Rewriting and emergent physics

Tempora mutantur, nos et mutamur in illis.
(Times change, and we change with them.)

— Latin Adage, 16t‍h-century Germany.

I want to understand how a certain kind of mathematical system can act as a foundation for a certain kind of physical cosmos.  The ultimate goal of course would be to find a physical cosmos that matches the one we're in; but as a first step I'd like to show it's possible to produce certain kinds of basic features that seem prerequisite to any cosmos similar to the one we're in.  A demonstration of that much ought, hopefully, to provide a starting point to explore how features of the mathematical system shape features of the emergent cosmos.

The particular kind of system I've been incrementally designing, over a by-now-lengthy series of posts (most recently yonder), is a rewriting system —think λ-calculus— where a "term" (really more of a graph) is a state of the whole spacetime continuum, a vast structure which is rewritten according to some local rewrite rules until it reaches some sort of "stable" state.  The primitive elements of this state have two kinds of connections between them, geometry and network; and by some tricky geometry/network interplay I've been struggling with, gravity and the other fundamental forces are supposed to arise, while the laws of quantum physics emerge as an approximation good for subsystems sufficiently tiny compared to the cosmos as a whole.  That's what's supposed to happen for physics of the real world, anyway.

To demonstrate the basic viability of the approach, I really need to make two things emerge from the system.  The obvious puzzle in all this has been, from the start, how to coax quantum mechanics out of a classically deterministic rewriting system; inability to extract quantum mechanics from classical determinism has been the great stumbling block in devising alternatives to quantum mechanics for about as long as quantum mechanics has been around (harking back to von Neumann's 1932 no-go theorem).  I established in a relatively recent post (thar) that the quintessential mathematical feature of quantum mechanics, to be derived, is some sort of wave equation involving signed magnitudes that add (providing a framework in which waves can cancel, so producing interference and other quantum weirdness).  The geometry/network decomposition is key for my efforts to do that; not something one would be trying to achieve, evidently, if not for the particular sort of rewriting-based alternative mathematical model I'm trying to apply to the problem; but, contemplating this alternative cosmic structure in the abstract, starting from a welter of interconnected elements, one first has to ask where the geometry — and the network — and the distinction between the two — come from.

Time after time in these posts I set forth, for a given topic, all the background that seems relevant at the moment, sift through it, glean some new ideas, and then set it all aside and move on to another topic, till the earlier topic, developing quietly while the spotlight is elsewhere, becomes fresh again and offers enough to warrant revisiting.  It's not a strategy for the impatient, but there is progress, as I notice looking back at some of my posts from a few years ago.  The feasibility of the approach hinges on recognizing that its value is not contingent on coming up with some earth-shattering new development (like, say, a fully operational Theory of Everything).  One is, of course, always looking for some earth-shattering new development; looking for it is what gives the whole enterprise shape, and one also doesn't want to become one of those historical footnotes who after years of searching brushed past some precious insight and failed to recognize it, so that it had to wait for some other researcher to discover it later.  But, as I noted early in this series, the simple act of pointing out alternatives to a prevailing paradigm in (say) physics is beneficial to the whole subject, like tilling soil to aerate it.  Science works best with alternatives to choose between; and scientists work best when their thoughts and minds are well-limbered by stretching exercises.  For these purposes, in fact, the more alternatives the merrier, so that as a given post is less successful in reaching a focused conclusion it's more likely to compensate in variety of alternatives.

In this series of physics posts, I keep hoping to get down to mathematical brass tacks; but very few posts in the series actually do so (with a recent exception in June of last year).  Alas, though the current post does turn its attention more toward mathematical structure, it doesn't actually achieve concrete specifics.  Getting to the brass tacks requires first working out where they ought to be put.

Contents
Dramatis personae
Connections
Termination
Duality
Scale

Dramatis personae

A rewriting calculus is defined by its syntax and rewriting rules; for a given computation, one also needs to know the start term.  In this case, we'll put off for the moment worrying about the starting configuration for our system.

The syntax defines the shapes of the pieces each state (aka term, graph, configuration) is made of, and how the pieces can fit together.  For a λ-like calculus, the pieces of a term would be syntax-tree nodes; the parent/child connections in the tree would be the geometry, and the variable binding/instance connections would be the network.  My best guess, thus far, has been that the elementary pieces of the cosmos would be events in spacetime.  Connections between events would, according to the general scheme I've been conjecturing, be separated into local connections, defining spacetime, and non-local connections, providing a source of seeming-randomness if the network connections are sufficiently widely distributed over a cosmos sufficiently vast compared to the subsystem under consideration.

I'm guessing that, to really make this seeming-randomness trick work, the cosmos ought to be made up of some truly vast number of events; say, 10⁶⁰, or 10⁸⁰, or on up from there.  If the network connections are really more-or-less-uniformly distributed over the whole cosmos, irrespective of the geometry, then there's no obvious reason not to count events that occur, say, within the event horizon of a black hole, and from anywhere/anywhen in spacetime, which could add up to much more than the currently estimated number of particles in the universe.  Speculatively (which is the mode all of this is in, of course), if the galaxy-sized phenomena that motivate the dark-matter hypothesis are too big, relative to the cosmos as a whole, for the quantum approximation to work properly —so one would expect these phenomena to sit oddly with our lesser-scale physics— that would seem to suggest that the total size of the cosmos is finite (since in an infinite cosmos, the ratio of the size of a galaxy to the size of the universe would be exactly zero, no different than the ratio for an electron).  Although, as an alternative, one might suppose such an effect could derive, in an infinite cosmos, from network connections that aren't distributed altogether uniformly across the cosmos (so that connections with the infinite bulk of things get damped out).

With the sort of size presumed necessary to the properties of interest, I won't be able to get away with the sort of size-based simplifying trick I've gotten away with before, as with a toy cosmos that has only four possible states.  We can't expect to run a simulation with program states comparable in size to the cosmos; Moore's law won't stretch that far.  For this sort of research I'd expect to have to learn, if not invent, some tools well outside my familiar haunts.

The form of cosmic rewrite rules seems very much up-for-grabs, and I've been modeling guesses on λ-like calculi while trying to stay open to pretty much any outre possibility that might suggest itself.  In λ-like rewriting, each rewriting rule has a redex pattern, which is a local geometric shape that must be matched; it occurs, generally, only in the geometry, with no constraints on the network.  The redex-pattern may call for the existence of a tangential network connection —the β-rule of λ-calculus does this, calling for a variable binding as part of the pattern— and the tangential connection may be rearranged when applying the rule, just as the local geometry specified by the redex-pattern may be rearranged.  Classical λ-calculus, however, obeys hygiene and co-hygiene conditions:  hygiene prohibits the rewrite rule from corrupting any part of the network that isn't tangent to the redex-pattern, while co-hygiene prohibits the rewrite rule from corrupting any part of the geometry that isn't within the redex-pattern.  Impure variants of λ-calculus violate co-hygiene, but still obey hygiene.  The guess I've been exploring is that the rewriting rules of physics are hygienic (and Church-Rosser), and gravity is co-hygienic while the other fundamental forces are non-co-hygienic.

I've lately had in mind that, to produce the right sort of probability distributions, the fluctuations of cosmic rewriting ought to, in essence, compare the different possible behaviors of the subsystem-under-consideration.  Akin to numerical solution of a problem in the calculus of variations.

Realizing that the shape of spacetime is going to have to emerge from all this, the question arises —again— of why some connections to an event should be "geometry" while others are "network".  The geometry is relatively regular and, one supposes, stable, while the network should be irregular and highly volatile, in fact the seeming-randomness depends on it being irregular and volatile.  Conceivably, the redex-patterns are geometric (or mostly geometric) because the engagement of those connections within the redex-patterns cause those connections to be geometric in character (regular, stable), relative to the evolution of the cosmic state.

The overall character of the network is another emergent feature likely worth attention.  Network connections in λ-calculus are grouped into variables, sub-nets defined by a binding and its bound instances, in terms of which hygiene is understood.  Variables, as an example of network structure, seem built-in rather than emergent; the β-rule of λ-calculus is apparently too wholesale a rewriting to readily foster ubiquitous emergent network structure.  Physics, though, seems likely to engage less wholesale rewriting, from which there should also be emergent structure, some sort of lumpiness —macrostructures— such that (at a guess) incremental scrambling of network connections would tend to circulate those connections only within a particular lump/macrostructure.  The apparent alternative to such lumpiness would be a degree of uniform distribution that feels, to my intuition anyway, unnatural.  One supposes the lumpiness would come into play in the nature of stable states that the system eventually settles into, and perhaps the size and character of the macrostructures would determine at what scale the quantum approximation ceases to hold.

Connections

Clearly, how the connections between nodes —the edges in the graph— are set up is the first thing we need to know, without which we can't imagine anything else concrete about the calculus.  Peripheral to that is whether the nodes (or, for that matter, the edges) are decorated, that is, labeled with additional information.

In λ-calculus, the geometric connections are of just three forms, corresponding to the three syntactic forms in the calculus:  a variable instance has one parent and no children; a combination node has one parent and two children, operator and operand; and a λ-expression has one parent and one child, the body of the function.  For network connections, ordinary λ-calculus has one-to-many connections from each binding to its bound instances.  These λ network structures —variables— are correlated with the geometry; the instances of a variable can be arbitrarily scattered through the term, but the binding of the variable, of which there is exactly one, is the sole asymmetry of the variable and gives it an effective singular location in the syntax tree, required to be an ancestor in the tree of all the locations of the instances.  Interestingly, in the vau-calculus generalization of λ-calculus, the side-effectful bindings are somewhat less uniquely tied to a fixed location in the syntax tree, but are still one-per-variable and required to be located above all instances.

Physics doesn't obviously lend itself to a tree structure; there's no apparent way for a binding to be "above" its instances, nor apparent support for an asymmetric network structure.  Symmetric structures would seem indicated.  A conceivable alternative strategy might use time as the "vertical" dimension of a tree-like geometry, though this would seem rather contrary to the loss of absolute time in relativity.

A major spectrum of design choice is the arity of network structures, starting with whether network structures should have fixed arity, or unfixed as in λ-like calculi.  Unfixed arity would raise the question of what size the structures would tend to have in a stable state.  Macrostructures, "lumps" of structures, are a consideration even with fixed arity.

Termination

In exploring these realms of possible theory, I often look for ways to defer aspects of the theory till later, as a sort of Gordian-knot-cutting (reducing how many intractable questions I have to tackle all at once).  I've routinely left unspecified, in such deferral, just what it should mean for the cosmic rewriting system to "settle into a stable state".  However, at this point we really have no choice but to confront the question, because our explicit main concern is with mathematical properties of the probability distribution of stable states of the system, and so we can do nothing concrete without pinning down what we mean by stable state.

In physics, one tends to think of stability in terms of asymptotic behavior in a metric space; afaics, exponential stability for linear systems, Lyapunov stability for nonlinear.  In rewriting calculi, on the other hand, one generally looks for an irreducible form, a final state from which no further rewriting is possible.  One could also imagine some sort of cycle of states that repeat forever, though making that work would require answers to some logistical questions.  Stability (cyclic or otherwise) might have to do with constancy of which macrostructure each of an element's network connections associates to.

If rewriting effectively explores the curvature of the action function (per the calculus of variations as mentioned earlier), it isn't immediately obvious how that would then lead to asymptotic stability.  At any rate, different notions of stability lead to wildly different mathematical developments of the probability distribution, hence this is a major point to resolve.  The choice of stability criterion may depend on recognizing what criterion can be used in some technique that arrives at the right sort of probability distribution.

There's an offbeat idea lately proposed by Tim Palmer called the invariant set postulate.  Palmer, so I gather, is a mathematical physicist deeply involved in weather prediction, from which he's drawn some ideas to apply back to fundamental physics.  A familiar pattern in nonlinear systems, apparently, is a fractal subset of state space which, under the dynamics of the system, the system tends to converge upon and, if the system state actually comes within the set, remains invariant within.  In my rewriting approach these would be the stable states of the cosmos.  The invariant set should be itself a metric space of lower dimension than the state space as a whole and (if I'm tracking him) uncomputable.  Palmer proposes to postulate the existence of some such invariant subset of the quantum state space of the universe, to which the actual state of the universe is required to belong; and requiring the state of the universe to belong to this invariant set amounts to requiring non-independence between elements of the universe, providing an "out" to cope with no-go theorems such as Bell's theorem or the Kochen–Specker theorem.  Palmer notes that while, in the sort of nonlinear systems this idea comes from, the invariant set arises as a consequence of the underlying dynamics of the system, for quantum mechanics he's postulating the invariant set with no underlying dynamics generating it.  This seems to be where my approach differs fundamentally from his:  I suppose an underlying dynamics, produced by my cosmic rewriting operation, from which one would expect to generate such an invariant set.

Re Bell and, especially, Kochen-Specker, those no-go theorems rule out certain kinds of mutual independence between separate observations under quantum mechanics; but the theorems can be satisfied —"coped with"— by imposing some quite subtle constraints.  Such as Palmer's invariant set postulate.  It seems possible that Church-Rosser-ness, which tampers with independence constraints between alternative rewrite sequences, may also suffice for the theorems.

Duality

What if we treated the lumpy macrostructures of the universe as if they were primitive elements; would it be possible to then describe the primitive elements of the universe as macrostructures?  Some caution is due here for whether this micro/macro duality would belong to the fundamental structure of the cosmos or to an approximation.  (Of course, this whole speculative side trip could be a wild goose chase; but, as usual, on one hand it might not be a wild goose chase, and on the other hand wild-goose-chasing can be good exercise.)

Perhaps one could have two coupled sets of elements, each serving as the macrostructures for the other.  The coupling between them would be network (i.e., non-geometric), through which presumably each of the two systems would provide the other with quantum-like character.  In general the two would have different sorts of primitive elements and different interacting forces (that is, different syntax and rewrite-rules).  Though it seems likely the duals would be quite different in general, one might wonder whether in a special case they could sometimes have the same character, in which case one might even ask whether the two could settle into identity, a single system acting as its own macro-dual.

For such dualities to make sense at all, one would first have to work out how the geometry of each of the two systems affects the dynamics of the other system — presumably, manifesting through the network as some sort of probabilistic property.  Constructing any simple system of this sort, showing that it can exhibit the sort of quantum-like properties we're looking for, could be a worthwhile proof-of-concept, providing a buoy marker for subsequent explorations.

On the face of it, a basic structural difficulty with this idea is that primitive elements of a cosmic system, if they resemble individual syntax nodes of a λ-calculus term, have a relatively small fixed upper bound on how many macrostructures they can be attached to, whereas a macrostructure may be attached to a vast number of such primitive elements.  However, there may be a way around this.

Scale

I've discussed before the phenomenon of quasiparticles, group behaviors in a quantum-mechanical system that appear (up to a point) as if they were elementary units; such eldritch creatures as phonons and holes.  Quantum mechanics is fairly tolerant of inventing such beasts; they are overtly approximations of vastly complicated underlying systems.  Conventionally "elementary" particles can't readily be analyzed in the same way —as approximations of vastly complicated systems at an even smaller scale— because quantum mechanics is inclined to stop at Planck scale; but I suggested one might achieve a similar effect by importing the complexity through network connections from the very-large-scale cosmos, as if the scale of the universe were wrapping around from the very small to the very large.

We're now suggesting that network connections provide the quantum-like probability distributions, at whatever scale affords these distributions.  Moreover, we have this puzzle of imbalance between, ostensibly, small bounded network arity of primitive elements (analogous to nodes in a syntax tree) and large, possibly unbounded, network arity of macrostructures.  The prospect arises that perhaps the conventionally "elementary" particles —quarks and their ilk— could be already very large structures, assemblages of very many primitive elements.  In the analogy to λ-calculus, a quark would correspond to a subterm, with a great deal of internal structure, rather than to a parse-tree-node with strictly bounded structure.  The quark could then have a very large network arity, after all.  Quantum behavior would presumably arise from a favorable interaction between the influence of network connections to macrostructures at a very large cosmic scale, and the influence of geometric connections to microstructures at a very small scale.  The structural interactions involved ought to be fascinating.  It seems likely, on the face of it, that the macrostructures, exhibiting altogether different patterns of network connections than the corresponding microstructures, would also have different sorts of probability distributions, not so much quantum as co-quantum — whatever, exactly, that would turn out to mean.

If quantum mechanics is, then, an approximation arising from an interaction of influences from geometric connections to the very small and network connections to the very large, we would expect the approximation to hold, not at the small end of the range of scales, but only at a subrange of intermediate scales — not too large and at the same time not too small.  In studying the dynamics of model rewriting systems, our attention should then be directed to the way these two sorts of connections can interact to reach a balance from which the quantum approximation can emerge.

At a wild, rhyming guess, I'll suggest that the larger a quantum "particle" (i.e., the larger the number of primitive elements within it), the smaller each corresponding macrostructure.  Thus, as the quanta get larger, the macrostructures get smaller, heading toward a meeting somewhere in the mid scale — notionally, around the square root of the number of primitive elements in the cosmos — with the quantum approximation breaking down somewhere along the way.  Presumably, the approximation also requires that the macrostructures not be too large, hence that the quanta not be too small.  Spinning out the speculation, on a logarithmic scale, one might imagine the quantum approximation working tolerably well for, say, about the middle third of the lower half of the scale, with the corresponding macrostructures occupying the middle third of the upper half of the scale.  This would put the quantum realm at a scale from the number of cosmic elements raised to the 1/3 power, down to the number of cosmic elements raised to the 1/6 power.  For example, if the number of cosmic elements were 10¹²⁰, quantum scale would be from 10⁴⁰ down to 10²⁰ elements.  The takeaway lesson here is that, even if those guesses are off by quite a lot, the number of primitive elements in a minimal quantum could still be rather humongous.

