Theism, naturalism and simplicity
December 20, 2012 — 8:19

Author: Alexander Pruss  Category: Concept of God Existence of God  Tags: , , ,   Comments: 39

When one’s book in sexual ethics is coming out (shameless self-promotion), one’s thoughts naturally turn to the philosophy of science. 🙂 A standard line of thought is that naturalism is a simpler theory than theism in that it only posits one kind of entity, the natural world, while theism posits that and God.

A standard theistic response is to concede the point but say that theism wins out through greater explanatory power. Trent and I have, however, been exploring a different line of thought: One measures the simplicity of a theory (with “simplicity” understood in such a way that it is an intellectual merit of a theory that it be simple) primarily by looking at the simplicity of the theory’s explanatorily fundamental posits (this has some structural resemblance to Huemer’s work) rather than at claims explained by the theory.

For instance, suppose that according to our best physics certain laboratory conditions not occurrent in nature produce a Zeta particle. Alien scientists, who are the only ones ever to have the technology for this, are facing a great natural disaster they cannot avert that will destroy their civilization. As one last hurrah for science, they plan to produce a Zeta before the disaster. Unfortunately, at the last minute, they find that an extremely expensive part, which there is no time to repair, has only probability 1/2 of functioning.

Consider the theories: (S) They will succeed in producing a Zeta due to the part functioning and (F) They will fail in producing a Zeta due to the part malfunctioning. Theory S posits the instantiation of a new kind of particle that F does not. If explained phenomena also count towards the complexity of a theory, S is more complex. But that just seems wrong: S and F are on par simplicity-wise. Besides, if S were more complex than F, then if all other intellectual merits are equal–which they sure seem to be–then we should take S to be more likely than F. But that would violate what seems an unproblematic instance of the Principal Principle–F and S should have the same probability.

Or consider the Five Minute Hypothesis (5M) that the universe came into existence five minutes ago, fully formed, with fossils and light traveling apparently from distant but never actual stars, versus the Big Bang Hypothesis (BB). One might say: 5M wins out on simplicity grounds–there are many kinds of things it doesn’t require us to believe in: world wars, Sumerians, dinosaurs, big bangs, etc.–but loses out on explanatory grounds. But I think that’s conceding too much to 5M. BB is not only explanatorily superior to 5M, but it is at least as good in terms of simplicity–the explanatorily fundamental state of the universe posited by BB appears simpler than that posited by 5M: it doesn’t include oak trees, dogs, people, continental plates, planets, stars, females, males (see, my initial self-promotion is on topic!) or galaxies.

But now go back to the philosophy of religion angle. On a filled out naturalism, the fundamental explanatory state is some brute physical facts, say the Big Bang. On a filled out theism, the fundamental explanatory entity is a perfect being, which then out of the goodness of its will causes the Big Bang. If perfection is a non-gerrymandered concept, then the theistic fundamental posit appears simpler, and so theism appears the simpler theory.

However, there is a wrinkle. I suspect that when counting the complexity of a theory one doesn’t always get to completely discount the complexity of the explanatorily non-fundamental parts of the theory. Explanation comes in degrees. Deterministic explanations are more explanatory of the outcome than low-probability stochastic explanations. The Big Bang state is not determined by God’s nature. So its complexity contributes somewhat to the complexity of the filled out theistic story. But given that the theistic fundamental posit is much simpler than the naturalistic fundamental posit, this may only make the simplicity contest a toss-up. Or at least it’s now a much more involved question which story is more complex.

Comments:
  • Ralph

    This just smacks of desperation.

    December 21, 2012 — 18:05
  • jordan.nwc

    Ralph,
    Do you have an argument for that? Or is this type of irrelevant attack simply based on a feeling?
    If the latter, then I guess… thanks for sharing?

    December 22, 2012 — 5:14
  • T

    Of course the relevant simplicity is the simplicity of the fundamental posits. Otherwise, we should reject the whole of science, physical theory, evolutionary theory, etc–because they introduce new kinds of entities. Wasn’t the point made by Swinburne long ago? I’m not sure what I’m missing so that all these other examples and arguments are necessary?

    December 23, 2012 — 10:22
  • Justin

    Why is a perfect being a simpler fundamental explanatory state than the big bang?

    December 23, 2012 — 12:05
  • The Big Bang state has many free parameters–and there are also the laws which include a number of fundamental contingent claims. Many complex claims are needed to state this.
    An Anselmian perfect being requires a single and simple posit: There is a maximally perfect being.
    Of course, the Anselmian story requires background necessary truths of axiology from which it follows what qualities are or are not perfections. But this is paralleled by the fact that the Big Bang story requires background necessary truths of mathematics.
    A Swinburnean deity also requires a simple posit: There is a being that is omniscient and omnipotent.

    December 23, 2012 — 12:10
  • Justin

    Alex: I see. Does the fact that the big bang is a physical event, whereas the perfect being (I assume) is nonphysical, count towards the simplicity of the big bang as a fundamental explanatory state?

    December 23, 2012 — 12:28
  • My intuition goes the other way, actually. 🙂

    December 23, 2012 — 12:35
  • Justin

    Alex: That’s interesting. Why so? I would have thought that a theory that explains physical stuff just in terms of more physical stuff is more simple – all things being equal – than one that appeals to some nonphysical stuff. it’s hard to evaluate judgments about what differences in kind matter in evaluating theoretical simplicity, but I would have thought that the difference between physical things and nonphysical things is a difference in kind that matters. Is it because what needs to be explained – on your view – isn’t just physical stuff?

    December 23, 2012 — 12:46
  • Justin:
    Maybe the way to put the worry is this: Maybe a nonphysical thing, as such, isn’t any more complex than a physical thing, but a nonphysical thing that can cause physical effects is, as such, more complex than a physical thing that can cause physical effects.
    That’s an interesting suggestion.
    But this supposes some kind of idea that causation goes “more simply” between things of the same sort.
    And I am not sure I see this as a matter of simplicity. Why should a physical thing’s having the causal power to move a rock be any simpler than a nonphysical thing’s having the exact same causal power?
    Maybe, though, there is a greater “uniformity” in the physical thing that can move a rock than in the nonphysical one. I am not sure that’s a matter of simplicity, but maybe of something like naturalness.
    Also, even between physical things, causation can bridge large gaps in kind. While motion can cause motion, electric charge can also cause motion and motion can cause electric charge. Electric charge and motion are not very much alike. Causation even bridges the ontological gap between substances and accidents: I, a substance, can cause the movement of a rock, which is an accident of the rock, though I admit that that’s controversial.
    Notice, too, that if one accepts a principle on which uniformity between cause and effect is simpler, then some design explanations in biology will, at least thus far, be simpler than corresponding evolutionary ones: it will be simpler to suppose that a living designer causes life or that an intelligent designer causes intelligence than to suppose that nonliving molecules cause life or mutations in nonintelligent beings cause intelligence. So it’s a somewhat dangerous principle for the naturalist.

