How has mathematics gotten so abstract?
6 hours ago
- #set-theory
- #infinity
- #mathematics
- Mathematics evolved from a natural science to an abstract discipline, moving away from reliance on physical intuition.
- Zeno’s paradox, involving infinite subdivisions of motion, was resolved by calculus showing infinite sums can converge to finite values.
- Giuseppe Peano formalized arithmetic with axioms, defining numbers recursively via a successor function, independent of physical reality.
- Set theory provides a foundation for numbers as ordered sets, extending to infinite ordinals and cardinals.
- Infinite ordinals like ω exhibit non-commutative addition (1 + ω ≠ ω + 1) and hierarchical structures (ω, ω+1, ω·2).
- Cardinality measures set size via one-to-one mappings, revealing different infinities (e.g., ℵ₀ for naturals, larger cardinality for reals).
- Cantor’s diagonal argument proves reals are uncountable, showing higher cardinality than naturals.
- Debates exist over the reality of infinite sets, with some mathematicians rejecting them entirely.