[I wrote this post 2 weeks ago, but didn't get around to posting it due to being in China, with a slow Net connection..]

En route from Seoul to Hong Kong, exhausted from a 5 hour night's sleep following a 4 hour night's sleep, over-jazzed by too much strong coffee (which I rarely drink), stomach-sickened by ordering and consuming random dishes in a Korean restaurant via pointing at random hieroglyphs on the menu and hoping vainly for the best ... head full of Mark Twain's wacky biography which I just finished ... irrationally nervous due to having left my oldest son in Japan to tour around on his own for a week before his Japanese class in Kanazawa starts (yes, he's mature enough to handle himself ... and Japan is a damnably safe place, aside from the risk of spending all your money ... but even I the ultimate anarchist parent can't help a bit of worry) ... dulled almost but not quite to a stupor by a relentless series of software technology oriented business meetings (all with wonderful and interesting people, but still, there's only so much meeting I can take) ... I picked up Lee Smolin's book on quantum gravity, which I bought for my physician-cum-maverick-physics-theorist father-in-law years ago but never read myself ... and while reading a totally irrelevant passage and eating the oddly sour fish that passes for food on Korean Air, some very simple and obvious ideas popped into my mind, and I realized to my surprise that, via converging together several streams that have been tumbling through my head for years, I'd happened upon what appeared to be the correct probabilistic logic of unified quantum gravity.

I'm eager to write up this logic in a paper, but, I've promised myself not to undertake anything serious -- except tasks critical for Novamente as a business, or in order to fulfill obligations already incurred -- until the OpenCog Prime (http://www.opencog.org) wiki pages are done (maybe another 20-30 hours of work, but hours for concentrated writing/editing work are very hard to come by these days due to the combination of business obligations and ongoing research projects needing supervision and/or feedback).... But I'll indulge myself in a brief blog post on the topic as a stopgap - partly to ensure the idea doesn't escape from my mind tonight when I finally slip into the deep sleep my body's been craving for 72 hours or so....

Twain's bio was a fascinating read, by the way. Three things among many others struck me, viewing his life-story from a selfish view in terms of its potential lessons for my own life. One is the way he spent a load of his time on stuff other than writing -- business of various sorts, as well as lecturing, traveling and so forth. But these "distractions" didn't seem to detract from his productivity as a writer as much as I would have thought -- they filled his head with stimulation and ideas, and no doubt made his writing more interesting than if he'd just sat home writing all day. Second is the romance he found in business pursuits ... which reminded me a bit of Rimbaud, who gave up poetry as a very young man after too few years as a writer, and wasted his twenties chasing African gold, ultimately dying from poisoning attained via wearing gold under his undies to hide it from thieves. Rimbaud, due to his premature death among other issues, failed to transform his digressive life experiences into art. I can see in my own psychology the excitement that the business world held for these people: it does stimulate parts of the mind that creative art and science don't touch. Finally I'm struck by the amount of real trash literature Twain produced. I'm reminded of Danilo Kis's (a truly great Serbian writer -- thx to Predrag Janicic for waking me up to him) comment that he didn't write his complete works, only his selected works. Twain was not like that. Twain's best work was awesome, his worst work was terrible. He could have omitted a good 50% of his production and his legacy would be greater not less. Philip K. Dick had the same property: there's Ubik ... and then there's Dr. Futurity.... The lesson for me is, I suppose, not to worry too much about spending time on apparently digressive pursuits (like writing this blog post, um) -- so long as they're feeding the creative engine one way or another -- and given the limited time I have for creative pursuits, to try hard to be more like Kis than Twain or Dick, and filter out crappier works before I take the time to produce them.

Another striking thing about Twain was the way he foresaw the power of machinery to alleviate human suffering. A lesson that seems obvious these days but was surprisingly poorly understood in his times, even though the industrial revolution was in full swing and new mechanical inventions of all sorts were pouring out of human minds at an amazing rate. If you haven't read it, his Connecticut Yankee in King Arthur's Court -- arguably the first American SF novel -- is a hilarious and deeply insightful premonition of the promise and peril of advanced technology. As a time travel fable, it's got Back to the Future beat by a long shot, without need of paradoxical absurdities beyond those intrinsic to human nature.

And now... what about quantum gravity...

