Differential Propositional Calculus • 17

Differential Propositions • Tangent Spaces

The tangent space to $A$ at one of its points $x,$ sometimes written $\mathrm{T}_x(A),$ takes the form $\mathrm{d}A$ $=$ $\langle \mathrm{d}\mathcal{A} \rangle$ $=$ $\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.$ Strictly speaking, the name cotangent space is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.

Proceeding as we did with the base space $A,$ the tangent space $\mathrm{d}A$ at a point of $A$ may be analyzed as the following product of distinct and independent factors.

$\mathrm{d}A ~=~ \displaystyle \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n$

Each factor $\mathrm{d}A_i$ is a set consisting of two differential propositions, $\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},$ where $\texttt{(} \mathrm{d}a_i \texttt{)}$ is a proposition with the logical value of $\lnot\mathrm{d}a_i.$ Each component $\mathrm{d}A_i$ has the type $\mathbb{B},$ operating under the ordered correspondence $\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.$ A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type $\mathbb{D},$ whose sense may be indicated as follows.

$\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.$

Viewed within a coordinate representation, spaces of type $\mathbb{B}^n$ and $\mathbb{D}^n$ may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Resources

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