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Effective Field Theories: A Case Study for Torretti’s Perspective on Kantian Objectivity

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Current Debates in Philosophy of Science

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Abstract

Those enlightened philosophers of physics acknowledging some manner of descent from Kant’s ‘Copernican Revolution’ have long found encouragement and inspiration in the writings of Roberto Torretti. In this tribute, I focus on his “perspective on Kant’s perspective on objectivity” (2008), a short but highly stimulating attempt to extract the essential core of the Kantian doctrine that ‘objects of knowledge’ are constituted, not given, or with Roberto’s inimitable pungency, that “objectivity is an achievement, not a gift.” That essential core Roberto locates in the Kantian notion of apperception, or self-activity, manifested in cognition in the idea of combination (Verbindung) or composition, which, Kant tells us, “among all ideas … is the one that is not given through objects, but can only be performed by the subject itself, because it is an act of self-activity” (B 130). I first rehearse Roberto’s proposal for how an imaginative interplay between sensibility and understanding can be fashioned via the productive imagination or power of reflective judgment (of the third Critique). In this way, the notion of composition in general, unfettered from needless period constraints issuing in “pure forms of sensibility” and “pure concepts of the understanding”, can be seen as the intellectual motor for the “free creation” of concepts celebrated by Einstein and others, furnishing structural scaffolding required to articulate and display physical objects and processes, a conceptual panoply that “cannot be fished out of the stream of impressions”. Roberto emphasizes that historical case studies are needed to evaluate his proposal, suggesting one himself, the continuous conceptual development inaugurated by Riemann’s Habilitätionsschrift (1854) resulting, some hundred years later, in the fiber bundle formalism of modern differential geometry and topology. I sketch a related suggestion, that the gauge groups of modern particle physics are the outcome of a similar line of conceptual advance, a structural scaffolding saving the phenomena of high-energy experiment within the framework of ‘effective field theory.’

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Notes

  1. 1.

    E.g., (2008, p. 89): “the 4 × 3 table of the categories … a millstone around Kant’s neck”.

  2. 2.

    Kant (1997), a translation of the 1781 (‘A’) and 1787 (‘B’) editions.

  3. 3.

    Torretti (2008, p. 90) cites Kant’s 1797 letter to Tieftrunk which refers to the “creative interplay between the sensibility and understanding” in the Critique of Judgment as “the schematism of the power of judgment”.

  4. 4.

    Cassirer (1929). All translations are my own. The corresponding text in the published translation by R. Manheim (1957) follows the semi-colon.

  5. 5.

    (1929, p. 349): “Und dies gilt nicht nur für die mathematischen Begriffen, sondern es stellt einen Wesenszug aller echten begrifflichen Strukturen dar. … daß ein neuer ideeller Bezugspunkt für dasselbe auggestellt wird. Indem das Besondere, das zuvor Auseinanderstrebende sich nach diesem Bezugspunkt richtet, wird ihm in dieser Einheit der Richtung eine neue Einheit des “Wesens” aufgeprägt – wobei ebendieses Wesen selbst nicht ontisch, sondern logisch, al seine reine Bestimmung der Bedeutung, zu nehmen ist.” Cf. (1957, p. 303)

  6. 6.

    Weyl (1923) is the comprehensive presentation; for discussion see Ryckman (2005) and the refences cited there.

  7. 7.

    Torretti (2008, p. 92) offers a similar suggestion: In place of “the pure forms of sensibility” the constituting subject has available “the intuitive ‘form’ in every Erlebnis [that] merely adumbrates the mathematical notion of a – should we say four-dimensional? – coordinate patch”.

  8. 8.

    See Hawkins (2000, Chapters 11 & 12) and Eckes (2013). ‘Lie algebra’ is the term coined by Weyl in 1934 lectures at the Institute of Advanced Study for what Lie, Killing and Cartan referred to as an “infinitesimal group”. Weyl showed that it designates a linear vector space structure at the identity of the Lie group from which most, not all, of the information of the group can be derived.

  9. 9.

    Weyl (1954, p. 628 & p. 627): “(T)he constructions of physics are only a natural prolongation of operations [the] mind performs in perception, when, e.g., the solid shape of a body constitutes itself as the common source of its various perspective views. These views are conceived as appearances, for a subject with its continuum of possible positions, of an entity on the next higher level of objectivity: the three-dimensional body. Carry on this ‘constitutive’ process in which one rises from level to level, and one will land at the symbolic constructions of physics. Moreover, the whole edifice rests on a foundation which makes it binding for all reasonable thinking: of our complete experience it uses only that which is unmistakably aufweisbar.” “…The words ‘in reality’ must be put between quotation marks; who could seriously pretend that the symbolic construct is the true real world?” In fact, Weyl uses the term ‘symbolic construct’ to encompass not merely the symbolic universe in which physical systems, states, transformations and evolutions are mathematically defined in terms of manifolds, functional spaces, algebras, etc., but also the symbolic specification of idealized procedures and experiments by which the basic physical quantities or observables of the theory are related to observation and measurement. It reflects an insistence, reinforced by quantum mechanics, that physical quantities (beginning with ‘inertial mass’) are not simply given, but “constructed” See especially Weyl (1931, p. 76).

  10. 10.

    The terms “local” and “global” are a bit misleading since the fields and their transformations ostensibly “live in” an internal dynamical space but the local gauge transformations are functions of the space or space-time coordinates at the given point.

  11. 11.

    Gauging a global symmetry mandates (to restore invariance of the field Lagrangian) introduction of a gauge-covariant derivative; the new derivative is required to transform in a manner that introduces a new (gauge) field; the gauge field provides the form of the interaction forces of a matter field. The same mathematical expressions appear with only minor changes in the different quantum fields of the SM, e.g., in place of the phase of the electron field in QED, there are generalized phases associated with the wave functions of multicomponent matter fields.

  12. 12.

    A symmetry of a system is said to be “spontaneously broken” if its lowest energy state is not invariant under the operations of that symmetry. This is an extremely important concept in the weak interaction as the bosons introduced by gauge symmetries are massless, like the photon; their masses arise from the “spontaneous breaking” of the SU(2) × U(1) symmetry through couplings to the scalar Higgs field.

  13. 13.

    Technical detail: Symmetry breaking allows the full electroweak gauge symmetry SU(2) × U(1)Y (‘Y’ for weak hypercharge) be replaced by the electromagnetic subgroup U(1)EM at low energies. The “broken” U(1) gauge group is a linear combination of the original U(1) and a subgroup of SU(2).

  14. 14.

    Besides the limitations to be mentioned, the measured values of some 26+ free parameters in the SM lack theoretical explanation, nor does the SM tell us why it contains three families of fermions.

  15. 15.

    Bain (2013) is an overview written for philosophers.

  16. 16.

    The Compton wavelength is the wavelength of the quantum wave associated with a particle of mass m;the mass of the electron is me.

  17. 17.

    Georgi (2009) provides a rigorous treatment, informally exposited here.

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Ryckman, T. (2023). Effective Field Theories: A Case Study for Torretti’s Perspective on Kantian Objectivity. In: Soto, C. (eds) Current Debates in Philosophy of Science. Synthese Library, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-031-32375-1_4

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