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In Book I, third section, first chapter of Science of Logic, Hegel makes it rather clear, I find, that the measure that he is talking about -- at least in its aspect as specifying measure -- is a choice of (physical) units, hence is a choice of gauge. See §714:

A measure taken as a standard in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth’s diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.

What I am looking for some secondary literature that would highlight and pick up this thought of the (specifying) measure being related to gauge choice. Is there any?

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  • I'm unsure whether there's any relation of gauge theory to Hegel's logic, although I wouldn't be surprised if there is, given that, per Weyl, Kant's own theory of experience owes a lot to Leibnizian notion of symmetry as articulated by the principle of identity of indiscernibles (PII). Leibniz uses this principle to justify, e.g. his relational theory of space-time. Kant himself doesn't accept the principle (aswell as Leibniz's theory of space-time; Kant is an inertial-frame relativist, cf. first chapter of his Metaphysische Anfangsgrunde) but there seems to be a link between his... [1/n]
    – user73173
    Commented Apr 9 at 2:44
  • ...notion of intuition (Anschauung; it has nothing to do with "intuition" in the sense of "insight" or "mystical experience" or even "grasping something intuitively"; the term, as often is with Kant, is taken from scholastic vocabulary - intuitio - that is alien to the modern reader) and the notion of a symmetry. In one of his earlier papers (from 1768) Kant argues for absolute space from chirality. He later gave up the idea quite quickly, realising that his proof was fallacious, but it seems that he understood quite well, for someone living in XVIII th century, the significance of... [2/n]
    – user73173
    Commented Apr 9 at 2:52
  • ...the idea of symmetry etc. In the aforementioned chapter of the Metaph. Anfangsgrunde we find a proof which basically relies on the idea of invariance under a transformation, as later articulated by Klein etc. Another major premise for this reading of some of Kant's theory is found in the First Analogy of Experience of the first Critique where Kant basically articulates the idea that conservation laws are associated with symmetries (of course this idea isn't exactly new, it's articulated perhaps more explicitly already in Huygens). Carl Weizsäcker wrote an article about this. [3/n]
    – user73173
    Commented Apr 9 at 3:05
  • The discussion abover gives, I hope, a solid grounding for the thesis that Kant anticipates in some way later developments, ultimately related to Weyl and gauge theory. And he probably even influenced some of the developments or was influenced by them (he was definitely influenced by mathematicians such as A.G. Kästner, who taught Gauss, and J.H. Lambert - and he definitely influenced many later mathematicians, including Gauss, Riemann, Poincare etc. and Weyl himself). For example, the notion of a group was associated by people like Poincare with Kantian forms of intuition. [4/n]
    – user73173
    Commented Apr 9 at 3:14
  • Outside of this historical background, I never really thought much about these matters in more mathematical terms. There's an article by Marco Giovanelli: "Leibniz, Kant und der moderne Symmetriebegriff" which discusses these matters in broad terms, though. I'd also look at Weyl's philosophy book from 1929 and his "Raum-Zeit-Matterie", especially when he speaks Kantianese. There's also an article "Kant vs. Legendre on Symmetry" which seems quite good but I strongly disagree with the author that Kant failed to comprehend the mathematical significance of the problem and delegated it to... [5/6]
    – user73173
    Commented Apr 9 at 3:31

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