How compatible are the beliefs of atheists and Buddhism?

Referencing MischievousSage's answer I agree that the Buddha taught Buddhism in a way that's compatible with belief in the supernatural (and maybe taught it to people who had such beliefs a priori).

I think that Buddhism is also compatible (or that there are forms or versions of Buddhism compatible) with atheist beliefs.

In particular most of the central core of Buddhism, e.g. ...

• four noble truths
• noble eightfold part
• three-fold training
• harmlessness
• non-attachment
• three marks of existence
• see e.g. Dhamma Lists for more

... don't require belief in God.

In fact in contrast to some theistic religions I think that Buddhism teaches that your salvation depends on your own efforts ... it's not enough, in most forms of Buddhism, to pray for a divine intervention that will save you (there is Taking Refuge, but that's maybe not quite the same thing).

Though I agree with MischievousSage's answer I hope you don't mind if I also disagree with it e.g. to say that Buddhism is compatible with atheist or agnostic beliefs.

1. The central doctrines of Buddhism (see e.g. the lists above) don't mention God etc.

2. Saying that "the supreme beings are Buddhas" is I think true, but doesn't mean that Buddhas need to be viewed as gods (whatever that means), though they can be: e.g. you can admire (or reverence) your (earthly) teacher (or doctor) without viewing them as a god; and you might see the Buddha as "perfect" (e.g. in the "completed" sense of the word "perfect") without seeing the Buddha as "divine" (whatever that word is supposed to mean).

   There are definitions which don't involve God -- e.g. the Dhamma Lists, referenced above, describes "Buddha" as "both the historical Buddha and one's own innate potential for Awakening".

3. Yes there are stories of supernatural powers. I think it's possible to be agnostic ("without knowledge") about such subjects: to suspend belief or disbelief, or to treat it as an article of faith. Some people say that acquiring supernatural/psychic powers is possible via Buddhist practice, but is not the goal. Psychic powers aren't apparently routinely demonstrated to lay people, and the principles and practice of Buddhist may be important whether with or without them.

   I think there are forms of Buddhism which do expect or require or assume a belief in miracles (e.g., in Theravada, the ability of arahants to fly), that doesn't seem to be true of all schools of Buddhism; a story like The Real Miracle seems to me to downplay the importance of miracles.

   Also, believing that some people demonstrate supernatural (psychic) abilities, and belief in God, are two different topics.

4. Many of the suttas referenced in MischievousSage's answer aren't exactly arguing that God exists: instead I think they're arguing that kamma exists, and that a nihilist view like "it doesn't matter what we do, there are no consequences" would be a 'wrong view'.

5. I think there's a (at least one) sutta in which the Buddha does explicitly talk about (and refute the story of there being) a "creator God". If I recall/paraphrase that sutta correctly, I think he says, "Imagine an empty world, then a god comes along and finds himself alone and thinks 'I must have created all this', and then there are more Gods and eventually the Gods' time in heaven comes to an end." When I read that story I didn't take it as teaching that those events and personages literally happened in that way. Instead I took it as a story intended for people who are predisposed to believe that there is a creator God (perhaps because they've previously been taught that), and the story is telling them that even if there were such a God, that God would be subject to ignorance/delusion and to impermanence.

I checked the answers to What teachings do all schools of Buddhism share? -- and I think those answers confirm that "God" is not an important part of Buddhism (one of them says, "Whether Theravada or Mahayana, we do not believe that this world is created and ruled by a god at his will").

In all those answers there was (only) one line-item which someone identified as controversial, i.e. the Four Bases of Supernatural Powers. According to Wikipedia's Iddhipada article, "a major source of information on the iddhipāda is in the Samyutta Nikaya, ch. 51" -- or see e.g. here for further details about that, but I guess I'm going to end this answer here.

FYI belief in miracles and/or God[s] isn't the only belief that some people (non-believers) find difficult. Even so perhaps you needn't reject the whole of Buddhism, for having a doubt about one aspect of it.

From my perspective as a Soto Zen follower, Buddhism is extremely logical. The only faith required is that Buddhism as a system has merit and is worth investigating. Most of the "supernatural" elements are artistic embroideries to help shake people out of a common way of thinking, or are inherited from other religions that were more supernaturally focused.

For example, Buddhist texts talk of "hell" and "nirvana", which people in a theist society equate to the Christian heaven and hell. But that isn't the intent of what the Buddhists are saying. Both hell and nirvana are mental states, and are not externally imposed. If you say something you regret to your spouse, you'll likely enter a hell state (regret, shame) for a little while until you burn off that karma (action). If you are coding something awesome and you lose all track of time and your surrounding to where only you and the code exist... no mind, no thoughts of the past, no thoughts of the future, just connection with the code - that's nirvana. Like a hell state, it's temporary. You spill coffee on your computer, and boom, you're now in a different state.

In a single day, you'll live millions of lives in millions of states. Thus reincarnation can be considered the act of dying and starting over, or it can be interpreted within the course of a single life. I'm not the person I used to be, and in the future, I'll be someone else. That is a form of reincarnation.

So in short, I think Buddhism can be very compatible with Atheism. It's just a matter of overcoming all of the mystical sounding labels and getting to the true essence of the teaching - which I am over-simplifying, btw.

In my opinion most forms of atheism and agnosticism are incompatible with Buddhism.

In Buddhism the supreme beings are Buddhas.

The Buddha did indeed believe in superhuman powers (iddhis) and performed many miracles (like walking on water as if land). He also believed that a Creator of the universe really did exist, he just believed that the Creator mistakenly believed himself to be Supreme and was not worth worshipping, but certainly really does exist and had many good qualities.