Study of the emergence of quasiparticles seems indicated.

Nabla

They [quaternions] are relatively poor 'magicians'; and, certainly, they are no match for complex numbers in this regard.

— Roger Penrose, The Road to Reality, 2005, §11.2.

In this post, I'm going to explore some deep questions about the nature of quaternion differentiation.

Along the way I'm going to suggest some reasons Penrose's assessment quoted above may be somewhat off-target.  I'm quite interested in Penrose's view of quaternions because he presents a twenty-first century form of the classical arguments against quaternions, with an (afaics) sincere effort at objectivity, by someone who patently does appreciate the profound power of elegant mathematics (in apparent contrast to the vectorists' side of the 1890s debate).  The short-short version:  Not only do I agree that quaternions lack the magic of complex numbers, I think it would be bizarre if they had that magic since they aren't complex numbers — but I see clues suggesting they've other magic of their own.

If I claimed to know just what the magic of quaternions is, it would be safe to bet I'd be wrong; the challenge is way too big for answers to come that easily.  However, in looking for indirect evidence that some magic is there to find, I'll pick up some clues to where to look next, the ambiguous note on which this post will end... or rather, trail off. 

Before I can start in on all that, though, I need to provide some background.

Contents
Setting the stage
Quaternions
Doubting classical nabla
Considering quaternions
Full nabla
Partial derivatives
Generalized quaternions
Rotation
Minkowski
Langlands

Setting the stage

When I was first learning vector calculus as a freshman in college (for perspective, that's about when Return of the Jedi came out), I initially supposed that the use of symbol ∇ in three different differential operators —  ∇,  ∇×,  ∇·  — was just a mnemonic device.  My father, who'd been interested in quaternions since, as best as I can figure, when he was in college (about when Casablanca came out), promptly set me straight:  those operators look similar because they're all fragments of a single quaternion differential operator called nabla:

∇

=

i ∂ ∂x

+

j ∂ ∂y

+

k ∂ ∂z

.

If you know a smattering of vector calculus, you may be asking, isn't that just the definition of the gradient operator?  No, because of the seemingly small detail that the factors i, j, k aren't unit vectors, i, j, k — i, j, k are imaginary numbers.  Which has extraordinary consequences.  I'd better take a moment to explain quaternions.

Quaternions

A vector space over some kind of scalar numbers is the n-tuples of scalars, for some fixed n, together with scalar multiplication (to multiply a vector by a scalar, just multiply all the elements of the vector by that scalar) and vector addition (add corresponding elements of the vectors).  An algebra is a vector space equipped with an internal operation called multiplication ("internal" meaning you multiply a vector by a vector, rather than by a scalar) and a multiplicative identity, such that scalar multiplication commutes with internal multiplication, and internal multiplication is bilinear (fancy term, simple once you've seen it:  each element of the product is a polynomial in the elements of the factors, where each term in each polynomial has one element from each factor).

Whatever interesting properties a particular algebra has are, in a sense, contained in its internal multiplication.  So when we speak of the "algebraic structure" of an algebra, what we're talking about is really just its multiplication table.

Quaternions are a four-dimensional hypercomplex algebra.  They're denoted by the symbol ℍ (after Hamilton, their discoverer).  Hypercomplex just means that the first of the four basis elements is the multiplicative identity, so that the first dimension of the vector space can be identified with the scalars, in this case the real numbers, ℝ.  Traditionally the four basis elements are called 1, i, j, k; which said, hereafter I'll prefer to call the imaginaries i₁, i₂, i₃, and occasionally use i₀ as a synonym for 1.  The four real vector-space components of a quaternion, I'll indicate by putting subscripts 0,1,2,3 on the name of the quaternion; thus, a = a₀ + a₁i₁ + a₂i₂ + a₃i₃ = Σ a_ki_k.

Quaternion multiplication is defined by i₁² = i₂² = i₃² = i₁ i₂ i₃ = −1, where multiplication of the imaginary basis elements is associative (i₁(i₂ i₃) = (i₁ i₂) i₃  and so on) and different imaginary basis elements anticommute (i₁ i₂ = − i₂ i₁  and so on).  The whole multiplication table can be put together from these few rules, and we have the quaternion product (take a deep breath):

ab	=	(a0b0 − a1b1 − a2b2 − a3b3)
		+ i1 (a0b1 + a1b0 + a2b3 − a3b2)
		+ i2 (a0b2 − a1b3 + a2b0 + a3b1)
		+ i3 (a0b3 + a1b2 − a2b1 + a3b0)	.

This is that bilinear multiplication table mentioned earlier, where each element of the product is a simple polynomial in elements of the factors.  If you stare at this a bit, you can also see that when a and b are imaginary (that is, a₀ = b₀ = 0), the real part of the product is minus the dot product of the vectors, and the imaginary part of the product is the cross product of the vectors:  ab = a×b − a·b.

A few handy notations:   real part  re(a) = a₀;   imaginary part  i‍m(a) = a − a₀;   conjugate  a^* = re(a) − i‍m(a);   norm  ||a|| = sqrt (Σ a_k²).

Quaternion multiplication is associative.  Quaternion multiplication is also non-commutative, which was a big deal when Hamilton first discovered quaternions in 1843, because until then all known kinds of numbers had obeyed all the "usual" laws of arithmetic.  But what's really interesting about quaternion multiplication — at least, on the face of it — is that it has unique division.  That is, for all quaternions a and b, where a is non-zero, there is exactly one quaternion x such that  ax = b, and exactly one quaternion x such that  x‍a = b.  In particular, with b=1, this says that every non-zero a has a unique left-multiplicative inverse, and a unique right-multiplicative inverse.  These are actually the same number, which we write

a−1

=

a*  ||a||2

(the conjugate divided by the square of the norm).  So  a a⁻¹ = a⁻¹a = 1.

Division algebras are very special; pathological cases aside, there are only four of them:  real numbers, complex numbers, quaternions, and octonions.  (Yes, there are hypercomplex numbers with seven imaginaries that are even more mind-bending than quaternions.  But that's another story.)

To solve equation ax=b, we left-multiply both sides by a⁻¹, thus  a⁻¹b = a⁻¹(ax) = (a⁻¹a)x = x; and likewise, the solution to x‍a=b is  x = b a⁻¹.  We call right-multiplication by  a⁻¹  "right-division by a", and write  b / a = b a⁻¹; similarly, left-division  a \ b = a⁻¹ b.  (Backslash, btw, is such a dreadfully overloaded symbol, I can somewhat understand why I haven't seen others use it this way; but I'm quite bowled over by how elegantly natural this use seems to me.  It preserves the order of symbols when applying associativity:  (a / b) c = a (b \ c).)  Naturally, I won't write division vertically unless the denominator is real.

Okay, we're girded for battle.  Back to nabla.

Doubting classical nabla

Our definition of nabla, you'll recall, was

∇

=

i1 ∂ ∂x1

+

i2 ∂ ∂x2

+

i3 ∂ ∂x3

.

This operator is immensely useful; people have been making great use of it, or its fragments in vector calculus, for well over a century and a half.  But three things about it bother me, the first two of which I've seen remarked by papers written in the past few decades.

Handedness

My first bother follows from the fact that quaternion multiplication isn't commutative.  This was, remember, a dramatic new idea in 1843; the key innovation that empowered Hamilton's discovery, because what he wanted — something akin to complex numbers but with arithmetic sensible in three dimensions — requires non-commutative multiplication.  But if multiplication isn't commutative, why should the partial derivatives in the definition of ∇ necessarily be left-multiplied by the imaginaries?  Why shouldn't they be right-multiplied, instead?

I've seen modern papers that use together both left and right versions of nabla.  Peter Michael Jack, who's had a web presence for (I believe) nearly as long as there's been a web, has suggested using a prefix operator for the left-multiplying nabla and a postfix operator for the right-multiplying nabla.  Candidly, I find that notation dire hard to read.  The point of prefix operators (which Hamilton championed, the better part of a century before Jan Łukasiewicz) is to make expressions much simpler to parse, and mixing prefix with postfix doesn't do that.  Another notation I see used in at least one place modernly is a subscript q on the left or right of the nabla symbol to indicate which side to put the imaginary factors on.  I'm not greatly enthused by that notation either because it uses a multi-part symbol.  I have an alternative solution in mind for the notational puzzle; but I'll want to make clear first the whole of what I'm trying to notate.

Truncation

My second bother is that traditionally defined nabla isn't even a full quaternion operator.  It only has the partial derivatives with respect to the three imaginary components.  Where's the partial with respect to the real component?  In the 1890s debate, the quaternionists said quaternions are profoundly meaningful as coherent entities, and the vectorists said scalars and vectors are meaningful while their sum is meaningless.  Now, I'm quite sympathetic to the importance of mathematical elegance, but come on, guys, make up your mind!  Either you go full-quaternion, or you don't. A nabla that only acts on vector functions is just lame.

There's a good deal of history related to the question of imaginary nabla versus full nabla.  The truncation of nabla to the imaginary components throughout the nineteenth century may have been partly an historical accident.  Consideration of the four-dimensional operator seems to have started just before the turn of the twentieth century, and modern quaternionists I've observed use a full-quaternion operator.  I'll have more to say about this history in the next section.

Meaning

My third bother is a byproduct, as best I can figure, of my persistent sense of arbitrariness about the nabla operator.  (This is the difficulty I've never seen anyone else remark upon.  Perhaps I'm missing something everyone else gets, or maybe I've just never looked in the right place; but then again, maybe people are just reluctant to publicly admit something doesn't make sense to them.  That might explain a lot about the world.)  It was never obvious to me why it should be meaningful — or, if you prefer the word, useful — to multiply the partial derivatives by the imaginaries in the first place.  It's clear to me why you'd do that if you were defining gradient, because gradient is meaningful for any number of dimensions, and doesn't depend in any way on the existence of a division operation.  But quaternions do have unique division, in fact it's rather a big deal that they have unique division, and the usual definition of ordinary derivative involves dividing by Δx.  So why are we multiplying by the imaginaries, instead of dividing by them?

Considering quaternions

Some of my above questions have historical answers, which also bear on the challenge raised by Penrose in the epigraph at the beginning of this post.

By Chapter 11 of The Road to Reality, where Penrose makes that remark (and where he also candidly describes the question of quaternions' use in physics as a "can of worms"), he's already described some marvelous properties of complex numbers, culminating with one (hyper‍functions) only published in 1958.  Which raises an important point.  Complex numbers have been intensely studied by mainstream researchers throughout the modern era of physics, yet Penrose's crowning bit of complex 'magic' wasn't discovered until 1958?

Compare that to how much, or rather how little, scrutiny quaternions have received.  Hamilton discovered them in 1843; but Hamilton was a mathematical genius, not a great communicator.  Quaternions, so I gather, remained the archetype of a baffling abstract theory until Einstein's General Theory of Relativity took over that role.  The first tome Hamilton wrote on the subject, Lectures on Quaternions, daunted the mathematical luminaries of the day; his later Elements of Quaternions, published incomplete in 1866 following his death in 1865, wasn't easy either.  The first accessible introduction to the subject was Peter Guthrie Tait's 1867 Elementary Treatise on Quaternions.  Quaternions got a big publicity boost when James Clerk Maxwell used them (for their conceptual clarity, rather than for mundane calculations) in his 1873 Treatise on Electricity and Magnetism.  And then in the 1890s quaternions "lost" the great vectors-versus-quaternions debate and their use gradually faded thereafter.  There simply weren't all that many people working with quaternions in the nineteenth century, and as world population increased in the twentieth century quaternions were no longer a hot topic.

Moreover, exploration of nabla got off on the wrong foot.  Hamilton seems to have first dabbled with it several years before he discovered quaternions, as a sort of "square root" of the Laplacian, at which time naturally he only gave it three components; and when he adapted it to a quaternionic form it still had only three components.  He didn't do much with it in the Lectures, and planned a major section on it for the Elements but, afaict, died before he got to it.  James Clerk Maxwell was a first-class mind and a passionate quaternion enthusiast, but died at the age of forty-eight in 1879 — the same year as William Kingdon Clifford, who was only thirty-three, another first-class mind who had explored quaternions.  The full-quaternion nabla was finally looked at, preliminarily, in 1896 by Shunkichi Kimura, but by that time the quaternionic movement was starting to wind down.  Yes, quaternions were still being used for some decades thereafter, but less and less, and the notations get harder and harder to follow as quaternionic notations were hybridized with Gibbs vector notations, further disrupting the continuity of the tradition and undermining systematic progress.  Imho, it's entirely possible for major insights to still be waiting patiently.

A subtle point on which Penrose's portrayal of quaternions is somewhat historically off:  Penrose says that although to a modern mind the one real and three imaginary components of quaternions naturally suggest the one time and three space dimensions of spacetime, that's just because we've been acclimated to the idea of spacetime by Einstein's theory of relativity; and quaternions don't actually work for relativity because they have the wrong signature (I'll say a bit more about this below; see here).  But as far as the notion of spacetime goes, the shoe is on the other foot.  Hamilton expected mathematics to coincide with reality (a principle Penrose also, broadly, embraces), and as soon as he discovered quaternions he did connect their structure metaphysically with the four dimensions of space and time.  Penrose is quite right, I think, that ideas like this get to be "in the air"; but in this case it looks to me like it first got into the air from quaternions.  So I'm more inclined to suspect quaternions suggested spacetime and thereby subtly contributed to relativity, rather than relativity and spacetime suggesting a connection to quaternions.  The latter implies an anachronistic influence that must be illusory (for relativity to influence Hamilton would seem to require a TARDIS); the former hints at some deeper magic.

The point about quaternions having the wrong signature has its own curious historical profile.  Penrose expresses very much the mainstream party line on the issue, essentially echoing the assessment of Hermann Minkowski a century earlier who, in formulating his geometry of spacetime, explicitly rejected quaternions, saying they were "too narrow and clumsy for the purpose".  The basic mathematical point here (or, at least, a form of it) is that the norm of a quaternion is the square root of the sum of the squares of its components, √(t²+x²+y²+z²), whereas in Minkowski spacetime the three spatial elements should be negative, √(t²−x²−y²−z²).  But here the plot thickens.  Minkowski, who so roundly rejected quaternions, defines a differential operator that is, structurally, the four-dimensional nabla.  As for quaternions and relativity, Ludwik Silberstein (a notable popularizer of relativity, in his day) did use quaternions for special relativity — except that, to be precise, he used biquaternions.

Biquaternions (which Hamilton had also worked with) are quaternions whose four coefficients are complex numbers in a fourth, independent imaginary.  Or, equivalently, they're complex numbers whose two coefficients are quaternions in an independent set of three imaginaries.  Either way, that's a total of eight real coefficients.  Biquaternions do not, of course, have unique division.  However, there are some oddly suggestive features to Silberstein's treatment.  His spacetime vectors have only four non-zero real coefficients (of the four quaternion coefficients, a₀ is real while a_k≥1 are imaginary, so that Σ(a_ki_k)² = a₀²−||a₁||²−||a₂||²−||a₃||²; while other biquaternions he considers have imaginary a₀ and real a_k≥1).  Moreover, he prominently uses the "inverse" of a biquaternion, defined structurally just as for quaternions,

a*

||a||2

, notwithstanding the technical lack of general biquaternion division.

Silberstein's approach contrasts with the quaternionic treatment of special relativity by P.A.M. Dirac, published in 1945 as part of the centennial celebration for the discovery of quaternions.  Dirac used real quaternions on the grounds that since the merit of quaternions is in their having division, it would be pointless to use biquaternions which are of no particular mathematical interest.  His mapping of spacetime coordinates onto real quaternions was unintuitive.  But looking at the oddly familiar-looking patterns in Silberstein's treatment, and Minkowski's operator which is hard not to think of as a full quaternionic nabla, one might well wonder if there is something going on that defies Dirac's claim about the importance of unique division.  Perhaps we've been incautious in our assumptions about just where the deep magic is to be found.

There are two pitfalls in this kind of thinking, which the inquirer must thread carefully between.  On one hand, one might assume there is some unknown deep magic here, rather than trying to work out what it is; this not only would lean toward numerology, but if there really is something to be found, would miss out on the benefits of finding it.  On the other hand, one could derive some superficial mathematical account of the particular mathematical relationships involved, based on math one already knows about, and assume this is all there is to the matter; which would again guarantee that any deeper insight waiting to be found would not be found.  (Current mainstream thinking, btw, falls into the latter pitfall, essentially reasoning that geometric algebras are useful in a way that quaternions are not, therefore quaternions are not useful.)  Is there any situation where it would really be time to give up the search altogether?  Well, yes, one does come to mind — if one were to arrive at some deep insight into why one should really believe there isn't some deep magic here.  Which might itself be some rather deep and interesting magic.

Frankly, I don't even know quite where to look for this hypothetical deep magic.  I sense its presence, as I've just described; but so far, I'm exploring various questions in the general neighborhood, patiently, with the notion that if these sorts of great insights naturally emerge from a large, broad body of research (as they have done for complex numbers), the chances of finding such a thing should improve as one increases the overall size and breadth of one's body of lesser insights.

Which brings me back to the particular point I'm pursuing in this post, the full quaternion nabla.

