Calculus, as we know it today, is the culmination of an intellectual journey spanning more than two millennia and drawing from diverse geo-cultural traditions. Within the confines of the Western canon, this journey begins with Zeno’s paradoxes (5th century BCE), which questioned whether continuous motion is possible, since it seems to require completing an infinite number of intermediate steps in finite time. Because motion and change defied intelligibility, Zeno argued that only what is unchanging can truly be said to exist, a claim that shaped two millennia of scientific and philosophical thought.

Over the following centuries, Greek mathematicians like Eudoxus and, later, Archimedes turned to what seemed like a feat of mathematical magic to calculate the areas and lengths of geometric figures. Their method broke problems into infinitely many vanishingly small parts, solved them individually, and then ‘integrated’ the results into a finite solution. Yet magic is never without a pact with something beyond: in this case, it invited into mathematics the unruly guest of infinity, both the infinitely large and the infinitely small. While endless progression (like counting) is, to a certain extent, intuitive, the notions of actual infinity (an infinite totality existing all at once) and infinitesimals (nonzero quantities smaller than any finite magnitude) pose indeed deep mathematical and philosophical challenges.

In the 17th century, Newton and Leibniz independently synthesized these ideas into the calculus we know today. While its computational power has far exceeded early expectations and underpins modern physics, its foundational concepts—the continuum, infinity, and the infinitesimal—remain open to both philosophical and mathematical inquiry.

"This workshop will introduce core concepts such as functions, limits, derivatives, integrals, and series. To do so, we will first revisit and deepen one of our earliest mathematical abilities: counting, though this time venturing into what the mathematician David Hilbert called Cantor’s paradise, a realm teeming with infinite infinities, arranged in an ecstatic, never-ending ascension. From there, we will trace the evolution of number systems, from whole numbers to numbers that are “less than nothing,” numbers that are not “rational,” continuous number systems that can be endlessly divided, and finally to “imaginary” numbers that complexify the linear order of previous systems with a lateral twist. Along the way, we shall toll the bells of PAF to unveil the elegance of Fourier analysis, a theory that allows us to synthesize sounds of any timbre from ever-ringing pure tones. If time allows, we’ll close with a brief glimpse into the beauty of differential geometry."

The lectures will be given by Gabriel Catren with the assistance of Manuel Eguía, Jonathan Lorand and Emily Roff.

The LIFE project is an experimental scientific education initiative carried out at PAF since 2018. It aims to collectively explore new ways of producing, sharing, and enjoying scientific thought — and, amid the current state of the world, of putting it at the service of life.

Further information and practicalities of attendance here.