Is there an ur-philosophy behind the philosophy of Pythagoras and Lao Tze Tung?

Question

Chapter forty-two of the Dao De Jing has the following:

The Dao engenders One

One engenders Two

Two engenders Three

And Three engenders the many myriad things

This looks rather like how Pythagoreanism is explained in Plato. The One, the Dyad and so on. Does this mean that both Pythagoros and Lao Tze Tung were referencing some primodorial ur-philosophy or that somehow it arose in one of these places and somehow over the distance on continents they managed to influence the other. Or is the most likely story is that they arose independently. Is it possible to make a plausible case for any of these options?

The above is glossed in the translation by Richard Ames and David Hall as

The Dao gives rise to continuity

Continuity gives rise to difference

Difference gives rise to plurality

And plurality gives rise to the manifold of everything that is happening.

Answer 1

There's a theory, going back to the Renaissance, but most closely associated recently with author Aldous Huxley, that all valid philosophies relate back to a single common core, independent of traceable common origins. This is called the perennial philosophy. If it interests you, Huxley has a book of that same name that explicitly references both ancient Greek and ancient Chinese philosophy.

If you're looking for actual historical relationships, it's not generally believed that ancient Chinese and ancient Egyptian/Greek/Roman philosophy had any direct influence on each other. They are usually considered to have developed independently, despite the coincidence in time periods. (On the other hand, the line of influence in philosophy from Egypt to Greece to Rome is well-attested, although not uncontroversial, due to its political and racial overtones in modern discourse.)