So you might wonder about the meaning of the term evolution.

Since the term evolution is abstract, it’s definition will be abstract: a process is evolutionary if and only if it increases complexity. Generally, this means that the complexity of the most complex things is increasing. Less complex things may still exist. This means that evolutionary processes build stratified layers of complexity – they build complexity hierarchies. Of course, the weight is now on the term complexity. And, fortunately, there are explicit ways to cash that out. Different types of evolution will obviously use different complexity metrics (and that, indeed, is exactly what makes them different types of evolution).

Within biological evolution, the arrow of complexity hypothesis states that: “the complex functional organization of the most complex products of open-ended evolutionary systems has a general tendency to increase with time.” (Bedau, 1998: 145) And biological evolution does support various arrows of complexity. You might say this is Kantian purposiveness without purpose. But it would be distracting to get into that. On to the metrics:

Bower says that the complexity of an organism is the number of distinct cell types it contains (1988: 101). He argues that evolution tends to increase the complexity of the most complex (species of) organisms. Adami et al. (2000) equate the complexity of organisms with the complexity of their genomes; they define the complexity of a genome to be the amount of information it encodes about the environment in which it has evolved. Generally speaking, this genomic complexity has always been steadily increasing.

Within chemistry, one might simply define the complexity of an element to be its number of protons. Within molecules, more structural definitions can be used. Over time, ever more complex elements have progressively appeared in our universe. Thus the complexity of the most complex elements has been increasing.

At the most general level, Chaisson says that the complexity of a system is “the rate at which free energy transits a complex system of given mass”; it is “the free energy rate density, alternatively called the specific free energy rate, expressed in units of energy per time per mass” (2001: 134). Chaisson shows – with impressive clarity – how the complexity of the most complex things have been steadily increasing.

Another way to look at physical complexity is to use Dennett’s levels (1991). He distinguishes between the physical, design, and intentional levels. The history of our universe started with just the physical level; design levels emerged (chemical and biological); and then intentional levels emerged (psychological, social). Dennett has also applies his levels to other types of universes like cellular automata. And, close to Dennett’s ideas, I’ll give a shout out to Jaker op Akkerhuis’s operator hierarchy (2008). (Though I admit I find Akkerhuis very hard to understand.)

One very general measure of complexity (and probably the best) is Bennett’s notion that complexity is logical depth (1988). The complexity of a structure is the amount of computational work required to generate the structure. This can be measured formally in terms of the run times of programs that generate the structures. For cosmological evolution, something like logical depth is a good measure. The process atheist says that cosmological evolution is increasing the logical depth of universes.

It’s interesting to note that logical depth maps very closely onto Leibniz’s notion of perfection. (And Leibniz, remarkably, even offered his analysis of perfection in terms of binary strings! I love Leibniz!) Leibniz offers a quantitative analysis of perfection: he says perfection is quantity of essence (1697: 86). Leibniz often says that perfection has two dimensions: it is a product of variety and order (Monadology, sec. 58; Theodicy, sec. 207; Discourse on Metaphysics sec. 6). Order is like algorithmic regularity and variety is like algorithmic randomness. Hence Leibniz’s concept of perfection is like logical depth.

Once we get out into the infinite, more powerful measures are needed. Kyburg (1961: 392-393) says that the complexity of a theory is measured by the number of quantifiers in the shortest version of the theory. Another and probably better approach is to use something like the Kleene-Mostowski hierarchy. Given any axiom system (any theory), expressed in the predicate calculus in prenex normal form, the complexity of the theory is the number of alternating blocks of the same type of quantifier. Thus the complexity of a universe is the complexity of the simplest theory of which the universe is a model. The process atheist says that this (or some similar) metric of complexity is steadily increasing as structures are produced one after another by metaphysical evolution.

Adami, C., Ofria, C. & Collier, T. (2000) Evolution of biological complexity. Proceedings of the National Academy of Sciences 97 (9), 4463 – 4468.

Bedau, M. (1998) Philosophical content and method of artificial life. In T. Bynum & J. Moor (Eds.) (1998) The Digital Phoenix: How Computers are Changing Philosophy. Malden, MA: Basil Blackwell, 135-152.

Bennett, C. (1988) Logical depth and physical complexity. In Herken, R. (1988) The Universal Turing Machine: A Half-Century Survey. New York: Oxford University Press, 227-257.

Bower, J. (1988) The Evolution of Complexity by Means of Natural Selection. Princeton, NJ: Princeton University Press.

Chaisson, E. (2001) Cosmic Evolution: The Rise of Complexity in Nature. Cambridge, MA: Harvard University Press.

Dennett, D. (1991) Real patterns. Journal of Philosophy, 27-51.

Jagers op Akkerhuis, G. (2008) Analysing hierarchy in the organization of biological and physical systems. Biological Reviews 83, 1-12.

Kyburg, H. (1961) A modest proposal concerning simplicity. The Philosophical Review 70 (3), 390-395.

Leibniz, G. W. (1697/1988) On the ultimate origination of the universe. In P. Schrecker & A. Schrecker (1988) Leibniz: Monadology and Other Essays. New York: Macmillan Publishing, 84-94.

Within many familiar systems, loveliness is very very rare. It’s very rare within the models of simple physical theories and even more rare within the models of complex physical theories (e.g. the solutions to the equations of string theory). This can be illustrated with the cellular automaton known as the game of life. You can read lots about the game of life on the web. I’ll just give a very quick presentation of the relevant points.

