Don Page is one of the world’s leading experts on theoretical gravitational physics and cosmology, as well as a previous guest-blogger around these parts. (There are more world experts in theoretical physics than there are people who have guest-blogged for me, so the latter category is arguably a greater honor.) He is also, somewhat unusually among cosmologists, an Evangelical Christian, and interested in the relationship between cosmology and religious belief.
Longtime readers may have noticed that I’m not very religious myself. But I’m always willing to engage with people with whom I disagree, if the conversation is substantive and proceeds in good faith. I may disagree with Don, but I’m always interested in what he has to say.
Recently Don watched the debate I had with William Lane Craig on “God and Cosmology.” I think these remarks from a devoted Christian who understands the cosmology very well will be of interest to people on either side of the debate.
Open letter to Sean Carroll and William Lane Craig:
I just ran across your debate at the 2014 Greer-Heard Forum, and greatly enjoyed listening to it. Since my own views are often a combination of one or the others of yours (though they also often differ from both of yours), I thought I would give some comments.
I tend to be skeptical of philosophical arguments for the existence of God, since I do not believe there are any that start with assumptions universally accepted. My own attempt at what I call the Optimal Argument for God (one, two, three, four), certainly makes assumptions that only a small fraction of people, and perhaps even only a small fraction of theists, believe in, such as my assumption that the world is the best possible. You know that well, Sean, from my provocative seminar at Caltech in November on “Cosmological Ontology and Epistemology” that included this argument at the end.
I mainly think philosophical arguments might be useful for motivating someone to think about theism in a new way and perhaps raise the prior probability someone might assign to theism. I do think that if one assigns theism not too low a prior probability, the historical evidence for the life, teachings, death, and resurrection of Jesus can lead to a posterior probability for theism (and for Jesus being the Son of God) being quite high. But if one thinks a priori that theism is extremely improbable, then the historical evidence for the Resurrection would be discounted and not lead to a high posterior probability for theism.
I tend to favor a Bayesian approach in which one assigns prior probabilities based on simplicity and then weights these by the likelihoods (the probabilities that different theories assign to our observations) to get, when the product is normalized by dividing by the sum of the products for all theories, the posterior probabilities for the theories. Of course, this is an idealized approach, since we don’t yet have _any_ plausible complete theory for the universe to calculate the conditional probability, given the theory, of any realistic observation.
For me, when I consider evidence from cosmology and physics, I find it remarkable that it seems consistent with all we know that the ultimate theory might be extremely simple and yet lead to sentient experiences such as ours. A Bayesian analysis with Occam’s razor to assign simpler theories higher prior probabilities would favor simpler theories, but the observations we do make preclude the simplest possible theories (such as the theory that nothing concrete exists, or the theory that all logically possible sentient experiences occur with equal probability, which would presumably make ours have zero probability in this theory if there are indeed an infinite number of logically possible sentient experiences). So it seems mysterious why the best theory of the universe (which we don’t have yet) may be extremely simple but yet not maximally simple. I don’t see that naturalism would explain this, though it could well accept it as a brute fact.
One might think that adding the hypothesis that the world (all that exists) includes God would make the theory for the entire world more complex, but it is not obvious that is the case, since it might be that God is even simpler than the universe, so that one would get a simpler explanation starting with God than starting with just the universe. But I agree with your point, Sean, that theism is not very well defined, since for a complete theory of a world that includes God, one would need to specify the nature of God.
For example, I have postulated that God loves mathematical elegance, as well as loving to create sentient beings, so something like this might explain both why the laws of physics, and the quantum state of the universe, and the rules for getting from those to the probabilities of observations, seem much simpler than they might have been, and why there are sentient experiences with a rather high degree of order. However, I admit there is a lot of logically possible variation on what God’s nature could be, so that it seems to me that at least we humans have to take that nature as a brute fact, analogous to the way naturalists would have to take the laws of physics and other aspects of the natural universe as brute facts. I don’t think either theism or naturalism solves this problem, so it seems to me rather a matter of faith which makes more progress toward solving it. That is, theism per se cannot deduce from purely a priori reasoning the full nature of God (e.g., when would He prefer to maintain elegant laws of physics, and when would He prefer to cure someone from cancer in a truly miraculous way that changes the laws of physics), and naturalism per se cannot deduce from purely a priori reasoning the full nature of the universe (e.g., what are the dynamical laws of physics, what are the boundary conditions, what are the rules for getting probabilities, etc.).
In view of these beliefs of mine, I am not convinced that most philosophical arguments for the existence of God are very persuasive. In particular, I am highly skeptical of the Kalam Cosmological Argument, which I shall quote here from one of your slides, Bill:
- If the universe began to exist, then there is a transcendent cause
which brought the universe into existence. - The universe began to exist.
- Therefore, there is a transcendent cause which brought the
universe into existence.
I do not believe that the first premise is metaphysically necessary, and I am also not at all sure that our universe had a beginning. (I do believe that the first premise is true in the actual world, since I do believe that God exists as a transcendent cause which brought the universe into existence, but I do not see that this premise is true in all logically possible worlds.)