Full nabla

From a purely technical perspective, it isn't difficult to define four versions of the full quaternion nabla, differing only by whether each imaginary acts on its corresponding partial derivative by left-multiplying, right-multiplying, left-dividing, or right-dividing.  The only remaining — purely technical — question is how to write these four different operators in an uncluttered way that keeps them straight.  Since the traditional nabla has three partial derivatives and is denoted by a triangle, I'll denote these full nablas, with four partial derivatives, by a square.  To keep track of how the imaginaries are introduced, I'll put a dot inside the square, near one of the corners:  upper left for left-multiplying by imaginaries, upper right for right-multiplying, lower left for left-dividing, lower right for right-dividing.  (This operator notation affords coherence, as the dot is inside so there's no mistaking it for a separate element, and, as a bonus, should also be easy to write quickly and accurately by hand on the back of an envelope.)

Let  a = f(x).  Noting that for imaginary i_k,  1/i_k = −i_k, the full-quaternion nablas are

●

a

=

∂a ∂x0

+

i1 ∂a ∂x1

+

i2 ∂a ∂x2

+

i3 ∂a ∂x3

●

a

=

∂a ∂x0

+

∂a i1 ∂x1

+

∂a i2 ∂x2

+

∂a i3 ∂x3

●

a

=

∂a ∂x0

−

i1 ∂a ∂x1

−

i2 ∂a ∂x2

−

i3 ∂a ∂x3

=

(

●

a

)

‌*

●

a

=

∂a ∂x0

−

∂a i1 ∂x1

−

∂a i2 ∂x2

−

∂a i3 ∂x3

=

(

●

a

)

‌*

and when we expand  a = a₀ + a₁i₁ + a₂i₂ + a₃i₃,

●        a

=

(

∂a0 ∂x0

−

∂a1 ∂x1

−

∂a2 ∂x2

−

∂a3 ∂x3

)

+

i1

(

∂a1 ∂x0

+

∂a0 ∂x1

+

∂a3 ∂x2

−

∂a2 ∂x3

)

+

i2

(

∂a2 ∂x0

−

∂a3 ∂x1

+

∂a0 ∂x2

+

∂a1 ∂x3

)

+

i3

(

∂a3 ∂x0

+

∂a2 ∂x1

−

∂a1 ∂x2

+

∂a0 ∂x3

) .

Here, the left-hand column is the partial with respect to x₀, and the rest is the fragmentary differential operators from vector calculus:  the rest of the top row is minus the divergence, the rest of the diagonal is the gradient, and the remaining six terms are the curl.  When we reverse the order of multiplication for the right-multiplying

●

, the imaginaries commute with the scalars and with themselves, but anticommute with each other — so everything stays the same except that the sign of the curl is reversed.  We have

●	=	∂ ∂x0	−	div	+	grad	+	curl
●	=	∂ ∂x0	−	div	+	grad	−	curl
●	=	∂ ∂x0	+	div	−	grad	−	curl
●	=	∂ ∂x0	+	div	−	grad	+	curl	.

By taking differences between these nablas, one can isolate the partial with respect to x₀, and the curl, and... the gradient minus the divergence.  One cannot, however, separate the gradient from the divergence this way, which raises the suspicion that the gradient and divergence are, in some profound sense, a single entity.  There may be some insights waiting here into the intuitive meanings of these various fragments of the full nabla.

Wait.  Wasn't part of the point of the 1890s debate that the quaternionists maintained the whole quaternion was in a profound sense a single entity?  Why are we still talking about the meanings of fragments of this thing, instead of the whole?  And while we're at it, why is it in any way meaningful to multiply-or-divide the partial derivatives by the basis elements?

Partial derivatives

From here, the path I've been following breaks up, with faint trails scattering off in many directions.  No one trail immediately suggests itself to me as especially worth a protracted stroll, so for now I'll take a quick look down the first turn or so of several, getting a sense of the immediate neighborhood, and let my back‍brain mull over what to explore in some future post.

Possibly, in my quest for the deeper meaning of the nabla operator, I may be asking too much.  With the caveat that this may be one of those situations where it's right to ask too much; some kinds of results must be pursued that way; but it's worth keeping in mind that, idealism as may be, there's always been a strong element of utility in the nabla tradition.  Starting with, as noted above, the pre-quaternion history of nabla, the choice of operator has been in significant part a matter of what works.

A secondary theme that's been in play at least since Shunkichi Kimura's 1896 treatment is total derivatives versus partial derivatives.  Without tangling in the larger question of coherent meaning, Kimura did address this point explicitly and up-front:  why write

●

a

=

∂a ∂x0

+

i1 ∂a ∂x1

+

i2 ∂a ∂x2

+

i3 ∂a ∂x3

rather than

●

a

=

da dx0

+

i1 da dx1

+

i2 da dx2

+

i3 da dx3

?

Kimura, after noting that the two forms are interchangeable when the x_k are independent, chose partial derivatives.  And reached this choice by considering the utility of the two candidate operators in expressing some standard equations, and adopting the operator he finds notationally more convenient.  It figures this would be the operator using partial derivatives, which are more technically primitive building blocks and thus —one would think— ought logically to provide a more versatile foundation.

An (arguably) more definite form of the total/partial question appears in modern quaternionic treatments of Maxwell's equations ([1], [2]), with the peculiar visible consequence that the definition of full nabla in these treatments has a stray factor of 1/c on the partial with respect to time (x₀).  On investigation, this turns out to be a consequence of starting out with the total derivative with respect to time, supposing (as I track this, three and a half decades after I took diffy Q‍s) the whole is time-dependent.  Expanding the partials,

d d‍x0

=

∂ ∂x0

+

∂x1 ∂ ∂x0 ∂x1

+

∂x2 ∂ ∂x0 ∂x2

+

∂x3 ∂ ∂x0 ∂x3

.

Now, the partials

∂xk≥1

∂x0

are the velocities of propagation along the spatial axes, which for Maxwell's equations are taken to be the speed of light, c.  This factor of c therefore shows up on three out of four partials, but not on the partial with respect to time; for convenience —that again— one defines an operator with a factor of 1/c on it, which eliminates the extra factors of c on three of the partials, but introduces a 1/c on the partial with respect to time. 

And then there is the matter of orienting the partials.  Which I'm still foggy on, how the imaginaries get in there and thus whether they multiply or divide, on the left or on the right.  I see treatments just splicing the imaginaries in with at most a casual reference to orientation in an algebra, which early classroom experience has conditioned me to treat as someone who understands it all and doesn't take time to explain every little thing (I've been in that position a few times myself); but over time I've started to suspect that the folks acting so in this case might not really understand it any better than I do (I've been in that position, too).

Generalized quaternions

Quaternions lost out on the concrete front to vector calculus.  But they also lost out on the abstract front.  Mathematicians took Hamilton's idea of using axioms to define more general forms of numbers and reason about their properties, and ran with it.  Linear algebra.  Clifford algebras.  Lie and Jordan algebras.  Rings.  Groups.  Monoids.  Semi-groups.  People who want special numbers won't go as far as quaternions, and people who want general numbers won't stop at quaternions.

Yet, generalized quaternions — quaternions whose four coefficients aren't real numbers — have occasionally been employed.  Why?  On the face of it, generalized quaternions don't have the specific properties that make real quaternions unique.  Are they used, then, out of some perceived mystical significance of quaternions, or is there actually something structural about quaternions, aside from their unique mathematical properties as a division algebra, that they can confer even in the generalized setting?  I do not, of course, have a decisive answer for this question.  I do have some places to look for small insights building toward prospects of an answer.

The places to look evidently fall into two groups, those that look within the scope of real quaternions and those that look at generalized forms of quaternions.  In looking at real quaternions the point is to understand what they have to offer beyond mere unique division, that might possibly linger after the unique division itself has dropped away.  I'll have more to say, further below, about real than generalized quaternions; I'm simply not familiar with much research using generalized quaternions as such, as most researchers either stick with real quaternions or drop quaternion structure.

On the generalized-quaternions front, I've already mentioned Silberstein; but, tbh, all I get from Silberstein is the question.  That is, Silberstein's work suggests to me there's something of interest in generalized quaternions, but doesn't go far enough to identify what.  There are some well-known generalizations that go off in different directions from Silberstein; besides geometric algebras, which are enjoying some popularity atm, there's the Cayley–Dickson construction, which offers an infinite sequence of hypercomplex algebras with 2ⁿ components, each losing just a bit more well-behavedness:  complexes, quaternions, octonions, sedenions, and on indefinitely (though usually not bothering with fancy names beyond the sedenions).  So far, I haven't felt any of those sorts of generalizations were retaining the character of quaternions; so that, whatever merits those generalizations might enjoy in themselves, they wouldn't offer insights into the peculiar merits of quaternions.

As it happens, I do know of someone who continues further in what appears to be the same direction as Silberstein.  But there's a catch.

The work I'm thinking of was done about sixty years ago by a Swedish civil engineer by the name of Otto Fischer.  He wrote two books on the subject, Universal Mechanics and Hamilton's Quaternions (1951) and Five Mathematical Structural Models in Natural Philosophy with Technical Physical Quaternions (1957).  It happens I can study the earlier book all I want, because my father bought a copy which I've inherited.  Fischer indeed did not stop at real quaternions nor biquaternions.  He moved on to what he called quadric quaternions — quaternions whose coefficients are themselves quaternions in an independent set of imaginaries, thus with six elementary imaginaries in two sets of three, and sixteen real coefficients — and thence to double quadric quaternions, which are quadric quaternions whose sixteen coefficients are themselves quadric quaternions in independent imaginaries, thus twelve elementary imaginaries in four sets of three, and 256 real coefficients.  If what is needed to bring out the secrets of generalized quaternions is a sufficiently general treatment, Fischer should qualify.

Looking back now, Fischer's work looks a bit fringe; but it didn't look so extra-paradigmatic at the time.  The 1890s vectors-quaternions debates were in the outer reaches of living memory, about as far removed as the 1950s are today; and work on quaternions had been done by some prominent physicists within the past few years.  In particular, Sir Arthur Eddington, who had tinkered with quaternions, had only recently died.  Fischer's work was — deservedly — criticized for its density, but afaict wasn't dismissed out of hand, as such.

In any case, my current interest is on the periphery of things, rather than in the center of prevailing paradigm research; so I can afford to tolerate a certain off-beat character in Fischer's work — up until Fischer gives me a reason to think I've nothing further worthwhile to find in it.  And Fischer comes across as competent and quite self-aware of the density and indirection of his work, which he seeks to mitigate — though there's a real question as to whether he succeeds.

What I really want to understand about Fischer's work is, having provided himself with such an immense array of generalized quaternionic structure, what does he use it for?  There are some clues readily visible in the preface and final sections of the book; somehow he seems to be associating different quaternion subsets of his general numbers with different specialties, and he's playing some kind of games with "pyramids" of differential operators.  To really get a handle on it all, I fear it may be necessary to confront the book in full depth from page 1, which I've tried far enough to realize it's the single densest mathematical treatment I've encountered (though he does take very seriously his own advice to "begin at the beginning", else I'd hold out no hope at all of making sense of it).

So, studying Fischer's work may be one source of... eventual... insights into the puzzle of generalized quaternions.  It certainly isn't a short-term prospect; but, there it is.

Before getting back to real quaternions in the next section, I'll digress to remark that Fischer reinforces a belief I've held ever since I really started researching the history of quaternions — in 1986 — that what we really need in mathematics is a certain type of software.

By my reading of the history, the vectorists in the 1890s debate really did have one important practical point in their favor:  if you have to deal with the algebra by hand, it seems it'd be vastly easier to not make careless errors when following the rectangular regimentation of matrix algebra than the spinning vortices of quaternion algebra.  (Recalling from my earlier post, the equivalence between matrix and quaternion methods is akin to the equivalence between particles and waves — with quaternions playing the part of waves.)  That is, if you try to do quaternion algebra, involving breaking things down into components, on the back of an envelope you're awfully likely to make a mistake; so I immediately imagined having a computer help you get it right.  (I didn't imagine a graphical user interface, btw, as that technology really didn't exist yet for personal computing.  Looking back, I find myself ambivalent about GUIs; sure, they can be sparkly, but they don't always help us think clearly; we're so busy thinking of how to use the graphics, we forget to think first and foremost about the logical structures we'd like to interface with.)

Thinking about this idea, I eventually decided the underlying logical structures one wants would be essentially proofs, so that in a sense the software would be a sort of "proof processor", by loose analogy with the "word processor".  Achieving the fluidity of back-of-the-envelope algebra was always key to my concept; my occasional encounters with "symbolic math" software have given me the impression of something far too cumbersome for what I envision.  Facilely moving between alternative paths of reasoning should be easy; symbol definition would seem to call for something halfway between conventional "declarative" and "imperative" styles.  I also imagined the computer trying, in its free moments, to devise context-sensitive helpful suggestions for what to do next — without trying to take control of the proof task away from the human user.  I've never been a fan of fully-automated proof, as such; in the early days of personal computing (as a commenter on another of my posts reminded me) we anticipated computers of the future would enhance our brain power, not attempt to replace it, and the enhancements weren't to be just increasing our ability to look things up, either.

Where does Fischer come into this?  Well, Fischer not only deals with massive grids of coordinates, his notation looks extremely idiosyncratic to me, using different conventions than anything else I've seen.  Perhaps a typical 1950s Swedish civil engineer would find much of it quite conventional.  But, unless you spend all of your time in one narrow mathematical sub‍community, studying mathematics is a pretty heavily linguistic exercise, because every sub‍community has their own language and one is forever having to translate between them.  Wouldn't it be nice to be able to just toggle some controls and switch between the way one author (such as Fischer) wrote, and the conventions used by whichever other author you prefer?

Btw, this software I'm describing?  Not a minor interest.  Not just a past interest.  I still want it, all the more because, even though I've always felt it was doable and would be immensely valuable, afaict we're no closer to having it now than we were thirty years ago.  Never assume that what you think is needed will be provided by somebody else.  Think of it this way:  if you can see it's doable and would be valuable, presumably you'd be more likely that most people to make it happen; so if you aren't going to the effort to make it happen, that's a sample of one suggesting that nobody else will go to the effort either.  I've also never felt I could properly describe this software in words, so even if I was gifted with a team of programmers to implement it I couldn't tell them what to do; so I figured if it was going to happen I'd have to do it myself.  Only, it looks like a huge project, so for one person to implement it would require a programming language with unprecedentedly vast abstractive power.  By some strange coincidence, designing a programming language like that is something I've been striving for ever since.

Rotation

Another trail that, sooner or later, clearly needs to be explored is the relationship, at its most utterly abstract, between quaternions and rotation.

Hamilton was looking at rotations, from the start.  Quaternions, as noted, stand in relation to matrices as waves to particles; in some profound sense, quaternions seem to be the essence of rotation.  The ordinary understanding of quaternion division is that a quaternion is the ratio of two three-vectors, and the non-commutativity of quaternion multiplication then follows directly from recognition that rotations on a sphere produce different results if done in a different order.  Even Silberstein, who was using biquaternions rather than real quaternions and was working in Minkowski rather than Euclidean spacetime, was doing rotations, which in itself suggests that what's going on is more than meets the eye.

This is a tricky point.  The relationship between quaternions and rotation is readily explained, indeed rather trivialized, in terms of peculiarities of rotation in three-dimensional Euclidean space.  This is very much the canonical view, the one embraced by Penrose.  Real quaternions become a single case in a general framework, and are then easily dismissed as merely an aberration that loses its seeming specialness when the wider context is properly appreciated.

The weakness in this reasoning is that it depends on the choice of general framework.  This would be easier to see if the framework involved were alternative rather than mainstream.  Suppose there were two different general frameworks in which the specific case (here, quaternions) could be fit; and in one of these frameworks, the specific case appears incidental, while in the other framework it appears pivotal.  It would then be hard to make a compelling case, based on the first framework, that the specific case is incidental, because the second framework would be right there calling that conclusion into question.  If the first framework is the only one we know about, though, the same case can be quite persuasive.  To even question the conclusion we'd have to imagine the possibility of an alternative framework; and actually finding such an alternative could be a formidable challenge.  Especially with the possibility hanging over us that perhaps the alternative mightn't really exist after all.

Investigating this trail seems likely to become an intensive study in avoiding conceptual pitfalls while dowsing for new mathematics.

Minkowski

A narrow, hence more technically fraught, target for mathematical dowsing is Minkowski spacetime.  Minkowski's decisive condemnation of a quaternionic approach —"too narrow and clumsy for the purpose"— is a standard quote on the subject, cited by quaternion opponents and proponents alike.  If there is an alternative general framework to be found, after all, it'd have to handle Minkowski.

Without actually wading into this thing (not to be undertaken lightly), I can only note from a distance a few features that may be of interest when the time comes.  The mechanical trouble in this is evidently to do with the pattern of signs, which seems reminiscent of the multiple variants of nabla (though the pessimist in me insists it can't be quite that easy); which, logically, oughtn't be applicable to the situation unless one were really already dealing with a derivative.  Off hand, the only way that comes to mind for derivatives to come into it is if the whole physical infrastructure is something less obvious than what Minkowski was doing — which, yes, is cheating; and cheating (so to speak) is likely the only way to end up with a different answer than Minkowski did, so this might, just conceivably, be a hopeful development.

Langlands

I wondered whether even to mention this.  The geometric Langlands correspondence lies at the extreme wide end of mathematical dowsing targets; about as poetic as mathematics comes (which is very poetic indeed), and at the same time about as esoteric as it comes (yea, verily).

