We have started by accepting a single axiom on the validity of axioms, and all of the relevant circularity. We must now be careful though, for we currently have no criteria for which other axioms we accept. In fact in our current state we are allowed to choose any and however many axioms we want. Don’t feel like arguing for a particular proposal? Just take it as an axiom!
This epistemic explosion puts no limits on what we can claim as fundamental truth. As with it’s opposite, epistemic nihilism, there isn’t actually anything philosophically wrong with this. We are still effectively too close to the circular trap to be able to make that kind of judgement. However, like epistemic nihilism, accepting an epistemic explosion just doesn’t seem practical or useful. If we take a nihilistic view then there is nothing more we can say, whereas if we take an explosive view then we can say literally anything. Either way, we’re philosophically finished. Frankly, that kind of view is just boring.
So, we will add a second axiom to our set, based on a fairly well-known principle: Occam’s Razor. A lot of the historical details around Occam’s Razor are rather fuzzy, including who said it first, whether Ockham said it at all, and why the spelling has changed. These questions are not really relevant to the underlying principle though, which is often stated as “entities must not be multiplied beyond necessity” (Wikipedia claims this formulation is due to John Punch).
The razor has many other various formations, but they all basically boil down to “simpler is better”. Or more practically, if you’ve got two possible explanations for a thing that do an equally good job on all points, pick the explanation that only requires a paragraph, not the explanation that requires thirty pages, two diagrams and an appendix. With all that in mind, we can formulate our second axiom:
Axiom 2: The fewer axioms you need, the better.