Showing posts with label Quantum nonlinearity. Show all posts
Showing posts with label Quantum nonlinearity. Show all posts

Friday, May 29, 2020

Signalled Diffusion Book V: Complex Drift-Diffusion Signalling


Book V of my "Signalled Diffusion" project can be picked up here. All the other books can be picked up from this post. Below I reproduce the introduction to Book V. 

Introduction

This book rehashes and enhances some of the mathematics that can be found in Gravity and Quantum Non-Linearity and Gravity from Quantum Non-Linearity. In particular it redoes the mathematics leading to the complex non-linear equation numbered in the text as 15.1. It also probes a bit further into the nature of the sub-microscopic signalling regime that gives rise to this equation. As I have said on several occasions this is a highly speculative personal project that in all likelihood amounts to little more than a walk down an obscure track in the woods to nowhere special. But one takes a walk not because it necessarily is going to end up somewhere significant but instead one enjoys the walk for its own sake. It is our duty and yet also our pleasure to enjoy the highways and byways of creation and creativity.

Friday, September 06, 2019

Signalled Diffusion Book IV: Real Numbers vs Complex Numbers



Book IV of my Signalled Diffusion project can be down loaded from this post. The introduction to Book IV is reproduced below.  The books so far are listed below along with their links:

BOOK I:     Foundations
BOOK III:  Drift-Diffusion


***

Introduction

Richard Dawkins and Brian Cox set the scene

The general consensus among scientists and layman alike is that quantum mechanics seems a strange and unintelligible way for the physical world to work; in particular it seems outrageous that this world should appear to do more than one thing at a time in that particles seemingly move in multiple directions at once. Further confounding our common sense physical expectations are those well-known apparently random discontinuous leaps of the system state vector when an attempt is made to measure a physical variable. Evangelical atheist Richard Dawkins puts his explanation on this state affairs:   

Physics appears to be a complicated subject, because the ideas of physics are difficult for us to understand. Our brains were designed to understand hunting and gathering, mating and child-rearing; a world of medium sized objects moving in three dimensions at moderate speeds. We are ill-equipped to comprehend the very small, the very large; things whose duration is measured in picaseconds or gigayears; particles that don't have position; forces and fields that we cannot see or touch, which we only know of only because they effect things that we can see and touch.

According to this we are creatures only “designed” (I’m sure that’s not meant literally!) for biosphere survival and therefore have no right to expect to understand the deeper mysteries of the cosmos. There is no reason why, for a jumped up primate, the cosmos should be anthropomorphically comprehensible and meaningful. The cosmos hasn’t been put there for this purpose; in fact you might hear Dawkins say that the cosmos is unlikely to have been put there for any purpose or reason at all; for Richard Dawkins it’s a fundamentally absurd cosmos and you can’t expect human beings to plumb the depths of this absurdity.  The cosmos is what it is without purpose; take it or leave it.

In regard to the specific enigma of quantum mechanics it is perhaps no surprise that parallel universe theory is a popular interpretation of quantum mechanics; it is one way of restoring the notion that reality isn’t ambiguous and in fact follows a single deterministic path of evolution; here the thought is that apparent quantum ambiguity is a sign there is a lot more unambiguous solid reality out there of which human consciousness is simply unaware. The parallel universe idea seems to provide two clarifications for the price of one: a) The totality of parallel universes is both unambiguous and deterministic b) the random leaping is just a perspective effect of us observers, confined as we are to just one of the parallel universes and only able to see just one of the many possible universes that ride side by side.

The intuitive agreeableness of the parallel universes interpretation of quantum mechanics is hinted at by Brian Cox in a BBC article entitled Brian Cox: 'Multiverse' makes sense (See also here):

“That there’s an infinite number of universes sounds more complicated than there being one,” Prof Cox told the programme.
“But actually, it’s a simpler version of quantum mechanics. It’s quantum mechanics without wave function collapse… the idea that by observing something you force a system to make a choice.”*1
Accepting the many worlds interpretation of quantum mechanics means also having to accept that things can exist in several states at the same time.
But this leads to another question: Why do we perceive only one world, not many?

The suggestion here is that the many worlds interpretation makes sense because it cuts out the asymmetrical contingency inherent in the idea that the reduction of the quantum state vector entails the cosmos making an arbitrary selection singled out from among the many possible choices. In the light of Richard Dawkins views one might question why the cosmos should make even some sense by conforming to the aesthetics of symmetry. But if we run with the symmetry idea as per Brian Cox then symmetry considerations suggest that all possibilities ought to be out there somewhere, thus doing away with any seemingly arbitrary and contingent treatment being preferentially given to a very limited range of possibilities. Max Tegmark has gone even further with the elimination of arbitrary contingency in favour of the hyper-symmetry of his mathematical universe. In Tegmark’s mathematical universe Tegmark extends reification to all possible mathematical constructions; in short nothing, absolutely nothing, is subject to special and seemingly arbitrary selection. However, in Tegmark’s model there remains the tricky philosophical issue of why there is something rather than nothing; in the selection of something rather than nothing we have another awkward asymmetry. There have been some attempts at addressing the counter intuitive ideas that it is possible to get something from nothing by redefining “nothing” in terms of the quantum void. But this could equally be construed as effectively redefining “something” in terms of the quantum void! All in all how you answer these difficult questions is probably very much influenced by your a priori world view preferences and what makes sense to you.

What purports to make sense is a very subjective affair: For someone like Richard Dawkins the cosmos need not make any sense at all. But if we are to appeal to just what makes sense then I have to confess that to me the “many worlds” view, apart from perhaps the vague aesthetic appeal of symmetry, doesn’t make much sense. Moreover, as Brian admits the problem with the many worlds suggestion is the question of why is it only one world is visible to us? Therein, I believe, is the observational clue to the problem’s solution – there is, in fact, only one real world and that world is highly skewed in favour of order*. This suggestion makes more aesthetic sense to me than does “many worlds”.  Making sense of things is how we attempt to join the dots of experience into a coherent narrative or a world view synthesis. But because world views are a very complex product of very complex and differing life experience, it is not very easy to submit them to the formal scientific process of prediction and test. Accordingly, world view synthesis must be carried out with caution and with epistemic humility; fundamentalists and evangelicals of all flavours please take note. Although as far as world view synthesis is concerned some people like Brian Cox clearly have a different mind-set to myself in this respect, it would be very wrong to think the worst of them; like so many of us Brian Cox is simply doing his best to bring intelligibility to the world; if he has a clear conscience we have no grounds to condemn him.