    December 25, 2012 — 14:49
  • Dianelos Georgoudis

    “A standard line of thought is that naturalism is a simpler theory than theism in that it only posits one kind of entity, the natural world, while theism posits that and God.”
    Theism, on subjective idealism, also only posits one kind of entity.
    And naturalism’s one kind of entity is deeply problematic, given its mind-body problem, as well as other hard problems, such as the deeply mathematical nature of the physical world. Or how quantum phenomena resist naturalization. If naturalism is true then reality is grounded on an unappealingly incongruent lot of fundamental posits. Does not look like a simple theory to me.
    In comparison, subjective idealism’s theism does not suffer from any metaphysical conceptual problems I can see. And is both very simple and very easy to understand, for it posits a metaphysics where the one kind of entity is intimately known by us.

    December 25, 2012 — 15:23
  • Eric Steinhart sent me this comment which for some reason he was unable to post:
    You write: “On a filled out naturalism, the fundamental explanatory state is some brute physical facts, say the Big Bang.” Why would that be the case? I doubt many naturalists with any physical sophistication would stop with the Big Bang (given inflationary cosmology, etc. etc.). But suppose there’s some initial basic physical state (as even many of the exotic cosmologies seem to require – see recent work by Mithani & Vilenkin). Why would this state be “brute”? Plenty of philosophers have argued it would have an explanation. It might be an axiarchic explanation (as Leslie & others argue). Or it might be some necessary principle that selects the forms that become concretely realized (as it seems Parfit argues). It might even be a version of the PSR that selects abstract for concrete instantiation. I see no reason why naturalism requires bruteness.
    You also write: “On a filled out theism, the fundamental explanatory entity is a perfect being, which then out of the goodness of its will causes the Big Bang.” What kind of explation is that? A “personal” explanation (rather than a “mechanical” explanation). Well then, say more about it. Are there psycho-physical laws connecting this person to the initial physical state? If no laws, then no explanation at all. Or perhaps there are axioms of some sort that aren’t like scientific laws. What are they? Might God play a role in the axioms of set theory? Or the axioms of quantum mechanics? Or the axioms of some multiverse theory (perhaps the string-theory landscape)? Or perhaps the perfect being runs an optimization algorithm (as Leibniz suggested, in Theodicy sec. 225), which selects an abstract possibility for actualization. Such algorithms can be formulated in fairly standard ways (see the work of the computer scientist Jurgen Schmidhuber). We do have theories of minds and how they act, so I suppose you’ll be offering us such a theory about how God acts. If not, then you’ve given no explanation at all.
    This is one of my big complaints against theism: it offers no theories with any explanatory power of any kind. It merely asserts: God did it. Absent some story that looks like a theory (in logic, in math, in computer science, in psychology, in physics, etc.), absent some story that involves statements that look like laws or axioms or algorithms, theism is vacuous. It’s magic. I know how to evaluate the complexities of theories (count quantifiers, look at the ranks of the models in ZFC, etc.). But theism doesn’t offer anything to evaluate for its complexity. Theism doesn’t even have a horse in this race.

    December 25, 2012 — 15:42
  • Eric:
    These are exactly the right kinds of questions to ask about my suggestions.
    1. I think causation is sufficient for explanation. I find plausible the Humean principle if A causes B, then that A happened explains that B happened. But I don’t think causation requires laws. So, explanation doesn’t require laws.
    2. Axiarchic explanation doesn’t seem to fit well with naturalism given the plausibility that the best world would contain a very intelligent being not limited by physicality. Principles that can exist independently of any entities and can explain positive outcomes without any causation seem dubious, but there are some philosophers (e.g., Rescher) who accept them. But most of all, even if such principle-based approaches are in the letter of naturalism, they aren’t in the spirit of it. One of the fundamental commitments of naturalists is that no physical state has a nonphysical cause. Allowing a physical state to have a nonphysical, albeit noncausal, explanation isn’t in the spirit of naturalism. So just stipulate that the kind of naturalism I am talking about isn’t of this sort. I am interested in the kind of naturalist who thinks that physical states have no irreducibly nonphysical explanations.
    3. You are right, however, to press for more details on the theism. I think one move I can make is to say: “There is a perfect being. The rest is a matter of the necessary truths of axiology.” And then I just need to hope that the necessary truths of axiology imply that a perfect being is a being of the sort theists posit. And unless we’re going to be nihilists, we all, naturalist and theist alike, need there to be necessary truths of axiology.
    4. A different move is to spell out the theory a little more. A perfect being is omnipotent (say, in the Pearce-Pruss sense of having a perfectly free and perfectly efficacious will), omniscient (say, for all p, the being believes p iff p is true, and the being believes only what it knows for sure) and chooses what to create on the basis of what are on balance good reasons, in such wise that if there is more than one action for which the being has on balance good reason, the being is capable of selecting among these. We might even add, though we currently have no way of making this precise (if only because cardinality considerations seem to give us no good way to make sense of probabilities in such a context), that the likelihood of the being’s performing an action covaries with the strength of the reasons for that action.

    December 25, 2012 — 16:01
  • Dianelos Georgoudis

    Eric: I wonder, if all of reality is based on an initial physical state, which itself is based on axiarchic explanation, in what is that explanation metaphysically grounded? I suppose a naturalist could suggest that the ground of all being is some axiarchic principle, and that given that we ourselves are products of the evolution of that axiarchically formed initial physical state we have cognitive access to that principle, and use it to track truths about metaphysics.
    If so, that view is basically theism’s view but takes out the personal ground and leaves only the axiarchic properties of God’s will hanging there as the foundational principle of reality. And a rather complicated principle this is, since it entails, say, the goodness of mathematics but also the goodness of love. Further, in my view, it presents us with a stressful, if not incoherent, worldview. Take for example what appears to be a deep fact of the human condition, namely one’s experience of the metaphysically ultimate as being of a personal nature (or, more precisely, as being no less than personal). What good is there in that?
    Anyway, suppose that theism is true. The non-theist can always take the theistic worldview, excise God from its center, and claim that what remains is the true view of reality. That will work, only, going back to Alex’s OP, that view will by necessarily be less simple than theism’s.

