Axiarchism and Paganism, Part 1 by Eric Steinhart

Today we continue our late-winter theme of “Order and Structure” with Eric Steinhart’s discussion of Axiarchism and Paganism.  This essay is broken into two parts.  In Part 1, which is posted today, Eric Steinhert lays out the basic motivations for axiarchismIn Part 2, which will be posted tomorrow, Eric discusses how axiarchism can be used to rationally justify the natural existence of a Pagan God and Goddess.

Axiarchism Introduced

Axiarchism is a philosophical theory which states that reality is ultimately defined by some kind of value.  Axiarchism has ancient roots.  It starts with Plato, then gets taken up by Plotinus.  It becomes further developed by Leibniz.  Most recently, it’s been advocated by two living philosophers, Nicholas Rescher and John Leslie.  

Axiarchism is flexible enough to be interpreted in several ways.    I’ll develop one way here which is both Pagan and naturalistic.  This interpretation of axiarchism includes various divine principles.  If you like, you can refer to them as deities, as gods and goddesses.  But these divine principles are not supernatural people.  On the contrary, they are natural powers.  They are patterns of natural activity.  But natural doesn’t mean material or physical.  For the axiarchist, nature is much deeper and much bigger than any material or physical reality.  Everything said here is perfectly consistent with our best science.  But consistency with our best science doesn’t entail the kind of extreme and often unjustified skepticism that unfortunately typically gets associated with naturalism.

Platonism and Mathematics

Axiarchism starts with a Platonic theory of existence.  Platonists say that mathematical objects, such as numbers, really exist.  For Platonists, numbers are not concepts in your head and they are not symbols written down on paper.  Numbers exist whether or not anybody ever thinks of them.  Thus numbers exist objectively.  Numbers exist whether or not any physical things exist.  They are independent of any physical reality, and they do not exist in any space or time.  They exist eternally.  Likewise, numbers don’t depend on any other things for their existence.  They exist necessarily. 

Some current physicists, like Max Tegmark, argue that all reality is purely mathematical.  Tegmark is a Pythagorean.  But Platonists (and therefore axiarchists) don’t think that everything is mathematical.  Besides numbers, physical things exist.  Physical things are defined by mathematical things.  Mathematical things serve as the forms or templates of physical things.  Mathematical things provide physical things with their shapes and structures.  Physical things are examples or instances of mathematical patterns.  Hence they are said to exemplify or to instantiate mathematical essences.  But how?

One way to think about the relation between the mathematical and the physical, between the abstract and the concrete, is based on computers.  Of course, this appeal to computation is mainly (but not entirely) metaphorical; these computers are not like the devices we have on our desks.  We could avoid this computer metaphor by using lots of technical philosophical and mathematical jargon.  For example, we could talk about bare particulars instead of computers.  Since we want to avoid that, we’ll just talk about computers.

The computational interpretation of Platonism starts by treating numbers as programs.  Every number can be expressed as a series of binary digits, a sequence of zeros and ones.  But a series of binary digits is just a program for a computer.  So the distinction between the abstract and the concrete is like the distinction between software and hardware.  Numbers are abstract programs which can be run by concrete computers.  When some computer runs a number, a physical thing comes into existence.  The nature of this physical thing is defined by the number.  The physical things running on these computers might be as small as quarks or as big as entire universes or systems of universes.

Divine Law

Platonists have long argued that mathematical objects exist necessarily.  They cannot fail to exist; it would be impossible for them not to exist.  But not so for concrete things.  They are not necessary; on the contrary, they are contingent.  Any concrete thing might not exist; it might fail to exist.  Concrete things don’t have to exist.  So, why are there any concrete things at all?  Why are there any physical things at all?  This is a famous question, first raised by Leibniz: why is there something rather than nothing?  More precisely, why is the set of concrete things populated rather than empty?  There needs to be some explanation for the existence of any physical things at all.  The explanation can’t be causal.  After all, causes are physical.  It has to be a deeper kind of explanation. 

Axiarchists have argued that the best explanation for the existence of any concrete things involves a natural Law.  This Law is deeper than any physical law, and it provides the reason or explanation for the existence of concrete things.  This Law logically brings all physical things into existence.  It is the ultimate sufficient reason for the existence of any physical things, including our universe as well as any other universes that exist.  Philosophers (like Leibniz) have given many arguments that there must be some ultimate sufficient reason lying behind all concrete things.  The success of science (which depends on finding reasons in nature) has been used to justify the existence of this ultimate sufficient reason.  

The truth of this ultimate Law is the power that brings all concrete things into being.  If nature is the totality of all concrete things, then this truth is the ultimate natural power, the power which produces nature itself.  Many theologians have thought of this kind of power as divine.  Axiarchists like Plato, Plotinus, and Leslie have explicitly referred to this power as divine.  Of course, this power is not personal.  It is not the Abrahamic God or any other personal deity.  This power is an impersonal force, which emanates or erupts from the Law.  The Law is the divine Source of concrete existence (a Source which exists prior to any concrete things).  Many Pagans have posited an ultimate Deity, a divine Source which serves as the ground of nature.  If axiarchism is given a Pagan interpretation, then the ultimate Law is this Source, and its truth is the divine power which it emanates.  But this Source is not a thing.  Rather, it is the abstract reason for all things.  It is like a mathematical axiom, except that it is deeper than every mathematical axiom.  It is the ground of things.

