Archive for the ‘Plagiodoxic thinking’ Category

Symmetry and Orderliness

February 8, 2014

One problem, two solutions, and three questions.

The problem is not new, the solutions are among many, the questions express my plagiodoxic view of the relationship between symmetry and orderliness.

The problem- 12 coins

Twelve coins identical in every respect except that one coin weighs slightly more of less than the others.
A beam balance.

To find:
The aforementioned coin, hereafter called the object coin, and whether it is lighter or heavier.

First solution:

Put 6 coins in each pan of the balance.
Weighing #1 One side goes up, the other down. Denote the coins L (light) or H(heavy) accordingly.
Take the 6 coins from either side, L or H, and put three in each pan.
Weighing #2 Either the coins balance or not.
If they balance, you know they are all correct, so however they are denoted, the object coin is the opposite denotation, either L or H. Set the correct coins aside and take up the other six coins, which are denoted all either H or L.
Put three of these coins in each pan of the balance.
Weighing #3
One side goes up, the other side down. All of these coins are denoted the same, either L or H. The movement of the pan containing the object coin will be consistent with its denotation, either up for L or down for H.
The movement of the other pan will be opposite the denotation of the coins in it, indicating that they are correct, neither light nor heavy. At this point you know that the object coin is one of the three that are left, and that it is heavy or light, according to its denotation.
Put one of the coins in each pan.
Weighing #4
The pans may balance, in which case the object coin is the one left out of this weighing. It is heavier or lighter according to its denotation.
The pans may not balance, in which case the object coin is the one that moved consistent with its denotation, up (L) or down (H).
You have isolated the incorrect coin and determined whether it was heavy or light with four weightings.

The second solution:

Put 4 coins in each pan of the balance, leaving 4 coins aside.
Weighing # 1
I If the pans balance, the object coin is among the set aside 4, and is either H or L. Denote all 8 of the coins you have weighed c (correct). Set them aside and take up the 4 unknown coins.
Weighing # 2
Put 2 of the unknown coins in one pan of the balance. Put a third unknown coin in the other pan, together with 1 correct coin.
Ia If the pan balances, the object coin is the remaining coin not yet weighed. Put it in one pan, and a correct coin in the other pan.
Weighing #3(Ia) If the object coin goes up, its L, down its H.
You found it with only three weighings.
Ib If the pans do not balance, the object coin is one of these three. Two of them are in one pan, the third is in the other pan with a correct coin. Denote them H or L according to how its pan moved.
Take the two coins that are together, put one in each pan of the balance.
Weighing #3(Ib)
If the pans balance, the object coin is the third coin, and it is light or heavy according to its denotation in weighing #2
If the pans do not balance, then the object coin is the one of these two that moved according to its denotation in weighing #2.
You have found the object coin in three weighings.

II If the pans do not balance
One side goes up, the other down. Denote the coins accordingly, H, L, or c-the 4 unweighed coins must be correct. It will help in the following if you also number the coins like this:
H1, H2, H3, H4, L5, L6, L7, L8. In this first weighing (of the second solution), you have determined that four coins are correct, four coins are not heavy- denoted L, and four coins are not light- denoted H.
Put coins H1 and H2 in one pan, and put H3 in the other pan. Put L5 in the pan with H1 and H2. Put L6 and correct coin in the pan with H3. In the balance pans, you have
H1-H2-L5 vs H3-L6-c
Weighing #2 One of three things can happen. The pans balance, the H1-H2-L5 pan goes up, or the H1-H2-L5 pan goes down.
IIa If the pans balance, the object coin must be one of the un-weighed coins, either H4, L7 or L8. Weighing #3 Put L7 in one pan, L8 in the other. If they balance, the object coin is H4-heavy. If they do not balance, the object coin is whichever of the two, L7 or L8, goes UP.
You have found the coin in three weighings.
IIb If the pan with H1-H2-L5 goes up, the object coin cannot be H1, H2, or L6- they moved opposite there denotation. Neither can it be H4, L7 or L8, which were not weighed. That leaves H3 and L5. Put either one in one pan and a correct coin in the other. If the pan moves according to the denotation of the coin, that is the object coin. Otherwise it is the other one.
You have found the object coin in three weighings.