    December 26, 2012 — 6:27
  • Dianelos Georgoudis

    Eric: I was struck by this bit you wrote: “If no laws, then no explanation at all”. On the one hand I agree, indeed this proposition strikes me as an analytic one, in the sense that any explanation entails some law-like knowledge. On the other hand though it can mislead people into thinking that presenting the laws is necessary for the validity of the explanation.
    What is an explanation? I think we all would agree that A explains B when given A one can see that B is likely. An example will be explanations of the physical sciences, which relate A and B through some kind of deterministic or probabilistic mathematical formula. Such explanations are commonly called “mechanical”. But another very common example in our everyday life are “personal” explanations, which relate one’s knowledge of a person’s character with one’s knowledge about how this character is expressed in a person’s will. For example, given A=“S loves animals” we explain B=“S moved to help the injured bird”. Can such “psychological insights” be formalized as “probabilistic laws”? Of course they can; any proposition of the form “If A then likely B” can. Is it useful to do so? No, for the psychological insight into A entails that law. Or, perhaps better, the psychological insight into A is richer in knowledge than any such laws.
    Perhaps a way to demonstrate the difference is by comparing personal explanations common in social life with the mechanical explanations common in the physical sciences. The explanatory structure of general relativity (or of quantum physics) is not present in one’s experiential (irreducible) knowledge of falling apples (or of burning logs). A great deal of work must be done to go from the immediate knowledge of such physical phenomena to the knowledge of the deep mathematical patterns present in them. In contrast, one’s (irreducible) insight in what loving animals is, entails all one needs to explain the will of a person who loves animals. My point is that the difference between so-called “mechanical” and “personal” explanations does not reside in the nature of the explanation itself, but in the nature of the different kinds of knowledge about the A’s and B’s. Crudely said, we speak of mechanical explanations when the A’s and B’s refer to one’s perceptions of the external world (i.e. of physical phenomena), whereas we speak of personal explanations when the A’s and B’s refer to our own being.
    Which brings me following bit you wrote: “This is one of my big complaints against theism: it offers no theories with any explanatory power of any kind. It merely asserts: God did it.” Theism does not simply assert that “God did it”. Rather it asserts that “given God’s perfection God would likely will X”. And such assertions are based on our insight into the character a perfect person would have – the same way that a scientific assertion is based on our experience of physical phenomena. At this juncture some naturalists claim that naturalism is intrinsically more testable, and thus more amenable to general agreement. But this claim is fallacious and only demonstrates naturalists’ tendency of conflating the physical sciences with their metaphysics. The physical sciences are indeed publicly testable in their claims about mathematical patterns present in physical phenomena. But naturalists disagree among themselves about how reality is at least as strongly as theists do; indeed the more the physical sciences advance the more they seem to disagree among themselves. And whereas, it seems to me, contemplation of one’s own being (or “enorasis” as the ancients called it) does present an efficacious albeit difficult and slow way towards general agreement, it does not seem to be the case that naturalists have access to any such metaphysical truth-tracking cognitive process.

    December 26, 2012 — 6:34
  • Dianelos Georgoudis

    Eric: A final point. You ask “Might God play a role in the axioms of set theory?“ Much more than that – God willed the axioms of set theory. After all, the very possibility of counting is created by God. I am afraid people easily forget that on theism God is the metaphysically ultimate, and thus grounds all that exists and all therefore about which knowledge is possible.
    At this juncture some might ask “Might then God have willed that 2+2=3?”, and the answer is “Obviously no. Given our knowledge of personal perfection, and our knowledge about the absurdity of 2+2=3, we can easily see that God would never have willed such an absurd thing.” On the other hand, for all we know, God might have willed a creation not based on string theory (assuming that the physical world is based on it). It is probable that in many contexts God has various good options, and chooses one of them by creative fiat. And in many other cases God might choose not to choose, but to let chance play a role. This, incidentally, is often what artists do in their creative enterprise. To be surprised is a good thing, there is delight and fecundity in it. Creation, even in the human experience, is not about following well given laws, but about creating truly new things, even new laws. So, it’s not like on theism everything must have its individual explanation.

    December 26, 2012 — 6:46
  • @Diane – You wrote: “a naturalist could suggest that the ground of all being is some axiarchic principle . . . that view is basically theism’s view but takes out the personal ground and leaves only the axiarchic properties of God’s will hanging there as the foundational principle of reality.”
    You got it. And this gets right to the atheistic/naturalistic complaint: why do theists insist on personalizing everything? Of course, personal explanations are more intuitively obvious to persons; but the fact that they are easier does not mean that they are true. If memory serves, Aristotle once thought that gravity was a kind of love and that falling bodies accelerate as they get closer to the earth because of their joy at arriving at their home. But that’s false.
    The charge, of course, is just that theism is anthropic projection (ah, Feuerbach…). As such, it’s just idolatry – the false projection of ourselves into the divine.
    And if somebody says “P is true because God wills P”, I’d say it’s simpler just to say that P is true.

    December 26, 2012 — 14:19
  • @Alex – You write: “Allowing a physical state to have a nonphysical, albeit noncausal, explanation isn’t in the spirit of naturalism. So just stipulate that the kind of naturalism I am talking about isn’t of this sort. I am interested in the kind of naturalist who thinks that physical states have no irreducibly nonphysical explanations.” Sure, you can stipulate that, but now the naturalism you’re talking about is a straw man – especially since the naturalists who are concerned with exactly these issues (e.g. Leslie, Rescher, Parfit) explicitly do affirm that physical states can have non-physical explanations. So I’m sticking with a naturalism that says that physical states have non-physical (and thus non-causal) explanations.
    I like very much your points about the “necessary truths of axiology”. I agree that theists and naturalists alike ought to affirm that there are such truths. And of course Leslie’s point is that at least some of these necessary truths are creatively effective, all by themselves. (ON a historical note, Leibniz’s doctrine of the striving possibles also suggests that necessary axiological truths are creatively effective). These truths are abstract principles: necessarily, for every abstract form F, if F has axiological property P, then there exists some concrete x such that F(x). Of course, you don’t have to agree with the Lesilian analysis of P (I don’t). The key point is that if there are such necessary truths of axiology, there’s no need to posit any God. The truths do all the work, all by themselves.
    Of course, I’d love to see the theistic claims spelled out in more detail – I suspect it would lead to a formalization of the notion of a perfect being. But I also suspect that such a formalization would be disastrous for theism. Give me any formalization of your perfect being, and I’ll give you a more perfect being (give me any set of truths…).

    December 26, 2012 — 14:35
  • The kinds of truths of axiology that I was thinking of are what one might call “conceptual” (I don’t really like the term). It’s truths like: “Having P has value V to degree D” or “Having P has value V to a lower degree than having Q has value V.”
    But it could also be that there are truths of axiology that such as “If P has some disvalue, then anything entailing P has some disvalue” and “Necessary existence has no disvalue”, from which it follows that there is a necessary being (cf. my Goedelian ontological arguments). But I wasn’t thinking of stuff like that–I was talking of more uncontroversial truths of axiology, like that knowing whether p has epistemic value to a greater degree than not knowing whether p.
    Anyway, depending on what primitives one is allowed, it’s easy to formalize a perfect being.
    With the one primitive Perfect(x):
    1. x is a perfect being iff Perfect(x). 🙂
    Somewhat less trivially, this second-order formalization using the second order primitive Perfection(P):
    2. x is a perfect being iff (P)(Perfection(P) → P(x)).
    Or using a partial-order axiology with the primitive Better(x,w1,y,w2) (“how x is at w1 is better than how y is at w2”) and quantification over worlds and over individuals not restricted to worlds:
    3. x is a perfect being at w iff (w’)(y) ~Better(x,w,y,w’).

