Understanding Fock space

Let’s start with a single particle Hilbert space $H$ with basis ${|alpharangle}$, where $alpha$ represents a complete set of quantum numbers. If we now take two particles of the same type, we must consider the space $$H_2=H_1otimes H_1,$$ and more generally $n$ particles live in $$H^{otimes n}=H_1otimes…otimes H_1.$$ The (free) Fock space is now defined by taking the direct sum $$mathcal{F(H)}:=bigoplus_{nge 0} H^{otimes n}.$$ I understand why we take the tensor products: we wish to describe the whole system of $n$ particles at the same time. However, it is not at all clear to me why we also need to take the direct sum. What is its physical interpretation?