Epistemology and the Scientific Method

What physics is

Physics is the attempt to describe, with the least possible number of independent assumptions, everything that can be observed in Nature, and to predict what will be observed next. Three commitments separate it from every neighbouring discipline. First, its statements are quantitative: they concern numbers obtained by measurement, and are meaningless without a statement of how those numbers were obtained and how much they may be trusted (Measurement, SI Units, and the Theory of Errors). Second, its statements are testable: a proposition belongs to physics only if some conceivable observation could refute it [Popper:1959]. Third, its statements are connected: isolated regularities are catalogued, but the goal is always the smallest set of principles from which the catalogue follows deductively.

This treatise takes the second commitment as a hard rule of scope. Theories of great mathematical beauty whose predictions have not been observed — string theory, supersymmetry, supergravity, grand unification — are not treated here, however influential they may be as research programs; they are named only once, in What We Observe but Do Not Understand, precisely in order to say honestly that the evidence for them is currently absent. What is treated is everything for which Nature has already voted: from the parabola of a thrown stone (Experiment: Free Fall and Projectile Motion) to the strain of \(10^{-21}\) that two colliding black holes imprinted on an interferometer (Experiment: Gravitational Waves).

Physics, mathematics, and philosophy

Mathematics supplies the language in which physical principles are stated and their consequences extracted. The relationship is asymmetric: mathematics owes physics nothing, while physics without mathematics cannot even formulate a prediction. Why abstract structures invented for their own sake — complex numbers, Riemannian geometry, group representations — repeatedly turn out to be exactly what Nature uses is a genuine puzzle, famously called the unreasonable effectiveness of mathematics [Wigner:1960]. This treatise responds to that effectiveness practically: Part II develops the mathematics in the generality it deserves — often in arbitrary dimension \(D = p + q\) and signature \((p,q)\) — while the physical parts instantiate it in the observed \(3+1\) dimensions.

Philosophy, by contrast, supplies the questions physics cannot answer from within: what counts as an explanation, whether unobservable entities exist, how theory choice works when data underdetermine it [Kuhn:1962]. We borrow from it sparingly and explicitly — and from one tradition in particular, for reasons set out in Section 1.3. Where an interpretive question has no observable consequence — as with the interpretations of quantum mechanics — this treatise states the competing readings and the experiments that bound them (Interpretations (Evidence-Anchored)), and goes no further.

Why this treatise follows analytic philosophy

The preceding section borrowed from philosophy; this one says from which philosophy, and why. The borrowing is not incidental. The rule that governs what may appear in this book — only physics for which Nature has already voted — is not itself a result of physics, since no experiment establishes it. It is a philosophical position, and a treatise that acts on one owes its reader the argument rather than the posture.

The choice cannot be avoided

Three questions must be settled before the first equation is written. What counts as evidence? What counts as having explained something? Which apparently physical questions are questions at all? Physics does not answer them from within: an experiment can decide between two theories, but no experiment can decide whether a proposition is the kind of thing an experiment could decide.

Every treatise answers all three regardless — in its selection of material, in what it is willing to call a derivation, and in the questions it declines to discuss. The only real choice is whether the answers are stated or merely enacted. Enacted, they are usually inconsistent: the same textbook will define temperature by the procedure that measures it on one page and speak of the wavefunction as a thing on the next, without noticing that the two paragraphs answer to incompatible theories of meaning. This treatise states its answers instead, and takes them from the analytic tradition — the line running from Frege's logic [Frege:1879] through the Vienna Circle [Carnap:1929] to the modern philosophy of science.

What the commitment amounts to

“Analytic” here names a method, not a creed, and the method reduces to five working rules.

1. Logical form, not grammatical form. A well-formed sentence need not pose a well-formed question. “What is the temperature of this single molecule?” is impeccable English and has no answer, because temperature is a property of an ensemble in equilibrium and of nothing else (Kinetic Theory of Gases). Before a question is answered it is worth asking what would count as an answer.

2. To explain is to derive. A phenomenon is explained when it is exhibited as a consequence of premises that are stated in full [Hempel:1948]. That is editorial rule 2 of this treatise in one sentence, and it is why an unproved claim here is marked as pending rather than covered by a plausible-sounding paragraph: the premise list is either complete or it is visibly not.

3. Content through procedure. A quantity that no procedure could measure carries no empirical content [Bridgman:1927]. Hence the fixed structure of every experiment box — apparatus, procedure, data with uncertainties — and hence the axiomatic status of the SI (Measurement, SI Units, and the Theory of Errors). The strong form of this rule, which identifies a quantity with one procedure, is too strong and was abandoned: thermodynamic temperature is fixed by the agreement of gas thermometry, radiation pyrometry and Johnson-noise thermometry, not by any one of them (Classical Thermodynamics). What survives is the demand that at least one procedure exist and that all of them agree.

