According to axiarchism, reality is ultimately defined by some kind of value. Axiarchism can be used to support a rational and naturalistic kind of Pagan theology. Part 1 of this article laid out the basic motivations for axiarchism. It showed how to use axiarchism to explain the existence of the Pagan ultimate Deity, the Source of all things. Part 2 continues the development of axiarchism. It solves some of the problems raised in Part 1, and it uses axiarchism to justify the natural existence of a God and Goddess.
A Second Problem with Axiarchism
Axiarchism implies that reality is maximally valuable. Reality is as good as it possibly can be. Reality is the way that it is because that way is the best way it can be. Unfortunately, this leads to a problem. The problem is that the very idea of the best seems impossible. The best is like the biggest. Every number is surpassed by some bigger number. Analogously, every number (every program) is surpassed by some better program. More concretely, if there were any best of all programs (or even many such programs), then running that program would produce the best of all universes. But there isn’t any such thing. Any universe is surpassed by some better universe. Given any universe, you can always define a bigger universe which contains more things with greater intrinsic values of their own. Although intrinsic value doesn’t depend on people, people do have intrinsic values. So any universe can be surpassed by some greater universe which contains more people who reach greater heights of self-realization or flourishing.
To solve this problem, axiarchists can turn to the mathematicians. The problem of the best program is like the problem of the biggest number. But mathematicians don’t define any biggest number. Mathematicians define a sequence of ever bigger numbers. For instance, the first infinite number is defined as the set of all finite numbers. Mathematicians define the counting numbers using two rules. The initial rule states that there exists an initial number zero. The successor rule states that every number is surpassed by a greater successor number. The number n is surpassed by its successor n+1. The first infinite number is the set of all numbers defined by the initial and successor rules.
Axiarchists can adopt this mathematical approach. Just as mathematicians define two laws for making numbers, so axiarchists will define two laws for the actualization of programs. Just as the two mathematical laws define infinity, so these two axiarchic laws define the best. The Selector is still the best. However, just as the meaning of infinity requires a detailed analysis, so also the meaning of the best requires a detailed analysis. According to the Pagan interpretation of axiarchism, this detailed analysis is part of the study of the internal nature of the Source. It is a part of Pagan theology. However, it is not a study of the consequences or manifestations of that divine meaning. This detailed analysis is provided by the initial axiarchic law and the successor axiarchic law.
The Two Axiarchic Laws
The first law is the initial axiarchic law, which involves an initial selector. Just as the initial number is zero. so the initial selector is the property of being least valuable. So the initial axiarchic rule states that for every program, if that program is one of the least valuable programs, then there exists some computer that runs that program. So, all the least valuable things exist. These aren’t bad or awful things. They are just minimally valuable, in the way that a penny is the least valuable type of monetary unit. Pennies aren’t bad, they just aren’t as valuable as nickels, dimes, dollars, or hundred dollar bills.
For axiarchists, the successor law involves a successor selector. Just as the successor rule for numbers defines bigger numbers in terms of smaller numbers, so the successor law for programs defines more valuable programs in terms of less valuable programs. This selector acts on programs that have already been selected. Since any program that has already been selected is running on some program, the successor law acts on those programs. Any program can be modified in many ways. Some of those modifications introduce errors, making the original program less functional or less excellent. Other modifications make the original program more functional or more excellent. The successor law involves a successor selector, which always selects the better versions of any program.
Any modification of a program which makes it better is an improvement. Of course, an improvement of an old program is a new program. If some program is running on some computer, then there are always some ways to improve that program. There are some ways to upgrade it, so that it becomes a better program. A better program generates more value when it gets run. On the basis of these ideas, axiarchists define the successor law like this: for any program running on some computer, for any way to improve that program, there exists some computer that runs that improvement. You can think of improvements as offspring, so that each computer has many better offspring.
These two axiarchic laws define a series of generations of computers, each of which is some concrete thing. The initial generation is the zeroth generation. It contains all the computers running the least valuable programs. The first generation contains all the computers running improvements of programs running in the zeroth generation. These are slightly better programs which, when run, generate more valuable things. But computers in the first generation are now surpassed by computers in the second generation. And so it goes. Each next generation contains all the computers running improvements of all the programs in the previous generation. If you think of improvements as offspring, then each least good computer in the zeroth generation serves as the root of a genealogical tree. Concrete reality is all the computers in all these generations. It is a forest of trees.
