What Brouwer might have thought about the fate of algebraic number theory

Brouwer's philosophy can be reinterpreted as a kind of computational process platonism. Harsh was his attitude to what he considered the Hilbertian kind of mathematics. But with hindsight he was right if not about classical algebraic number theory, about what it eventually became. What destroyed the beauty of algebraic number theory was the inordinate importance and value (no pun intended) given to the unnatural and objectionable construction of the p-adic numbers (it is better to focus, as did Gauss and Kummer, on finite approximations mod $p ^n$ or Dedekind domains) and most of all, the monster-objects of ideles and adeles.  Ideles and adéles are, in their massive redundancy and spirit of brute-force (harmonic analysis), like a generative AI approach to number theory. Critics note that using the idele group can overcomplicate proofs that are simpler in classical ideal theory. Introducing infinite products over all completions adds heavy categorical and topological machinery that obscures basic algebraic intuition. Classical ideal theory and localizations of a field often solve finite extension and divisibility questions directly without needing adèlic frameworks. The natural topology on the idele group is finer than the subspace topology inherited from the adèle ring. Treating idèles simply as a subset of adèles fails to make them a topological group, requiring extra care with definitions. Translating the idèlic framework back to the classical Artin map in class field theory can feel artificial unless local class field theory is already fully established.  Valuations themselves appear to us to be very unnatural and ugly devices and it leads us to question the philosophical and logical status and legitimacy of certain kinds of "completion"  operation. Also we can question if the p-adic number fields deserve to be called number fields at all since they have neither the order nor topology nor cardinality  (from the point of view of certain finitist or continuum based foundations) of the natural genera of "numbers"    It is much nicer when "measures" emerge naturally from rings of algebraic integers at hand (discrete valuation rings as localizations of Dedekind domains). Completions seem to have their best use not for the very big but for the very small, for locality.