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I find number theory to be closer to a natural, investigative science akin to Physics rather than pure math such as calculus. IANAM, but I doubt there was ever
by air7 7y ago
I find number theory to be closer to a natural, investigative science akin to Physics rather than pure math such as calculus.
IANAM, but I doubt there was ever a modern calculus theorem that was hinted by a brute force search.
One can argue that Calculus is "made up" and thus research in this field is finding deeper logical consequences for the initial axioms, while natural numbers just "have properties" and research is Number Theory is trying to uncover these properties by brute force and later trying to prove them, arguably quite unsuccessfully so far.
It's an interesting tidbit in the philosophical question of "Does math exist only in our head?"
- qubex 7y agoYour 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.
- dntbnmpls 7y ago> I find number theory to be closer to a natural, investigative science akin to Physics rather than pure math such as calculus. And yet the man who gave us physics ( newton ) also gave us calculus. Rather calculus was invented to help describe and model physics and the natural world.
- ianai 7y agoThe study of calculus that you’re referring to is real analysis. Never quite thought of it as pure math though as it’s very applicable to many things - hence the “analysis.” Number theory is actually an example of pure math having real world impacts despite its nature. It’s very much about studying math for maths sake - though there are also people interested in it for their own applications. They’ve gone the brute force route with number theory because proving things in it is hard - in the sense that a hard number theory problem takes centuries to solve otherwise.