Three threads need to be drawn together, into a single mathematical formalism. But there really seems no obstacle to doing so. (Except time of course, which is a distressingly rare commodity for me these days. Must confess to a bit of jealousy of my son as he wandered off from Tokyo to Kyoto, with no specific plan for spending his days, other than to amuse himself. I never intended to accumulate so many obligations -- keep companies running, organize conferences, pay other peoples' college tuition, close a mortgage, finish a book, pay child support each month, drive the kids to and from school, blah blah blah blah blah ... I once actually thought I'd live the life of the "free and easy wanderer" from the Chuang Tzu (on my mind as the flight I'm on approaches China), or maybe of Paul Erdos who freeloaded off one friend after another as he spent his life journeying around the world doing mathematics and taking drugs ... I never envisioned taking on all this responsibility for other people (kids, wife, ex-wife) and organizations (companies, non-profits, egads!) ... yet it's all wonderful, interesting stuff ... people and ideas I really care about ... so I'd really be an ass to complain ... it's a fantastic time to be alive ... yet not quite as fantastic as a few decades hence will likely be, when minds will be far more fully liberated from the horrifying/stultifying constraints of legacy human physiology ... but, well, anyway...)

Quantum gravity!

Thread one was invented by Saul Youssef, and I've written about it before. Check out the lovely bibliography he's assembled at

Brilliant, brilliant man. A hero of our time! Someone give that man a muffin!!

The observation here is that if you're willing to take the step of assuming probabilities are complex rather than real numbers, the basic rules of quantum theory fall right out. This is one of those things that seems shocking and weird at first, and then seems tremendously obvious after you read through the math. Three cheers for Saul Youssef!!

Thread two is something I came up with a couple years ago, and wrote up in a paper which I'm in the middle of submitting for publication. I sent the paper to a journal and they sent it back asking me to provide names and mailing addresses of eight referees able to review the paper. I've been lagging on that task along with a huge amount of other stupid paperwork that's accumulated during the last N years. I guess the editor couldn't think of anyone to send it to. The idea, anyway, is infinite order probability.

An ordinary probability is a probability of an event. A probability distribution is a function that assigns a probability to each one of a set of mutually exclusive outcomes of some event (the different values assigned to different exclusive outcomes must sum to one). A second-order probability is a probability distribution over probability distributions ... it's a function that assigns a probability to each one of a set of probability distributions. A third-order probability ... etc.

An infinite-order probability is a function that assigns a probability to each one of a set of infinite-order probability distributions. Sounds odd, but it's a mathematically consistent idea, as I showed in my paper. I also showed that these oddball entities are closely related to some much more familiar and intuitive mathematical entities, Markov matrices.

The third thread is causal networks. A foundational notion in general relativity is causality. The causal network of events, in relativity, tells you for any pair (A, B) of events, which ones have the property that A is causal for B. This has to do with the finitude of the speed of light: if A and B are too close in time and too distant in space, there may be no way for A and B to causally affect each other.

If A and B are not causally related, there may still be some event C so that C is causal for both A and B. In that case we may say that, probabilistically speaking, A and B are independent conditional on C. That is,

The causal network gives us a set of independence assumptions on the space of events.

General relativity is in essence a dynamic on causal networks: it tells you how a causal network at one time (plus some extra information) gives rise to a different, related causal network at a subsequent time.

Finally let's reflect on what Smolin (see Three Roads to Quantum Gravity) calls the "strong holographic principle." His reasoning for this principle is subtle and involves the Bekenstein bound and related results, which state that all the information about the interior of some physical region, may actually be thought of as being contained on the surface of that regions. (He explains this better than I could, so I'll just refer you to his book.)

What the principle says is: a la Nietzsche, there are only surfaces. Re-read Nietzsche's Twilight of the Idols and you'll see that he presaged quantum gravity here, in a similar way to how he presaged quantum theory proper in his vision (in The Will to Power) of the universe as composed of a dancing swarm of discrete interacting quanta. Kant posited phenomena and noumena, Nietzsche saw only noumena. Smolin also. Smolin views the universe as a collection of surfaces, each one defined as a relationship among other surfaces. Put in words like this, it sounds mystical and fuzzy, but there's math to back it up -- the words just hint at the mathematical reality.

