Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm to find it?

Take the example by Noam Elkies from the comments: let $V$ be the vector subspace of $\mathbb{R}$ over $\mathbb{Q}$ spanned by $\lbrace 1,e,\pi \rbrace$. Should we believe that $\dim(V)$ is either $2$ or $3$, even if we don't know which?