How has mathematics gotten so abstract?

Romantic math anecdotes and culture

• Several commenters share stories of talking about infinities, the halting problem, or linear programming on first dates, which later became long-term relationships; math talk is framed as an expression of passion rather than showing off.
• Some note the social risk of “lecturing” on a date, but argue being authentically enthusiastic often works.

Infinities, existence, and foundations

• A long subthread debates whether claims like “one infinity is larger than another” rest on unstated philosophical assumptions.
• One side argues standard education silently commits students to ZFC-style set theory and a notion of existence that includes non-constructible reals and non-constructive algorithms, which many laypeople would find unintuitive.
• Others respond that:
  • Courses do introduce axioms and proofs early, and later work just builds on that.
  • Given a formal system like ZFC, talk of larger infinities is straightforward, and different philosophies (formalism, constructivism, Platonism) are just different “games.”
• Constructivist perspectives are explained: existence = constructability; all mathematically relevant objects can live in a countable universe (e.g., within the naturals), so uncountable ≠ “more” in the same sense.
• There is back‑and‑forth over whether non-constructive existence (“there must be an object, though we can’t describe it”) is meaningful or merely a convenient way to talk about possible worlds.

Was math always this abstract?

• Some say math has been abstract from the start: even counting cows is already abstraction.
• Others emphasize historical evolution: early mathematics was tightly tied to practical tasks; zero, negatives, and complex numbers were once seen as absurd; set theory and Cantor’s infinities, then Zermelo and Bourbaki, pushed abstraction much further.
• Euclid’s Elements is cited on both sides: as an early pure axiomatic treatment, and as still grounded in geometric diagrams and physical intuition.

Math vs science and proof

• A large subthread disputes whether mathematics is a “science”:
  • One camp: math is a formal science of proofs in axiomatic systems; science is empirical and falsifiable, so conflating them fuels public confusion about “truth.”
  • Another camp: both are systematic inquiries; math is just non-empirical science.
• Several note that proofs can be wrong, humans are fallible, and community checking (or proof assistants) functions analogously to experiment and replication.

Abstraction, intuition, and pedagogy

• Commenters stress that mathematicians rely heavily on intuition; abstraction often clarifies rather than obscures once one has the right mental models.
• Some criticize online cultures (including parts of StackExchange) for being impatient with requests for intuition, even though good intuition is crucial and hard to teach.
• There’s debate over whether abstraction and jargon are “gatekeeping” versus necessary compression to communicate precisely within a complex field.

Abstraction’s utility and links to CS/physics

• Many celebrate abstraction as a ladder: each layer (e.g., limits → calculus → linear operators, algebraic structures like monoids, groups, vector spaces) enables unification and powerful new tools.
• Examples include:
  • Graph minor theory giving nonconstructive polynomial-time algorithms.
  • Category theory, lattices, and monoids informing programming languages and type systems.
  • Coding theory and error-correcting codes built on highly abstract algebra.
• Some physicists and applied folks say they value analysis and concrete tools but “lose” interest when abstraction feels detached from physical models; others argue history shows abstract math later becomes indispensable.

Other side notes

• Zeno’s paradox and the coastline paradox come up as illustrations of how subtle infinity and limits are.
• Alternatives like constructivism and ultrafinitism are mentioned, with skepticism about their ability to support modern physics.
• Several point out that many “simple” areas (e.g., linear algebra, convex analysis) are relatively recent, so not all low-level math was solved millennia ago.