ORGANIZE YOUR MIND --- THINK ABOUT NUMBERS -- Episode 2

                           THOSE INTERESTING PRIMES --- continued

   Prime numbers are like atoms. They are the building blocks of all integers. Every integer is either itself a prime or the product of primes. For example, 11 is a prime ; 12 is the product of the primes 2, 2, 3 ; 13 is a prime ; 14 is the product of the primes 2 and 7; 15 is the product of the primes 3 and 5 ; and so on. Some 2,500 years ago, Euclid gave a proof that "the supply of primes is inexhaustible. [ This has been discussed before, but bears repeating.]  
   Assume, said Euclid, that there is a finite number of primes. Then, one of them, call it P, will be the largest. Now consider the number Q, larger than P, that is equal to the product of the consecutive whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 . . . x P) + 1. From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into it ; each each division would leave a remainder of 1. If Q is not prime, it must be evenly divisible by some prime larger than P. On the other hand, if Q is prime, Q itself is a prime larger than P. Either possibility implies the existence of a prime number larger than the assumed largest prime P. This means, of course, that the concept of "the largest prime" is a fiction. But if there's no such beast, the number of primes must be infinite. 
   As of this writing, the largest known prime is a 909, 526-digit number formed by raising 2 to the 3, 021,377th power and subtracting 1. The prime was found on January 27, 1998, by the GIMPS project( Great Internet Mersenne Prime Search), in which 4,000 "primees" (prime number groupies) communicated over the Internet and pooled their computers for the hunt. Each of the 4,000 computers was assigned an interval of numbers to check. Roland Clarkson, a 19-year-old sophomore at California State University Dominguez Hills, was the lucky primee whose 200 Mhz Pentium-based home PC, after 46 days of running part time, examining the numbers in his assigned interval, proved the primality of 2 raised to the 3, 021, 377th power -- 1. 
   The hunt for large primes has come a long way since the seventeenth century, when Marin Mersenne, a Parisian monk, took time out from his monastic duties to search for primes. A number like 2 raised to the 3, 021th power ---1 that is the form 2 to the nth --1 is said to be a Mersenne number.  For a Mersenne number to be prime, n itself must be prime. Thus, since 2 raised to the 3, 021, 377th power --1 is prime, 3, 027, 377 must also be prime. But n being prime does not guarantee that the corresponding Mersenne number is prime. When n takes on the first four prime numbers, Mersenne primes are indeed generated : 

     For n = 2, 2 squared - 1 = 3
     For n =3, 2 raised to the third power -1 = 7 
     For n = 5, 2 raised to the 5th power -1 = 31
     For n = 7, 2 raised to the 7th power - 1 = 127 

   But when n is the fifth prime number, 11, the corresponding Mersenne number proves to be composite (2 raised to the 11th power -1 is 2,047, whose prime factors are 23 and 89) . In 1644, Mersenne himself claimed that when n took on the values of the sixth, seventh, and eighth prime numbers, namely, 13, 17, and 19, the corresponding Mersenne numbers, 2 raised to the 13th power - 1 (or 8,191), 2 raised to the 17th power - 1 (or 131, 071 ) and 2 raised to the 19th power - 1 (or 524,287) were primes. He was right. 

    The monk also made the bold claim that 2 raised to the 67th power -1 was prime.  The claim was not disputed for more than 250 years. Then, in 1903, Frank Nelson Cole of Columbia University delivered a talk with the unassuming title "On the Factorization of Large Numbers" at a meeting of the American Mathematical Society. Cole, who was always a man of very few words, walked to the board and, saying nothing, proceeded to chalk up the arithmetic for raising 2 to the sixty-seventh power. Then he carefully subtracted 1 [ getting the 21-digit monstrosity 147, 573, 952, 589, 676, 412, 927]. Without a word he moved over to a clear space on the board and multiplied out, by longhand : 

          193,707, 721 x 761, 838, 257, 287 

   The two calculations agreed. Mersenne's conjecture --if such it was --- vanished into the limbo of mathematical mythology. For the first time on record, an audience of the American Mathematical Society vigorously applauded the author of a paper delivered before it. Cole took his seat without having uttered a word. Nobody asked him a question. 
   

ORGANIZE YOUR MIND --- THINK ABOUT NUMBERS ---- Episode 1

                                 WHY THINK ABOUT NUMBERS ?

