The problem with “species” and “races” is that it is we (humans) that have invented them. Neither of them is real (ie, exist in reality). The problem with them is due to that reality is fundamentally continuous and thus can’t be partitioned into them. The problem with them is thus that they are impossible by that they lead into a paradoxical contradiction (ie, Russell’s paradox).
Now, Carl von Linné invented an orthogonal system of classification, meaning that it is diametrically opposed between every level (ie, species, genera, families, and so on), that avoided this paradoxical contradiction. This system may appear to solve the problem, but it doesn’t. It just avoids it by any possible route.
Unfortunately, this means that the Linnean system has infinitely many consistent solution of “species” and “races”, instead of the fundamental paradoxical contradiction (ie, Russell’s paradox). The system does thus solve the problem only if we can agree on one out of the infinitely many solutions, and agreement does unfortunately not appear to be the hall mark of humanity.
Fact is thus that “species” and “races” are concepts that are either paradoxically contradictory or ambiguous. It means that every logical reasoning that builds on these concepts is either contradictory or ambiguous. This is the fundamental problem for biological systematics – it is either contradictory or trivially true whatever it asserts.