Sunday, February 26, 2017

Vive Le Difference: A Review of From Bacteria to Bach and Back

Daniel C. Dennett

In this intelligently designed book, philosopher Daniel Dennett retraces the themes of his philosophical and scientific research & speculations which have afforded him so much acclaim over the past four decades. He develops a definitive version of his account of how human culture went from blind variation - "bacteria" - to extremely intelligent design - "Bach". Dennett also takes the time to extend, revise, correct and edit his thought, engaging fruitfully with many of his critics and admirers (not distinct sets to be sure). Luckily for the reader, all of intellectual heavy lifting is set out in Dennett's characteristic immaculate prose - which alone would be sufficient for me to recommend the book highly.

You may be asking "Who is Daniel C. Dennett and why should I care about his philosophical nonsense?". The most pressing reason is that Dennett's subject: consciousness. Dennett's purpose is to protect consciousness from her suitors - both those that treat her roughly (such as the Churchlands) and those who treat her with suspicious reverence (such as David Chalmers).

Dennett's methods and tools are much like Father Brown's in "The Absence Of Mr Glass", which contrasts Father Brown's uncommon common sense with the deductive powers of a certain (in both senses) Dr Hood. Dr Hood, incredibly brilliant but primed towards certain reactions by his background, looks at an improbable scene and "sees" a murder - a single thing of great moral importance. Father Brown looks at this same scene and "sees" a series of unlikely events, each of which has no moral importance but the accumulation of which is, after all, a man's life.

In the same way, Chalmers looks at Le Penseur and sees him "thinking" - a singular act of great moral importance - but Dennett looks at Le Penseur and sees him wheeling through an enormous number of possible activities. Dennett sees that Le Pensuer might be visualizing, sounding out ('murmuring syllables under his breath' in Ryle's unforgettable essay), doing simple algebra or arithmetic, deploying models of other minds ('What would Jesus do?' or 'When my father was young, he had it worse.' or 'Batman wouldn't be afraid.') or even attempting to be deliberately mentally random in order to shake his thoughts in new directions (like the dadaists in one direction or the Chan Buddhists in another). Each of these activities may or may not have any "moral importance" for its own sake. I would be sad to lose my memories of my parents and other mentors - they have moral importance to me - but I would not be so sad to forget an obscure scrap of algebra, real analysis or biology - I have books to grab these tools should I need them. I think that Dennett's point of view - that the thick, morally drenched world is made of the thin merely physical world - is not only substantially right but even right in most details. So let's get to discussing it.

Anyone who has read Dennett knows that though he is well-versed in philosophy, he considers many of its traditions to be shackles - as full of tricks and traps as it is of useful goods. Still, it is useful to see how Dennett sits in the philosophical traditions that spawned him. Further, anyone who has even heard of Dennett knows that "evolution" (whatever that is) plays an important role in his philosophy. This has to be gone over is some detail. Finally, anyone who engages critically with Dennett can sense that his affection for "memes" are the weak link - I share this view so I want to develop what Dennett means by "meme" and what are the holes in his exposition. And because I'm polite, I'll even do this in that order.

Plato

Begin at the beginning and go on till you come to the end: then stop.
- Alice In Wonderland

Daniel Clement Dennett III was born, like most human beings are, to parents. His father - whose name can be deduced from hints already given - was an OSS spy in Lebanon who knew Kim Philby and died under mysterious circumstances when Daniel the third was quite young. His mother raised him in Massachusetts & New Hampshire until he became of age. Dennett's philosophical curiosity and ability were noticed quite young. As an undergrad at Harvard, he was taken up by logician/mathematician/philosopher W V O Quine. For his graduate studies, he moved to England and studied with Gilbert Ryle, then in the midst of working on Plato's Progress.

