Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Monday, June 23, 2025

Bayes and God

 

Bayes: A man of the cloth


There is a long tradition of Bayes theorem being used in discussions about the probability of God. I've never been very keen on using Bayes to "prove" God's existence: I expressed my reservations in this short paper where I discussed the use of Bayes Theorem by Christians Roger Forster and Paul Marsden in their book Reason and Faith. In this connection, however, I noticed this post by Christian Blogger James Knight where once again we see God and Bayes appearing together. 

Well, in this instance I didn't want to get embroiled with the subject of God and Bayes, but in my correspondence with James I picked up on a technical issue which obliquely impinged upon his post. 

The theorem that interested me can be expressed as follows....

If 

                                                                        P(A) < P(A|B)

....then it follows that....

                                                                  P(B) < P(B|A)


....where P(A) and P(B) are the unconditional probabilities of A and B respectively and P(A|B) and P(B|A) are the respective conditional probabilities of A and B.

As per my practice in my paper on randomness I'm going to use Venn diagrams. But Such an approach implicitly assumes my frequentist interpretation of probability, an interpretation I won't attempt to justify here. 

In terms of a Venn diagram the relationship of A and B will in general look something like this....


Here the area labelled A represents the set of possible cases with property A and the area B represents the set of possible cases with  property B. This Venn diagram is imagined to reside in a large domain of a total number of possible cases of T.

Now, if N(A) = number of cases with property A, then the unconditional probability of A is given by P(A) where... 

                                                                            P(A) = N(A) / T

If the number of cases with property B is N(B) and the number of cases where B and A overlap is expressed as  N(A|B) = N(B|A), then the probability of A given B, P(A|B), equates to....

                                                                     P(A|B) = N(A|B) / N(B)

Now we postulate that:

                                                                            P(A) < P(A|B)

 Expressed in frequentist terms we can write that as.....

                                                                   N(A) / T  < N(A|B) / N(B).

We now multiply both sides of this inequality by N(B) and this gives......

                                                                    N(A) N(B) / T  < N(A|B)

Now divide both sides of the latter inequality by N(A) and this returns.

                                                                N(B) / T  < N(A|B) / N(A)

But N(A|B) = N(B|A) and so the above inequality becomes....

                                                                N(B) / T  < N(B|A) / N(A)

Expressed in terms of probabilities the latter inequality can be written as.....

                                                                        P(B) < P(B|A)

....and this inequality has thus been proved from our first postulate which was... 

                                                                            P(A) < P(A|B)

In other words:

                                                        P(B) < P(B|A)  =>   P(A) < P(A|B).


*****

James was concerned that the apparent symmetry of this result is contrary to his intuition that the general case is far from symmetric. However this intuition of asymmetry is backed up by the following special case where we have....


From this diagram we see that B=>A (i.e. B implies A with certainty). But clearly given A the probability of B, depending the relatives sizes of the two sets A and B, may be quite low. This may be the kind of asymmetry that James is thinking of. 

Thursday, March 06, 2025

The Aumann's agreement theorem paradox.


Different perspectives implies a likelihood of different experience
sets and conflicting probability estimates, therefore setting 
 the scene for potential disagreement. 


I was rather intrigued by James Knight's use of Aumann's agreement theorem in a blog post of his that can be found here:

The Philosophical Muser: Why Christians Disagree So Much

James' post was a response to a challenge put to him that If Christianity is true, why are there such a varied set of Christians who disagree and squabble about so much? Towards the end of his article we read this (My emphases)....


Given the state of humanity, I’d no more expect Christians to agree on everything than I would mathematicians to agree about politics, or opera singers to agree about economics. But, I do wish they would – and as I often argue – Christians SHOULD agree more, especially on objective things – and two Christians of any sex, ethnicity, denomination, should converge on more and more consensus if they were to sit by the fire, Aumann’s Agreement-style, and honestly, rigorously seek the truth together, like people who care about what is true."