    December 26, 2012 — 15:00
  • Dianelos Georgoudis

    Eric:
    “The charge, of course, is just that theism is anthropic projection”
    Why is that a charge? If theism is true then finding out how God is by using our own sense of perfection, is an effective epistemic principle. As is personally relating to that in whose image we are made. You can’t charge theists for using epistemic principles which they shouldn’t use if naturalism is true.
    “Of course, personal explanations are more intuitively obvious to persons; but the fact that they are easier does not mean that they are true.”
    Right; on the other hand, all other things being equal, what’s intuitively obvious is more probably true. And “easier” is an understatement. The theistic view of a personal creator is a very natural one, for we ourselves are persons, are moved by our values, and through our will bring about good states of affairs every day of our lives. In contrast, to say that an abstract axiarchic principle has creative powers comes close to being semantically incoherent. (One would think that, if anything, the ontological nature of the metaphysically ultimate must be in some sense “more concrete” than the reality it creates.)
    But there are other reasons to object to a metaphysics which grounds reality on some axiarchic principle. Significantly it appears to be unstable. For, as others have noticed, if such a creative principle were the ground of all reality, than that reality would immediately reduce to theism, since God is by definition the greatest possible being. Now one way out for the axiarchic principalist who does not wish to embrace theism is to argue that the greatest possible being is not personal – which, given our own perception of perfection, strikes me as a very difficult argument to make. Another way out is to argue that the creative powers of the axiarchic principle are limited in the right way, which is ad-hoc.
    Another problem is this. One can barely conceptualize a reality in which an axiarchic principle brings about actual existence, indeed an orderly existence, and one capable of evolving complex structures – since all of that is good. But to bring about life, and intelligence, and consciousness, and cognitive powers to do truth-tracking metaphysics – appears to be impossible without some kind of planning, i.e. without some kind of mind. Now (thinking on my feet) perhaps there is a way around this problem. Perhaps we should conceptualize the axiarchic principle as continuously but mindlessly pulling the actual state of affairs towards a more valuable state. Thus it starts by pulling existence out of non existence, then by making it orderly, then by making it capable of producing complex systems, once such state is reached nudging it towards life, then towards intelligence, then, somehow, towards consciousness, and finally towards its own self-revelation. But this view does not comport with what we know about our universe which follows the same fixed laws since its primitive beginning. A solution is to assume many worlds. Perhaps the axiarchic principle brings about many worlds and then mindlessly nudges forward and multiplies into existence only those which are more valuable. Our own world then is the outcome of one of these evolutionary strands which go back far before the visible universe. Indeed our world may be a work in progress. – Which is a fine (and fun) speculative metaphysics as it goes, but rather not simple.
    I am not well acquainted with Leslie, but it is clear that he doesn’t go the mindless path I described above. Rather he embraces a form of pantheism, which he describes as “the belief that nothing exists except divine thinking”. So he is no naturalist. Rather, it seems to me, Leslie reacted to naturalism’s many conceptual problems by moving more or less on a straight line towards theism (indeed towards a view which appears to be close to subjective idealism’s theism), without in the end embracing it. I understand he argues that his ultimately non-theistic view comports better with the world we know, perhaps by making more sense of the presence of evil. On the other hand it is hard for me to make sense of the idea of “divine thinking” without a “divine thinker”. Isn’t Leslie’s divine thinking conscious of its thoughts? Doesn’t divine thinking entail thinking about itself – doesn’t it know it exists? If it does then doesn’t it improve itself? Say, by becoming conscious? But then something which is conscious, thinks, has values, acts on them, contemplates and interacts with its work, knows about itself – isn’t that a person?

    December 27, 2012 — 2:52
  • Alex –
    Let’s focus on your “partial-order axiology”. I think we can just use counterpart theory to avoid the awkward 4-place better-than relation. So now we just have a partial order relation x

    December 27, 2012 — 12:27
  • Eric Steinhart

    Alex –
    Let’s focus on your “partial-order axiology”. I think we can just use counterpart theory to avoid the awkward 4-place better-than relation. So now we just have a partial order relation “x is better than y” (or perhaps “x is axiologically superior to y”). Your job is to prove that this partial order has a maximum. By parallel with set theory, I’d argue that any such maximum is impossible. (Permit me to wax poetic: if anything is unsurpassable, it is merely unsurpassability itself.)
    – Eric

    December 27, 2012 — 16:31
  • Eric Steinhart

    Diane:
    I don’t see why axiarchism requires a perfect being of any type. That’s the whole point. All that it claims is that there are metaphysical principles like “Necessarily, for every abstract form F, if F satisfies axiological property P, then there exists some concrete x such that F(x)”. That doesn’t require a perfect being at all.
    You write: “Perhaps the axiarchic principle brings about many worlds and then mindlessly nudges forward and multiplies into existence only those which are more valuable. Our own world then is the outcome of one of these evolutionary strands which go back far before the visible universe. Indeed our world may be a work in progress.”
    I endorse pretty much exactly this position (my website lists several articles either now out or forthcoming on it, plus a few talks I’ve given). You object that it isn’t simple; yet the principles behind it are very simple indeed. (The axioms of set theory are simple, despite their enormously complex consequences.)
    It’s necessary to distinguish Leslie’s axiarchism from his idealism (which he, strangely, calls “pantheism”). I don’t endorse his idealism at all. Whenever he does discuss his “divine minds” in any detail, he compares them to quantum computers.
    My own view is that abstract axiarchic principles bring into concrete existence an infinite hierarchy of increasingly complex computing machines. (Computers seem nicely situated between the abstract and concrete.) As they grow ever more complex, they run ever more complex universes, eventually bringing our universe into being. Concrete reality is an enormous distributed optimization algorithm.
    I have no objection to calling these computers “gods”. So I’ve often referred to my own view as a kind of polytheism. (“Ordinal polytheism”, since there are as many ranks of gods as there are ordinal numbers.)
    You quite correctly ask the question whether Leslie’s divine minds are persons; the same question can be asked about the gods of ordinal polytheism. I suspect that on your minimal definition of “person”, the computer network at Google may already be a person. And I see no reason to think that the development of extremely high intelligence or consciousness entails anything at even remotely like the psychologies of earthly animals. Look, I know what happens: say a word like “person”, and the result is an explosion of unjustified entailments (like “oh, so the gods have emotions…”). I don’t want the explosion of unjustified entailments.
    The gods I posit fall entirely within the scopes of logic, mathematics, computer science, and cognitive science. They harbor no mysteries. Falling within the scopes of these sciences, they are natural objects. They can be scientifically studied in the same way the string theory landscape or transfinite machines can be scientifically studied. So I sometimes refer to my position as “theological naturalism”.
    – Eric

    December 27, 2012 — 16:33
  • Eric:
    I don’t see it as my job to prove that there is a maximum. There is no need to prove an explanatory theory coherent (and no possibility if the theory contains a sufficiently large fragment of arithmetic or string manipulation). We can infer the coherence of an explanatory theory from its truth, and its truth from its being the best explanation.

    December 28, 2012 — 8:10
  • By the way, Eric, on the basic point of my post, it looks like you agree that it is the explanatorily fundamental parts that we look at in trying to figure out the complexity of a theory.
    I disagree, by the way, that the axioms of set theory are simple. The axioms of naive set theory are simple, but alas incoherent. The axioms of non-naive set theories like ZF, TST or NF are significantly gerrymandered (in rather different ways) to avoid inconsistency.