This book attempts to make sense of quantum theory by taking real number diffusion as far as possible before quantisation (which is achieved by introducing “i” into the diffusion constant) with the aim of using this approach to unpack the meaning of quantum mechanics. In the final chapter of this book I propose my own qualitative sense making narrative, narrative which I’ve wrapped round quantum theory in an attempt to render it humanly intelligible.  In my opinion positing quantum theory as just a variation on real number diffusion brings us to a very anthropomorphic understanding of quantum mechanics and relativity; the very opposite of the opinion expressed by Richard Dawkins. In fact may I express my intuition in advance of any clearer analysis that to me the cosmos looks suspiciously like the interior of a huge cognitive system as it seeks and selects solutions to general purpose goals. In this study goal seeking, or “purpose”, is an important sense making construct and trumps, say, the aesthetics of symmetry, especially if the latter is empty of meaning. I touch on some of these matters in more detail in the epilogue.



1* This a priori bias toward order is clearly a necessary condition for perceiving observers. And I would question whether it is entirely coherent to talk about a world without conscious cognating observers. 




Sir Kenneth Clark on Symmetry (Again)

It is an irony that the "Many Worlds" view majors on symmetry and closed
 endedness.  It achieves this trick by expanding the limits of existence so far as to 
eliminate uncertainty and the unknown: All things have a certainty of occurring within
 a mathematically defined envelope, an envelope which although infinite, which
nevertheless has known boundaries (but at infinity!)



ADDENDUM 12/09/19
Here's Sean Carroll on his enthusiasm for the multiverse. I've taken these quotes from a post on Uncommon Descent. Some of his views are in many ways  the opposite of mine although the second quote below may prove more amenable to my way of thinking.  (See here for more on Sean Carroll's views)

The Many-Worlds formulation of quantum mechanics removes once and for all any mystery about the measurement process and collapse of the wave function. We don’t need special rules about making an observation: all that happens is that the wave function keeps chugging along in accordance with the Schrödinger equation. And there’s nothing special about what constitutes ‘a measurement’ or ‘an observer’ – a measurement is any interaction that causes a quantum system to become entangled with the environment, creating a branching into separate worlds, and an observer is any system that brings about such an interaction. Consciousness, in particular, has nothing to do with it. The ‘observer’ could be an earthworm, a microscope or a rock. There’s not even anything special about macroscopic systems, other than the fact that they can’t help but interact and become entangled with the environment. The price we pay for such a powerful and simple unification of quantum dynamics is a large number of separate worlds.

Sean Carroll, “Splitting the Universe: Hugh Everett blew up quantum mechanics with his Many-Worlds theory in the 1950s. Physics is only just catching up” at Aeon

In my own research, I’ve gone even farther, arguing that the quest for quantum gravity is being held back by physicists’ traditional strategy of taking a classical theory (such as Albert Einstein’s general relativity) and ‘quantising’ it. Presumably nature doesn’t work like that; it’s just quantum from the start. What we should do, instead, is start from a purely quantum wave function, and ask whether we can pinpoint individual ‘worlds’ within it that look like the curved spacetime of general relativity. Preliminary results are promising, with emergent geometry being defined by the amount of quantum entanglement between different parts of the wave function. Don’t quantise gravity; find gravity within quantum mechanics.
That approach fits very naturally into the Many-Worlds perspective, while not making much sense in other approaches to quantum foundations. Niels Bohr might have won the public-relations race in the 20th century, but Hugh Everett appears ready to pull ahead in the 21st.

Sean Carroll, “Splitting the Universe: Hugh Everett blew up quantum mechanics with his Many-Worlds theory in the 1950s. Physics is only just catching up” at Aeon

Sunday, October 02, 2016

Dark matter, dark energy and matter; all the same thing?



The unexpected orbital velocity curve associated with galaxies. 

When she's not pushing de facto ID's dualist dichotomy of God did it vs. Natural forces did it, reporter Denise O'leary can be a useful read. Being part of North America's anti-academic-establishment, anti-liberal right wing lobby, 0'leary sniffs out the stuff that doesn't fit easily with current ideas, to the discomfiture of established science.  She means mischief of course, but the upside of that is that she reports those erratics and anomalies which are the fuel of scientific revolutions.

So, along these line I was very interested to see this post of hers entitled Understanding of dark matter  muddier due to new findings on the ID website Uncommon Descent. She quotes this article from Inside Science.  Viz:

Now researchers examining 153 galaxies find that by looking solely at where stars and gases in those galaxies are located, they could precisely predict the anomalous ways in which they moved. This may hint that dark matter is more strongly coupled to normal matter than currently thought. It could also indicate that dark matter does not exist and that another explanation is needed for the discrepancies that dark matter models were invoked to solve, said study lead author Stacy McGaugh, an astrophysicist and chair of astronomy at Case Western Reserve University in Cleveland.

I find that v. interesting because my own theory of gravity as a manifestation of quantum non-linearity would mean that matter and those galactic gravitational anomalies should go together. In the theory I've developed (see here) "dark matter" doesn't exist but quantum non-linearity produces a modified gravity which explains the anomalies. In fact this theory tenders explanations for both the apparent existence of dark matter and dark energy. As the theory in effect identifies "dark matter" and "dark energy" with matter itself, then it follows that matter and the putative effects of dark matter and dark energy are the effects of one realm rather than two potentially independent classes of matter. If dark matter and dark energy actually existed then further explanation would be needed to account for this association. But if matter, dark matter and dark energy are of a piece then the reason for the association would make sense.

All that's easy to say of course. I've only partially quantified my dark matter and dark energy theories. I must admit I am not heavily motivated to take it much further because I tend not to have much confidence in my own theories, those "left brain" creations. Steve Pinker calls the left brain the "baloney generator".   But nevertheless generating baloney can be fun!

Why I wrote my book: See here

Wednesday, March 09, 2016

Gravity from Quantum Non-Linearity: Edition 3


The third edition of "Gravity from Quantum Non-Linearity" can be downloaded from here,  Below I publish the preface:


***


Preface to third edition
Gravity from quantum non-linearity contains a shortened and more direct treatment of gravity than I presented in my book Gravity and Quantum Non-Linearity. However, the latter does attend to some details that need airing and which are not found in this current paper; in particular in the book there was a lot more attention given to the matter of conceiving a version of quantum mechanics based on a computer simulation of it, although the approximate nature of the resulting theory is also mentioned in this paper.