Mathematics in its final form is, of course, highly formal (I say "of course", but see my earlier remarks on axioms as a legacy of quaternions).  The ideas don't start out formal though; and there's always lots of material that hasn't yet worked its way across to the formal side.  Moreover, attempts to describe the poetry of mathematics for non-mathematicians, in my experience, ultimately fail because they're trying to do something that can't really be done:  they're trying to divorce the (very real) poetic nature of mathematics from the technical nature of the subject, and at last this can't really be done because the true poetry is that the elegance arises from the technicalities.

Poking around on the internet, I found a discussion on Quora from a few years ago on the question Can the Langlands Program be described in layman's terms?  There were some earnest attempts that ultimately devolved into technical arcana; but my favorite answer, offered by a couple of respondents, was in essence:  no.

My own hand-wavy assessment:  Robert Langlands conjectured broad, deep connections between the seemingly distant mathematical subjects of number theory and algebraic geometry.  Especially distant in that, poetically speaking, number theory is a flavor of "discrete" math, while algebraic geometry is toward the continuous side of things.  (I riffed on the discrete/continuous balance in physics some time back.)  An especially high-publicity result fitting within this vast program was Andrew Wiles's proof of Fermat's Last Theorem, which hinged on proving a conjecture about elliptic curves.

Why would I even bring up such a thing?  The Langlands program has gotten tangled up, in this century, with supersymmetry in physics; and the geometric side of Langlands is about complex curves.  In effect, Langlands biases mathematical speculations toward further enhancing the reputation of complex numbers.  So if one suspects physics may also lean toward the quaternionic, and one is also looking for interesting mathematical properties of quaternions, it seems fair game to ask whether quaternions can play into some variation on Langlands.

Storytelling

But when it was midnight Shahrazad awoke and signalled to her sister Dunyazad who sat up and said, "Allah upon thee, O my sister, recite to us some new story, delightsome and delectable, wherewith to while away the waking hours of our latter night."

— The Book of the Thousand Nights and a Night:  A Plain and Literal Translation of the Arabian Nights Entertainments, Richard Francis Burton, 1885.

In this post, I mean to further two purposes at once:  to expand my thinking on the evolution of sapient thought, and to deepen my understanding of Julian Jaynes's book The Origin of Consciousness in the Breakdown of the Bicameral Mind (1976).  The evolution-of-sapience part is my long-term interest; but the questions Jaynes raises are my current fuel and direction for exploring that evolutionary theme.  My evolutionary thinking emerges on the other side of this driven exploration with some fascinating new insights and a set of further investigations to pursue.

I've considered language evolution several times on this blog, lately in February of last year.  My unifying theme has been an extension of Eric Havelock's theory expounded in his Preface to Plato (1963), where he supposes that ancient Greek culture around the time of Plato had just undergone a profound transformation from orality, in which human culture is preserved in oral sagas, to literacy in which human culture is preserved in writing.  I conjecture the existence of a still-earlier phase of human culture, before orality, for which I took cues from the Pirahã culture recently studied in the Amazon.  The Pirahã have neither art nor storytelling; and their language, amongst a variety of other peculiarities, neither number nor time vocabulary, nor verb tense.  To provide a convenient handle on the idea, I've used working name verbality for the pre-orality phase of culture; and I've hazarded a guess that the transition from verbality to orality is marked by the appearance of art and new technologies at the start of the Upper Paleolithic, circa forty thousand years ago.

I blogged preliminary thoughts on Jaynes, on my first reading of the book, in March of last year.  Jaynes's thesis is that for some time leading up to about five thousand years ago, human minds functioned differently than the consciousness we experience today.  Instead, the human mind was, in Jaynes's terminology, bicameral, with the left and right brains (so-called) operating in a mode of partially independent coordination resembling schizophrenia, and this bicameral coordination has gradually broken down and been replaced by modern consciousness over the past five millennia.  He apparently sees the modern self as a character in a story the mind tells itself, a view I've encountered from others and with which I agree.  Jaynes presents a detailed case for his thesis; my top criticism, on first reading, was that a less radical explanation of the evidence could be afforded by the memetic hypothesis, which was not yet available when Jaynes first formulated his bicameral thesis (as memes were only proposed in the same year Jaynes published his book, 1976), and which [memetics] I have been refining myself on this blog (starting some time back).

Another theme from past posts that informs my view of mind evolution is my model of sapient mind.  Going in to this post, I'd expected my model itself to play a passive supporting role; instead, however, the discussions in this post have provided extensive feedback on implications of my model of mind.

As I tied up my thoughts on first reading Jaynes, I reckoned a second in-depth reading would be in order, coordinated with a systematic effort to reinterpret Jaynes's accumulated evidence as grist for a more detailed timeline of memetic and linguistic evolution following my verbality hypothesis.  That's what I mean to do here.

I was blindsided, while preparing what I thought would be the final draft of this post, by a flash of insight that I hadn't remotely seen coming — at least, not (heh) consciously.  Now, in these posts I mostly try to keep the path of my explorations intact — so it's clear how I got to where I did, and, also, so diverging paths not taken can be returned to some day; but it does sometimes happen that later turns of the path are related to earlier ones, so that the earlier turns should be marked for later recall, and I'll add forward references in the earlier discussion to cue the reader.  This time, though, the new insight sharply altered the complexion of points scattered across the whole discussion, causing a few to look prescient and several others oblivious.  Rather than compromise the whole depiction of the journey, I've left most of it untouched; with this one paragraph as a warning up-front, so the reader, when hit by that final turn, may hopefully suffer less mental whiplash from it than I did.  For what it's worth, the one thing earlier in this introduction that now grates on me is the list of peculiarities of the Pirahã language; that list was cherry-picked from a ready-to-hand longer list of oddities of Pirahã in a discussion with no obvious relation to any of this — it was about conlanging.  What grates is that there's something really important missing from the list, that looks obvious to me in retrospect (but while it's important here, it would have been entirely out-of-place in the conlanging discussion I cribbed from).  It really will all make more sense (I hope) when I come to it in due season, at the far end of the path; though meanwhile my proofreading of this whole post will be punctuated by winces where I squelch my impulse to inject forward-references to it.

One point I wish to be perfectly clear from the outset of this post:  I have great respect for Jaynes.  In the process of this post I will say some pretty harsh things about him, and do not care to be misunderstood on this point.  In a single lifetime, none of us can see everything; we need the testimony of those who have visited realms we have not, and Jayne's insights from his professional background, though I may question and criticize mercilessly, I do take very seriously.  I wouldn't otherwise be devoting such close attention to them.

Contents
Timeline
Sapient mind
Jaynesian consciousness
Words
Gods
Brains
Evolution
Breakdown
Vestiges
Jaynes as a whole
Frame story

Timeline

Here is my working theory, before undertaking my second reading of Jaynes, on what happened when.

Human sapience started, I conjecture, at the onset of the Paleolithic —the old stone age— circa three million years ago.  Dates assigned to these early milestones vary somewhat; at this writing, for instance, Wikipedia puts the Paleolithic onset at 3.3 million.  (I blogged on the emergence of sapience back yonder.)  Some modern thinkers on this —e.g. Daniel Dennett, Darwin's Dangerous Idea (1995)— have not only viewed language as the key distinguishing feature of sapience, but given language a causative or even definitive role in the process; however, I think language is more usefully understood as an effect rather than cause of sapience.  I've speculated sapience is some sort of algorithmic phenomenon, quite possibly positioned in a pocket of evolutionary search-space such that most evolutionary paths are diverted away from it, requiring some peculiar set of conditions for ignition and surrounded by mostly disadvantageous alternatives.  My best guess atm is that language may be a catalyst for sustained ignition:  that language naturally emerges in a sufficiently dense population of sapients and, perhaps, helps to create the survival advantage needed to drive further evolution of both sapience and language.  I also suspect the sapience engine may be related to the non-Cartesian theater of human short-term memory (more on that in the next section) — an algorithmic, rather than linguistic, view of the sapient mind.

I am, btw, not inclined to the Chomsky-esque notion of an elaborate universal grammar device genetically programmed into the human brain.  As usual, I've an Ockham-ish preference for a simpler theory.  I envision the sapience engine as a simple and robust —if rather evolutionarily hard-to-find— "chunking" device, accounting simultaneously for, on the cognitive side, formation of abstract ideas, and, projected onto the linguistic side, the tree-structured tendencies of human grammar.  Though it seems entirely plausible that genetic evolution, once it got sapience in its metaphorical teeth, would favor improved linguistic capacity, I still see the internal sapience engine —what the solitary sapient mind does in itself, rather than how it interacts with other sapient minds— as the root cause of the distinctive shape of human evolution.  I'd tend to ascribe technical features of human language to practical constraints of communication by a simple sapience engine rather than to incidental constraints of an elaborate language engine.  (Daniel Everett recommended a similar conclusion from his study of the Pirahã, Don't Sleep, There Are Snakes: Life and Language in the Amazonian Jungle, 2009:  "Language is a by-product of general properties of human cognition [...] constraints on communication that are common to evolved primates [...] and the overarching constraints of human cultures".)

From my first reading of Jaynes, his vision of the early human mind differs rather extremely from mine.  Because I envision the algorithmic core of human consciousness as essential to sapience, necessarily I would expect sapient early hominins some millions of years ago to have minds structured along very broadly the same lines as those of modern humans.  Jaynes views conscious humans as an anomaly (if not a pathology) that has developed only within the past five thousand years or so.  Since I view language as a natural consequence of sapience, and possibly a necessary part of its evolutionary advantage, my scenario also ascribes some form of language to sapient early hominins, again some millions of years ago.  Jaynes suggests, in candid disagreement with mainstream thinking, that language didn't emerge until the start of the Upper Paleolithic —the late stone age— some forty or so thousand years ago; the point in human development where I'm placing the start of orality.  On first reading, I found his attitude on this point quite refreshing, taking it for a cheerfully good-natured try at a dramatically unorthodox alternative to prevailing thought on a point that, honestly, we're all guessing at — our ancestors from ten or a hundred thousand years ago, let alone a million, having neglected to leave us any audio recordings of their interactions.  I approve (as I've remarked numerous times on this blog) of shaking up orthodoxy, to keep our thinking limber.  Nevertheless, in this case I'm disinclined to Jaynes's late date for the onset of language.

Jaynes's notion of modern consciousness —if I understood correctly on first reading— presumes the narrative self is generated by a monolithic device, whereas my notion of essential sapience engine, though possessing a center (non-Cartesian theater), is inherently distributed and thereby more flexible.  As mentioned in my earlier post, Susan Blackmore in The Meme Machine (2000) also described a monolithic notion of self, though she explained she no longer believed in it due to her study of memetics.  When she adopted a notion of self as a character in a narrative generated by the mind, apparently she reckoned that only the generated character was monolithic, not the generating mind.  Daniel Dennett too, e.g. in Consciousness Explained (1991), described a non-monolithic mind; Dennett's notion of mind, as I understand it, was radically decentralized, apparently lacking any internal structure to the device prior to memes being fed into it.  Differences aside, all these non-monolithic models of mind seem consistent with substituting memetics for bicamerality at the center of Jaynes's grand scheme.

The very fact that storytelling is involved in the modern notion of self implies, within my scheme, that early hominins, with a verbal rather than oral culture and therefore with no storytelling at all, would not have a self in quite the modern sense.  This seems to me a less radical prospect than Jaynes's bicameral mind, because I don't see the narratized self as central to the structure of the mind; again, my sapience device affords a moderate degree of coherence prior to any memetic programming.  There is a curious implication in this early lack of modern self, that, just as my conjectured sapience engine would support a softened form of modern conscious mind, it might also support a softened form of Jaynes's bicameral mind.  It's unclear to me just what form this would take, and I'm not immediately convinced it should play a significant role in the evolutionary scheme, but it's another thing to consider on a second reading.  One can't rule out this variant bicameral scenario without first properly understanding it; and one should also keep in mind that Jaynes was a psychologist, with (as alluded to earlier) expertise in quite a different realm of phenomena than my own background has offered me.

For the next several million years, through the entirety of the Lower Paleolithic and Middle Paleolithic, by my timeline human language was in its verbality phase, without —one supposes— art, storytelling, number, time.  The only model I have for this sort of language is Pirahã, and from its peculiar circumstances one expects Pirahã to be an atypical, even pathological, example.  Though I take Pirahã as an existence proof for language that lacks key features of orality, I have as yet no notion of the range of variation of verbal languages, let alone how in particular human languages before the age of orality might likely differ from the strange outlier Pirahã in an age of literacy.  Not only does Pirahã provide only one data point in the conjectured range of verbal languages, but I don't immediately see any strong reason to expect Pirahã to be a holdover from the age of verbality; it seems just as likely to have somehow reinitiated verbality (that is, they're both vanishingly unlikely events, with nothing but the existence of Pirahã to suggest that at least one of the two probabilities is non-zero).  It's a guess that the absences of number and time are really related to those of art and storytelling; absence of time seems, imho, intuitively as if it ought to be related to absence of storytelling, but it would not be nearly so hard to imagine absence of number as just a coincident peculiarity.  Derek Bickerton —Adam's Tongue (2009)— would populate these millions of years of human prehistory with a series of baby steps along the road to full-fledged language — steps that Jaynes must fit into the few tens of thousands of years of the Upper Paleolithic.  I've wondered whether one might be able to somehow do a partial forensic reconstruction of verbal language by studying grammatical peculiarities of the Basque language isolate, but it's not immediately clear that I could pursue that with my current means.

Orality started, in my timeline, at the onset of the Upper Paleolithic, the late stone age, perhaps forty thousand years ago (be the same more or less; at this writing, Wikipedia puts it around fifty thousand).  Or more precisely, in my scenario the start of orality brought about the onset of the Upper Paleolithic around that time, marked by a flourishing of art and technology.  I'd expect the sluggish information transmission of verbal society to support only a very gradual advancement of technology, and the introduction of orality to produce an immediate acceleration of technological progress.  The correlation between technology and art was already suggested by the conspicuous absence of art along with storytelling from Pirahã culture; one might, with some element of justice, say that art appears necessary to technological advance.  (A beta-audience for this text points out early cave paintings telling a story.)

Recently reported evidence dates Neanderthal cave paintings to sixty five thousand years ago (link).  Under my premise, this would indicate that the transition to orality was not specific to species Homo sapiens, making the transition (inasmuch as the species are separate) a memetic rather than genetic phenomenon, and suggesting that the genetic potentials of both species were able to reach the transition.  With my supposition that sapience itself is part of the genetic potential involved, and guessing sapience only developed once, rather than evolving convergently for both species, it would follow that even if sapience didn't start all the way back at the onset of the Paleolithic, it ought to be at least as old as the divergence between H. neanderthalensis and H. sapiens, circa four hundred thousand years ago.

An important point in my recent reasoning on sapience is that memetic evolution is several orders of magnitude faster than genetic evolution.  One might ask whether that applies to memetic evolution in verbal culture, or only later in oral/literate culture.  If the Lower and Middle Paleolithic must be excluded from memetic evolution, it would raise the question of whether the onset of sapience ought to be reckoned from the start of the Upper Paleolithic after all, akin to Jaynes's timeline (and thereby weakening my verbality premise).  However, the relatively slow start in the verbal phase may be compatible with rapid memetic evolution after all; note that genetic evolution on Earth got off to a very slow start, languishing in relatively primitive forms for at least three billion, perhaps as much as four billion, years before abruptly shifting into a higher gear with the Cambrian explosion.  That pattern would fit tolerably well with memetic evolution extending through all, or a significant part of, the Paleolithic, with the interval between Paleolithic onset and Upper Paleolithic onset suggesting a rate of progress roughly three orders of magnitude faster than genetic evolution.

The technical character of the verbality/orality transition is unclear.  According to my notion of a substantially fixed sapience engine, it seems the change ought to be linguistic, or rather (more precisely) ought to have a key manifestation in that form; but is the essential/signature linguistic element a provision for time?  For number?  Something else?  Atm —poised to undertake my second reading of Jaynes— I see no basis for a strong preference between these alternatives; conceivably, though, there might be a way to work backward to it:

Until my first reading of Jaynes, I hadn't had occasion to consider what the verbality hypothesis implies about the evolution of orality.  On consideration, though, it seems apparent that, starting from the dead stop of verbality, the high art of storytelling wouldn't spring fully formed but would, in fact, require an immense effort, presumably over a very long time (by human standards), to develop.  Indeed, the development of the narrative self would seem to be itself a contiguous part of the evolution of storytelling, while the evidence Jaynes presents for bicamerality might also inform my alternative premise with some glimpses of intermediate stages in the evolution of storytelling.  (I noted, in my earlier post, Jaynes remarked that the Iliad never describes human bodies as a whole, but rather as collections of parts; and I wondered at the time, what that says about the state of the art of storytelling.)

With a sufficiently detailed picture of the evolution of storytelling, then, one might imagine running it backward to deduce how it might have started.  Though one might also try to reason forward to the start of storytelling by imagining what verbal culture might have been like.  Evidently some modest level of technological skill was passed on from generation to generation, as glacially advancing technology characterizes the Paleolithic.  Speculatively, might the volume of technological knowledge, being passed from generation to generation, have simply grown large enough to draw the attention of sapient minds, which then did what sapient minds do — thought about it, thereby causing it to creep into their language?

Even after the transition from orality to literacy, the evolution of storytelling wouldn't stop; it would just adapt to the changing memetic environment.