    December 28, 2012 — 8:20
  • Eric Steinhart

    Alex –
    You write: “We can infer the coherence of an explanatory theory from its truth, and its truth from its being the best explanation.”
    Well, you don’t have a true theory, or best explanatory theory, or any theory at all, if your theory is logically impossible. If your theory posits a maximal object (a perfect being) that is like the set of all sets or like the biggest number, then you’re positing something that doesn’t exist. (One of my main objections to theism is an objection to Anselmianism, which I think is just incoherent. I have the same objection to some of the atheists, like Quentin Smith, or some pantheists, who have tried to make “being” or “reality” or “the world” into a maximal Anselmian type object.)
    You raise a very interesting point about simplicity. I come to the concept via my technical background (in logic, math, and computer science). So I’m familiar with all sorts of measures of complexity (and thereby simplicity). Thus for theories, count non-logical symbols, count quantifiers in prenex normal form, count alternating blocks of quantifier types in prenex normal form. Or measure information-theoretic quantities like logical depth, sophistication, etc. Since there’s a very good literature on the complexity of theories, I’ll stick with that. It would thus be nice to see if you might spell out your proposal in those terms.
    (And I recall that Swinburne, long long ago, tried to work out an information-theoretic way to quantify theory-complexity, when he was initially talking about the simplicity of theism. But I don’t have that text at hand.)
    I don’t see that ZFC (or its extension to classes a la VGB) are gerrymandered at all. In which axioms lies the gerrymandering? TST and NF are just crazy. But that’s off topic.
    Perhaps the main complaint is this: theists want to talk about simplicity, yet they don’t use the well-established formal literature on complexity; theists want to talk about a maximally perfect being, but they refuse to use mathematics; they want to talk about a divine mind, but they refuse to apply cognitive science to it; and on and on.
    So here’s the argument: If God exists, then God is consistently definable; if God is consistently definable, then God falls under the formal sciences (logic, mathematics, computer science, and their abstract applications in cognitive science and abstract biology); thus God falls under all those sciences; but any applications of those sciences to God yield contradictions; therefore, God does not exist.
    – Eric

    December 28, 2012 — 12:35
  • SK

    A couple quick notes:
    I’m not sure why one would apply cognitive science to the divine mind if the divine mind is purportedly not like those in the types of minds that normal cognitive science deals with (e.g. human or other animal minds). That is, unless we’re, for some reason, working within a framework where God’s mind is at least partially physical, and similar in architecture to human minds (and, I guess, can be accessed by humans).
    I’m also not sure what the purpose of these mathematical analogies is. If Anselm considered God to be a perfect being, why assume this perfection can extend to all possible infinities? If perfection works along such a numerical scale, isn’t it possible that there is some infinity (or finite value) at which maximal perfection is attained?

    December 29, 2012 — 0:17
  • Eric:
    “Well, you don’t have a true theory, or best explanatory theory, or any theory at all, if your theory is logically impossible.”
    Sure. But there is no need to prove logical possibility. Observe how the sciences proceed. Darwin doesn’t need to prove that it’s logically possible for an organism of one species to be a descendant of an organism of another. We infer possibility from truth here. The 19th century atomists did not need to prove the possibility of indivisible objects. They were within their rights to infer the possibility from the evidence for actuality (say, successes of particle-based thermodynamics).
    Even in physics, rigorous axiomatizations and possibility proofs can lag significantly behind the theories that they are axiomatizations and possibility proofs for. Newton’s second law of motion was a reasonable thing to believe for a long, long time before we had a mathematically rigorous account of derivatives (and hence accelerations).
    “If God exists, then God is consistently definable”
    I don’t know what your criteria of adequacy for a consistent definition are.
    But in any case, unless the conditional is a material one (in which case, I will maintain it’s true because I hold the consequent to be true, and you should maintain it’s true because you hold the antecedent to be false), I see little reason to accept it. I exist, but I don’t know that I am consistently definable. It does not seem that I am finitely definable. Plausibly, for any finite description D that does not make _de re_ reference to me, infinitely many individuals could satisfy D.
    I don’t know if electrons are non-trivially definable. We can, perhaps, specify their role in terms of the way they interact with particles and fields (even that’s not clear; for the role may depend on constants in the laws, and the constants may not be finitely specifiable), but it is far from clear that it is a necessary truth that anything that fulfills that role is an electron. It seems quite conceivable that there could be shmelectrons that behave just as electrons do in interactions with the entities of our world’s physics (but perhaps the shmelectrons behave differently from electrons in interactions with entities outside of our world’s physics).
    “Since there’s a very good literature on the complexity of theories, I’ll stick with that. It would thus be nice to see if you might spell out your proposal in those terms.”
    I wonder if this doesn’t take the direction of fit for theories of complexity the wrong way. When physicists, biologists or police officers discuss whether theory A or theory B in their discipline is simpler, they do not advert to theories of complexity. Rather, there is an intuitive judgment as to complexity, and epistemologists, philosophers of science and others then try to build theories of complexity that fit reasonably well with these intuitive judgments. Granted, these theories can help with the first order judgments in difficult cases. But there is no need for every reasonable complexity judgment to be such that we can spell it out in terms of the formal theories of complexity.
    It might be nice if we could. But I am not sure we can. Among the physicists, talk of simplicity seems closely allied to aesthetic talk of elegance. The kind of simplicity they want in a theory is an elegant simplicity. And I think most (though not Plato!) agree that our prospects for formalizing aesthetic concepts are poor.
    “I don’t see that ZFC (or its extension to classes a la VGB) are gerrymandered at all. In which axioms lies the gerrymandering?”
    We get to ZF by noticing that comprehension is incoherent. Many of the axioms (e.g., separation, axiom of union, axiom of pairs, etc.) are special cases of comprehension. Sure, they have some intuitive plausibility, but this plausibility is no greater than that of comprehension–as far as I can see the only thing they have going for them over comprehension is that while we have found an inconsistency in comprehension very quickly, despite a long time searching we haven’t found an inconsistency in ZF. This is a clear case of gerrymandering: we choose a bunch of axioms more complex than comprehension precisely to avoid the inconsistency we know of in comprehension.
    I am not saying this gerrymandering is necessarily objectionable. We may not have a better theory. But it is, nonetheless, gerrymandering, much as an unfortunate amount of 20th century epistemology involved attempts at definitions of knowledge gerrymandered to avoid ever more sophisticated Gettier cases.

