Pure mathematics has a remarkable habit of becoming useful. Often much later.

We tried to find counterexamples. Here are the ones still standing.

G. H. Hardy's A Mathematician's Apology (1940) states that number theory has no practical value, and that relativity doesn't either. Both examples were wrong within a lifetime: number theory became RSA cryptography, relativity became the correction GPS satellites apply to their clocks every day. The Introduction has that story in full.

Hardy is one of eight named, dated dismissals on record across the 1959 Mathematics Subject Classification's pure-math fields, restricted to those already established by 1960. All eight were wrong. Nineteen more fields were never dismissed by anyone and became essential anyway, at gaps from zero years to three centuries. Three have a real, documented application that's genuinely niche rather than mass-market — K-theory classifies topological phases of matter, several complex variables governs multidimensional filter and control theory, abstract harmonic analysis drives fast transforms on the rotation group — and one, set theory, has no external application for its distinctive research content (forcing, large cardinals), though its elementary form is the foundational language the rest of mathematics is written in.

What the full survey actually shows

Not “all pure math becomes useful” — that's a stronger claim than the data supports. What holds is narrower: every named dismissal on record has failed, eight times, independently, in fields that never spoke to each other. The gap between founding and first application ranges from zero years (potential theory was physics from its first page) to over three centuries (a sphere-packing question posed for its own sake in 1611 that fed lattice cryptography in the 2020s).