This is the only logical - perhaps I should say reasonable or practical - place to start with comparisons, but it is not easy. Ryle and Quine were quite different philosophers. From Quine, Dennett took the idea that philosophy is continuous with science. Not over, to the side or under, but simply an attempt to use the same kind of reasoning as seen in science to a different variety of question. But Quine was interested in not just intelligently designed systems - but in incredibly intelligently designed systems of terrifying logicality. Dennett has never shared Quine's interest in axiomatic systems or in "reduction" in the sense of replacing a successful theory with another theory with one less kind of relation.

It is from Ryle that Dennett absorbed the phenomenological point of view that was the precursor to his own. What is "phenomenology"? To tell a just-so story, phenomenology begins, like all of Western Philosophy, in Ancient Greece. To Aristotle, it was clear that the point of life - the aim that one should set one's mind toward - was a life of mild pleasure, physical and intellectual. The Great-Souled-Man would have a wife, a boyfriend, wine, slaves, money, land, a lyre and scrolls. There could not be many Great Souled Men and this life of mild pleasure could not always be achieved, to be sure. But it was the goal of the state and of private life to make this possible. When developing his grand synthesis of the Greek and Christian point-of-view, Thomas Aquinas could not accept this in total. But he could accept part of it - that there was an aim to life. This "intentionality" was properly aimed at God by Aquinas's lights, not stable worldly pleasure. The Catholic priest, proto-psychologist and philosopher Franz Brentano took the concept of intentionality and made it central to the description of the conscious mind - the "stream of consciousness" in William James's famous phrase. From this environment, Husserl took the concept of intentionality and attempted to construct the entire scientific method from reflection over the object of intention. And that is what phenomenology is: the study of the stream of consciousness when it is directed at something.

But Dennett is not a phenomenologist. He has described his position as "heterophenomenology" - the study of another person's stream of consciousness when it is directed at something. As his paradigm case of heterophenomenology in his book Consciousness Explained, he takes experiments on measuring the speed of mental rotation. And Dennett is not a Rylean - Ryle's insistence that the Ancient Greeks had a realistic view of mind as a thing used for action is simply unimportant to Dennett.

Dennett shares much with J L Austin and the subsequent school of "ordinary language philosophy". Like them, he uses analysis of ordinary use of words to defuse artificial theorizing about their meaning, a deflationary attitude towards axiomatization and a belief that history of philosophy is not its most important aspect. But he implicitly rejects their conservative attitude that he meanings in ordinary language are the distinctions worth making - he proposes that experimental science from Newton to Darwin to Einstein to Gibson can and have enriched us with new distinctions and concepts, so there's no special reason to suppose philosophers cannot do the same.

Dennett shares with the pragmatist postmodern Richard Rorty a skepticism that mere "conceptual analysis" in the mode of Bertrand Russell (or WVO Quine) could settle anything meaningful and a love of deflating and differentiating. But Rorty took the fact that meaning was language relative to "explain away" the idea of truth and Dennett desires to always explain and never to "explain away" - especially not truth. For Dennett, there may be many stories, many mirrors which are true as Rorty says - but there are many more which are not true. For Dennett, biological evolution plays an important role in explaining the subjective differences experienced by humans. For Rorty differences are not things that need explaining. Further, Rorty believes passionate political engagement is terribly important, Dennett is faintly uninterested in politics except in so far as it intrudes on his work.

There is an obvious connection between Dennett and the French postmodern Gilles Deleuze - both would agree with the formula "monism = pluralism", that subjective différence is grounded in biology and perhaps more (I'm no Deleuze expert). But Deleuze accepts a primitive Freudian psychology (this is giving Deleuze credit), believed it was important to have extreme leftist politics, a faintly dismissive view of science (except as metaphor) and a deep interest in the history of philosophy. On all of these counts Dennett is opposed - in addition to the deep distinction between Dennett's clarity and Deleuze's unclear style.