The exact "science" of Mathematics is a domain of knowledge incommensurable with politics and economics and no easy like-for-like comparison can be made. Mathematics is an activity, in fact a form of model building, which depends on very strictly agreed symbolic conventions and algorithmic procedures being followed. If in mathematics every one keeps strictly to the same conventions and procedures disagreement can't arise. The progress of mathematics bares this out; I'm not aware that mathematicians frequently and fundamentally disagree (except perhaps about un-proven conjectures). This of course is not so of politico-economics; disagreements about best economic policy and its political implementation abound. This why economics is a breeding ground for politicians and their political passions & power seeking; after all the only way to implement a particular contentious economic policy is to get political power (Hopefully by democratic means). But let's not think any better of those sanguine mathematicians over and against those battling politico-economists - the latter are dealing with very complex and epistemically tricky material which as we will see provides one reason among others why politico-economics breaks the assumptions of Aumann's agreement theorem and promotes the sharp divisions of power politics.....and that's before we consider those ever present very human psycho-sociological factors which one expects of complex adaptive systems like human beings. 

In fact I would rewrite the first sentence of the above quote as follows....

Given the natural state of human affairs, I’d no more expect Christians to agree on everything than I would politicians to agree about politics, or economists to agree about economics. 

***


Aumann's agreement theorem assumes we have a set of interlocutors who start with a common information base ("common priors") but then these interlocutors bring to the discussion table differing levels of knowledge in the form of conditional probabilities that all interlocutor believe to be a trustworthy contribution to the discussion (So-called "common knowledge" as opposed to "common information"). The interlocutors update their probabilities by mutual cooperative sharing of their differing conditional probabilities (*1).  They assume one another to be rational honest agents and that they can trust one another's probability estimates as they share them. According to the agreement theorem they will eventually converge on the same information set. See the following link for more on the agreement theorem: Aumann Agreement - LessWrong

Before I go any further let's get one thing straight. Most common sense people (which includes myself and people who believe there is such a thing as a single truth out there which stands over and above the slippery slopes of cultural relativism and critical theory) have an intuitive grasp of Aumann's theorem; that is, they understand that in an epistemically transparent world where evidence acquisition is not an issue and interlocutors are rational and honest, then agreement  about truth will inevitably emerge. So the agreement theorem proves what most common sense people already believe (Of course critical theorists and cultural relativists are likely to make heavy weather of this common sense truth). Aumann's theorem is a nice confirmation of what all reasonable people already know intuitively. But the article on Aumann's theorem that I have linked to above ends with this warning: 

The fact that disagreements on questions of simple fact are so common amongst humans, and that people seem to think this is normal, is an observation that should strike fear into the heart of every aspiring rationalist.

So, given the agreement theorem which is undoubtedly mathematically correct why is disagreement between humans so widespread? In this connection I made the following comment on James' blog entry. As a rule my comments never get past the Philosophical Muser's approval process and are therefore cancelled (The Philosophical Muser's concept of "free-speech" is qualified) So, rather than let my comments disappear into oblivion I thought this matter to be so important that it needs airing. What follows in the next section is based on the comment I added to James' post...

***

I think I agree with the general drift of your argument here but not in one or two of the details; especially, may I say,  you are missing the crucial point of the agreement theorem and vastly underestimating the epistemic issues impacting attempts to get agreement.

That cozy fireside talk seldom, if ever, arises. For a start whilst our interlocutors are locked in by the fireside they are not updating their experience or accumulating any further experience. They have to try and get agreement on the basis of the information they already have (in the form of priors and conditional probabilities). If this pool of information contains contradictions and they insist on sticking to their scripts they won’t necessarily reach full agreement even if they are rational.

Therefore our interlocutors are going to have to get off their backsides, get out on their bikes and find a set of consistent priors and conditionals. But that brings us to the main problem: This information can only come from statistics which result of a wide and long term experience of the cosmos. Moreover, any mature engagement with that cosmos requires thousands if not millions of bits of information. Single interlocutors, therefore can’t survey the whole lot; ergo, their experience is liable to being skewed and/or very partial. So, unfortunately our interlocutors, on top of all their other very human survivalist social traits, have to face the epistemic problem of systematic and random sampling errors.

The agreement theorem simply sets a lower limit on agreement time. That is:

Agreement time >= Aumann agreement time:

 

As I found out with my own AI Thinknet project AI systems also suffer from similar epistemic problems relating to sampling bias and partiality. After all, I think the YEC organization AiG have implemented their own YEC AI interlocutor presumably by training it with a bias on YEC texts.

As I’ve said before because of these fundamental epistemic limitations tribal identification & group think where one outsources epistemic help to the experience of a large group of minds is an adaptive trait and this factor shouldn’t be underestimated in terms of its potential epistemic utility. So what James refers to as “incentives, needs and agendas” have the potential to be adaptive whether we like it or not.

