Grothendieck's Categorial Revolution

I consider Alexandre Grothendieck the central figure in the history of category theory. Although he did not originate the central concepts of category, functor and natural transformation, he clarified the way of thinking appropriate to these concepts, and within a short time period, thoroughly imbued mathematics with this way of thinking. The best place to start in learning about Grothendieck is the 2-part biographical article, Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre Grothendieck, by Allyn Jackson:

Grothendieck described his way of thinking as listening to the voices of things in a beautiful passage which expresses the heart of category theory more aptly than any other passage I have seen.

I would call a piece of mathematics categorial if it adhered to this method, even if the notion of a category was not used at all. And I would not call a piece of mathematics categorial if it did not adhere to this method, even if the notion of a category was heavily used.

Grothendieck's Spirit

After revolutionizing mathematics, Grothendieck started to become more conscious of the urgency of existential risk and the massive suffering that humanity inflicts on itself. He resigned from his post at the IHÉS when the director went back on his word that he would no longer accept military funding (Jackson, p. 1198). Thereafter he attempted to give up mathematics to focus on tasks he believed to be far more urgent: the imperatives of survival and the promotion of a stable and humane order on our planet. (ibid., p. 1202). However well-meaning, he was not very effective at politics. Ultimately, as his relationship with the mathematical community deteriorated, he left it behind for good, eventually becoming a hermit afflicted with paranoia towards his former mathematical colleagues. During this period he wrote multiple spiritual works, many of which are lost, unreleased, or untranslated; the portion of Récoltes et Semailles which has been translated into English is amazing and I implore the wealthy reader to support the efforts of the Centre for Grothendieckian Studies to continue the work of publishing and translating his spiritual writings.

Despite Grothendieck's withdrawal from mathematics, and despite his death, his spirit lives on within the mathematical community, and especially within the Applied Category Theory community. Not only has there been a mass adoption of his mathematical concepts and ways of thinking, but his politics also seem to arise wherever these are most apparent. It is as if his insistence, upon leaving the IHÉS, to give a mathematics lecture only if arrangements were made for him also to give a political lecture (ibid, p. 1202) continues to haunt any mathematical activity associated with his memory.

One additional aspect of Grothendieck's spirit which lives on is his honesty and commitment to conceptual and pedagogical clarity. He was never out to solve problems just for the sake of getting to the answer, rather he sought ever deeper understanding of the fundamental truths a given problem was based on. By uncovering these fundamental truths, he would achieve much more than solving a single problem; he would precipitate a paradigm shift in an entire subfield. Indeed, he notes that much of his work in mathematics was marked by an attitude of service: service to the mathematical community of writing clear and complete expositions that make fundamental and foundational ideas widely accessible (p. 1208). This last aspect seems to illuminate the connection between Grothendieck's mathematics and his politics, and show why one, when done correctly, will always involve the other. To listen to the voices of things with the utmost attention, to look for the most fundamental reality, one cannot avoid political or spiritual questions, and one cannot deny existential risk and suffering.

Unfortunately, during his life, in the Parisian mathematical culture, Grothendieck was not able to satisfactorily unite his mathematics with his politics and his spirituality. Instead, he renounced math, as the way that it was being practiced was incompatible with his deeper values, ultimately writing a letter in 2010 withdrawing his consent to any further publication or translation of his works.

Those who continue his work today still seek the unification of mathematics with politics and spirituality that he was unable to achieve.