    December 29, 2012 — 9:54
  • Eric Steinhart

    Alex –
    You started off this sub-thread by talking about “necessary truths of axiology”; so questions of logical possibility seem appropriate. But now you’ve shifted to giving examples of mere empirical plausibility. That Darwinism, the atomic theory of matter, etc. are all empirically plausible is undeniable. So is naive set theory, so were the original internally inconsistent foundations of the calculus. (And both QM and GR are empirically plausible, but mutually inconsistent, and perhaps internally inconsistent, and physicists worry a great deal about both of those issues.)
    You are entirely right that you don’t have to prove logical possibility. I’m asking you to reply to what looks like an obvious formal objection, analogous to well-known paradoxes of maximality. You spelled out an axiom (3) that links perfection to maximality in an axiological partial-order relation; the natural objection is that the relation has no maximum, so that either there is no perfect being or the axiom is incorrect. Since there are already well-known objections to omniscience and omnipotence based on failures of maximizability (as well as issues concerning the lack of a best possible world), it seems crucial for any theist to address these issues, especially in light of your axiom (3).
    – Eric

    December 29, 2012 — 11:50
  • Dianelos Georgoudis

    Eric:
    “I don’t see why axiarchism requires a perfect being of any type.”
    If the axiarchic principle actualizes good things, then one would think it would actualize the best thing, or at least something quite close to it.
    “Necessarily, for every abstract form F, if F satisfies axiological property P, then there exists some concrete x such that F(x)”
    Isn’t that an excessive definition of axiarchism? I mean there is the axiological property of me not being bothered by a fly in the room right now. Does it follow that a concrete world exists where somebody quite like me is not so bothered?
    I’d think a more reasonable definition would be something like: The stronger an axiological property P is, the more probable it is that a concrete x actualizes its form.
    “You object that it isn’t simple; yet the principles behind it are very simple indeed.”
    It seems that simplicity is often in the mindset of the beholder. Perhaps a way to be more objective in one’s judgment of simplicity is to use a philosophical analogue of the Kolmogorov complexity. If one tries to describe the principles on which theism is founded and the principles on which axiarchism is founded in sufficient detail for a third party to be able to derive truths about theism or axiarchism, I bet one will need much less space for the former than for the latter.
    “My own view is that abstract axiarchic principles bring into concrete existence an infinite hierarchy of increasingly complex computing machines.”
    Computing machines are paradigmatic cases of thoughtful design, but the idea might work based on the context of genetic algorithms. On the other hand, the Darwinian algorithm requires a quite fine-tuned environment to work. Are you quite certain the principles of axiarchism are that simple? It seems to me they require not only a non-reducible brute-fact set of axiarchic properties, but also of goal-oriented computational properties.
    And I suppose the god-like computing machines would produce the parameters of a phenomenal physical world, in which case I don’t quite see why you dislike Leslie’s idealism.
    “I suspect that on your minimal definition of “person”, the computer network at Google may already be a person.”
    I don’t believe the Google computer network is conscious. If it were then it would have moral rights.
    “So I sometimes refer to my position as “theological naturalism””
    Myself I like “natural theology”. Anyway, I observe that free-thinking naturalists tend to arrive at worldviews which are conceptually similar to theism, or at least more similar to theism than traditional naturalism is.
    “[Theists’] job is to prove that this partial order has a maximum.”
    I think St Anselm’s definition of God should not be understood ontologically but epistemically, i.e. not making a claim about how God is but about how one should think about how God is. After all, considering that God is the most basic thing, I don’t think a proper definition of God is possible, nor that there exists an abstract idea or concept which properly describes how God is. Rather, St Anselm’s definition should be understood as follows: For all persons S, and all God-concepts G1 and G2 which S entertains, if according to S’s sense of perfection G1 is greater than G2, then it is more probable that S will form true beliefs about God using G1 than G2.
    Might it be the case that for any Gn there is a Gm such that S will judge that perfection(Gm)>perfection(Gn)? I don’t think so, but even if that were true it would be immaterial since one’s time for thinking about God is limited. As a practical matter I claim the following holds for the human condition: The greatest being one can conceive after some measured investment of effort is more than sufficient for one to be able to form the true beliefs about God which one requires for living a good life.

    December 29, 2012 — 17:39
  • Eric Steinhart

    Diane –
    You write: “If the axiarchic principle actualizes good things, then one would think it would actualize the best thing, or at least something quite close to it.”
    The Dedekind-Peano axioms entail the existence of an endless series of ever greater natural numbers; the ZF axioms entail the existence of an endless series of ever more inclusive sets. Yet, there is no biggest natural number, nor any most inclusive set.
    The axiarchic principle I gave does not specify property P. Various authors are free to specify it in various ways (and they have). I take it that any specification of P needs to be defensible (both philosophically and scientifically). You do raise the very good point that P needs to be developed and not just left vague.
    Note that Kolmogorov’s compressibility measure is no longer thought of as having anything to do with complexity. (And nobody has ever thought that random strings are complex.) Complexity measures are standardly thought of as “one-hump”, that is, as being low for both regularity and randomness, and high in between. But you’re on the right track here. Check out Bennett’s papers on “logical depth”, and go from there.
    I think of computers as models of certain types of iterative structures. They don’t have to be physical things. And they certainly don’t have to be artifacts.
    Having done a fair amount of work in optimization theory, I don’t understand words like “design”, “planning”, “fine-tuning” or even “intelligence” in the ways that most philosophers understand them. I reject all folk-psychological definitions of those terms in favor of purely computational definitions based on optimization. I’m happy to say (as Dennett also says) that evolutionary algorithms design things.
    Note that I’m happy to talk about intelligent design in both biology and cosmology. Only I don’t think it requires an intelligent designer. All it requires is a massively parallel optimization algorithm — a vast distributed computation. I think almost everthing in biology is intelligently designed and that the laws of physics are intelligently designed — but there’s no intelligent designer.
    If this all sounds crazy, it’s because computer science has changed the way I think about intelligence. I’m sure that there are plenty of intelligent processes that are not minds, and to which our folk-psychology does not apply.
    I think all sorts of distributed computing systems are intelligent, conscious, and so forth (e.g. forests, IBM’s Watson, Google’s server net, etc.). Yet I would not apply human folk-psychology to any of them. Nor would I see why mere consciousness would entail any moral status. If consciousness is, say, merely higher-order thought, then my MacBook Pro is conscious. Yet I deny it has any moral status.
    I’d use the term “natural theology”, except it tends to signify the use of science to confirm already established Christian faith. I’m not Christian. I seek theological alternatives to Christianity.
    – Eric