IIc The pan H1-H2-L5 goes down.
The object coin must be H1, H2, or L6. Put H1 in one pan, H2 in the other:
Balance means the object coin is L6, and is light.
Not balance means the coin that goes down is the object coin and is heavy.
You have found the coin in three weighings.

Thanks for bearing with me thus far. It’s a bit of a slog, and I may have goofed it up somewhere.
Summing up, the first solution found the object coin in four weighings. The second solution found the object coin in three weighings. Parenthetically, if you “knew” which coin was the object coin, it would take you two weighings to prove it.
The first solution is simpler to understand and follow. The second solution packs more information into each weighing.

Now for the questions.

Symmetry and Orderliness
The first solution begins by dividing the coins in half. The second solution divides the coins in thirds. Which of these two solutions to the problem would you say involves a higher level of symmetry?
As each process unfolds, less information is gained at each weighing in the first process, more in the second process. Which would you say is the more orderly process?
What can you say about the relation of symmetry and orderliness on the basis of your answers to these two questions?

Thank you for your attention.


The Algebra of Truth

October 4, 2010

Here is a true statement:  ” Liberals did not invent taxes.”

Here is another true statement: “Conservatives did not invent taxes.”

As far as I know, both of these statements are correct and historically accurate. They are both true.

Combined together, by simple addition: “Liberals did not invent taxes and conservatives did not invent taxes.” Another way to say it is “Neither liberals nor conservatives invented taxes.” But these two statements are not quite the same.

Each true statement, taken separately, can be described as a half-truth (½T). The question is how to put them together, by addition or by multiplication. Does a half-truth plus a half-truth yield a whole truth (½T + ½T = T) or does a half-truth times a half-truth yield a quarter truth (½T x ½T = ¼T²), only we also have multiply the units and get square-truth. What??

Multiplication can be thought of as a mathematical metaphor for two different kinds of experience, repeated addition or a blending of properties. So, 3 inches time 2 equals six inches, and 2 inches time 3 also equals 6 inches. But 3 inches times 2 inches yields six square inches. Fortunately, we have a clear physical picture of what a square inch is, so the blending of the two dimensions is not confusing. Square truth??

The power of language lies in its ability to create virtual worlds by the simple manipulation of grammar and vocabulary. That is true of mathematical language as well as cultural language. It is a power that is dangerous if the mind gets too absorbed and intoxicated by the unreal intensity that is easily generated, particular by modern technologies.

Back to square truth. Here is a word combination that flows strictly from the rules of algebra. Is there a way to put ordinary meaning into these words? The second combining statement above, the one that starts with “Neither..”, does this. This statement points to the possibility of something larger than the simple polarity of liberals and conservatives, putting these two contending groups into a larger historical framework. Like √-1, symbolized by i, we know the number exists though we cannot write a numeral for it.

Square truth. It’s just that for which the truth is the square root of! Like many radical ideas, it may not be expressible in numerals. Mathematicians call that irrational. Perhaps it is irrational to think that there is something beyond the polarities of argument about “who’s better than who” (see Seuss:The Big Brag (Yertle the Turtle)).

I have argued elsewhere in the blog that the mominalization of the adjective “true” into the noun “truth” is something that should not be done casually. I would go on and assert that “truth” is always partial, and therefore subject to the multiplication of fractions rules. As a direct consequence, as we learn more and more about the world, the fraction of knowledge we have, compared to the implied whole, gets smaller and smaller! That’s a little scary, and humiliating.

Lying in Virtuality

August 7, 2010

Talk!! Just writing the word down is a contradiction. But what a seductive contradiction.

Even in the parlance of the street, the contradiction is clear. “You talk the talk! But can you walk the walk?”

Philosophy is talk about talk. It is talk about the difference between “talking the talk” and “walking the walk”. In philosophy, all you can do is talk about it. In the actual world, you can shut up and walk away. But then, there is nothing left to say. The ones who get their power from talking are left speechless.

I just got up and walked away- to get a glass of water. You couldn’t know this without my telling you because we are not sharing real space and time. We are only virtually together.
I am here in my now and you are somewhere else in the future of my now. In a non-literate world, one without written words, we could only communicate when sharing space and time. Gradually, mechanisms developed for communicating across time and ultimately space, leading to written words. So now we can pretend we are sharing space and time and experience. That pretending creates a “virtual” world, a world of essences derived from walk-world but is still only in the talk-world.

