The Intrinsic Soul is a new-age and Eastern belief that a life-essence is a quality universal to all living things: the idea that life and souls exist together by definition. If it has a soul, it is alive. If it is alive, it has a soul. Every piece of organic matter possesses a spirit in and of itself. Holes in this theory begin to arise when one asks the question “What is life?”
Or, more accurately, “Give me an absolute dividing line between what makes something living and what makes it ‘dead.’” After all, proponents of this theory would agree that a human or a dog would possess this life essence, but that a rock or a hydrogen atom would not.

I prefer to think of the 'soul' as a wave-function. Everything physical has a 'soul' in that sense.

Asking at what point does inorganic matter becomes organic is like asking at what point does red blend into orange on the electromagnetic spectrum. There is no one point. There is no way of saying that everything to the left of this point is “red” and everything to the right is “orange” without applying frail and arbitrary human perception to the mix.

This is extreme reductionism and this is as much a product of your way of thinking as the conclusions that you make. That is, you assume extreme reductionism is correct, and then go on to make extreme reductionist conclusions. What if extreme reductionist assumptions are wrong? For example, what if components of a system do not completely describe a system? Let's use the wavefunction concept again. What if an apple, that is a particular apple that you last ate, had a wavefunction associated to it. Now, that wavefunction could still exist even though the apple does not, at least not in it's not digested form. The wavefunction represents the identity of the apple as a seed, unripe apple, ripe apple, and Mario digested apple, and so on. We might say that the apple is composed of atoms, and the atoms are composed of 'elementary' particles, etc, however are we are doing is describing in more elaborate terms the wavefunction, but we are never able to quantitize the wavefunction completely since there is always something remaining in the wavefunction that is left out (such as the history of the apple's elementary particles from their birth, etc). Whatever you want to call the 'wavefunction', this holistic concept of nature is in competition to an extreme reductionist set of assumptions with entirely different conclusions. Of course, there are many more options besides these two positions.

A starfish, for example, can be divided into five pieces, and each piece will grow into a new, wholly self-sufficient organism. In such a case, would one suggest that each new organism possesses one fifth of the original consciousness? Or perhaps would each fifth of the animal attain its own individuality and essence?

No. One might suggest that a starfish, a particular starfish, exists as its own unique identity that refers to that particular starfish in a unique time and space point, or one might suggest that a starfish exists as its own unique identity that spans a particular space-time. The identity of the starfish at a particular time and space is an identifiable expression of the larger identity (or wavefunction, if you prefer) that spans a particular space-time. The elements of the starfish (either at a specific space and time position, or at a spanse of space-time) are inconsequential to the identity of the starfish. They partially compose the wavefunction expression (or identity), but the elements themselves do not define the starfish, rather the identity of the starfish is what defines it, and that identity is holistic in nature. So, for example, the starfish can lose one of its tentacles and still be 'the starfish' from yesterday. It is just that 'the starfish' has a different number of tentacles than yesterday. The reason we can say it is the 'same' starfish as yesterday is because the identity is the same, just with the caveat that the starfish lost one of its tentacles, but didn't lose it's identity. The identity of the starfish is such that it spans spacetime with one set of days having five tentacles and the other days having only four tentacles. The identity (that spans a particular spacetime) refers to both the starfish with five tentacles and to the same starfish with four tentacles.

What sense does it make, then, to treat oneself as one unchanging soul, one unchanging essence, one unchanging consciousness, if the very seat of one’s sentience could have so easily been divided and redistributed at birth? If a brain can be separated into two independent people, what does this mean for the ontological question "Who am I?"

But, then why stop at personal identity? The same can be stated for the causal relationships as well. For example, if a window glass 'pops' out of a skyscraper, shattering onto a car below and causing the car alarm to blare, the reason for the window glass popping out of the window frame needs a cause. However, this scheme is disrupted with the extreme reductionist view. That is, there are many structural reasons as to why a window can pop out of its frame. Perhaps its due to the wind hitting the north side of the building day after day which weakens the glue inside the frame (I have no idea if this is how windows are mostly held inside their frames, just for example). Or, perhaps its because the construction company used Anderson windows instead of Pella windows. Or, perhaps its because Joe always put his hand on the window everyday for the last 20 years. Or, perhaps its because of slight tremors that put undo pressure on the window frame over the years. Or, perhaps its because the moon exerted more gravitational pressure in May than in previous months. Etc. Etc. We can easily visualize that one of these causes is more the cause than the others. However, this does not change the fact that the glass fell onto the car and the alarm to start blaring. Does the cause of the car alarm blaring change if the cause of the window popping is changed. For example, what if I tell you that Joe and Max both pushed on the window pane moments before to see if they could push the window pane out of the building as a joke to smash Fred's car? Would the cause of the alarm blaring be any different than if Joe and Max stopped a few seconds before but then an earthquake occurred that actually put excessive force on the window? I don't think so. The cause of the alarm blaring has nothing to do with how the window glass popped out of the building. It's not relevant, really, except only in the most extreme reductionist view where identity of cause does not exist. Similarly, it doesn't matter if two brains are separated at birth or whether you take an atom and exchange for another, etc. These issues are just descriptions of what composes the identity (such as the earthquake causing the window glass to pop is a description of what composes the cause of the alarm blaring), but they do not in anyway define the "I" or self. The only attribute that fully defines the identity of self is the 'self' itself. The things that compose the self (e.g., being born with brains connected at birth) certainly describe who you are, but they do not define who you are. Only 'you' can define who you are, and you do that by being 'you'. You are what you are, and what you are is who you are. It's tautological, but that's what identity relationships are (a=b, hence b=a), or, identity=self, and self=identity. Your soul (or wavefunction) is your identity.

Instinctively, culturally, egotistically and naturally we all assume our identities to be unique and unchanging. In an ocean of chaos we long to see ourselves as the bedrock, the unmoved and unchanging and eternal. However, as shown, the realism of the inviolate “I” is a hypothesis largely devoid of reductionistically compatible evidence or argumentation. One might argue that the nature of the self is a question far too metaphysical and complex to delve into with any kind of accuracy, and this is largely true, but with the important caveat that any kind of model of consciousness must be at least logically consistent. Thus, it would seem that any logically consistent model of consciousness must do away with the notion of an inviolate, unchanging “I.”

If you want to understand the matter, then to discount identity is not very fruitful. It is an extreme reductionist viewpoint that is naive and only leads to ridiculous statements which others have made (e.g., there are no cause-effect relationships). These are ridiculous assertions since they end up, like positivism before, going way out on a limb trying to create a language that somehow can account for phenomena in some epiphenomena terms. Inevitably, these languages breakdown because they cannot explain the successes of language that treats the phenomena as real. For example, a language that treats causal relationships as real is much more able to demonstrate why it is that biological evolution is more successful at explaining why we find specific fossils in the particular geological strata that we expect to find them. Languages that try and eliminate causal-effect (akin to identity relationships), inevitably fail miserably.

So, why do people do it? Why do people go so far out of their way to eliminate things that are very apparent (e.g., cause-effect, identity, etc)? I think it's because reductionism has been somewhat successful as a way of describing the world, however some go way overboard with the idea and then think extreme reductionism must be a more fundamental metaphysical description of the world (i.e., which it tries to be anti-metaphysical). In short, humans like extremes. The answer, as I see it, is that holism and reductionism are both needed to understand the world, and the question is not so much as reductionism trying to eliminate holism, but rather the question is how can holism best account for reductionism such that the reductionism success of science can be best understood.