I agree with you, Sean, that we learn our ideas of causation from the lawfulness of nature and from the directionality of the second law of thermodynamics that lead to the commonsense view that causes precede their effects (or occur at the same time, if Bill insists). But then we have learned that the laws of physics are CPT invariant (essentially the same in each direction of time), so in a fundamental sense the future determines the past just as much as the past determines the future. I agree that just from our experience of the one-way causation we observe within the universe, which is just a merely effective description and not fundamental, we cannot logically derive the conclusion that the entire universe has a cause, since the effective unidirectional causation we commonly experience is something just within the universe and need not be extrapolated to a putative cause for the universe as a whole.
However, since to me the totality of data, including the historical evidence for the Resurrection of Jesus, is most simply explained by postulating that there is a God who is the Creator of the universe, I do believe by faith that God is indeed the cause of the universe (and indeed the ultimate Cause and Determiner of everything concrete, that is, everything not logically necessary, other than Himself—and I do believe, like Richard Swinburne, that God is concrete and not logically necessary, the ultimate brute fact). I have a hunch that God created a universe with apparent unidirectional causation in order to give His creatures some dim picture of the true causation that He has in relation to the universe He has created. But I do not see any metaphysical necessity in this.
(I have a similar hunch that God created us with the illusion of libertarian free will as a picture of the true freedom that He has, though it might be that if God does only what is best and if there is a unique best, one could object that even God does not have libertarian free will, but in any case I would believe that it would be better for God to do what is best than to have any putative libertarian free will, for which I see little value. Yet another hunch I have is that it is actually sentient experiences rather than created individual `persons’ that are fundamental, but God created our experiences to include beliefs that we are individual persons to give us a dim image of Him as the one true Person, or Persons in the Trinitarian perspective. However, this would take us too far afield from my points here.)
On the issue of whether our universe had a beginning, besides not believing that this is at all relevant to the issue of whether or not God exists, I agreed almost entirely with Sean’s points rather than yours, Bill, on this issue. We simply do not know whether or not our universe had a beginning, but there are certainly models, such as Sean’s with Jennifer Chen (hep-th/0410270 and gr-qc/0505037), that do not have a beginning. I myself have also favored a bounce model in which there is something like a quantum superposition of semiclassical spacetimes (though I don’t really think quantum theory gives probabilities for histories, just for sentient experiences), in most of which the universe contracts from past infinite time and then has a bounce to expand forever. In as much as these spacetimes are approximately classical throughout, there is a time in each that goes from minus infinity to plus infinity.
In this model, as in Sean’s, the coarse-grained entropy has a minimum at or near the time when the spatial volume is minimized (at the bounce), so that entropy increases in both directions away from the bounce. At times well away from the bounce, there is a strong arrow of time, so that in those regions if one defines the direction of time as the direction in which entropy increases, it is rather as if there are two expanding universes both coming out from the bounce. But it is erroneous to say that the bounce is a true beginning of time, since the structure of spacetime there (at least if there is an approximately classical spacetime there) has timelike curves going from a proper time of minus infinity through the bounce (say at proper time zero) and then to proper time of plus infinity. That is, there are worldlines that go through the bounce and have no beginning there, so it seems rather artificial to say the universe began at the bounce that is in the middle just because it happens to be when the entropy is minimized. I think Sean made this point very well in the debate.
In other words, in this model there is a time coordinate t on the spacetime (say the proper time t of a suitable collection of worldlines, such as timelike geodesics that are orthogonal to the extremal hypersurface of minimal spatial volume at the bounce, where one sets 
) that goes from minus infinity to plus infinity with no beginning (and no end). Well away from the bounce, there is a different thermodynamic time 
(increasing with increasing entropy) that for 
increases with 
but for 
decreases with (so there becomes more positive as becomes more negative). For example, if one said that is only defined for 
, say, one might have something like
the positive square root of one less than the square of . This thermodynamic time only has real values when the absolute value of the coordinate time , that is, 
, is no smaller than 1, and then increases with .
One might say that begins (at 
) at 
(for one universe that has growing as decreases from -1 to minus infinity) and at 
(for another universe that has growing as increases from +1 to plus infinity). But since the spacetime exists for all real , with respect to that time arising from general relativity there is no beginning and no end of this universe.
Bill, I think you also objected to a model like this by saying that it violates the second law (presumably in the sense that the coarse-grained entropy does not increase monotonically with for all real ). But if we exist for 
(or for 
; there would be no change to the overall behavior if were replaced with 
, since the laws are CPT invariant), then we would be in a region where the second law is observed to hold, with coarse-grained entropy increasing with 
(or with 
if ). A viable bounce model would have it so that it would be very difficult or impossible for us directly to observe the bounce region where the second law does not apply, so our observations would be in accord with the second law even though it does not apply for the entire universe. I think I objected to both of your probability estimates for various things regarding fine tuning. Probabilities depend on the theory or model, so without a definite model, one cannot claim that the probability for some feature like fine tuning is small. It was correct to list me among the people believing in fine tuning in the sense that I do believe that there are parameters that naively are far different from what one might expect (such as the cosmological constant), but I agreed with the sentiment of the woman questioner that there are not really probabilities in the absence of a model. Bill, you referred to using some “non-standard” probabilities, as if there is just one standard. But there isn’t. As Sean noted, there are models giving high probabilities for Boltzmann brain observations (which I think count strongly against such models) and other models giving low probabilities for them (which on this regard fits our ordered observations statistically). We don’t yet know the best model for avoiding Boltzmann brain domination (and, Sean, you know that I am skeptical of your recent ingenious model), though just because I am skeptical of this particular model does not imply that I believe that the problem is insoluble or gives evidence against a multiverse; in any case it seems also to be a problem that needs to be dealt with even in just single-universe models.