The Creator (the Great Brahma) can be influenced by the Evil One (Mara, Satan) and is still under the cycle of birth and rebirth, but is a higher being with a stronger karmic influence, so the Creator is not Supreme like how Buddhas are Supreme.

In Buddhism Buddhas are superior to the Creator and other higher beings, are boundless, are all-knowing, and have superior superhuman abilities to even the highest beings in the universe (the Brahmas, Pajapatis, etc...).

As The Buddha said to a Brahma-being:

Thus I am not your mere equal in terms of direct knowing, so how could I be inferior? I am actually superior to you. - Brahma-nimantanika Sutta

In Buddhism, the Creator of the Universe (the Great Brahma or Maha Brahma) encourages people to follow the Buddha's teachings and praises the Buddha, it is the Evil One (Mara, the Buddhist equivalent of Satan) that tells people not to or distorts things away from the Buddha's teachings.

The Buddha had much more of an issue with atheists and skeptics of the time than Theists and believers of the time.

Wrong views, highlighted in many suttas seem to closely resemble the views of most atheists:

He has wrong view, is warped in the way he sees things: 'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no brahmans or contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' - Cunda Kammaraputta Sutta, (repeated in many suttas that this is a wrong view)

"Sariputta, when I know and see thus, should anyone say of me: 'The recluse Gotama does not have any superhuman states, any distinction in knowledge and vision worthy of the noble ones. The recluse Gotama teaches a Dhamma (merely) hammered out by reasoning, following his own line of inquiry as it occurs to him' — unless he abandons that assertion and that state of mind and relinquishes that view, then as (surely as if he had been) carried off and put there he will wind up in hell." - Maha-sihanada Sutta

Secularism Buddhism would be viewed as a wrong view.

In the Payasi Sutta, an arahant disciple Kumara debated with someone who has views resembling most modern day atheists:

Once everyone was seated, Prince Payasi said, "Reverend Kumara, I maintain that kamma does not have effects. I believe that there is no life after death, no world beyond our own. I think that angels and demons are things from a child’s dream." - Payasi Sutta

Kumara came up with all types of arguments against the atheist Prince Payasi.

The view that there is no afterlife is opposed in Buddhism numerous times, including in the Brahmajala Sutta:

Wrong view 51: "Here a certain ascetic or Brahmin declares and holds the view: ‘Since this self is material, composed of the four great elements, the product of mother and father, at the breaking up of the body is annihilated and perishes, and does not exist after death. This is the way in which this self is annihilated.’ That is how some proclaim the annihilation, destruction and non-existence of beings." - Brahmajala Sutta

So based on my research Theistic beliefs are more compatible with Buddhism. In Buddhism some Theistic beliefs are viewed as slightly wrong, but nearly as wrong as atheistic and materialistic beliefs.

So realistically the Buddha would view most Abrahamic religions as better or higher than most other religions but with slight errors. The story of Moses kind of reminds me of the story of the Great Steward in the Maha Govinda Sutta.

A lot of people often cite the Kalama Sutta but this is taken out of context as the Kalamas were non-Buddhists and The Buddha was addressing non-Buddhists, when The Buddha spoke to his own disciples he encouraged faith and right views.

Some believe that "Secular Buddhism" and other forms of Buddhism match into the prediction in the Ani Sutta:

"In the same way, in the course of the future there will be monks who won't listen when discourses that are words of the Tathagata — deep, deep in their meaning, transcendent, connected with emptiness — are being recited. They won't lend ear, won't set their hearts on knowing them, won't regard these teachings as worth grasping or mastering.

But they will listen when discourses that are literary works — the works of poets, elegant in sound, elegant in rhetoric, the work of outsiders, words of disciples — are recited. They will lend ear and set their hearts on knowing them. They will regard these teachings as worth grasping & mastering.

In this way the disappearance of the discourses that are words of the Tathagata — deep, deep in their meaning, transcendent, connected with emptiness — will come about." - Ani Sutta

Thus we see many people (specifically outsiders) out to replace Buddhism with flowery attractive elegant words rather than the Buddha's words, matching in nearly exactly to the prediction.

In the future Buddhism will probably be eliminated by people replacing the Buddha's original teachings with different newer forms.

I think at least some of the Mahayana writers were self conscious atheists.

This is from the Fudo-Myo Sutra and is one of my favorite quotes:

He is the markless Dharmakaya, identical with [all-encompassing] space itself, thus he has no dwelling. His only dwelling is in the minds and thoughts of living beings.

ref: http://www.chinabuddhismencyclopedia.com/en/index.php?title=The_Sutra_Spoken_by_the_Buddha_on_Arya-Acalanatha

So even in a system of Bodhisattvas, there is a subset of people who are skeptical about all the supernatural and see the Bodhisattvas (and all the other being unsubstantiated by evidence) as role models, pedagogical tools, rhetorical devices and so on.

This does not change the fact that there is a huge group of lay followers and historical monastics who clearly took the celestial Buddhas, Pure Lands, the realms, a semi-sentient Karma and a permanent, rebirthing soul as literal truths and true on account of faith.

In Basic Teachings of the Buddha, by Glenn Wallis (his old work that predates his current project), the author at points argues that the historical Buddha was an atheist and a nihilist that had no choice but to speak in the theological terms of the culture around him.

My own analogy is this-- it's like if you wanted to explain how evolution worked to a theist Christian, but didn't really care if the audience was atheist or not. Such a hypothetical teacher might teach about evolution as designed by God. In the same way the historical Buddha used the language of Karma, Dharma, Rebirth, Brahma to discuss a program of morality, restraint as part of a program to end suffering.

Buddhism is compatible as there is no notion of an almighty.

Buddhism revolves around the practice of Morality, Control over the Mind and Wisdom. So it is compatible regardless of what you call your self. May it be a Atheist, Christian, Muslim, etc.