It is not uncontentious to say that philosophical problems are timeless. To some it looks like an excuse for examining problems which in fact can have no answer because there is something wrong in considering them as ‘problems’ in the first place.
However, the subject of philosophy certainly acts as if philosophical problems are timeless. Certain topics may be more of a central concern at a particular time, but that is mainly a function of fashion.
The central topics and questions come round again and again. Rarely is it the case that a matter considered by philosophy is wholly dismissed, or the way it was once treated regarded as valueless. Quite the contrary.
Philosophers find themselves going back to philosophers of the past at the least to use their ideas on certain topics as starting points, but often much more than that. A book that considers the nature of justice will naturally find itself looking to see what Plato had to say. The problems of induction and causation normally involve discussing Hume in depth. The starting point for considering the nature of mind is often Descartes.
It’s far from clear that progress is made in philosophy as in some other subjects. In this sense philosophy is quite unlike science – a chemist would rarely find any value in checking to see what another chemist said about something a hundred years ago.
So one may wonder what is the point of philosophy in this case if it does not definitively solve problems. As suggested already philosophical problems arise when we start to think deeply about our most fundamental beliefs. When we do so we often find that we neither fully understand the content of those beliefs, nor have any clear justification for holding them. For a certain kind of mind this is perplexing and the problems will not go away through the acceptance of glib answers or in response to a dismissive frame of mind.
We may not be able to present final solutions, nevertheless we can come to a conclusion that is a result of the best thinking on a certain matter.
I would conclude that philosophical problems are timeless by virtue of their profundity, generality and, as a consequence of that, the uncertainty surrounding the very methods by which they may be best approached. The result is that the problems do not die, nor do the ways of attempting to solve them or at least deal with them.
One thing is pretty certain: the issue of whether philosophical problems are timeless is itself a philosophical problem.