The game of life is played on a grid composed of square cells, like a chessboard. A clock is ticking (all cells can hear it). A cell is either ON or OFF (alternatively, LIVE or DEAD, or 1 and 0). Cells blink ON and OFF like lightbulbs, according to a rule each cell computes every time the clock ticks: (1) a cell counts its ON neighbors; (2) if it is ON and has 2 or 3 ON neighbors, then it stays ON, else it turns OFF; if it is OFF and has 3 ON neighbors, then it turns ON, else it stays OFF.

The figure below illustrates how cells change according to the life rule.

The game of life supports regular patterns, such as the glider, which appears to move across the space of the life grid. The glider is shown below.

It’s even possible to construct universal Turing machines and self-reproducing patterns in the game of life. But the game of life is rare within the class of similar cellular automata.

The rule for the game of life can be expressed in a small table. The table is shown in the figure below.

Since there are 16 slots in this table, and each can take the value 0 or 1, there are 2^16 different variants of the game of life. That’s 65536 variants. These are all the possible classes of life-like universes. Some of these variants are shown below.

Out of these, very few support any physical content at all. Perhaps a dozen or so support moving patterns. And only one is known to support patterns that compute and that reproduce (namely, the game of life itself). Within an extremely large number of physical systems, or purely mathematical systems, loveliness is vanishingly rare. Hence, that any actual universe is lovely, when almost all possible universes are not, is extremely surprising.

To say a fact is “surprising’ is far from merely subjective. Surprising facts almost always carry information. That’s because carrying information is itself a lovely feature. And that’s why our brains are highly sensitive to loveliness: when, out of the endless background of noise, you are given a signal carrying information, you’d better pay attention. It’s a good rule to follow in any uncertain environment.

First, I want to stress, contra-Eric, that “atheism” is the best and most accurate general term we have for non-theists.

Eric complains that “atheism” characterizes him negatively in terms of what he does not believe (as though that were more important than the many positive things he does believe). He also worries that the term atheism has too much baggage which leads people to make misleading assumptions about his metaphysics (e.g. that he is a materialist when he is not).

What this ignores is that a concerted and potentially successful effort is underway to make clear that the only thing atheism itself strictly means is that one lacks belief in any personal gods and that it need not mean anything else.

This most stripped down, strictly negative, “dictionary” meaning for atheism is the one we should emphasize for several reasons.

First of all, it is accurate. Regardless of whether previous atheists all got lumped in with materialism or nihilism or existentialism or communism or any other questionable “-ism”, the term is the clearest, broadest, most natural, and most familiar candidate available for contrasting our shared position with theism. While the word is taboo in many places, it is a more natural catch-all than other words with more content that would divide people too much for a true classification scheme. I am perfectly fine with the proliferation of atheist metaphysics and atheist ethical groups and political groups, etc. But on the most general level of categorization, an atheist is an atheist if she simply lacks belief in, or worship of, personal gods.

This bareness and simplicity of “Dictionary Atheism” unifies us as a competing bloc against the large contingent of theists, while also being empty enough to not constrain different atheists from having different particular views about epistemology, ethics, metaphysics, politics, etc. (just as theists do). If agnostic atheists, gnostic atheists, and deists can all find common cause as opponents of theism–i.e., as atheists–then there is a unified front against theism that comes in handy on a central issue, even if particular kinds of atheists diverge from each other in any number of particulars from there.

Eric also complains about going to atheist meetings where

everybody was mainly there just to be hostile towards religion (by which they pretty much all seemed to mean the conservative Christianity of the American religious right). I don’t see how hostility is ever helpful. I dislike hostility from atheists as much as I dislike it from religious fundamentalists. I have no interest in participating in a group whose primary purpose is the hatred of some other group. I’d prefer to build positive bridges and to expand the rational community of hope.

While I agree that it is dangerous for atheists only to be associated with the negative and to only organize as an opposition group and never as people who do anything positive, there is nonetheless still room for a level of organization on which we are an opposition group. The hegemony of irrationalistic, faith-based thinking and institutions is real. And it requires not only positive alternatives but some degree of hostile (but always non-violent) opposition to faith’s damaging ethical and political consequences.

Atheists have been treated with an enormous amount of hostility over the ages. A good part of atheists’ reputations as generally angry and hostile is actually due to religious projection of these traits into us as part of their vilification of us. Where atheists are angry and focused on their anger, often it is either a phase of getting out pent up and repressed frustrations based on cultural, familial, and ecclesiastical abuse over their personal dissent from religion. They deserve this outlet, even if they should then be steered into constructive channels from there.

Other times atheist anger is a proper and calibrated response to injustices, including the unfairness of manifestly unqualified religious leaders and institutions being held up as authorities to the detriment of morality and knowledge.

Do angry atheists sometimes go over the top? Yes, regularly activist atheists are as unnecessarily intemperate in their rhetoric as activist liberals or activist conservatives or activist environmentalists or activist feminists or Tea Party activists or members of any other activist groups. But does that mean they should never organize specifically around their opposition to unwarranted religious power over hearts, minds, and governments? No way. We can encourage activist atheists to temper their rhetoric without counter-productively conflating the whole endeavor of being an opposition movement with being a hate group. Conflict is part of life. We should be as respectful as possible but that does not mean treating conflict as an inherent evil to be avoided at all costs.

An atheist opposition movement is very necessary. Atheists (and other God-ignoring scientists and academics) have for a couple of centuries now been doing the positive, constructive science, metaphysics, ethics, cosmology, etc. that Eric wants us to focus on. And yet all the positive advances in god-indifferent ideas have not by themselves been able to supplant numerous theistic and superstitious ideas or institutions in the cultural mainstream.

Countless natural and social scientists and philosophers have advanced our knowledge in ways which de facto undermine theism and religious beliefs and institutions and should naturally have made them irrelevant and obsolete. And yet, those institutions chug on (and often even grow) because all the positive alternatives in the world make no difference to the mainstream culture as long as they only exist in the parallel universe of academia and science labs.