Sean, at one point your referred to some naive estimate of the very low probability of the flatness of the universe, but then you said that we now know the probability of flatness is very near unity. This is indeed true, as Stephen Hawking and I showed long ago (“How Probable Is Inflation?” Nuclear Physics B298, 789-809, 1988) when we used the canonical measure for classical universes, but one could get other probabilities by using other measures from other models.
In summary, I think the evidence from fine tuning is ambiguous, since the probabilities depend on the models. Whether or not the universe had a beginning also is ambiguous, and furthermore I don’t see that it has any relevance to the question of whether or not God exists, since the first premise of the Kalam cosmological argument is highly dubious metaphysically, depending on contingent intuitions we have developed from living in a universe with relatively simple laws of physics and with a strong thermodynamic arrow of time.
Nevertheless, in view of all the evidence, including both the elegance of the laws of physics, the existence of orderly sentient experiences, and the historical evidence, I do believe that God exists and think the world is actually simpler if it contains God than it would have been without God. So I do not agree with you, Sean, that naturalism is simpler than theism, though I can appreciate how you might view it that way.
Best wishes,
Don
The contents of Dr. Jefferys’ pdf make sense (of course) but I don’t see the same error being made in Dr. Page’s example, as I interpret it. Another way I see of describing it is:
There is a machine which spits out a single ping-pong ball from an internal bin when its button is pushed (used for a lottery). The ball has a number on it, from one to one million (it happens to be the number 42). We know there exist a million ping-pong balls with different numbers in that range which might have been loaded into the machine’s bin, but we don’t know how many were actually loaded.
H_1 = all one million were loaded.
H_2= only the ones numbered 1-1000 were loaded.
H_3=only the one with number 42 was loaded.
P(42|H_1)=1/one-million, etc.
Then if somehow you had valid prior probabilities for the H’s, could you not do Dr. Page’s calculation to update those priors?
In the pdf example, P(E)=1 regardless of the hypothesis (incorrectly), since it was observed. In Dr. Page’s case, P(42) only =1 under and due to H_3.
But it might have been 43 instead, given that there were one million balls to chose from. I guess the problem or lack thereof depends on whether the issue is that a specific number was loaded, or whether the issue was that only a single ball was loaded.
So unless we picked the number 42 for H_3 before pushing the button, its probability should be one-millionth for both H_1 and H_3. Light dawns on the jungle.
Hi Paul
Regarding reports of miracles, which I am defining here as ‘obvious departures from widely held models of expectation in the macro world’ or for statistical physics as ‘extremely high sigma probability departures’ or ‘violations of all known physical laws’, I have heard of people in Africa who claim to have been healed by sangomas/witch doctors.
Regarding the supernatural rather than the outright miraculous, I used to work with an excellent systems engineer with a very wide electronic/mechanical/ hardware/software skills set who was married to a medium. He was not a Christian. He believed his wife’s gift was substantive and supernatural.
I am aware that I may be being dismissed as naive and simplistic, but Christians are used to that. Being voted as unpopular is one thing, being proven wrong is another. Christians might say you are not seeing the forest for analyzing the trees, and are playing too much on your strong points, i.e. analytical abstraction. Most people probably won’t vote someone with a very strong bias to analytical abstraction into power. They are more concerned with day-to-day relational judgement.
My basic philosophical point hasn’t changed throughout. I don’t think all this philosophizing from a human perspective and these attempts to rigorously assign probabilities to theses is likely to penetrate into the rational/mathematical frameworks God works with, or even into the totality of his relational mindset.
Pure Boltzmann brain=stochastic outcome, but all apparent meaning derived by BB unreliable. Conscious BB extremely unlikely, but reliable deductive engine in conscious BB much less likely still.
I believe that ultra-rationalism for final truth is a self-deceived condition.
I would say that one may defend Christianity rationally up to a point, and that one may use rationale to define limits on what is reliably accessible to rationalism.
I believe there is something in man which God calls to. I don’t believe we understand it.
Trust in the LORD with all your heart, and do not lean on your own understanding.
(Pro 3:5)
Deep calls to deep……. (Psalm 42:7)
Attempting to find a human focused Godlike purpose in the broad Cosmos is certain to lead to failure. Once one begins to examine even our solar system, the immense inefficiency of the energy flow from the sun, were its purpose being to energise life on earth, would be the first and most obvious design flaw. The miniscule amount of the Sun’s energy that fuels earth is an inefficiency scale indicator for what can be found when expanding ones imagination outwards to our galaxy, and then further beyond that to the whole universe itself. To believe that the affairs of one tiny amount of matter for an equally tiny span of time were the “purpose” of the universe would require a cosmos sized amount of faith, if in fact the scale of the claim could truly be comprehended.
