Conclusion

We have presented Chu spaces, as $A\times X$ matrices, from a perspective that emphasizes their rows as representing their elements. Elsewhere we have developed the analogy with topology by emphasizing their columns, which can be understood as generalized open sets.

The advantage of the row perspective is that topology is not as widely appreciated as it could be, with the result that most people find it easier to think about the elements of a set than about the open sets of a topological space. Not only are open sets relatively unfamiliar, but they also transform strangely, namely backwards.

Functions $f:{\cal A}\to{\cal B}$ are naturally presented in terms of elements, namely by listing the values of f(a) for all elements $a\in A$. This is then an A-indexed list of values of ${\cal B}$ each represented in ${\cal B}$ as a word of length Y. A function is therefore representable by a word of length $A\times Y$, and the space ${\cal A}{\relbar\mkern-9mu\circ}{\cal B}$ of all continuous functions from ${\cal A}$ to ${\cal B}$ is represented by a Chu space of width $A\times Y$. This is all the information we need to give about ${\cal A}{\relbar\mkern-9mu\circ}{\cal B}$ in order to equip it with exactly the right structure for it to soundly interpret linear implication.

One topic we did not treat is the Stone gamut, described in detail elsewhere [Pra95]. The Stone gamut coordinatizes transformational mathematics, which we understand as dealing with transformable objects and its associated logic of transformation, defined in terms of (di)natural transformations. There are two dimensions. In the horizontal direction range from the discrete, namely sets, to the coherent, namely complete atomic Boolean algebras or dual sets. In the vertical direction objects are classified according to the $\Sigma$ by which they are represented.

This picture locates sets at one ``edge'' of the mathematical universe, and dual sets at the other, with all other structures in between. One can think of sets as made up of discrete atoms or particles (in the abstract sense of being dual to waves rather than in the more concrete ``particle zoo'' sense of particle physics) and dual sets as consisting of coherent waves. In the middle are the square Chu spaces, mens sana in corpore sano, such as finite-dimensional vector spaces, Hilbert spaces (suitably transforming), complete semilattices (join or meet is immaterial), locally compact Abelian groups, and finite chains with bottom.

Abstracts of the above and other papers by the author and colleagues, together with links to their postscript versions, may be found on the web site http://boole.stanford.edu/chuguide.html.