An instrument does more than place a number between an observer and the world. It couples to a system, responds over time and preserves selected distinctions. Understanding that physical process is part of understanding the observation.
The apparatus joins the system
Consider an ideal voltage source of 6 volts with an internal resistance of 1,000 ohms. A voltmeter connected across its terminals has a finite input resistance. If that resistance is also 1,000 ohms, the two resistances form a divider and the displayed terminal voltage is 3 volts. With a 10,000-ohm meter, it is 60/11 volts, approximately 5.455 volts.
Both readings follow from the same source model. The meter has changed the circuit it measures. Calling the lower reading an illusion would miss a real interaction: the loaded terminal voltage genuinely differs from the ideal source voltage. The first task is to state which quantity is being sought.
If the resistance values are known, the ideal source voltage can be inferred from Vₛ = Vₘ(1 + Rₛ/Rₘ). That correction is model dependent. An unknown internal resistance, nonlinear source or time-varying load requires additional information. Writing an inverse formula does not remove the need to justify its inputs.
A present indication may contain a recent history
Now consider an idealized thermal sensor initially at 20 degrees Celsius. Its surroundings suddenly become 30 degrees. Suppose its response obeys a first-order relaxation with a one-second time constant. Its indication after t seconds is 30 − 10 exp(−t), giving approximately 26.32 degrees at one second and 29.50 degrees at three seconds.
The sensor is functioning exactly as stipulated. Its transient reading is neither an instantaneous sample of the new surroundings nor an arbitrary error. It reflects the sensor's state and coupling. A measurement made during that transient must account for the response dynamics or explicitly define the quantity as the sensor's own temperature.
Averaging several fast readings does not automatically remove this lag. Those readings may share the same systematic departure and may be temporally correlated. More samples can describe the transient more clearly without turning it into an equilibrium measurement.
The display is a further reduction
After the physical response, electronics can filter, sample, threshold or quantize the signal. A stored value can therefore lose distinctions at several stages. A temperature change may be too brief for the sensor to follow, fall between samples, or remain below the display's rounding increment. These are different limitations with different remedies.
The existing essays on exposure and sampling and digital readout isolate two such stages. Increasing the number of displayed digits cannot recover a fluctuation already suppressed by the physical response.
It helps to describe the whole chain: source, coupling, sensor state, signal processing, retained record and inferred quantity. A change at one stage can alter the final result while leaving other stages intact. That decomposition suggests targeted tests rather than an undirected demand for a “better instrument.”
Use the interaction to design a check
In the circuit example, changing the meter's input resistance creates a controlled comparison. If the source model is adequate, the readings should follow the predicted loading curve with a common source voltage and internal resistance. Fitting each reading with unrelated source values would abandon that test.
For the thermal sensor, varying the waiting time tests the predicted relaxation. Independently changing the sensor's thermal coupling can test whether the proposed time constant has a physical explanation. Such checks ask how the apparatus responds, rather than treating its output as an uninterpreted verdict.
What must a completed record retain?
In SMM/DTT, a completed observation should specify the accessible readout and its connection to the generating process. The instrument examples show why the retained number alone can be insufficient: one also needs the loading condition, observation time and calibration model.
The expanded book's distance from admissible comparisons demonstrates how a family of allowed readouts can define a mathematical distance. Identifying those comparisons with physical apparatus requires an additional construction. The mathematical readout rule and the instrument that realizes it should be connected explicitly.
Continue to calibration for a case where two reference readings are reproduced exactly but the instrument still gives a biased answer between them.
Sources and further reading
- JCGM, Developing and using measurement models, GUM-6:2020. The circuit and thermal-response examples are original idealized calculations.
- What counts as an observer?