Collapsing awareness back to the human scale, I do believe in a heaven and a hell, as these are properties of our human consciousness which exist within us as hopes and fears. As self awareness has extended the possibilities for consciousness with expanding brain size and complexity, desire to control destiny inevitably becomes entangled with the unpredictable indifference of nature, giving rise to hopes and fears. Managing fear for explanation of the failings of human effort inevitably, due to self awareness, loops to explaining the environment in personified terms. The shear brutality of nature expands that environmental personification into an immense being, a God.
This argument against an all encompassing being is not based on assumptions universally accepted. It depends in similar fashion to biblical beliefs upon the testimony of science theorists as against religious theists, but this testimony is a product of observation, calculation, and inspired thinking rather than ancient stories, superstition and assumptions. The conclusion is that our world has evolved to be what it is as the universes energy intensity steadily dissipates. The difference in belief structures is that if I choose to look, I too can see the evidence of the structure of the universe, I can personally witness the machinery of nature at work. To accept that a God is the creator of the universe and that universe’s singular purpose is to give human beings the field in which to interplay with right and wrong as a test towards achieving ultimate immortality, would require a very blinkered view of observable reality. To those who choose to believe in a God I suggest, you should “believe that if it gives you comfort”.
Simon: so, you’ve encountered other people making claims similar to the kind of claims you make. Do you believe those other claims?
You wrote that “Being voted as unpopular is one thing, being proven wrong is another.” What would it take to prove Christianity wrong to your satisfaction? Is that possible?
You also said that “Most people probably won’t vote someone with a very strong bias to analytical abstraction into power.” Do you think it’s a good thing that people who with an analytical bias will not get votes?
You write “ultra-rationalism for final truth is a self-deceived condition”. What is self-deceiving about it?
You finished with a call to trust God (the Christian one, I guess) and not try to understand things. Would you accept such a call to trust another god (rather than the Christian one) without thinking too analytically about it? If not, why would you expect people reading here to trust your god?
Hi JimV,
If this is what you are describing as Don’s experiment, it is a most strange idea (given that he says he is a Christian and supposedly believes in sin, free will, and all those other thing that go along with it, though he does say he is somewhat unorthodox).
He wrote:
“H_3: The first sentient experience (my actual one) has measure 1, and all the remaining ones have zero measure.
“H_3 corresponds to the model I mentioned in the quoted paragraph above, that only my present sentient experience exists. This gives P(E_1|H_3) = 1, whereas P(E_1|H_1) = 0.000001 and P(E_1|H_2) = 0.001. H_1 is a crude analogy for cosmological theories in which Boltzmann brain observations dominate, and H_2 is a crude analogy for theories in which the observations of ordinary evolved observers dominate.”
Now if your interpretation is correct, he’s denying free will because he’s saying that P(E1|H3)=1, or put in the terms of logic (as in my paper) H3–>E1. If H3 is true, then E1 is the only possible observation he can make.
But I keyed off his comment: “H_3 corresponds to the model I mentioned in the quoted paragraph above, that only my present sentient experience exists.” This is why I mentioned that he’s really saying that E1 has been observed, that’s the only present sentient experience of his that exists, so that he may have been committing the error I pointed out in my paper.
Here: I’ve just taken out a quarter and tossed it. It came up heads, and my present sentient experience is that I am looking at a head on the quarter. Does Don seriously think that my sitting here looking at a head was inevitable given his theory of the world? If so, why bother to do physics? For, one ought to be able to look at H3 and “predict” the outcome of any physics experiment by using P(E1’|H3)=1 or H3–>E1′ where E1′ is the physicists’s (Don’s) “next sentient experience” and evaluating this. Can Don do this? I doubt it.
I submit that this is pretty stupid.
So I am in agreement with Daniel’s earlier comment:
“Don, not to beat a dead horse here, but H_3 isn’t a hypothesis. You couldn’t come up with any functional form that represented it. It doesn’t assign any one measure to your probability space in the absence of data. It only specifies a class of hypotheses, namely the set H_3_i where i ranges 1 to a 1,000,000. It doesn’t predict which of the one million sentient experiences is your “present” one.”
I think that Don needs to rethink this line of argument. It’s going nowhere.
Hi Paul
I believe in supernatural realms, good and evil. I have what I consider to be a Biblical view on which is which.
An analytical bias probably describes me, most people who know me say so. But someone who only ever analyses with the mind is I think not functioning in the entirety of their humanity. They may also be slow and pedantic in making everyday decisions. To use your understanding is one thing, to lean on it for security and for all important decisions is another.
The hard line rationalist wants to minimize/eliminate all personal judgement and get to the ‘facts’. He does this by seeking immutable laws and consensus on them. I am fine with doing that up to a point. I see Christianity as the most plausible overall worldview with my rational mind. Competing views such as other religions and scientific naturalism, do not stack up for me.
However, it is one thing to think and analyse, it is another to continually and exclusively put your final faith in doing those things alone. ‘I trust nobody and nothing except through my mind’, would be, I think, a lonely and hard place.
‘You write “ultra-rationalism for final truth is a self-deceived condition”. What is self-deceiving about it?’
