Monday, 29 July 2013

Maths Terminology Query

I have five objects (O1, O2, O3, O4 and O5) and three machines to measure them with (M1, M2 and M3).
I know that M1 weighs the objects and have been told that M2 and M3 are also scales.  With M1 I can place the objects in order of weight which is, conveniently the same order as the numbering I've given them:

OM1: O1 then O2 then O3 then O4 then O5

When I measure the objects with M2 I get a different order.  M2 gives the same reading for O2 and O3 and the same reading for O4 and O5.  I have a partial order:

OM2: O1 then both O2 and O3 then both O4 and O5

Although it is possible that M2 measures something other than weight I'm quite happy to accept that it does measure weight but is less sensitive than M1.

With M3 I get a third order:

OM3: O1 then O3 then O2 then O4 then O5

This, again, is different from OM1.  But the difference between OM1 and OM3 is different from the difference between OM1  and OM2.  And this difference tells me that M3 isn't measuring weight.  O3 can't be both heavier and lighter than O2: the machines must be measuring different things.

What is it that makes OM3 support the conclusion that M3 is measuring a different quality to M1?  It isn't that the order is different, OM2 is different.  It isn't any of the things usually used to characterise orders (OM1 and OM2 are both linear orders, OM2 isn't).

So what is the nice, neat, mathematical term for the type of difference between OM1 and OM3 that lets me reach my conclusion?