Mathematics is hard for mathematicians to understand too

Understanding vs “getting used to” mathematics

• Several commenters distinguish between formal understanding (being able to follow a definition/proof) and deep internalization that makes ideas feel natural.
• Many describe understanding as a gradual, neurological “getting used to” patterns, not a single epiphany.
• Examples from functional programming (monads, adjoint functors) and advanced category theory are used to illustrate that definitions can be simple while their significance and connections take years to sink in.
• Some emphasize foundations (predicate logic, set and model theory, close reading) as giving a unifying mental framework that makes later math more comprehensible.

Breadth, depth, and specialization

• Math is portrayed as extremely old and vast; even professionals only deeply understand narrow subfields.
• Simple-to-state open problems (e.g., Collatz, Busy Beaver) are popular because most open problems require years of background just to parse the statement.
• Historical shifts in terminology (e.g., “abelian”) and styles (minimalist proofs vs contextual ones) add to the difficulty of reading older or cross-field work.

Notation: diversity, power, and pain

• Many complain about fragmented notation (sets, tuples, intervals—especially national variants), and the lack of a “single manual” akin to a programming-language spec.
• Others argue that such diversity is inevitable in a field as broad as math, and compare it to multiple programming languages and data-structure names.
• Strong defense of dense symbolic notation: it compresses repeated rewriting, supports pattern recognition, and often makes hard ideas easier, not harder, than wordy or code-like forms.
• Some insist notation is “the easy part”; others reply that overloaded, inconsistent symbols and the inability to even look up a symbol are serious entry barriers.

Gatekeeping vs necessary formalism

• One side suspects esoteric language and syntax in STEM (especially statistics and some math) function as gatekeeping or status signaling.
• Others counter that specialized language is primarily for precision and efficiency; cryptic papers are often genuinely describing cryptic ideas.
• There is debate over whether dry, formal presentation is needless hazing or a necessary way to remove ambiguity; several claim modern textbooks increasingly balance rigor with motivation and context.

Teaching, motivation, and examples

• Many criticize math education for poor motivation: definitions and lemmas are presented without explaining why they’re natural or important, unlike typical physics pedagogy.
• Repeated exposure, multiple proofs, and applications of a theorem are seen as what eventually make it “digestible.”
• Several call for more numerical or concrete examples, especially in research papers, to complement abstract formulations.

Proposals to “fix” notation and use tools

• Some wish to replace symbols and Greek letters with named functions and types, plus hyperlinked references to definitions and proofs.
• Pushback notes that verbose notation becomes unreadable and unwieldy for real proofs; formal proof assistants (Lean, Coq) show how ugly fully explicit versions can be.
• Others suggest AI/automation will increasingly handle tedious proof search, calculations, and optimization, while humans focus on conceptual understanding and structure.

Meta‑point: what math is for

• Several comments echo the article’s theme: the ultimate goal isn’t just proving theorems but fostering human understanding of structures and ideas.
• Notation, rigor, and automation are seen as tools; the real challenge is building and communicating intuition across an ever-expanding landscape.