     If one plays around with numbers, one doesn't have to concern herself with the concrete world and all its contradictions. Toying with number proofs involves perfect abstract generality. Here's an example. It concerns the primes, which---as you probably remember from high school --- are those integers that can't be divided into smaller integers without remainder. One proposition about primes states that there is no largest prime number. { What this means of course is that the number of prime numbers is infinite. But at the time this proposition was first advanced, mathematicians danced around using the term "infinite."} Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2, 3, 5, 7, 11, . . . Pn) is exhaustive and finite : ( 2, 3, 5, 7, 11, . . . , Pn) is all the primes there are. Now think about the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime ? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2, 3, 5, . . ., Pn) , because by dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2, 3, 5, . . . , Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2, 3, 5, 7, . . . , Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction.  By removing false assumptions, we conclude that there is no largest prime. This, so far as our minds can currently stretch, is a fact. 

         THE WONDERFUL WORLD OF PRIME NUMBERS

   As we've already seen, the prime numbers are integers like 2, 3, 5, 7, 11, 13, and 17, which are evenly divisible only by themselves and the number 1. We happen to have ten fingers, and our number system is conveniently based on ten digits. But the same primes, with all the same properties, exist in any number system. If we had twenty-six fingers and constructed our number system accordingly, there would still be primes. It's easy to conceive of a culture that doesn't use base 10. We have plenty of examples. Computers use a binary system, and the Babylonians had a base-60 system, vestiges of which are evident in the way we measure time(sixty seconds in a minute, sixty minutes in an hour) . Cumbersome as this sexagesimal system was, it, too, contains the same primes. So does the octary system that Reverend Hugh Jones, a mathematician at the College of William and Mary, championed in the eighteenth century as more natural for women than base-10 because of women's experience in the kitchen working with multiples of 8 (32 ounces in a quart, 16 ounces in a pound ). [ Come on, girls, let's hear it for Rev. Jones !!!! ] 
   G.H. Hardy, a number theorist, believed numbers constituted the true fabric of the universe. In an address to a group of physicists in 1922, he took the provocative position that it is the mathematician who is in "much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'."  But "a chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it ; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly we scrutinize it . . . 317 is a prime,not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way." 

BUILDING A LOGICAL MIND --- Episode 6

                CAN WE LIVE IN A COMMONSENSE WORLD
                YET "KNOW" MANY ABSTRACT TRUTHS ? 

      Never before have there been so many gaping chasms between what the world seems to be and what science tells us it is. "Us"meaning laymen. It's like a million Copernican Revolutions all happening at the same time. As in for instance we "know," as high-school graduates and readers of Newsweek, that time is relative, that quantum particles can be both there and not, that space is curved, that colors do not inhere in objects themselves, that astronomic singularities have infinite density, that or love for our children is evolutionarily programmed, that there is a blind spot in the center of our vision that our brains automatically fill in. We "know" that our thoughts and feelings are really just chemical transfers in 2.8 pounds of electrified pudding. We "know" we are mostly water, and water is mostly hydrogen, and hydrogen is flammable, and yet we are not flammable. We "know" a near-infinity of truths that contradict our immediate commonsense experience of the world. And yet we have to live and function in the world. So we abstract, compartmentalize : there's stuff we know and stuff we "KNOW" : I "know" my love for my child is a function of natural selection,but I know I love him, and I feel and act on what I know. Viewed objectively, the whole thing is extremely schizoid ; yet the fact of the matter is that as subjective laymen we don't feel the conflict. Because of course our lives are 99.9% concretely operational, and we operate concretely on what we know, not on what we "KNOW." 
   Understand that we talking about laymen like you and me, not about giants of philosophy and math, many of whom had well-documented trouble navigating the real world----Einstein leaving home in his pajamas,  Kurt Godel (genius logician and mathematician) unable to feed himself, and so on. To appreciate what the inner lives of great scientists / mathematicians / metaphysicians are like, we need only sit in a quiet spot and try to form a truly rigorous and coherent idea --- as opposed to a fuzzy or Newsweekish idea --- of what we really mean by "omnipotent," "three," or "finite but unbounded." I.E., to try to do some disciplined or directed ABSTRACT thinking. There's a very definite but inarticulable strain on the seams of the mind involved in this kind of thinking. One of the quickest routes to to this feeling is to try to think hard about dimension. There is something we "KNOW," which is that spatial dimensions beyond the Big 3 exist. Some can even construct a tesseract or hypercube out of cardboard. A weird sort of cube-within-a-cube, a tesseract is a 3-D projection of a 4-D object. The trick is imagining the tesseract's relevant lines and planes at 90 degrees to each other, because the 4th spatial dimension is one that somehow exists at perfect right angles to the length, width, and depth of our regular visual field. We"KNOW" all this, but now try to rally and truly picture it concretely. You can feel. almost immediately, a strain at the very root of yourself, the first popped threads of a mind starting to give at the seams. 

