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Your intuition is not all that wrong, but it’s not entirely correct either. Number Theory is basically the exploration of the deep consequences of a set of (ul
by qubex 7y ago
Your intuition is not all that wrong, but it’s not entirely correct either.
Number Theory is basically the exploration of the deep consequences of a set of (ultimately arbitrary but apparently obvious) axioms. The exploration of those consequences arises from the fact that they are ultimately too plentiful, ramified, and incompletely provable (per Godel’s Incompleteness Theorems & Turing’s Halting Problem).
- rumanator 7y ago> Number Theory is basically the exploration of the deep consequences of a set of (ultimately arbitrary but apparently obvious) axioms. Isn't that true for all fields of pure math?
- chess93 7y agoAlmost all fields of pure math use a much stronger set of axioms called ZF and essentially everyone also accepts the axiom of choice (making it ZFC). The axioms in ZF are reasonable but the axiom of choice is surprisingly controversial for an axiom. There are some unintuitive consequences of the axiom but even more unintuitive consequences without it or with the negation of it. Number theory uses a much smaller set of axioms. It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.
- lisper 7y agoNot entirely arbitrary. The Peano axioms turn out to be a really good model for a lot of real-world processes.
- qubex 7y agoThat’s why I meant by “ultimately arbitrary but apparently obvious”: mathematically, they are just one combination in an infinite set of internally consistent axiom sets, but apparently obvious insofar as they very accurately model our universe. Whether other universes could exist where such axioms do not obtain is something of an imponderable question.