But is each of these Smolin surfaces definitively known? No. Each one is probabilistically known. And if each of these surfaces is to be thought of as a relationship between other surfaces, then this means each of these surfaces is most directly modeled as a hyperset (see my prior blog posts on these mathematical constructs). (This is not how Smolin models things mathematically, but I doubt he'd be opposed, as he's used equally recondite math structures such as topoi.) So these surfaces should be modeled as probabilistic hypersets -- aka infinite-order probability distributions.

But what kinds of probabilities should be involved in these distributions? Clearly, Youssef has taught us, these should be complex probability distros -- or in my variation, infinite-order complex probability distributions.

The inescapable conclusion is: The physical universe is a dynamically evolving causal network defined on an infinite-order complex probability distribution.

You read it first here, folks.. ;-O

Or, to put it a bit more conservatively: A useful, perhaps critical language for modeling quantum gravity phenomena is the logic of causal networks on infinite-order complex probability distributions.

There are fun connections here with the psychology of self-awareness and free will, as I've discussed in a couple previous blog posts (follow the links). According to those blog posts, a good way to model reflective awareness would be using infinite-order real probability distributions; and a good way to model will would be using causal networks on these distributions. What quantum theory introduces is the complex-number probability aspect, which makes everything counterintuitive and weird.

I hope I can really find time -- amidst the manifold obligations of middle age plus the not incidental life-task of creating superhuman AI, plus other distractions like bioinformatics and fiction and music and what-not ... and family and the occasional personal entertainment -- to write these ideas up carefully, because I really do think they have deep potential.

There seem to be more connections lurking here: the logic of causal networks seems somehow inescapably tied up with Clifford algebras, providing a tie-in with my algebra of multi-boundary forms (my only publication in a physics journal so far, but it's really a math paper). Presumably one can go from causality somehow straight to discrete Clifford algebras using some kind of axiomatic derivation, and from there to the various beautiful algebraic symmetries underlying modern physics ... Gell-Mann's "Eightfold Path" and its kin ... but anyway, the flight's about to land and the stewardess wants me to put away my laptop, so the blog post is gonna end .. I'll post it online when I get back to the hotel assuming there's functional internet there ... the inimitable Yan King Yin (famous on various AI email lists) is picking me up at the airport and I'm curious to meet him, although I'm so worn out I'm not sure I'll be lucid enough to milk the occasion's potential for lively AI discussions....

How about the bloody Yverse (see previous blog post on this)? Each Smolin surface ... each relation in the network of interdefining hyperrelations ... defines its own multiverse: a quantum multiverse relative to its own perspective. The network of surfaces (aka relationships) is then a Yverse. QED.

Another day, another dozen digressions ... it's SOOOOO tempting to take a few days and formalize the logic of causal nets over complex infinite-order distros, but instead (inbetween biz meetings and AI research meetings and conference speeches and meals with AI colleagues) I'll spend my "spare" hours in the next few weeks on the OpenCogPrime documents ... a very tedious matter of taking about 50 wiki pages from the Novamente wiki site and editing them down into OpenCogPrime rather than Novamente Cognition Engine pages... yecch...

(Any wealthy patrons out there want to hire me a secretary, a housecleaner and a scientific assistant? I can't promise the Singularity will be accelerated by a few years but it's a definite possibility. For sure a lot more fascinating math, art and science would be generated were I to be thus endowed. (And, getting back to Twain, I wonder what additional great works he would have produced if some of his rich friends had decided to fund him, sparing him the financial anxiety that led him to waste years of his life on various harebrained business schemes. Yeah, they provided grist for his creative mill ... but there's such a thing as too much grist and not enough time to mill.) But I can't complain too much (er, OK, wait, I guess I am...) ... whenever I get TOO frustrated at the realization that 50% of my really good ideas and creations will remain forever unarticulated or un-worked-out-in-detail because I've failed to be born rich or become rich (so far), I remind myself of my favorite Haruki Murakami character, the Mongolian Skin-Peeler ... a World War II torture artist who tortured Chinese prisoners of war by slowly peeling their skin off ... as I see Ulaanbaator on the video screen of the plane as it approaches Hong Kong (haha, I'm a bad boy and failed to shut off my laptop when instructed ... how very non-Oriental of me!!) it's hard for me not to feel thankful that I'm not one of his victims ... I've got my epidermis attached to my dermis, woo hoo! ... and I at least have time to work out a nontrivial percentage of the cool ideas and creations that course through my overheated brain...)