What I said. Ultimately rationalism undermines itself with the unreliability of it’s own precepts. It is a position of personal judgement in reality. No final objectivity is available to the mind alone.
To inject a bit of humor into this conversation, go here:
http://www.explainxkcd.com/wiki/index.php/1505
Without wishing to get personal, yes, people can hypothesize about why other people want to believe in God. We can move it into evolutionary psychology, or we can postulate a strong personal need and/or vulnerability to deception. I certainly at times get comfort from God, and at other times challenge. However I could postulate that a reductionists need for physical hard facts is driven in part by a difficulty in evaluating, relating to and trusting other people. Who is truly aware of their own motives?
The prehensions of ‘theories.’ It is evident, how- ever, that the primary function of theories is as a lure for feeling, thereby providing immediacy of enjoyment and purpose. Unfortunately theories, under their name of ‘propositions,’ have been handed over to logicians, who have countenanced the doctrine that their one function is to be judged as to their truth or falsehood. Indeed Bradley does not mention ‘propositions’ in his Logic. t He writes only of ‘judgments.’ Other authors define propositions as a component in judgment. The doctrine here laid clown is that, in the realization of propositions, ‘judgment’ is at very rare component, and so is ‘consciousness.’ The existence of imaginative literature should have warned logicians that their narrow doctrine is absurd. It is difficult to believe that all logicians as they read Hamlet’s speech, “To be, or not to be: … ” commence by judging whether the initial proposition be true or false, and keep up the task of judgment through- out the whole thirty-five lines. Surely, at some point in the reading, judg- ment is eclipsed by aesthetic delight. The speech, for the theatre audience, is purely theoretical, a mere lure for feeling. Again, consider strong religious emotion-consider a Christian medi- tating on the sayings in the Gospels. He is not judging ‘true or false’; he is eliciting their value as elements in feeling. In fact, he may ground his judgment of truth upon his realization of value. But such a procedure is impossible, if the primary function of propositions is to be elements in judgments.
By AN. Whitehead
Since Don seems to agree with the summary I posted:
“We observe a universe derived from consistently simpler laws, therefore in lack of absolute determining evidence, we should bias ourselves towards simpler theories. Since we also observe miracles (such as the resurrection) that would coincide with the existence of a god, we note that god has more explanatory power than naturalism which would require undue complexity (more free parameters) to explain true miracles. Meanwhile, god may conceivably remain quite simple while still explaining these miracles.”
I believe the following are probably the key issues moving forward in this discussion:
1) If both naturalism and theism contained within themselves equally viable explanations and predictive power for the ‘miracles’ we see, shouldn’t we bias ourselves towards naturalism? This is not to speak on whether either worldview does in fact act as a better explanation, but if we momentarily assumed equal validity, I think this serves as a good exercise to see if we may actually be on the same page about simplicity. For instance, we may all be actually agreeing about the application of Occam’s, just not which theory has the better explanatory power.
2) What makes the miraculous claims of Jesus different from other historical and modern supernatural claims in regards to the evidence available? I belief this is important as I presume Don would forfeit his Christianity if the resurrection of Jesus were sufficiently invalidated and that we should reconsider naturalism if it was sufficiently upheld.
If Don could respond to this when he returns, I think that’d be great. And best of wishes in the workshop.
Dr. Jefferys, thanks for your patience and for teaching me a valuable point about conditional probabilities. After painfully thinking through the example I gave above, “light dawned” that Dr. Page and myself may have indeed been making the error that you and Daniel Kerr have elucidated, depending on how H_3 is defined (as being pre- or post-measurement).
However, I am still having difficulty understanding how you interpret this thought experiment as being some kind of naturalism vs. supernaturalism argument (if that is what your last comment implies – as usual I may have misunderstood). My interpretation is that Dr. Page was trying to give a contrived example where Bayesian priors could overcome likelihood, not that he actually thought the example was meaningful in itself. I of course agree with the general thrust of your argument that there is no good reason in my opinion for anyone’s priors to be so biased against naturalism as to rule out the strong evidence we have.
Hi JimV, I was just responding to what Don wrote (and you clarified).
I certainly hope that Don just used this as an example, since it’s pretty flawed.
My consistent point (and Brent’s) has been simply that you can’t have the same pieces of evidence support two contradictory hypotheses. If it supports one it has to undermine the other, and vice versa. In particular, if a piece of evidence (the resurrection, for example) were to be reliably observed, it would refute the purely naturalistic hypothesis since it would confirm the existence of a god. Thus, evidence that supports the naturalistic hypothesis (we observe that physics predicts correctly in our lab experiments, for example, over and over), must undermine the supernatural hypothesis, if only by a little. I’ve explained this in earlier posts and won’t repeat the details.
This means that each independent piece of data that we observe that supports what physics predicts raises the likelihood ratio of naturalism against supernaturalism, and as we continue to observe more and more such data with no exceptions, that likelihood ratio gets as large as we wish and the posterior probability of naturalism as close to 1 as we wish, regardless of the priors. This is why I say that in this case, priors don’t matter, so long as they aren’t chosen to rule out any hypothesis under consideration.