The axiarchic laws define a cyclical process of creation in which computers beget computers. The biological analogy is appropriate, since the programs running on these computers are analogous to genetic codes, and they reproduce like asexual organisms. The axiarchic laws define an evolutionary algorithm. This is evolution by rational selection. Rational selection ensures that there is an arrow of value. Along any lineage of computers, the value of the computation increases. So there are two patterns in this axiarchic process of creation. The first is the circle while the second is the arrow. These are entirely natural patterns in the expression of the power of the Source. They are manifestations of that power. If that power is divine, then these patterns are also divine.
Axiarchism and the Pagan God and Goddess
Although these individual computers all reproduce asexually (and, indeed, their reproduction is really only a purely logical process), the circle and the arrow are emergent patterns of the entire process. One way to interpret these emergent patterns, which is consistent with some versions of Paganism, is to think of the circle as a feminine pattern and the arrow as a masculine pattern. This interpretation is obviously based on well-known features of human and animal sexuality. Of course, you might object that these features are superficial, and far from biologically universal. Here the correct reply is that this sexual interpretation of these powers is an example of analogical predication, which has long been used in Western theology. We use analogies and metaphors to refer to the divine aspects of reality, in order to be able to understand them better and to relate to them more intimately.
On this sexual interpretation, the circle is the Goddess while the arrow is the God. And this means that the Goddess and God are not supernatural people. They are not concrete things that somehow live in outer space. They are not merely things among things. Rather, they are ways in which divine power makes itself manifest in the production of concrete things. These two divine powers are far deeper than any type of personality. Obviously enough, sexuality itself is deeper than any type of personality. Most organisms that reproduce sexually are not people. And, even within people, sexuality is an impersonal force. But perhaps sexuality is far too crude in this context. A more abstract approach, which is also therefore deeper, treats the masculine and feminine as expressions of the polarity of love. As the axiarchic laws express their power, two perfectly harmonized poles of action emerge; these poles can be thought of analogically as a divine loving couple.
One of the religiously relevant consequences of this computational axiarchism is that the axiarchic laws apply to all computations. Universes are computations, but so are protons and pulsars and planets and puppies and people. The circle and the arrow are patterns at work in the generation of all these computations. To use some older theological language, these patterns are immanent in all computations. Your body is a computation, so the circle and arrow brought it into existence, and are active within it. The Goddess and God are at work in your brain and body, in every cell in your body, in every molecule in every cell, every atom in every molecule, every particle in every atom, all the way down, however far down it goes. Of course, these patterns are active in all natural things.
Axiarchism and Naturalistic Paganism
For Naturalistic Pagans, this means that religious symbols and rituals are ways of becoming more intimately aware of the natural powers in our bodies. The Wheel of the Year illustrates these powers at work. The sun represents the arrow while the earth represents the circle. The arrow rises and falls; but the circle always renews the arrow. These powers, the Goddess and God, are directly active in our lives, from particles up through the atoms and so on, all the way up to our brains and bodies. These are powers that inspire rather than coerce. The circle urges us forward while the arrow urges us upwards. Working together, these powers ensure that every human life is a wheel that rolls uphill.
Your present life is a computation which can be improved in many ways. The successor law implies that your life will be surpassed in every possible way. And so your life is surpassed by other lives. These are your better future lives. Where do they live? Not in our universe, which is already running its own program. Your better future lives will run in better future universes. Many Pagans already subscribe to some type of reincarnation theory, and axiarchism leads to a naturalistic conception of reincarnation.
This reincarnation theory involves many lives in many universes. Karma is just the system of ethical rules which maps lives on to lives. This concept of reincarnation even involves a naturalistic definition of the soul. This definition is as old as Aristotle, and it is always surprising that more people do not refer to it. On this Aristotelian definition, the soul is the form of the body. The soul is to the body as a program is to a computer. Your soul is a number running on your body. But your body is already a number running on some network of deeper computations. All of this is perfectly naturalistic. It is consistent with our best science. It involves nothing paranormal or occult. On the contrary, the logical and mathematical aspects of axiarchism rigorously exclude all superstition.