(While you're at it, imaginary patron, recruiting Novamente LLC a CEO with lots of game or virtual world industry experience would be nice. I think I'm doing a decent job as CEO, with help from Bruce and Cassio and Wendy and my other wonderful colleagues, but it would be nice to have a sufficiently complete management team that I could spend 80% of my Novamente-time on science rather than business. And you may as well recruit us a kick-ass project manager too, so Cassio can help me out with research and retire from project management. (Ok, dream on, Ben.... And remember the Mongolian Skin-Peeler....). And while you're at it, throw in maybe $1M per year so that I can actually fund a team of kick-ass programmers to build a thinking machine ... in case you haven't heard I have a pretty detailed and well-argued design for one, but it's getting built bloody fucking slowly due to lack of funding, and because it's not the sort of thing where partial progress yields exciting incremental results, any more than building 30% of a human brain would yield a 30% functional human... but I dididididididigress ;_)

(I think my wife is really, really tired of hearing about the Mongolian Skin-Peeler. He seems to occupy an unjustifiably prominent role in my emotional topography. Read The Wind-Up Bird Chronicle.)

Mark Twain, I add, would have been a hell of a blogger; far more entertaining than me. He wrote a dozen letters each day back then in the pre-digital dark ages.

Time to get off the plane.

The observation here is that if you're willing to take the step of assuming probabilities are complex rather than real numbers, the basic rules of quantum theory fall right out. This is one of those things that seems shocking and weird at first, and then seems tremendously obvious after you read through the math. Three cheers for Saul Youssef!!

Thread two is something I came up with a couple years ago, and wrote up in a paper which I'm in the middle of submitting for publication. I sent the paper to a journal and they sent it back asking me to provide names and mailing addresses of eight referees able to review the paper. I've been lagging on that task along with a huge amount of other stupid paperwork that's accumulated during the last N years. I guess the editor couldn't think of anyone to send it to. The idea, anyway, is infinite order probability.

An ordinary probability is a probability of an event. A probability distribution is a function that assigns a probability to each one of a set of mutually exclusive outcomes of some event (the different values assigned to different exclusive outcomes must sum to one). A second-order probability is a probability distribution over probability distributions ... it's a function that assigns a probability to each one of a set of probability distributions. A third-order probability ... etc.

An infinite-order probability is a function that assigns a probability to each one of a set of infinite-order probability distributions. Sounds odd, but it's a mathematically consistent idea, as I showed in my paper. I also showed that these oddball entities are closely related to some much more familiar and intuitive mathematical entities, Markov matrices.

The third thread is causal networks. A foundational notion in general relativity is causality. The causal network of events, in relativity, tells you for any pair (A, B) of events, which ones have the property that A is causal for B. This has to do with the finitude of the speed of light: if A and B are too close in time and too distant in space, there may be no way for A and B to causally affect each other.

If A and B are not causally related, there may still be some event C so that C is causal for both A and B. In that case we may say that, probabilistically speaking, A and B are independent conditional on C. That is,

P(A & B | C) = P(A | C) P(B |C)

The causal network gives us a set of independence assumptions on the space of events.

General relativity is in essence a dynamic on causal networks: it tells you how a causal network at one time (plus some extra information) gives rise to a different, related causal network at a subsequent time.

Finally let's reflect on what Smolin (see Three Roads to Quantum Gravity) calls the "strong holographic principle." His reasoning for this principle is subtle and involves the Bekenstein bound and related results, which state that all the information about the interior of some physical region, may actually be thought of as being contained on the surface of that regions. (He explains this better than I could, so I'll just refer you to his book.)

What the principle says is: a la Nietzsche, there are only surfaces. Re-read Nietzsche's Twilight of the Idols and you'll see that he presaged quantum gravity here, in a similar way to how he presaged quantum theory proper in his vision (in The Will to Power) of the universe as composed of a dancing swarm of discrete interacting quanta. Kant posited phenomena and noumena, Nietzsche saw only noumena. Smolin also. Smolin views the universe as a collection of surfaces, each one defined as a relationship among other surfaces. Put in words like this, it sounds mystical and fuzzy, but there's math to back it up -- the words just hint at the mathematical reality.