In this virtual world, we also lie to each other. Of course, people can be deceptive face to face, but they can also be discovered or suspected. We can also be honest, face to face. But the written word is always a lie. That may seem a strong thing to say, so let me follow up on it a little.

“I just got up and walked away..” I wrote several lines above here. While that was a true statement, it was not the whole story. It was not everything I did or thought. It did not place my action in any kind of meaningful frame of reference. I left that all up to you. If I had set about to fill all that in, I would still be writing about the glass of water! To completely exhaust the subject, I would need to write forever for after I finished writing about the glass of water I would have to write about the writing about the glass of water and… Let’s not go there. Only computers are stupid enough to chase their own tails into logical oblivion.

So I employ my own internal editor to decide what to write and what not to write. Now, I am an honest fellow and altruistic in the extreme. So in the editing that I do while I am writing this for you, you can be assured that I am not forwarding my own personal agenda or lust for social prominence, or any other kind of lust. Trust me on this one!

One can communicate the real world only in a limited way. You can throw a stone only so far. But one can communicate virtually a great deal more. Even so, what is left out of the communication is greater than what is included. How the communication is shaped is determined by the internal editor of the communicator. That editor is subject to the conscious and unconscious impulses of the human being doing the communicating. Like all other human beings, the communicator lives in social setting that is, one way or another, a dominance hierarchy. Our communication skills have been shaped and honed not just to send signals about our environment, but also to facilitate our struggles for our desired place within that hierarchy.

Whenever we speak or write, we have as part of our agenda how that speaking or writing affects our struggles within the social hierarchy. And it is so much easier to lie when I don’t have to look you in the face! Trust me on this one!

I’m not sure all of this hangs together very well. Maybe you have something to say?

No defenders of “Truth”?

March 7, 2010

I am not terribly surprised. But it is surprising. “Truth” is a very popular word. You can say the most awful thing about some one and should you be queried why, the most common defense is “It’s the Truth!” Scientists claim it. Relgionists claim it. But no one will defend it?

I have raised the question that one should be careful in the use of language, suggesting that some easily and commonly made word conversions, like nominalizations, are correctly critisizable on the basis of whether or not they map something that actually happens in the world. Here is a another opinion:

“Once the human intellect creates  symbols from reality, those symbols or words can be manipulated and catalogued to increase our understanding of reality. “(The Trivium: The liberal arts of logic, grammar and rhetoric : Sister Miriam Joseph  (Paul Dry Books Edition 2002)) p.24

I think this statement is true, but I’m sure it is not the whole story.

Unfortunately, when you try to tell the whole story, the language begins to bend around on itself, like the proverbial snake biting its own tail. We can find our way out of the confusion with a map.

In fact, that is where I got started in this whole business, talking about maps with students in the classroom. First, draw a map of how to get from home to school. Then go on to written messages versus maps, visual and verbal problem solving, right brain and left brain, algebra and geometry, grammar and vocabulary of maps. It took two class periods altogether. Toward the end I would ask who drew a perfect map. Usually no response. Sometimes “What do you mean by perfect?”

My response: If it solves the problem of instructing someone how to get to your place, it is perfect. Now several people volunteer that their maps were perfect in that sense. But still imperfect in two basic ways: uniformity of scale, and suppression of detail. So I point out that the utility of the map for solving a real world problem is directly dependent on those very distortions. A map that was free of all distortions and that fully recorded every detail of existence, were it possible to construct such a map, such a map would be useless for guiding the person to your place. It would be like pushing them out the door and saying Try!

So is the map true? Most student said yes. If it solved the problem, it was true.

The ethical question comes up when you ask “What if you made this map and somebody used it and got lost because of some distortion you put on the map, and suffered harm because of getting lost. Whose fault is it? Now there is some debate

But if the map were the Truth, there is no debate. It’s the traveler’s fault for getting lost.

A map, which is a statement in visual language, can be true. A statement in verbal language can be true. The word “true” is an adjective, and as such, in my limited understanding of such matters, in the jargon of Aristotelian thought, is an accident that is a property of a substance, like the blue color of the mineral azurite. The nominalization of the adjective to the noun, “Truth”, brings forth a substance, something that exists in and of itself.

But would Aristotle approve of this switching from accident to substance? In conversation with a knowledgeable friend, I was reminded that nominalization was introduced into English by the Norman Invasion (1066 CE).  Before that, the Anglo-Saxons who were uttering the forerunners of English had no such word as truth! No wonder they lost.