Dennett's greatest philosophical influences (except for Ryle) are certainly Wittgenstein - like all post-war philosophers - and Wittgenstein's student GEM Anscombe, who was the first to subject intentionality to a withering Wittgensteinian criticism. But Wittgenstein and Anscombe do not really form a school - not one bigger than themselves anyway - and even then Dennett does not share their dismissive view of experimental science.

One can see from this brief survey that Dennett is not precisely in any school of philosophy - other than generally "the guys what study the mind". So what is Dennett's story? How does he go From Bacteria To Bach And Back?


All stories begin in the middle but some middles are very far away. I'm going to be very vague - for more information see Maynard Smith and Szathmary's The Major Transitions In Evolution. Before there were living things there were autocatalytic reactions, which I will call "circular processes", since they go from state A-> B-> ... -> A. One example is the RNA replication proces A->B->A. Any such circular process is provably impossible in or near thermal equilibrium, so these process must have been receiving energy. From the sun, from heat vents, from other chemical process, from the churning of tides, etc. To the extent that these differences in energy source etc. are perceptible, different theories become testable. One example of a thermal process would be a chunk of RNA - a ribozyme - that organizes another chunk of RNA to grab an atom of iron then stick it near an oxygen molecule. The iron would then oxidize and heat would be released, creating energy that the ribozyme could then use to reproduce itself. This ribozyme would be respiring - it would have a metabolism and heredity. Of course, if it was too good at making heat, it might want to be near another ribozyme that does nothing but take in heat and reproduce itself - that second ribozyme would help the first by dissipating heat. The balance between active respiring ribozymes and passive heat dissipating ribozymes would be spatially asymmetric - too many passive ribozymes and there won't be enough heat, too many active ribozymes and there won't be enough oxygen. A rybozyme can have some control over what rybozymes they are near by, for instance,  literally linking together. These RNA strands are the precursors to life as we know it.

These RNA strands are differentiated - they have different chemical properties. The differentiation between long RNA strands is potentially infinite - rybozymes are non-periodic crystals. As noted earlier, ribozyme -> ribozyme chains of creation can be circular. As they respire, they therefore have a metabolism. Finally there are (chemical) laws that govern the relative density of their activities - there are combinations of rybozymes that do well (replicate) in their environment and those that do poorly (run out of energy or build up so much energy as to disintegrate). Non-periodic difference, circular replication and blind selection are the building blocks of "natural selection", Darwin's most important contribution to the theory of biological evolution. If a rybozyme tends to be attached to another rybozyme, their fates become linked, so that they are the first "genes" and the RNA strands the first "organisms". After this, the second "major transition", to use Maynard Smith's terminology, was the discovery by a ribozyme that it doesn't have to create energy for catalysis via other rybozymes, but could use a more chemically active amino acid. Once the amino acid was discovered, the ribozyme became the ribosome - no longer a laborer but a capitalist. They would hold the knowledge capital of which amino acids to choose, the amino acids would create the proteins and the proteins would create, among other things, ribosomes. Again, a circular process is achieved and evolution by "natural selection" occurred.

Dennett now skips in the story with a mere "and so on in that fashion" all the way to the most recent major transition of evolution - the invention of language. I think Dennett feels he has covered this territory in other books, however I still find this disappointing. The distinction between the cell - with its milieu interieur - and the naked ribosomal process is not merely thermodynamic (though it is rooted in that) it also fundamentally altered the law of motion (it is the forth "major transition in evolution"). The primitive bacteria was just a bubble of protein protecting an RNA strand or two - letting in a bit of raw materials, keeping out harmful chemicals & foreign RNA. It would have been very useful to compare and contrast this first evolutionary discovery of an milieu interieur with another one - the discovery of the mind. But I digress.

