"I know that the Chaldaeans and Indian sages were the first to say that the soul of man is immortal, and have been followed by some of the Greeks, particularly by Plato the son of Ariston." (Pausanias, Description of Greece 4.32.4)

Now that's an arresting passage. An ancient author - albeit one who wrote a half-millennium after Plato's death - stating outright that Plato derived one of his most important doctrines from India. But the fact that an ancient author claims something does not, of course, make that thing true, and very few classicists think that Plato derived any of his core doctrines from non-Greek sources. Ancient claims to the contrary reflect an impulse as old as Greco-Roman civilization: to see the ancient cultures of the east as a source of deep, if sometimes dangerous, wisdom.

The Greeks and Romans assumed that Plato traveled widely, and there is no particular reason to doubt that he did. We don't really know where he went, since our sources about his life - with the partial exception of the spuriously autobiographical Seventh Letter - are late and unreliable. Egypt was then (depending on the time of Plato's visit) either part of the Persian Empire or under the rule of the native 30th Dynasty. In either case, Egypt was - though not nearly to the extent that it would be in the Hellenistic and Roman periods - connected by trade with India. It is conceivable that Plato could have encountered someone who had been to India, or someone who knew someone who had.

But there is no direct evidence - in Plato's own works or those of his contemporaries - that any Greek writing before Alexander's conquests had a substantive understanding of Indian religion or philosophy. There were of course rumors and reports about the far east - one thinks of Herodotus' gold-digging ants, said to live in the deserts of northern India - and a few authors active during or before Plato's lifetime produced works purporting to describe Indian customs. With the exception of Herodotus, these authors survive only in excerpts. They do not seem, however, to have been especially accurate; the most (in)famous of them, Ctesias, was apparently responsible for the myth of the skiapods, men who hopped around on a single enormous foot, and then (when wearied by hopping) used their feet as umbrellas as they napped. What, if anything, Ctesias had to say about Buddhism is unknown, but it is unlikely to have been inspiring.

Nor is there any internal evidence for Buddhist doctrines in Plato's works - or so Richard Stoneman concludes in his recent book on the Greek Experience of India. There was the potential for real intellectual cross-fertilization between the Greek and Indian traditions; Stoneman, for example, thinks that the philosophy of Pyrrho of Elis was deeply influenced by Buddhism. Pyrrho, however, was born a generation after Plato, and supposedly accompanied Alexander to India. He was, in other words, exposed to Indian philosophy in a way that only became possible in the wake of Alexander's conquests.

During the Hellenistic period, a considerable number of Greeks in the Indo-Greek kingdoms would convert to Buddhism (the Questions of King Menander are the most famous product). But in the Mediterranean world, Buddhism remained an ill-understood religion, known - if at all - through the distorted mirror of Manicheism or the late antique fable of Barlaam and Joasaph. If Plato knew anything substantial about the Buddha or Buddhism, in other words, he kept it to himself.

You may be interested in /u/artfulorpheus post:

• Were there any interactions between Buddhism and Europe prior to 1000AD?

It doesn't quite have the same emphasis as /u/toldinstone's post (/u/toldinstone is a Classicist looking East, /u/artfulorpheus is an Indologist looking West) and doesn't directly deal with things as early as Plato, but /u/Toldinstone writes, "there is no direct evidence [...] that any Greek writing before Alexander's conquests", so you can think of /u/artfulorpheus picking up where /u/toldinstone leaves off.

I’m going to take a different route from others who have commented and write more on the history of philosophy and the possible connection between Indian/Buddhist and Greek philosophy, which I am more familiar with. In Platonism the idea of reincarnation is central to Platonic ontology and epistemology. In the Phaedo, written by Plato, which recounts Socrates last dialogue with Cebes and Simmias before his death, Plato presents arguments in favor of reincarnation. As stated by Plato in the Phaedo, “The living come from the dead, just as the dead come from the living; and if this is true, then the souls of the dead must be in some place out of which they come again.” In the Phaedo, Plato’s argument for the immortality of the soul, and its reincarnation in a body, is as follows: “If generation were in a straight line only, and there was no compensation or circle in nature, no turn or return into one another, then you know that all things would at last have the same form and pass into the same state, and there would be no more generation of them.” Therefore, the immortality of the soul, as well as its reincarnation in a body, is necessary for the continuation of life and existence.

