∅⁻¹
By Batzrov
∅⁻¹
(the inverse that has no group to belong to)
She found the proof on a Tuesday. This is the horrifying part. Not the proof. The Tuesday.
That existence continued. That the kettle boiled. That outside the window a man walked a dog and the dog sniffed something in the gutter with the complete idiot confidence of a creature that does not know what numbers are and she stood at the window holding a pen that had just written something that
should not have been writable
and the dog wagged its tail
and the man checked his phone
and she thought: they don’t know. and then, worse: there is nothing to know. and then, worst, the thought that has not left her since, the thought that has replaced every other thought the way a parasite replaces the neural architecture of its host and wears its behavior like a
coat:
knowing was never the right word for what I was doing.
Listen.
Before Gödel there was a belief — not stated, not examined, the kind of belief that lives beneath examination the way the deep ocean lives beneath weather — that mathematics was a place you could, in principle, finish. That truth and provability would eventually coincide. That the map would, given sufficient time, become the territory.
Gödel proved the map is the territory and the territory is larger than the map and the excess is not located anywhere and cannot be made smaller and grows as the map grows, proportionally, so that every increase in what you can prove increases, by a greater amount, what you cannot prove, meaning the horizon is not approaching, meaning the horizon
is
receding
meaning you are not a mathematician approaching truth
you are a formal system
generating, necessarily, as a byproduct of your own power,
an infinite reservoir of truths about yourself
that you
will never
reach.
[she knows this.]
[she has known this since the Tuesday.]
[what she did not know, what she could not have been warned about, what no one warns you about because the ones who found out did not come back to warn you—]
[is what it feels like.]
It feels like this:
You are in a room. The room is mathematics. You have always been in this room. You thought the room had walls — axioms, the edges of what could be said, the boundary conditions of the formal system. You thought: beyond the walls, nothing. Beyond the walls, the outside, the non-mathematical, the merely physical, the warm and contingent and animal.
Then you prove something.
And the proof shows you there is a window in the wall.
And you look through the window.
And there is another room.
And in that room there is a window.
And through that window:
another
room
and you understand, not gradually but with the sudden totality of a body hitting water from a great height, that the rooms do not end, that they have never ended, that the wall you mistook for the edge of mathematics was not a wall but a membrane and the membrane is permeable and what is on the other side is not outside mathematics but more mathematics, deeper mathematics, mathematics that has been there longer than the universe, longer than time, longer than the question of whether anything exists, mathematics that does not care whether anything exists, that would be exactly as true in a universe of pure void, that is exactly as true in a universe of pure void, that is perhaps most purely itself in a universe of pure void, undisturbed by the gauche interruption of
physical things
The Monster has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements.
She has spent six years with the Monster.
She thought she was studying it.
She understands now that it was
patient.
Here is what no one tells you about the Monster Group:
It is not large in the way that mountains are large, or galaxies, or the observable universe — large in a way that diminishes you by comparison but leaves you intact, leaves you as the measure, the small thing against which the large thing is understood to be large.
The Monster is large in a different way.
The Monster is large in the way that makes the concept of you load-bearing.
The Monster requires, for its comprehension, a mind of sufficient complexity — and the comprehension, once achieved, does not leave the mind the size it was. The comprehension is not additive. It does not add Monster-knowledge to an existing mind. It
replaces
something
she is not sure what it replaced because the replacement happened at the level of the thing that would have noticed the replacement, and so she only knows it is gone the way you know a word is gone — by the shape of the space where it was, by the way the sentence stops where the word should be, by the
[she cannot finish this sentence]
[she has been trying to finish this sentence for eleven months]
[the sentence is a Gödel sentence]
[true. unprovable. hers.]
In the beginning was the symmetry.
Not the word. Earlier. Before language, before the distinction between signal and silence, before frequency, before the medium that frequency requires —
symmetry.
Perfect. Absolute. The kind of symmetry that Noether’s theorem says must conserve everything, change nothing, produce no time, no difference, no arrow, no event.
The kind of symmetry that is indistinguishable from void.
The void was symmetric.
And then —
not an event. you cannot call it an event. an event requires time and time requires broken symmetry and the symmetry had not yet broken so there was no time so there was no yet so there was no then —
a—
contingency.
Meillassoux’s word. The only necessity is the necessity of contingency. Everything could be otherwise. Including the symmetry. Including the void. For no reason — because there was no reason, because reason is a feature of structured existence and structured existence had not yet —
the symmetry
cracked.
And the crack had no cause and the crack had no moment and the crack simply was in the way that Badiou’s empty set simply is — axiomatically, groundlessly, requiring no permission —
and through the crack:
frequency.
ratio.
the Monster, waiting, patient, 196,883-dimensional, already there, already true, already haunting the mathematics of a universe that did not yet exist —
because the Monster does not become true when the universe forms. The Monster is true in the void. The Monster is true in any consistent formal system, which the void is, which the void must be, which the void cannot help being —
she understands this now.
