Web of C-lief: conjectures vs. model assumptions vs. scientific beliefs

A sketch of the theoretical computer science Web of C-lief weaved by the non-contradiction spider.

In his 1951 paper on the “Two Dogmas of Empiricism”, W.V.O Quine introduced the Web of Belief as a metaphor for his holistic epistemology of scientific knowledge. With this metaphor, Quine aimed to give an alternative to the reductive atomising epistemology of the logical empiricists. For Quine, no “fact” is an island and no experiment can be focused in to resole just one hypothesis. Instead, each of our beliefs forms part of an interconnected web and when a new belief conflicts with an existing one then this is a signal for us to refine some belief. But this signal does not unambiguously single out a specific belief that we should refine. Just a set of beliefs that are incompatible with out new one, or that if refined could bring our belief system back into coherence. We then use alternative mechanisms like simplicity or minimality (or some aesthetic consideration) to choose which belief to update. Usually, we are more willing to give up beliefs that are peripheral to the web — that are connected to or change fewer other beliefs — than the beliefs that are central to our web.

In this post, I want to play with Quine’s web of belief metaphor in the context of science. This will force us to restrict it to specific domains instead of the grand theory that Quine intended. From this, I can then adapt the metaphor from belief in science to c-liefs in mathematics. This will let me discuss how complexity class seperation conjectures are structured in theoretical computer science and why this is fundamentally different from model assumptions in natural science.

So let’s start with a return to the relevant philosophy.