So, even without factoring in those many awkward human social foibles (which potentially have adaptive value), epistemic challenges alone are very likely to lead to agreement failure. My guess is that disagreement due to epistemic issues is the biggest factor in disagreement. The only antidote I see for this is epistemic humility. But the trouble with this is that when faced with utterly convinced  group-think such as we find in AiG & Trumpite brands of Christianity epistemic humility & tentativeness is read as weakness. Hence, a certain amount of vehemence is demanded in the heat of argument.


***

Olber's paradox was a famous theorem in astrophysics. This paradox shows that under plausible assumptions the night sky should not be black but a continuum of bright star light; the fact that this isn't the case pointed to the need to revisit the underlying assumptions; it was a profound piece of theoretical thinking which lead the way to our understanding of an expanding universe. I contend that likewise Aumann's theorem  prompts us to think a bit deeper as to why it's not a real world model; in particular it urges us to think about both our epistemological limitations and the complexities of socio-psychology which strongly influence the acquisition of knowledge. With respect to the latter we are prompted to investigate the adaptive value of group think & group belonging along with its potential downsides and tradeoffs. Because Aumann and his successors are making us think harder about human affairs then like Olber's paradox its pedological value can't be underestimated.

The upside of group think is that it widens the number of experiencing agents contributing to the conversation and this increases the amount of incoming evidence. It's true, however, that the instincts behind group think have a big potential downside as group-think can lock in error such as we see among cultists and fundamentalists who exploit the adaptive instincts; in this context the survival of the group identity takes precedence over further evidential updates. Aumann's theorem prompts us to study the cost/benefit balance entailed by joining an epistemic group with a strong sense of cohesion and collective identity. In this sense Aumann's paradox is as profound as Olber's paradox. 

***

I would want to rewrite the second half of the quote at the beginning of this post which I took from Philosophical Muser along these lines....

Christians are expected to make heavy weather of agreement, even about on objective things – and two Christians of any sex, ethnicity, denomination will not necessarily converge to a consensus if they were to sit by the fire, Aumann’s Agreement-style attempting to get convergence; disagreement is likely even if they honestly, rigorously and rationally seek the truth together, like people who care about what is true."

The Agreement theorem tells us that in principle agreement is possible if we get our priors and evidences right, but therein lies the epistemic challenge of gathering huge amounts of data some of which may present accessibility problems; this epistemic challenge necessitates that the quest for knowledge becomes a social symposium and this cues in all the foibles of the sociological dynamic. That these human and epistemic factors can make agreement problematical should always be at the back of our minds and therefore our difficulty in conforming to Aumann's theorem SHOULD be the basis of an attitude of epistemic humility rather than thinking that Aumann's theorem underwrites an attitude of epistemic arrogance; in my books this classifies as an abuse of the theorem. Agreeing to disagree until more information comes to light should not make us shudder.

But let me repeat and finish with this warning..... 

The only antidote I see for inevitable disagreement is epistemic humility. But the trouble with this is that when faced with convinced group-think such as we find in AiG & Trumpite brands of Christianity, humility is read as weakness. Hence, a certain amount of vehemence is demanded in the heat of argument.

Disagreement, sharp disagreement in fact, seems to be the natural state of human affairs.


Footnotes

*1 A conditional probability has the form "The probability of A given evidence B is P"; formally expressed as P(A/B). Here B is the evidence relevant to the truth of A. 

Thursday, November 21, 2024

Let's Carry on Carriering Part II




In Part I of this series we were left with Richard Carrier's entirely plausible assertions; here is a summary of them...


That which is logically impossible can never exist or happen......It’s really hard to fathom what one could even mean by saying logical contradictions can obtain in the real world,  (that is, contradictions are unintelligible notions - I agree!)

The most nothingly state of nothing that can ever obtain, is a state of affairs of zero size lacking all properties and contents, except that which is logically necessary.

Nevertheless, the very notion that logically necessary things necessarily exist, necessarily entails logically impossible things never exist. Because one of the things that necessarily exists, is the absence of logically impossible things.


Nothing wrong with any of that as far as I can see. He then goes on to conclude that if the demand for absolute nothingness necessarily leaves a hard kernel of logical necessities, the absence of these necessities would entail a logical contradiction. Again, I agree. But Richard then draws this conclusion:


Therefore we no longer need gods to explain why there is something. That there would be something is logically necessary....... it is logically necessarily the case that there will be something, even without gods


But as I pointed out in Part I in stating the above Richard appears to have missed the obvious: 

1. Theists, whatever they may think they are telling us are not actually saying that the universe came from nothing since their starting point is, of course, a god of some sort.

