Beyond Limits: Infinitesimals, Abstract Algebra, Category Theory, Synthetic Differential Geometry, and Automatic Differentiation

For centuries, mathematicians have been fascinated by the infinitesimal—quantities smaller than any measurable number yet not quite zero. These elusive entities powered the original breakthroughs of calculus in the hands of Newton and Leibniz, only to be set aside in the 19th century as mathematics pursued the rigor of limits. But the idea of the infinitesimal never truly vanished. This book tells the story of its remarkable return, tracing how a once-suspect intuition was reborn with modern precision and transformed into one of the most powerful unifying ideas in contemporary mathematics.

The journey unfolds across four interconnected landscapes: hyperreal numbers, which give infinitesimals a rigorous modern home; dual numbers, where differentiation becomes elegant arithmetic; synthetic differential geometry, which rebuilds geometry itself on infinitesimal foundations; and automatic differentiation, where these ideas drive the computational engines behind machine learning, scientific simulation, and finance. Along the way, the book develops the necessary algebraic foundations and situates these systems within a broader mathematical context. Together, they reveal how an ancient intuition has become both a cornerstone of modern theory and a practical tool shaping today’s computational world.