    December 29, 2012 — 20:28
  • Dianelos Georgoudis

    Eric:
    You write: “I take it that any specification of P needs to be defensible (both philosophically and scientifically).”
    I suppose this means that the axiarchist must specify P in such a way that the axiarchic principle that results from it would give rise to a world which fits with the generally accepted philosophical and scientific truths that hold in our world. If so, please observe the following epistemological difference with theism: The theist does not try to find a God-concept which will fit the world we know. Rather she starts with her basic knowledge of perfection, and checks to see whether the metaphysical ground which embodies that perfection will give rise to the world we know. Thus, I suggest, if a theistic and an axiarchic metaphysical theory are equally successful, theism’s success will be more significant.
    “Note that Kolmogorov’s compressibility measure is no longer thought of as having anything to do with complexity.”
    I think the kind of “simplicity” we are looking for here entails that an economical principle will have a rich explanatory scope. Which, it seems to me, fits well with Kolmogorov’s idea. (That random strings are not considered complex is irrelevant in this context – rather, both a simple theory and a complex theory will be expressed by a string which looks random, albeit the latter string will be longer than the former.) And I fail to see why Bennett’s emphasis on the workload factor rather than on the size of the algorithm is appropriate. Consider the subcase of the historical evolution of the physical sciences. The ratio between explanatory scope and theory becomes clearly larger, i.e. the Kolmogorov complexity of the newer theories relative to their explanatory scope becomes lower. On the other hand the computational complexity of applying the newer theories grows fast, perhaps even faster than their explanatory scope.
    “I think almost everthing in biology is intelligently designed and that the laws of physics are intelligently designed — but there’s no intelligent designer.”
    It seems you hold that the distributed optimization algorithm, which produces powerful universes like our own, is conscious, intelligent, and so on. But then I don’t understand why you won’t call it an “intelligent designer”. Perhaps because it lacks libertarian free will?
    “I seek theological alternatives to Christianity.”
    What I find so very significant is that if one thinks just about the greatest conceivable being, one sees that God would undergo kenosis and incarnation in order to suffer creation with us. Thus I would argue that the basic idea of Christianity is entailed in theism.
    Incidentally, I wouldn’t say that Christianity is an “already established faith”, because there is not one Christianity, and because Christians search and investigate all the time, and because Christian philosophy and theology are evolving.

    December 30, 2012 — 16:47
  • Eric Steinhart

    Diane –
    You are certainly right that there are many differences between axiarchism and theism. But I didn’t get why theism will be more significant if successful. Surely not because it promises us some future good; axiarchism can promise that too. I find it hard to think of anything that theism can do that axiarchism can’t.
    (But let’s be careful: when I stated that axiarchic principle, I’m afraid you may have mistaken it for some physical principle, operating in our universe; I would never say that. I take it that axiarchic principles bring universes into existence. Perhaps they do this in ways that govern counterpart relations or other modal relations. But axiarchic principles are certainly not like physical laws. I don’t take the value involved in axiarchism as being human-centered, and I certainly do not think that value is pleasure, or happiness, or any other human psychological category.)
    I’m glad, once again, to see that you’re starting to think of complexity in information-theoretic terms. But the paragraph you wrote did not make any sense to me. There are some good introductory resources on these issues on the net.
    You write: “It seems you hold that the distributed optimization algorithm, which produces powerful universes like our own, is conscious, intelligent, and so on. But then I don’t understand why you won’t call it an “intelligent designer”.”
    Because there is no intelligent designer. Design does not require any designer at all, that’s the whole point. A distributed algorithm runs across a plurality of machines, none of which may be running the algorithm itself, or even a part of the algorithm. The algorithm is a feature of their collective activity, over space and time. (Ok, perhaps “life on earth” is the intelligent designer of itself, or “the earth” is running the algorithm, but those claims seem like pretty thin metaphors.)
    You write: “What I find so very significant is that if one thinks just about the greatest conceivable being, one sees that God would undergo kenosis and incarnation in order to suffer creation with us.”.
    Well, you’re a Christian! I am not. But I am indeed happy that “Christians search and investigate all the time” and that “Christian philosophy and theology are evolving”. So far, at least in the US, the whole religion seems to be evolving into atheism, and much more rapidly than I ever would have thought.
    – Eric

    December 30, 2012 — 20:51
  • Eric:
    “Since there are already well-known objections to omniscience and omnipotence based on failures of maximizability (as well as issues concerning the lack of a best possible world), it seems crucial for any theist to address these issues, especially in light of your axiom (3).”
    Theists can, should and do respond to such specific worries as the omniscience and omnipotence ones.
    My response to the omnipotence worries is the Pearce-Pruss account. 🙂
    As for omniscience, I assume you mean Grim-style arguments. There are real worries there, but they do not seem to be specific to omniscience. These worries appear to me (after careful examination) to be variants of the heterologicality and liar paradoxes, and are worries for anyone who has an unrestricted notion of truth (whether of a proposition or a sentence), or of the satisfaction of a predicate, or of the possession of a property, or of a number of other semantic properties or predicates. These worries are too broad for me to see them as specifically worries about omniscience. In any case, I have things to say about these paradoxes. These things are controversial, of course, but they do cohere with omniscience–and they’re not the only answers that do that.
    I agree that there is no best world, but not for the usual reason of thinking that for any world there is a better. On the contrary, I think there are infinitely many of maximally valuable worlds. The usual constructions for making a better world fail, however, because of widespread incommensurability. Thus, adding more happy mathematicians to a good world need not make the world better simpliciter, since while it may make it better in respect to one kind of value, it is apt to make the world less good in terms of aesthetic values such as elegant simplicity.
    I don’t have an a priori proof of the coherence of the maximality claim. If I did, I’d probably have a very nice ontological argument. But the a posteriori reasons for accepting theism give one reason to think the maximality claim is coherent. 🙂

    December 30, 2012 — 21:00
  • Dianelos Georgoudis

    Eric:
    You write: “ But I didn’t get why theism will be more significant if successful.”
    Both theism and axiarchism explain creation on axiological properties P (albeit theism grounds them in the character of God, and grounds their creative powers in God’s will). But axiarchists are free to design P, whereas theists aren’t. The validity of St Anselm’s definition (which virtually all theists accept) entails that we possess an innate, basic cognitive faculty for perceiving axiological properties. Thus the epistemological ground on which theists must build their metaphysics is given, or at least less flexible, compared to axiarchism. For example, the axiarchist may claim some features of P which grossly violate our innate sense of value, and justify that choice on the fact that these features successfully explain truths about the world we observe around us. Similarly, the physical scientist is on the same grounds free to violate (and famously has violated) our sense of what’s plausible in the physical realm. Indeed, as you write, the axiological properties are not circumscribed by any human psychological notions. The theist does not have that freedom. Thus if a theistic metaphysics is deemed to be as successful as an axiarchic metaphysics, its success will be more significant. In the same way that the success of completing a puzzle without the freedom of manipulating its building blocks is more significant.
    “ when I stated that axiarchic principle, I’m afraid you may have mistaken it for some physical principle”
    No, not at all. But I do have a question. I understand the axiarchic principle creates algorithmic computers which run universes (in all their dimensions: physical, mental, aesthetic, whatever), which universes instantiate to some degree the axiological properties of the principle. (And the whole thing works automatically as it were, i.e. without libertarian free will.) But I don’t know how on your view the dynamics of optimization work. I suppose the axiarchic principle itself is immutable and serves as the standard for the optimization. So, does the axiarchic principle have the power to improve the computers, or do the computers run the universes as experiments and improve themselves? Probably the latter. But then wouldn’t the computers be moved by the optimization goal to evolve themselves into a God-like being of *one* conscious identity? Their distributed nature would not be any hindrance. (If I am on the right track here, axiarchism has several parallels with process theology.)
    “But the paragraph you wrote [on complexity in information-theoretic terms] did not make any sense to me.”
    I am looking for a philosophical analogue of the information-theoretic sense of complexity. And it seems to me that Kolmogorov’s idea which centers on the size of the algorithm comports better with the philosophical notion of simplicity than Bennett’s idea which centers on the algorithm’s computational complexity. Computational complexity relative to a particular goal is important to a software engineer who wants to build useful machines. Indeed in the computer world the efficiency of an algorithm is much more important than its size. But it is irrelevant to the philosophical notion of simplicity, it seems to me. Finally, I used the recent history of the physical sciences to demonstrate that what scientists look for is simplicity in Kolmogorov’s sense and not in Bennett’s. For example, the idea of the TOE is to have a formula that fits on a T-shirt but has the power to explain all physical phenomena. Physicists win the Nobel price on the ratio between the size of what their theory explains and the size of their theory. Nobody worries about the computational complexity of applying the theory.
    “Well, you’re a Christian! I am not.”
    Our own metaphysical views are irrelevant to the claim that the basic idea of Christianity, namely the incarnation of God, is entailed in theism. Non-theistic philosophers are perfectly capable of theological thought as evidenced by the theologically sophisticated argumentation they engage in.
    “So far, at least in the US, the whole religion seems to be evolving into atheism”
    This it is a natural reaction away from the dogmatic/obscurantist features of religion. Fashion certainly plays a role too. But of course I meant the movement in the philosophical and theological discourse. I think it is accurate to say that we live through a renaissance movement in philosophy or religion, whereas traditional (i.e. “scientific”) naturalism is having a hard time. It will take some time before these movements in the frontlines of knowledge filter down to the popular mindset. (Great ideas move slowly outside of the specialists’ circle. Almost a hundred years after the discovery of quantum mechanics its relevance to metaphysics is lost even to many philosophers.)