Which brings up the always underlying historical question:”When did it come to be the way it is now?” Could Aristotle conceivably approve the moving of an idea from one kind of Platonic entity (accident) to another (substance)?

The substance , being substance, can be possessed by some, and not by others. Those that possess it are exalted in power and held blameless for the execution of that power, no matter how hurtful. Those that do not possess it are debased and held responsible for their own misery.

One could become cynical about it all. I prefer to remain a skeptical optimist. I am skeptical of all maps, visual, verbal, mathematical, whatever, knowing that they can only imperfectly capture the actuality of the existent world. I am optimistic that the human community will continue its quest to create ever more precise and subtle maps to solve the problems that bear down on us in our actual existence.

The virtual world created by words has enthralled us for millenia.

Once upon a time....

But somewhere there is a baby crying.

Enough of this eclectic plagiodoxic rambling!

Improper truth

February 25, 2010

The word “truth” is an improper nominalization. Read that as parallel to this statement: The number 3/2 is an improper fraction.

One learns about fractions, proper and improper, somewhere in grade school, along with how to convert the improper ones into proper ones. After grade school, no one worries about their propriety anymore.

In other words, “improper”, in the sense I am trying to use here, does not mean you should never, never do it. It’s more like “it isn’t quite what it seems to be- so be careful”.

Consider these two statements:

  • The sky is blue.
  • There is blueness in the sky.

In the second statement, the adjective “blue” has been changed to the noun “blueness”. This is the process of nominalization, the conversion of a word, usually a verb or an adjective into a noun.

The core of my assertion is that the conversion of the adjective “true” into the noun “truth” produces something that is not quite what it seems to be, so be careful!

Back to the blue sky. Here is a photo of a really blue sky.

Blue Sky at the Four Corners

Here is something else that  is blue.

Moki in hand

The object I am holding here is known in the trade as a moki marble. Usually they are white. I bought a bag of them in a shop in Moab, Colorado, on the same trip that I took the Four Corners photo of the Blue Sky.

The following statements are valid:

The sky, in the photo, is blue.

The moki marble, in the photo, is blue.

No problem.

In the photo, there is blueness in the sky.

In the photo, there is blueness in the moki marble.

Now we have a problem.

Geologically speaking, the moki marble is an azurite concretion, a small stone weathered out of a larger mass of sandstone as a spherical body because a spherically shaped zone within the sandstone was particularly tightly cemented, oddly, in this case by the blue mineral azurite, and so this zone was resistant to erosion. That means that the blueness in the moki marble is due to a substance, the azurite, that can actually be physically separated out and possessed by someone. So the abstraction and conversion of the adjective blue into a noun blueness is paralleled by the real-world action of dis-aggregating  the azurite from the sand.

Can the same scenario play out with regard to the blue sky? Or the blueness of/in the sky? Is the sky blue because it contains a blue substance?

Well, no. For a fuller story of the blueness of the sky, I emphatically recommend Peter Pesic’s delightful little book Sky in a Bottle (MIT Press, 2005). Art, religion and science chase each other around. “The sky is blue! Calculate Avogadro’s  number”.  “The night sky is black. Calculate the age of the Universe!”

There is no substance  in the sky that makes it blue the way the azurite is a substance that makes the moki marble blue. Even so, both the statements about each having blueness are true statements. But is either statement the truth? What can we say that is true-blue?

Along the timeline of existence words came into being, phyllogenetically, some sooner than others, just as they do in our individual lives, ontogenetically. “Truth”, a noun, came from “true”, an adjective, by the process of nominalization. That the word “true” came before the word “truth” is indicated by how they are defined. “Truth” is defined by using the word “true”, but “true” is defined in a way that does not use the word “truth”.Therefore, “true” came before “truth” chronologically. Just like sand comes before sandstone.

We know that statements can be true, or false, just as we know, in a more personal way, that people can be honest or not in what they say to us. But is there a substance-like thing that can be properly called by a noun-name, the truth? If it is substance-like, like the azurite in the stone, it can be segregated out and possessed by some, and not by others.

Perhaps I should pause here to see if you have any comments to make?

