Question

What are the two fundamental problems of measurement?

Answer 1

the first problem is justification of the assignment of numbers to objects or (phenomena.).

The second problem Concerns the specification of the degree to which this assignment is unique.

First Fundamental Problem: The Representation Theorem
The early history of mathematics shows how difficult it was to divorce arithmetic from particular empirical structures. The ancient Egyptians could not think of 2 + 3 , but only of 2 bushels of wheat plus 3 bushels of wheat. Intellectually,it is a great step forward to realize that the assertion that 2 bushels of wheat plus 3 bushels of wheat equals 5 bushels of wheat involves the same mathematical considerations as the statement that .2 quarts of milk plus 3 quarts of milk equals 5 quarts of milk.
From a logical standpoint, there is just one arithmetic of numbers,not an arithmetic for bushels of Wheat, and a separate arithmetic for quarts of milk. The first problem for a theory of measurement is to show how various features of this arithmetic of numbers may be applied in a variety of empirical situations. This is done by showing that certain aspects of the arithmetic of numbers have the same structure as the empirical situation investigated. The purpose of the definition of isomorphism to be given is to make the rough-and_ready intuitive.idea of "same structure" precise. The great significance of finding such an
isomorphism of structures is that we may then use many of our familiar computational methods, applied to the arithmetical structure, to infer facts about the isomorphic empirical structure.

The first fundamental problem of measurement may be cast as the problem of showing that any empirical relational system which purports to measure by a simple number given property of the elements in the domain of the system is isomorphic (or possibly homomorphic) to an appropriately chosen numerical relational system.