There is no single known set of physical laws that universally applies to every situation. Physicists instead construct physical models, which are collections of assumptions about particular situations, and deduce the mathematical consequences of those assumptions. These assumptions are called postulates; they are used as the rules of the game and function like mathematical axioms, the starting assumptions of a mathematical argument. Whether the conclusions of a model follow logically from its assumptions, and whether those conclusions accurately describe reality, are two entirely different questions; the first question is answered by logical proof, the second by experimental verification.
the roles of postulates
Postulates supply the principles from which a model's predictions follow. A framework, such as classical mechanics, supplies general principles; a model applies them to a particular situation. Classical mechanics describes motion using definite positions, velocities and forces. A model of a falling ball might use Newton's laws of motion and assume constant gravity and negligible air resistance. The latter assumptions simplify this situation; they are not general laws. Measurements, such as the ball's starting height, supply additional inputs.
Different frameworks can give incompatible predictions. In the double-slit experiment, electrons pass through a barrier with two narrow openings and land on a screen. A model of independent classical particles predicts the sum of the two single-slit landing patterns, assuming opening one slit does not change their motion through the other. Instead, when no measurement distinguishes which slit each electron passes through, the screen shows alternating bands of frequent and rare arrivals, called an interference pattern. The pattern builds up even when electrons are sent one at a time. Quantum mechanics, which calculates probabilities of measurement outcomes, predicts these bands by combining contributions from both slits before calculating landing probabilities. The classical deduction can be logically valid while its assumptions fail to describe the observed pattern.
Physicists can also combine compatible principles from several frameworks. Classical electromagnetism describes electric and magnetic fields, quantities assigned throughout space that determine electromagnetic forces. Combining its field description with quantum principles and special relativity, which relates space and time for observers moving at different constant velocities, leads to quantum electrodynamics (QED), a theory of light interacting with electrically charged matter. QED gives both matter and the electromagnetic field quantum descriptions. It does not retain the classical assumption that light can exchange arbitrarily small amounts of energy.
the roles of experiments
Some experiments help physicists formulate postulates. In the photoelectric effect, light ejects electrons from a metal. Experiments found that, below a minimum light frequency, increasing brightness did not eject electrons under the conditions studied. Frequency means the number of wave cycles per second. These observations helped motivate the proposal that light transfers energy in individual packets, called photons, with energy proportional to frequency. A photon must supply enough energy to free an electron. Greater brightness supplies more photons, but does not increase each photon's energy.
Other experiments test a model's accuracy in a particular situation. A model that neglects air resistance predicts the same acceleration, the rate at which velocity changes, for a dropped feather and a dropped hammer. In air they fall differently; in a vacuum, where air has been removed, they fall together. The Apollo 15 demonstration tested this prediction on the Moon. The comparison shows why specifying the conditions is part of testing a model.
Experiments also supply inputs. Measuring gravitational acceleration lets us predict how far a ball will fall in a given time; measuring its actual fall then tests that prediction. Fitting a model to measurements and predicting new measurements are different achievements. Every comparison must account for measurement uncertainty, the quantified limits on how precisely a measured value is known. Agreement supports a model under the tested conditions; it does not prove that its postulates hold everywhere.
the roles of logical derivation
Logical derivation establishes what must follow once the assumptions and inputs are specified. Suppose a ball starts at rest and falls with constant downward acceleration \(g\), with air resistance neglected. After time \(t\), its downward speed is \(gt\). Because that speed increases uniformly from zero, its average is \(gt/2\). Distance equals average speed multiplied by time, so the distance fallen is
\[d=\frac12gt^2,\]
where \(d\) is distance, \(g\) is acceleration and \(t\) is elapsed time. With \(g=9.8\) metres per second squared, meaning speed increases by 9.8 metres per second each second, the model predicts a fall of 19.6 metres in two seconds. This result follows exactly from the stated inputs. Its accuracy for a real ball depends on those inputs and assumptions.
A derivation can also produce a probability. Quantum mechanics uses a wave function, a mathematical description of a system's state, and a postulate called the Born rule to calculate probabilities of measurement outcomes. If a model assigns a detector a 50% chance of registering each independently prepared particle, it predicts an average of 500 detections over repeated batches of 1,000 particles. It does not require exactly 500 in each batch or determine the next outcome. Experiments test these probabilities through repeated measurements.
the roles of physical quantities
Physical quantities describe measurable properties; which are inputs and which are derived depends on the model. In classical mechanics, mass is usually an input and kinetic energy, energy associated with motion, is calculated as \(K=\frac12mv^2\), where \(m\) is mass and \(v\) is speed. Relativity also associates mass with rest energy, energy measured when an object is at rest: \(E_0=mc^2\), where \(c\) is the speed of light in vacuum. Mass can therefore be calculated from rest energy, although relativity does not derive every particle's mass.
The International System of Units (SI) provides shared measurement standards. Its seven base quantities include mass, length and time, measured in kilograms, metres and seconds. Base quantities are choices for organising units, rather than a framework's fundamental ingredients. Derived units combine base units: both \(\frac12mv^2\) and \(mc^2\) have energy units, called joules, with \(1\,\mathrm{J}=1\,\mathrm{kg\,m^2/s^2}\). Modern SI standards use fixed values of physical constants to define units and calibrate instruments.
Physicists identify corresponding quantities by matching predictions where frameworks overlap. In general relativity, gravity is described through the geometry of space and time. For weak gravity and slow motion outside a spherical body, it recovers the classical acceleration \(a=GM/r^2\), where \(G\) is Newton's gravitational constant, \(M\) is the body's mass and \(r\) is distance from its centre. Using the same measured \(G\), with units \(\mathrm{m^3/(kg\,s^2)}\), requires \(M\) to have units of kilograms so that \(a\) has units of \(\mathrm{m/s^2}\). This connects relativistic mass to its classical measurement meaning.
SI was established in 1960, after these frameworks. Their postulates express relationships independent of unit choice. Physicists ensure compatible units by using shared standards, matching the quantities being compared and including conversion constants such as \(c^2\). Compatible units alone do not ensure accurate predictions; experiments test those.
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