- Differentials and partial differentials over the real numbers work because one can pick two real numbers that are arbitrarily close to one another. The difference between any two real numbers can be made as small as desired. If you have a real function of a real variable then you characterize the change in the function’s real value as the value of the argument changes. This can be represented as the tangent to the curve representing the function at a given point.
In the Boolean domain there are only two values and they are always one step, unity, apart. There is nothing to differentiate. There is no variation in the spacing of the arguments or the function values. There are no curves in the Boolean domain. There is nothing to differentiate.
My last post is really just a note-to-self reminding me to get back to work on differential logic, my memory being jogged by a number of posts on the Azimuth Blog. But if I could nudge a few people to reflect on what the logical analogue of differential calculus ought to look like, that would be a plus.
The short answer to your objection is we don’t need limits in discrete spaces. We follow the example of the finite difference calculus, using logical analogues of the enlargement operator and the difference operator
A differential is a locally linear approximation to a function, that is, a linear function which approximates another function at a point. In boolean spaces, we know what the functions are, we know what the linear functions are, and all we need is a notion of approximation to define differentials. Yes, there are numerous tricky bits to work out in boolean spaces, but I worked those out in the array of expositions at many different levels of abstraction and detail to which I have linked before, as again below.
- Differential Logic and Dynamic Systems