BUILDING A LOGICAL MIND --- Episode 5 x

                             Oops, I Gotta Go Back To Induction 
             { Plus, I Forgot To Discuss Abstract Thinking } 

                            ABSTRACT THINKING 

   The dreads and dangers of abstract thinking are a big reason why we now like to stay constantly busy and bombarded with stimuli from TV, iPods, radio, smart phones, the social media on the internet, and the search engines on computers. We sit for hours staring at computer screens and we walk down crowded sidewalks of large cities with electronic "buds" plugged into our ears. Abstract thinking tends most often to strike during moments of quiet repose, such as, for example, in the early morning before the alarm goes off. You might lie there in the quiet of your bedroom without the slightest doubt that the floor will support you when your feet eventually touch down. But, then you start thinking that it's at least theoretically possible that some flaw in the floor's construction or its molecular integrity could make it buckle, or that even some aberrant bit of quantum flux or something could cause you to melt right through. I mean, it's not logically impossible or anything. Obviously you're too smart to be much concerned that the floor might give way when you finally do get out of bed. It's just that certain moods and systems of thinking are more abstract, not just focused on whatever needs or obligations you're going to get out of bed to attend to. This, of course, is just a silly example. The abstract question you're lying there considering is whether you are truly justified in your confidence about the floor. The initial answer, which is yes, lies in the fact that you've gotten out of bed in the mornings thousands --- actually well over tend thousand times so far, and each time the floor has supported you. It's the same way you're also justified in believing that the sun will come up, that your spouse or companion will know your name, that when you feel a certain set of sensations it means you are getting ready to sneeze, etc., etc. Because they've happened over and over before. The principle involved is really the only way we can predict any of the phenomena we just automatically count on without having to think about them.  And the vast bulk of daily life is composed of these sorts of phenomena ; and without this confidence based on past experience we'd all go insane, or at least we'd be unable to function because we'd have to stop and deliberate every last little thing. It's a fact : life as we know it would be impossible without this confidence. Still, though : Is the confidence actually justified, or just highly convenient ? This is abstract thinking, with its distinctive staircase-shaped graph, and you're now several levels up. You're no longer thinking just about the floor and your weight, or about your confidence concerning same and how necessary to basic survival this kind of confidence seems to be. You're now thinking about some more general rule, law, or principle by which this unconsidered confidence in all its myriad forms and intensities is in fact justified instead of being just being a series of tics or reflexes that propel you through the day. Another sure sign it's abstract thinking : You haven't moved yet.  It feels like tremendous energy and effort is being expended and you're still lying perfectly still. All this is just going on in your mind. It's extremely weird ; no wonder most people don't like it. It suddenly makes sense why the insane are so often represented as grabbing their head or beating it against something. If you had the right classes in school, however, you might now recall that the rule or principle you want does exist ---- its official name is the Principle of Induction.  It is the fundamental principle of modern science. Without the Principle of Induction, experiments couldn't confirm a hypothesis, and nothing in the physical universe could be  predicted with any confidence at all. There could be no natural laws or scientific truths. The Principle of Induction states that if something x has happened in certain particular circumstances n times in the past, we are justified in believing that the same circumstances will produce x  on the ( n + 1 )th occasion.  The Principle of Induction is wholly respectable and authoritative, and it seems like a well-lit exit out of the whole problem. Until, that is, it happens to strike you (as can occur only in very abstract moods or when there's an unusual amount of time before the alarm goes off) that the Principle of Induction is itself merely an abstraction from experience----and so now what exactly is it that justifies our confidence in the Principle of Induction ? This latest thought may or may not be accompanied by a concrete memory of several weeks spent on a relative's farm during childhood. There were four chickens in a wire coop off the garage, the brightest of whom was called Mr. Chicken. Every morning, the farm's hired man's appearance in the coop area with a certain burlap sack caused Mr. Chicken to get excited and start doing warmup-pecks at  the ground, because he knew it was feeding time. It was always around the same time t every morning, and Mr. Chicken had figured out that t(man + sack) = food, and thus was confidently doing his warmup-pecks on that last Sunday morning when the hired man suddenly reached out and grabbed Mr. Chicken and in one smooth motion wrung his neck and put him in the burlap sack and bore him off to the kitchen. Memories like this tend to remain quite vivid, if you have any. But with the thrust of the story, lying here in your bed, being that Mr. Chicken appears now actually to have been correct---according to the Principle of Induction --- in expecting nothing but breakfast from that (n + 1)th appearance of man + sack at t. Something about the fact that Mr. Chicken not only didn't suspect a thing but appears to have been wholly justified in not suspecting a thing---this seems concretely creepy and upsetting. Finding some higher-level justification for your confidence in the Principle of Induction seems much more urgent when you realize that, without this justification, our own situation is basically indistinguishable from that of Mr. Chicken. But the conclusion, abstract as it is, seems inescapable : what justifies our confidence in the Principle of Induction is that it has always worked in the past, at least up to now. Meaning that our only real justification for the Principle of Induction is the Principle of Induction, which seems shaky and question-begging in the extreme. 

