Serial Musings #3

I may be biting my tongue when I say this, but the emphasis on morphisms in category theory seems to be an interesting way to formalize this, especially as it is a function-oriented theory, which are processes applied to objects.

The difficulty here is to get rid of dependence on objects, or at least to be able to redefine them in a way that favours the flow of processes rather than static conceptions of immutable blocks of characteristics. Even so, some conception of identity is necessary, not as a timeless permanence or immutability, but as a transformation of oneself (in)to oneself over (some) time (period).

It seems to me that there is an interpretation of predicate logic in which predicates are as applications of functions over the subjects to which they apply. I may be wrong on this one, I honestly do not know.

In this way, we may attempt to define any entity as a nullary function that always returns the codified entity. Identity would be a singulary function on that entity that returns the entity itself, enabling the codification of time-dependent transformations. And if we start to consider time, we can make identity a binary function that depends on the time interval under consideration (or even ternary to account for the initial and final time interval under consideration).

It is important to stress that a process-oriented theory will fundamentally depend on time, but this does not betray our basic intuition: thinking depends on time, so it is no problem to keep operating with an underdefined concept of time. Especially because the difference between thinking and the thing thought is immanent in thinking itself, therefore, with the passage of time, corrections can be made, and we can achieve a better-defined concept of time (if not at least asymptotically).

The main problem to be faced by identity as function is the Humean problem of induction. Put simply, this is the impossibility of being sure about inductions, because things do not necessarily have to remain as they are. We can be convinced that they will be as they are – as formulated in Carnap's masterly book on probability – but to be convinced is not the same as to be certain (of the future).

Here comes into play the rigid designators explored by analytical philosophers (notably by Kripke). It seems that these do not depend on an immutable essence to maintain the link from name to thing (a link which, as we stated above, is internal to thought); rather, it seems these designators must only respect a threshold of (i)mutability.

How to determine this limit seems to be a specialized issue in each case (that is to say, a different threshold for each designator).

However, the proposal is made. Somehow, solving the problem of induction seems to lead to the resolution of the difficulties with the use of identities, or in other words, solving the problem of induction will free the notion of identity from the clutches of essentialisms.

Another aspect that seems to be essential: getting rid of geometric axiomatic based on the dot. Being able to develop spatial reasoning independent of the identification caused by the notion of dot will provide reasoning closer to the intuition of thresholds needed here.

Perhaps the notion of region, or space, or field, or environment, or something similar to these notions, is sufficient to formulate an expanded conception of identity that is independent of rigid equality. Here once again we are touching on the union between category theory and a functional approach: two morphisms are equal when their results are equal, even if the transformations operated in each one/by each one are different. It is something like a pragmatic equality, perhaps more appropriately called equivalence (literally: same value (at the end of the process)).

This resolution may betray our self-perception, it may hurt our ego, our pride, the feeling we have of ourselves, but it is not wrong just because of that.

The notion of an identity function (of oneself to oneself) is curiously similar to the notion of a bundle of perceptions from empiricism. A crucial point of divergence between the two proposals lies in the threshold of (i)mutability from which a bundle of perceptions ceases to be what it was and becomes another one. Would this be something similar to the passage from woman to mother? We do not know how to answer this for the time being, but the reflection is certainly valid, if not necessary.