But is each of these Smolin surfaces definitively known? No. Each one is probabilistically known. And if each of these surfaces is to be thought of as a relationship between other surfaces, then this means each of these surfaces is most directly modeled as a hyperset (see my prior blog posts on these mathematical constructs). (This is not how Smolin models things mathematically, but I doubt he'd be opposed, as he's used equally recondite math structures such as topoi.) So these surfaces should be modeled as probabilistic hypersets -- aka infinite-order probability distributions.

But what kinds of probabilities should be involved in these distributions? Clearly, Youssef has taught us, these should be complex probability distros -- or in my variation, infinite-order complex probability distributions.

The inescapable conclusion is: The physical universe is a dynamically evolving causal network defined on an infinite-order complex probability distribution.

You read it first here, folks.. ;-O

Or, to put it a bit more conservatively: A useful, perhaps critical language for modeling quantum gravity phenomena is the logic of causal networks on infinite-order complex probability distributions.

There are fun connections here with the psychology of self-awareness and free will, as I've discussed in a couple previous blog posts (follow the links). According to those blog posts, a good way to model reflective awareness would be using infinite-order real probability distributions; and a good way to model will would be using causal networks on these distributions. What quantum theory introduces is the complex-number probability aspect, which makes everything counterintuitive and weird.

I hope I can really find time -- amidst the manifold obligations of middle age plus the not incidental life-task of creating superhuman AI, plus other distractions like bioinformatics and fiction and music and what-not ... and family and the occasional personal entertainment -- to write these ideas up carefully, because I really do think they have deep potential.

There seem to be more connections lurking here: the logic of causal networks seems somehow inescapably tied up with Clifford algebras, providing a tie-in with my algebra of multi-boundary forms (my only publication in a physics journal so far, but it's really a math paper). Presumably one can go from causality somehow straight to discrete Clifford algebras using some kind of axiomatic derivation, and from there to the various beautiful algebraic symmetries underlying modern physics ... Gell-Mann's "Eightfold Path" and its kin ... but anyway, the flight's about to land and the stewardess wants me to put away my laptop, so the blog post is gonna end .. I'll post it online when I get back to the hotel assuming there's functional internet there ... the inimitable Yan King Yin (famous on various AI email lists) is picking me up at the airport and I'm curious to meet him, although I'm so worn out I'm not sure I'll be lucid enough to milk the occasion's potential for lively AI discussions....

How about the bloody Yverse (see previous blog post on this)? Each Smolin surface ... each relation in the network of interdefining hyperrelations ... defines its own multiverse: a quantum multiverse relative to its own perspective. The network of surfaces (aka relationships) is then a Yverse. QED.

Another day, another dozen digressions ... it's SOOOOO tempting to take a few days and formalize the logic of causal nets over complex infinite-order distros, but instead (inbetween biz meetings and AI research meetings and conference speeches and meals with AI colleagues) I'll spend my "spare" hours in the next few weeks on the OpenCogPrime documents ... a very tedious matter of taking about 50 wiki pages from the Novamente wiki site and editing them down into OpenCogPrime rather than Novamente Cognition Engine pages... yecch...

(Any wealthy patrons out there want to hire me a secretary, a housecleaner and a scientific assistant? I can't promise the Singularity will be accelerated by a few years but it's a definite possibility. For sure a lot more fascinating math, art and science would be generated were I to be thus endowed. (And, getting back to Twain, I wonder what additional great works he would have produced if some of his rich friends had decided to fund him, sparing him the financial anxiety that led him to waste years of his life on various harebrained business schemes. Yeah, they provided grist for his creative mill ... but there's such a thing as too much grist and not enough time to mill.) But I can't complain too much (er, OK, wait, I guess I am...) ... whenever I get TOO frustrated at the realization that 50% of my really good ideas and creations will remain forever unarticulated or un-worked-out-in-detail because I've failed to be born rich or become rich (so far), I remind myself of my favorite Haruki Murakami character, the Mongolian Skin-Peeler ... a World War II torture artist who tortured Chinese prisoners of war by slowly peeling their skin off ... as I see Ulaanbaator on the video screen of the plane as it approaches Hong Kong (haha, I'm a bad boy and failed to shut off my laptop when instructed ... how very non-Oriental of me!!) it's hard for me not to feel thankful that I'm not one of his victims ... I've got my epidermis attached to my dermis, woo hoo! ... and I at least have time to work out a nontrivial percentage of the cool ideas and creations that course through my overheated brain...)