Language starts - in Dennett's story - with a discretization of knowledge in order to make it transmittable. This is called "signalling". Discretization is a trick that evolution has been forced, by Pontryagin's Principle, into over and over again (unfortunately, Dennett does not use Pontryagin's Principle to explain the repeated discovery of discretization). The simplest way of remembering Pontryagin's Principle is that the shortest path between two points is teleportation, but if this confuses there are plenty of stories to be told with it. Let's look at a bacterium. It wants to stay a certain temperature, not too hot and not too cold. If it's in a temperature gradient, then it should head in the direction of the temperature it prefers. By Pontryagin's Principle, the best way to do this is to go as fast as it can in the right direction, then stop entirely once there. The control is digital - not because it is made of digital DNA/RNA/Protein parts but because that is the best possible control.

With signals discretization starts all over again. The first chemical signals between ribozymes looking to cooperate might have been single ribozymes. The same discovery might have been made by ribosomes looking to cooperate by sending out amino acids. In either case, an arms race would have instantly taken off, with signals becoming more complex so as to be more difficult to fake. Cells can also gain from cooperation - those signals went through the same arms race. Organisms may desire to cooperate - once again the arms race takes off. By the time hominins began evolving, it's probable that they had - if anything - overly complex signal making apparatus. We cannot assume, for instance, that voice box moved in the throat that this was for "language".

But what is a language? A language is a signalling system - a structure capable of making and translating incredibly complex signals. Some of these (such as words) have a certain structure, others (such rude hand gestures) do not. Those with a logical structure are considered language proper, those without are mere signals. Language and signals are both used to transmit information from the speaker to the listener. The words of language have meanings (somehow) and the sentences of language can be - among other things - true or false.

The study of the structure of language is called "linguistics".

The study of the structure of truth values is called "logic".

For Dennett, the study of meaning in language is (a special case of) "memetics".



Science must begin with myths and the criticism of myths.
- Karl Popper

If it is anything, the study of "memetics" must be the study of the "meme". The prototypical meme is the word "meme", intelligently - or not-so intelligently, as he has come to regret its use - designed by biologist Richard Dawkins in his book The Selfish Gene. This is confusing, so I'm going to look at another, less loopy case - the speakable term "macro".

You might think that "macro" is descended from the Ancient Greek word "makros" meaning long. This is a mistake. The speakable term "macro" is a collection of phonemes. It is not the phonemes ˈmæk.ɹoʊ because in England, Russia and South Korea it is pronounced differently but identifiable as the same. A US economist can say ˈmæk.ɹoʊ.iː.kə.nɑ.mɪks and a UK economist can say ˈmæk.ɹəʊ.iː.kə.nɑ.mɪks with no loss of meaning. They can talk about a procedural macro or a keyboard macro with perfect understanding. Further, the speakable term "macro" isn't an instantiation of a collection of phonemes. It could be, by a remarkable coincidence, that nobody on God's Grey Earth spoke the collection of phonemes "macro" on the 26th of February 2017 at 17:00:00.00 at all, but the speakable phrase "macro" didn't go away. The speakable phrase "macro" can be a word (as in the shortened version of macroinstruction) or a part of a word (as in macroinstruction).

Dennett argues that what "macro" is must be a kind of meme. A meme is a mentally representable distinction. Not all memes are speakable phrases (what Dennett, with some imprecision, calls "words"). Colors are memes or perceived through memes (depending on how you want to define things). To you and me, the sky and the sea are blue - to Homer the sky is sun bright and the sea is wine dark. The sky and the sea didn't change their electromagnetic frequency, what changed is the distinctions that are made.

But this isn't enough. That there are mental distinctions is unavoidably true, what Dennett would mean when he says "macro" is a meme is that it is a small collection of mentally representable phonological distinctions that is in competition with other small collections of mentally representable phonological distinctions. Now that we understand that "macro" is a mental distinction, we can talk about what it does to its users. When early programmers call something a "macroinstruction" they meant that it was an instruction that was large - large as in made of many instructions. This was not the only possible speakable term. The meme "macro" was in competition with, for instance, the meme "meta" (which is also a speakble term). It is equally descriptive - in the given intellectual niche - of macroinstructions to call them metainstructions - instructions made of instructions. One can easily imagine that "meta" may have actually fought against "macro". Its possible that at sometime in the 40's or 50's, two distinct engineers developed the same idea - one calling it a macroinstruction and the other a metainstruction. In another possible world, metainstruction won and we talk about syntactic metas and there is a constant struggle between users of vi and Emets. The vomit puns in this struggle alone wouldn't be worth it.