                        

BUILDING A LOGICAL MIND---Episode 4

 RANDOM OR REPRESENTATIVE SAMPLING IN 
                     STATISTICAL INDUCTIONS 

   Assume you are in a business economics class of a hundred students. One day the teacher asks how many in the class favor the tax exemption for business lunches. All but two students raise their hands. The teacher concludes that most students at the university are in favor of tax exemptions for business lunches. This is the inductive method, but it's neither fair nor accurate since the selected sample is not representative of the student body as a whole. It excludes students in art, drama, physics, philosophy,  many of whom might oppose the tax exemption for business lunches. So what is the right way to arrive at an accurate conclusion ? There are two ways : (1) We can randomly select a large enough number of students---say 300 in a school made up of a thousand students --- and solicit their opinions. Random sampling prevents any bias from influencing the outcome. (2) We can select a group of students representative of the entire population. Let's assume, for example, that the university breaks down the following way : 35 percent business ; 15 percent engineering ; 20 percent liberal arts ; 15 percent physical sciences ; 10 percent fine arts ; 5 percent other. If you wanted to find out what the students as a whole think about tax-exempt business lunches, but couldn't interview every student, you could select a hundred out of the thousand students at the school, keeping the percentages in this group the same as the percentages of students in the various colleges : 35 percent from business ; 15 percent from engineering ; 20 percent from liberal arts, and so on. Such a method will lead you to a fairly accurate conclusion about the student body's opinion on tax-exempt business lunches. If statistics are to have any relevance at all, they must be a result of random or representative sampling. 
   Although induction leads to truths, especially in science, for the most part it leads to a high probability. So we must use induction cautiously. After all, it was not that long ago that people thought frogs fell from the clouds since they appeared out of nowhere every time it rained. 

                                              DEDUCTION 

   Induction helped our remote forebears take control of their destiny and make sense out of a bewildering world. Each truth established by observation became a small step on the stairway human beings were building to lift themselves out of the purely animal world. Induction also made possible something else --- another way to create steps, another way to arrive at reasonable conclusions about the world : DEDUCTION.  Fr the first time, they could make reliable generalizations about the natural world; they could begin classifying it. Some things were animals ; some were plants ; some were minerals. Among those things that were animals, some were insects, some were reptiles, some were mammals. Among the mammals, some were land-dwelling, some were water-dwelling. And the classification would continue until there were 15,000 species of mammals alone. Creatures with x characteristics belong in this species ; creatures with y characteristics belong in that species. Humans were, in effect, putting their world into categories ---categories that eventually led to biological classification of shared characteristics. Grouping creatures into various classes can be very useful, for it enables humans to reason deductively --- that is, to draw an accurate conclusion from two general premises. 
   Assume for example that you, as a biologist, know very well that "all warm-blooded, egg-laying, winged and feathered vertebrates are birds. " Then one day you come across a penguin, the first one ever seen. Upon examination you discover that even though it doesn't resemble any bird you've ever seen, it is a warm-blooded, egg-laying, winged and feathered vertebrate. You conclude : "This creature must be some kind of bird." The creature is now less mysterious since you can name it, which is another way of saying categorize it. Induction led us to the generalization about birds ; deduction led you to the conclusion that this new-found creature is also a bird. 