And even if we were able to mainstream knowledge about alternatives to theism and theistic religion, this will not by itself win over people who feel no compunction to abandon their existing beliefs. So what if there would be an evolutionary metaphysics on the market that some philosopher says for some arcane reason is better than theism? If someone has deep personal, cultural, familial ties to her theism that go unchallenged, she feels as though she has no reason to ever seriously consider this new alternative. A negative attack on theism is important for people to understand why they need to leave what is comfortable to them to begin with.

And while I agree with Eric that atheists should engage in philosophical speculation and develop robust alternative accounts about metaphysics and ethics, there are dangers in defining ourselves as primarily “evolutionists” and “evolvers” as he suggests.

First and foremost, in the popular mind there are few rules if any constraining speculative thinking. People think that when it comes to speculation, we are free to believe whatever we want as our “faith-based choice”. Atheists who center themselves not on a call for more epistemological rigor but rather on a speculative metaphysics would be to the average person “Just another speculative, faith-based belief system, only as valid as any other, including theism”. Even worse, we would be cast as “just another religion” which can be waved away as easily as one waves away a foreign culture’s beliefs as “maybe good for that culture but not at all a challenge to my own religion”.

We would be “just another faith” since we are no longer just speculating metaphysically but defining and identifying ourselves by a metaphysical hypothesis that is insufficiently grounded in evidence to justify the strength of our commitments.

I think an atheist metaphysician may feel well enough justified in believing in cosmological evolution to qualifiedly believe in it and hypothesize about it. But to make it a defining feature of one’s identity and a foundational belief that influences one’s entire way of thinking in a controlling way risks being formally identical enough to other, irrationalistic faiths for the other faiths to see an opening and say, “See ‘Evoluitonism’ is just another faith—no more justified than ours—and therefore there is no need to abandon the faith you already are attached to for something that is only another faith.”

And not only this, but the preexisting faiths already cozy up nicely to people’s anthropomorphizing, superstitious brains in a more naturally intuitive way than evolutionary thinking does. In the competition among leaps of faith, faith in a purposeful person in the sky offering love and eternal life would likely trounce a vague, indifferent, counter-intuitive, unguided process like cosmic evolution which still leaves the question open “where did the first thing come from since it could not have evolved out of any prior thing?” I think if it was faith in personal gods vs. faith in all-encompassing metaphysical evolution, a populace which already will not even accept the scientifically sure evolution by natural selection (but constantly demands either creationism or personalistic theistic evolution) will certainly choose the personal gods over the impersonal evolution principle. It’s a losing proposition.

And not only would evolutionary metaphysics persuade hardly anyone in the cultural mainstream, there would be a much worse consequence: advocating it and centralizing our identities based on it would lose us the moral high ground. We would no longer credibly be able to define ourselves as those who stand for rational scrupulousness and against faith. I am for speculation but not for the elevation of speculative beliefs to a greater role and influence than their degree of sureness warrants. This is what separates me first and foremost from the faith-based thinker and I want to keep it that way.

And, finally, the worst consequence of defining ourselves as “evolutionists” is that it would play right into the fundamentalists’ hands. They desperately want to drag down the theory of evolution to the status of “mere, faith-based speculation” and a “prejudicing worldview”.

By trumpeting speculative metaphysical evolutionary theories as foundational to our thinking and identifying as “evolvers”, we would be actively helping them in their efforts to confuse all evolutionary thinking with “just another unscientific, faith-based worldview”. We would be undermining the crucial effort to establish in the public mind the truth that evolution by natural selection is a scientific theory that should be rationally compelling to all people regardless of faith commitments and that should be capable of undermining many religious beliefs which its objective truth objectively falsifies.

Publicly equivocating in our use of the concept of evolution and extending it liberally and ideologically to solve every other cosmological, metaphysical, and ethical problem—and doing all of this as part of building an inevitably partisan community identity—would add all sorts of counter-productive baggage to the theory of evolution that it does not need and which fundamentalists have been working hard to saddle it with for decades.

Evolution’s primary association should only be with scientific neutrality, not with any one community, not with wishful speculation, and not with overreaching, all encompassing accounts of everything in existence.

The worst consequence to fear from Eric’s suggestion is that fundamentalists are able to disingenuously claim that evolution by natural selection is not the outcome of the best, most objective science, but is only supported by a metaphysical prejudice towards an evolutionary cosmos. If evolution becomes a matter of faith, a dogma we are caught looking to sneak under every nook and cranny of existence, then we give ammunition to the lie that even biological evolution is only believed in because of that preexisting dogmatic faith that evolution is everything.

And once evolution is tied up with faith, then it loses any standing to challenge people’s existing faiths. And, again, when given the choice between their own, preexisting arbitrary faith attachments on the one hand and an alternative set of arbitrary faith attachments on the other, the vast majority feel justified in sticking with their own arbitrary beliefs and ignoring the alternatives.

Your Thoughts?