Simon wrote: “I believe in supernatural realms, good and evil. I have what I consider to be a Biblical view on which is which.”
My question was “do you believe such claims?” I can’t work out from your response whether you do or not. Perhaps you do believe everything anyone tells you about a supernatural event?
As an aside, I guess it is important for people considering Christianity to realise that orthodox Christianity is much more than just belief in the existence of God, though: there’s all that stuff about angels and demons, as well as miracles like the resurrection. (I don’t know whether Prof Page believes in angels and demons. I’m aware that more liberal Christians tend not to.)
I noticed that you did not answer the question on how it would be possible to prove Christianity false to your satisfaction. Is that because nothing would change your mind? If that’s the case, why criticise others for failing to change theirs?
The position you describe as “ultra-rationalism” sounds like what’s been described as a Straw Vulcan (by analogy with a “Straw Man”). This is not a position I hold. I haven’t said that it’s bad to have emotions or to trust people other than myself. The reason I do not trust God is not because God isn’t me, but because there is no God to trust. Aside: This is the problem with the apologist’s statement that “you have faith that your spouse loves you”. Faith is pretending to know things you do not, but in the case of a spouse, you hopefully have good evidence (and so know, assuming your belief is true) that your spouse both exists and loves you.
What I would say is that I want my emotions to reflect reality. As it is written, ” If the iron approaches your face, and you believe it is hot, and it is cool, the Way opposes your fear. If the iron approaches your face, and you believe it is cool, and it is hot, the Way opposes your calm.”
I thought I’d better try to contribute to the main discussion having taken up so much space talking to Simon. wrt Bill Jefferys’s last comment, the relevant page from the Secret Bayesian Conspiracy is Conservation of Expected Evidence, where a similar point is made, “If you argue that God, to test humanity’s faith, refuses to reveal His existence, then the miracles described in the Bible must argue against the existence of God.”
(To link the two threads, by Prof Page’s lights, Simon’s description of miracles he’s done or heard of are evidence against the existence of God, if I’ve understood correctly.)
I was interested in Daniel Kerr’s comments about simplicity metrics, but my lack of pure maths lets me down. What would the universe have to be like for information theoretic comparisons between physical theories to lead us down the garden path?
Hi Paul
Missed this:
‘What would it take to prove Christianity wrong to your satisfaction? Is that possible?’
Seems to me God sometimes answers prayers and heart cries in spectacular and unexpected ways. Just did this weekend on a personal issue. So the question just seems strange to me. I don’t seriously expect it to happen. It would be an abstract thought experiment. It was largely the resurrection that gave me grounds in the first place to seriously consider Christianity, so I suppose that if it were conclusively or near conclusively disproved by some clear evidence? A large chunk of solid evidence proving that Exodus never happened, as many claim, would shake things a lot.
Another little bit of humor:
https://www.youtube.com/watch?v=nBuDoNl0XQI
Paul
“If you argue that God, to test humanity’s faith, refuses to reveal His existence, then the miracles described in the Bible must argue against the existence of God.”
Sounds more Douglas Adams than Bible.
God nowhere refuses to reveal his existence. Naturalism formalizes unbelief.
If God revealed himself to you unmediated, that would be you totally undone; the end of you.
Faith is designed to align you with the precepts of a realm where you can exist enjoyably eternally without being a problem to everyone else there. It is not God’s attempt to find a reason to exclude people. Miracles, in certain contexts, can help people to believe. They can also be explained away to the satisfaction of some. When God spoke to his incarnate Son, some say it thundered, John 12:29.
Oh please, millions of years of evolution didn’t make him a great football player. Let’s face it. We are a race of fat lazy apes. If evolution had anything to do with it, it would be because he was a genetic anomaly. Really, when is the last time you actually seen a man in real life that looked like he could become a professional athlete? Thankfully, the same thing can’t be said about women really, but it doesn’t take a rocket scientist to figure out why women have evolved more recently into becoming more physically desirable. There is no environmental reason why they would need breast outside of their gestation period, and there are no other mammals that have them outside of it…
Paul asked, “I was interested in Daniel Kerr’s comments about simplicity metrics, but my lack of pure maths lets me down. What would the universe have to be like for information theoretic comparisons between physical theories to lead us down the garden path?”
My issue is that the notion of a bit relies on the existence of something to process it. Usually it’s a Turing machine. A Turing machine “knows” what to do with a bit, a bit only has content because of what a Turing machine “does” with a bit. Even assuming the universe is a Turing machine or can be modeled by one, there’s lots of schemes to encode the information from formal logic, to set theory, to mereology, to arithmetic and so on.
Peano’s arithmetic is expressive enough to encode ZFC and vice verse, however the fundamental bits of information for the two schemes are quite different. ZFC takes sets while Peano’s arithmetic takes numbers. You can model ZFC in a base 2 modular arithmetic by representing each set as a countable string of 0s and 1s where each digit represents a possible element in ZFC. The empty set would be the infinite string of 0s. In this conversion you’re assuming a representation of the information, but ZFC on its own deals with sets and propositions, which are represented in themselves as logical atoms and operators. ZFC doesn’t know what to do with a countably long binary string unless its defined in terms of sets.