(While you're at it, imaginary patron, recruiting Novamente LLC a CEO with lots of game or virtual world industry experience would be nice. I think I'm doing a decent job as CEO, with help from Bruce and Cassio and Wendy and my other wonderful colleagues, but it would be nice to have a sufficiently complete management team that I could spend 80% of my Novamente-time on science rather than business. And you may as well recruit us a kick-ass project manager too, so Cassio can help me out with research and retire from project management. (Ok, dream on, Ben.... And remember the Mongolian Skin-Peeler....). And while you're at it, throw in maybe $1M per year so that I can actually fund a team of kick-ass programmers to build a thinking machine ... in case you haven't heard I have a pretty detailed and well-argued design for one, but it's getting built bloody fucking slowly due to lack of funding, and because it's not the sort of thing where partial progress yields exciting incremental results, any more than building 30% of a human brain would yield a 30% functional human... but I dididididididigress ;_)

(I think my wife is really, really tired of hearing about the Mongolian Skin-Peeler. He seems to occupy an unjustifiably prominent role in my emotional topography. Read The Wind-Up Bird Chronicle.)

Mark Twain, I add, would have been a hell of a blogger; far more entertaining than me. He wrote a dozen letters each day back then in the pre-digital dark ages.

Time to get off the plane.

## 12 comments:

I think that grokking the whole of this post will resemble the one-thousand-year digestion (or something like that) of that Star Wars monster. Meanwhile... what in hell is a Mongolian Skin-Peeler? :)

You might be able to get somewhat rich does by making a pet in SecondLife that does vaguely interesting things. Whether it's built on the Novamente platform or not.

I'm reminded that I recently read (in The Modern Mind: An intellectual history of of the 20th century) that in the 1930s the

Journal of the American Chemical Societypublished many of Linus Pauling's papers unrefereed, because they could find no one to referee them!But you probably knew that already from co-authoring the man's biography (Linus Pauling: A Life In Science And Politics is still on my reading list!)

Someone had better sound a negative note here:

Complex probability makes no sense as a foundational concept.

Neither do infinite-order probabilities, really. Ben defines an infinite-order probability recursively, as a probability distribution over infinite-order probabilities. I do not see how this can be anything but a formal pseudo-concept, i.e. a generalization of a formalism intended to represent some reality, in such a way that the generalized formalism is no longer capable of representing the reality.

I might say in passing that a lot of math-intensive crypto-metaphysical speculation in the present consists of doing this - of taking some classical formalization of a known aspect of reality, formally generalizing it to the point of disconnection from reality, and then saying 'but maybe that's how reality is!' The great wellspring of inspiration in this regard is quantum mechanics, but there are other sources too.

A curious property of these infinite-order 'probabilities' is that they have an indexical aspect. Being themselves an element of the space on which they are a distribution, they must each assign

themselvesa 'probability'. Presumably Ben develops this stuff in his paper, perhaps by analogy with other formal theories of ill-founded sets, but (as must be abundantly obvious) I don't expect it to be relevant to anything real.As for the final ingredient in this synthesis, 'causal networks' - that's not a problematic concept, but it's a super-generic one. At least it provides an almost-reality-based grounding for the other two ingredients.

So to sum up this critique, Ben, I think the part where you really go off the rails is with the 'infinite order' stuff. It would not be surprising to learn that quantum gravity can be formulated in terms of complex probabilities over causal networks. Every real-world quantum theory already has some element of causality in it, so you could even say that the theories we have already fit that description, for a sufficiently vague definition of 'causal network'.