Sapient mind

As I hope to effectively transplant Jaynes's research from his model of mind to my own model of mind, if I'm to carry it off plausibly, I need to be clear on what my model of mind is.  I've described this model before, in my post on sapience and language.  In outline:  I envision human short-term memory, with its seven-plus-or-minus-two chunks of information, as a sort of "non-Cartesian theater" forming the centerpiece of the mind.  The audience watching the theater is a vast population of agents within the mind, each one representing and promoting —essentially, embodying— a thought.  When a thought is promoted loudly enough —for which an important element is relating to other thoughts that are currently well-promoted, especially the ones now on-stage— the thought may be promoted onto the stage, becoming one of the lucky seven-plus-or-minus-two.

(A degree of freedom here is that I'd entertain, if necessary, a model in which the theater is something other than short-term memory; but short-term memory was the inspiration for my model, and remains my favored hypothesis.) 

Daniel Dennett spent a great deal of effort, in his 1991 book Consciousness Explained, debunking the notion of a "Cartesian theater", in which the audience watching the theater is a monolithic consciousness.  If the monolithic observer is immaterial (a soul), you've got Cartesian dualism; if the monolithic observer is material, it's apparently a mind, thus of the same type as the mind within which the theater and observer occur, so you've got an infinite recursion.  However, neither objection applies to a model of mind in which the audience of the theater is a massively distributed sea of agents.

Presumably the agents in the audience also have some limited direct interaction (whispering to each other during the performance, as it were); and occasionally these interactions might rise to the level of sub-communities.  The model would seem to be continuously deformable into a spectrum of unusual configurations, in which a large subcommunity could come to operate on a similar scale to the primary theater.  One might wonder whether, in cases of split personality, alternate personalities would have normal-sized, or smaller-than-normal, short-term memories (on which, I have no strongly favored hypothesis).  Conceivably, some such unusual configuration might resemble Jaynes's bicamerality.

Whatever philosophers (such as Descartes) may have said about the mind, some common figures of speech have no difficulty portraying the mind as coherent-but-separable.  Having difficulty choosing between two alternatives, one might say, "I'm of two minds.  Part of me says [A], but another part says [B]."  In my experience, this usage feels entirely natural, neither remarkable nor problematic, unless we're told to notice it.

The theater evidently provides a good loom (or the frame for one) with which to weave a narrative self.  Dream states may be examples of configurations in which the primary theater is in retreat, raising some question of how far the coherence of the primary theater is, or is not, directly related to the narrative self.  Researchers tell us that dreaming takes place during REM sleep, so that we don't really "wake up out of a dream" with things happening around us contributing to the dream — the dream has to have happened earlier — but I envision the dream process as a sort of idling function similar to the self-loom, producing fragments that the primary loom may catch up and weave in as it's coming on-line when we wake.  The dreams we "remember" would be those woven into the self-narrative, and could therefore have been selected based on events taking place as we wake; this ought to relate to the mechanism whereby agents are promoted onto the stage based on relevance to external stimuli.  The resulting patched narrative, with stitching from the dream fragment to the situation on waking, would look like waking up to that situation out of the dream.

Note that I'm using the term "narrative" to describe a semantic understanding of a process that happens, not a linguistic rendition of the process.  The loom I'm describing is prior to (or, deeper than) language.

It may be crucially important that (as anyone with a pet cat or dog has likely noted) non-sapient animals dream.  If dreaming is an idling function of what I've been calling the self-loom, then the loom would seem to be properly part of the mind prior to sapience.  Presumably an animal mind would have an agent-promotion system akin to the one that advances agents to our non-Cartesian stage, leading to questions about the size of an animal's short-term memory that I have difficulty imagining how to investigate.  The relations between promotion device, loom, and sapience ought to bear on the development of the narrative self.

Another likely manifestation, btw, of the self-loom is the illusion that we hesitate before acting on a sudden stimulus.  Something abrupt happens, we hesitate, and then we act; or rather, that's the story we end up with.  But that hesitation should be an artifact of the loom.  The stimulus has to propagate through our nervous system, and when you're talking about fractions of a second, that propagation time is significant.  As the loom weaves a story after the fact to describe what happened in terms of a synthesized self, it can't ascribe that time delay to propagation through the nervous system because that's not part of the story world (it would be breaking the fourth wall).  So, to explain the time delay in terms of a directly present self, the loom says that the self hesitated.

Key mysteries in the recipe for sapience would appear to be how new "chunks" of information —abstract thoughts, or agents— are formed, and how the theater-like organization arranges itself.  (As my thinking on this has advanced, I've lost my erstwhile interest in trying to build a sapient mind; the social consequences of building sapiences would be fraught, and I suspect, in any case, that the highest likelihood of producing a healthy, well-balanced sapience would be by the traditional method — biological reproduction followed by many years of child-rearing.)

Jaynesian consciousness

Jaynes envisions consciousness as a construct, with a narrative, so it's not entirely unlike my view of the narrative self.  His notion, which he assembles slowly and carefully over his first three chapters (counting his Introduction), is apparently more structurally detailed than mine, apparently specific to consciousness rather than appealing to a more general theory of sapience, based on a sophisticated notion of metaphor which he presents in his third chapter and associates with language.

Some caution is wanted against possible confusion between Jaynes's approach to consciousness, which does involve narratization; and my approach to mind, which does include consciousness.  To avoid getting tangled, I'll try to consistently use consciousness for Jaynesian consciousness, and reserve narrative self for the memetic notion I'm using within my non-Cartesian-theater-based theory of mind.

The first word of Jaynes's Introduction is "O".  How many modern books can you think of that begin with the word "O"?  How may modern books even use the word "O" (other than in quoting something from a bygone age)?  The terms "prevenient counsel" and "introcosm" seem to belong in a paragraph that starts with the word "O".  The word "introcosm", btw, isn't in any of the unabridged dead-tree dictionaries I have ready access to, and has almost no profile on the internet; one of the top Google matches I got was someone asking what it meant ten years ago on Yahoo Answers —because they'd encountered the word in Jaynes's book— and apparently nobody else knew either.  From hints here and there, I'll guess it's psychology terminology borrowed into English from Spanish, perhaps tracing back to Juan Luis Vives, the sixteenth-century father of modern psychology.  My favorite definition so far, from Steven Kotler: "the universe within, the infinity glimpsed down the rabbit hole of mind."

Jaynes's Introduction is a whirlwind tour of different approaches taken historically to the question of consciousness.  Based on my impressions from first reading, I was particularly keen to look for discrepancies between what question Jaynes asks, and what questions the other historical approaches ask.  That rather delightful first paragraph of Jaynes's, I find, defines his question:  where does the introcosm, the internal world of the conscious mind, come from?

On my first reading, I suggested Jaynes was rather hard on past theories.  On second reading, less so, with ambivalence on one theory in particular.  Most of the historical approaches he describes are too broad to qualify as paradigms in the Kuhnian sense, and he offers plausible criticisms of them for his purpose.  However, near the bottom of his list when he comes to the major twentieth-century school of behaviorism, which does qualify as a paradigm, it seems to me there may be some confusion between what Jaynes is interested in, what the paradigm is interested in, and what the behavioral approach —viewed in historical context, which is after all the context in which Jaynes presents it— applies to.  Kuhn noted that a scientific paradigm defines what questions can validly be asked, and behaviorism expressly declines to ask about the introspective view of the mind; in historical context, behaviorism's defining role seems to be, to learn about the mind by studying it strictly from the outside.  Perhaps, as Jaynes portrays, behaviorists explicitly denied that consciousness exists (on which Jaynes's training informs while mine does not); perhaps, also, Jaynes's perception of them may have been influenced by his own interests.  As may be, it seems a valid point by Jaynes that if your objective is to understand the origins of consciousness, behaviorism won't help you.

It didn't register on me till partway into the following chapter, that one of those general approaches Jaynes had ticked off in his whirlwind tour is partly implied by the narrative self.  I'll come back to that.

Back near the start of his Introduction, Jaynes notes that a variety of metaphors have been used to describe consciousness, varying with the popular imagination of the day (such as one from the nineteenth century that makes consciousness sound very like a steam engine).  Starting with early metaphors describing consciousness as a sort of vision (which, I note, the above definition of introcosm also does).  I didn't much note the early mention of metaphor on first reading; it seems more significant on second reading, though, foreshadowing where Jaynes will take his explorations later in the book.

(I've been calling, btw, the Breakdown of the Bicameral Mind a book, because honestly in plain English that's what you call a thing made up of consecutively numbered pages bound in a cover with a collective title for the whole.  But Jaynes calls it an essay.  His essay is nineteen chapters divided, after the Introduction, into three parts with six chapters each, and he calls the parts books (numbered I–III), hence presumably his avoidance of the term book for the whole bound volume.)

Following his Introduction, Jaynes's Book I is meant to follow the path by which he arrived at his beliefs, and his first chapter of Book I is about what consciousness is not.  He suggests, at the end of the chapter, that this is an essential start to making his case, because if he can't show it's believable that an entire civilization could be made up of people who aren't conscious, the rest of what he has to say will fall flat.

The significance of the chapter from my perspective is quite different, due apparently to Jaynes focusing on consciousness while I'm focusing on the narrative self.

Each of the things Jaynes says consciousness is not, to me is a feature of the story we tell about the narrative self, therefore it's an element of storytelling about which we could ask, when did this element first enter the storytelling tradition?  That is, each of them is potentially something to try to place somewhere on my timeline.  Major items on his list:  reacting to things doesn't require consciousness; concepts do not require it, nor learning, nor thinking, nor reason; and consciousness has no specific location (he notes that while we place it in the head, Aristotle placed it in the upper chest).  Along the way, he also notes that the notion of the mind as a blank slate (evidently related to several of the items on his list) is present in Aristotle but didn't really catch on until John Locke in the seventeenth century.

However, while Jaynes reasons carefully about each individual thing on his list to show it doesn't need consciousness, for me it's immediate that none of those activities would require the narrative self because, simply put, the narrative self isn't involved in any of them until after the fact.  The narrative self is like a fictional character in an historical novel, witnessing real historical events but unable to affect them because, after all, the character wasn't really there, but instead its participation was invented later by the storyteller.  This, somewhat ironically, puts the narrative self roughly into one of Jaynes's rejected general views of consciousness from his Introduction:  the fifth in his whirlwind tour, which he calls the "helpless spectator", a witness to events but unable to change them.  Afaict it differs from the helpless spectator, because this fictional character is modeled on something else —a sapience engine— that really was a participant in the historical events.  The sapience engine doesn't behave quite like a narrative self would, and this mismatch leads to some artifacts in the storyline (like the illusion of hesitation, mentioned above); but mostly the story hangs together pretty well.  Even though any engagement of the narrative self in those real events is, strictly speaking, an illusion.

Jaynes presents what he's doing in this chapter as disproving misconceptions —his word— about consciousness.  The reason these things are features to me, elements potentially to be placed on the timeline rather than misconceptions, is exactly because I'm embracing the essentially fictional and after-the-fact nature of the narrative self.

Despite the perspective shear between Jaynes's focus on consciousness and mine on the narrative self, Jaynes and I do have something curiously in common here, in the big picture of what we're doing:  our ideas are both quite mind-bending to consider at scale.  Jaynes openly acknowledges, in concluding the chapter, that the idea of an entire civilization of people who aren't conscious is extraordinary.  However, even though I've described my alternative as "less radical", in a sense the narrative self goes even further than bicamerality in this regard:  if the narrative self is a fiction that imperfectly approximates after-the-fact the actual performance of a sapience-engine, and to the extent this narrative fiction is the essence of what we think of as a "person", then while Jaynesian consciousness offers the prospect of whole ancient civilizations with no conscious people, the memetic narrative self proposes that the civilization we're in now has, in a certain sense, no people at all.

Having explored in-depth his key thesis that entire civilizations of people could function without consciousness, Jaynes develops in Chapter I.2 his theory of what consciousness is.  From my perspective, there are two kinds of content in this chapter, differing in their relevance to my agenda:  how consciousness is formed, and features of the form it takes.

The features of consciousness described here are of less interest to me than the ones in the previous chapter.  Those in the previous chapter were, in my terms, features of the narrative self that don't accurately reflect the approximated sapience-engine — which made them of particular interest to me since they were evidently part of the storyline of the self, and I'm interested in exploring the development of that storyline.  They were of less interest to Jaynes, as they bore neither on the character of the approximated mind nor, apparently, on the process of forming consciousness, both relevant to Jaynes's agenda.  The features in this chapter are of more interest to Jaynes as he means them to accurately reflect the underlying mind, which he wants to understand better; but I already have a model of the sapience-engine.  I'm not looking for new insights there, open though I'd hope to be to enhancements —or, more unfortunately, contradictions— to my model.  So features of consciousness that Jaynes considers correct, i.e., accurate reflections of the mind, would likely have less bearing on my agenda.

Jaynes and I do, broadly, share an interest in how consciousness —or the closely related narrative self— is formed.  Jaynes acknowledges that consciousness is an incomplete projection of the actual mind; but he can't invoke memetics to explain how the projection works because he developed his ideas before memetics was proposed by Richard Dawkins's The Selfish Gene (1976, the same year Jaynes's Bicameral Mind came out).  Jaynes therefore devises another means of projection from the actual mind to the consciousness:  metaphor, for which he provides an elaborate theory, with a cross-connection to language that makes of language a major tool for probing the historical development of consciousness.  This results in a remarkably dense chapter, a fact that Jaynes explicitly acknowledges at the end of it.

Jaynes coins a flurry of terms to describe facets of metaphor.  Any metaphor projects characteristics of one thing onto another thing; the thing projected from, he calls the metaphier, the thing projected onto, the metaphrand.  The metaphier, he says, is always more known than the metaphrand, as we're projecting the metaphor in order to say something about the metaphrand.  Not everything about the metaphier is relevant to the metaphor; so the particular features of the metaphier being projected are the paraphiers, and the features of the metaphrand they project onto are the paraphrands.  An example he uses is "the snow blankets the ground":  metaphier a blanket on a bed, metaphrand a layer of snow on the ground; paraphiers, he suggests, warmth, protection, sleep to be followed by waking, projected onto paraphrands of the snow keeping the earth snug while it sleeps till spring.  He also has a term analog for a specialized type of metaphor, in which the metaphrand is meant to correspond part-by-part to the metaphier, as with a map corresponding part-by-part to the mapped territory.  Even on the second reading I found myself wondering, why do you think this is important enough to warrant all these new terms?

Jaynes lists six features of consciousness:  spatialization, exerption, the analog 'I', the metaphor 'me', narratization, and conciliation.  Essentially, these are an array of operations for generating aspects of the introcosm (cf. the epigraph from Locke at the start of the Wizard Book, also seeking to enumerate thought-generating operations).  After intense study, I still can't figure out the difference between the analog 'I' and the metaphor 'me'.  He spends about a page explaining the term conciliation — as I understand it, representing something in the conscious mind by shoehorning it into a form we're already familiar with — and then scarcely uses the term conciliation again in the volume (though it gets iirc at least one passing mention in Book III, and another in the 1990 Afterword).

Jaynes develops his treatment of metaphor in terms of language.  Metaphor, he says, is the primary means by which language builds new vocabulary.  By the end of the chapter, he takes this a step further and claims that metaphor is a linguistic process, i.e., that metaphor cannot exist without language — which implies, since he maintains consciousness is formed by metaphor, that consciousness cannot exist without language.  I don't see evidence supporting this in his treatment, perhaps because (as remarked earlier), in difference from most authors I've read on this subject, I'm inclined to think of the algorithmic processes of the sapient mind as prior to, and thus generating, language rather than vice versa.  However, Jaynes and I end up in similar places after all, Jaynes theorizing that consciousness is generated through linguistic metaphor, and I that the narrative self is a product of storytelling.

There appears to be an interesting comparison and contrast here between the machineries envisioned, for processes within language versus prior to language, in Jaynes's model of mind, Dennett's, and mine.  As remarked earlier, Dennett's model appears, by my lights, to lack any internal structure to the device prior to memes being poured into it; but at the linguistic level, where Jaynes has his extensive inventory of consciousness-generating processes, I have as little to say about detailed processes as Dennett does.  Whereas below the level of language, I'm the one of the three who suggests algorithmic structure while Jaynes and Dennett impose no structure at that level.

The distinction between high- and low-level processing also plays in to my earlier point that the narrative self is fiction, versus Jaynes's view of consciousness as metaphor.  Jaynes makes clear, as his discussion branches out in later chapters, that he sees the metaphor projection of consciousness as suppressing bicameral function of the mind, and thereby fundamentally altering the way human beings behave.  Under my approach, the evolution of the narrative self would obviously affect people's behavior (as would any aspect of the evolution of storytelling), since it's part of what we think, and what we think affects what we do.  So Jaynes and I agree on this much, that these phenomena, consciousness/self, have real consequences for human behavior.  There is no contradiction here with my observation that the narrative self is a helpless spectator, witnessing events but unable to change them — because this is the difference between thinking, and being thought about.  That is, the narrative self does not actually think, because its supposed role in thought is an after-the-fact fiction, with actual thought being done by the sapience-engine; but the sapience-engine does actually think about the narrative self, constructing it and reasoning about it, and the sapience-engine is a real actor in the world, so its thinking about the narrative self has real consequences.  This distinction, though, is possible for me because I assign structure to algorithmic processing below the level of language; while the distinction is apparently not available to Jaynes because he doesn't consider algorithmic processing below the level of language.  Without lower-level algorithmic processing, he continues to view consciousness as an active participant in thinking.  His position is not obviously inconsistent (though lacking some telltale explanatory power noted earlier, such as for the hesitation effect); but, with Jaynes already acknowledging that thought does not require consciousness, it seems to me much simpler to keep the two cleanly separated by making the conscious mind always thought about rather than thinking.

