From
by Bas C. van Fraassen --
This 'problem of identical particles' and the options it allows for interpretation can be used as paradigm to motivate Ladyman's exploration, following Weyl, of invariance as definitive clue to structure. In the above 'identical particle' debate, the salient scientific fact is that the quantum mechanical state is permutation invariant. This may be glossed as :
only those features which are permutation invariant belong to structure -- if there are other features they belong to 'content'.
Any such other features there may be characterize what bears that structure, in a way which goes beyond what science describes.
Reading the scientists who write about this, we certainly see ample precedent for this gloss, though it remains generally quite unclear which options are being taken up. Thus Weyl:
Objectivity means invariance with respect to the group of automorphisms.
(Weyl, 1952, p.132; cited Ladyman 1998b, p.237)
Which topic and it consequences Weyl places in historical perspective:
the founders of modern science [...] discarded the sense qualities, on account of their subjectivity, as building material of the objective world which our perceptions reflect. But they clung to the objectivity of space, time, matter, and hence of motion and the corresponding geometric and kinematic concepts [....] But soon the objectivity of space and time also became suspect. (Weyl 1931/1950, p. 17; cited Ladyman 1998b, p. 237)
But this is treacherous ground, for "objectivity" has many antonyms. "Objective" can contrast with "subjective", "relative", perspectival", "unreal" and "scientifically insignificant". Thus, if electrons have 'haeccity' to individuate them, that is not invariant, hence (pace Weyl) not objective -- but it is neither subjective, nor relative, nor unreal in that case. On the other hand, length (being not invariant under Lorentz transformations) is relative, but not subjective, unreal, or even scientifically insignificant. And so forth.
Let me suggest a specious argument, of which no writer on this subject is guilty, but which might lurk behind an unsuspecting reader's comprehension:
A symmetry of an object is a transformation that leaves the object the same -- identical to itself. The paradigm examples are reflection through an axis of bilateral ('mirror image') symmetry, or a rotation through 360 degrees. But in physics we keep as many symmetries as are required by the dynamics. As this group gets larger, there is much in the original representation that is no longer invariant; then it appears as if the symmetries do not preserve all the structure there was in the object. But a symmetry of an object is a transformation that leaves the object the same -- identical to itself. So this appearance must be illusory or deceptive: there is nothing to the object except what is invariant. The representation 'clothed' the reality with appearances.
Stated so bluntly, this sophistry will not take in anyone. The correct response is of course to note that a symmetry is a transformation that leaves the object the same in all relevant respects. What are the relevant respects -- what are the inessential aspects, the irrelevant parameters that symmetries can vary -- is equivalent to the question of which transformations are the symmetries. But relevance is contextual. A parameter may be relevant in the solution of one problem and not in another. Two isomorphic groups can differ from each other; they just do not differ as groups, there is no difference if we take only the group operations into consideration. Symmetry, isomorphism, relevant sameness are all context-dependent notions.
References
Ladyman, James. 1998a. "Structural realism: epistemology or metaphysics?", Studies in the History and Philosophy of Science 29A: 409-424.
Ladyman, James. 1998b. Structural Realism and the Model-Theoretic Approach to Physical Theories. Dissertation, University of Leeds.
Weyl, Hermann. 1952. Symmetry. Princeton: Princeton University Press.
Weyl, Hermann. 1931/1950. The Theory of Groups and Quantum Mechanics. tr. H. P. Robertson, New York: Dover Publications.
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