We can now return to the loopy "meme" meme case if we want. The phoneme "meme" was chosen more or less intelligently by Richard Dawkins with more or less the meaning I have given it above. But the speakable phrase doesn't care about Dawkins and quickly found a new niche. The speakble phrase now also means Internet memes. The meme "meme" is selfish and cares not one whit that this isn't its intended place - no more than a gene instantiated in Dawkins's body cares one whit about Dawkins's desires.

There are two objection to memes that are left undeveloped by Dennett in From Bacteria to Bach and Back. He points out that most of the objections don't hold water. Dawkins can intelligently design "meme" and Ragnar Frisch can intelligently design "macroeconomics". But so what, farmers have intelligently designed chickens - flightless birds - to have gargantuan flight muscles (called "breasts" in a quite memetically pleasing but inaccurate analogy to human mammaries). This is no proof against Darwinism or genetics. The real objection is that genes are potentially immortal and selfish - a genetic disease wants (Dennett extensively discusses and praises the practice of using phrases like "wants" here - even though these reasons and desires are not represented by any genetic diseases mind) to be in as many animals as possible, never mind detriment to the creatures it lives in. The metaphor to a businessman who wants to own shares in every business to hedge against his self-destructive capacities is a sort of pragmatic deduction from the well-established and uncontroversial models of genetics. Sex and dying of old age are examples of biological phenomena clarified by this metaphor. They are both attempts by genes to disassociate themselves from other genes that they work poorly with or are parasites. Sex and death are to a gene is like a contract with an exit clause.

But with the "meme" meme, Dawkins does the opposite. He starts with the immortal/selfish metaphor and then proposes scientific work be done. We have every reason to be stricter with the metaphors that precede new sciences than with metaphors to describe established work. What memes want is to be instantiated, so if picking up meanings will help they will. It does not follow that we are as mayflies to the meme, that when we are of no more use to the meme, we die out as the mayfly dies out so its genes may associate more freely. Dennett and Dawkins attempt to push this, but I find their examples to be not entirely convincing. Many of their examples are far too complex. Religions are not memes - neither are bacteria genes. The 124 year Wars Of Religion period was not caused by a memetic difference in the way that sickle cell anemia is a caused by a genetic difference. A person with sickle cell anemia cannot stop having sickle cell anemia, but a person can change religion.

The other objection is that the social sciences have every reason to be suspicious of biology and biological explanations. It is not that they have not yet been tried - rather they were extensively tried from 1851 to 1911. The almost entirely negative influence of Herbert Spencer (who was not at all a Darwinian, of course) was not a trickle, but the dominant flow of social science for decades. One can see it captured in amber in Marshall's Principles and other economic and sociological books of its time. Franz Boas and others did an enormous amount of intellectual labor to free sociology from the shackles of biological and pseudobiological metaphors. Dennett writes as if fear of biology was tied to a need for skyhooks and miracles, not a worry that an extensively mined field bottomed out a century ago. In order to convince people of the ripeness of their fruit, memeticists need to prove they have done this intellectual labor. The best and only way to counter this criticism is to show that working with memes is too fruitful. They must run when other theories may walk.

This book is one of many instances of Dennett working in this direction.

 Daniel Dennett

And so...
- Mayor Poopenmeyer, Futurama

The direction taken, the concepts discussed and the wonderful clarity of argumentation make this an excellent book, but I don't think it's good as a first Dennett book nor do I think it is his best. A neophyte Dennettier should start with Conciousness Explained and Darwin's Dangerous Idea, his best and most important books, after which reading him in any order is acceptable. In the absence of time for that, I still highly recommend this to anybody, no matter what their opinion on Dennett & his work is and even if they don't have a background in it.