On further reflection, it seems an advantage to the particulars of my low-level treatment that they factor out the chunking aspect of processing, by which ideas are formed, addressing only certain other kinds of processing — some ways that ideas, once formed, arrange themselves into a coherent mind.  So I'm able to say something useful about low-level processing without being dragged into the central mystery of what Terrance Deacon, following C.S. Pierce, called symbolic thinking.  In contrast to which, Jaynes (and, for that matter, Locke) tried to describe idea formation at a high level — which seems to me a doomed attempt as it gives an unavoidably oversimplified account of something I suspect is at the heart of sapience.

Words

Jaynes is particularly interested in the changes of meaning by which words gradually go from referring to concrete things, to referring to abstracts to do with mind or spirit.  He touches on various words, e.g. English be and obey, Greek soma and wanax, but focuses especially on words for mind or spirit illuminated by the Iliad, which he considers the earliest written material we understand well enough to sift closely for the sorts of subtleties he's after.  The Iliad is an oral story preserved in written form possibly as late as the eighth century BCE (about 28–2700 years ago; Jaynes figures 29–2800), describing alleged events from at least four centuries before that (32–3100 years ago).  Jaynes focuses principally on seven words:  thumos, phrenes, noos, psyche; kradie, ker, etor.  But here I hit a major procedural snag.  How am I to make use of Jaynes's work?

Jaynes acknowledges a bias problem with this material; he describes translation of abstract terms in ancient texts as "a Rorschach test in which modern scholars project their own subjectivity with little awareness of the importance of their distortion."  (Btw:  when Jaynes says subjectivity, he means thinking in a conscious manner.)  That's why he starts with the Iliad rather than something older.  But how then can we judge the bias of Jaynes's interpretation, without first learning ancient Greek and extensively studying the text of the Iliad ourselves?  The best available answer, trite though it is, would seem to be:  carefully.  With each of Jaynes's observations about these words, one has to consider the likelihood of opportunities for Jaynes to misread the evidence in that particular case.  The same goes, of course, for mainstream thought on these words, which Jaynes so colorfully describes being led astray by its expectations.

I also see an opportunity here to make use of another major body of etymological work — again, of course, keeping in mind likelihood of misconstructions:  Proto-Indo-European (PIE).  Far and away the most broadly cross-correlated reconstruction of an early proto-language, PIE is for that same reason far more likely to reflect real trends in early language, and probes significantly further back into the originating period of consciousness/self than the Iliad; PIE is believed to have been spoken in a period from roughly 6500 to 4500 years ago, more or less, which would have it falling out of use more than a thousand years before the events in the Iliad (let alone their written recording, another half millennium or so later).

And here is something I already know about PIE (once reminded of it), from years of dabbling in conlanging:  PIE does something funky with verb tense.  I've seen authors say it doesn't even have tense, just aspect.  Wikipedia portrays PIE tense as present versus past, but the deeper one goes into that, the weirder it gets.  I'm reminded of a remark (somewhere in the Conlangery podcast) that all these technical grammatical terms become wobbly when one starts looking at multiple languages.  But evidently those reconstructing PIE have had particular difficulty working out what to do with verb treatment of time, which in my framework is already something to watch out for, a likely area of volatility as verbs wouldn't have treated time at all until the advent of orality.

Jaynes also mentions related work by Bruno Snell, a couple of decades before Jaynes's Bicameral Mind, on the gradual development of awareness of mind from Homer through Aristophanes.  Snell's apparent view of the process as a growing awareness rather than a change in mind-function would naturally make his work uninteresting to Jaynes, who mentions Snell in a footnote to explain, in effect, why he won't have any more to say about him; to me, though, Snell seems well worth investigating hereafter, as a different, philological take on the matter and a counterpoint to Jaynes — my own view of the process, as development of story, being somewhat different from (afaik pending further study) either.

Gods

At the core of bicamerality, as Jaynes envisions it, is the god-voices generated by the right side of the brain.  He develops this idea at length, that the human characters in the Iliad, Achilles and Agamemnon and on, are told what to do by gods; and that the humans have no self-awareness, no consciousness, no introspective world.  Now, this is an interesting point, because lack of a narrative self seems to me quite a separate question from no inner world.

By contrast, Havelock maintained that abstract forces are a literate notion, while oral traditions require actors, for which the Greek gods, he noted, are exceptionally well-suited.  That's a distinctly narrative view of gods.  (I'm reminded, at this point, of something I read about pre-Christian Slavic mythology — that when arrested by Christianity its development was partway though a natural evolution from lesser spirits to greater gods.  That too seems a potential source of some insight, worthy of further investigation, into the developmental track of storytelling.)

Under my model of sapience, which presupposes an underlying algorithmic mind-structure that doesn't vary much across these changes of mindset, the gods, like the modern narrative self, are just another thing thought about.  So that the existence of an inner world thought about, and perhaps used for planning in a not-so-alien sense, would be quite separate from whether the characters in that fictional world are presented as being self-aware in the modern sense.  In this way, self-awareness appears to be a key story element, whose development for the story one would expect to be extraordinarily difficult to distinguish from the development of self-awareness by the storyteller.

What would a mind be like if primed by an earlier form of storytelling, without self-awareness built into it?  Presumably the answer ought to be:  rather like the minds of the human characters in the Iliad.  And possibly also rather like Jaynes's bicameral mind.  Jaynes, being a psychologist, was understandably much concerned with what it would be like to be bicameral.  For my part, with zero training in psychology, the psychology of minds in earlier stages of the development of storytelling initially seemed a far more intractable puzzle than working out a sequence of development of overt storytelling features.  Central though the reality, or unreality, of bicamerality is to Jaynes, my own central concern in this regard was finding which aspects of bicamerality are compatible with my model of sapience, and whether any incompatible aspects are especially likely.  It only came on me gradually that I need the internal psychology to unravel the compatibility questions.  Keeping in mind that imaginability though perhaps necessary can't be sufficient — however far Jaynes might have successfully imagined bicamerality, and however compellingly communicated to the reader, doesn't require it to have actually been so.

Jaynes has a persistent problem with confirmation bias, which imho he doesn't do terribly well at guarding against; I remarked this on my first reading.  The overall force of his argument builds through many specific details that, taken individually, could afford alternative explanations — some of which, in fairness, are much easier to see if one has an alternative explanation on-hand for the general phenomena he's noting (which, of course, I do).  For example:  In Chapter I.3, he addresses the potential objection to his theory that bicameral civilization should lead to chaos in any but a rigidly structured society, which is not what the Iliad depicts.  He notes recent translation of Linear B, which turns out to depict a rigidly hierarchical society, and he concludes that the authors of the Iliad simply ignored this aspect of Mycenaean society.  Apparently the available Linear B texts are all administrative records, which to me makes it unsurprising they'd present the society as rigidly structured.  But it also struck me that Jaynes had just argued (on the previous page, no less), in reply to another objection, that the authors of the Iliad, in describing pervasively interventionist gods, were just describing the world as they knew it, not exercising some sort of poetic license.  So Jaynes wants the authors' descriptions of gods to be simply describing the world as they knew it (which would be my guess also; it's quite consistent with an evolution of storytelling), but then when the poem isn't consistent with the kind of ancient society he wants, he figures they're systematically leaving that out of the epic.

Brains

Though above I contrasted Jaynes's, Dennett's, and my models of mind at the linguistic versus algorithmic levels, there's also a still-lower level one could consider; the neurological, the realm of brain hemispheres and regions, commissural fibers, modules, and whatnot.  Dennett has some things to say about the neurological level; Jaynes has a great deal prominently to say about it, as he conjectures a particular neural mechanism responsible for generating god-voices.  I'm the one of three who has had nothing to say at the neurological level, focusing instead on the algorithmic level with an eye to how it can give rise to higher-level structure.

Jaynes claims the brain is specialized to support bicamerality, with both hemispheres separately capable of understanding language, but only the left hemisphere in control of externally producing language, while the corresponding right-hemisphere facility is set up to produce an internal god-voice to tell the left-hemisphere what to do when a decision is needed.  He attaches much significance to the fact that language is controlled by a single hemisphere while most other "important" facilities, he says, are redundant to both hemispheres; he claims some great evolutionary pressure must have been at work to create this asymmetry — which falls rather flat for me, as I see no reason why this asymmetry should be motivated by supporting bicamerality rather than, say, because language is too delicate to allow multiple sources of control, or, if the right-hemisphere facility must have some driving functional purpose, why the purpose mightn't be something else, related perhaps to short-term memory (which may also require singular control, implying the asymmetry would be far older than sapience).  Jaynes proposes, as the channel carrying god-voices from right to left, the anterior commissure, and spends some time describing cases of commissurotomy, surgical cutting of the connections between the hemispheres (as a treatment for epilepsy); interestingly to me, though passing very quickly and casually in Jaynes's treatment, "all patients show[ed] short-term memory deficits".

To what extent are bicameral god-voices, or similar phenomena, reconcilable with my model of mind?  The audience for my non-Cartesian theater is a sea of agents that may sometimes coalesce into rather large structures, each of which one might wish to treat either as a coherent group of agents or as a single especially-large agent.  If the theater effectively resides in the left hemisphere, small agents in the right hemisphere may find it more difficult than their left-hemisphere colleagues to achieve individual promotion to the stage, so they may have better luck if they form coalitions with other right-hemisphere residents; and there's also a second language facility on the right that could allow such a coalition to take on a relatively self-like form.  One consequence could be an actor, from the right hemisphere, appearing on the stage with more-or-less the aspect of a bicameral god.  More generally, one might expect the hemispheres to specialize in qualitatively different sorts of agents, the left perhaps for smaller and the right for larger (whatever that would actually come out as in practice).  Perhaps a major advantage of the asymmetric theater is simply that it offers nonuniform granularity of thoughts.

Evolution

Jaynes says bicamerality is the last step in the evolution of language.  Of course, my theory posits a different relation between language and mind; and the Iliadic mind, whatever it was like, is not the last step for me; but this language-development view does give us some common ground, enhancing the likelihood that Jaynes's reasoning may have bearing on my timeline.  Jaynes advocates group selection; he has in mind a bicameral civilization as a hive-like phenomenon with great numbers of people coordinated by the voices of gods, and he envisions this evolving at the group level rather than the individual level — which, I admit, strikes me as rather ironic since Jaynes's essay came out in the same year with Dawkins's The Selfish Gene which has, as a major theme, debunking group selection.

I've mentioned Jaynes's theory that language didn't even start until the Upper Paleolithic, nominally forty thousand years ago.  His basic justification for this claim is that if language had started millions of years earlier, he would expect lots more technological and cultural progress over all that earlier time; for which I have an alternative explanation, with my verbality hypothesis, fitting neatly with the long incubation time of genetic life before the Cambrian explosion.  Going in to my second reading of Jaynes, my guess was that the steps he envisions in the development of language should be a mix of some steps that for me belong in the verbal phase of development, and some that belong to the development of storytelling in the oral phase.

Jaynes's major milestones in the development of language:

• Intentional calls (as opposed to silent signals), then separation of modifiers (e.g. intensifiers) from what they modify.  Both prior to the Upper Paleolithic.
• Commands.  He figures this causes the Upper Paleolithic onset, which I've attributed to the phase shift from verbality to orality.  It seems an awfully dramatic consequence to attach to something as simple as expressing a command, until one realizes that Jaynes is (and I am too, for that matter) considering how the conceptual framework we construct affects what we can think.  This harks back to the old debate over the Sapir–Whorf hypothesis; Jaynes evidently puts the framework at the language level and I at the algorithmic, but we both figure there's some room for the bounds to be stretched and thus to evolve over time and generations.
• Animal nouns.  Causes appearance of animals in cave paintings, between twenty five and fifteen thousand years ago.  It's not immediately obvious to me what sort of development in storytelling might correspond to this observable effect.
• Thing nouns.  Causes appearance of various new things — technical innovations, like barbed fishhooks.  Here I'm insufficiently clear even on what event he has in mind.

Jaynes suggests that with language would come auditory hallucinations, providing continuity of attention in the absence of consciousness.  Which is what the theater does in my model; and since I consider the theater necessary for the self-loom, and the loom necessary for dreaming, and since the cat now lying draped across my foot clearly dreams, I don't buy that language is needed for continuity of attention.  I might buy that a larger theater (i.e., short-term memory) would make it easier to think through complex tasks, and remain focused on detailed procedures for a long time.

• Names.  Jaynes puts these quite late on his timeline, around twelve to ten thousand years ago.  This is curious since Everett describes the Pirahã as having names.  Jaynes sees this as a major step in our conceptualization of other people, and therefore a major step in the development of bicameral god-voices.  I'm skeptical on this point, as animals who aren't remotely sapient clearly recognize other individuals.  Jaynes says, plausibly, that names increase the intensity of our thinking about people, just as animal nouns increase the intensity of our thinking about animals.  I'm interested in why he thinks it's such a late development.

Shifting from the linguistic to the archaeological, Jaynes notes the emergence from around seven thousand years ago of great agricultural civilizations.  He figures some tremendously powerful device —bicamerality— is needed to enable this.  At this point, though, I find his case rather weakened by comparison with current events.  Large numbers of people motivated to follow the directives of a fictional being does not require some radically different type of mind.  Not to put too fine a point on it, if people want badly enough to find a strong leader with their interests at heart, they can be appallingly vulnerable to someone who tells them what they want to hear.  They'll convince themselves that this demagogue has their best interests at heart, despite ample evidence of intense selfishness, and will believe all manner of reality-defying claims because the demagogue said them.  Not only is this demonstrating a tremendous capacity of modern conscious populations to be ring-led, but it's fair to say the demagogue's devotees, who generally want a strong leader with their interests at heart, are following the leader they imagine the demagogue to be, rather than the actual person.  That in itself makes me doubt that any radical hypothesis such as bicamerality is needed to explain ancient civilizations organized around god-kings.

Jaynes offers three basic features of ancient civilizations that he maintains only make sense given bicamerality. 

His first feature, treatment of gods as the rulers of civilizations, I found tbh thoroughly unconvincing as evidence for his theory, and (at first blush) not overtly insightful for the evolution of storytelling, either.  Major explicit archaeological evidence for this feature is "houses of the gods" placed prominently at the center of ancient cities, in the positions, as Jaynes notes, where one might expect to find the dwelling of the prince or ruler.  That expectation struck me as based on an unjustified bias in favor of an atheistic social structure.  The most spectacular edifice in a medieval European city was apt to be their cathedral.  Of the evidence Jaynes describes in that part of his discussion, the only point that seemed to me to invite explanation (I remarked on this also on first reading) was the way the Incan empire was conquered so ridiculously easily by Spanish conquistadors in 1532.  That peculiar episode doesn't seem to need an extreme explanation such as bicamerality —religion has plenty of control over conscious people today— though it might, perhaps, demonstrate that different stages in the development of storytelling can alter how receptive individuals are to certain kinds of memes.  On reflection, that's not a surprising phenomenon.

The second feature he presents as evidence is custom treating the deceased as if they were still alive.  There are certainly plenty of examples of that; but again, I see no demand for bicamerality here.  The motive for such practice could be as simple as a state of denial, or, more in line with my scenario, a narrative presentation of people as forces not limited in time; a natural result, perhaps, of attempting to position people, as speech participants, in a conceptual framework that has had time added to it (fitting neatly with the theory that time is the essential ingredient for the transition from verbality to orality).  Jaynes claims that a wide variety of early civilizations used the same word for dead people as for gods — depending on the tricky question of how to interpret somewhat abstract terms in a language further removed and less triangulated than PIE.  Jaynes's interpretation of such words being apparently as much of a Rorschach test for his biases as more mainstream researcher's for theirs.

The third feature is the extensive religious use of human idols.  This does seem to want psychological explanation.  Here too, Jaynes reasons that the observed phenomenon demands explanation and only bicamerality will do, while I see no call for any such extreme explanation.  Nonetheless, the use of human idols does seem likely to fit into the evolution of storytelling; and offers the enticing prospect of concrete archaeological evidence going much further back in time than linguistic evidence can — though the details most of interest to me aren't of interest to Jaynes, details of sequence of changing practice over time.

I'm quite open to the possibility that the human mind, when conditioned with an earlier stage of orality (i.e., an earlier form of storytelling), might engage in more hallucination as part of its normal function; conceivably even hallucination as pervasive as Jaynes suggests (though I do wonder if the effects involved could take some intermediate form less explicit than what we would call hallucination... or if, conversely, what we call hallucination is differently perceived because of our different conceptual framework).  The two main aspects of Jaynes's theory I'm actively disinclined from are his insistence on a radically altered mind architecture for his bicameral man, and his casual dismissal of religion as a highly impactful idea system rather than a symptom of this radically altered mind-architecture.  This sometimes makes it quite awkward to sort out what to think about some of Jaynes's supposed evidence, as I may find myself neither agreeing nor disagreeing, in that while Jaynes's interpretation of the evidence may seem quite unjustified, it may at the same time seem credibly like evidence of some different-but-related possibility that Jaynes isn't acknowledging.

As a major test of his theory, he then tries it out on writings from civilizations he figures were bicameral; his main choice is between hieroglyphic/hieratic and cuneiform, and for a starting point he rejects hieroglyphic/hieratic as far less accessible.  As elsewhere in his linguistic explorations, though, while I agree with his concern that mainstream modern scholars fill in the unknowns in translation with their own conceptual biases, I find Jaynes is just as guilty of filling in the unknowns with his bicameral theories.  It seems one would like to go into an in-depth translation effort already aware of a wide range of possibilities, to reduce the risk of lapsing unaware into one or another of them.  How to do that without pouring a large increment of one's life into the effort, is not immediately clear to me (but there's a lot of that sort of problem floating around in this).

