Observation, measurement, and physical quantities

In brief

Two people watch the same stone fall from a wall. One says it fell "for a while." The other says it fell "for two and three tenths seconds." Only one of these statements can be checked by a stranger in another country using different equipment and get the same answer. That difference, between an impression and a reproducible number, is the entire starting point of physics. A physical quantity such as length, time, or mass is not defined by how it feels to a particular observer. It is defined by a stated procedure of comparison that anyone can repeat and that will keep giving the same result. This entry treats that procedure, not the name of the quantity, as the real definition, because it is the procedure that lets physics be a public, checkable activity rather than a private report.

The full treatment

First look: comparing two sticks

Suppose you want to know whether a table is longer than it is wide, but you have no ruler. Lay a length of string along the table's length, mark it, then lay the same string along the width. If the mark falls short, the width is longer. You have just measured a physical quantity, length, without a number, using nothing but a comparison procedure. The string is a stand in, a physical intermediary you carry from one object to the other so the comparison does not depend on memory or judgment. This is the seed of all measurement: replace "which is bigger" with a procedure that gives the same verdict no matter who performs it.

Building the idea: what a procedure must supply

For a comparison procedure to define a genuine physical quantity, it needs three things. First, repeatability: the same object measured the same way twice gives the same result, within the limits of the method. Second, transportability: the comparison can be carried from place to place, as the string was carried from the table's length to its width, so that objects far apart in space or time can still be compared. Third, additivity or ordering: quantities of the same kind can be laid end to end, or ranked, so that "twice as long" or "longer than" has an operational meaning, not just a verbal one. A property that fails all three, such as "how beautiful the table looks," is not a physical quantity in this sense, however real the impression may be.

The formal move: operational definition

Percy Bridgman gave this idea its sharpest statement: a physical concept is nothing more or less than the set of operations used to measure it. Time, in this view, is not first a mystery and then a number, it is defined by the operation of counting repetitions of some chosen periodic process, a swinging weight, the vibration of a crystal, the rotation of the earth, and comparing that count against a reference. Length is defined by the operation of laying a rigid reference object end to end, or its modern equivalent, against the thing measured. Symbolically, if Q is a quantity, its measure is a pair, a numerical value n and a unit u, produced by applying a stated procedure P to an object, so that P applied to the object equals n times u. Neither n alone nor u alone is a measurement. Only the pair, tied to the procedure that produced it, counts.

Why this ordering matters

Notice what has not appeared yet: no named law, no formula relating quantities to one another, not even units fixed by international agreement. Those come later. What comes first is the discipline of stating, for any quantity you intend to use, exactly what someone would have to do to measure it. This discipline is why two physicists on opposite sides of the world can report "the rod is 1.2 meters long" and mean the same claim, checkable by the same kind of procedure, rather than trading impressions that cannot be compared at all.

Lineage

The demand that physical claims be tied to measurable operations runs through the history of experimental science. Galileo insisted on timing falling and rolling bodies with water clocks and his own pulse, rather than trusting Aristotelian argument about what heavy bodies "ought" to do, and his insistence on repeatable comparison over authority is often marked as the start of modern physics. Lord Kelvin's remark that when you can measure what you are speaking about and express it in numbers you know something about it, but when you cannot, your knowledge is meager and unsatisfactory, became a working motto for nineteenth century experimental science. Percy Bridgman formalized the philosophy in the twentieth century as operationalism, arguing explicitly that a concept is defined by its measuring operations and nothing more.

The strongest case for it

Grounding physics in operational definitions is what makes physics a shared, cumulative enterprise rather than a set of private observations. Because "one second" and "one meter" are tied to stated procedures, a measurement made in one laboratory can be checked, disputed, or confirmed anywhere else, and results accumulate instead of resetting with each observer. This is also what allows physics to detect when two seemingly different properties are secretly the same quantity, because if two different operations always agree, in every case tested, that agreement is itself a discovery worth explaining, not an assumption.

The strongest case against it

Strict operationalism has real limits. Some quantities central to modern physics, such as a quantum wavefunction, are not themselves directly measured by any single operation, only their statistical consequences are. Taken too literally, defining a concept purely by one measuring procedure would force physics to treat two different procedures that happen to agree, such as gravitational mass measured by a balance and inertial mass measured by acceleration under force, as two unrelated concepts until proven otherwise, when in fact their agreement is one of the deepest clues in the field. A common misconception is to think a bare number is a measurement. It is not: "the length is 3" means nothing without stating the unit and, ultimately, the procedure that unit encodes.

Where it stands now

The practice of defining physical quantities by stated, repeatable procedures is universal in experimental physics and uncontested as a working method. Philosophers of science still debate whether operationalism is correct as a full theory of what physical concepts mean, particularly for quantities that are inferred rather than directly measured. That debate does not touch the working discipline this entry teaches: before you use a quantity, you should be able to say exactly what someone would do to measure it.

Test yourself

You are given an unfamiliar liquid and asked to determine, without any thermometer, whether it is "hotter" than a second liquid, in a way a stranger could repeat and get the same answer. Describe a comparison procedure that defines this ordering, then explain what additional step you would need before you could say one liquid is "twice as hot" as another rather than merely hotter. Finally, state one physical property for which no repeatable comparison procedure exists yet, and explain why that property is not currently a physical quantity in the sense this entry defines.