Saturday, February 4, 2017

From Hot To Cold?

EDIT: D'oh! Added a bunch about energy hypersurfaces that makes it all clearer and more correct!

Imagine you have a room with a tank of water. The ambient temperature of its room is, say, 20 C. Right now the water in the tank is still and all the same temperature (thermal & mechanical equilibrium). Below the tank is a heating element (right now this element is off), above the tank a refrigeration unit (right now this unit is off). You watch me twist knobs so that the following is true: the temperature of the layer of water on the heating element will rise (perhaps slowly) to some value greater than 20 C but the temperature of the layer of water near the refrigeration unit will remain constant at 20 C. What will happen? Of course, the water will convect. This is called "Rayleigh-Bénard Convection". Essentially, I am moving energy out of into the plate and out of the refrigerator, losing some to the inward turning motion of the fluid.



None of this is controversial. There is a temperature difference and a flow. In the study of Rayleigh-Bénard Convection, this temperature difference gives the key qualitative facts about the flow. The temperature difference controls the flow (holding other things constant). But the temperature difference does not cause the flow, even holding other things constant. The continuous energy input to maintain the temperature difference causes the flow. How is this different?

There's an economic analogy - money supply changes may control trade in a monetary economy, but it does not cause it. Desire for income causes trade. But analogy isn't enough.

Look outside the room. There's a column of air that reaches all the way up into space. Let's say the air is also still and at constant temperature (because the Earth rotates this isn't quite true, but the conditions can be obtained in a lab). There is a very distinct pressure difference in this air column. The air on the ground has exactly 1 atmosphere of pressure. This pressure declines exponentially until air molecules get so scarce that it's best to just call it a vacuum. There is a pressure difference. Yet there is no flow - air is not constantly being sucked into space. Mechanical equilibrium is consistent with potential differences.


You might say that in this case it is because the pressure potential is balanced by the gravitational potential. This won't work. In Rayleigh-Bénard Convection, the temperature potential is balanced by the gravitational potential (this is the key to analysis, in fact). But in one there is mechanical equilibrium and in the other there is not.

What happened in the Rayleigh-Bénard case then? Simple: in Rayleigh-Bénard Convection there is net energy movement. The opening paragraph was comparative statics: we move from a static condition of thermal equilibrium to a static condition of linear temperature difference. In both the thermal equilibrium and Rayleigh-Bénard Convection cases, molecular chaos gives rise to large scale order. In the thermal equilibrium cases, the order is completely described by mechanical equilibrium. In Rayleigh-Bénard Convection, there is a much richer set of large scale equilibrium states - and it is impossible to tell which one you'll end up in when starting from the mechanical equilibrium state!

To do comparative statics, we need to go back to statics again. Let's go over Boltzmann's H-"Theorem"* which tells us what we need. To do this we need to bring in some statistical mechanics concepts. We start with a "microstate". A "microstate" is a big data structure that tells us everything about a physical system to the finest detail (for historical reasons, we will use a vector as our data structure). A "macrostate" is then a function of microstates. An interesting macrostate tells us something about the system. So-called "sufficient" macrostates tell us everything that we can know about a system. In the classical setting, the sufficient macrostates are exactly the classical thermodynamic potentials, as proven by Mandelbrot. I will ignore this and just give a clear example


Look at this square. Your location (a two dimensional vector of integers) gives your microstate. The macrostates are given by capital letter. In the above drawing, there are 625 microstates and 6 macro states. This is an "energy hypersurface", each square is a possible state with the same energy. In the Rayleigh-Bénard Convection example, there is a rise in energy of the system. This takes the system to a new energy hypersurface with more squares. To make things round, we'll make it 8 new macrostates with 10,000 microstates. This table gives the breakdown:

Macrostate Side Length Number Of Microstates H
A' 1 1 0
B' 1 3 .123...
C' 2 12 .278...
D' 3 33 .391...
E' 6 120 .536...
F' 12 456 .686...
G' 25 1875 .844...
H' 50 7500 1

Again, slowly. There are 8 macrostates and 10,000 microstates (add them up if you don't believe me). I normalize the entropy so that the largest macrostate has unit entropy**. Entropy is now, as Boltzmann promised, a measure of uncertainty. If I measure a system to be in macrostate A', I know exactly the microstate. If I measure a system to be in macrostate H', I know only that it is in one of 7,500 microstates. Notice that state [1,1] means different things in the original drawing and the higher energy hypersurface. I note these new states with a prime. This is a different data structure than a pure vector, but in the usual theory one does this just with vectors.

Now we need a transition rule. One possible rule is that the system can move in any cardinal direction that doesn't exit the system. What happens when I run this system starting in macrostate A'?


It climbs up to the maximum entropy macrostate, just as Clausius intended. This is Boltzmann's H Curve! This device was introduced in his letter to Nature in 1895. It's a great illustration of his goals and vision.

Wait, can I do this, just make up a transition rule? Well, I'm a mathematician, of course I can. Boltzmann did not have the necessary mathematical tools to see when this was physically realistic though. The first person to give a microphysically realistic situation that results in a dynamic like this (the technical term is 'Markovian') was Yakov Sinai in 1963 - 70 years later. From simulations and improvements in mathematical theory, we know that there ain't no rule sayin' a dog can't play soccer isn't quite a physical law saying a physical system has to behave like this. Some complex systems have weird conserved quantities and fail to behave like the H curve above ("thermalize"). But we will assume we have such a system for now.


Just looking at the above figure, we can see H does not always increase. What happens in the long run? To illustrate this, I let the simulation go a long time (in comparison with the number of microstates), then downsampled for legibility:




We spend most of our time at the maximum entropy state, sometimes dipping down into lower equilibrium states. But given we are in an non-maximal entropy macrostate, we expect to go straight back to the maximum entropy macrostate. This can be a source of flow! In the literature, these are called the "Onsager reciprocal relation". We can check to make sure that the amount of time is the number of states in each macrostate.

Perfectamundo!



Now we can come back to explaining the existence of flow in Rayleigh-Bénard Convection experiments. It starts off like the drawing above. In thermal equilbrium, the tank of water is in the maximum entropy macrostate - which in the drawing is macrostate F. When I changed the boundary conditions so that there is net energy flow, the fluid moved to a new energy hypersurface. Let's say that it moved it to macrostate A'. The number of potential states changed with the entropy. The fluid moved from the mechanical equilibrium macrostate F to macrostate A' (by fiat) to the convective macrostate H' (by H-"Theorem").*** In doing so, the entropy increased. This is the first H curve I drew. It's all as Gibbs could have told you: flow is a fundamentally statistical mechanical property.

* Why the scare-quotes? Well, for one, Boltzmann didn't prove this even by his own standards of rigor. For two, though my model can be tied up with the Perron–Frobenius theorem, the physical ideas are not always amenable to strict mathematical analysis. Boltzmann's statistical mechanics ideas were not at all easy to be really formalized. Finally, there are many H-Theorems with different proofs.

**Normalizing the entropy is equivalent to choosing Boltzmann's constant. There is something unphysical about my states. Each state is exponentially larger than the previous (about five times bigger) but in real mechanical systems we expect states to grow faster - Stirling's approximation is H ~ n log(n) rather than H ~ log(n). So in my example, H is roughly linear, but really we expect almost all the H on the maximum entropy state. I couldn't think of an easy geometric way of illustrating this (though I think Knuth gives one in "Why Pi?"). This doesn't effect the qualitative conclusions.

*** Caveat: the Rayleigh-Bénard Convection maximum entropy macrostates are no longer unique which makes the drawings more complicated without changing the essential point.