The various cultural organizations he describes —such as land owned by gods, with kings as their "tenant farmers"— seem like plausible ways to develop a story of the world if one is building it around deities as the actors needed for oral sagas, making these various cultural organizations stages in the evolution of storytelling.  The particular manners of writing he quotes, in which gods are described as speaking/uttering/commanding, may seem to him compelling evidence of pervasive auditory hallucinations, but seem to me entirely plausible forms of expression for a culture in an intermediate stage of inventing manners of storytelling description.

Jaynes offers particularly a bicameral alternative explanation of the ancient Egyptian notion of the ka, which he presents as badly in need of reconsideration.  Here again, he is at his most credible when criticizing mainstream translators for injecting modern concepts into ancient writings, which they seem likely to do; I'm inclined to take him seriously when he doubts the conventional interpretation of the ka.  (Wikipedia's account of the ka presents a strikingly confident picture, which I don't trust at all because Wikipedia nurtures a consistent bias toward mainstream thought.)  I'm doubtful, though, of Jaynes's contention that these ancient languages were very concrete; my sense from Havelock was that while literate abstraction does not occur in oral society, there is a different sort of oral abstraction, so that it seems Jaynes could be confusing oral-abstract with concrete and thus missing an essential element for grasping concepts such as ka.

For example, Jaynes mentions a Sumerian proverb, "Act promptly, make your god happy"; or at least, that's a common translation.  As an alternative (and with a nod to the extreme uncertainty of abstract translations so far back), Jaynes suggests, "Don't think; let there be no time between hearing your bicameral voice and doing what it tells you."  I observe, though, that if one supposes such voices, be they either hallucinated or conceptualized, are simply a natural impulse of the mind, not fundamentally different from our impulses except for how it's thought about as shaped by more primitive storytelling technology, one might try —for example— "Don't hesitate to follow the dictates of your conscience."

Breakdown

In the later chapters of Book II, Jaynes focuses on the process by which consciousness replaced bicameralism, especially in the second millennium BCE (4000–3000 years ago), describing its causes (Chapter II.3) and evidence of the process (Chapters II.4–II.6).  This material contains more of direct interest to me, since my hope going in was to reinterpret the changes in terms of storytelling technology; though Jaynes's natural efforts to view everything in relation to bicameralism can seem more of a distraction for my purposes.  For example, he discusses trade early in Chapter II.3, noting (or, claiming) that early trade between bicameral kingdoms was not an interaction between individual people, and as I approached the end of the chapter that was one of the points that had stuck in my mind; so that when, in concluding the chapter, he summarized the causes he'd named of the breakdown of bicameralism, I was surprised to see he didn't mention trade, instead (on careful inspection) naming the larger causative point under which he'd mentioned trade, "the inherent fragility of hallucinatory control". Which is the difference between my interest in changing behavior, versus his interest in properties of the internal state of bicameralism.

Another change he describes in that chapter that stuck in my mind —and also not mentioned in his summary— is the emergence of warfare:  apparently, in the preceding millennium villages didn't have defensive walls.  He also notes immense viciousness of Assyrian laws and warfare when Assyria resurged late in the second millennium, which he ascribes to breakdown of social order because the previous social order had been by bicameralism that was failing.

Tbh, the first of Jaynes's arguments I actually found impressive was at the very start of Chapter II.4 (basically, halfway through the essay).  It seems, proportionate to its impact, especially at hazard from Jaynes's persistent difficulty in presenting a false dilemma between mainstream thought and his bicameral theory.  By this point he's already described monumental depiction of Hammurabi, Babylonian king of law-giving fame in the early second millennium BCE (3750 years ago), standing before his god seated on a throne, listening diligently to his god's instructions being delivered in a business-like way — according to Jaynes, a typical, matter-of-fact portrayal of a bicameral king being instructed by his god through normal hallucination in a smoothly functioning bicameral theocracy.  Jaynes starts Chapter II.4 with the monumental depiction of Tukulti-Ninurta I, Assyrian king half a millennium later and ostensibly the first to style himself "King of Kings"; starkly contrasting with Hammurabi, for the first time in history, according to Jaynes, in two respects:  he kneels before the throne — and the throne is empty; also, according to Jaynes, a straightforward portrayal that the gods, who guided orderly bicameral societies through hallucination, have ceased to appear.

Jaynes presents this contrast forcefully, making it quite a stunning revelation:  the gods disappeared, and these monuments directly tell us so if we're able to understand the message.  Jaynes was apparently deeply impressed, put that into his presentation of it, and it comes through.  But, not to get carried away with the false dilemma here, how would this fit into a memetic alternative to bicameralism?  Religious hallucinations happen even today.  The Pirahã apparently have group hallucinations.  Religions, and similar ideologies, exert tremendous force on modern, literate populations.  The evidence presented, in the contrast between Hammurabi and Tukulti-Ninurta I, suggests something momentous shifted, something to do with social order, how one conceptualizes the world, and perhaps even hallucinations; but none of that seems to require a drastic rearrangement of internal architecture as implied by the bicameral hypothesis.  The challenge seems to me to be in understanding how people in these ancient societies conceptualized the world, which, under the memetic hypothesis, should follow a continuous path of development from the verbal-oral transition toward the oral-literate transition.  I'm thinking I should reread Havelock, which just possibly I might get more out of now; and Snell.

Jaynes notes a number of phenomena that arose after, in his view, bicamerality broke down.  First on his list is prayer, begging a god to speak; he reports that recovering schizophrenics occasionally do this as their hallucinated voices retreat from them.  Then he mentions angels, part-bird beings that, he says, start to appear as part of a distancing from gods:  earlier, individuals have gods that can speak to the greater gods, then the same scenes are shown but the greater god is absent as with Tukulti-Ninurta I, and then, angels.  Tbh I don't see how this progress follows naturally with the breakdown of bicameralism, but if one supposes these beings are conceptualizations of aspects of the world it seems this may afford a more natural interpretation of the sequence as a memetic evolution.

Then he describes demons, malevolent entities.  Evil, he says, didn't exist earlier.  As usual, he presents this development as evidence of bicameralism, which I found quite unconvincing:  if these beings are part of the way one conceptualizes reality, and really terrible things are happening in the world (which seems a pretty good summary of that millennium), it doesn't seem remotely surprising that malevolent beings would be introduced into the conceptual mix.

He notes the retreat of the gods, most of whom used to be on Earth, to heaven.

He makes detailed note of the emergence of divination, distinguishing four kinds as gradual steps on the way to developing an analog space in which to consider alternative behaviors of the self — exopsychic, he calls them.  In order of progression toward consciousness in Jaynes's view:  omens, sometimes-bizarre supposed cause-effect connections, so divining from miscellaneous events; sortilege, the casting of lots, so divining from random events; augury, reading from natural processes (such as oil or wax poured in water, or the arrangement of entrails of a sacrificed animal); and at last spontaneous divination, reading from whatever next catches one's eye.  He notes in the second and third steps the use of the right hemisphere, which is good at spatial relations, and use of metaphor.  The first three he says were occasionally known in Mesopotamia in the mid-second millennium BCE, but became major trends later; the first two in the early first millennium BCE, the third in the late first millennium BCE.  Spontaneous divination he doesn't find in Mesopotamia but figures must have been there, noting its description in the Old Testament; and he notes it was popular in Europe into the Middle Ages.

With the force of Jaynes's presentation, it took some time for me to register, as a subliminal sense of discomfort worked its way into the open, that all the while he uses Hammurabi as an exemplar of a bicameral-theocratic king, the law code Hammurabi is famous for concerns penalties for crimes by individual people that don't altogether fit with Jaynes's portrayal of a bicameral society.

Closing out his discussion of Mesopotamia, he notes scattered signs of emerging consciousness:  A change in tone of personal messages from Hammurabi —whose letters are apparently quite factual, or so Jaynes construes— to messages a thousand years later from an Assyrian king.  (Which, I note, is very much a change in mode of storytelling.)  Initiation of detailed annals of events, which Jaynes construes as spatialization of time.  Versions of the epic of Gilgamesh from, seemingly, different eras, wherein the earlier lacks the interior perspective of the later.

A rare glimpse of Jaynes's own internal state occurs in his discussion of spontaneous divination, as he describes applying the technique as he writes that section.  On his first try, he "reads" that he is getting too speculative, and on his second try, that he has to tie together a bunch of miscellaneous threads.

Jaynes's historical analysis of Greek literature in Book II, looking for evidence of the transition from bicamerality to consciousness (he likes for this the term transilience), is at once particularly relevant to my own search for patterns of change over time, and a particularly clear case of the perspective flaw in Jaynes's approach.  He considers the changing treatment of most of the terms from his earlier discussion, which is just the sort of thing I want; calling these terms preconscious hypostases, by which he means, terms used to stand for the elements of internal state that will eventually be assembled into consciousness.  (Btw:  pronounce hypostasis with accent on the second syllable, similarly to hypothesis; shifting to the third syllable in the adjectival form, hypostatic, as hypothetical.)  He hypothesizes that these hypostases pass through four phases of development: an objective phase of literal meaning about the external world; internal phase of literal meaning internal to the person; subjective phase where they become abstract spaces where feelings/thoughts can be "put"; and synthetic phase where all the hypostases are assembled into a singular conscious self.  And this sequence of phases is where his method loses its way.  The intermittent evidence can't testify in detail to the whole sequence, so Jaynes's reconstruction of the changing meanings of the words is inspired by his hypothesis; which demonstrates that the historical evidence is consistent with the hypothesis, that it can be interpreted so as to line up with the hypothesis, rather than actively supporting the hypothesis.  Making his reconstruction more difficult for me to apply to a variant hypothesis, since his reconstruction is partly founded on his hypothesis.

Jaynes's first, "objective" phase of preconscious hypostases seems imho especially dubious, because it is grounded in his much-invoked supposition that early writings are very concrete.  If the nature of abstractions was shifting at that time —and especially if it was continuously shifting, an elaboration from Havelock's thesis— one might plausibly expect the vocabulary of these intermediate-oral abstractions to be nearly indistinguishable, at our great conceptual remove, from a vocabulary we would describe as "objective".

Jaynes dates the crucial transilience to consciousness in Greece to roughly 600 BCE (2600 years ago), noting a tremendous blossoming of Greek literature thereafter.  This would be about two centuries before Plato — recalling that (to my understanding) Havelock reckoned the great shift of Greek culture from orality to literacy had just recently happened when Plato was writing.  It seems that Jaynes and Havelock both placed a tremendous shift in Greek thought in this era, the difference between them being the character they assigned to the shift.

Jaynes remarks of several texts, as he discusses them, that they could be understood as describing the transition from bicamerality to consciousness.  There is some ambiguity in this suggestion, between a text from which one can learn of the transition, and a deliberate recounting of the transition.  The latter —deliberate, though one ought not in Jaynes's framework to say conscious, recounting— raises a more general point about Jaynes's theory in relation to mine.  Supposing that some of these texts were deliberate recountings of the matter, why would one undertake to tell that story?  A closely allied question is, why religion?  Jaynes says the books of the Pentateuch were assembled out of "nostalgic anguish for the lost bicamerality of a subjectively conscious people." (Chapter II.6, p. 297)  This seems to me, on reflection, a rather piecemeal approach to motivations.  I tend to posit a basic human impulse to describe, explain — to storytell; which is admittedly not altogether adequate since the Pirahã apparently lack such an impulse, but does stand in essence for storytelling as a coherent phenomenon rather than a hodgepodge of separate effects.  And this coherence leads shortly to my difference from Jaynes:  once one starts to think of the whole sequence of development in terms of a coherent phenomenon of storytelling, it seems clear that the technology of storytelling would have to be invented, developing gradually over time; and the introduction of this powerful new factor into one's understanding of the situation softens, without eliminating, the ideas Jaynes is applying exclusively.  That is, Jaynes appears to be assembling his model of the evolution of human thought from just two pillars —bicamerality and consciousness— whereas it seems to me a smoother model of the evolution ought to be afforded by building into it an explicit role for the technological development of storytelling.

Despite my objections to Jaynes's method in trying to defend his thesis, and my contention that his core insight signifies something somewhat different, less radical, than what he extrapolates from it, I'm fascinated by the insight itself.  On a few occasions scattered through the essay — I have in mind atm three in particular— the core insight shines through, dazzlingly.  I remarked above on the depiction of Tukulti-Ninurta I kneeling before an empty throne.  Another dazzling moment occurs in Jaynes's discussion of the Old Testament as a record (deliberate or no) of the stormy bicameral-conscious transition; though, curiously, the impact of it failed to reach me on my first reading, as to some extent did all three moments of dazzlement, perhaps because I had to get through the whole of the essay once, to get the measure of Jaynes's overall vision settled in my mind, before I could see these particular moments from Jaynes's perspective.

What I suspect to be the deepest core of Jayne's insight is revealed in Chapter I.4, The Bicameral Mind; an epiphanic recognition of commonality between experiences of modern schizophrenia and ancient writings.  Jaynes, after several paragraphs' detailed description of a schizophrenic episode in which a man visiting the Coney Island beach was commanded by an auditory hallucination to drown himself, writes,

The patient walking the pounded sands of Coney Island heard his pounding voices as clearly as Achilles heard Thetis along the misted shores of the Aegean.  And even as Agamemnon "had to obey" the "cold command" of Zeus, or Paul the command of Jesus before Damascus, so Mr. Jayson waded into the Atlantic Ocean to drown.  Against the will of his voices he was saved by lifeguards and brought to Bellevue Hospital, where he recovered to write of this bicameral experience.

— Julian Jaynes, The Origin of Consciousness in the Breakdown of the Bicameral Mind, Book I, Chapter 4.

This passage not only failed to grab me on my first reading, but its effect on me was delayed even on second reading, finally drawing me back to it after some half dozen additional chapters.

Vestiges

Jaynes devotes Book III to "vestiges" of bicamerality, effects left over from bicamerality that linger even to this day.  He considers religion to be, itself, such a vestige; which I honestly find not just unsupported, but implausible.  Jaynes's basic method in promoting his thesis is to show that it can offer a coherent interpretation of the evidence, and at times he does make it all feel rather persuasively coherent — but, set down Jaynes for a while and go immerse yourself in the traditional interpretation of the history of religion, and it's quite coherent, too.  Religion feels like a thing in itself, not a vestige of something else.  This, I observe, is a common property of Book III:  the effects he describes are, to varying degrees, not suggestive of bicamerality unless one starts with the bicameral hypothesis and looks for things that would then be related to it.  In fairness, Jaynes presents his primary evidence for bicamerality in earlier chapters, and states up front in Book III that he hopes through these later vestiges to send illumination backward to "some of the darker problems of Books I and II"; so he's not really claiming these effects as further directly persuasive evidence.

Jaynes considers oracles a vestige, an effect caused by the loss of ubiquitous bicamerality, and outlines a six-stage process as bicamerality retreats.  His stages:  (1) locality oracles, awe-inspiring places that, in the early post-bicameral age, would still allow individuals to get in touch with their bicameral voices.  (2) prophets, individual people who were still in touch with their bicameral voices after members of the general populace weren't.  (3) trained prophets, taught with increasing difficulty to reach a bicameral state.  (4) possessed oracles, taught to reach a frenzied state from which their voices would speak to others, but not to them.  (5) interpreted possessed oracles, where additional specialists would be needed to figure out what was said in the frenzy.  (6) erratic oracles, less and less teachable, less consistently accessible, less interpretable even by specialists.  This sequence is of some potential interest to me since it's change of behavior, for which Jaynes offers some evidence of chronological progression; but the presentation of the stages as stages in a progression, and interpretation of them relative to bicamerality, seems rooted in Jaynes's chosen thesis.

Jaynes particularly notes that possession, which occurs starting from stage four of his oracular progression, is distinctly different from bicamerality in that the possessed individual doesn't remember it afterward.  Jaynes doesn't feel fully able to fit this into his theory of the internal workings of the mind, though he feels it ought to fit somehow.  Moving outward to increasingly un-bicamerality-like effects, he notes negatory possession, where the subject doesn't want to be possessed (briefly touching on Tourette's Syndrome), and glossolalia.  These effects seem worth some careful attention in my own efforts to reconcile the hallucinatory aspect of Jaynes's ideas into my view of the mind.

Then Jaynes discusses music and poetry — an important topic also for Havelock, as his theory was reinterpreting Plato's remarks on the subject from The Republic.  Jaynes's timeline calls for the bicameral gods to have poetized their instructions; then, as bicamerality begins to break down, for poets to have musical accompaniment, using the music to stimulate areas in the right hemisphere, adjacent to where god voices are generated; then later poets to be unaccompanied by music as poetry becomes a left-hemisphere concern.  He notes Plato describing poets being possessed by the muse.  It strikes me once again, reading Jaynes's treatment, that Jaynes wants a rather extreme rewiring of the brain to be implied by, broadly, the conceptual difference between perceiving the muse as an aspect of the poet versus separate from the poet.  This relates to my earlier remarks about the narrative self being thought about but inherently not a participant in thinking; which I see as a key error in Jaynes's theory, that he supposes the self participates in thinking when it cannot do so.  However, he does make a pretty good case (even if it's not quite what he had in mind) that what we think can somehow affect gross brain architecture, cross-hemispherically.  I often prefer (as one might notice from a sufficiently large cross-section of this blog) to find some way to bypass controversial problems, as a sort of Gordian-knot-cutting; with this thought/architecture entanglement, though, integrating Jaynes's observations into my storytelling timeline seems to require some sort of provision for the entanglement, and it's not immediately obvious how to do this while avoiding commitment on psychological/neurological questions way outside my areas of expertise.

