Curiosities: two pairs of ideas that intrigue/trouble me

Troubles with theories

Evidence alone can’t determine which scientific theories we should believe since more than one theory will often be consistent with the available evidence.

In mathematics, there exist unintended (“nonstandard”) models of formal theories. An example theory where this is the case is Peano Arithmetic. The intended (“standard”) model is the set of (countably infinite) natural numbers (0, 1, 2, 3, …) and operations thereon that we know and love. But there are non-standard models of Peano Arithmetic that are uncountably infinite. That’s weird. In fact, anyfirst-order logic theory which has a countably infinite model also has an uncountably infinite model (upward Löwenheim–Skolem Theorem).

Boundedness of selves in spacetime

We often think of our minds as bounded by our skull. This has been challenged by the extended cognition thesis (Andy Clark and David Chalmers). The gist: we’re happy to accept that we can have beliefs that we’re not conscious of at any given time; they lie dormant until called upon for, say, an argument. Selves are more than what we are conscious of. But we also scribble stuff in notebooks and (these days) apps, set reminders, etc. These notes and reminders are similarly beyond consciousness but also thoroughly outside our heads. They seem essential for cognition.

There’s a problem with 4D block universe conceptions of spacetime: if all of time – past, present, and future – already exists at points in this 4D geometry then how do we consciously experience the passage of time? Assuming a material conception of conscious experience, each individual experience is scattered across spacetime and individually frozen. No passage. Natalja Deng (2019) points to an solution. Rather than trying to work out how these individual experiences can lead to an experience of passage, “recognize that the fundamental experiential unit is itself temporally extended, and use this to explain how there can be an experience of a temporally extended content.”