4. Convention separated from content. Parts of any theory are decisions rather than discoveries [Poincare:1902], and the stipulations that tie a formalism to measurements — Reichenbach's coordinative definitions [Reichenbach:1928] — must be declared as such. Which coordinates, which gauge, which metric signature, which unit system: none of these is measured. The empirical content is what stays invariant when they are varied. This is why the SI is imposed by axiom and labelled a convention, and why Part II is written for signature \((p,q)\) instead of quietly building \(3+1\) into the mathematics.

5. Triage before answer. Every question is first sorted: empirical, to be settled by measurement; conventional, to be settled by stipulation and then recorded; or defective, to be repaired or dropped. Delivering that verdict is a result, not an evasion — “the aether's rest frame” was disposed of in the third category (Experiments: Light, the Aether, and Time).

Why this tradition and not another

Three reasons of principle, and a fourth of practice.

It was built out of this material.

Analytic philosophy is the only school formed in direct response to the crises of physics and mathematics between 1880 and 1930. Mach's critique of absolute space [Mach:1883] is part of the prehistory of general relativity; Einstein's operational dissection of simultaneity is conceptual analysis performed by a physicist (Lorentz Transformations), and he stated the methodology explicitly [Einstein:1921]: “insofar as the propositions of mathematics refer to reality, they are not certain; and insofar as they are certain, they do not refer to reality.” Frege's new logic, and the paradox that struck it, opened the foundational crisis (Section 3.6); the Vienna Circle assembled around relativity and the new quantum theory [Carnap:1929]; Reichenbach's philosophy of space and time [Reichenbach:1928] is also a technical treatment of the conventionality of simultaneity. The working vocabulary of a careful physicist — operational definition, auxiliary hypothesis, underdetermination, model error as distinct from statistical error — is that tradition's vocabulary. Adopting it costs no translation.

Its results occasionally become physics.

The traffic runs both ways, and this is the strongest single argument for the choice. Einstein, Podolsky and Rosen raised a philosophical objection to the completeness of quantum mechanics [Einstein:1935]; Bell converted the objection into an inequality between measurable correlations [Bell:1964] (Entanglement and Bell Tests); experiment then decided it, against local hidden variables (Experiment: Bell Tests). A tradition that can promote a metaphysical dispute into a laboratory result has demonstrated the only credential this treatise recognizes.

It audits itself to destruction.

Its central doctrine was proposed, refuted and abandoned in public. The verification principle fails its own test; Duhem showed that no hypothesis meets experience alone [Duhem:1906] and Quine generalized the holism until the analytic–synthetic distinction went with it [Quine:1951]; Popper replaced verifiability by falsifiability [Popper:1959]; Kuhn supplied the historical objection [Kuhn:1962]; Lakatos replaced the single falsifiable theory by the progressive or degenerating research programme [Lakatos:1970]; and from inside mathematics Gödel ended the dream of complete formalization [Goedel:1931]. The full sequence is set out in Section 3.7, with what remains of it in Section 3.7.3. What matters here is the pattern: a book whose own rule is that no assertion stands without its derivation cannot consistently adopt a philosophy that exempts itself from that treatment.

It yields the decisions this book must make.

Beyond provenance and self-audit, the tradition supplies standards for choices that recur on every page: whether a subject is admissible (testability); whether an explanation is finished (derivation from stated premises); how a superseded theory relates to its successor — as a limit case with a stated domain of validity rather than as an error, so that Newtonian mechanics survives inside relativity and quantum mechanics [Nagel:1961] (Relativistic Dynamics and The Postulates of Quantum Mechanics); and how a theory relates to its mathematics, namely as a class of mathematical models together with a rule mapping some of their features onto measurements [Suppes:1960]. That last one is literally the architecture of this book: Part II builds the structures, and the physical parts supply the interpretation and the measured numbers.

What the alternatives would cost

Four alternatives deserve naming. None is dismissed as foolish; each is declined for a stated reason.

1. A priori metaphysics — determining the structure of Nature by reason in advance of measurement. Its track record in physics is the argument against it: Aristotle's natural places, Descartes' vortices, Kant's Euclidean space and absolute simultaneity as preconditions of any possible experience, all refuted by measurement (Part IV). Where a priori reasoning has succeeded in physics it did so as symmetry argument (Lagrangian Mechanics) — and symmetry claims are empirical, as parity discovered when an experiment found it violated (Experiment: Parity Violation). Accepting this alternative would license precisely what editorial rule 1 excludes.

2. Phenomenological and hermeneutic traditions. Their questions are real ones: how a scientific object is constituted in experience, how a practice is situated in its history. But they yield no procedure for deciding whether a given claim belongs in a physics book, and no account of when a derivation is complete. They are not false; they are the wrong instrument for this particular job.

3. History- and sociology-first accounts [Kuhn:1962] [Feyerabend:1975]. As description these are substantially right, and the five-station cycle of Section 1.4 is an idealization that says so. Feyerabend's demonstration that no proposed rule of method has gone unviolated in some successful episode is accepted here without reserve. The reply is that this treatise legislates presentation, not discovery: how a result was found may be as anarchic as the record shows, but what is asserted in a book still has to come with its evidence, its premises and an admission of what remains unproven.