2. Richard doesn't tell us much about this irreducible kernel of logical necessity and in particular why it might not be inclusive of that dreaded backdrop of divinity which many theists claim has the property of aseity even if, I concede, the ontological arguments that have been advanced so far are flawed. 

Of course, Richard isn't going to buy this divinity line, but I just want to point out that his conclusion (based on the necessity of logical truths) that "Therefore we no longer need gods to explain why there is something" doesn't yet stand on a firm logical foundation. I have the distinct feeling that Richard is motivated by a strong a priori desire to conclusively eliminate divinity from the inquiry come what may. 


***

Now let's continue where I left off in Part I.  Below is Richard's proposition 3. This proposition makes use of what Richard defines as "Nothing", that is "nothing" spelt with a cap N. When all mere logical possibilities have been removed, "Nothing" is the hard core of logical necessities that are left - removal of Nothing would be a logical contradiction whereas the non-reification of mere logical possibilities (or contingencies) is not a logical contradiction. But apart from this we know very little about the details of "Nothing"; Richard doesn't supply any of these details. All he tells us is that "Nothing" has no content and governed by no rules or laws except the laws of logical necessity. That's a very big "except" there! He doesn't yet tell us what these laws of logical necessity are but he's going to "ask what predicted observations this hypothesis entails and how well it accords with what we see". I'm looking forward to his conclusions! 

Without further ado here is Richard's proposition 3 (My emphases are in bold):

***

· Richard:  Proposition 3: If there was ever Nothing, then nothing governs or dictates what will become of that Nothing, other than what is logically necessary.

This is true by definition, once you accept Proposition 2. So there is no logically consistent way to deny this Proposition without also denying Proposition 2. In fact Proposition 3 is just a restatement of Proposition 2 with respect to the specific absence of “rules” and “properties.” It is logically entailed by that absence, that when there is Nothing, there are also no rules or properties that dictate what will happen to that Nothing or what that Nothing will do.

Which also means the total absence of physical laws. So all cosmology papers arguing for a universe from nothing are invalid for the condition of Nothing, as those papers depend on the existence or operation of certain physical laws or properties. See, for example, this point as made in 1987 by W.B Drees in “Interpretation of The Wave Function of the Universe,” International Journal of Theoretical Physics 26. Only if some such paper proved the physical laws or properties they depend on are logically necessary would they become applicable to Nothing. They could, for instance, someday show how denying that that physical law applies to any state of affairs (even a Nothing-state) entails a logical contradiction. But I am not aware of that having been done.

My Comment: ....yes, and neither am I. Again, I largely agree. In fact, this point has already been made by Richard in Part I, and I agreed: Current physics, even at its most minimalist, assumes the existence of transcendent laws which are not nothing. They are in fact descriptive contingencies about the reified patterns of behavior we observe in the cosmos. 

However, I would like to raise a query here: If the laws of physics are, as far as we know, descriptive contingencies (i.e. they are not logical truisms as far as we know) about cosmic patterns of form and behavior, do they meaningfully exist if they had no reified instantiations to describe? I contend that these descriptive laws have no meaningful existence if they had no instantiations which they usefully describe. Take for example Newton's laws: As we know these laws don't work for atoms and high velocities; that is, they are not fundamental truths of nature. And yet in their domain of applicability these laws are still a very useful descriptive device. They have no fundamental or transcendent existence and so their existence is conditioned on their ability to describe the reified cosmic reality we observe around us, and within limits these laws still work very well. Ergo, Newton's laws are only meaningful because of their descriptive success, and they have no fundamental transcendence. Likewise, I propose that all laws of physics have no transcendent existence; they are only meaningful if they successfully describe a reified reality; they are very human centred computational devices.  

In his comment above Richard says that someday it might be shown "how denying that physical law applies to any state of affairs (even a Nothing-state) entails a logical contradiction. But I am not aware of that having been done." Neither am I aware of this having been done. But the crucial point here is this: Richard is hinting that a proof of the logical necessity of our cosmos may yet be found: I'll be coming back to this admission shortly. 