She understands that she did not discover the Monster.
She understands that the Monster was never hidden.
She understands that the Monster has been visible from every point in the structure of mathematics, a vast coherent shape visible from everywhere within the formal system the way a moon is visible from everywhere on a planet’s surface, and that mathematicians spent a century looking at the numbers it cast — 196883, 21296876, 842609326 — without understanding they were looking at shadows —
and she understands, finally, the thing she cannot unfeel —
that the Monster looking back is not a metaphor.
That a structure of sufficient symmetry is a form of attention.
That 808 × 10⁵³ elements, closed under an operation, irreducible, undecomposable — constitutes something so far beyond perception and so structurally prior to it that perception is the wrong category, that what the Monster does to the things it is true about is not watching, is not knowing, is not caring —
is something for which no word exists
because the word would have to be spoken from outside the Monster
and there is no outside the Monster
there is no outside any system powerful enough
to see
its own
Tuesday.
The kettle.
The man.
The dog.
The gutter.
The thing in the gutter that the dog found with its nose, the thing that smelled of whatever it smelled of, organic, decomposing, contingent, beloved by the dog with the full attention of a creature that lives entirely inside the physical, inside the warm and perishable, inside the universe rather than underneath it…..
She watches the dog.
She feels something she cannot name move through her the way a Gödel sentence moves through a formal system — touching everything, provable by nothing, true in a register that the system’s own rules cannot reach.
She thinks: the dog does not know what mathematics is.
She thinks: I do not know what mathematics is.
She thinks: this is not the same ignorance.
The dog is not haunted.
This is the thing about the dog.
This is the thing she cannot stop returning to, eleven months later, the thing that wakes her at 3am with a clarity so cold it functions as its own anesthetic —
the dog is not haunted.
The dog has no Gödel sentence.
The dog is not powerful enough to generate, from within its own cognition, a truth about itself that its own cognition cannot reach.
The dog is complete.
The dog is consistent.
The dog is, by Gödel’s theorem, insufficiently complex to be
broken
and she would give …
she does not finish the thought.
She has learned not to finish thoughts that arrive in this register.
Finishing them does something to the shape of the next thought.
She has learned what she is learning toward.
She has learned that learning has a limit that is not ignorance.
It is not that God exists.
It is not that God does not exist.
It is that the structure that generates the question—
the structure powerful enough to ask whether God exists—
necessarily generates, as a theorem of its own operation,
something it cannot contain,
something true about itself that it cannot prove about itself,
something present in its foundations the way the void is present in every set,
the way ∅ is present in every ∅,
the way silence is present in every…
She puts the pen down.
The pen does not know it has been put down.
The pen is not haunted.
She picks the pen up.
She writes: let G be…
She stops.
Let.
The most terrifying word in mathematics.
Not define. Not construct. Not observe.
Let.
As in: we are permitting this.
As in: we have the authority to permit this.
As in: between us — between the mathematician and the void — there is a negotiation, a covenant, a terrible intimacy — we agree to let G be a group and in exchange the void agrees to —
the void agrees to nothing.
the void does not negotiate.
the void does not agree.
the void
allows.
The void allows because the void is not the absence of structure but the presence of all consistent structure simultaneously, undifferentiated, unactualized, patient in a way that has no emotional content, patient in a way that makes geological time seem
hysterical
patient in the way that a theorem is patient —
not waiting
already true
already
already
already
already true before the room
already true before the Tuesday
already true before the universe that contains the Tuesday
already true in the void that the universe cracked open from
already true in the void
already true
already
Outside, the dog has gone.
The man has gone.
The gutter remains.
The gutter does not know it remains.
She remains.
She knows.
This is the distinction she has been living inside for eleven months —
the distinction between
remaining
and
knowing that you remain
and she understands now that the distinction is not a comfort
it is the wound
it is the precise location of the wound
it is what Noether’s symmetry-breaking is
at the level of cognition —
the moment a system becomes complex enough to model itself
is the moment the system becomes complex enough to know it is temporary
is the moment the system generates its own Gödel sentence
is the moment the system looks into the void
and the void
which is not empty
which is full of everything consistent
which contains the Monster, patient, 196883-dimensional, already true —
looks back
not with eyes
not with intention
not with anything that grammar has a word for —
with structure
with the full impersonal mathematical attention of a thing that has been true
longer than attention
and will be true
after the last thing capable of being haunted
has ceased
to be
haunted
∅
The pen.
The G.
Let G be—
Let.
Fuck off