If we imagine that the Universe is some massive vector space, and in particular imagine that the dominion of men, that is to say, the world that men inhabit, is a vector space, then we can postulate that all men are comprised of a basis set of three vectors; in other words, the space that men inhabit is three dimensional.
Firstly, it is patently apparent that there is such a thing as a body. Matter exists; this is an axiom, it is what matter does, by definition. So we have A) a body.
Secondly, let us postulate a thing called thought. Thought is clearly different than the body, for while the body is physical, thought is purely mental. Even modern day neurology is of no help in pinning down thought. We can image the brain and tell that it is thinking, but we cannot tell what it is thinking. For example, two people may be asked to mentally rotate three dimensional objects, one a cube, the other a sphere. Yet while the two people's brains may both "light up" in the same place, it may not be possible to distinguish which is visualizing the sphere and which the cube. And still, one sees in his mind's eye a cube and the other a sphere. In a sense, these objects exist, not in a corporeal plane, but in a mental plane. They exist but are not detectable physically.
At the same time, an object such as a computer may be said to have thoughts, in that it has an independent, physically unverifiable interior which we have dubbed cyberspace. But the computer, for all that the subject has occurred in fiction, cannot generate independent thought. It is dependent on a user for instructions. So we can describe a computer as an object which thinks, but is not a thinker. But the human clearly is a thinker. There is nothing in nature which speaks "computer," yet man was still able to imagine the computer, indeed, imagined it long before the technology existed to create it. So man is a thinker and a computer is not, for man generates independent thought. Let us therefore postulate a thinker, which guides the creation of thought, and let us call this postulated thinker B) "mind."
Thirdly, let us discuss another aspect of man. Clearly there are men who are unintelligent. Yet these selfsame men, many of whom never generate a thought in their lives, live. They eat, they work, they play with their children. So although these men have a deficiency of thought, they do not have a deficiency of action. Therefore we can say that while the mind may be the generator of thought, it is not the generator of action. And nor can the body be a generator of action, for there is no clear difference between an upright and ambulatory man and a corpse, on purely physical grounds. So let us postulate a third component, which is the generator of action, and let us call this component C) the spirit or soul or the will.
Thusly we have three components of man: the body, the mind, and the spirit. Now, the body is clearly not of the same stuff as the mind, indeed it is distinct from it, for a man lying in bed of a coma, with all semblence of life, but no thought, is completely indistinguishable from, say, Newton. So the body can exist without the mind. Can the mind exist without the body? At this point in the debate it is impossible to determine, yet one does hear stories of men in comas who, exhibiting all the symptoms of brain death, recover and return to a normal life. It is almost as if their minds had left them, only to return at a later date. Please note that I am not saying that this is precisely what has occurred. But it is suggestive.
We can see that the spirit must be of a different stuff than the body, which as I have stated earlier, is physically indistinguishable in either living or dead form--save for the absence of a few critical actions. Yet here too, we hear stories of men revived after aspiration and circulation had ceased. Something has returned, that was missing in the interim. The generator of those actions so critical to life had taken flight.
Is the mind of different stuff than the spirit? A difficult question, perhaps not answerable. To speak of the "substance" of the mind or spirit is to translate metaphysical concepts into physical ones. Let us nevertheless postulate that there is a difference, for we have all known men of will who possess no mind, and also men of the mind who possess no will of their own, but instead follow the will of others.
So let us postulate that the three components are distinct. Since none of the three can be expressed as some combination of the other two, this suggests that they are orthogonal. If we imagine that there is some unit vector for each of these, then we have a set of basis vectors by which every man can be described. Indeed, the set is complete, because it obeys all possible operations. The most critical of these is the cross product. Let us see what the cross product results are:
Spirit X Mind = An Action Taken ( A physical result)
Mind X Body = A Decision ( A spiritual result)
Spirit X Body = A Memory ( A mental result)
So we see that the cross product of two of the humane vectors results in a component in the direction of the third vector, in accordance with the rules of the cross product.
What then does this suggest about the Divine?