But one thing I can be clear about is this: If physical laws are merely algebraic algorithms which in a compressed form successfully describe our reified patterns of observation, then we won't ever find logical necessity in our current physics which describes those patterns in terms of equations: Descriptive equations cannot be compressed to nothing; they necessarily start with a kernel of contingent digital information. Any attempt to prove their logical necessity with further equations may lead to incrementally more succinct descriptions but it will also lead to an "equations all the way down" regress.  We will not get logical necessity out of our current method of description which uses descriptive equations.

***


· Richard: Proposition 4: If nothing governs or dictates what will become of Nothing (other than what is logically necessary), then nothing (other than what is logically necessary) prevents anything from happening to that Nothing.

This is again true by definition. It’s what follows with logical necessity from saying nothing governs what happens to Nothing; because Nothing contains nothing, not even rules or properties that would limit what Nothing can do. So you cannot deny Proposition 4 without denying Propositions 1, 2, or 3.

My comment: But Nothing doesn't contain nothing; it contains what is logically necessary and just what is logically necessary hasn't been clarified by Richard.  In fact, as we have seen above Richard himself has even admitted that the following may be the case.....

Only if some such paper proved the physical laws or properties, they depend on are logically necessary would they become applicable to NothingThey (i.e. physicists) could, for instance, someday show how denying that that physical law applies to any state of affairs (even a Nothing-state) entails a logical contradiction. But I am not aware of that having been done.

That is, Richard is telling us that although he doesn't know if anyone has done the work, as far as he is aware our physical regime may be a logical necessity - in short, Nothing could conceivably be very content laden. (As I've said above if this was true however, this could not be proved via descriptive equations because this always leads to an incompressible kernel of digital information of an "equations all the way down" regress).  Therefore, as far as Richard is concerned it is conceivable that Nothing is far from being empty; for if our physics is a logical necessity as Richard at least allows it means that Nothing is not devoid of rules or properties that would limit what Nothing can do. So, by Richard's own logic it is at least conceivable that Nothing is very much Something.  That Something could be very significant indeed as it not only explains why absolutely nothing is a logical impossibility, but conceivably also explains an immense and highly organized cosmos.

***


Richard: This entails that the assertion ex nihilo nihil, “from nothing, comes [only] nothing,” is false. Because that is a rule, and Nothing contains no rules. No such rule can therefore exist when there is Nothing, so as to govern that Nothing. Therefore it cannot be the case that only nothing comes from Nothing. In fact we cannot even establish that it is likely that only nothing will come from Nothing.

My Comment: Nothing contains no rules? But as we've just seen Richard has admitted that Nothing could contain rules that are an outcome of logical necessity. 

On the basis that theists are starting from a sentient entity (which they presumably believe has the logical property of aseity, even if that can't be explicitly proved) their starting point is not nothing in the absolute sense. I think Richard and I would actually agree that "absolutely nothing" is likely to be a logical contradiction and therefore it would follow that ex-nihilo creation is not just false but an unintelligible notion; that may also apply to ex nihilo nihil. Of course, I can't expect Richard to believe an a priori sentient being with the property of aseity to be the starting point, but it seems to me that he is building up to an idea that the source of creation (= Nothing) is far from being a trivial entity. 

***


Richard: The only way to challenge this is to disprove Proposition 4. And the only way to disprove Proposition 4 is to prove that it is logically necessary that only nothing come from Nothing. I know of no such proof. None has ever been produced. Not even after over two thousand years of philosophy. There is not only no proof that it is necessarily the case that ex nihilo nihil, there is no proof that that’s even an expected outcome.

My Comment: Once again, I have no objections here. But bit by bit it is becoming clearer that Richard's notion of Nothing is potentially a very sophisticated object indeed and could well be the seat of the origins many of us are seeking. As for ex nihilo nihilif as seems likely to be the case absolute nothingness is an incoherent notion, then in terms of absolute nothingness the phrase ex nihilo nihil will also be incoherent. But this phrase does make some sense if we are talking about the patterns of behavior of our own world. In our world antecedent conditions constrain the possibilities open to future conditions.  Therefore, although I don't accept that there is any such thing as a logically true law of "cause & effect", the dynamic of our world is such that "cause & effect" is valid as a synthetic heuristic.  Hence, in our world something always comes from an antecedent something where that antecedent something is a blend of initial conditions and dynamical rules that at least constrain the possibilities which arise from these antecedents. So, if we have neither initial conditions nor the dynamical rules nothing can be inferred as an outcome; that is, from nothing comes nothing in this relative sense. It is conceivable that this synthetic rule is not always true, but it is a good heuristic.

