsilon” to mean a very small difference in an informal setting. To use a recent example from discussing a calculus midterm that we were going to make two versions, one of the professors said something like “the midterms will be within epsilon of each other” and no one batted an eye.
Because professional mathematicians understand and depend on the technical usage, there’s little risk of the technical sense becoming diluted by such quasi-humorous, figurative allusions to the technical jargon, which can serve as a means of in-group bonding. When outsiders do it, hover, it’s no longer clearly an allusion to something else, and risks being mistaken for a distinct technical usage in its own right, in addition to losing the slight humor/bonding value.
Another mathematical in-term that has been subject to similar abuse by outsiders is the word “isomorphic”. When a mathematician speaks to a colleague of all the local cafeterias being isomorphic, this is clearly hyperbole—but it’s only clear if one understands the actual meaning and normal context of the word.
From what I’ve seen of cafeterias on large college campuses, it isn’t actually hyperbole to say that “all the local cafeterias are isomorphic”. They’re technically distinct, but under a transformation that preserves all relevant properties, they can all be mapped to each other; they are the same up to isomorphism.
Do you have evidence that people are actually misunderstanding what either of these terms mean?