    December 31, 2012 — 5:42
  • Eric Steinhart

    Diane –
    You write: “Kolmogorov’s idea which centers on the size of the algorithm”
    That is incorrect. Kolmogorov defines the algorithmic information content of a string of bits as its most concise description divided by its least concise description. That is, the length of the compression of the string divided by the length of string itself. The algorithm which compresses the string is typically invariant. When you compress a file using ZIP on a Mac or PC, you are using Kolmogorov compression.
    You write: “Bennett’s idea which centers on the algorithm’s computational complexity.”
    This is not correct. Bennett defines the logical depth of a string of bits as the smallest number of computational steps (e.g. by a universal Turing machine) to reconstruct the string from random input.
    You write: “Computational complexity relative to a particular goal is important to a software engineer who wants to build useful machines.”
    The computational complexity relevant to software engineering has nothing to do with either Kolmogorov or Bennett’s metrics.
    You write: “I used the recent history of the physical sciences to demonstrate that what scientists look for is simplicity in Kolmogorov’s sense and not in Bennett’s.”
    If that’s true, then they are looking for simplicity in the data, not in the theory. The data is the least concise description (e.g. a table of astronomical data) while the theory is the most concise description (e.g. astronomical laws).
    But, as is well-known, scientists aren’t looking for better compression algorithms. If I took all the data of the stock market and compressed it into the smallest possible file, the result would not be a theory of anything.
    Kolmogorov’s measure merely indicates the amount of entropy in some data. It has no relevance to anything you’ve discussed.
    And there’s a large literature on complexity of scientific theories, in logic and philosophy of science, most of which measures complexities by counting quantifiers in the theory, or alternating blocks of quantifier types, or ranks of models in ZF, or things like that.
    – Eric

    December 31, 2012 — 9:11
  • Eric Steinhart

    Alex –
    You write: “I don’t have an a priori proof of the coherence of the maximality claim. If I did, I’d probably have a very nice ontological argument. But the a posteriori reasons for accepting theism give one reason to think the maximality claim is coherent.”
    Ok, you’ve significantly changed your tune here, but I think that’s the start of a defense of maximality.
    – Eric

    December 31, 2012 — 9:27
  • Eric Steinhart

    Alex, by the way, I have no real objection to local Anselmianism – God is that than which no greater is actual. But I can’t agree with global Anselmianism – God is that than which no greater is possible. That’s too much. – Eric

    December 31, 2012 — 10:52
  • Dianelos Georgoudis

    Eric:
    I feel pretty certain that the Kolmogorov complexity of a string is the length of the minimal (algorithmic) description of that string. (And “string” stands for any description of some piece of information.) So I stand by my claim that Kolmogorov’s idea centers on the size of an algorithm. Compression programs such as ZIP are useful for computing an upper bound of the Kolmogorov complexity of a string (even though one should add the size of the program to the size of the compressed file), but I am not sure how else they apply. Indeed Kolmogorov complexity does not entail the use one particular algorithm. Suppose strings S1 and S2, algorithms A1 and A2, and compressed strings C1 and C2. If the system (A1,C1) is the smallest which will produce S1, and the system (A2,C2) is the smallest which will produce S2, then the respective sizes of the two systems define the Kolmogorov complexities of S1 and S2. (Here I assume a consistent language.)
    I am less familiar with Bennet’s idea, but since, as you write, it’s about the number of computational steps an algorithm requires for completing some task, it is certainly relevant to the computer engineer.
    “The data is the least concise description (e.g. a table of astronomical data) while the theory is the most concise description (e.g. astronomical laws).”
    Here a theory consists of astronomical laws plus initial conditions. And is a concise (but not neccessarily the most concise) description of the relevant data. So, among two alternative theories wouldn’t we choose as simpler the one which is the more concise?
    “If I took all the data of the stock market and compressed it into the smallest possible file, the result would not be a theory of anything.”
    I wonder about that. It seems to me that if a theory exists in that data set, then its optimal compression method instantiates it.
    Anyway, the epistemological problem we are discussing is a serious one. Philosophers keep arguing about simplicity in a very arbitrary fashion. So, for example, Alex in the OP speaks of the number of kinds of entities, but Occam’s razor only speaks of the number of entities. I once encountered the argument that the multiverse (which atheists use to counter the fine-tuning argument) is simpler than a one-universe theory, since it does not require the limitation that only one universe exists. By the same measure modal realism is the simplest possible theory, whereas others might it’s the most complex possible theory. You appear to think that by taking God out of a theistic metaphysics one may get to a simpler theory. I think that idealism is much simpler than dualism, others disagree. There is great confusion.
    One possible solution to this problem is to call “simpler” that metaphysical theory which is smaller in size. There are a host of tricky problems (including the fact that reality contains not only quantitative but also qualitative data which are not amenable to algorithmic/mechanical production, or the fact that we don’t know the entire data-set the theory should produce), but I think the general idea is right.

    January 2, 2013 — 3:05
  • Dianelos Georgoudis

    As for the potential paradoxes of the maximality claims, to my knowledge they only to quantitative properties. But, as the ancients said, not in quantities lies perfection (“ouk en to pollo to eu”). Indeed, my own perception of greatness is almost exclusively about quality and not quantity.
    Incidentally, I very much like Eric’s suggestion that “God is that than which no greater is actual”, since for me perfection entails the will for improving.

    January 2, 2013 — 3:17
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