BUILDING A LOGICAL MIND --- Episode 3

  UNDERSTANDING THE WAY REASONABLE MEN AND 
                                       WOMEN THINK 


                                             INDUCTION

Men like Aristotle considered the science of reason an ethical exercise that would best guide human beings in their relationships with other human beings, and, by extension, a society's relationship with other societies. It was during his time, the fourth century B.C., that most of the rules of reason we still follow today were established.

A good way to learn these rules is to transport yourself back a few hundred thousand years and consider the way reason itself evolved. Our ancient ancestors were able to wrestle their way out of a hostile environment and eventually dominate it because of reason. One of the simplest ways they reasoned was inductively, that is, by drawing conclusions from observing the repetition of events. Ever since they could remember, our early forebears observed that the antlers of deer fell off every winter ; once they began herding sheep, they noticed that most sheep lived only sixteen years. Because of such repetition in events, they could make generalizations about the world : deer antlers fall off every winter ; sheep die on the average at sixteen years of age. This ability to make generalizations helped them gain control of their world because they could then make predictions and plan their existence  in nature, rather than just to react to it. Because of induction, they learned when to plant, harvest, prepare for a cold winter, migrate, fish, and hunt. Farmers still rely on inductions made centuries ago : plant underground crops when the moon is waning; plant above-ground crops when the moon is waxing. Noting this repetition of events in nature not only helped the primitive humans take some control of their destiny, but also made them acutely aware that there are things in the world called truths. If there were truths about the seasons and about poisonous plants, then there must be other truths as well. The world, then, is not only manageable, it is understandable because of these certainties. Arriving logically at these certainties eventually became a science. Scientists would rely on the inductive method of thought to confirm the truth that water freezes at thirty-two degrees Fahrenheit, that the earth completes its rotation approximately every twenty-four hours, that light objects and heavy objects fall at the same rate of speed.

Induction, then, is the method of arriving at a probable truth by relying upon the repetition of the same fact to lead you to a generalization or conclusion --- no matter how mundane : every time I've used that soap my hands have broken out. There must be something in the soap that makes my hands break out. Although the inductive method of reasoning is a very useful way to arrive at conclusions, it can be abused, as it often is by the propagandist. However, Aristotle and his colleagues have set out some rules by which to judge the accuracy of any induction. 

      HOW TO TEST THE ACCURACY OF ANY INDUCTION

1.) SUFFICIENT EVIDENCE 

   I've owned my Volkswagen less than a week and I'm already having trouble with the transmission. A friend at my gym is likewise having trouble with the transmission in her Volkswagen. I conclude that all Volkswagens have bad transmissions. In arriving at this conclusion, I am moving inductively ---from the observation of specific examples to a generalization --- but I'm arriving at a conclusion too hastily. The thousands of other Volkswagen owners may never have had trouble with their transmissions. So while induction can be an accurate way of arriving at conclusions, they must not be drawn too hastily. The rule : In all inductive arguments the evidence must be sufficient to warrant the conclusion. 

   

BUILDING A LOGICAL MIND---Episode 2

LOGIC IS THE FORMAL NAME OF THE SCIENCE THAT ATTEMPTS TO REDUCE THE PROCESSES OF CORRECT THINKING TO RULES 



                               PROPAGANDA vs. REASON 

   In Mein Kampf, Adolph Hitler demonstrated his complete understanding that one can manipulate an audience by appealing to its emotions and prejudices. He explained that he learned from an early age that the proper employment of propaganda is a real art. He didn't believe that considerations of humaneness and beauty counted in the battle, and that neither could they be used as standards to judge propaganda. In his experience, all propaganda must be popular in tone, and must keep its intellectual level to the capacity of the least intelligent among those at whom it is directed. Hitler was convinced that propaganda can always succeed because most people let emotions and feelings rather than sober consideration determine their thought and action. 

   Now, contrast that defense of propaganda, with this defense of using reasoning, not propaganda, to influence belief and behavior : 

     Because he believes the proper way to influence others is to bring those persons to see for themselves te rightness or the justness of the claims he presents, the advocate who chooses argument as his instrument treats his readers or listeners not as things to be manipulated, but as persons to be reasoned with, as responsible, rational beings whose judgment deserves respect and whose integrity must be honored. Modes of persuasive appeal which seek to circumvent or benumb the understanding are disrespectful of the individuals addressed ; they degrade the listeners or readers by endeavoring to produce the automatic, instinctive sort of response characteristic of animals, rather than the considered, judgmental sort of response humans alone are capable of making.  Argument, in contrast, is respectful of people and those distinctive qualities of reason, understanding, and reflection which mark them off as "human." Instead of addressing the biological individual,it addresses the person as thinker. 
    
