One basic question in metaphysics is this:

How many gods exist?

Atheism can be defined as the view that there are 0 gods.

Monotheism is the view that there is just 1 god.

Polytheism is the view that there are 2 or more gods.

Thus all of the various answers to the metaphysical question above are included in these three categories.

The term polytheism, however, is a very broad category that includes many different and conflicting answers to the question above.

Manichaeism – the view held by Augustine before he converted to Christianity, is the view that there are two (major) gods, a good god and an evil god:

Mani’s teaching dealt with the origin of evil, by addressing a theoretical part of the problem of evil by denying the omnipotence of God and postulating two opposite powers. Manichaean theology taught a dualistic view of good and evil. A key belief in Manichaeism is that the powerful, though not omnipotent good power (God) was opposed by the semi-eternal evil power (Satan). Humanity, the world and the soul are seen as the byproduct of the battle between God’s proxy, Primal Man, and Satan.

http://en.wikipedia.org/wiki/Manichaeism

A polytheist might also believe that there are three gods, or four gods, or a dozen gods, or one hundred thirty-six gods, or… I don’t know if there is an actual religion that proposes this, but it is theoretically possible that there is an infinite number of gods.

Therefore, atheism and monotheism are just two possible answers to the question ‘How many gods exist?’, and there is actually an infinite number of different and conflicting answers that could be given. The category of polytheism, unlike atheism and monotheism, lumps together an infinite number of different and conflicting views.

Although monotheism represents a single view as to the the number of gods that exist, there are, of course, many different types of god that might exist, and thus many varieties of monotheism. I won’t try to define the term ‘god’ here, but will specify one necessary condition for something to be a god: it must be a person. Humans are persons, and we don’t think of humans as being gods, so being a person is NOT a sufficient condition for being a god. But one must be a person in order to be a god.

Thus, I do not consider ‘pantheism’ to involve belief in the existence of one or more gods. Pantheists believe that the ultimate source of the world is an impersonal force. This belief is NOT belief in a god. A pantheist could be an atheist and believe that there are zero gods. Or a pantheist could be a monotheist and believe that there is just one god (so long as that one god has its source of existence in a great impersonal force). Or a pantheist could beleive in two, three, four, or a hundred thirty-six gods.

Back to monotheism. How many different kinds of gods are there? Richard Swinburne points to three basic characteristics of persons that are the basis of his analysis of the word ‘God’ and the claim ‘God exists’: freedom, power, knowledge. Persons can make choices and decisions with various degrees of freedom or free will. Persons have the power to change things and have various degrees of power of various kinds. Persons have beliefs about themeselves, others, and the world, and these beliefs can be true or false, and persons can have various degrees of ignorance and knowledge.

The God of western theism is supposed to have an infinite or unlimited degree of freedom, power, and knowledge. So, lesser sorts of gods are lesser because they possess only a finite degree of one or more of these basic characterstics. I would add one more critical characteristic to the basic three: longevity. A god that exists only for a few seconds is not likely to have much impact on the world. The longer a god exists, the more opportunities the god has to impact and influence the world (or to create a world). So, the element of time or longevity seems rather important.

Swinburne divides up degrees of the various characteristics into just two: infinite or finte. Since there are only two possible degrees of a characteristic on his schema, we can represent the various kinds of deities in terms of a truth table, where TRUE means that the characteristic in question is infinite, and FALSE means that the characteristic is finite:

Based on this very simple and straightforward classification of gods, there are 16 different types of gods. A god of type 1 is the sort of god that western theism believes exists. Such a god has infinite or unlimited freedom, infinite power, infinite knowledge, and exists for eternity (has existed infinitely in the past and will exist infinitely into the future). Thus, on this classification schema, there are 16 different varieties of monotheism.

How many different varities are there of the view that there are two gods? If both gods are of the same type, if someone believes in ‘twin’ gods, there are 16 different possible pairs of ‘twin’ gods. But there is no necessity in believing that both gods must be of the same type. So, we must also consider the various combinations of types of gods, where the two gods are of different types.

Any combination of two different objects can be placed in two different orders. Object A and object B can occur in two orders: A-B or B-A. Both are permutations of one combination, the combination of A and B. Thus, for pairs of objects, we can determine the total number of possible permutations, and then divide by 2 to arrive at the number of different combinations. We already know that there are 16 possible combinations of gods that are ‘twins’, so we need only figure out how many combinations there could be of pairs of different types of gods and then add 16 to that number.

For permutations of a pair of dissimilar gods, the first god can be from any one of the 16 types. The second god selected, however, must differ in type from the first, so there are only 15 possibilities for the second god. Thus, the number of permutations of two dissimilar gods (based on there being 16 different types of gods) would be 16 x 15 = 240. But we are interested not in the number of permutations (which include different orderings), but only in the number of combinations, so we divide by two: 240 / 2 = 120. There are 120 different combinations of pairs of gods that are dissimilar, plus 16 pairs of gods that are ‘twins’, so there are 136 different combinations of gods possible for the view that two gods exist.

So, there is only one version of atheism (there are simply 0 gods of any sort), and there are sixteen versions of monotheism, and there are 136 versions of belief in two gods. As the number of gods slowly increases, the number of different versions/combinations of gods increases exponentially.

I’m not entirely happy with the very simple categorization of gods that we have derived from Swinburne’s analysis of ‘God’. Having only two degrees for each characteristic seems excessively stingy. Think for example, about the idea of a god possessing a ‘finite degree of power’. Think of all of the various possibilities that this category encompasses.

