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There is an explanation of Buddhist conditionality in the Bundles of Reed Sutta (SN 12.67) below, together with an analogy.
Name-and-form (mind-body) is a condition for consciousness. Consciousness is a condition for name-and-form. If any one of these ceases, the other will cease too. This is like two bundles of reed leaning on each other.
However, the rest are linearly dependent. Name-and-form is a condition for the six sense fields, the six sense fields are conditions for contact, and so on.
But the dependency model here does not mean that they arose sequentially in time. What is more likely the case, is that they arose together. And they will cease together.
Please see SN 12.2 for the definition of what name-and-form, consciousness etc. refer to.
However, please remember that these refer only to the clinging aggregates (SN 22.48), and not the non-clinging aggregates. A living arahant does not have the clinging aggregates, because there is no longer clinging. However, the living arahant still has functioning aggregates. This is discussed in Iti 44 (also please read Ven. Thanissaro's comments in the footnote to the Iti 44 translation).
There's another bundles of reed relationship between ignorance (avijja) and fermentation (asava). Please read this question for details.
From SN 12.67:
Well, Reverend Sāriputta, are old age and death made by oneself? Or by another? Or by both oneself and another? Or do they arise by chance, not made by oneself or another?”
“No, Reverend Koṭṭhita, old age and death are not made by oneself, nor by another, nor by both oneself and another, nor do they arise by chance, not made by oneself or another. Rather, rebirth is a condition for old age and death.”
“Well, Reverend Sāriputta, is rebirth made by oneself? Or by another? Or by both oneself and another? Or does it arise by chance, not made by oneself or another?”
“No, Reverend Koṭṭhita, rebirth is not made by oneself, nor by another, nor by both oneself and another, nor does it arise by chance, not made by oneself or another. Rather, continued existence is a condition for rebirth.”
“Well, Reverend Sāriputta, is continued existence made by oneself? …” … “Is grasping made by oneself? …” … “Is craving made by oneself? …” … “Is feeling made by oneself? …” … “Is contact made by oneself? …” … “Are the six sense fields made by oneself? …” … “Well, Reverend Sāriputta, are name and form made by oneself? Or by another? Or by both oneself and another? Or do they arise by chance, not made by oneself or another?”
“No, Reverend Koṭṭhita, name and form are not made by oneself, nor by another, nor by both oneself and another, nor do they arise by chance, not made by oneself or another. Rather, consciousness is a condition for name and form.”
“Well, Reverend Sāriputta, is consciousness made by oneself? Or by another? Or by both oneself and another? Or does it arise by chance, not made by oneself or another?”
“No, Reverend Koṭṭhita, consciousness is not made by oneself, nor by another, nor by both oneself and another, nor does it arise by chance, not made by oneself or another. Rather, name and form are conditions for consciousness.”
“Just now I understood you to say: ‘No, Reverend Koṭṭhita, name and form are not made by oneself, nor by another, nor by both oneself and another, nor do they arise by chance, not made by oneself or another. Rather, consciousness is a condition for name and form.’
But I also understood you to say: ‘No, Reverend Koṭṭhita, consciousness is not made by oneself, nor by another, nor by both oneself and another, nor does it arise by chance, not made by oneself or another. Rather, name and form are conditions for consciousness.’
How then should we see the meaning of this statement?”
“Well then, reverend, I shall give you a simile. For by means of a simile some sensible people understand the meaning of what is said. Suppose there were two bundles of reeds leaning up against each other.
In the same way, name and form are conditions for consciousness. Consciousness is a condition for name and form. Name and form are conditions for the six sense fields. The six sense fields are conditions for contact. … That is how this entire mass of suffering originates. If the first of those bundles of reeds were to be pulled away, the other would collapse. And if the other were to be pulled away, the first would collapse.
In the same way, when name and form cease, consciousness ceases. When consciousness ceases, name and form cease. When name and form cease, the six sense fields cease. When the six sense fields cease, contact ceases. … That is how this entire mass of suffering ceases.”
“It’s incredible, Reverend Sāriputta, it’s amazing! How well spoken this was by Venerable Sāriputta! And we can express our agreement with Venerable Sāriputta’s statement on these thirty-six grounds.
