Intersubjectivity as an Alternative to Realism and Relativism

a+0=a∧≢a

This outrageous claim is, in the terms of pure mathematics, indefensible. Indeed, mathematically speaking, it is incorrect — but why it is incorrect is important and worth exploring, because it highlights an ontological gap between abstract mathematical objects and real world states of affairs.

In pure mathematics, adding 0 to 1 yields a result that is equivalent to the original 1. That is, the 1 obtained as the result of 1+0 is identical to the 1 we started with, which can be expressed as 1+0≡1 , emphasizing their equivalence under addition.

Therefore, in purely mathematical terms, 1+0=1∧≡1.

In the real world, however, this is not the case.

Consider the example of having one apple and, for a moment of planck time, we add nothing to the apple. The apple’s fundamental structure has not been changed by us, but it is not fundamentally identical due to the continuous evolution of physical systems at all times. Furthermore, the fact of quantum decoherence means that the apple before we’ve added nothing to it and the apple after we’ve added nothing to it cannot be entirely equivalent, as the apple has interacted with its environment. Whether we add nothing to the apple or add something, the apple at t_n and t_n+x are not entirely equivalent at the quantum or physical level.

For any state of affairs in the world, this remains the case. Therefore, while it is mathematically incorrect, the statement a+0=a∧≢a correctly describes our world.

This, then, demonstrates a tension between the world of pure abstract mathematics and the instantiated world of states of affairs. What, then, is the gap between the two, and why is this important?

The statement a+0=a is a codification of a process that has happened to a, with the doer of that process being the subject and a being the object under question. We thus have bifurcated a from its context and specifically the context that has done something (here adding 0 to a) to it. This is imposing a dialectical relationship and is the fundamental structure of theories of objective and subjective relationships.

In the real world, all states of affairs are non-objects subject to the manifold of spacetime, which is itself a subject of states of affairs. Thus we can state that duration, change, relationship, identity, and variability are themselves conceptual means of understanding intersubjectivity — that is, any statement of states of affairs must abstract away some elements of the states of affairs to be examined (thus imposing an objective status on that subset). This can be considered the choice function of the subject who is attempting to map out the chosen state of affairs, and it will by necessity exclude some facts of the world.

That this is necessary to understand the world is an empirical truism, but what is more interesting and less often considered is the ontological implications of this fact. It suggests that objective states of affairs do not exist, and instead that intersubjective relationships between constantly continuous states of affairs is the ontological reality of our world.

Alfred North Whitehead’s process philosophy suggests that occasions of experience are themselves misconstrued as snapshots of the world, when in actuality the world is nothing but continual process.

While Whitehead is famous for being on the wrong side of the debate with Einstein on relativity, the observations above suggest that his process philosophy deserves a second look. For if pure mathematics does not describe our world because our world lacks ontologically fundamental objects, it also suggests that our world consists only of apparent objects as epiphenomenal appearances of the underlying process that is fundamental.

Furthermore, we can say that, ontologically speaking, states of affairs do not have an objectively real existence nor are states of affairs merely subjectively constructed and do not exist independent of any psychological state — realism and relativism are not wrong, per se, but are a foundational category error: ascribing reality to objective states when states of affairs are in fact intersubjective.

Heraclitus was right.