However, it is highly unlikely that Plato derived such ideas from Indian philosophy, and especially Buddhism. If we’re looking for Indian influence on Greek philosophy, and here it is still unlikely, we could possibly look towards Pythagoras, who Plato likely derived the idea of reincarnation and the immortality of the soul from. According to Herodotus in his Histories, the idea of reincarnation began in Egypt: “The Egyptians were the first to declare this doctrine, too, that the human soul is immortal, and each time the body perishes it enters into another animal as it is born. When it has made a circuit of all terrestrial, marine, and winged animals, it once again enters a human body as it is born. Its circuit takes three thousand years. Some Greeks have this doctrine, some earlier and some later, as if it were peculiar to them. I know their names, but do not write them.” Moreover, Porphyry, in The Life of Pythagoras, writes that Pythagoras introduced the idea of reincarnation and the immortality of the soul into Western philosophy: “It was especially well known by all that first he declares that the soul is immortal; then it changes into other kinds of animals; in addition that things that happen recur at certain intervals, that nothing is absolutely new, and that all things that come to be alive must be thought akin. Pythagoras seems to have been the first to introduce these opinions into Greece.”

Pythagoras, for much of his early life, studied extensively in Egypt, which sat at the juncture between Mediterranean and Asian civilizations. Pythagoras also travelled to Babylonia, and spent some time there, where it is possible but not likely that he came into contact with Indian philosophy. Thus, it is possible (but not likely) that Pythagoras learnt of reincarnation from a second hand source, who was familiar with Indian philosophy, in either Egypt or Babylonia. Although Pythagoras lived before the alleged time of the Buddha, the idea of reincarnation was likely already established in Indian philosophy. Considering that there are so many conflicting accounts of Pythagoras’ life however, we cannot say for certain who Pythagoras learned of reincarnation from, and whether or not this person learnt of reincarnation from Indian philosophers. While Pythagoras is credited by several writers with having introduced the idea of reincarnation and the immortality of the soul, Plato was largely responsible for popularizing the idea in Western philosophy through his dialogues.

Ultimately, it would be unwise to definitively assert that Pythagoras, and by extension Plato, learnt of reincarnation from Indian philosophers, and particularly Buddhism. What is most likely, given the available historical records, is that the idea of reincarnation which Pythagoras learnt of in Egypt, was already well known and established in Egypt and neighboring civilizations, and whether or not these civilizations adopted this idea from Indian philosophy is of course open to further investigation. It is possible that different cultures and religions came up with their own conception of reincarnation independently of one another, or perhaps the idea really did originate in India and was introduced over the centuries into other cultures. However, this is mere speculation and should be taken as a factual assertion as to the origin of the idea of reincarnation and the immortality of the soul. I hope this helps you develop a more informed understanding of Platonic philosophy and the influence of Egypt, and possibly India, on Western philosophy. If anybody has any corrections they'd like to offer, feel free to do so. This is my first time posting.

Edit: For another possible connection between Greek and Indian philosophy, which is also a disputed connection, we can look to Ammonius Saccas, who was the teacher of Plotinus, the founder of Neoplatonism. There is scholarship that supports the idea that Ammonius Saccas was of Indian origin. Those who believe that Ammonius was of Indian origin point to the efforts of Plotinus to travel to Persia and India after the death of Ammonius, which indicates that Plotinus may have been aware of Persian and Indian philosophy through Ammonius. Supporters of the theory that Ammonius was of Indian origin also highlight the similarity between the names Saccas and Sākyas, an ancient Indian clan. However, Ammonius Saccas left no philosophical writings behind and no memoir, and so he is a very obscure figure in the history of philosophy and little is known for certain about who he was and where he came from, asides from the fact that he was the teacher of Plotinus.

Works Cited:

Curd, Patricia. (2011). A Presocratics Reader: Selected Fragments and Testimonia. Second Edition. Hackett Publishing Company, Inc.