The epistemology of abstractionism

I examine the nature and the structure of basic logico-mathematical knowledge.
What justifies the truth of the Dedekind-Peano axioms and the validity of Modus
Ponens? And is the justification we possess reflectively available? To make progress
with these questions, I ultimately embed Hale's and Wright's neo-Fregeanism in a
general internalistic epistemological framework.
In Part I, I provide an introduction to the problems in the philosophy of mathematics to motivate the investigations to follow. I present desiderata for a fully satisfactory
epistemology of mathematics and discuss relevant positions. All these positions turn
out to be unsatisfactory, which motivates the abstractionist approach. I argue that
abstractionism is in need of further explication when it comes to its central epistemological workings.
I fill this gap by embedding neo-Fregeanism in an internalistic epistemological
framework. In Part 11, I motivate, outline, and discuss the consequences of the frame-
work. I argue: (1) we need an internalistic notion of warrant in our epistemology
and every good epistemology accounts for the possession of such warrant; (2) to avoid
scepticism, we need to invoke a notion of non-evidential warrant (entitlement); (3)
because entitlements cannot be upgraded, endorsing entitlements for mathematical
axioms and validity claims would entail that such propositions cannot be claimed to
be known.
Because of (3), the framework appears to yield sceptical consequences. In Part 111,
I discuss (i) whether we can accept these consequences and (ii) whether we have to
accept these consequences. As to (i), I argue that there is a tenable solely entitlement-
based philosophy of mathematics and logic. However, I also argue that we can over-
come limitations by vindicating the neo-Fregean proposal that implicit definitions can
underwrite basic logico-mathematical knowledge. One key manoeuvre here is to acknowledge that the semantic success of creative implicit definitions rests on substantial
presuppositions - but to argue that relevant presuppositions are entitlements.