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The best way to impress a philosopher is to tell him/her about ultrafilters. A filter on $\mathbb N$ is a set of subsets of $\mathbb N$ closed under intersections and taking super-sets. A maximal filter (under inclusion) is called an ultrafilter. There are plenty of those but nobody saw them since their existence depends on the axiom of choice. For every filter $\omega$ one can define the concept of convergence of sequences of real numbers: a sequence $b_n$ converges to $b$ if for every $\epsilon$ the set of $i$'s such that $|b-b_i|\le \epsilon$ is in $\omega$. If $\omega$ is an ultrafilter, then every bounded sequence of real numbers has unique $\omega$-limit. It is not true if $\omega$ is not an ultrafilter. The smallest interesting filter (called Fréchet filter) consists of all sets with finite compliments. The limit corresponding to that filter is the ordinary limit studied in Calculus. You can start from the Fréchet filter and add sets to it to produce bigger and bigger filters. Each filter gives you a Dirichlet-like density. If $\omega$ is an ultrafilter, then all sets will have density (between 0 and 1). Otherwise there will always be sets without density assigned.