Words and numbers: Aristotle and Ifrah

September 30, 2009

I have been playing with some time-lines, working on a 11×17 sheet that ranged through the last 10,000 years. It occurred to me to show on this graph the time periods covered by some books I have been reading lately. That is, to show the range of time that each book covered. My immediate reading was  The Trial of Socrates: I F Stone, but the first book I lined out on the graph was The Horse, the Wheel and Language: David Anthony. Then I tried to put on one of my favorites,  The Universal History of Numbers: Georges Ifrah, but it turns out that people have been symbolizing numbers a lot longer than we have been symbolizing word-sounds, so that I could not put them on the same graph.

The first recorded numbers were notches on sticks. The first written numbers were quite possibly words, rather than number symbols, which, in turn were quite possibly also the first words. Since that time, words and numbers have drifted apart into their own academic departments. Too bad!

According to Ifrah, the Brahmins who shepherded the number system that became ours across the threshold from oral to literate cultures asserted that the numbers had no history. They were simply given to the Brahmins by the gods. Plato said much the same thing about the spoken language as he and the other Greeks accomplished the same task for the spoken language.

We can follow the history of numbers through the mysteries of e, i, pi, 0 and -1. But what about the history of language development? By the time 2400 years ago when Aristotle recorded the rules of grammar, logic and rhetoric, the spoken language had been evolving for perhaps 800,000 years. Most of us ordinary people are still laboring to catch up with Aristotle.

Only in the time since then we have been able to look at written math symbols, apply logic to their manipulation, invent algorithms and equations like Euler’s.

Before venturing into some philosophical exploration of the meaning of all that one should recognize that natural selection did not produce these linguistic goodies for the sake of philosophy. Rather, there was some practical problem that they resolved in some way. And the resolution of practical problems only has to reach the standard of practicality. They don’t have to be “The Truth”.


January 2, 2009

Somewhere along the time-line of the Earth, extending from 4.7 billion years ago until now, everything you know about, care about, like or dislike, love or hate or are indifferent about came into existence. Nothingness may have existed somewhere forever, but no thing on the Earth that is present now in some form existed in that form at the formation of the planet. Every earthy thing came into its earthy form somewhere along that line. When you become convinced of the correctness of this proposition, a great many question spring up in any discussion or debate. When did the IT that you are debating begin? From whence did it come? What were its precursors in the time before? What were the causes, contingencies and coincidences of its formation?

At a New Year’s/birthday party, the assertion came out: “Virtuality was born in 1948, with the advent of television.”

While it is clear that “virtuality”, as an earthy thing, had to come into existence from its predecessors at some time, I think AD 1948 is way too late. It has erupted into the forum of debates at parties only much more recently, to be sure. But that is because it has only drawn attention to itself after being exponentially magnified by the digital revolution.  Before the digital revolution, virtuality was constrained, living only in the pages of novels, the scripts of plays, the exhortations of poets. In this guise it was frequently mistaken for “truth”, that is, something better than the imperfect grit of the world as it actually exists.

But it has been around here in that earlier form for a long time. And even before the printed page and literacy began molding the human mind, there was the oral world of myth and magic doing the same job in a simpler way. Pavlov’s dog showed the way. By the simple association of the sound of a bell with the presentation of food, Pavlov could prompt the dog to salivate simply at the sound of the bell- a virtual presentation of virtual food. Of course, he also found out that you couldn’t fool even the old dogs indefinitely.

The first practitioners of virtuality manipulation were the storytellers using the controlled bells of their voices to invoke the spirits around the campfires of our ancestors. That is also where we all start our journey through virtuality toward actuality, believing in the “truth” of the stories we are told. But from time to time, things fall apart- there’s a novel for you! Perhaps we are living in a Hegelian dialectical moment, when virtuality taken to extremes will succumb to its antithesis.

History of Civilization: a summary for the 21st Century

December 19, 2008

History begins, according to scholarly historians, with the written record. This statement is, in itself, evidence of the first phase of the history of civilization, the establishment of the hegemony of the written word. That is, it is an example of the expression of the primal power struggle that is characteristic of human societies, amongst others. In this case, the power struggle was between an established power elite whose power was based on oral skills and an aspiring elite-Plato and his group- whose skills were developing with the then new technology of writing.

The first major phase of our history involved just exactly that: the overthrow of the older, oral based culture with the newer written based culture. The rise the monotheistic “religions of the Book” were an integral part of the struggle.

