## An Essay on The Ruler: What is Measurement?

Consider the ubiquitous 12 inch ruler (or its multiples, the yardstick, tape measure,or any metric equivalent) and how useful it is in everyday life. What properties produce such wide usage?

1. A ruler is independent of its construction.

Rulers come in many forms; wood, plastic, fabric and metal. Their sizes and fractional divisions are made for a variety of applications. But these are only variations on the general form. They all implement the idea of a straight line.

2. A ruler implements the idea of a unit.

The fundamental idea contained in the structure of a ruler is that of a unit - the inch. The inch is a universally agreed upon quantity of length:

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It is treated as a fact, yet is really a fiction because any specific representation of this length is conditioned by some degree of error.

3. The unit is the same everywhere along a ruler.

The abstract unit-inch is of equal length across any part of the ruler. These "inches"are exchangeable. A reordering of the twelve inch-units produces the same ruler. This has great utility because it is not always possible to start measuring from the end of the ruler.

4. The inch is the concretization of the unit.

Concrete representation solidifies the idea of the inch ­ all "inches" are the same. Hence we can connect concrete inches end-to-end and by doing so "construct" the total measures of inches by concatenation. We go beyond merely asserting the exchangeability of units, we demonstrate it through their concrete realization.

Obtaining total length by concatenating inches demonstrates that additivity holds. Since combining inches by physical concatenation gives the result predicted by additivity, rulers implement what Physicist Norman Campbell called fundamental measurement.

6. Rulers employ the natural number system

Numerals are used to designate sequence and order as in 1, 2, 3, ... We count by means of the ordered numerals to obtain the total.

7. Rulers implement a correspondence between units and numbers.

The markings on a ruler implement a one-to-one correspondence between theordered natural numbers and the unit-inches. The left-most edge or marker of the ruler is the beginning and indicates the absence of any units, i.e., none. The natural numbers mark the unit-inches as successive amounts concatenated to the right. Though the natural numbers themselves are ordinal, their combination with inch-units makes them interval/linear on a ruler.

The ruler is a sophisticated device. It contains a history of ideas and concepts that are fundamental to measurement. The crucial concept is that the abstract mathematical properties of a ruler transcend the concrete instrument. So long as social science lacks rulers, our mathematical manipulations can never transcend their context.

Mark H. Stone
Chicago, Illinois

Stone M.H. (1996) An essay on the ruler: what is measurement? Rasch Measurement Transactions 10:2 p. 502.

Stone M.H. (1996) An essay on the ruler: what is measurement? Rasch Measurement Transactions 10:2 p. 502.

An essay on the ruler: what is measurement? Stone M.H. … Rasch Measurement Transactions, 1996, 10:2 p. 502

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