Check out a new Page: Zooming up and down
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.
The aforementioned coin, hereafter called the object coin, and whether it is lighter or heavier.
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.
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.
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.
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 human community is exploding and imploding at the same time. What we know about the world outside of ourselves, and inside of ourselves, grows exponentially every day. But the part of the human community that knows this knowledge grows daily further away from the vast unknowing majority of the human community whose daily labors produces the wealth that sustains the whole community.
But there is no “whole community”.
Humanity was cradled in community. Then it exploded out over the face of the earth, diversifying into innumerable communities that, discovering the roundness of the globe, now converge implosively upon each other in the global market. But the global market is not a community. It is an arena within which the diverse communities of humanity encounter each other with both brotherhood and fratricide.
Science, the machine of worldly knowledge, forms its own community. But in their hubris, the community of science eschews the questions community responds to: how to live a life and die a death that both have meaning.
Religion responds to these questions. But, being vane and jealous, religion fractures the human community into competing orthodoxies.
Along the shore of the unfathomable sea, an old man walks along the beach, mulling all this over, playing with the rippling waters with his bare feet, watching for the waves that might sweep in, waiting for the final turning of the tide. He holds in his hand a bauble, a toy, an intricate new device that can serve an old purpose. Put a note in a bottle. Cast it into the sea. Wait for an answer.
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.
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?
A couple of weeks ago the very prolific Callan Bentley at Mountain Beltway raised some speculative questions inspired by the concept of mineral evolution, the idea that the mineral suite of the Earth has changed over time. I’ve been mulling over this proposition since, more or less in the back of my mind as more pressing personal happenings have been going on.
First, there is a terminological issue. The word “evolution” has meant very different things to different people over the centuries. Like most, if not all words, you cannot be assured that the etymological roots are anything other than a guide to the history. In today’s world, the word automatically brings up the name Darwin, even though it meant very different things to Erasmus than it did to his grandson Charles, who used it only sparingly.
Taken as its roots imply-unfolding in time-, the concept of mineral evolution would follow directly from logic of thermodynamics and an expanding Universe. Particular minerals, such as the one shown above, will form in environments wherein they are stable, or at least closer to equilibrium than whatever was there before. The parameters of the environment, as far as minerals are concerned, are temperature, pressure and composition. Consequently, in a Universe where energy is constantly dissipating, temperatures will constantly fall, and new, conceivably unique patterns of temperature, pressure and composition come and go. Predictably then, mineral suites will change with time, with or without life!
Which takes me to Kelvin.
Geologists typically know Kelvin in the context of the age of the Earth debate. Geochemists and others whose curriculum involves thermodynamics know that aspect of his work: entropy, Kelvin scale, so forth. Never the twain should meet! But, in fact, they do. Albritton, in his book The Abyss of Time points out that the late 19th century debate between Kelvin and the geologists centered not so much on the age of the Earth as on the nature of geologic history. Kelvin’s argument was not with Darwin the geologist, but through Lyell, with Hutton and his memorable phrase “no vestige of a beginning…” that we have all read, somewhat uncomprehendingly.
We read Hutton with difficulty, but not because of his writing style. You can check it out for yourself ( Theory of the Earth) . What makes him difficult to understand is that he was writing in a thoroughly Aritotelian frame of reference that has eternalism at its base. Furthermore, he had no chemical atomic theory, no way to distinguish between fire, heat and phlogiston. No oxygen in his vocabulary. What he did have was the awesome ghost of Newton and the image of an eternally revolving system of Sun and planets. In short, at least according to historians of Geology like Martin Rudwick (Worlds Before Adam), no sense of geologic history as a series of unique events unfolding in a unique, non-repeating sequence. This was the Huttonian foundation for Lyell’s cyclical uniformatarianism which made such difficulties for his admirer, Charles Darwin.
The mineral at the top of this page illustrates mineral evolution in the sense that it almost doesn’t exist. The conditions of temperature, pressure and composition that it requires are rarely achieved in the Earth, so it occurs in only one or two places on Earth. Fortunately for me, few people prize it as much as I do, so the price for this small example was within my wife’s birthday budget for me when I turned 65. I wore it as an ear stud until she passed.
Most of what I know factually about the Bucks Batholith is in the literature referred to. My fantastic ideas about how it came to be the way that it is can correctly be left to expire on their own.
But if you are interested in what happens inside an intruding body of magma, particularly near the top, there is a fact or two here you might find interesting and that are not reported in the literature to my knowledge. Plus a bit for the paleomag buffs.
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.
But somewhere there is a baby crying.
Enough of this eclectic plagiodoxic rambling!
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.
Here is something else that is blue.
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.
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?