By this author’s account, in Kant’s 1780s writing there is a role for beauty and genius in mathematics, but in the 1790s he rejects this in the Critique of Judgement.
In science, Kant seems to regard Newton not as a genius but as someone whose achievements were the result of diligence. There are no geniuses in mathematics as it consists of rules to be learned (ref 7, 8). Before this period, genius was not so tightly bound up with taste and imagination (ref Paul Menzer 19, Giordanetti 20).
At the end of Section 1 of the CJ Kant states
Even if the given presentations were rational, they would stil be aesthetic if, and to the extent that, the subject referred them, in his judgement, solely to himself (to his feeling).
The author asks whether the objects of imagination must be sensible or whether they can be abstract. Kant speaks of the “representation of a object of judgement of taste as a given representation”, but mathematical representations may be thought of as constructed – the author understands Kant’s object of beauty as an empirical and sensible object, and adds his own ideas about ineffability and infinite complexity. I’m unsure about this – I thought that the Beauty/Sublime distinction was about being able to grasp the form of an object. A mathematical object however could be defined by “finite rules”. (Really?) He does note the infinite potential of a mathematical object (he’s focusing on a polyhedron, etc., here. Which aspects of a proof are decided by rules is, I suppose, Breitenbach’s way in).
There follows a discussion of subjective vs objective purposiveness (which were made distinct in Kant 1790). In 1772/3 Kant describes proofs as beautiful for their shortness, completeness, etc.