Science today is trapped between blurred motives and urgent demands. We invest billions expecting both paradigm-shattering insight and deployable technology, yet our funding systems seldom distinguish between those disparate aims. The Edge of Purpose argues that, in today's ecosystem, conflating curiosity-driven inquiry with technology development stifles each. Tracing the long arc from Aristotle to AI, the Element shows how post-war policy hardened an outdated linear model and how even refined schemes (e.g., Pasteur's Quadrant) still neglect knowledge that is valuable without being useful. A complementary 'Knowledge-Level' (KL) scale is proposed, parallel to Technology Readiness Levels, to map the maturation of understanding itself. Re-aligning incentives around distinct KL and TRL tracks, while building bridges for serendipity, can reinvigorate discovery, accelerate innovation, and restore public trust. Clear purpose, not bigger budgets alone, is the key to unlocking science's full creative power. This title is also available as open access on Cambridge Core.