According to Leibniz, God selects our universe for actualization because God knows that it is the best of all possible universes. But how does God know that? It’s not sufficient to say that God just knows it. God’s knowledge of the best universe requires an explanation. And Leibniz gives one: God runs a search algorithm. The search algorithm starts with a sorting algorithm. God compares possible universes with respect to goodness:

The infinity of possibles, however great it may be, is no greater than that of the wisdom of God, who knows all possibles. . . . The wisdom of God, not content with embracing all the possibles, penetrates them, compares them, weighs them one against the other, to estimate their degrees of perfection or imperfection, the strong and the weak, the good and the evil. It goes even beyond the finite combinations, it makes of them an infinity of infinities, that is to say, an infinity of possible sequences of the universe, each of which contains an infinity of creatures. By this means the divine Wisdom distributes all the possibles it had already contemplated separately, into so many universal systems which it further compares the one with the other. The result of all these comparisons and deliberations is the choice of the best from among all these possible systems. (Leibniz, Theodicy, sec. 225)

The output of this sorting algorithm is an ordered series of equivalence classes of universes. All universes in the same class are equally good. Leibniz uses an architectural metaphor to illustrate the output of the sorting algorithm. The totality of possible universes is like a library. The library is organized into levels. Each level is an equivalence class of possible universes. Higher levels of the library have better universes. Leibniz refers to the library as the Palace of the Fates. Leibniz describes this Palace in a story in which the goddess Athena takes a priest Theodorus for a tour of the mind of God:

You see here the Palace of the Fates . . . Here are representations not only of that which happens but also of all that which is possible. God, having surveyed them before the beginning of the actual universe, classified the possibilities into universes, and chose the best of all. . . . These possible universes are all here, that is, as ideas [in the mind of God] . . . . They went into other rooms, and always they saw new universes. The halls of the Palace rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful universes. Finally they reached the highest hall which was the most beautiful universe of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. That is because among an endless number of possible universes, there is the best of all . . . but there is not any one which has not also less perfect universes below it: that is why the pyramid goes on descending to infinity. (Leibniz, Theodicy, secs. 414-417).

According to this Leibnizian story, the divine designer runs a search algorithm that involves two steps: first, it sorts the universes; second, it actualizes the universe that comes out on top in the sort. But if that’s all the divine designer does, then it isn’t God. All the design arguments show is that there is some computer running a trivial search algorithm. The computer is not a person; it isn’t conscious; it doesn’t have free will. It isn’t God. It isn’t even any of the Gods of the Philosophers. (Or, if it is, the term “God” is so trivial that it’s not worth arguing about, not worth worshiping, not really worth much of anything.) The designer is simply an abstract mathematical machine. It converges to a solution (to a final or halting state). The solution is the universe.

Design arguments are wonderful! They justify the existence of a platonic computer that is in no way divine. Atheists ought to use design arguments against theists. Why don’t they?

The theistic version of the Fine Tuning Argument goes like this: (1) The Fine Tuning Argument is sound. (2) If the Fine Tuning Argument is sound, then there is a Tuner. (3) Therefore, there is a Tuner. (4) If the Tuner is God, then God exists. (5) The Tuner is God. (6) Consequently, God exists.

The atheistic version of the Fine Tuning Argument goes like this: (1) The Fine Tuning Argument is sound. (2) If the Fine Tuning Argument is sound, then there is a Tuner. (3) Therefore, there is a Tuner. (4) If the Tuner is not God, then God does not exist. (5) The Tuner is not God. (6) Consequently, God does not exist.

The fourth premise in the atheistic version follows from the definition of God. If God is a tuner, then God is essentially a tuner. It’s not possible for God to exist and for God to fail to be a tuner. The theist, on learning that the Tuner is not God, can’t say that God’s busy doing other things or that fine-tuning the universe falls outside of God’s job description. On the contrary, it’s part of what it means to be God.

The fifth premise is what the atheist has to justify. The atheist has to prove that the Tuner is not God. This can’t be done by just blithely saying that the Tuner might not be God. It isn’t enough to just point to some other alternative possibility. Mere possibility is not what is asked for here. The atheist has to demonstrate that the Tuner is not God.

The way to do this is to present an alternative that is clearly a better explanation for the fine tuning than God. The argument goes like this: (1) The Tuner has a feature F if and only if F is required for tuning the universe (say, for life). (2) If the Tuner were God, then the Tuner would have additional features. (3) Therefore, the Tuner is not God.

Finely-tuning universes for life doesn’t seem to be very demanding. The assumption behind all the fine tuning arguments is that tuning involves setting the values of some finite number of numerical parameters. If there are n parameters, then their value ranges are the coordinate axes of an n-dimensional space. Each point in this space has the form (v1, . . . vn). The Fine Tuning Arguments always seem to assume that these values are plugged in to some fixed system of equations E. Thus E(v1, . . . vn) is the form of some possible universe. It’s usually also assumed that the equations apply to some initial conditions. They act as an operator on the initial conditions i. Any initial condition is a member of some set of initial conditions I. So the full form of a universe is (E(v1, . . . vn))(i). This form has to be tested for life. Here’s the algorithm:

    algorithm FineTuneForLife() [
        for every point (v1, . . . vn) in the space (P1, . . . Pn) do [
            for every i in the set of initial conditions I do [
                if (the form (E(v1, . . . vn))(i) permits life),
                    then make a universe with that form;]]]

Either there is some machine (some computer) that can run the algorithm FineTuneForLife, or else there is not. If there is no such machine, then perhaps the Tuner is God. If there is such a machine, then the Tuner is not God. The Tuner is merely a computer. More precisely, the Tuner is the simplest machine that can run the algorithm FineTuneForLife.

I doubt that any finite state machine can run FineTuneForLife. I’m not sure that any Turing machine can run FineTuneForLife. But Turing Machines are not very impressive; there are far more powerful transfinite computers. The outer limits of computation are said by some mathematicians to be equivalent to the constructible hierarchy of sets (also known as L). If there are models of our physics that fall within the constructible hierarchy of sets (also known as L), then there is a machine that can run FineTuneForLife. So the atheist has to defend this claim: there are models of our physics that lie in L. And I think that’s a claim that is very easy to defend.

It’s worth pointing out that FineTuneForLife is a simple brute force search. There are almost certainly very powerful ways to optimize that search. More optimal searches usually require less powerful computers.