4. Instrumentalism and constructive empiricism [vanFraassen:1980]. This is not a rival method but an answer, within the same tradition, to a question the treatise deliberately leaves open: whether the unobservable structures of a successful theory are real or are devices for organizing observations. Where that dispute acquires observable consequences it is treated as experimental (Experiment: Bell Tests); where it does not, the competing readings are reported and the matter is left there (Interpretations (Evidence-Anchored)).

The price of the choice

The tradition's failures are part of the reason to trust it, and they are inherited along with the method.

1. No sharp line of demarcation. Holism guarantees that any prediction can be rescued by adjusting an auxiliary hypothesis, so testability is a policy applied case by case with the reasons given, not a decision procedure. The dark sector is treated as physics because it has independent quantitative signatures in several channels (The Dark Sector: Evidence Without Explanation); programmes with no such signature are named once, to record that absence honestly (Quantum Gravity: The Honest Status and What We Observe but Do Not Understand).

2. No theory-free observation. Every number in this book was produced by an apparatus modelled with theory, so the observational base is not neutral ground (Measurement, SI Units, and the Theory of Errors). The response is disclosure — apparatus, procedure, calibration and uncertainty budget in every experiment box — not the pretence of raw data.

3. No complete formalization. Axiomatization is a tool with a proven ceiling [Goedel:1931]. It is used where it earns its keep (The Postulates of Quantum Mechanics and Axiomatic Quantum Field Theory) and not imposed elsewhere for appearances.

4. No theory of discovery. Logical reconstruction is not history: derivations here are ordered for validity, not chronology, and Planck's radiation law was a successful fit years before anyone could say why it worked (Black-Body Radiation and Planck's Hypothesis). Where the order of discovery carries the lesson, it is recorded in the experiment chapters rather than smuggled into the proofs.

How the choice shows on the page

The commitment is not decorative; it is the reason this book has the shape it has. The numbers below are those of the five rules in Section 1.3.2.

• Phenomenon boxes state what was observed, with its uncertainty — rule 3.

• Experiment boxes carry a fixed field structure, so that the procedure giving each term its content is on the page and not in the reader's imagination — rule 3, made procedural.

• Definitions and axioms are declared before use, and conventions are labelled conventions rather than presented as findings — rule 4.

• Every phenomenon carries its derivation, or a visible pending derivation marker collected in a front-matter list — rule 2, and the bookkeeping that holism demands.

• Sources are cited so that a reader can retrieve them and check the claim independently; where a source cannot be retrieved, that fact is recorded rather than hidden — rule 2 applied to the evidence base.

• Where no observation discriminates between competing readings, the treatise stops (Interpretations (Evidence-Anchored)) — rule 5.

The scientific method as used in this treatise

The working cycle of physics, as this treatise practices it, has five stations.

1. Phenomenon. Something is observed, reproducibly, with stated apparatus and stated uncertainty. In the text, such facts are boxed in the Phenomenon environment.

2. Model. A mathematical structure is proposed whose solutions mimic the phenomenon. Its assumptions are listed explicitly.

3. Derivation. The phenomenon is proven to follow from the model. This treatise's first editorial rule is that this step is never skipped: every Phenomenon carries its derivation inline, in Appendix A, or — when not yet written — as a visibly marked pending derivation.

4. Prediction. The model is pushed beyond the data that motivated it, into statements that could fail.

5. Experiment. The prediction is confronted with Nature, in an experiment reported with apparatus, procedure, data, and uncertainty — the fixed structure of every Experiment box in this book. A failed confrontation sends the cycle back to station 2 with the failure as new input.

Galileo's inclined planes [Galilei:1638] already exhibit the full cycle, and Newton's Principia [Newton:1687] is its first complete monument: observed regularities (Kepler's laws), a model (three laws of motion and universal gravitation), derivations, and predictions reaching to the tides. The cycle has not changed since; only the precision has, from Galileo's pulse beats to the \(10^{-18}\,\mathrm{m}\) displacement sensitivity of a gravitational-wave interferometer.

Primitive notions

Every deductive structure must start from undefined terms. Physics is no exception; the following notions are taken as primitives, sharpened but never fully defined by the theories built on them.

How this treatise is organized

Three threads are interwoven throughout. Mathematical physics (Part II) develops the language once, in general dimension, so that no later part need pause for it. Theoretical physics (Parts III to XIII) builds the models and carries out the derivations. Experimental physics lives inside the theory parts as dedicated experiment chapters placed immediately after the theory each experiment tests, and is indexed globally by the List of Experiments in the front matter. Two conventions bind all three threads: SI units are used axiomatically everywhere (Measurement, SI Units, and the Theory of Errors), and no observed phenomenon is left without its derivation or an explicit admission that the derivation is pending.