Examples
- The Weil conjectures, which used an analogy between algebra and geometry to develop the machinery of algebraic geometry used for counting numbers of solutions of equations over finite fields.
- The Vojta conjectures, which use an analogy between complex function theory and arithmetic to give arithmetic predictions about the structure of solutions of equations over the ring of integers and the field of rational numbers.
Quotations
Weil (1979), quoted in @1997mazurConjecture
Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when this illusion dissolves: the presentiment turns into certainty; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time. Metaphysics has become Mathematics, ready to form the material of some treatise whose cold beauty has lost the power to move us.