Two books leaning on each other:
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The difference is that the law of causality is a general principle whereas paticcasamupadda is a special case of it's application.
Special because it's about something specific.
The law is like an abstract mathematic forumulae of xyz that can be applied to any calculation whereas dependent origination is more like doing physics where one uses the forumulae to think about certain circumstances substituting xyz for the real values of 123, special case not generalized.
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Re. the first point.
There's a truism according to scientific logic, "correlation doesn't imply causation" -- i.e. that "if I do X and then Y happens" that doesn't mean that "X caused Y".
The answer to "what's the cause?" can become debatable in fields like sociology or politics.
Sometimes the question is important -- people don't want to spend money on changing a condition, if that condition doesn't cause the problem or if changing the condition will not solve the problem.
Sometimes the question is counter-productive -- people ask the question to raise doubt and argue against doing something useful.
And the question can be carried to great length -- people might argue about whether something is a proximate cause or a root cause.
I think that simple theories of "causation" also tend to imply a chronological sequence -- "if A caused B, then A happened before B" -- which isn't well-suited to answer questions about a perpetual cycle like for example, "Which came first, the chicken or the egg?"
I think that Buddhist doctrine explicitly avoids this controversy, by not always emphasizing a causal (or linear) sequence. Instead things "co-arise", they happen together.
That might be similar to a complex scientific view. Consider fire for example ...
... and so far as I know, a scientific answer is "both" -- the latter being called "pyrolysis" (which precedes combustion). So the heat and the inflammable nature "co-arise" and once they start they continue.
A similar loop -- sometimes pathological -- might happen in psychology, "I drink because I'm unhappy and I'm unhappy because I drink", or the ways in which some enmities perpetuate.
But perhaps Buddhism doesn't try to argue about "Which came first?" -- may simply observe that they arise together, and might stop together.
I think this is so in the Noble Truths. The Second Truth is that craving is the origin of suffering, but I think the converse is true also i.e. that suffering causes craving (and so it perpetuates), what's clever is the Third Truth explaining how to stop the cycle.
I think it's also true in the analysis of the 12 nidanas -- I try to read the Wheel clockwise as if each causes the next, but I think that's simplistic -- there's two-way interaction or feedback especially between adjacent nidanas.
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Science deals with the physically measurable. Buddhist teachings focus on the end of suffering. Both discuss causality, but in different ways.
The scientific method emphasizes reproducibility of results for a given hypothesis. In schools we can all drop large and small weights to observe they fall at the same speed. However, we have to be very very careful about using words such as "scientific law of causality". We have to be careful because scientific hypotheses change. In particular, Newton asserted that gravity is a force. Modern science asserts instead that mass bends space-time in such a way that falling objects move at the same rate. So there is no law of causality. Science is a simply a body of hypotheses that are verifiable under specified conditions. As we learn more, our hypotheses change. For example we still haven't figured out dark matter and dark energy. Science is our current best guess for the physically measurable.
In contrast, the Buddha taught about suffering, its origin and its end. With regard to suffering, the Buddha also discusses causality and views (i.e., hypotheses). Here, he discusses bad kamma arising out of bad actions.
MN60:24.7: So they give up their former ethical conduct and are established in unethical conduct.
MN60:24.8: And that is how wrong view gives rise to these many bad, unskillful qualities—wrong view, wrong thought, wrong speech, contradicting the noble ones, convincing others to accept untrue teachings, and glorifying oneself and putting others down.
MN60:25.1: A sensible person reflects on this matter in this way:
MN60:25.2: ‘If there is no causality, when this individual’s body breaks up they will keep themselves safe.
MN60:25.3: And if there is causality, when their body breaks up, after death, they will be reborn in a place of loss, a bad place, the underworld, hell.
Science relies on the reproducibility of physical measurement. Buddhism relies on the reproducibility of subjective measurement of suffering. Both science and Buddhism assert reproducibility of experience. However, science is not really concerned with the end of suffering. And Buddhism isn't really concerned with the rate of falling objects.