Axiarchism leads to a very rich concept of nature. Just as mathematicians have defined transfinite numbers and other transfinitely complex mathematical structures, so axiarchists can take their concepts of programs and improvements into the transfinite. Nature is an absolutely infinitely rich totality of ever better computations. Although no universe is the best of all possible universes, nature itself is the best of all possible totalities. Anselm said that the Abrahamic God is that than which no greater is possible; but the axiarchist says that nature is that than which no greater is possible. Nature is so rich that every part of nature is simulated by some other part of nature. It is so rich that any definition of nature merely describes some smaller part of nature. Mathematically speaking, this means nature satisfies reflection principles. Nature has the status of a proper class. Nature is the pleroma, the unsurpassable manifestation of divine power. It is the unfolding of the meaning enfolded in the ultimate sufficient reason, which is the Source of all things.
Resources: An early version of axiarchism is presented in Leibniz’s essay, “On the ultimate origination of the universe”. You can find it in many editions of his works. One of the first modern approaches to axiarchism is by Nicholas Rescher, in his book The Riddle of Existence. Sadly, it’s out of print. John Leslie gives a very accessible presentation of his axiarchism in his book Immortality Defended. He gives a more detailed and technical presentation of his axiarchism in his book Infinite Minds.
Eric Steinhart is a professor of philosophy at William Paterson University. He is the author of four books, including the forthcoming Your Digital Afterlives: Computational Theories of Life after Death. He is currently working on naturalistic foundations for Paganism, linking Wicca to traditional Western philosophy. He grew up on a farm in Pennsylvania. He resides with his wife in New York City. He loves New England and the American West, and enjoys all types of hiking and biking, chess, microscopy, and photography.
This Wednesday, we hear from B. T. Newberg: “The Gadfly: A Socratic Interrogation of Naturalism”.
Given that axiarchism seeks to explain the existence of this universe as a necessary consequence of objective mathematics, I’m disappointed part 2 didn’t even begin to demonstrate how the “best” computation/automaton/selector isn’t an entirely arbitrarily judgment. Does it mean the most moral, the most prolific, the most robust, the most empty, the simplest, the biggest, etc.?
Utterly perplexed by this…
1. It is impossible for numbers not to exist, even if there are no concrete things to be numbered… why? Seems to me like nothing would be nothing. I don’t see why numbers would have to exist if there was nothing to be numbered. Just like hotness would not have to exist if nothing was hot, and Brandonness wouldn’t have to exist if nothing was Brandon. Seems like that’s just as plausible an account as presuming these things exist without any concrete thing to instantiate them.
2. Why does Parfit get to posit that some computer exists to run everything that goes through the selector? This is said to entail existence of concrete things but it sounds like mere hand-waving.
3. So the selector supposedly has to be the best… best at what? There is never just “the best”, it’s all always best at something. (this may have been the gist of Jonathan’s comment)
4. How is any of this naturalistic? It seems supernaturalistic in the purest, most traditional sense of the word, of positing something outside space and time which infuses the physical universe with some kind of form or pattern.
5. All of this is promised to rationally justify the natural existence of a Pagan God and Goddess, but the link to those deities is tenuous indeed. It appears to hinge entirely on a choice of “analogical predication” or metaphor. Yes, we can use metaphors – so what? What is new about that? How is this any more “rationally justified” than any other use of metaphor ever in the history of language?
I think I better stop there. Thoroughly confused.
Hi B.T. – Sorry to confuse! Your questions are all very good.
(1) Yes, axiarchists are all Platonists. Platonists say that mathematical objects really do exist. Many, many arguments have been made for Platonism elsewhere. Of course, Platonism might be wrong. Not everybody agrees with it. But axiarchists do.
(2) Axiarchists say it’s a law of nature that if some pattern satisfies the selector, then there exists a concrete thing that is an example or instance of the pattern. That’s the idea they use to explain the existence of concrete things. The point of axiarchism is to offer a lawful explanation for why there is something rather than nothing. Again, maybe they’re wrong, but they are offering a lawful hypothesis.
