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:
(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.
the imperatives of survival and the
promotion of a stable and humane order on our planet.
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
(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 attitude of service
: service to the
mathematical community of writing clear and complete
expositions that make fundamental and foundational ideas
widely accessiblelisten 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.