 yields the mathematical machinery of quantum mechanics. This is not particularly new; it is part of the mathematical folklore. Hermann Weyl outlined the lifting program by first considering an attribute on a set, which defined the set partition or "grating" [43, p. 255] of elements with the same attribute-value. Then he moved to the quantum case where the set or "aggregate of n states has to be replaced by an n-dimensional Euclidean vector space" [43, p. 256].13 The appropriate notion of a partition or "grating" is a "splitting of the total vector space into mutually orthogonal subspaces" so that "each vector −→x splits into r component vectors lying in the several subspaces" [43, p. 256], i.e., a direct sum decomposition of the space, where the subspaces are the eigenspaces of an observable operator. Weyl's grating metaphor also lends itself to (our own example of) seeing measurement of the, say, 'regular polygonal shape'of an 'indefinite blob of dough'as it randomly falls through a opening in a grating to take on that 'polygonal shape' (with the attribute-value or eigenvalue being the number of regular sides λ = 3, 4, 5, 6). Figure 5: Imagery of measurement as randomly giving an indefinite blob of dough a definite eigen-shape. Note how the blob of dough is objectively indefinite between the regular polygonal shapes and does not simultaneously have all those shapes even though it might be mathematically represented as the set {N,, . . .} or the superposition vector N++ . . . in a certain space. 13Note the lift from sets to vector spaces using the basis principle where the cardinality n becomes dimension n. 13 3.3 What is a vector space attribute? A set attribute is a function f : U → R (where the set of values is taken as the reals). The inverseimage f−1 (r) ⊆ U of each value f(u) = r is a subset where the attribute has the same value, and those subsets form a set partition. Given a basis set {bu}u∈U of a vector space V over a field K, we can apply a set attribute f : {bu}u∈U → K to the basis