|My cheat sheet says the answer is 42|
Did you know that we are nothing but biological machines governed by physical processes which, according to the leading materialist philosopher Sean Carroll, are completely modelled by the above equation? (Hat tip Jayarava)
Now I wouldn’t claim to understand the various parts the equation, let alone the equation as a whole, but I’m pretty sure, assuming the Church-Turing-Deutsch principle to be true, that the whole equation consists of a concatenation of Turing Machines and nothing else. The Turing Machine is a mathematical structure (not normally or necessarily instantiated as an actual physical device) that completely describes the concept of ‘mechanism’ to any level of complexity, including computers and all phenomena that can be modelled by computers.
This immediately flags up two yawning gaps in this model's claim to be a complete description of the world. This equation cannot deal with (i) qualitative phenomena (qualia) nor (ii) anything that involves meaning and semantics (intentionality), because Turing Machines can only process Boolean, quantitative and syntactical information, and have zero capabilities with intentionality and qualia. In fact, no matter how many zillions of Turing machines you concatenate, a zillion times zero is still zero.
And no matter how may megazillions of Turing Machines you concatenate or link in a network, you won't get any mind-like 'emergent properties' emerging from them, because emergent properties emerge from the mind of the observer, not from the data. The materialists' claim that mind is an emergent property of mechanism seems to be an example of the logical fallacy of petitio principii or circular reasoning.
Is the bunch of cherries an emergent property of the 13x15 pixel array, or does it emerge from the mind of the observer?
Incompleteness and incoherence
So Sean Carroll's equation may be accurate, but it is incomplete.
It may accurately describe all known physical processes, but it says nothing about non-physical processes such as the experience of qualia (and most significanty, from the Buddhist viewpoint, the pervasive experience of dukkha) neither does it address intentional awareness - such basic features of our world as attention and aboutness.
The incoherence of materialism
Materialism claims that the basis of all phenomena is matter. This is incoherent and unscientific and has been demonstrably so since the Michaelson-Morley experiment.
Michaelson and Morley proved that the foundations of electromagnetic physics are based on processes, not substances. Prior to their experiment it had been assumed that light waves and the rest of the electromagnetic spectrum propagated in the same manner as sound and water waves, through a substrate. Michaelson and Morley proved that this hypothetical substrate (known as the 'luminiferous aether') did not exist. So light waves are pure 'disembodied' processes which function without any material support whatsoever, with no need for even the wispy and tenuous hint of matter or substance provided by the aether.
Subsequently, quantum mechanics dealt the death blow to the 'substantialist' interpretation of physics by showing that fundamental particles aren't 'things' at all, but are processes. Electrons and protons etc only appear as 'things' at the moment of measurement. (They are reified by the observer). When left to themselves they propagate through space as probability waves, which are of course processes. Consequently, Sean Carroll's equation is not about substances and 'things in themselves', it deals with relationships and dynamics, in other words processes.
The incoherence of substance dualism
Substance dualism is an erroneous attempt to counteract materialism by claiming that there are non-material things and substances, such as souls and ectoplasm. This fails as a model of the mental world for much the same reasons as materialism fails as a model of the physical world. Mental phenomena, like physical phenomena, are ultimately processes rather than things. The root mind is known is Buddhism as the 'mental continuum' or 'mindstream', and like Heraclitus' river is never the same thing for two successive instants.
The Hard Problem
So the Hard Problem of consciousness, which is normally stated as 'how does the mind interact with the body?', could be restated from a Buddhist viewpoint as 'how do mental processes interact with mechanistic processes?' One possibility, as suggested by Henry Stapp is by intentionality in the form of attention acting via the Quantum Zeno effect.
See also Buddhist Philosophy