The changes in this edition are as follows:

1.   Numerous small changes have been made to the text.

2.  A section has been inserted (See section 3) which examines the role played by the second term on the right hand of equation (2.8.1)

3.  In the previous version of this paper and also in my book I referred to the space-time metric expressed in (6.17.1) as the Schwarzschild metric. However, I’ve since realized that my space-time metric, unlike the Schwarzschild metric, is isotropic and I have made the appropriate long-overdue changes to the text. This, of course, implies further differences between the theory I am proposing and the ideas behind the standard Einstein equation.

4.  I have, I hope, improved the clarity of the discussion which surrounds the gravitational metric.

Finally my usual disclaimer: This paper is entirely speculative and makes no heavy claims about being the solution to the gravity vs. quantum mechanics problem. As a hobbyist I do this sort of work for its own sake and don’t necessarily expect a successful outcome. As I always say; one must endeavor to enjoy the journey because the destination may not be up to much. At least one can come out of it with some interesting explorations achieved.  

Saturday, October 17, 2015

Twere to consider too curiously, to consider so.

Could MI6 have an interest in my little story? Probably not, because I try not to live in the real world.

I spotted the following material online over a year ago, but as I'm currently short of time for blogging I thought I would tell this odd little tale in lieu of my otherwise absence from the blogosphere......

This paper on solid state physics written by Chinese authors  Xiang-Ying Ji and Xi-Qiao Feng is rather curious. Why so? .....because my book Gravity and Quantum Non-Linearity is referenced in it. Honours like this don't often come my way, so it's time to play it for all it's worth and bask in the glory.  Here's where in the paper's text the reference occurs: 


The reference 14 attached to their biased diffusion equation (4) appears in their reference list as follows:


I wonder if these guys know who I am and what I'm about because there is a bit of "but" in all this! The equation in my book Gravity and Quantum Non-Linearity  that is nearest in form to equation (4) above is equation (3.13) below:
If you can read equations like this then you will understand that apart from a couple of differences my equation (3:13) is more or less of the same form as Xiang-Ying Ji and Xi-Qiao Feng's equation (4). Those differences are in the constants and also the appearance of the last term on the right hand side of (3.13). 

But .. and here's that "but" I spoke of..... notice  I have "i" ( that is, root minus one) in front of the diffusion constant. When you add this complex factor to the diffusion term it changes an ordinary biased diffusion equation into a wave equation. The other thing to notice is the appearance of the last term on the right hand side - this is to compensate for the fact that in the kind of diffusion represented by (3.13) the diffusion is a process where in the underlying random walk the stepping agent bifurcates and steps both left and right at the same time. To prevent such a system violating conservation laws the third compensating term on the right hand side must be subtracted.  (This is not a point that appears in my book, but is something that has occurred to me more recently). It is this feature that gives my equation its relativistic character.

Xiang-Ying Ji and Xi-Qiao Fengstop stop short (wisely perhaps) of this "relativistic complexification" of biased diffusion - after all, their paper is really all about the ordinary diffusion of real particles in materials and not about a quantum equation or its extension into quantum gravity. It is perhaps a little strange that they should reference some weird and eccentric amateur theory of quantum gravity when the derivation of the non-imaginary form of their equation (4) is pretty standard fare! But I'm honoured that our esteemed Chinese friends have chosen to use this reference! I wonder if there are some security issues entailed here? I'll leave MI6 to decide on that one!

Tuesday, July 07, 2015

Melencolia I Part 6: Quantum Leaps


Part 6 of my Melencolia series can be found here. I reproduce the introduction to this paper below:

***

1     Introduction
In this short paper I make a proposal as to the nature of quantum leaps. These “leaps” are the apparent discontinuous changes of the quantum mechanical Hilbert vector, a vector which otherwise moves continuously according to a deterministic wave equation. I have come down in favour of the view that these leaps are literal rather than apparent. The following paper is largely a qualitative discussion of a subject which could no doubt bear a lot more rigorous quantitative formulation. However, in this Melencolia I series my sights are really on the evolutionary and OOL questions and I hope I have enough in this paper on quantum leaping to assess its impact on my general objectives. But having said that I’m not quite sure just where this paper leaves my ideas about a declarative model of evolutionary computation. For on the face of it my proposal on quantum leaping seems to hamstring the searching that would be needed to find the configurations of life: This is because the “leaps” would, apparently, clear the quantum signalling field before it could make any worthwhile discoveries. One little consolation, however, is that in conceiving matter as a combination of a coherent object and a shadowy gravitational field I find some scope for fixing the energy problem thrown up by proposing literal quantum leaps.

The general idea that guides the Melencolia I series is the view that intelligence is a process, a process with a general declarative structure, of search, reject and select. Thus, the life generating processes are, in this context, viewed as intelligence at work and therefore open to observational scrutiny. This very much contrasts with the views of the de-facto ID community who envisage intelligence as a kind of black box very distinct from natural processes. This black box gets little or no analytical treatment from the de-facto IDists.  In contrast one thing that encourages me to pursue the endogenous ID proposal is the fact that the our current understanding of the mind suggests conscious cognition is very much bound up with the material organisation of the brain;  That is, we do not see “mind” down at the low neuronal level;  these low level elements are wholly impersonal. But at the high level personality becomes apparent. Likewise we don’t see cosmic intelligence/personality operating at the low particulate level, but we may only see it in the big picture. This is not to say that current molecular views of the mind are the full answer; for example, we may eventually have to feed into to the mix the ideas of people like John Searle, Roger Penrose or whoever.

As I continue to use this series to explore the processes that generate life there is, I feel, little chance I’m following anything like the right path. But as I always say: Enjoy the journey while you can because the destination may not be up to much! And below, the journey so far…..

Also relevant are these links:

Tuesday, August 26, 2014

The General Language of Reality

This yellowing paper back started something for me!

This post on Uncommon Descent piqued my interest: It is about the question of whether animals really have language and the first sentence of the post, probably written by Denise O’Leary, reads:
And you just know how desperate they are to get to “yes”
I think you will find that Denise O’Leary’s thought behind this sentence is likely to ruin run along these lines “Uh Oh, here we go again; materialists trying to blur the distinction between animals and humans in their efforts to spread Darwinist propaganda!,

I’m afraid I have to say that once again I find myself at odds with Denise. In my opinion it is very likely that animals do have language albeit at an elementary precursor stage. Why do I say this?