But the very success of the enterprise generated problems, for the establishment of “the Book” raises directly the question of “Which Book?” The answer to that question was, obviously, “Our Book!” The struggles to establish one Book have followed down to the present time in what can be called the second phase of our history. That struggle was to establish the hegemony of a particular Book over all others. I call this the struggle to establish the hegemony of a monolithic orthodoxy. This is the struggle that dominated the 20th Century, and still goes on.

But the struggle is in vain, for the world view that any orthodoxy displays is but a virtual world created in the mind by the flow of words. The words may map the terrain of reality with great effect for solving particular problems, but those maps are not the reality that actually exists! There is always some aspect of the existence that escapes or is ignored by the word-makers, and ultimately that reality will dismantle the theory the orthodoxy is based on.

What to do?

One cannot give up and go back.  The asymmetry of time forbids that.

One can go forward, understanding that orthodoxies are but special cases of a more general phenomena- plagiodoxy. Further, that multiple plagiodoxies give better perspectives on existence than any one of them. We may not be able to solve the problems we are faced with, which the sales pitch of the orthodoxies claim to do. But we will stand a better chance with a clearer image of the problems themselves.

Eclectic Plagiodoxy- the map metaphor

October 25, 2008

From Martin Rudwick:

The Great Devonian Controversy:  The University of Chicago Press (1985)
P 454    “Bookish people with no practical experience of mapping often assume that a map is an unproblematic replica of reality, or merely a miniaturized version of what one would see from the air. Those who make intensive use of cartography know on the contrary that any map is a pervasively conventional representation. They also know that an indefinite number of different maps of the same area can be made for different purposes, yet all may be equally valid representations of the same natural reality. Even where such maps prove mistaken, they are always corrigible; but it makes no sense to talk of ever achieving a uniquely “perfect” representation, or a complete “correspondence” with reality, since different kinds of maps are designed for different uses, and there is no limit to the further representations that may be needed for new and unforeseen purposes.”

As field geologists know, any map is both a collection of data and an expression of theory. The theory is one answer to the underlying question that drives the science.

Global society today is caught up in a transformational process, the sharp point of the wedge of transformation being economic/financial. The maps/theories that have served as guides to the future for the past several decades are failing us. It is not that Atlas has finally shrugged. But rather he seems be showing himself as an old fool.

Pay heed to what Rudwick is saying. It is unsurprising to any one that maps are never “perfect”, because most of us can remember when we first started making simple maps, and frequently gave up on the process as not being attuned to our individual talents. Most people would rather write notes than draw maps.

But the notes, being just the expression of the other half of our intellect applied to the same reality, are no more perfectible than the maps. If we could remember our first stumbling attempts to express ourselves in words, we would be more humble about our word production efforts. Our grasp of the virtual nature of the world created in words, spoken, written or printed, would be more direct.

Correspondingly, we would have a greater appetite for alternative word-maps. We would look for different slants on the world. We would see the breakdown of our word-virtual image as an opportunity to look through the metaphor into reality itself a little more clearly.

But we are stuck with the fact that our memories are focused on the product of those first stumbling efforts and we can dredge up the feelings of frustration and inadequacy that surrounded that production only by great effort of internal examination. It seems to our minds that our knowledge of the world, expressed in words, actually precedes our experience in the world. In the current unfolding of historic time it is clear that this is in fact not so. It is simply an artifact of the data, the fact that our earliest memories are expressed in words.

Eclectic plagiodoxy- Let’s look for a diversity of maps , based on different slants.


August 28, 2008

The tag line says “Not monolithic orthodoxy”.

This is not a definition of plagiodoxy, but a description, stated in the negative. Negativity gets a bad press.

But if our most powerful problem solving tool-science- is founded on the principle of falsification, it seems to me this leads us directly to re-evaluating the negative. We can, after all, state with much greater confidence that the world is not flat than we can say that it is spherical. Further, we can say with greater confidence that the world is not spherical than we can say that it is an oblate spheroid of revolution. Even further, we can say with more confidence that it is not a smooth oblate spheroid of revolution, working our way through falsification away from mathematical abstraction toward a description of the Earth’s shape as it actually exists, at least as it existed at the time of measurement, plus or minus error.

A longer tag line could read:”Thinking on the slant about the world as a heterogenous system, far from equilibrium”.