Let’s just focus on one simple sort of power: the power to lift an object of a certain weight. An ant has a finite amount of this power. A large ant can, perhaps, lift an object that weighs one ounce. A human infant can only lift an object that weighs about one pound or perhaps a few pounds. One of the strongest human beings who ever lived could lift an object weighing about 1/2 of a ton. (http://www.youtube.com/watch?v=wWUNjTHHIxY) Perhaps someday a human being will be able to lift an object that weighs one ton (on earth under normal gravity).

But we can imagine a being that could lift an object that weighs two tons, or three tons, or one hundred tons, or one thousand tons, or one million tons, or one trillion tons. All of these possibilities from the power of an insect to lift an object weighing one ounce, to the power of an imaginary person to lift an object weighing trillions of trillions of tons are included in the broad category of having a ‘finite degree of power’.

So, I don’t think it would be at all excessive to add a few more categories of degrees of power, and the same goes for the other characteristics of persons. I suggest a system of four categories of degrees:

Sub-Human

Human

Superhuman

Infinite/Unlimited

An ant has sub-human power, at least in terms of lifting objects. A person who can lift an object that weighs 1/2 ton is still within the range of human power, in terms of lifting objects. A person who can lift an object that weighs one billion tons, but not an object weighing one trillion tons would have superhuman power, in terms of lifting objects. A person who can lift any weight whatsoever, would have infinite power, in terms of lifting objects.

We generally think of gods as being either superhuman or infinite in various respects, but this is not an absolute requirement. The gods of the greeks were superhuman in their power, but were often quite human in other ways. Greek gods could be tempted to do things that were foolish or stupid. So, gods can have a human degree of a given characteristic, perhaps a god can be even be sub-human in some respects. But let’s toss out the sub-human degree of the four characteristics of persons, since gods are generally conceived of as having at least a human level of these characteristics. That still gives us a little more specificity than the Infinite vs. Finite categories of Swinburne. We at least have divided the Finite category into two sub-categories: Human vs. Superhuman.

Given this small ammendment to the above categorization of types of gods, there are three possibile degrees for each of the four basic characteristics of persons. That means that the number of types of deities is increased to 3 x 3 x 3 x 3 =

9 x 9 = 81. So, I propose a categorization of types of gods that includes 81 different types, not just the 16 types from the above overly simple classification.

If atheism is defined as the view that there are 0 gods, then there is still just one version of atheism.

But since there are 81 different types of gods on my proposed system of classification, there would be 81 varieties of monotheism.

How many versions of belief in two gods would there be?

We know that there would be 81 different pairs of ‘twin’ gods, where each god was of the same type as the other god in the pair. But there could also be pairs of dissimilar types of gods. First we need calculate the number of possible permutations of such pairs (which includes different orderings), then we divide the number of permutations by two, to arrive at the number of different combinations of disssimilar gods.

For the first god of a pair, we have 81 different possibilities from which to choose. But since the second god of the pair cannot be of the same type as the first, there are only 80 possible choices for the second god. Thus, the total number of permutations of two dissimilar gods (when choosing from 81 different types of gods) is 81 x 80 = 6,480. Since we are only interested in the number of combinations, and don’t care about different orderings, we must divide this number by two: 6,480 / 2 = 3,240. So, there are 3,240 different combinations of two dissimilar gods. We already know that there are 81 different combinations of two similar gods (gods of the same type), so we add these two numbers together: 3,240 + 81 = 3,321 different combinations of two gods.

This shows how with the slightly ammended system of classification, where we allow for three degrees of possession of a basic characteristic of a person, the exponential increase of versions/varieties of views as the number of gods grows is even more radical than the exponential increase that we saw with the initial overly-simple system of classification.

Versions of atheism: 1.

Versions of monotheism: 81.

Versions of bi-theism (belief in two gods): 3,321.

How many versions are there of tri-theism (belief in three gods)?

Given that there are 81 different types of gods, there would be 81 different combinations of three gods where all three were of the same type.

But there are two other kinds of combinations of three gods. One other kind of combination is where all three gods were of different types (none being of the same type). Finally, the remaining kind of combination would have two gods of the same type plus one god of a different type.

To figure out the number of combinations of three gods where all three are of different types, we can first determine the number of permutations of three gods there are when there are none of the same type, and then divide that number by six, because for each combination of three gods where the gods are all different types, there are six different permutations.

These are the permutations of the combination of A and B and C:

1. ABC

2. ACB

3. BAC

4. BCA

5. CAB

6. CBA

To determine the number of permutations of a series of three gods, where all three are different, and there are 81 different types of gods, we have 81 choices for the first god, 80 choices for the second god, and 79 choices for the third god. Thus the number of permutations for a series of three disimilar gods is 81 x 80 x 79 = 511,920. Since there are six permutations for every combination of three different gods, we must divide the number of permutations by six to get the number of combinations: 511,920 / 6 = 85,320 combinations of three gods, where all three are of different types of gods.

Now we need to determine the number of combinations of three gods there are when two of the gods are of the same type. We already know that there are 81 different pairs of gods that are ‘twins’, that are both of the same type. Each of these pairs can be modified to form various combinations of three gods by adding one more god of a different type than found in the pair. Since one of the types of gods has been used in forming the pair of twin gods, that leaves only the remaining 80 types to choose from in order to form the various combinations of three. Therefore, each of the 81 pairs of twins can be modified in 80 ways to form a combination of three gods, where two are of the same type and the other god is of a different type: 81 x 80 = 6,480 combinations of gods where two are of the same type and the other god is a different type.

Now we just add the number of each of the three different kinds of combinations together: 81 combinations (where all three gods are of same type) + 85,320 combinations (where all three gods are of different types) + 6,480 combinations (where two of the gods are the same type and one is different) = 91,881 combinations of three gods.

There are 91,881 different versions of tri-theism (the belief that three gods exist).