Well, that is a very long story which I hope to tell one day in full. Its beginnings go back at least to the time I started programming a simple word association network on a spectrum home computer with ideas taken from Edward De Bono’s book “The Mechanism of Mind”. This project, which I called project "X", got more and more complicated as I upgraded first to an Amiga500 and finally a Windows PC where I programmed the project in C++.  It was this project that actually prompted my rather amateurish foray into quantum theory and the private publishing of my book Gravity and Quantum non-linearity. To cut a long story short it was clear to me that the association game had a very general structure about it, so general that it even echoed aspects of quantum mechanical ideas. I actually allude to the project in these web items:


I have recently read the book “Don’t Sleep, There Are Snakes” by ex-Christian SIL missionary Daniel Everett. He challenges the ideas of Chomsky and proposes that there is not so much a language instinct as there is a general cognitive ability on which language runs. If my ideas are valid then I partly agree with Everett. Intelligence has an underlying general structure, so general that it is even reflected in quantum mechanics. Language arises out of this very general cognitive structure as a specialised application of it. However, having said that I suspect that the general “De Bono thinking surface” has to be tweaked by setting up its adjustable parameters so as to make it work well for a specialised job like complex  language and this is probably why the human brain has “Broca’s area”. But because language is rooted in such a general cognitive “machine” animals are likely to display a proto-language use. So, I probably position myself somewhere between Chomsky and Dennett. But more about this another time. For the moment I’ll leave you with the first paragraphs of an essay I started writing several years ago where I intended to go into more detail:

By 1987, the year my paper on probability was accepted, I had also been playing around with a word association program I had written in Basic on a Spectrum computer. My interest in association networks had found the beginnings of a theoretical base after reading Edward De Bono’s book “The Mechanism of Mind”. So, having got the probability issue behind me with a paper in the British Journal for the Philosophy of Science I stood back for a bit and took stock. As I pondered the work I had done with probability and randomness I fancied I saw a connection between the latter and the ideas underlying the association network software I had written for my spectrum computer. Moreover, the association network seemed to be at least a faint of echo of the process of human thinking (as De Bono intended it). In particular it reflected somewhat the way thought flows from one conscious impression to the next. My mind at that time was alive with ideas of artificial intelligence and my work in probability had thrown some light on the subject. One day at the beginning of August 1987 I wrote some notes on AI, notes that started with the following rather grandiose lines:

“When developing a thinking machine we need to ask ourselves a question: What is a very general representation of reality?  What is the basic form of a notation that can be used to describe any situation? Once we have found this very general notation we can then start to construct a machine that models reality using it.”

The idea here was that if there is a very general representation of reality, so general that it applies to any situation, then this representation amounted to a universal way of thinking about things. Thus, if that representation is modeled on a machine then perhaps we would have in our hands a way to model thinking and intelligence. As for the answer to the question of the general representation to use, that seemed obvious to me: The association network formalism gave me that language, and the isomorphism that that formalism had with the models I was using to explore randomness and probability suggested to me that this formalism was the one to use.  Let me explain how this works.....

…perhaps another time! Since I last wrote the foregoing I think there has been a subtle shift in my thinking; The general language of reality is not so much a representation of reality as reality itself!

Thursday, August 14, 2014

Melencolia I Part 3: Sharpening the Focus.


The previous parts of this series can be seen here:

I was interested to see this blog post by evangelical atheist Larry Moran where he refers to a post on the IDist web site Uncommon Descent by a Gordon E. Mullings (aka “Kariosfocus”). In the referenced post Mullings publishes a quote from an IDist who bemoans the fact that the academic establishment simply hasn’t been moved by what to him is the clear and convincing case for his version of intelligent design.  As often happens with the members of a  somewhat beleaguered and marginalized subculture he just can’t believe that this stone walling is down to a dispassionate intellectual rebuttal but rather a wilful ulterior-motive laden rejection of what to him is oh so obvious. So, all too typically we find that Mullings’ quotation puts it down to a “heart problem” rather than a “head problem”. The intended hint here is that those who continue in an informed rejection of Mullings’ version of ID probably do so with bad consciences if not black hearts. This tendency to believe in the depravity of one’s detractors, as I have pointed out on my blogs, is a background attitude which provides fertile ground for conspiracy theorism, should it take root. But to be fair, Mullings and his friends have often been unpleasantly abused, although frankly I don’t think that some of Mullings’ somewhat paranoid denunciations have helped calm things down.

For my own part I find it difficult to take sides here: Polarization has killed off dispassionate debate. Many atheists are probably thoroughly annoyed by the North American IDist’s claim that they have scientific authority to push their case for God via Dembski’s explanatory filter (*1). But conversely atheists also attempt to use scientific authority to rule theism out of court (See here). In any case Mullings often fails to do justice to himself with his paranoid behavior and I doubt there is much respect for all his technical efforts – these are just brushed away by evangelical atheists without thoughtful engagement as just so much window dressing and laughable sophistry. But there is in my opinion a deep flaw in the kind of IDist thinking Mullings stands for: Viz: He talks about the computational resources of the cosmos being incapable of locating life using a “blind search”, but he appears not to consider the obvious solution that the cosmos may have sufficient ongoing immanent divine resourcing to presume it to be a blind search; In fact the Wiki entry that comments on Dembski’s “Universal Probability Bound”, (an idea enthusiastically promoted by Mullings) spots the way out:

Dembski appeals to cryptographic practice in support of the concept of the universal probability bound, noting that cryptographers have sometimes compared the security of encryption algorithms against brute force attacks by the likelihood of success of an adversary utilizing computational resources bounded by very large physical constraints. An example of such a constraint might be obtained for example, by assuming that every atom in the known universe is a computer of a certain type and these computers are running through and testing every possible key. However, universal measures of security are used much less frequently than asymptotic ones.[6] The fact that a keyspace is very large is useless if the cryptographic algorithm used has vulnerabilities which make it susceptible to other kinds of attacks.

For a theist like myself the weakness in Mullings' argument is that divine intelligence is, presumably, well able to express itself in a much better than “blind” search. Like many IDists of his persuasion Mullings gives every impression of being against the idea that the cosmos has the wherewithal to “naturally” generate life. This is, I believe, an outcome of the polarised state of the debate where respective sides have tacitly taken onboard a “God vs Nature” dichotomy along with the implicit conclusion that it’s an exclusive choice between “natural forces” and God. This dichotomy is reinforced by Dembski’s explanatory filter which necessarily imposes a sharp distinction between “natural causes” and “intelligent agency”.

