Are there real examples of strongly emergent systems? A strongly emergent system is a physical system whose properties are not reducible to causal relationships and interactions between the elements of this system. That is, the whole is not the sum of its parts. However, are there real examples of strongly emergent systems?
 A: A strong piece of evidence that all physical phenomena are in principle reducible to constituent interactions is: For all symmetries and conservation laws (such as energy and momentum) that are strictly satisfied by the known microscopic interactions, no macroscopic violation is known in any physical system, however complex. This includes biological organisms, which have had billions of years of opportunity (and a strong incentive) to find a way to "cheat" conservation of energy, say, but have not done so.
It is generally impossible in practice to predict specific behaviors of complex systems, but it is very plausible that these behaviors are nevertheless mathematically determined by constituent interactions and that it is "merely" too difficult to solve the equations (and obtain accurate initial conditions). An important and successful test of this view is: When, fortuitously, some property of the underlying interactions is readily mathematically composable (i.e., tractable reductive reasoning can constrain the overall system behavior in some way) -- as with symmetries and conservation laws -- the resulting constraints do match observations.
A: In order to answer this you need to fill out the meaning of the word 'reducible'. A concept such as 'a heap' can be applied to a heap of coins, yet a single coin is not a little heap, nor does it have a property 'heapness' in addition to its other properties.
A more thought-provoking example is afforded by computer programming languages. Is a language such as Python or Java 'reducible' to assembler code or logic gates? Arguably the answer is either 'no', or if it is 'yes' then we deduce that the term 'reducible' is being used in a way that does not give much insight into the phenomenon in question (the high-level language). The concept does not capture the relations between levels of description. For more on this you might find this discussion helpful:
Does physics explain why the laws and behaviors observed in biology are as they are?
Finally, in quantum entanglement we have a whole which is not the sum of its parts, in the sense that the state of the whole simply cannot be expressed as if the parts each had self-contained properties. It is hard to cash out the term 'reducible' in such a way that it says entangled systems are entirely reducible. So if that is the case then any entangled pair of particles is a counter-example which was asked about.
