What is the difference between counting and measuring? Are counting and measurement the same thing?
I think that they are different since in my mind the idea of counting pertains to discrete objects while the idea of measurement pertains to continuous objects. I also think of counting as something which can be done directly with numbers while measurement requires a unit.
 A: To count means to determine the cardinality of some finite set. Technically, since the natural numbers are usually defined as sets, that means to determine the natural number such that There is a bijection with the given finite set.
Measuring is not a single defined operation. A broad class of operations can be called measuring. Technically a measure on a set of objects $X$ is a function $m:X\rightarrow \mathbb R_{\geq0}$ satisfying some requirements.
A few immediate consequences of this fact are:


*

*you can measure anything as long as you have a measuring function for it while counting is restricted to finite sets

*you can define several measuring functions for the same set of objects. Changing units is barely scratching the surface of what is possible

*measure functions can assume any non-negative real value. Historically, this is why real numbers were introduced.


Note: Usually in mathematics $X$ is the set of subsets of another set, for instance plane geometric figures are the subsets of the plane. I omitted that part to keep the answer simpler and more general.
