 Mathematical theories of measurement (often referred to collectively as “measurement theory”) concern the conditions under which relations among numbers (and other mathematical entities) can be used to express relations among  objects.[2]  In order to appreciate the need for mathematical theories of measurement, consider the fact that relations exhibited by numbers—such as equality, sum, difference and ratio—do not always correspond to relations among the objects measured by those numbers. For example, 60 is twice 30, but one would be mistaken in thinking that an object measured at 60 degrees Celsius is twice as hot as an object at 30 degrees Celsius. This is because the zero point of the Celsius scale is arbitrary and does not correspond to an absence of  temperature.[3]  Similarly, numerical intervals do not always carry empirical information. When subjects are asked to rank on a scale from 1 to 7 how strongly they agree with a given statement, there is no prima facie reason to think that the intervals between 5 and 6 and between 6 and 7 correspond to equal increments of strength of opinion. To provide a third example, equality among numbers is transitive [if (a=b &amp; b=c) then a=c] but empirical comparisons among physical magnitudes reveal only approximate equality, which is not a transitive relation. These examples suggest that not all of the mathematical relations among numbers used in measurement are empirically significant, and that different kinds of measurement scale convey different kinds of empirically significant information.