To many in the atheist community, and to many in the Hindu community, the notion of Hindu atheism sounds like an oxymoron.

How could someone possibly be a Hindu (a demarker of religious identity) and also an atheist (whose worldview revolves around the negation of these identities)?

This is understandable, considering that many of the major global religions that have become successful through proselytization, include belief in a personal God as the major prerequisite to join their club. For instance in Christianity, the Nicene creed which laid out the foundation of Christian orthodoxy states

"We believe in one God,
      the Father almighty,
      maker of heaven and earth,
      of all things visible and invisible.

And in one Lord Jesus Christ,
      the only Son of God,
      begotten from the Father before all ages,
           God from God,
           Light from Light,
           true God from true God,
      begotten, not made"

The Nicene creed goes on to lay further criteria for this new theological system- and that being the immaculate conception, the death and resurrection of Christ, his ascent to heaven and so on.

All christians alive today, regardless of denomination, believe in this creed, be it Eastern orthodox, Greek Orthodox, Catholic, coptic or of the hundreds of protestant denominations that thrive within the west today. To be a christian would be to believe in this doctrine, and there is no way around it.

Similiarly, the other major religion on the planet, Islam also requires the recitation of the Shahada to be initiated into the faith.

“I bear witness that there is no God but God (Allah – i.e. there is none worthy of worship but Allah), and Muhammad is the Messenger of Allah.”

Clearly it is not possible to be an atheist under these two faith systems as it would clearly contradict the most fundamental prerequisite doctrine. Therefore, atheism in the west (including the middle east) emerged as a result of a rebellious rational uprising against traditional orthodoxy of these faith systems which held it back in many instances.

The classical world was not like this however. In many cases, there was theological tolerance and mutual acceptance of each other's gods and goddesses, even when people of different cultures met. Megasthenes was known to have written about the mythological parallels between the stories of Krishna and that of Hercules. How they both slayed a multiheaded snake, and carried a giant mountain etc. In fact Megasthenes referenced Heracles as "Hari-kula-es" meaning he who is of the clan of Hari.

Even in ancient India, despite the local Gods and Goddesses initially being different, the theological acceptance let them merge together as the same deity and allowed people to worship local Gods in other areas as their own. For example, the idea of a common "Shaktism" ensured that people from all across Bharatavarsha could worship the local Goddess as a manifestation of parvati without any difficulty.

In fact this theological tolerance was not merely limited to a God/Gods of a specific variety. Monotheism, polytheism, monism, dualism, qualified monism, qualified dualism, agnosticism, atheism and many other strains of philosophical persuasions were included in Hinduism as seperate Darsanas (school of thought). Each of these darsanas had several millenia old history. In fact, 6 of the 10 ancient darsanas did not believe in God- the Carvakas, Jains, Buddhists, Mimamsas, Sankyas, Yogas. So clearly it is possible to be a Hindu atheist if the majority of Hindu schools of thought are atheistic themselves.

Among them, the most radical were the Carvakas who rejected the authority of the vedas, belief in Gods, belief in afterlife and karma etc. They were materialists in the same way that modern atheists claim to be.

In fact, the Nasadiya sukta of the Rig Veda states

"But, after all, who knows, and who can say
Whence it all came, and how creation happened?
the gods themselves are later than creation,
so who knows truly whence it has arisen?

Whence all creation had its origin,
the creator, whether he fashioned it or whether he did not,
the creator, who surveys it all from highest heaven,
he knows — or maybe even he does not know."

This level of philosophical sophistication was found in the oldest of times, in the oldest of texts among the Indian corpus of literature. The text that is supposed to be authoritative for the asthika darsanas of hinduism is itself sceptical about claims of divinity.

Now, this isn't a question of theological or philosophical consistency. It is a question of socio-political considerations. Atheism as a movement distinct from occidental religion arose due to the nature of occidental religions in keeping atheists seperate and outcasted from their societies. However atheism in India, especially among the Hindu society doesn't operate on the same considerations and must therefore be understood to be a movement embedded deeply within Hindu society and not seperate from it.