"For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?"
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Courant is pushing back against the temptation to treat mathematics as a museum piece: something you can define, categorize, and admire from behind glass. His line has the brisk impatience of a working mathematician watching outsiders (and a few insiders) argue about “the nature of math” while refusing to do any. The provocation is deliberate: the question “What is mathematics?” sounds philosophical, but Courant insists it’s answered the way you learn what music is-by playing, not by theorizing about sound.
The intent is partly democratic and partly disciplinary. “Scholars and laymen alike” collapses the hierarchy: even experts don’t get to bypass practice, and non-experts aren’t barred by credentials. Yet the doorway he opens is narrow: you enter through “active experience,” not armchair reflection. The subtext is a warning about sterile formalism and definitions divorced from use. Courant, whose career bridged the classical, problem-driven tradition and the 20th century’s surge in abstraction, is arguing that mathematics is less a set of doctrines than a way of thinking under constraint: making structures, testing them, feeling where they bend, discovering that rigor is earned in the doing.
Context matters: Courant helped shape modern applied mathematics and math education, and he wrote at a moment when foundational debates (logicism, formalism, intuitionism) could make math seem like a courtroom drama about axioms. His sentence refuses the trial. Mathematics, he implies, is a lived craft. If you want its definition, pick up the tools and let the work define itself.
The intent is partly democratic and partly disciplinary. “Scholars and laymen alike” collapses the hierarchy: even experts don’t get to bypass practice, and non-experts aren’t barred by credentials. Yet the doorway he opens is narrow: you enter through “active experience,” not armchair reflection. The subtext is a warning about sterile formalism and definitions divorced from use. Courant, whose career bridged the classical, problem-driven tradition and the 20th century’s surge in abstraction, is arguing that mathematics is less a set of doctrines than a way of thinking under constraint: making structures, testing them, feeling where they bend, discovering that rigor is earned in the doing.
Context matters: Courant helped shape modern applied mathematics and math education, and he wrote at a moment when foundational debates (logicism, formalism, intuitionism) could make math seem like a courtroom drama about axioms. His sentence refuses the trial. Mathematics, he implies, is a lived craft. If you want its definition, pick up the tools and let the work define itself.
Quote Details
| Topic | Learning |
|---|---|
| Source | Richard Courant and Herbert Robbins, What Is Mathematics? (1941). |
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