***


RichardIt won’t do to say “but we don’t see that rule being violated anywhere now,” because we do not observe Nothing anywhere—everywhere there is something (an expanded spacetime, with contents and properties, governed by now-existent physical laws)—so none of our observations apply to Nothing. In fact, as Nothing entails the total absence of “contents and properties and physical laws,” the very reason we do not observe a violation of ex nihilo nihil is that those extant properties and laws now prevent “just anything” from happening. The only nihil we observe is actually a thing: propertied spacetime. And that thing, being existent, now limits what can happen.

 My Comment: I think the foregoing is Richard's way of saying something very similar to my last comment. True, we don't observe this enigmatic object he calls "Nothing" and which I think in the final analysis is very much Something, in fact Something very significant indeed. 

***

RichardEven insofar as we do observe the violation of ex nihilo nihil, indeed all the time now, in the spontaneous creation and destruction of virtual particles resulting from quantum indeterminacy, this is a highly constrained and ordered violation. It’s governed by limits, laws, and rules. You don’t just get rabbits and deathstars popping in and out, much less then sticking around. Yes, there actually is a calculable quantum probability on present physics of a rabbit or a deathstar popping into existence spontaneously; but it’s an absurdly small probability, because what can and can’t happen now is constrained by the possibilities allowed and disallowed by the specific spacetime we inhabit and its qualities

 My Comment: No problems with that either. In particular we can agree with Richard's comments about our own highly (statistically) constrained universe, so constrained, in fact, that it means we can rule out many patterns on the basis of their extremely low probability. The information that allows us to do this is implicit in the laws of physics which we think we know. 

But from here things start to go badly wrong for Richard (my emphases in the following)...

***

RichardBut when there is Nothing, there is no spacetime (much less the specific kind we inhabit) other than a dimensionless point of it, and no governing qualities. So, indeed, there can be not just Boltzmann brains but a Boltzmann anything on present physics (as I’ve discussed before). But when even the constraints that make such things unlikely don’t exist anymore, all Boltzmann things necessarily become far more probable—not less. An actual Nothing is therefore even more likely to randomly create rabbits and deathstars. This is a logically necessary fact, that follows necessarily from the fact that when there is Nothing, that which keeps the probability of such outcomes low no longer exists, and therefore nothing remains to keep that probability so low. It doesn’t follow that it’s therefore then a likely outcome. It may indeed still be an absurdly low probability (and I dare say surely is). But it will be so only if, and only because, it is logically necessarily so. And not because of any other rules, laws, or physics.

The principle point is that Proposition 4 entails the probability of Nothing spontaneously becoming anything is not zero. It logically cannot be zero. As it only could be if something existed to stop that happening. And by definition nothing exists when there is Nothing to stop that Nothing from becoming something else. And note that whatever then happens will also be totally uncaused, except insofar as it is caused by Nothing itself. Because whatever happens will be uncaused by anything whatever except the logically necessary fact that Nothing cannot limit what comes to exist. As being Nothing, it lacks any forces or constrains to limit what happens.

 My Comment: Lots of problems with that. This is where Richard's thinking comes off the rails spectacularly. This train crash is aptly summed up in this logically pathological statement: 

The principle point is that Proposition 4 entails the probability of Nothing spontaneously becoming anything is not zero. It logically cannot be zero

Why? My interpretation of what he is trying to tell us here is that Nothing has no rules and therefore the probability of something arising from Nothing is finite, albeit very small. At this point Richard's flawed concepts of probability and randomness kicks in and he uses it to do an enormous amount of heavy lifting. He takes it for granted that if you've got a probability then you've got a randomness generator/creator. No, you haven't!

OK we've admitted the logically necessary existence of Nothing, but we know very little about Nothing; it may have logically necessary rules about which we know nothing; essentially it is an Unknown. In fact, as we have already seen Richard himself has admitted that the cosmos itself in all its rule laden regularity and order just may be an outcome of some unknown logical necessity. Moreover, it may be ... and I'm gingerly looking left and right at this point.... that Nothing is inclusive of a very significant sentient object; no prizes for guessing what I mean by that. In fact that would be no surprise to a philosophical idealist like myself - to me the constructions of physics make no sense without the a priori existence of an up and running sophisticated conscious cognition that conceives, perceives and creates (not necessarily in that order) a rational world on its mental matrix.

But, and this is Richard's whopper of a train crash....just what does he think he's doing with statements like this? (again, my emphases).....