It is ironic that my own position is probably nearer that of the atheists Viz: That the cosmos has generated life …. although I propose that the cosmos has generated life because it is providentially resourced to do so. Where I differ from these atheists is that I see the generation of life as a remarkable non-trivial property of the cosmos. In fact Mullings and myself probably have common ground in agreeing that any  “law and disorder” description of the cosmos always leaves us with irreducibly startling features; we can never trivialize those features away with law and disorder explanations. The ontology of law and disorder posited by the physical sciences is, in the final analysis, complete when full description is reached; attempts to push it any further leads to a regression of nested explanatory contexts and this is mathematically akin to attempts to compress an already compressed data set; further attempts at compression gain no further reduction in the data string. The highly unrepresentative configurations of life are startling and remarkable, but attempts to explain them with law and disorder scenarios leads to logical conditions that are no less startling and remarkable. As Sir Patrick Moore once said: Our science is strong on detail but weak on fundamentals. We have to seek a very special kind of explanatory narrative if we are to stop this regress: An extraordinary universe requires an extraordinary explanation. But that’s another story I won’t talk of here; let’s leave theology out of it for the moment and just stick to physical science.

The bald fact is that Mullings and his colleagues have failed to get their message across; I put that down in part to their framing of the question of life with what appears to be a dualist “God did it vs Nature did it” dichotomy. This dualism is a gift to evangelical atheists who understandably are prompted to attack the weakness in this dualist IDism; namely, that there is little evidence for a "God-of-the-gaps-did-it" view: For it is clear that whether Mullings likes it or not, intended it or not,  at  least one atheist sees Mullings' case as a "God of the gaps" model. As Larry Moran says in his post:

There is no evidence for the existence of a creator who meddles in the affairs of living organisms.

(This statement, by the way, betrays a misunderstanding of how evidence is actually used; see here for details). If this “interfering” deity is what Mullings and his colleagues are really promoting – and they give every impression of doing so – then it is no wonder that the academic establishment, Christian and atheist alike, takes such a dim view of their efforts: While I personally don’t rule out the anomalous and the miraculous the fact is such things are few and far between by their very definition and amount to positing an erratic ontology that makes it impossible to provide high standards of evidence for; scientific epistemology has great difficulty with an ontology of erratics. So, if as the North American IDists appear to claim life was generated by the one-off interferences of an eminent intelligence, an intelligence which could well be little green or gray men inside the cosmos, then that is going to generate a very soft science, a kind of super-attenuated archeology in fact.

In order to back up his case for a “meddling” deity Mullings posts this graphic panel:

This graphic by Gordon E Mullings perpetuates the North American "Wow!" signal argument for an interfering Intelligence eminent to the processes of nature. (Click to enlarge)

Living structures if described in terms of their self-maintaining activities can be relatively simply defined without, as Mullings rightly says, specifying all the coordinates of their configuration. Moreover, compared to all the imaginable possible atomic arrangements it is clear that this class of self-maintaining structures, even the simplest of them, is negligibly small and so on a purely chance selection no member of this set would likely be found even given 10150 trials! But the trouble with the argument here is that Mullings doesn’t define what he means be “all relevant possibilities” and this vague reference hides a big issue.

To see this let’s take for example a construction set like Lego. Clearly the total possible ways of arranging the bricks in a big Lego set is a huge number and utterly dwarfs the set of constructions that do “something useful”.  But there is a big difference between the total number of conceivable arrangements of Lego bricks and the number of arrangements that come to light should we start moving around the Lego bricks according to some constraining developmental regime, perhaps with aim of searching for “something useful”. These two sorts of “relevant possibilities”, namely the total conceivable possibilities and the possibilities that come to light as a result of constrained change, are very different in character. Ostensibly it might seem that any configuration of atoms is possible and violates no physical law, but - and this is the big “but” -  physical law is also about change and given the regime of change in our cosmos it is clear than not all configurations pop up in a given time; so in the developmental sense the “relevant possibilities” are very different from the conceivable “relevant possibilities”.

If Mullings had taken this distinction onboard I suspect his panel would be very different in content. In fact I have yet to see any serious considerations by his IDist subculture of how physical constraints enhance the search for organic forms; when they have considered this topic they usually think terms of the “dynamic fallacy” whereby they believe the problem is to counter suggestions that the organic biopolymers are somehow coded directly into the physical regime. Better perhaps is their concept of Irreducible Complexity, but even that they have managed to screw up on. But if rightly defined irreducible complexity would, however, be a big evolution stopper, and Mullings and co would then have the last laugh. However, there is one other big issue I have with Mullings which may make all this irrelevant. This issue is Mullings promotion of Dembski’s probability bound; namely, that the cosmos has an upper limit of only being able to locate at best a 1 in 10 150 instance.

I would propose that potentially the cosmos has a much, much greater power of searching out rare cases than Dembki’s probability bound suggests and this follows from certain quantum mechanical considerations.  We can begin to appreciate this by first looking at the classical analogy of quantum mechanics – namely, elementary diffusion. One dimensional diffusion can be simulated using a system of nodes laid out on the x-coordinate. If the diffusion distribution is represented by Y(x) then the system of nodes can simulate the diffusion simply by signaling one another with wY(x) where x is the position of a node and w is the analogue of the step probability in random walk. The result is a simulation of the elementary diffusion equation:

Equation 1

 Where e is the distance between nodes and v is the velocity of the signal. (More detail about this topic can be read in my private publication “Gravity and Quantum Non-Linearity”)
The number of nodes working in parallel to simulate this equation is potentially proportional to L/e where L is the length of the x coordinate.  But this is just for one dimension. If we have N particles then the diffusion distribution is represented by Y(x1,…xjxN), a quantity defined over a space of N dimensions. The appropriate diffusion equation is then:
Equation 2

…where for simplicity I have assumed that e, w, and v are independent of coordinate, although in general this is not the case.

If for the sake of argument we assume that each coordinate has length L then the total number of nodes will be (L/e)N. These nodes effectively act as our processors and so potentially the simulation could utilize the power of (L/e)N nodes. If N ~ number of particles in the observable cosmos we can see straight away that the number of processors is so huge that potentially it will return a processing power far in excess of the strictures of Dembski’s probability bound; this bound only raises 10 to the power of a 3 digit number, whereas in (L/e)N, the power N, assuming a figure similar to the number of elementary particles in the observable universe, is far in excess of 3 digits.