It’s also worth pointing out that human beings already know how to finely tune toy universes for artificial life. Small numbers of human beings, working for only a few decades, have already produced impressive software universes that contain life-like structures. So, how hard can it be to run something like FineTuneForLife? Does it really require an intellect whose power transcends that of every logically possible machine? The successes of artificial life suggest that fine tuning isn’t all that hard. If it can be done by a machine, then God does not exist.

The Effectiveness Argument goes like this: (1) Mathematics is effective in science. (2) The best explanation for this effectiveness is that physical reality is grounded in mathematical reality. (3) So, by inference to the best explanation, all physical reality, including our universe, is grounded in mathematical reality – in pure mathematics.

The second premise in the Effectiveness Argument is supported by a variety of writers. Dipert (1997: 332) argues that “the very possibility of a clear understanding of the world requires the possibility that it is a simple mathematical structure”. Steiner (1998: 4 – 5) puts it even more powerfully like this:

The strategy physicists pursued . . . to guess at the laws of nature, was a Pythagorean strategy: they used the relations between the structures and even the notations of mathematics to frame analogies and guess according to those analogies. The strategy succeeded. . . . The success of the Pythagorean strategy might lead the reader to conceptual Pythagoreanism, the view that the ultimate properties or ‘real essences’ of things are none other than the mathematical structures and their relations. More radically, one might adopt metaphysical Pythagoreanism, which simply identifies the Universe or the things in it with mathematical objects or structures. (Some physicists write as though an elementary particle just ‘is’ an irreducible group representation, or even that the entire universe is.)

Steiner (1998: ch. 4) brilliantly discusses many examples in which the pythagorean strategy of identifying physical things with mathematical things is successful. His cases include: Maxwell’s study of electromagnetism; Schroedinger’s study of wave mechanics; Dirac’s study of the positron; Schwarzschild’s solution for the equations of general relativity (i.e. black holes); Heisenberg’s study of the symmetries of nucleons; Kemmer’s study of pions; Gell-Mann’s and Ne’eman’s study of particle systems with unitary spin and the consequent discovery of quarks; Einstein’s inference of the field equations for general relativity; the Heisenberg-Born-Jordan derivation of matrix mechanics; Schroedinger’s derivation of the Klein-Gordon equation; the derivation of the Yang-Mills equation; the study of analytic continuations in crossing symmetries.

As a continuation of Steiner’s reasoning, Tegmark (1998: 44) says: “the usefulness of mathematics for describing the physical world is a natural consequence of the fact that the latter is a mathematical structure.” Accordingly, Tegmark (1998: 46-47) simply collapses the distinction between mathematical and physical existence:

One might say that wherever there is light, there are associated ripples in the electromagnetic field. But the modern view is that light is the ripples. One might say that wherever there is matter, there are associated ripples in the metric known as curvature. But Eddington’s view is that matter is the ripples. One might say that wherever there is physical existence, there is an associated mathematical structure. But according to our TOE [theory of everything], physical existence is mathematical existence. (The italics are Tegmark’s.)

Dipert, R. (1997) The mathematical structure of the world: The world as graph. Journal of Philosophy 94 (7), 329-358.

Steiner, M. (1998) The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.

Tegmark, M. (1998) Is ‘the Theory of Everything’ merely the ultimate ensemble theory? Annals of Physics 270, 1-51.

Wigner, E. (1960) The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics 13, 1-14.

The ground doesn’t even come close to matching any of the descriptions of any of the gods of mythology or even philosophy. So it’s hard to see why affirming the existence of the ground would be offensive to atheists. And yet, looking at the various responses to my last post, it seems that many atheists object. Why?

The ground is not mysterious. Leibniz has given us a great deal of information about it. First, it exists necessarily. Second, the very fact that the ground is not involved in physical relations is informative. To say that the ground is not involved in physics means that the ground is an abstract object. Abstract objects are objects that don’t participate in any spatial, temporal, or causal relations. Mathematical objects are abstract.

Mathematical objects include things like sets, numbers, vectors, functions, and so forth. If mathematical objects exist, then they aren’t physical. And, just to be clear, they aren’t concepts. Numbers aren’t things in your brain. There are infinitely many numbers; but you don’t have infinitely many things in your brain. If numbers exist, then they have an entirely objective existence that doesn’t depend on you or your thoughts at all. If you think math is all in your head, then you can’t do math.

Mathematical objects obviously play roles in science (especially in basic physics, which is intensely mathematical). The Quine-Putnam Indispensability argument says that since mathematical objects are needed for science, they exist. If you believe in quarks and gravity, you ought to believe in math. If there are no numbers, what sense would it make to use equations to describe the physical world? And, of course, you can’t just say you believe in some mathematical objects and not others. You get the whole system or none of it. And, looking back to Leibniz, mathematical objects exist necessarily.

These ideas about mathematics are known in metaphysics as platonism. Platonists affirm the objective reality of a world of mathematical entities. And, even though we can’t see or touch mathematical objects, we obviously know a lot about them. Indeed, math is the most stable and enduring part of human knowledge. Math involves proof – everything else is uncertain. Of course, platonism, like every part of philosophy, is controversial. But platonism comes with enormous benefits. Why not use them?

You can use platonism to complete Leibniz’s Sufficient Reason Argument. The following logic justifies the thesis that the ground is a mathematical object: our universe is mathematically structured; the best explanation for the mathematical structure of our universe is that it is generated by a mathematical object. It’s reasonable to believe the best explanation. Accordingly, there is some mathematical object that generates our universe. And that object is the ground.