Selectors and the Computational Interpretation

Of course, while this theology may be interesting, it does not help to clarify the meaning of the Law.  What might this ultimate Law look like?  Any such Law has to start with purely abstract objects and end with some concrete things.  It has to be a Law that entails that concrete things exist.  The contemporary philosopher Derek Parfit uses the concept of a Selector to define this law.  His reasoning can be summarized like this: Numbers have various features.  For instance, some numbers are even while others are odd.  Some are prime while others are divisible.  The Selector acts like a filter on numbers.  Those numbers that pass through the Selector serve as the programs for concrete things.  On this view, the ultimate Law looks like this: For any number, if that number passes through the Selector, then there exists some computer which runs that number as its program.  

How would this Law work to create physical things?  Suppose the Selector is the property of being prime.  Prime numbers pass through the Selector.  This means that for every number, if that number is prime, then some computer exists which runs that number as its program.  Since there are some prime numbers, there are some computers which run them as their programs.  This Law entails the existence of physical things.  It explains why there are some physical things rather than none.  However, this Law does not cause any physical things to exist; on the contrary, it logically implies that they exist.

Of course, the notion that primeness is the Selector is far too easy.  But we know that some numbers, when run on computers, define physical systems like cellular automata.   The game of life is a well-known example.  So the Selector might be the property of defining a cellular automaton.   If that’s right, then the Law looks like this: for any number, if that number defines a cellular automaton, then there exists some computer which runs that number as its program.  Some cellular automata contain internal patterns of activity which are self-reproducing structures.  So the Selector might be the property of defining a cellular automaton which contains internal self-reproducing patterns.  

Some cellular automata contain internal patterns which are capable of universal computation.  These internal patterns are universal Turing machines.  So the Selector might entail the existence of computations which run internal computations.  Computations can be stacked on top of computations.  Computations running on top of other computations are often referred to as virtual machines.  Perhaps the most fundamental quantum mechanical processes in our universe are basic computations.  Things like protons are virtual machines stacked on those lower-level computations.  Things like atoms and molecules are higher level virtual machines stacked on top of lower level machines.  If this is right, then human brains and bodies are very high-level virtual machines.  But virtual machines, no matter how high, are still computations that fall under the laws of computing.

A Problem with Axiarchism

According to this computational interpretation, axiarchism provides an explanation for the existence of physical things, including our universe and all the things running inside of it. But this version of axiarchism has problems.  It involves a Law based on some selector.  One problem with the notion of a Law based on a Selector is that almost any Selector seems arbitrary.  Why this Selector rather than some other Selector?   If the Law is the ultimate sufficient reason for all things, then the Law itself can’t involve any arbitrariness.  What selects the Selector?  If the Selector is the reason for things, what is the reason for the Selector?  It seems the Selector depends on some deeper Super-Selector, which then depends on some Super-Duper-Selector, and so it goes.  The result is an endless regression of Selectors, which leaves the whole system powerless.  

Fortunately, axiarchists can provide a solution to the problem of the regress of Selectors.  They argue that the only way to avoid arbitrariness is for the Selector to be the best.  On this view, every program generates some goodness or excellence when it is run.  Programs can therefore be ranked according to how much goodness they generate.   Goodness doesn’t mean human pleasure.  It doesn’t even have anything to do with people (after all, universes don’t need to include people, and for a long time, even our universe didn’t include any people).  For axiarchists, goodness is an objective property of programs.  Goodness is intrinsic value, the value any thing has just because of its nature.  Thus every number has some degree of intrinsic value, which it produces if it runs on a computer.

Programs (numbers) have different degrees of intrinsic value.  Some are better than others.  Perhaps some of these programs are better than all others.  They are the best of all possible programs.  Axiarchists now say that the Selector is the property of being the best.  This Selector is not arbitrary.  Given any set of options or choices, it’s rational to select the best and irrational to select anything less than the best.  Moreover, this is a necessary property of rationality: reason necessarily selects the best.  So the Selector selects itself.  Why is the Selector the property of being the best?  Because the property of being the best is itself the best property.  The best Selector is the property of being the best.  The self-selection of the best means that the infinite regress of Selectors never gets started.


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.  Part 2 will show how axiarchists solve this problem, and it will continue the development of axiarchism.  Part 2 will also show how axiarchism can be used to rationally justify the natural existence of a Pagan God and Goddess.  Axiarchism can therefore serve as the logical foundation for a rich Pagan theology. 

The Author

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.


Tomorrow, Eric concludes with “Axiarchism and Paganism, Part 2”.

One Comment on “Axiarchism and Paganism, Part 1 by Eric Steinhart

  1. This is very interesting. I’ve been pondering the relationship between numbers and physical objects. I’ve been tempted by Platonism away from a simplistic naturalism that says that particles, energy, and spacetime are the only real things. Numbers, ideas, qualia, all seem to defy this kind of naturalism.

    The justification for self-selection at the end seems circular. The Selector is selected because it is the best. What selects the properties that makes something the best? In other words, according to what criterion is the Selector selected, and who selected that criterion? If the Selector selected the criterion, what is to prevent another selector from selecting itself through its own criterion?

    I also don’t share the drive to call the ultimate law by the name God or to call it divine. It doesn’t, in my opinion, add anything of value to the philosophy and probably adds a level of confusion with all the millions of ideas that have been attached to that name.

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