But 'infinite order probabilities' sound to me like a

shallowsort of generalization. All these years after Cantor and Peano, in a sense there is nothing more obvious than taking a mathematical operation and iterating it, even infinitely many times, in order to produce new concepts. But that doesn't mean that what results is profound, any more than the set {{{...}}} is necessarily the answer to the riddle of being, somehow - though I can't say how, exactly, it 's just gotta be, because it's -infinite!There is a well-known book from the 1920s, Dunne's

An Experiment with Time, in which the grand idea is that time can be observed from outside, but then that observer must inhabit its own time, and then there will be a third time from which the second time can be observed, and so on. Perhaps you will agree that if one has no particular apriori attraction to whatever metaphysical assumptions made this mode of thought appealing to Dunne, then that particular example of an infinitely iterated move - 'posit a second time outside the first' - seems merely an exercise in mental gymnastics, rather than a profound revelation. I submit that the same goes for infinite order probabilities.If I try to imagine the feeling behind your sense of revelation, I think of the first original sentence I ever produced in Korean. It was 'I will go shopping by bus'. Not an earthshattering announcement but I was very pleased with myself, that I had mastered the combinatorics of Korean to the point that I could generate original, syntactically correct utterances.

I figure that your sense of revelation resulted from doing the same with all the current frontier thoughts you were thinking. You managed to synthesize them all into a single concept which is at least formally (if rather vaguely) well-defined, so for a moment it seemed like the answers to everything must inhabit that synthesis, if only its implications could be worked out. Whereas I think that at best what you've produced is another curiosity for the mathematical ontologist, something which might belong in a parable by Hofstadter but which is not going to be the answer to anything in real life.

Speaking to the galleries now, I wouldn't say that this flaming of Ben's theory of everything reflects badly on his work in AI. That involves actual finite algorithms and actual code. I think what we're seeing is a sort of recreational/inspirational creative activity which then gets brought down to earth when one returns to working with real computers and real programming languages. Ideally one's philosophy would also avoid speculative excess, but if the choice is between excess and lack, it's better to have excess. As the saying goes, too much is always better than not enough.

Mitch: the concept of a "formal pseudo-concept" doesn't make much sense to me. I can imagine, long ago, someone arguing to me that negative numbers are just a formal pseudoconcept. At first, they probably seemed just as nonsensical as complex or infinite-order probabilities.

But anyways, the speculations in this blog post obviously were not written up in such a way as to withstand skeptical criticism! Whether I will ever take time to try to write them up in such a way (or whether if I do so I'll be successful), of course remains to be seen...

For sure, my wild speculations are not 100% accurate in physics, AI or any domain ;-) ... in physics I have never progressed beyond wild speculations and interesting-looking math formalisms, whereas in AI I've gone a lot further due to having expended orders of magnitude more time... and, some of my AI speculations survived the concretization process, others not...

Whether infinite-order probabilities are an interesting generalization or not is another question, separate from any of my wild physics-related speculations. I think they will prove to be, but I have to do more math to prove it, and I lack the time right now.

IMO Barwise and Etchemendy showed that hypersets are an interesting generalization of sets, via showing they are a good way to model the semantics of self-referential statements. My feeling that infinite-order probabilities are useful is in the same vein.

Mitch: Complex probabilities DO make sense, just as much as quantum logic makes sense, or the use of quantum state vectors to describe reality makes sense.

The counterintuitiveness of quantum mechanics implies that somewhere in the formalism of QM is going to be something counterintuitive. Whether one situates it in the probability formalism or somewhere else, it's still gonna be there.

As complex probabilities are mathematically consistent, if you find them a "senseless" foundation, the problem probably lies with the evolved biases of your human brain rather than w/ the formalism... '-)

I tend to agree with Mitchell. Something that is infinite cannot be compute (in a finite time). We live in a finite universe. If the universe is to be computable (which is still unproven, and perhaps unprovable), then there can be no infinites.

How do you get from n to n+1 if it requires an infinite number of steps?

Cannot 2nd order probabilities be converted to 1st order? If you have 2 distributions, one applying with p1 and the other with 1-p1, then you can make a weighted average of them, which is a single distribution. Carry the process on recursively and you can reduce nth order probability to 1st order, for any n.

I don't doubt that there are some interesting mathematical things one could say about the 1st order probabilities that arise from this sort of reduction, for large enough n. To me, therefore, that would be the interesting question: what does a first order probability distribution look like that is the result of reduction from large n, for given properties of the higher order distributions?

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