How could the ground be a mathematical object? What does that even mean? To start to answer this question, consider a cellular automaton like the game of life. Such cellular automata can have incredibly rich physical content. Any instantaneous stage of some game of life can be encoded in a bit string – a sequence of 0s and 1s. Another way of putting this is to say that the stage supervenes on the bit string. A game of life is a series of stages; so any game of life supervenes on a series of bit strings. The bit strings aren’t arbitrary. They are generated by the iteration of a function which encodes the causal laws for the game of life. The function is Turing-computable. So every game of life supervenes on a sequence of bit strings generated by the iteration of an abstract Turing machine. The abstract Turing machine is itself just a function from numbers to numbers.

Many writers have thought hard about the possibility that our universe supervenes on the iterations of some abstract Turing machine. This is the computable universe hypothesis. (See, for instance, the work of Ed Fredkin or Jurgen Scmidhuber). I suspect our universe is too complex to supervene on the iterations of a Turing machine. But there are far more complex abstract machines. So the hypothesis that the ground is a kind of mathematical object is perfectly intelligible.

It’s easy to see why a theist might object to the mathematical ground – it directly competes with the theistic god! If the mathematical ground exists, then the theistic god doesn’t exist (or at the very least, that god is cosmologically unemployed).

Should atheists object to the mathematical ground? If so, why? Platonism gives atheists an enormously powerful metaphysics – a world of abstract, eternal, transcendental, necessary objects. But none of them are gods. And that world is knowable by reason (it’s the very peak of rationality). You’d expect atheists to embrace that.

So I’m wondering: can atheists do math?

These first cause arguments are debates about the structure of contingent physical existence. It’s fun to reason backwards in time along causal chains, but that reasoning remains entirely within the system of contingent physical things. The deeper questions are these: Why is there a universe rather than no universe? Why are there any physical things rather than no physical things? Why are there any contingent things rather than no contingent things? No first cause argument (or atheistic reply) even tries to answer those questions.

Aquinas’s Third Way and Leibniz’s Sufficient Reason Argument are much deeper arguments. Theists and atheists both ought to study them carefully. Leibniz’s Sufficient Reason Argument is especially interesting. Here it is:

(1) Neither in any single thing, nor in the total aggregate and series of things, can the sufficient reason for their existence be discovered. (2) Let us suppose a book entitled The Elements of Geometry to have existed eternally, one edition having always been copied from the preceding. (3) Although you can account for the present copy by a reference to the past copy which it reproduces, yet, however far back you go in this series of reproductions, you can never arrive at a complete explanation; (4) You always will have to ask why at all times these books have existed, that is, why there have been any books at all and why this book in particular. (5) What is true concerning these books is equally true concerning the diverse states of the universe, for here too the following state is in some way a copy of the preceding one (although changing according to certain laws). (6) However far you turn back to antecedent states, you will never discover in any or all of these states the full reason why there is a universe rather than no universe, nor why it is such as it is. (7) You may well suppose the universe to be eternal; yet what you thus posit is nothing but the succession of its states, and you will not find the sufficient reason in any one of them, nor will you get any nearer to accounting rationally for the universe by taking any number of them together; (8) The reason must therefore be sought elsewhere. (9) Things eternal may have no cause of existence, yet a reason for their existence must be conceived. . . . (10) Hence it is evident that even by supposing the universe to be eternal, the recourse to an ultimate reason for the universe beyond this universe . . . cannot be avoided. (11) The reasons for the universe are therefore concealed in some entity not in the universe, which is different from the chain or series of things, the aggregate of which constitutes the universe. (Leibniz, 1697)

As it stands, this argument has some well-known problems (it isn’t really even an argument, it’s just a proto-argument). But the argument can be rebuilt in ways that make it incredibly strong. Rebuilding it is mostly tedious logic. There’s no need to do that here. I’m going to assume that some rebuilt version of the argument is sound. What’s most interesting about this argument is what it says about existence.

Leibniz says the universe is the totality of physical things. It’s a spatially, temporally, and causally closed system. The entire universe is contingent – it might exist but it might not exist. Why does it exist? Why is there a universe at all? You can’t answer that question by appealing to anything that is internal to the universe. You can’t answer that question by appealing to any entity that participates in any spatial, temporal, or causal relations. This isn’t an inference back in time to a first cause. If there is a first cause, then it falls within the scope of the question. If there’s a first cause, it’s just another part of the universe – and thus it needs to be explained. And Leibniz is perfectly happy to say that the universe has always existed – no first cause at all. Leibniz says that “the reasons for the universe are concealed in some entity not in the universe”. Call this entity the ground.

Contrary to theists, the ground isn’t any concrete god. It isn’t the god of the Old Testament. It isn’t the creator of Genesis. It isn’t Yahweh or El-Elyon. And it isn’t any of the gods that have appeared in any of the mythologies of old paganisms. It isn’t Zeus or Thor. All those old gods are just concrete physical things – they participate in spatial, temporal, and causal relations. And since creation seems to entail causing an effect at some time, the ground isn’t a creator at all. For the theists, it just gets worse. Since the ground doesn’t participate in spatial, temporal, or causal relations, it can’t be a person. The ground doesn’t have any psychology. The ground doesn’t perceive the universe or intervene in it. The doesn’t have any thoughts, no beliefs, no desires. And the ground isn’t the god of deism. After all, that god is a first cause. The ground is deeper than all those gods.

What about the gods of the philosophers? Well, the ground exists. So it can’t be Plato’s form of the good; it isn’t the One of Plotinus. All those old philosophical gods are somehow beyond existence. And the ground isn’t Tillich’s ground of all being; on the contrary, it’s just the ground of the physicality of our universe. What about Spinoza’s god? I have to confess that I don’t entirely understand what that god is supposed to be – which makes me doubt that it’s Spinoza’s god. Anyway, the argument from evil entails that the ground certainly isn’t all-powerful and all-good and all-knowing. So the ground can’t be the big 3O god of classical theism. Leibniz’s argument doesn’t seem to support theism at all.

