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<p>Noether’s first theorem demonstrates that continuous symmetries give rise to conserved quantities (under appropriate conditions). This fact tempts many to hold that symmetry principles explain conservation laws. Yet there is a puzzle: the derivation goes both ways. So why does symmetry explain conservation when the derivation is bidirectional? Lange (Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38(3):457–481, 2007, Laws and lawmakers: Science, metaphysics, and the laws of nature, 2009) provides an answer: symmetry principles are <i>meta-laws</i>, and meta-laws explain first-order laws just as first-order laws explain facts. Using a “non-standard” Lagrangian Smith (Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 39(2):325–345, 2008.) claims that conservation of angular momentum can hold without rotational symmetry, providing a counter-example to Lange. In this paper, I show that Smith’s non-standard Lagrangian fails to serve as a counterexample. However, that doesn’t leave Lange’s account unchallenged. I argue that the debate between Lange and Smith ultimately revolves around an ambiguity which, once clarified, leads to a dilemma. <i>Which</i> symmetry principle explains? Is it the symmetry of the action or the symmetry of equations of motion? If the former, then the symmetry is no more stable than conservation laws. Hence, we lose the desired explanatory direction. If the latter, the symmetry lacks explanatory relevance and fails to exhibit greater stability than conservation laws. However one disambiguates ‘symmetry’, it remains mysterious why symmetry principles explain conservation laws.</p>

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Are symmetry principles meta-laws?

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  • 发表日期：

    2025年5月22日

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  • Shelly Yiran Shi

摘要

Noether’s first theorem demonstrates that continuous symmetries give rise to conserved quantities (under appropriate conditions). This fact tempts many to hold that symmetry principles explain conservation laws. Yet there is a puzzle: the derivation goes both ways. So why does symmetry explain conservation when the derivation is bidirectional? Lange (Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38(3):457–481, 2007, Laws and lawmakers: Science, metaphysics, and the laws of nature, 2009) provides an answer: symmetry principles are meta-laws, and meta-laws explain first-order laws just as first-order laws explain facts. Using a “non-standard” Lagrangian Smith (Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 39(2):325–345, 2008.) claims that conservation of angular momentum can hold without rotational symmetry, providing a counter-example to Lange. In this paper, I show that Smith’s non-standard Lagrangian fails to serve as a counterexample. However, that doesn’t leave Lange’s account unchallenged. I argue that the debate between Lange and Smith ultimately revolves around an ambiguity which, once clarified, leads to a dilemma. Which symmetry principle explains? Is it the symmetry of the action or the symmetry of equations of motion? If the former, then the symmetry is no more stable than conservation laws. Hence, we lose the desired explanatory direction. If the latter, the symmetry lacks explanatory relevance and fails to exhibit greater stability than conservation laws. However one disambiguates ‘symmetry’, it remains mysterious why symmetry principles explain conservation laws.