An actual Nothing is therefore even more likely to randomly create rabbits and deathstars. This is a logically necessary fact, that follows necessarily from the fact that when there is Nothing, that which keeps the probability of such outcomes low no longer exists, and therefore nothing remains to keep that probability so low.

Probability and randomness - which are not identical concepts - are both highly sophisticated notions; they are not logical necessities that we can use to underwrite some half-cock notion of creation from Nothing. 

Firstly probability. Probability is function of an observer's information level, the well-known ratio of possibilities, possibilities that as far as the observer's knowledge/information is concerned are in the running as possible outcomes, or possible states of affairs. Therefore, probability only makes sense in the context of a cognating object, an object with sufficient cognitive sophistication to apprehend the options that are in the running for reification.  What probability certainly is not is a dynamic capable of generating anything. If somebody presents me with two inverted cups on a table and tells me that if I select the right cup a tasty rum-fudge is waiting for me then in the absence of further information as far as I am concerned the probability of the rum fudge being under each cup is 1/2. That probability is not a dynamic which generates anything; in this instance the dynamic has already happened - namely, the player who set up the puzzle has already done the deed and my subjective probability of a 1/2 doesn't generate anything new until I select a cup and my personal information changes.

Secondly randomness: Randomness is a class of pattern and a very sophisticated pattern at that: It is a pattern which foils any attempt to generate it using small space, short execution time parallel algorithms with a better statistic than evens. I've done my own independent work on this subject to make sure I thoroughly understood it. (Professional mathematicians have also solved this problem no doubt more professionally than myself, but the conclusion is the same). True, a random source is a generating dynamic but it is a very sophisticated dynamic. 

Richard appears to have conflated randomness and probability. The existence of probability doesn't necessarily imply randomness and vice versa the existence of randomness doesn't necessarily imply a probability. Probability is not a dynamic, it is a measure of the information available to a sufficiently cognating agent about circumstances that could well be very static and very finite. Randomness is a pattern generating dynamic but if the results of the dynamic are stored and memorized in a book of random numbers that output is no longer probabilistic to the holder of the book; but the configuration of numbers is still random.

But the main point is this: Randomness is no simple logically true axiomatic object that we can take for granted as a creation dynamic; it is by definition a highly complex phenomenon with certain mathematical properties and unless we are party to the right (large) algorithms the best we can do is use probability calculus, a calculus which betrays our minimal information about the highly complex object in hand.  Again, randomness is not identical to probability, but the algorithmic epistemic intractability of randomness makes it difficult to know and hence we may well have to fall back on probability calculus when dealing with randomness. 

Richard seems totally unaware of his huge fallacious leap of logic here. For him it seems from the admission of a probability (that is a lack subjective information) he wrongly concludes that it logically follows he then has a randomness generator on his hands. That is a false deduction. Fair enough, if Richard wants to postulate synthetic randomness as The Creator, the godlike dynamic that has given rise to the cosmos, that's up to him: But such a postulate is not known to be a logical truism. In his case it seems to be bound up with a personal quest to contradict theism and this motivates him to try getting past us the misconceived notion that from the admission of a probability it follows that we have a randomness generator/creator. 

***

RichardOf course, what could then come to exist includes time, space, contents, and properties. And indeed this is true even of rabbits and deathstars. By the very definition of those terms, you can’t spontaneously create those things without also creating a spacetime manifold in which they can exist, complete with laws and properties. For instance, an inalienable property of a rabbit is that it has a nonzero width. And for it to be alive requires change (an active metabolism), which requires a nonzero expanse of time. As well as all the laws of physics needed to realize the rabbit and hold it together, from atomic bonds to inverse square laws, even the basic forces and particles of the Standard Model. Otherwise, it would entail a logical contradiction to say anything else that Nothing spontaneously generated could aptly be called “a rabbit.”

My Comment:  I largely agree with all that, but I raise an objection to the very first sentence which reads:

Of course, what could then come to exist includes time, space, contents, and properties.

...I have a feeling this statement has built into Richard's implicit fallacy that the potentialities of probability and randomness can be used like logical truisms.  This fallacy turns the "I-don't-know" information measure of probability into a dynamic on the assumption that "I-don't-know" entails randomness which it doesn't. 

