on Grothendieck’s hyper-platonism
Even the early Platonic dialogues—before the doctrine of the Forms was fully developed—taught that mathematical truths had an abstract, independent, real existence. In the Meno, a Socrates gets an innumerate slave to ‘remember’ the Pythagorean theorem.
Much later in the Platonic corpus, mathematical objects play a more direct role. In the Timaeus, solids like the tetrahedron & icosahedron serve as the models for the elements, they are the ‘unchanging, uncreated forms’ that ground tangible objects in motion like water & fire.
Scholars from the Tübingen School who sought to reconstruct Plato’s ‘unwritten’ / esoteric doctrines, found a more specific place for mathematical objects in Plato’s ontology.
Hans Joachim Krämer in _Plato & the Foundation of Metaphysics_ claimed that the fully elaborated Platonic metaphysics had *four principal levels of reality*: sensible objects in motion; mathematical natures; universals; & finally principles above being.
& of course even before Plato, Pythagoras taught his cult that ‘all is number’—an obscure doctrine but one that seemingly avows a belief in the independent existence of abstract mathematical objects.
Throughout history—really until the middle of the 20th century—most philosophers were ‘mathematical platonists,’ affirming the existence of mathematical objects & truths.
As Gottlob Frege put it in “17 Key Sentences on Logic”:
“ ‘2 times 2 is 4’ is true & will continue to be so even if, as a result of Darwinian evolution, human beings were to come to assert that 2 times 2 is 5.
“ ‘2 times 2 is 4’ is true & will continue to be so even if, as a result of Darwinian evolution, human beings were to come to assert that 2 times 2 is 5.
Every truth is eternal & independent of being thought by anyone & of the psychological make-up of anyone thinking it."
But today, nominalism reigns: according to the 2020 PhilPapers survey of academic philosophers, most philosophers believe that mathematical objects, structures, relations etc do not ‘exist’ at all,
that mathematicians don’t ‘discover things about reality’ so much as they construct valid sentences according to the rules of a formal language.
Recently I’ve become fascinated by the story of Alexander Grothendieck (1928-2014), an eccentric visionary who essentially created algebraic geometry by uncovering the deep structures
underlying both the ‘discrete, discontinuous analysis’ of algebra & the ‘continuous variation’ of geometry.
As Grothendieck put it in _Recoltes & Semallies_ or _Harvests & Sowing_, his massive unpublished manuscript from the 1980s:
“a synthesis between these two worlds, hitherto adjacent & tightly connected, but nonetheless separated: the “arithmetic” world, in which (so-called) ‘spaces’ with no notion of continuity live, & the world of continuous size, where ‘spaces’ in the proper sense of the term live.”
After a brilliant 20 year career in mathematics, Grothendieck abruptly withdrew to a small village in the foothills of the Pyrenees, hiding his location, erasing his contact information, moving frequently, & breaking his correspondence.
Grothendieck became a ghost, dedicated first to eastern & then to Christian mysticism… but all the while composing huge mathematical texts that are still being studied, such as _Pursuing Stacks_ (1983), a 600 page exploration of a generalized foundation for homotopy theory.
Michael Atiyah said of Grothendieck: “No one but Grothendieck could have taken on algebraic geometry in the full generality he adopted & seen it through to success.
It required courage, even daring, total self confidence & immense powers of concentration & hard work. Grothendieck was a phenomenon.”
Grothendieck writes about deep, mysterious 'rivers' into which geometry, topology, & analysis flow; he compares mathematicians to builders & pioneers but perhaps most strikingly to children who search unconsciously, unafraid, in totally naïve & wondrous ways.
To bring the thread back to mathematical platonism, Grothendieck clearly believed in the existence of mathematical objects, & went even further than that in _La Clef des Songes_ (1987):
In this passage, mathematical truths are but a minuscule, superficial part of God’s nature, a tiny window opened by human reason. Mathematical objects don’t just have an independent existence, but they point to something even more grand, though more obscure—the face of God.
In this sense, Grothendieck seems to go beyond the ‘normal’ mathematical platonism of a Frege & gesture toward an even more profound metaphysical reality that lies beyond numbers…
Ineffable, mute vastnesses of which the rich structures Grothendieck found, his sheaves & motifs & topoi, can only delineate the barest traces.
/fin
/fin
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