Onwards, then, to the atheists. Assuming that the ground isn’t one of those old-fashioned religious or theological entities, what would it be? Well, the ground isn’t any physical thing or structure or event. The ground isn’t the big bang or the cause of the big bang. It isn’t space-time or some quantum field or some black hole or any other exotic physical thing. It isn’t any physical thing at all. It’s important to understand the scope of this assertion.

It may very well be true that our observable cosmos, including everything that we can measure or empirically detect, is a simulation running on some alien super-computer. But if that’s true, then our observable cosmos isn’t the universe – it’s just the part of the universe that we can observe. The whole universe is a much bigger place. If our universe is running on some alien super-computer, then the Leibnizian question applies to that super-computer and to the aliens that made it. Why do those contingent physical things exist? The ground isn’t the super-computer or the alien civilization. The ground explains the aliens and their artifacts. Perhaps our universe contains many smaller cosmic domains (as in inflationary cosmology, or Smolin’s fecund universe hypothesis). If it does, then that entire multiverse is a contingent thing. Why is there a multiverse rather than no multiverse? The multiverse needs to be explained. If our universe is a big foam composed of lots of cosmic bubbles, then the ground explains that foam.

Given all this metaphysics, here’s the test question: Should atheists affirm or deny the existence of the ground?

Leibniz, G. W. (1697) On the Radical Origination of the Universe. In P. Schrecker & A. M. Schrecker (Trans.) (1988) Leibniz: Monadology and Other Philosophical Essays. New York: Macmillan, 84-86. The translation is slightly edited for consistency.

The theists say: (1) If the reasoning in the cosmological argument is correct, then God exists. (2) The reasoning in the cosmological argument is correct. (3) Therefore, the argument proves that God exists.

The atheists say: (1) If the reasoning in the cosmological argument is correct, then God exists. (2) The reasoning in the cosmological argument is not correct. (3) Therefore, the argument fails to prove that God exists.

Atheists and theists both agree on the major premise: if the reasoning in the cosmological argument is correct, then God exists. Why the agreement? Why grant that the cosmological argument is an argument for God? Sometimes atheists do point out that it might not be an argument for God – it might be an argument for something else. But I’ve never seen that possibility seriously explored. And it’s too bad.

On the one hand, atheists can attack theism by showing that the classical arguments for God are logically flawed. On the other hand, atheists can attack theism by showing that those very same arguments are arguments for things that are not God. Which attack is deeper?

I think it’s clear that the second line of attack is much deeper – it’s much, much more threatening. When your enemies attack your arguments, well, you can always deal with that. But when your own arguments turn against you, you’re in big trouble.

So I’m going to encourage atheists to look at the classical arguments to see what else they might be used for. Fix them up, make them all shiny, and use them to drive to some new place. For an illustration, stay tuned . . .

It’s far more natural to think that these arguments justify the Simulation Hypothesis.  The Simulation Hypothesis says that our universe is a software process running on some deeper computational substrate.  The physicist Ed Fredkin refers to this substrate as “the Engine”.  Using his term, our universe is to the Engine as software is to hardware.  Our universe is analogous to a video game running on the Engine.   The old Cosmological and Design arguments are very nice arguments for the Engine.

The great advantage of the Simulation Hypothesis is that it isn’t mysterious.  Computer science involves precise mathematical definitions of computing machines (e.g. finite state machines, Turing machines, transfinite machines).  Software engineering is a rational discipline.  We know how to construct computer simulations of physical universes (including our universe).  We know how to develop video games.   There is a large and fascinating literature on computational world-design.

Many people have written about the Simulation Hypothesis, including David Chalmers and Nick Bostrom.  Bostrom maintains a website on the hypothesis.  Many of these writers suggest that simulation means that we are being simulated by some superior society.  The idea is that there is some Alien Society.  The engineers in that Alien Society built computers.  They programmed them to run an artificial universe.  This version of the Simulation Hypothesis is familiar from old movies like The Matrix and The Thirteenth Floor.  Dawkins mentions it: “Science fiction authors . . . have even suggested (and I cannot think how to disprove it) that we live in a computer simulation, set up by some vastly superior civilization.” (The God Delusion, p. 98)  And Harris mentions it too: “If intelligently designed, our universe could be running as a simulation on an alien supercomputer.” (Letter to a Christian Nation, p. 73)

But the Alien Society interpretation isn’t the only way to understand the Simulation Hypothesis.  The Engine need not be an artifact.  It might be entirely natural – a product of some sort of evolution.  After all, human beings are very powerful computing machines.  And we are natural products of natural evolution.  Alternatively, if you’re a Platonist, or Neoplatonist, you might want to think of the Engine as a purely mathematical machine.  It would be a machine existing eternally in the non-physical system of Platonic Forms.   On this interpretation, some Platonic objects emanate universes.  They are the relatively abstract grounds of physicality.  They exist eternally and necessarily.

On any interpretation, the Simulation Hypothesis involves the idea that the Engine is running some world-actualizing algorithm.  Perhaps it is just doing a brute-force iteration through some set of possible universes.  Perhaps it is using some optimization algorithm to maximize some feature of universes – it is conducting a rational search for maximally regular universes, or universes that contain the most internal computation.  Perhaps it is using some genetic algorithm to evolve universes – it might be evolving universes for maximum logical depth or for some other type of internal complexity.