However our simulation isn't doing anything useful. In order to make it build things we need to add a potential term, V, thus:
Equation 3

Here I have taken the quantity e inside the summation and made it a variable that depends on coordinate, but this will not substantially change the enormous number of processors involved. The significance of this change will become clearer shortly.

The potential terms V(x1….xN)Y is way of “pumping in” signal in some regions but extracting it in other regions in order to conserve the distribution for correct normalization. This term, as we know, can be arranged so that there is a tendency for particles to stick together, thereby acting much like the attachments of some kind of construction set. If so, then this will mean that the distribution has tendency to accumulate around certain coordinates values representing the case where particles coagulate. To be sure however, this accumulation of the distribution would be very small if the space is very large and the size of the particles in terms of their surrounding potentials are small; this is because under these conditions there is a very small chance that the particles will find such configurations. But here’s the crucial point: These configurations will effectively be flagged and marked by a raised distribution at those points.

We can considerably enhance this simulation in favour of organized configurations by turning equation 3 into its quantum mechanical equivalent (See again Gravity and Quantum Non-Linearity). This is easily done by simply replacing the real signal wY of ordinary diffusion with a complex number signaling of form iwY thus giving us the multidimensional Schrodinger equation: (*2)

Equation 4

…where w, v and ei are adjustable constants that can be set to give us the usual quantum mechanical factors of Planck’s constant and mass: Notice, however, that the factor ei which I placed inside the summation can now be seen as a way of simulating different particle masses.
The reason why this equation is a huge improvement over ordinary real diffusion is that the wave nature of the resulting solutions has the effect of cancelling out (when normalized) huge fields of bland randomness, thereby enhancing the relative presence of those peaks of coherent and organized structures. In short quantum signaling is way of insuring that organization and coherence become disproportionality represented. This is just the sort of constraint we need to help enhance the search for life.

So, the points I would like to leave us with at this stage are:

ONE: The quantum mechanical processing potentially involves huge numbers of processors
TWO: Bland fields of randomness are canceled out.
THREE: Organized and highly coherent structures are flagged with raised distributions.

But there is a little problem:  In the Wiki entry about Dembski’s probability bound of 10150 we read that Seth Lloyd has come up with a bound which has a power of similar order of magnitude as that of Dembski. Lloyd and Dembski aren’t sloppy workers, so how does this square with the enormous processing power I’m proposing where the power of ten isn't merely a three digit number but is itself a power of ten?

The answer may be this: The sort of quantum mechanical vision I've sketched out above entails quantum systems maintaining the ambiguity of their prospecting signals over huge regions of space. The trouble is, at the macroscopic level we don’t observe an ambiguous reality – the apparent discontinuous jumps of the state vector always seem to contrive an unambiguous macroscopic reality. This “collapsing” of the state vector has the effect of clearing away the work of huge numbers of nodes (*2). So in this light I suspect that Lloyd and Dembski are probably right is reducing the apparent processing power to a mere three digit power of ten. But the enormous computational potential of QM is nevertheless clear; perhaps we simply don’t perceive it because human consciousness is only ever handed unambiguous macro states that have been “collapsed” into a far less ambiguous state; who knows what goes on in between these discontinuities?

I find I can’t dismiss the powerful computational potential of quantum mechanics: QM looks to me as if it is meant to use that potential. In fact, let me speculate a bit: Those prospective signals look like the means by which possibilities are probed before a selection decision is made – in short, an important aspect of intelligent activity; seek, find, reject and select. So, if my guess is right then in quantum mechanics are we watching intelligence at work at the low level, much as we might see the low level neural signaling of human consciousness, if we look very closely; the trouble with the low level view is that it so easily misses the big picture.

An extraordinary cosmos requires an extraordinary explanation: For me the idea of the cosmos being intelligence in action is commensurately extraordinary with the extraordinariness of the universe itself. However, people’s intuitions about what is extraordinary are not necessarily going to concur and may actually conflict. But my excuse is that I’m in uncharted waters here and I can only present what are my own personal speculations and intuitions about the nature of the universe. This is no basis for accusing those who don’t agree with me about God as having bad consciences and perhaps even dark hearts, or as is the habit of fundamentalists of accusing disbelievers of “suppressing the Truth in unrighteousness”!  This sort of epistemic arrogance leads to tribalisation and at the extreme end can even be used to legitimize genocide.

...to be continued (This is a work in progress)


Footnotes
*1 This filter is a useful heuristic within the cosmos, but it falls over badly in a theological sense when used as an argument for God; with its clear cut "natural forces" vs. "intelligent agency" distinction, God competes with his own creation as an agent of causation.
*2 In linear quantum mechanics there is symmetry between momentum space and geometric space, but in this node simulation of QM it is meaningless to talk about momentum space being composed of a set of mutually signaling nodes. In this simulation picture momentum emerges as a synthetic function of the gradient of the wave function.

Notes
On indistinguishability: The indistinguishability of fundamental particle makes the idea of the distinguishability of substance problematical. I suggest that quantum physics doesn't create any “substance” (whatever “substance” means) but rather creates configurations taken from the platonic realm. Distinguishability is a result of distinguishability of configuration and therefore identity of substance is bound up with identity of form.
On Fine Tuning: In the quantum mechanical equation in this article V is a given; so how do we know from the outset that a given V will support organic forms? The answer to this question may be along the lines of the Church-Turing thesis; namely, that there is such a thing as the universal construction set, a set that will build anything (or compute anything). Thus construction sets can be judged on whether or not they are universal construction set complete. We could define a “fine-tuned” construction set as one of those construction sets that is universal construction set complete. My guess is that there is probably an infinite number of universal construction sets. Conceivably, the space of all construction sets could itself be explored with a signaling system.

Friday, January 13, 2006

Physics and The Wild Web

What draws some people to develop fundamental theories of physics even though they are well beyond the circles of the professional physics community? The odds against success seem overwhelming. Professional initiation into much advanced mathematics is required just to get to the frontiers of current physics, let alone succeed in pushing the boundaries of those frontiers into new areas. And yet have look at some of the titles from the Web enabled Print On Demand publishers. In their science sections you will find a plethora of books from “Do It Yourself” theorists, many of who radically challenge the basis on which current physical theory is founded. Added to this are the numerous offbeat physics papers to be found published directly on the Internet. This phenomenon is probably most prevalent in the USA with its go getting, “anything is possible”, frontiersman ethos. Where I live (the UK) people are less inclined to try anything so ambitious. This may be because a trace of medieval ambiance still lingers here and people are more likely to accept their station in life. If one wants to do different in the UK, it helps to be eccentric and it helps even more if you don’t care.