(3) I don’t know why the best has to be the best at something.
(4) So nature is just what exists in some spatial or temporal relation to us? How would you defend that? You’ve surely got no evidence that there’s nothing else in nature. If our universe is natural, then other universes are natural too. (I’m not saying other universes exist, just that, if they do, then they’re natural). But other universes aren’t in our space-time. Are string theorists studying the supernatural? Plenty of purely physical theories posit things not in the space-time of our universe. There are many, many good arguments that there are things not in the space-time of our universe. Nature is not centered around us, and certainly not centered around us spatio-temporally. Nature is big.
(5) Good point. The link to the God and Goddess is tenuous. I tend to think I’d do that part differently now. But we’re all explorers here, trying to understand new ways of thinking about the divine as it appears in nature. I’ll be working more on this in the future.
Thanks for the reply. Hope my tone didn’t sound too snippy. I was just really confounded.
3) I mean, what would the meaning of “best” be if not as the superlative of some particular scale? And those scales might be radically contradictory, like the best hero and the best villain. Without a “best at”, it’s hard for me to see what the meaning of best would be. Infinity does have a meaning without any concrete instantiation, because it is linked to a numerical scale. Best by itself isn’t linked to any particular scale, though.
I think maybe our different intuitions on this may be due to our starting points. I can see “best” by itself as meaningful within Platonism, where ideas themselves are able to have some kind of reality independent of a concrete instantiation (right?). So, maybe I see where you are coming from.
4) There are different ways of defining naturalism, so I don’t want to disagree per se. I know others who would probably accept the way you’re framing it here. Personally, I would push back on that logic, though. It seems like if there are other universes (and there may well be), and one could posit that in those universes things may be radically different enough that things can exist outside space or time (by which I mean outside any kind of space-time reality, not just our particular continuum), then one could also postulate that some universe may exist where the traditional Judeo-Christian God (or any other deity) exists and has the power to reach into our universe. In other words, that logic seems to make it possible to make any notion whatsoever, even the most explicitly supernatural, appear naturalistic. That’s how it appears from where I’m sitting right now anyway. I’m open to learning otherwise. 🙂 Note: Regarding string theory, my understanding is that strings are not outside space-time, they just involve additional dimensions beyond our familiar three.
No, you’re not being snippy, you’re asking excellent questions. (And of course my article was short, so not every issue could be addressed!)
(3) Yes, axiarchism (and many,many other theories in this area) require a scale of values. That scale is objective, and not based on humans (especially not based on human pleasures or pains). It’s “intrinsic value”, the value something has in itself, just because of its nature. One way to formalize intrinsic value is to use mathematical-computational theories of complexity, such as Bennett’s notion of logical depth. I like that approach, but it’s not the only one. Still, you’re right, more needs to be said here.
(4) Naturalism (as I conceive it) is pretty restrictive. But it’s not restricted by giving a list of what does or doesn’t exist, since it’s logically impossible to say that you’ve got *everything* on your list. So there are various marks of naturalism. A naturalistic theory is consistent with science (where science includes logic and mathematics). A naturalistic theory involves only the kinds of things that are fully open to scientific study (no mysteries). A naturalistic theory involves only lawful principles or operations (no miracles, no agents capable of violating natural laws). So there’s really no way anything like the Judaeo-Christian God could exist naturally. Most things in mythology or traditional religions would also be ruled out. But there’s still plenty of room for science (and logic and math) to grow in unexpected ways, and nature is defined in ways that don’t depend on humans.
4) So, how would, say, a Platonic form outside any kind of space-time be the kind of thing fully open to scientific study? It seems like it would be in principle unverifiable and unfalsifiable.
The forms are open to mathematical study – they’re mathematical objects. Science depends on math (it assumes the truth of math), but indeed math is not empirically verifiable or falsifiable; so, science assumes unverifiable and unfalsifiable truths; therefore, verifiability and falsifiability aren’t very useful foundations for science. Far too much “naturalism” is just warmed-up positivism, and positivism is inconsistent.
So, how is “Best” a mathematical object?