Cosmology

Mathematics and Its Notations

mathematics
notation
metawright
cascade
convergence
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cosmology

Every people that counts believes for a while that it invented counting. The record has met many peoples, and it has never once met a different arithmetic.

The Theorem and the Hand

The Archive files mathematics at two levels, and it keeps them apart because they answer to different authors.

The first level is what is true: the axioms of a conceptual realm, and what can be proved from them. A Metawright settles this, once, and every universe authored beneath that Metawright inherits it. Nobody in the Material layer chose it, voted on it or could have chosen otherwise.

The second level is how it is written: the marks, sounds, gestures or vibrations a people uses to hold a mathematical fact still long enough to pass it on. That is a notation, and a notation is made. It belongs to the people who made it, at the time and place they made it, and it carries their senses, their materials and their habits.

The record's shorthand for the distinction is the theorem and the hand. The theorem is discovered. The hand is invented. A history of mathematics is a history of hands.

What the Metawright Settles

The scope of physical law places the mathematical regime at the top of its ladder: the widest scope any lawful statement carries. A Metawright settles which mathematics there is. Formwrights instantiate lawful templates out of it, the Archewright assembles the archetypes a family of universes draws on, and Worldwrights build inside what remains. Physics sits inside mathematics, never the other way round.

Above the regime, Conseleme fixes only what any mathematics must satisfy. It does not choose among mathematics. And number itself begins at the Metawrights, who are individuals and can be counted; the tiers above them are neither one nor many, so a sum taken over them is not a hard sum but a category error.

Two things follow, and the record states both plainly. Within a regime, mathematics is discovered, not agreed: a theorem is true before anyone proves it and after everyone forgets it. And mathematics is not Ensemble-wide: a proof carried from one regime into another would arrive there as an assertion, with nothing underneath it. No instrument has been carried that far, and the record holds no observation of another regime's mathematics.

Found Many Times

The evidence the Archive has for the first level is the evidence it has for every tier above the Material layer: convergence. Peoples with no contact, no common ancestry and no shared vocabulary keep arriving at the same results. They count the same primes. They find the same ratios in a vibrating string. They meet the same impossibilities, and they meet them in the same places.

The instruments that came home from Counterpane, the neighbouring archecluster whose Archewright drew a different quantum kernel, brought back a different physics. They brought back no different arithmetic. That is what the scope ladder predicts: an archecluster is a narrower thing than a regime.

Convergence is why a first contact can begin before anyone shares a word. Two peoples who have learned nothing of each other's speech can still establish that the other side counts, and what it can prove. Mathematics is the one vocabulary they hold in common before either has learned the other's, and the reason is that neither of them wrote it.

Many Hands

The hands differ more than the theorems ever have. A few that the record holds:

The Chthonari. The Chthonari feel the world through rock rather than hear it through air, and for them a ratio between two integers and a harmony between two frequencies are close to the same fact. Their mathematics is carried in the Undersong, in tuned resonance rather than written symbols. Federation mathematicians who have studied under Chthonari tutors arrive at the same theorems by a different road: what took them years of abstraction is, for a Chthonari student, nearer to hearing.

The plainmark. The Orbital Habitats Compact's plainmark counts decks and berths with tally strokes, and those strokes are read by every people on a habitat whatever language they speak. They work because number is not a language. A tally is the barest notation there is, and the barest notation travels furthest.

Earth. The numerals the Solar System carried into space were not one people's. They were assembled over many centuries by several peoples, each adding what the others lacked: a place for zero, a positional order, a symbol for an unknown. The record notes this only to make the general point. No notation in its holdings was made by one hand, and none of them is the mathematics it writes down.

What a Notation Can Do

A notation cannot make a theorem true or false. What it can do is make a theorem cheap or expensive to find. A people whose hand makes a result easy to see will find it early. A people whose hand hides it may not find it for centuries, and will then find exactly the same thing.

That is why the order in which peoples discover mathematics differs so much, while the mathematics they discover does not. It is also where translation goes wrong. A notation carries habits of thought with it, and a translator who reads the habit as the content can mistake a different hand for a different truth. The Corps' xenolinguists treat a mismatch in mathematics as a mismatch in notation until it has been shown otherwise, and it has never been shown otherwise.

What the Record Does Not Claim

The record does not say which notation is best. Each was made for a body, a medium and a need, and the tally on a hatch is as good a hand as the Undersong for the work it does.

It does not say that mathematics is the same everywhere. It says that mathematics is the same everywhere the record has been, which is inside one regime, and that past the regime's edge the question cannot be put in a form an instrument could answer.

It does not hold that anything is made of mathematics. A being exists under a regime, as something its mathematics describes; the regime is not a place in which anything lives. So a creature of pure number, a mind living in the axioms, or a pattern alive in a proof is not a rare thing still waiting to be found. It is a category error, the same kind as counting the Upper Structure. What looks like one is something physical that mathematics describes well, and the record files it as that.

And it does not say what a Metawright is like. It says what a Metawright does, from inside the work, the way the Archewright is read only in the handwriting.