Fundamental physics, it seems, is fair game for a variety of outsiders who come to the subject with a mixture of motives and backgrounds. One can find, for example, electrical engineers who believe an extension of Maxwellian electromagnetism provides the key to the problem of gravity. The are also practically minded technologists hunting for the Holy Grail of star travel – the anti-gravity drive; why let the laws of physics get in the way of technological goals? Less practical are the New Agers trying to get a mystical handle on the fundamental laws as they seek spiritual enlightenment in physics. There are self-proclaimed geniuses, egotists whose totalizing theories rewrite physics to its last word. There are quasi-paranoiacs who despise academia and believe Relativity and Quantum Mechanics to be the product of a conspiracy of deception. There are, I think, even some professionally trained maverick scientists working independently – the distinction between crank and genius has never been clear-cut, as exemplified by the great Isaac Newton himself. An exhaustive taxonomy of the kind of worker we are talking about here is quite a study, but all in all this is physics with attitude, often bad attitude. Some of these workers carry their physics forward with ill humor and have a complete and unwavering conviction that they alone are right. Unshakeable self-belief is the survival strategy that keeps them going against the odds, and guards against any crisis of confidence. Self-awareness is a trait that sits uneasily with high confidence.

Whatever their temper and frame of mind these self-motivated theorists nevertheless share, with the greatest theorists and mythmakers of the past, the time honored aspiration to compress a profusion of complexity and mystery into relatively simple logical narratives and to perhaps discover deep meaning therein. Moreover, contemporary physics is suffering an intellectual logjam as a glut of concepts founder on the relation of Gravity and Quantum Mechanics. As professional practitioners attempt to resolve this issue they are disappearing rapidly over the intellectual horizon with Byzantine depictions of reality not conducive to a favorable social perception of physics. Many independent workers are aware of these problems and are exploiting physics in its hour of need as they short cut the thick undergrowth of professional theoretical physics. The subtext is: “Enough is enough. If you can’t come up with intellectually economic models of reality, then we will”. The feel-good factor that comes from knowing that you have a chance of undercutting the experts with bargain basement theory is not to be underestimated. Moreover, it is probably true that convincing one’s self of the groundbreaking importance of one’s work enhances one’s self image and thus the ego gets a boost. But then who doesn’t strive to feel good about themselves in this way? And who doesn’t know the strife and grief caused when social kudos and ego clash, with all the concomitant mental stress?

If it sounds as though I fancy myself as an authority on this subject then maybe that’s because I speak from the insights of “done that, got the T-shirt” experience. Yes, my T-shirt says “Cranko-Physics Fringe” and I tell my story of freelance physics and what it’s like to be closeted away struggling with difficult ideas from a perspective inside that closet. The fundamental physics bug got me around 1990 (although the roots go even further back). At the time I was totally absorbed minding my own business writing a piece of search engine software with no idea where this work was ultimately going to lead me. This software really depended on the idea of simulating word association. For example, “red” is associated with “blood”, “blood” with “liquid”, “liquid” with “wet” and so on. In fact, every word is embedded in a network of associations, where each node of the network is a word, thus giving rise to a so-called “semantic net”. Thus, activating “red” in one of these networks will not only activate “blood” – it will also activate “ink” or “Traffic Light”, or anything else that is commonly red in color. When my piece of software was complete, offering a word to the network resulted in simulated signals being radiated out to a halo of potential targets. However, if two words were offered to the network, like say “red liquid”, then this resulted in two haloes of signals which intersected and overlapped in a Venn-diagram like way, and effectively reduced the list of potential output solutions to the input problem. Thus, for example, an input of “red liquid” would return “blood” as a possible solution – it could also return “ink” as ink is sometimes red, but it would exclude “Traffic Light” as the latter has little to do with liquid. An understanding of probability was crucial to this project as “solutions” to “problems” were returned with an assigned probability. Some of the ideas I deployed here were based on a notion of probability I had developed and published in the June 1988 edition of The British Journal for the Philosophy of Science. Ultimately I brought this software to bear on the problem of searching text for meaning rather than literal character patterns. But before I could get a practical product up and running I dropped the project. Something exciting had caught my attention. Perhaps I should have ignored what I had seen, but for me it was like stumbling across a gold mine.

Most physics freelancers, I suspect, bring to bear the insights of their particular walk of life. I was no exception. If I had been an electrical engineer I would have applied induction. If I had been employed by the Jet Propulsion Laboratory I would have discovered an inertial discrepancy. If I played the violin I would have become a String Theorist. If I had been a cook I would have seen some connection between self-raising flour and the expanding universe. But I was a programmer and probability theorist so I fancied I saw a connection between my fields of probability in a semantic network of nodes, and the wave fields of Quantum Mechanics. Moreover, my fields, like quantum wave fields, had an “output” in the form of a “field reduction” – in this case a reduction to the Venn-diagram like intersections. Furthermore, these “intersections” could be activated as the input to a new problem. In fact, I perceived a kind general “computation” serially structured as halo-intersection-halo-intersection-halo…etc, and this seemed to mirror the alternation in quantum mechanics between wave development and wave reduction. Here was a process with a strong time asymmetry; a computation that disposes of possible outcomes in favor of other outcomes cannot be wound backwards.

Quantum Mechanics seemed to be carrying out the same kind of computational task that I found in my semantic net. Was there some mileage to be had in this similarity with Quantum Mechanics? Was there a profound clue here about the nature of reality? I thought there must be: I found the whole vision of the declarative programming model that my semantic net conjured up a very compelling metaphor imputing meaning to the enigma of quantum theory. It was at least as compelling as the action principles so beloved by some mathematical physicists. Many scientists do not expect to connect in some way with the objects of their study – quite the opposite, in fact, they expect to enter a world that gets more and more alien, inhuman, difficult and meaningless the more it is removed from the level of street and furrow. This ethos must surely impede the rationale and hope that drives them. But if one suspects that humanity is set up to connect with physical enigmas then the fuel of motivation is more readily found to help drive the project of physics forward – that and a little reverse engineering.

