Profiles feuds between such figures as Gerolamo Cardano and Tartaglia, Pierre de Fermat and René Descartes, Sir Isaac Newton and Gottfried Wilhelm Leibniz, and David Hilbert and L. E. J. Brouwer, discussing their respective arguments, their motivation, and the mathematical legacies of the discoveries over which they fought.

Tartaglia versus Cardano. Solving cubic equations -- Descartes versus Fermat. Analytic geometry and optics -- Newton versus Leibniz. Credit for the calculus -- Bernoulli versus Bernoulli. Sibling rivalry of the highest order -- Sylvester versus Huxley. Ivory tower or real world? -- Kronecker versus Cantor. Mathematical humbug -- Borel versus Zermelo. The "notorious axiom" -- Poincaré versus Russell. The logical foundations of mathematics -- Hilbert versus Brouwer. Formalism versus intuitionism -- Absolutists/platonists versus fallibilists/constructionists. Are mathematical advances discoveries or inventions?.