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Throwing things away — page 2

Progress made by deleting detail: the map that becomes a graph, the shape that becomes a number.

The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

Which axioms hold on which frames. A table of frames against modal axioms, each cell decided by checking the axiom under every valuation.

Logic

The axiom is the shape of the graph

Add one operator meaning necessarily and the choice of which axioms to accept stops being a matter of taste. Each candidate axiom is true of exactly those worlds-and-arrows diagrams whose arrows have a stated property, and a logic is a class of graphs.

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