Jaynes devotes Chapter III.4 to hypnosis, describing it in considerable depth and promoting it as another vestige of bicamerality, with the interesting twist that another person, the operator, takes the place of the bicameral voice.  When Jaynes asks whether hypnotized subjects have elevated right-hemisphere activity, predicts they should if the bicamerality hypothesis is correct, and describes evidence that they do, he loses me at the second step because I don't see why his bicamerality hypothesis should imply elevated right-hemisphere activity in this situation.  If the role of the right hemisphere in bicamerality is supposed to be production of the bicameral hallucination, and there isn't any hallucination involved in hypnosis because the operator provides the authorization instead, then wouldn't the hypothesis predict an unelevated (perhaps even depressed) level of right-hemisphere activity?  More broadly, Jaynes claims as he concludes the chapter that the alternative to a bicameral explanation is to suggest that the various aspects of hypnosis are all exaggerations of ordinary phenomena, and this he dismisses as not explaining, but explaining away, hypnosis.  I take his point to be that the purpose of viewing each aspect of hypnosis this way is that the viewer doesn't want to believe in hypnosis.  Some observations about this:  Undoubtedly some people who embrace such reasoning would do so for that purpose, but presumably not all, and in any case, if the reasoning is valid it shouldn't matter why it was suggested.  Jaynes's reason for suggesting various reasoning, after all, is to support his preferred hypothesis, which doesn't necessarily make his suggestions wrong (though it does, admittedly, make it especially needful to view his suggestions with careful criticism).  And just because one views the various aspects of hypnosis as exaggerated fragments of ordinary brain function, rather than as scattered fragments of some fundamentally alternative mode of brain function, does not apparently prevent hypnosis from being a coherent phenomenon.  I'm conjecturing that whatever state of thinking occupied the part of prehistory where Jaynes places the bicameral mind, it would be some coherent mode within the range of basic functions, and so would hypnosis be.  If it's imaginable that fragments of bicameral function would reform, with some pieces missing, into a different cohesive phenomenon of hypnosis, then it seems (to me, anyway) at least as plausible that fragments of ordinary modern brain function could also form into a coherent phenomenon of hypnosis, and if they could do that, why not form into some other configuration(s) with further similarities to bicamerality?

The mixture of features he describes in hypnosis, differing in interesting ways from the mixtures in other phenomena he's discussed, ought to be useful to me in exploring what refinements of my model of mind would be most useful to accommodate some of his insights.

At almost-last, in Chapter III.5 Jaynes discusses schizophrenia.  Which, he says, was not a thing in the bicameral age, was perceived during the breakdown of bicamerality as being god-touched, and only later came to be treated as an illness.  Defining schizophrenia medically is, he notes, a can of worms, but he reckons the florid unmedicated condition is uniquely similar to bicamerality; unsurprisingly, since he apparently modeled his notion of bicamerality on what he knew of schizophrenia.  As he describes patients reporting that their hallucinated voices would interfere with their conscious thinking by getting to the thoughts faster, this seems to me like support for both modes, the hallucination and the conscious thought, being simply alternative presentations of the underlying thought rather than, as I understand Jaynes to claim, profoundly different means of generating the thought in the first place.

Religion he views as a pale echo of bicamerality in the state of individual minds, whereas I see it as a tremendously powerful force in the memetic environment (the noösphere).  (Jaynes's father, btw, was a Unitarian minster, which likely implies he would would have picked up an extensive knowledge of religion and developed immunities to religious dogma.)

He also notes that schizophrenics are able to keep doing the same thing for a very long time without getting bored, and are apt to focus on details to the exclusion of big picture; which are also facets of the way he envisions his bicameral man.  Though, from my own perspective, it sounds slightly reminiscent of the autistic spectrum, which seems internally just about diametrically opposite from bicamerality.

His final chapter, III.6, is about science, which he treats as an offshoot of religion.

On one hand, I already didn't agree with his treatment of religion as a remnant of something earlier rather than an evolutionary development in its own right, and I similarly view science as a further evolutionary development in its own right.  It seems to me the shifts along the path from religion to science make more sense without his bicameral hypothesis; which is related to his hyperfocus —with respect to religion, science, hypnosis, etc.— on authorization, an aspect for him of our bicamerality-supporting neural structures.  Which I just don't see as that low-level; not that people don't look for social approval and all that, but it's never seemed to me to be more than one impulse among many.

On the other hand, I've felt all along (through both readings) there's something awkwardly off-kilter about Jaynes's approach to scientific methodology.  Some of that may be unavoidable since he's exploring areas whose inherent depth of complexity earns them the somewhat-pejorative descriptor "soft science"; but the evidential flaws I've been noting in Jaynes's treatment are, with hindsight, consistent with symptoms of treating scientific hypotheses as authoritative pronouncements rather than conjectures within a bundle of alternatives.

Jaynes as a whole

Jaynes's 1990 Afterword looks at his theory as a whole; conveniently for me since, for my objective here, coming to the end of his essay I need to sum up my understanding of his ideas into something applicable to elaborating my own theory.  He notes that he has, properly, not just one hypothesis, but several; which is true for me also, of course, as I've got my ideas about (at a quick inventory) sapience and mind; language; evolution; stages of culture; and the memetic structures of religion and science.  Jaynes's inventory of hypotheses puts first "consciousness is based on language", which he, rightly I think, spends the most time on.  This is the very point on which my treatment most differs from his (and from others'):  he says other researchers have failed to separate introspection from other cognitive processes that aren't done by consciousness; but again, as has gradually come out over the above discussion, I don't agree that anything is done by consciousness.  I'm seeing the conscious mind as an illusionary construct of the same order as hallucinated gods; which also implies that, even if Jaynes were right about the past ubiquity of hallucinated gods, the shift from there to ubiquitous consciousness would simply not be as foundational as Jaynes describes it.

He also, by my lights, overplays the idea of metaphor.  Just as I hold thinking prior to consciousness, likewise prior to language.  His metaphors are also, I think, too directional.  When we need a word for something, and stretch a word we already had to cover the new case, our choice of word says something interesting about how we're thinking, what meanings we find similar to each other, but it's not wholly a mapping from the previous meaning to the new one; stretching the meaning of a word to cover more is just stretching the meaning of the word to cover more.

The tail end of his Afterword strikes a peculiar note, as he claims emotions are consciousness of affect —consciousness of biochemically organized behavior, of things like fear, shame, sexual excitement— which cannot happen in bicameralism precisely because it is consciousness of these things.  He makes his case that techniques had to be developed, after the breakdown of bicameralism, to end emotions so they didn't just get stuck in a positive feedback loop.  He tells an impressive tale of how the first tragic play in Athens, The Fall of Miletus, was so upsetting to the populace that it shut down the city for days, after which it was banned, burned, and its author banished never to be heard from again.  For another example of out-of-control emotion, he offers the story of Oedipus, who, he says, is alluded to in the Iliad and Odyssey, where apparently he killed his father, married his mother, subsequently realized it and felt shame —an affect— then got over it and lived on with his mother and their children to the end of his days; whereas later, in a more conscious age, the story was retold with Oedipus feeling guilt, an emotion, and going completely off the rails, tearing his eyes out etc.  And thirdly Jaynes claims that sexual fantasy was also invented during this time.  This is another of those Jaynesian details I'm not sure what to do with, both because I'm not sure how it might be integrated with my scenarios, and because I'm not sure how much is really there to integrate.  Noting in this case, for example, that Jaynes's impressive tale of the first tragic play in Athens isn't particularly consistent with the current mainstream account of what happened.  (Wikipedia's inherent vulnerability to mainstream bias also means, conveniently in this case, it's tolerably likely to accurately depict mainstream thought.)

Jaynes's entire scenario seems to me to have too many parts; starting from his basic premise of bicameralism he then looks for diverse causes for its occurrence, its breakdown, and the emergence of various other things in its place, whereas my scenario calls for the continuous operation of a single process of memetic evolution.  Ideally, anyway.

Overall, Jaynes has both narratized time and (I think) storytelling starting post-bicameralism, whereas I've had them starting nominally forty thousand years ago, when he has language start.  He even describes the advent of consciousness in the first half of the first millennium BCE (3000–2500 years ago) as a "cognitive explosion", much as I've compared the advent of art and storytelling to the Cambrian Explosion, except of course I put the explosion more than thirty-five thousand years earlier.  So if I mean to account for his evidence in my scenario —and it does seem to me my alternative has some advantages worth exploring— I should have a working hypothesis for how storytelling evolves through the period where Jaynes put bicameralism and its breakdown.

I should also be considering how my model of mind might interact with some form of hallucinatory phenomena, and the neurology thereof.

Frame story

A frame story is a story within which a story is told.  A modern example occurs in the movie The Princess Bride, where a grandfather offers to read a book to his sick grandson, and most of the movie is the story he reads.  A classical example is the story of Scheherazade, saving herself and healing her king's mind with her stories night after night for a thousand and one nights, quoted in the epigraph at the top of this post.  There's another reason frame stories matter to this final section, though; read on.

The further we try to push back into the period Jaynes would call bicameral, the less we have to go on.  The invention of writing is itself evidence of something; of some step in the evolution I want to reconstruct; but having archaeological evidence of ancient writing doesn't necessarily imply having a clue what it means.  And even as one thinks one has a clue, one could be severely mistaken.  Jaynes rightly objects to interpreting ancient writings as using alternative metaphors to describe modern thoughts; but it is far more difficult to puzzle out a use of ancient words and idioms to describe the sort of thoughts one would have if those were the sorts of thoughts, words, and idioms one was accustomed to.  You don't have to buy into Jaynesian bicameralism for that sort of translation to be a horrendously difficult challenge.

An alternative searchlight into that period is, as mentioned earlier, the reconstructed PIE (Proto-Indo-European) language, estimated to have fallen out of use more than a thousand years before the events of the Iliad.  By Jaynes's timeline, the PIE speakers should be fully bicameral.  The handling of time in PIE is, indeed, more primitive than in its more modern descendants, which had (for example) to devise their own approaches to handling future tense from which one gathers the mother tongue had no such device.  My sporadic readings on the matter suggest early PIE didn't have any tense at all, just aspect; but this still seems to be a gradual evolution of treatment of time in an age that, by Jaynes's reckoning, ought not to be treating time at all.

Strikingly, Pirahã has aspect.  Which apparently puts it, in that sense, on a par with early PIE, circa 6000 years ago.  So if I'm right in putting the advent of storytelling some 34 (or more) thousand years before that, then evidently tense as such is spectacularly nowhere near the first event in the orality timeline, yet aspect stretches back all the way into verbality.  Granting, aspect in Pirahã could be a difference of that modern anomaly from ancient verbal languages; but even so it ought to imply that aspect is not inherently non-verbal.  Tense, with its apparent late onset, could be the first event in the phase of things that Jaynes is perceiving as the breakdown of the bicameral mind:  first past tense that gets the ball rolling, then future tense that brings the roof down.  Unless, of course, the introduction of past tense is the start of storytelling, more like where Jaynes puts the start of the decline of bicameralism, verbality ends there, and the event I've been figuring for the verbality/orality transition is something else again, such as Jaynes's envisioned start of language.  I prefer to pursue the 40-thousand-years-of-storytelling hypothesis, for now; in which case, it would seem storytelling has to be initiated by something subtler than tense.  A really close contrast of aspect between Pirahã and PIE seems indicated, whether it uncovers some significant difference, or not.

Reconstructed PIE also has a complex system of pronouns; I mention this because, from what I understand, Pirahã appears not to have its own pronouns, but instead a system of pronouns borrowed from another local language, suggesting that, just possibly, pronouns are characteristic of orality rather than verbality.  This could tie in to the question of when the concept of personhood developed.

Besides reconstructed linguistic development, I have a couple of other sources of inspiration available to me for how storytelling may have developed starting from, as I'm conjecturing, the Upper Paleolithic onset (though language reconstruction, notably featuring PIE, seems the most solid of the three).  As a second source, there is some internal reasoning to be done about what sorts of players could have inhabited ancient people's conceptual landscape — gods, spirits (see my earlier remarks on pre-Christian Slavic mythology), and of course the self, of which Jaynes too makes much.  One might add a gloss at this point for my suspicion, earlier in this post, that the notion of an afterlife (what Jaynes calls "the living dead") may result from trying to reconcile the concept of personhood with the concept of time.  As a third source, there are Jaynes's accounts of what was going on over the millennia, with the large caveat that Jaynes is a plainly biased interpreter of whatever direct or indirect evidence he can find.  He apparently pins his entire scenario on one putative event, the bicameral age, for which of course there can only be indirect evidence, and then the rest of what he says happened is an exercise in demonstrating that the evidence could be interpreted consistently with that one event.  So one can't altogether trust that what Jaynes says happened necessarily did happen, varying from case to case; though it can still inspire, and should be allowed for.

The dramatis personae of the introcosm would, as noted, include gods, spirits, and the self.  Some thought might be given to ordering these elements, relative both to each other and to other features such as past and future tense.  Written records are in some sense a character in the story as well, the psychological consequence of their inherent stability being, to my understanding, at the root of Havelock's notion of literacy.

A side notion I've been contemplating, in the vein of less radical variants on bicameralism, is that earlier forms of orality might encourage a less traumatic form of hallucinatory effect than what Jaynes proposes.  This seems implicit in describing the conscious mind as on the same order as hallucinated gods.  Jaynes evidently bases his vision of ancient bicamerality on modern schizophrenia, which, as he notes, is a debilitating pathology that interferes with the conscious mind; but, even if modern schizophrenia isn't something that would be anomalous in any era, we needn't expect ancient hallucinations to have been so disruptive.  If the effects of hypnosis can change over decades because of what subjects expect it's like to be hypnotized —as Jaynes reports— why shouldn't the effects of god-hallucinations change with mental framework over centuries?  In conjecturing a continuous evolutionary development of the oral mind, a less traumatic form of hallucination should better fit the theory.

Before trying to fully sort out quite what to do with Jaynes's version of events —on which I've certainly a better handle, at this point— and, likewise, his notions of internal phenomena both hallucinatory and conscious, I do want to reread Havelock, and study Snell.

...except...

at this point I hesitated.  (Yes, as promised at the top of this post, I'm about to get blindsided.)

There seemed nothing more to do, but I wasn't enthused with the state of things.  I'd finished rereading Jaynes's essay, accumulating some solid insights into implications of my model of mind.  I'd looked at the situation as a whole after the rereading, adding a solid insight about the invention of tense and planning what to investigate next.  Yet I was left with less notion that at the start, of what caused the onset of orality or even when it happened.  My theory, meant to provide coherent shape to the development of human culture, was instead becoming rather shapeless.

Realizing, however, these speculative exploratory posts can't all strike gold, and with no further inspiration apparent, I set about final polishing on the draft, preparing to post.

And that's when I got blindsided.

Remember that list of peculiarities of Pirahã?  No number or time vocabulary, no verb tense.  In my final summing up I added pronouns as sort-of-missing.  But Pirahã is notorious in linguistics for something else, which only dawned on me (wham!) when I stopped trying to push forward and spent some downtime polishing the draft.  There's a technical language property called recursion; Noam Chomsky has tagged this property as universal-and-unique to human language.  And Daniel Everett, after deep study of Pirahã, says Pirahã isn't recursive.  Thereby, if correct, blowing Chomsky's cherished theory out of the water.  Over which, Noam Chomsky — one of the most influential linguists of the modern age — has called Daniel Everett a "pure charlatan".  (Well, okay, my citation is in a newspaper in Brazil, so, he called Everett a "charlatão puro".)

Recursion is the ability to nest sentences (or some similar grammatical category) inside each other, potentially to unlimited depth.  An Old West hero and villain, in de rigueur white and black hats, are playing poker, and the villain says, dramatically aside with an evil snicker, "Little does he know that I know what's in his hand."  The hero says, aside, "Little does he know that I know that he knows what's in my hand."  Villain, "Little does he know that I know that he knows that I know what's in his hand."  And this can go on as long as both players have the stamina for it.  And that's recursion; the ability to say things about saying things.  The ability to tell a frame story.

Which brings me back to the Sapir–Whorf hypothesis, that language affects how we think.  I maintain stoutly that language is a secondary effect of our sapient thoughts.  We project our ways of thinking into our language, and if they don't fit, we bend the language to the purpose, whether that means stretching the vocabulary or stretching the grammar.  Naturally, once the language has been bent for a way of thinking, it may communicate that way of thinking to those who use the language, since communicating thought is what language does.  The language thus becomes a common pool for a shared cultural mode.  It can't do much to prevent us from ways of thinking we've acquired, though it might make it harder for us to communicate them.

My immediate point is that recursion is a symptom, not an underlying cause.  Saying about saying is a symptom.  The underlying cause is thinking about thinking.

Which is the best answer I can offer to Jaynes's question:  where the introcosm comes from.

This, then, is my sketch of the evolution of storytelling; four points, on which to pin the whole:

• invention of framing, of thinking about thinking, projected into language as recursion; forty (or more) thousand years ago.
• invention of past tense (by my tentative impression, about six thousand years ago).
• invention of future tense (a bit later, I take it; best not even guess till I've studied up a bit on the subject).
• literacy; onset in Greece about 2600–2400 years ago.

Sketchy, but no longer shapeless; inviting new avenues of investigation, including a tantalizing long interval between the first two points.  And likely providing new guidance on what to attend when studying Havelock and Snell.