Again, probability is not a dynamic which creates anything; it is just a measure of a cognating observer's information about a pattern. Randomness is a particular class of chaotic pattern we define mathematically in terms of algorithmics. True, some kind of dynamic is required to create/generate a random pattern, but because this dynamic is of maximum complexity in terms of its size and/or the number of algorithmic execution steps needed to create the pattern, very finite cognating agents like human beings find it epistemically difficult to cope with randomness with anything better than the statistics of probability calculus.  But again: One thing is clear: We have no reason to believe that randomness is a logical necessity. As far as we are concerned it is as contingent as the laws of physics. 

Richard has unconsciously assumed that probability logically entails randomness and that this union constitutes a logical truism that can come to his rescue as an axiomatic agent bringing forth the origin of the universe via logical necessity. No, it's the other way round. Probability and randomness are definable only within the contingent context of a given sophisticated up and running cosmos. 

Nevertheless, I can still go along with Richard's concept of Nothing. But whatever Nothing is he hasn't succeeded in locating its creative powers in probability and randomness. What we do know is that Nothing = Uknown.

***


RichardWhich means, every possible thing that can arise from Nothing—there being no logical fact nor any other thing to prevent it arising—will in effect be a “universe” in the broadest sense. Even just a rabbit, will actually be a rabbit within some “universe” necessary to materialize a rabbit. No matter what other thing you try to describe as a logically possible outcome of a totally random process, it will in effect either be a universe, or logically entail a universe to contain it. Which will of course include really bizarre universes, including static universes with no (or almost no) time, universes with only one dimension, and so on. But it is logically necessarily the case that no thing can exist without the existence of at least one dimension to contain it; otherwise it “never exists” and “exists nowhere,” which by definition means it does not exist (and thus cannot ever have been “produced” to exist). See my discussion of the Argument from Nonlocality for this point.

So everything that can logically possibly come to exist is, or entails (and thus comes with), a universe of some sort.

Which gets us to the next steps in reasoning…


·         Proposition 5: Every separate thing that can logically possibly happen when there is Nothing (other than Nothing remaining nothing) entails the appearance of a universe.

As just demonstrated.


My Comment: As just demonstrated? Bunk! If Nothing is capable of generating random patterns, which if of sufficient dimension will contain a subset of highly organised universes then Nothing is a pretty sophisticated object, in effect Richard's own creation myth/story. I can't begrudge him his faith in this story: It is the myth with which he's trying to make sense of the world, something many of us are also trying to do. But Richard's story is not the logical truism that he's cracked it up to be; especially so if he's resorting to a cack-handed use of probability and randomness. 

***

RichardAnd:

·         Proposition 6: If there is Nothing, then there is nothing to limit the number of universes that can logically possibly appear.

Unless you can come up with some logical proof showing it is logically necessarily the case that when there is Nothing, only some number n of universes can spontaneously arise. I know of no such proof. Good luck finding one.

My Comment: We simply don't know enough about Nothing (and Richard certainly hasn't enlightened us!) to be dogmatic about just what it prohibits, allows and creates. I don't disagree with Richard's general concept of "Nothing", which clearly equals "Something" and something significant at that, but it is not clear just what kind of stuff this very significant Something will create or not create. In fact "Nothing = Unknown" 

I think Richard needs to come up with a logical proof showing why it is logically necessarily the case that when there is Nothing (=Something) it necessarily spontaneously randomly creates universes out of nothing. He may be right and it does, but I know of no such proof; certainly not a proof involving probability and randomness. Good luck finding one Richard old son! 


***


 As I've already implied, I think Richard is right in identifying this unknown logical truism he understates by calling it Nothing. Nothing is a world of logical truisms. But may I venture to suggest another world, the platonic world of logical possibility rather than logical necessity. This world is the world of art and it seems that Nothing indulges itself in this contingent world of potential creativity. 

Richard says this:

.....every possible thing that can arise from Nothing—there being no logical fact nor any other thing to prevent it arising—will in effect be a “universe” in the broadest sense. 

Since we know next to nothing about Nothing (=Unknown) we simply don't know what kind of universes Nothing prevents or allows to arise. Least of all can we logically derive this class of cosmic potentiality from probability and randomness. 

Richard's logical error of assuming that given a probability we can then jump to conclusions about the necessary existence of random pattern generators perpetuates itself throughout the rest of his article. One doesn't have to be theist to understand that this is an abuse of probability theory and randomness. I suspect he's anxious to promote a preconceived interest-based ideology and this motivates his error. 


(Read Acts 17:22-28 on the Unknown God)


...to be continued