                           Watch For An Important Contrast
                           That This Commentator Makes 

     Humanity is something we gain for ourselves only insofar as we willingly grant it to those about us. The more skilled we become in the use of emotion, prejudice, or suggestions as instruments of persuasion, the farther we depart from the ideals which ought to govern our relations with our fellows.       

   

BUILDING A LOGICAL MIND ---Episode 1

                  HUMANS ARE ANIMALS THAT REASON 

    Reasoning is as natural and familiar a process as breathing, but it is also a skill in which indefinite improvement is possible for anybody who is not a genius. Unfortunately, too many people engage in illogical thinking and cling to it in the face of all the evidence that proves them wrong. 
   The logical person, being presented with an answer different from her own, says, "Hmmmmm. Let me follow the steps of the other solution and see if they work out. Well, what do you know ? They do !" The illogical person in the same situation, typically says, "I'm so-in-so years old, and I've known the answer since 19-something-or-other, and I don't have to work out the solution because I already know it." 
   Conundrums and riddles aside, this is one of the most common situations in everyday life---hanging on tightly to what one "knows" instead of opening one's mind and letting in the fresh air of simple logic. 
   We can use the precepts of defined approaches to the truth and get some idea of how to sue these precepts in everyday life and problem-solving. 
   Logic isn't something you leave behind in college. At least, we shouldn't. Because if we do, we'll have difficulty just getting through the day, not to mention accomplishing even the most modest of our goals. Living logically eliminates many of life's errors. 
   Having just completed reading How We Think, by the philosopher / educator John Dewey, I'm impressed with his suggested steps for establishing a logical approach to arriving at the heart of a problem and solving it. 

     The first step is to become aware of the problem. The second is to define it and analyze it, establishing its parameters. The third step is to approach it rationally from different angles, considering a number of options and various working hypotheses for its solution. And the last step is to select a solution and verify its effectiveness. 

                                    THINK FOR YOURSELF

   We shouldn't just read problems when we come across them in print : we should solve them for ourselves. Here's a good one to start with : A man is looking at a portrait on a wall and says : 

     "Brothers and sisters I have none, but this man's father is my father's son." 

     At whose portrait is he looking ? Before giving the answer, let's analyze the riddle, approach it rationally, select a solution and verify its effectiveness. 

     Let's call the man speaking "John" and the man in the portrait "Mr. X" and then phrase the sentence in a more normal conversational way without changing its meaning : 
     John says : "I'm an only child, and Mr. X's father is my father's son." 
     If John were to solve the puzzle himself, he could go on to say : "And just who is my father's son ? As I have no brothers, it can only be me ! Then Mr. X's father is me, and Mr. X is my son." 
     If we solve the problem ourselves, we must say that if John is an only child, and Mr. X's father is John's father's son, then Mr. X 's father must be John. And if Mr. X's father is John, Mr. X is John's son. 

       SEPARATE THE PROBLEM FROM THE SYMPTOM

Let's look at each of these common life situations and decide which illustrates a symptom and which illustrates a problem :

1) You're forty years old now, and you have to hold a book farther away in order to read it clearly.

2) You're forty years old now, and you get tired much more easily than you used to. 

3) You have two children now, and your husband spends less time with you than he did before you first got pregnant. 

4) You have two children now, and you weigh twenty pounds more than you did before you first got pregnant. 

    The answers ? #1 and #3 are problems ; #2 and #4 are symptoms. Here's why : 

1) A change in the ability to focus is a problematic, but normal part of the aging process. 

2) Getting tired much more easily at the age of forty is not normal and should be considered a symptom instead.

3) If you now have two children, it's nearly impossible for your husband to spend as much time with you as he did before for the simple reason that you are probably offering him less time. And if he spends time with the children himself, that's even more time away from the two of you. The situation may be a problem, but it isn't a symptom of anything that isn't inherent in the circumstances. 

4) Giving birth to children doesn't increase your weight, and if you weigh too much now, it can only be a symptom of another problem. 

                 DEFINE AND ANALYZE THE PROBLEM

   A large part of this is admitting the problem exists. But there's a key element to be remembered, an important part of using logic successfully in life. We must learn to approach problems rationally and not emotionally, or we won't be able to solve them.  Hating a problem won't solve it. 

