= Solution
\b[Theory change defeats an inference from scientific success to final, unrestricted truth. It does not show that successive theories make no progress in accuracy, explanation or warranted knowledge.] Calling two theories false says too little about how their claims compare.
The strongest pessimistic argument is historical: theories once regarded as successful and well supported have later lost central commitments. If successful predecessors could misidentify what exists, the success of a present theory cannot by itself guarantee its full ontology. <Larry Laudan> uses the historical record to challenge a simple connection between success, reference and approximate truth.<A Confutation of Convergent Realism by Larry Laudan>[sections 3–5, success, reference and the demands on a convergent realist account]{ref} The challenge is serious for <scientific realism>, particularly when it treats an entire accepted framework as a single package of warranted beliefs.
But the slogan conflates several claims: that an unrestricted theory contains a false assertion, that every one of its assertions is false, and that its successes were accidental. The first does not entail either of the others. A theory can be wrong outside its proper domain while accurately representing relationships within it. Establishing that domain is itself a scientific achievement.
The <velocity-addition formula> makes the difference precise. For collinear <velocities>, <special relativity> gives
$$
u_{\rm rel}=\frac{u+v}{1+uv/c^2},
$$
whereas the <Galilean transformation> yields $u_{\rm G}=u+v$. Their difference is
$$
\boxed{u_{\rm G}-u_{\rm rel}
=\frac{(u+v)uv/c^2}{1+uv/c^2}.}
$$
For $|uv|/c^2\leq\varepsilon<1$,
$$
|u_{\rm G}-u_{\rm rel}|
\leq |u+v|\frac{\varepsilon}{1-\varepsilon}.
$$
This is a controlled agreement, including the case $u+v=0$, where a relative-error formula would be undefined. At $u=v=0.6c$, the classical value $1.2c$ differs dramatically from $15c/17\simeq0.882c$. At small <velocities> the same comparison explains why ordinary classical calculations work so well. The relativistic rule and its low-speed relation to the classical rule appear in <Albert Einstein>'s treatment of velocity composition.<On the Electrodynamics of Moving Bodies by Albert Einstein>[section 5, composition of velocities]{ref}
The example does not prove that <special relativity> is the final account of nature. It shows that a successor can explain an earlier theory's restricted success and predict when it fails. One obtains a stronger account of the evidence than either blanket rejection of the older theory or unqualified belief in it. The surviving mathematical relation also need not preserve the older interpretation of absolute time.
<Structural realism> develops this possibility by locating continuity in interpreted relations rather than every proposed entity. <John Worrall> uses continuity between <Augustin Fresnel>'s <optics> and later <electromagnetism> to motivate such a response.<Structural Realism: The Best of Both Worlds? by John Worrall>[discussion of the Fresnel–Maxwell transition and preserved mathematical relations]{ref} Its attraction is discrimination: some tested structure can survive an ontological change. Its difficulty is specifying which structure deserves belief before the next change, rather than selecting the survivors afterwards. An equation without a physical interpretation and domain is too weak a candidate.
An empiricist can accept the improvement in prediction and systematic representation while withholding belief about unobservables. That shows that progress does not logically require <scientific realism>. The realist can reply that independent tests, novel successful predictions and quantitatively explained limiting relations support cautious belief beyond the immediately observed. Neither side should treat historical examples as a deductive proof of its complete position.
The original claim is therefore defensible only in a thin sense: successive unrestricted theories may each contain falsehoods. Used to deny progress, it is misleading. Theories can become more accurate, discriminate more possibilities and explain why earlier approximations succeeded. Theory change should make scientific confidence selective and revisable, rather than erase the distinction between a useful approximation and an arbitrary mistake.
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