From ZFC’s point of view, the amount of information necessary to encode the binary string 10, for example, is the number of characters in ZFC’s alphabet necessary to define it. This would really be quite lengthy to write out as it’s the cardinality of the set, {null, {null}} written in terms of a decimal representation with {} and {null} as its corresponding 0 and 1. Clearly the full statement in ZFC is quite lengthy even in its most simplified representation. You would get the same information content as evaluated from arithmetic though if you somehow “subtracted” the information necessary to specify this arithmetic in set theory. Every binary string requires the same exact information to encode the representation scheme in ZFC, so one can accurately calculate the “Peano information” of a Peano string as represented in ZFC.
Now imagine you have more than just Turing machines for processing information. For a lack of an intuitive example, I would give whatever is behind the “random number generation” of a quantum operator, whatever decides is the exact position of a particle in a given stationary wavefunction during a certain measurement. The process can’t be a Turing process for then it would have a deterministic hidden variable theory. This process would have a different concept of a “bit” as well. I don’t know if it would necessarily be convertible to a “Turing bit” in the same sense a “Peano bit” was convertible to a “ZFC bit” as above. Keep in mind this is highly speculative on my part but I think is a big problem when considering cosmological “truth,” and using simplicity as a metric for “truth.”
me again Paul
‘My question was “do you believe such claims?” I can’t work out from your response whether you do or not. Perhaps you do believe everything anyone tells you about a supernatural event?’
No, it varies. The sangoma stuff I don’t know. They certainly command a lot of fear in parts of Africa; people believe in the effect of their curses more than anything.
You seem to feel your stab at faith was misguided and deluded, or maybe insincere, and you want your feelings to be genuine. Why not just tell God (in case he’s there and you’re wrong) all that? Seems to me many atheists are angry with God or a concept of God they have.
Ockham’s razor tells us that we should adopt the simplest theory that is compatible with our present knowledge, until evidence compels us to adopt a more complex theory. So, in my way of thinking, we would expect new evidence that does not conform to the theory we presently hold, to move us in the direction of a more complex theory, not a less complex theory. This is probably not a binding principle; I can think of a possible exception in the rejection of the Ptolemaic model in favor of the simpler Copernican one, and there may be others, but usually theories grow in complexity, not shrink. On the other hand, it could be pointed out that the earlier model didn’t have a compelling evidentiary basis in the first place and was simply adopted because of hidden assumptions that lacked foundation (i.e., “Earth is obviously stationary”) and a consequent inability to comprehend the bigger picture.
Now, I can see how we might find evidence to go from a theory N which holds that “naturalism explains everything” to a theory S that states “naturalism explains most things, but some events are due to supernatural forces”. I can’t really see how, if we start from belief in S, any form of evidence would compel us that S doesn’t work, and that we must adopt N. (If I’m right about that, then S is not falsifiable.)
In light of my interpretation of Ockham’s razor, then, it seems more credible to me that N-to-S is a progression from simpler to more complex, not the other way around.
Richard
I think I am inclined to agree with you.
But is it not possible that N reduces down to some simple (compared to Standard Model and General Relativity) GUT equations, but these turn out to be mathematically ‘permeable’ to (at least some of) the supernatural aspects of S?
OK I confess. This is me doing reductionist thinking.
That last one was is in case Don and Stephen Hawking work it all out over the weekend.
After Bill Jefferys and Daniel Kerr continued to misunderstand my example, I think JimV partially cleared it up, but there are still some things I should have explained about my example with the three hypotheses. This gets longer than I hoped, so if everyone realizes how important the prior probabilities are in determining the posterior probabilities, they can skip down to the final paragraph and my conclusion that one cannot rule out the Resurrection purely by likelihoods, but if you have the illusion that it is sufficient to consider the ratios of likelihoods, read on to see how this can be grossly wrong.
First, the example was not meant to be of hypotheses I actually believe in (though H_2 was meant to represent the best of the three), but an example to show how prior probabilities can have a huge effect on the posterior probabilities. H_3 was so absurd that many of you could not believe that I was proposing it, and indeed I was proposing it only as an example of a hypothesis for which I would assign a very low prior probability. None of the three hypotheses were meant to represent hypotheses for theism or miracles, but just three qualitatively different hypotheses one might make in ordinary science. (I was trying to show how using both priors and likelihoods it might be reasonable to rule out both the Boltzmann brains hypothesis H_1 and the crazy hypothesis H_3 that only one observation has nonzero probability.)
So let me try to explain my example in more detail. To get actual finite positive probabilities that were not enormously close to zero or one, I made the highly unrealistic assumption that there were only a million possible observations possible. (I personally tend to think that the most basic observations in our universe are sentient experiences, which I also call conscious perceptions, meaning all that one is consciously aware of at once, but you can take whatever you think an observation is. One alternate model would be the integer that represents the content of the internet, or perhaps of the preprint arXiv used by physicists. I just am assuming a discrete set of possible observations, each distinguished by its content.)