   Okay, let's define the problem, using the above examples. 

1) You're forty years old now, and you have to hold a book farther away in order to read it clearly.

2) You're forty years old now, and you get tired much more easily than you used to. 

3) You have two children now, and your husband spends less time with you than he did before. 

4) You have two children now,and you weigh twenty pounds more than you did before you first got pregnant. 

   Now, compare your answers with these :

1) You're simply growing older.

2) Unless you're ill, you're not exercising enough and perhaps eating the wrong foods. To say here that you're just "not as young as you used to be" is to avoid recognizing the problem. If you were a hundred years old, it would be different. But you're not. 

3) There is less time now to spend with each other. Saying "he doesn't appreciate me" is an emotional, not a logical response. There were no other circumstances detailed in the original problem to indicate that likelihood.

4) Unless you're ill, you're eating too much and exercising too little. Saying here that "motherhood changes a woman;s figure, " is, again,  avoiding the recognition of the problem and the need to solve it. Motherhood may well cause changes in a woman's figure, but it doesn't make her weigh more. 



   

A LAND WHERE WEALTH ACCUMULATES AND PEOPLE DECAY --- Episode 1

           A LITTLE MORE HISTORY OF WEALTH & POLITICS 

   The terrorist attack on New York City in September 2001 came only a year after serious candidates in America's millennial presidential election had described how money and wealth in the United States were crippling democracy. Politics, they had said, was being corrupted as the role of wealth grew.
   Other critics had found a reemergent plutocracy ---defined as government by or in the interest of the rich ---challenging popular sovereignty as it had in the late nineteenth century. Scholars also pointed out that the reigning theology of domestic and global markets bore disturbing resemblance to the survival-of-the-fittest canons of that earlier Gilded Age.
   None of these circumstances were changed by the destruction of the World Trade Center. The increasing reliance of the American economy on finance is an even more obvious vulnerability. If September's stock market decline briefly shaved another trillion dollars from U.S. financial assets, national politics continued to wear its "for sale" sign. The United States remained what comparisons had clearly shown : the most polarized and inequality-ridden of the major Western nations. 
   In 2002, as in 1999 and 2000, these predicaments did not represent the American political and economic norm, which has been for such developments to be restrained by suspicions of the rich. Deviations from such wariness mostly have come during optimal periods of broad-based prosperity in which economic opportunities far outweighed these qualms. The early nineteenth century, for example, in the frontier settlement decades, humming with bargain-priced government land sales ---"doing a land office business" became a common phrase in the 1830s---empowered millions of new small landowners. New World openness in acreage or jobs became a beacon, drawing millions of emigrants from European embarkation ports. Stephen Girard and John Jacob Astor, America's richest men, were immigrants who had built fortunes WITH THE HELP OF JEFFERSONIAN POLITICS. Wealth in their hands symbolized opportunity. 
   The other great example came in the quarter century after World War II when the middle class pushed its share of national wealth and income to record levels. The skepticism of the rich imprinted by the Great Depression guided politics and public policy through the 1960s. 

   These were the two eras in which wealth and opportunity clearly nurtured democracy. Yes, the top 1 percent of Americans had a very large slice, but it was smaller than the aristocracy of Europe. 

YOU SAY YOU WANT A REVOLUTION--WELL, YOU KNOW, WE ALL WANT TO CHANGE THE WORLD ---Episode 5

                                    THE ISSUES OF U. S. ECONOMIC DECLINE 

   Louis Hartz, the historian best known for his discourse on the liberal tradition in America, pointedly wondered back in the middle of the twentieth century what would become of American exceptionalism ---and the optimism of the electorate --- if the United States was forced to rejoin world history after a 150-year vacation from it. In the seventies, eighties, and early nineties, many scholars and pundits were convinced that this Hartzian hour was at hand. 

   Attention to decline, whether in Europe or the United States, has had its own rhythm. Each nation's early worriers, reacting to decline from an absolute zenith in share of world trade or manufacturing, have been premature, in practical terms, by some four or five decades. Yet their analyses are a useful jumping-off point. The stage at which considerable popular concern has developed --- in the 1890s, for Britain, in the 1980s in the U.S. --- has usually involved a stalling of previous advances for the working class while the upper classes enjoy a glittering cosmopolitan zenith: Britain in 1900-1914, the United States of the 1980s, 1990s, and millennium.

