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Post by dalehileman » Thu Dec 22, 2005 6:33 pm

In Webster I just ran across one of the lengthiest definitions I’ve ever encountered:



metric...3: a mathematical function that associates with each pair of elements of a set a real nonnegative number constituting their distance and satisfying the conditions that the number is zero only if the two elements are identical, the number is the same regardless of the order in which the two elements are taken, and the number associated with one pair of elements plus that associated with one member of the pair and a third element is equal to or greater than the number associated with the other member of the pair and the third element

Edited to remove superfluous phrase per Ken below
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Post by Erik_Kowal » Thu Dec 22, 2005 7:02 pm

Not only that, but WTF is it actually saying?
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Post by Ken Greenwald » Thu Dec 22, 2005 9:15 pm

Dale, There’s a lot longer than that if you look around. If you recheck what you wrote against what Merriam-Webster wrote, you will find that you made a booboo [[now ocrrected]]. The concept of a metric in math is really fairly simple and the only confusion to a non-math person may be in the 3rd condition. But I will concede that M-W, although technically correct, did go about defining METRIC in a most atrocious way!

There are three simple properties that must be satisfied by what in math is known as a METRIC – a nonnegative function g(x,y) describing the ‘distance’ between neighboring points for a given set.

1) The distance between a point and itself is zero: g(x,x) = 0 [we all know this must be true]

2) The distance between two points is symmetric (the distance from X to Y is the same as the distance from Y to X): g(x,y) = g(y,x) [we also know that this must be true]

3) The triangle inequality must hold (or in familiar terms, the shortest distance between two points is a ‘straight’ line): g(x,y) + g(y,z) is greater than or equal to g(x,z) [if there are three points the distance between any two must be equal to or shorter than the distances added up if we take a detour to a third point OR the sum of the lengths of 2 sides of a triangle is always greater than the length of the third - and when there is equality, the triangle has collapsed into a straight line.]

One may ask, why bother defining this thing called a METRIC when it seems so obvious that it must always be satisfied? Well, in math we must define all the terms we use and this term is used for 'spaces' other than our normal Euclidean space, where things may not be that obvious. In normal Euclidean space the ‘distance’ between two points is a straight line. In the spherical geometry of a coordinate system laid out on the surface of the earth, for example, the ‘distance’ between two points is not a straight line but a curve on the surface of a sphere. Nevertheless the 3 above conditions do hold and this spherical geometry does comprise a ‘metric space.’ However, in the areas of math called topology and differential geometry, it is a simple matter to construct sets which are NONMETRIC, although without getting many folks lost in mathematical symbolism it is not that easy to explain. But trust me that 'nonmetric spaces' do exist and are ‘useful’ and in fact they popped up in physics in attempts at reformulations of general relativity in days of yore back in the 1970s and 80s, when I was up on this stuff, and I’m sure they are still useful in physics and other areas today.
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Ken – December 22, 2005
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Post by Edwin Ashworth » Wed Dec 28, 2005 11:24 am

There seems to be a similar set of conditions governing bus (and train) frequencies:

1) You will still be waiting at the same bus-stop after 40 minutes.

2) During this time, 5 buses will have passed in the opposite direction.

3) It will take you longer than if you had walked.

Oh, and cancelling is of course to be performed as often as possible.


Incidentally, Ken, do you know if there has been any agreement on the precise usage of the words "mapping" and "function" in recent years - whether many-to-one relations were allowed into either category seemed to be a matter of which textbook you were using, ten years ago.
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Post by Ken Greenwald » Thu Jan 05, 2006 2:44 am

Edwin, As far as I am aware a mapping and a function are the same thing and many-to-one is just fine, and always has been. I checked several of my texts, the newest of which is the monster 2003 CRC Concise Encyclopedia of Mathematics (3242 pages), and the oldest of which is a set theory book from 1964 (and I bought that new - Oy vey!) - they all say the same thing. I can’t see why a perfectly fine many-to-one function (e.g. y = x^2) would be excluded?

In checking the OED, however, they did say something extremely strange, which I had never seen before and which doesn’t make a hell of a lot of sense. They say:

MAPPING: A correspondence by which each element of a given set has associated with it one element (occasionally, one or more elements) of a second set.

The ‘occasionally, one or more elements’ part seems to me to defy what the whole point of what is generally meant by a function/mapping. A function is defined neatly as: “a set of ordered pairs no two of which have the same first element” or if you prefer: a map from A—>B is an object ‘f’ such that for every A an element of B, there is a unique object f(a) an element of B.” And unique implies only one. So I don’t know what they’re talking about as far as their 'occasionally' goes.
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Post by Andrew Dalby » Sat Jan 07, 2006 7:42 pm

Edwin Ashworth wrote: There seems to be a similar set of conditions governing bus (and train) frequencies:

1) You will still be waiting at the same bus-stop after 40 minutes.

2) During this time, 5 buses will have passed in the opposite direction.

3) It will take you longer than if you had walked.
You forgot (4) If you start walking, a bus will always overtake you at a point from which it's humanly impossible to beat it to the next stop.
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Post by Erik_Kowal » Sat Jan 07, 2006 8:30 pm

Not to mention 5): When a bus does eventually come, either its sign will be displaying 'Not in service' or 'Private hire', or it will be so full that no new passengers will be allowed to board.
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Post by Edwin Ashworth » Sat Jan 21, 2006 12:39 pm

Oh good, it's not just Oldham.
Thanks, Ken. To think I used to like maths because there was usually just one correct answer you had to worry about. We need a maths czar.
You can have my vote.
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