If one did use sentient experiences, I have no good idea of how many there actually can be; I would assume surely an infinite number of different ones if one allowed brains of arbitrarily large complexity (which would probably require different laws of physics, or at least different coupling constants, from what we have in our part of our universe, though if our universe is a multiverse, there presumably are regions with brains much more complex than ours; I might assume that the measure for this is not overwhelmingly larger than for our part, or else it would be surprising that the complexity of our sentient experiences is not even much larger than what we observe).
I am only consciously aware of a very small part of the information that my brain processes. I would be interested if there are estimates of how many bits there are in what I am actually aware of at a typical time when I am awake and alert. I might guess that it is of the order of a thousand, but it might be lower than one hundred or as large as in the millions. (or it could even be infinite, if there are actually a continuum of different sentient experiences, though for a Bayesian analysis without having to bin observations, for simplicity I shall assume there are a countable number of discrete possible observations, which I shall put into one-to-one correspondence with the positive integers). If there are a thousand bits in a typical human sentient experience, this would lead to about 2^1000 possible human observations, much larger than the one million I used in the example just to keep the numbers more manageable there.
Under the hypothesis that Boltzmann brains dominate, one might guess that most of the possible observations would be roughly equally probable, say all of the 2^1000 possible human ones if indeed they can be described by a thousand bits, or all of the million in my simplified example, in which I represented the hypothesis of domination by Boltzmann brains by H_1. On the other hand, under the more usual assumption that `ordinary observers’ dominate (ones that are assumed to arise by a long period of evolution), then it it plausible to assume that most of the probability would be for much more ordered observations, say compressible to one hundred bits (to make another wild guess at the number), giving, say, 2^100 ordered human observations, which I represented by the hypothesis H_2, in which only one thousand out of the million total possible observations had nonzero (and, for simplicity, equal) probabilities. Of course, the ratio of one million to one thousand is only 10^3, which is much less than the ratio of 2^1000 to 2^100, which is 2^900 or about 10^271, so I enormously reduced the difference between a Boltzmann brain dominated hypothesis and an ordinary observer dominated hypothesis in representing these as H_1 and H_2.
I chose my third hypothesis not to represent any seriously considered hypothesis but rather to show what one would get with a maximum likelihood hypothesis. The maximum possible likelihood is unity, and so a maximum likelihood hypothesis (with no further constraints on the hypothesis) is one that gives unit probability for the observation used to test the theory. This of course requires zero probability for all other alternative possible observations (not simply because I got whatever observation I did, but because I am choosing the hypothesis to maximize the probability of the observation that I did get). We all know this is crazy, but I want to show why I think it does not lead to a high posterior probability for this hypothesis when we do a Bayesian analysis.
I regard the craziness of H_3 to be reflected in the complexity of the observation E_1 used to test the theory, which I regard as more complex than either the very simple hypothesis H_1 (all observations equally probable) or the observation H_2 (all observations of ordinary observers equally probable, but no probability for observations that ordinary observers would not have). H_3 is more complex than H_2 essentially by the specification needed to pick out the observation E_1 from the set of possible ordinary observations. If there are 2^100 ordinary observations (or 1000 in my simplified model), then one might expect roughly half of them (2^99 or 500 respectively) to be simpler than E_1, so there might be about that many hypotheses more complex than H_2 but less than H_3.
Now even if one can order hypotheses by their complexity and use Occam’s razor to assign lower priors to more complex theories, there is still the further subjective ambiguity of how to do this mapping from complexity to priors. One simple, but ad hoc, way to do this mapping is to assign the nth simplest theory the prior probability 1/2^n, which of course obeys the normalization condition of summing to unity if n runs from 1 to infinity for a countably infinite number of hypotheses or theories. Then one might take the prior probability of H_3 to be that of H_2 multiplied by 1/2^(2^99), which is extremely tiny. In my simplified model, I made the prior much larger, P(H_3) = 0.000001.
But even in my simplified model with much less suppression of the prior from the complexity of the maximum likelihood hypothesis H_3, I still got that its posterior probability P(H_3|E_3) was only about 0.00001, far smaller than P(H_1|E_1) = 0.00892 and P(H_2|E_1) = 0.99107. Thus if one assigns a sufficiently lower prior probability to a complex hypothesis like H_3, even though it has the maximum possible likelihood of unity, its posterior probability can be a lot lower than that of other simpler hypotheses that have lower likelihoods. In fact, if I used my very crude estimate that there are 2^99 possible observations simpler than E_1 and set the prior probability of the nth simplest hypothesis to be 1/2^n, I get that the posterior probability of the maximum-likelihood hypothesis H_3 is of the very crude order of 1/2^(2^99) ~ 1/10^191,000,000,000,000,000,000,000,000,000, which is a decimal point followed by about 191 billion billion billion zeros and then a 1, which is extraordinarily tiny, showing how crazy the maximum likelihood hypothesis H_3 is.
I was not drawing any theistic conclusions directly from this, but just illustrating the fact that in general the prior probabilities are very important, and not just the likelihood ratios (which would have greatly favored the crazy hypothesis H_3). If I do apply this to considerations of the hypothesis that there is a God who resurrected Jesus, it seems to me that since this hypothesis has high likelihood (so that IF it were true the probability of the reports of seeing Jesus alive again after His crucifixion would be high), one can only say that its posterior probability is low if one assigns a low prior probability to the resurrection hypothesis.