But one must be wary: Kepler was compelled by what he perceived to be a connection between the five regular solids and the relative sizes of the planetary orbits – a hunch which, of course, proved to be spurious. More recently another compelling concept that proved to be wrong was the neat idea that the four letters (A, T, G, C) and three place words of the genetic code does not need to include a word separator if only 20 of the 64 possible word combinations are used. By strange coincidence this elegant and seductive logic was actually supported by the observation that 20 equates to the number of amino acids used by the genetic code to build protein chains. With this sort of thing at the back of my mind I certainly had doubts about a solitary foray into Quantum Theory. However, there was nothing for it but to give my own metaphor a chance and follow the path that had opened up before me. It was risky; in all likelihood it would prove to be a garden path. I am cutting a very long story short when I say that I developed (or should I say “reverse engineered”?) a form of quantum mechanics along similar lines: Nodes signaling Nodes with complex signals – and to incorporate the effects of relativity it was necessary for these “quantum signals” to squeeze and contract the separation between nodes, and thus apparent “space-time curvatures” dropped out quite unexpectedly and naturally: Gravity, it seemed, was staring me in the face. In time some bits of Einstein’s equation emerged. That was a bonus - if you can get Einstein’s name somewhere in your work then that puts you in the cranko-fringe premier league. Though I tend to despise the conceited theoretical totalizers who presume to clear the board completely, I found to my horror that I was starting go that way myself. One thing lead to another and it wasn’t long before I fancied I had “discovered” explanations for “dark matter” and the positive cosmological constant.

Needless to say someone coming along claiming to have, in one decisive action, blazed a trail (or at least a garden path) from the tiny quantum world through to the cosmological constant, solving the problem of gravity on the way, goes down like a feather sandwich with your average academic. I can’t say I blame them - the whole field is awash with ideas, and approaches from unaccredited upstart theorists must seem as unwanted distractions from time wasters. Nevertheless, I generally support the sterling work of many academics and unlike some other freelancers I don’t see myself as a competitor. But even so, given the enigma of gravity I say it ought to be all hands deck and freelancers should be welcomed - the more the merrier, because an outsider might just rumble the solution (or should I say “a solution”?) by daring to do different: Collect together enough monkeys and perhaps one of them will come up with something. And it is just possible that the experts could be looking in the wrong place; their tight knit and well networked community might actually be a disadvantage because a spurious perspective, if it takes hold, is likely to lock itself in. In fact, when one hears String Theory aficionados claim that theirs is the “only game in town” it’s not good news for physics, because if they are wrong, they are likely to stay wrong. Moreover, this goes to show that bad attitude and a lack of self-awareness are not only to be found amongst freelancers. Perhaps the String Theorists have tied themselves down with the most sophisticated mathematical trap the world has yet seen. If a pair of magic scissors, in the form of new 21st century mathematics, doesn’t turn up to help them cut the knots of String Theory, then they’ve got their work cut out because, as one internet correspondent has quipped, “The theory is brighter than we are!”. Perhaps one day the strings dancing in atoms will be as much a non-issue as the Byzantine angels dancing on a pinhead and the extra dimensions of String Theory will seem like Aristotle’s Quintessence. In any case what do they mean “The only game in town”? There are some very capable and respected theoretical physicists like Roger Penrose who seem to be playing another game altogether. (That’s actually “Sir Roger” to the likes of me – like I said, we are still feudal in the UK)

However, let me return to my humble story. Is this story really one of an obscure persona stealing a march on the experts and in one swoop solving the greatest scientific problem of all time? Well, I endeavor to be self aware enough to understand that there are lots of pretenders to that! Nevertheless, I still think my story does at least have some human interest value, if not scientific interest, for it is a story of the human struggle to understand and testament to the extraordinary ability of the human mind to fit a theoretical narrative around experiential complexities – at least for a while – for in science a spurious theory can fit some of the experiments some of time, but it can’t fit all of the experiments all of the time. Scientific chickens come home to roast in the real world and in my case my theory has yet to receive a roasting from that world. I take some consolation, however, in my philosophical view that all theories are likely to be limited simulations with a sell-by-date, even though it is a Newton or an Einstein who annunciates them. Limitations or not, theories are very useful sense making aide-memoirs organizing the complexities of a cosmos built around what I believe to be a Divinely ordained grand rationality, a rationality that makes that cosmos amenable to human theorizing. In this sense my theory is scientific; its thesis may prove to be of limited applicability and organizing power, but it is science in as much as it is at least a protem comprehension of my own perspectives, a comprehension that is open to logical and experimental challenge.

The story I tell is of a very personal engagement with a rational Cosmos. It recounts my own version of the time-honored strivings for understanding and the human aspiration to integrate cosmic variety into relatively simple narratives. I present it first and foremost as the story of my own quest into the unknown, regardless of its scientific status. I try to be self-aware enough to accept that the likelihood of me producing anything of long-term scientific value is small. However, at the very least my theoretical proposal does fit the perceptions that have come to my attention, and given my perspective on the Universe, it is my best shot. In the final analysis this personal project may have nothing whatsoever to do with official science, but it still remains as at least a partially successful attempt to create a unifying “myth” around some of the disparate raw texts generated by our society, texts which are deemed to contain the “facts”.

Thanks to the technological innovations of the Print On Demand (POD) companies, book printing nowadays is not just for best selling exclusive elites, and the canonical publishing process can be circumvented. Like other technological changes this has resulted in a shift of controlling interests, which in turn has caused some hard feelings. However, balancing potential Internet exposure against both the high feelings generated by non-canonical publishing, and the improbability of a single draught manuscript even being looked at, I decided to publish my story in a POD book called “Gravity and Quantum Non-Linearity”. A limited number of free copies of this book are available for those who might wish to seriously review it. However, don’t necessarily expect any more than an idiosyncratic excursion into mathematics and physics. If it’s going to be the theory that will ultimately sweep the board then it’s just as well I wrote that paper on probability, because I think I am going to need all the “luck” I can get. If you want to find out more about that luck (or lack of it) try “Google=Quantum+Non-Linearity”, because, like all freelancers, “I’m feeling lucky”. You’ve got to; otherwise you wouldn’t get out of bed in the morning.

c. Timothy V Reeves, June 2005

With many thanks to AuthorsOnline

Note: To access the probability article try Google=Reeves+probability