   Indeed, the wave of books assessing decline in Holland and Britain published in Europe and North America during the 1980s and 1990s suggests that full, open, and informed retrospect may even require the passage of of fifty (or 150) years. Even then, the disagreement among scholars can be fierce. Volumes published during a leading power's sunset have to be indirect or oblique, like Paul Kennedy's The Rise and Decline of the Great Powers, or limited to one evident dimension, like the English books of the 1890s sounding alarm bells about the American or German economic threats.

   Nevertheless, for U.S. purposes, the record of the three decades between 1970 and 2000 was replete with American moods, circumstances, and debates familiar from later trajectory of two previous leading world economic powers. The sullenness of the workforce, especially men. The concern with globalization and hitherto domestic investment flowing overseas. The growing awareness of the rich and conspicuous consumption, and THE HINTS OF PLUTOCRACY.  Other parallels include anger at the corruption of officeholders and anxiousness to democratize politics and increase popular electoral participation as well as a finger-pointing at financiers and incipient attention to increasing taxes on the rich to pay for popular social programs, pensions, health insurance, and the like. 

   The Dutch and British precedents that follow can be put alongside the short chronology of U.S. popular frustration and off-and-on Middle American Radicalism. It makes sense to begin with the broadest frustration : the desire for institutional, moral, and economic revitalization. 

   In either a reactionary, democratic, or some mixed form of government, "revitalization" has been a common agendum in leading economic powers once the public starts to identify national decline or the corruption of formerly vital political institutions . In ancient Greece and Rome, Plato and Plutarch wrote of the need to escape plutocracy. 

YOU SAY YOU WANT A REVOLUTION---YOU KNOW WE ALL WANT TO CHANGE THE WORLD---Episode 4

     MORE RECENT EXAMPLES OF THE KIND OF PUBLIC            OUTRAGE THAT MIGHT SPUR A REVOLUTION ---cont

Of Bill Clinton's Democratic 1992 primary opponents, Massachusetts senator Paul Tsongas talked of competitiveness, exaggerating that "the cold war is over---Japan and Germany won." Former California governor Jerry Brown blistered the elite, charging that, "The ruling class has lost touch with the American people. They have lost touch because they float in a world of privilege, power, and wealth." Clinton himself was said to have become furious after reading a New York Times analysis that the top 1 percent had received over half of the additional income generated in the United States between 1977 and 1989.

The race between Clinton, Perot,and Bush sparked a five-point jump in the percentage of eligible-age Americans voting --- up to 55 percent in 1992 from a meager 50 percent in 1988. At least temporarily, this suspended the argument that nonvoters had become the nation's most important party as turnout kept sinking among low and middle-income Americans whose previous inclinations, at least, had been Democratic. But in 1996,when Perot had become old hat, turnout dropped back to 50 percent. 

Clinton's first years in office seemed to worsen the disillusionment. He retracted his middle-income tax-cut promise and, by 1994, the scandals touching the White House and his personal life together with ongoing weakness in the economy --- median household income was stagnant --- made him an albatross for Democrats in midterm elections. "Washington" itself by this point had become a focus of public contempt, with trust in the capital gang dropping to record (19 percent) lows.  Some 57 percent told pollsters that "lobbyists and special interests" controlled Washington, not the president or Congress. In 1993, polling for the Boston-based Americans Talk Issues Foundation reported the citizenry so contemptuous of Congress that one-third of those sampled thought the offices might as well be auctioned to the highest bidders. Half thought Congress could be chosen randomly from a list of eligible voters. The emergence of rightwing "militias" in states from Michigan to Montana was still another sign of popular frustration.

Middle American Radicalism had one more late -twentieth- century moment in the sun. As popular insurgent Buchanan beat the eventual nominee,Senator Robert Dole, in the 1996 Republican primary in New Hampshire, one issue caught hold --- Buchanan's appeal to the middle class with criticism of corporate chieftains whose pay had risen to two hundred times that of workers. Even Dole started speaking about "greedy CEOs" and Clinton called together a hundred of them to discuss the matter. But as left-leaning Mother Jones magazine said a half year later, "after Pat Buchanan shocked the political establishment by prying open the Pandora's box of slow growth,wage stagnation, globalization, and increasing inequality, the lid is back on." After New Hampshire, Buchanan faded, and although Dole and some Republicans made off-and-on comments about how median family income had stagnated in 1993-94 while male earnings had continued to fall, their counterpoint hardly mattered. By mid-1996 the economy in general and the stock market in particular were visibly on the rise.