Obviously, the Engine has some degree of computational power – it has the power to run some universe of some complexity.  Its power can be measured using standard technical scales (e.g. the cardinality of its state set; the cardinality of its halting conditions – does it halt after finitely many states, or can it run out to transfinite limit cardinals).

Any Engine running something as complex and valuable as our universe has to have some degree of benevolence.  This benevolence need not be anything like human morality.   It is far more likely that it is simply an abstract orientation towards the maximization of certain computational values like logical depth or intensity of information processing.  The Engine would have its own axiological imperatives, and these would all be defined in computational terms.  Perhaps these would be game-theoretic – think of the evolution of cooperation in the prisoner’s dilemma.  Life on earth might have very high computational value because living things encode programs for their own construction.  And evolution shows that they are recursively self-improving.  Now, recursion is computationally deep.  Perhaps the Engine aims to maximize recursive depth or recursive intensity of some type.

What about humans?  Are we valuable because we have consciousness?  Maybe.  But there might be even stranger features of human being that make us computationally valuable.  Our immune systems are capable of running genetic algorithms – they run their own simulations of Darwinian evolution to design antibodies.  This means that, in our immune systems, an evolutionary algorithm is stacked on top of an evolutionary algorithm.  Perhaps this sort of stacking is a primary computational value: more valuable systems support higher stacks of virtual machines, simulations running inside simulations.

Perhaps the Engine is intelligent.  Here again, its intelligence has a precise computational analysis.  And the intelligence of the Engine need not be anything like human intelligence.  It might not have a mind.  It wouldn’t have to be conscious or have emotions or desires or beliefs or understanding.  It need not be psychological in any recognizable sense.  It might be a vast statistical machine (like Google Translate).

So the Engine has some degrees of power, benevolence, and intelligence.  Heaven forbid, it looks like God!  Well, no, it doesn’t.  Its power and benevolence and intelligence are all defined in purely naturalistic ways.  And those qualities are not maximal in the way they are for God.  On the contrary, those qualities are defined in precisely technical ways exactly up to specific cardinal numbers.  There is nothing at all mysterious about the Engine.  It is an entirely lawful entity defined using mathematics and computer science.

But where did the Engine come from?  You know the drill: if the Engine created our universe, then what created the Engine?  Here’s a nice quote from Dawkins:

Science fiction authors . . . have even suggested (and I cannot think how to disprove it) that we live in a computer simulation, set up by some vastly superior civilization.  But the simulators themselves would have to come from somewhere.  The laws of probability forbid all notions of their spontaneously appearing without simpler antecedents.  They probably owe their existence to a (perhaps unfamiliar) version of Darwinian evolution: some sort of cumulatively ratcheting ‘crane’.  (The God Delusion, pp. 98-99)

Dawkins is using the Alien Society interpretation of the Simulation Hypothesis.  And that’s fine.  But on the naturalistic interpretation, the Engine is itself a product of a deeper type of evolution.   The Engine is the product of an evolutionary algorithm.  It is the product of some kind of recursively self-improving algorithm.  Perhaps that algorithm is just the deepest algorithm there is.  It is necessary and eternal.  Just as the fact that 1+1=2 does not require an explanation, so the deepest evolutionary algorithm does not require any explanation.  It’s part of the harmony of things.  Here’s a lovely quote from Leibniz:

For it is necessary to refer everything to some reason, and we cannot stop until we have arrived at a first cause – or it must be admitted that something can exist without a sufficient reason for its existence, and this admission destroys the demonstration of the existence of God.  Yet what is the ultimate reason for the divine will?  The divine intellect.  For God wills the things which he understands to be best and most harmonious and selects them, as it were, from an infinite number of possibilities.  Yet what provides the reason for the divine intellect?  The harmony of things.  What the reason for the harmony of things?  Nothing.  For example, no reason can be given for the ratio of 2 to 4 being the same as that of 4 to 8, not even in the divine will.  (Leibniz, in Rescher’s edition of The Monadology, p. 148)

Did Leibniz just admit to being atheist?  Well, that’s a topic for another day.  The point here is that mathematical necessity itself might be the ultimate sufficient reason for the existence of each and every actual physical universe.  Some purely mathematical things, namely, abstract computers, actualize selected possible universes.  The naturalistic interpretation of the Simulation Hypothesis is ultimately mathematical.   It therefore shades off into the Platonic interpretation – they aren’t really different.

On the Platonic interpretation of the Simulation Hypothesis, the Engine is like a number.  It exists in a sequence of engines just like the sequence of numbers.  There is an initial engine.  This is the zero-engine.  For every engine, there is a greater engine.  The sequence of engines is a recursively self-improving sequence just as the sequence of numbers is a recursively self-increasing sequence.  The rules for the existence of engines are just like the rules for the existence of numbers or like the axioms of pure set theory.  Engines, like numbers or sets, are eternal and necessary mathematical objects.

However you decide to work it out, the Simulation Hypothesis is an interesting non-theistic alternative to theism.  It has positive metaphysical and ethical content.  It may even provide an entirely naturalistic theory of life after death.

Guest Contributor Eric Steinhart is an associate professor of philosophy at William Paterson University and the author of More Precisely: The Math You Need To Do Philosophy, On Nietzsche (Wadsworth Philosophers Series), and The Logic of Metaphor – Analogous Parts of Possible Worlds (Synthese Library, Volume 299). Professor Steinhart has explained many of his views on metaphysics, philosophy of religion, and Richard Dawkins in an audio interview with The Pale Blue Dot. Abstracts to his papers on the philosophy of mathematics, philosophy of religion, metaphysics, the metaphysics of persons, Nietzsche, and analogy and metaphor can all be found here (in some cases with links to the papers themselves).  All of his guest posts at Camels With Hammers are archived here.