From the early Bronze era the most respectful burial custom was the cremation of the dead. We meet that custom in the Homeric epics and, throughout, the ancient Greek history we have examples that when a katharsis was needed, cremation was used. Read more on funerary and lamentation Greek customs here. The reason for cremation’s use is not, only, the environmental and hygienic efficiency and economy but a matter of metaphysical importance.

The Christian dogma, in the West, - in the East the Orthodox Church does not accept the concept of purgatory, but rather the “condition of waiting”, therefore prayers are only used for comforting the dead throughout their ‘waiting’ time – accepted from the early years the concept of purification (i.e. Origen). In the writings of Gregory of Nyssa (4th century) we read that no one can “approach God till the purging fire shall have cleansed the stains with which his soul was infested” (the earliest source that mentions the term ‘purgatory’ is of Benedictine Nicholas of Saint Albans to the Cistercian Peter of Celle in 1176). For the Catholic Church, purgatory, “is a place or condition of temporal punishment for those who […] are, not entirely free from venial faults”.

The Hellenic view of purification has two main differentiations from the Christian: a) there is no purification of the soul from any evil in this or the afterlife, b) and the concept of purification is not a punishment of any short.

Little more analytically: The soul of the dead Patroclus cannot pass to Ades because the other souls disallow him the entrance, for that reason ‘waits’ and plead to Achilles:

θάπτε με ὅττι τάχιστα, πύλας Ἀΐδαο περήσω. / τῆλε μἐέργουσι ψυχαί, εἴδωλα καμόντων, / οὐδὲ μέ πω μίσγεσθαι πέρ ποταμοῖο ἐῶσιν, / ἀλλαὔτως ἀλάλημαι ἀνεὐρυπυλές ϊδος δῶ. / καὶ μοι δς τὴν χεῖρ’ ὀλοφύρομαι. οὐ γὰρ ἔταὔτις / νίσομαι ἐξ Ἀΐδαο, ἐπήν με πυρός λελάχητε.

Thus, in the Homeric ages the burial (θάπτε με) was the cremation of the dead as, Patroclus, at the end clarifies (πήν με πυρός λελάχητε). The soul of Patroclus is not accepted to enter Ades because without cremation of the body is unpurified.

Now let as look at the case of Lake Acheronta and the third river called Pyrephlegethonta (Πυριφλεγέθοντα) as has been described in Phaido. It is important to understand the different concept and principle of the katharsis in the Christian and Hellenic view. Socrated, in Phaido (113d), articulates the reason for the Acheronta’s existence: διὰ πυρός κάθαρσις τῶν κακῶς βεβιωκότων and after οἰκοῦσί τε καὶ καθαιράμενοι τῶν τε δικημάτων διδόντες δίκας πολύονται, εἴ τίς τι δίκησεν. The underworld purification in the Hellenic concept had no connection with punishment (the κολαστήριον) but rather a well accepted and desired state of any Greek soul.

Today neo-Hellenes and individuals who believe on the concept of soul’s immortality and they wish for “κύκλου τ’ ἇν λήξαι καὶ ναπνεύσαι κακότητος, requesting from the contemporary Government to accept egalitarianism and cremation of the dead to be executed in Greece (see detailed article here).