reduction

Floor 3, Ce qui est su · introduced in Chapter 2, L’espace en creux

Which absences are the same fact seen twice?

An absence reduces to another when the book proves it from the other without reproving it; the reductions sort the absences into three clusters that meet only through bridges.

Plate 3.11The absences and the reductions between them, in three clusters, with the bridges that are their only meetings.

Definition(Reduction, common source)

An absence $A$ reduces to an absence $B$ , written $B\Rightarrow A$ , if the book proves $A$ from $B$ by an argument that does not reprove the content of $B$ . Two absences have a common source $C$ if both reduce to $C$ . When no reduction in either direction and no common source is known, the two are unrelated here.

In classical logic every theorem implies every other, so the relation has to mean something finer. These relations describe proofs, not truth, and unrelated here is a statement about present knowledge, not a theorem of independence.

Remark(Clusters)

The reductions fall into three clusters. The group of order 168: Galois’s window, the spinor window and the spin bundle, the real form, the plane and the line, Fano’s axiom and the Hurwitz bound, with common sources in Dickson’s list and the sign of $\mu=2h-2+\sum(1-1/m_i)$ . The octonions: Hurwitz’s theorem, the stop at the plane and the Jordan algebras, with common source the doubling lemma. Hilbert space: Gleason and Kochen–Specker, with common source the overlap of contexts. The separation of $\PSL(3,4)$ from $A_8$ stands alone.

Between clusters no reduction is known. They meet through bridges, each with its status: the 1344 automorphisms of the octonions that permute the units $\pm e_i$ induce all 168 automorphisms of the Fano plane of the units, with a kernel of 8 sign changes, a built bridge; the Bloch sphere and the celestial sphere are one $\Proj^1(\C)$ , a built bridge; and the three of Gleason’s threshold and the three of $\mathrm{PG}(2,2)$ share only a name.

Proposition(Gleason implies Kochen–Specker)

For each $d\ge3$ , Gleason’s theorem implies the Kochen–Specker theorem.

Proof

A valuation would be a frame function, hence of the form $e\mapsto\langle e,\rho e\rangle$ , which is continuous on the unit sphere. The sphere is connected, so a continuous function with values in $\{0,1\}$ is constant; but on any context it takes the value 1 once and the value 0 at least once.

Example

Two reductions meet at the group of order 168 and are complementary: Galois’s window at $p=7$ produces the double life $\PSL(2,7)\cong\PSL(3,2)$ , and the Sylow structure of $\PSL(2,r)$ excludes every other isomorphism between a plane group and a line group. Neither uses the other.

Fano’s axiom reduces to the statement that the Fano plane lives only in characteristic 2. So the double life joins characteristic 2, the plane, to characteristic 7, the line, and neither side can be moved: the plane side by Fano’s axiom, the line side because $r(r^2-1)/\gcd(2,r-1)=168$ only for $r=7$ .

Built from: absenceimprintbridge
Objects: the seven pointsthe seven linesthe sky
In the volume: XIVWhy Octonions