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Ancient Chinese Inventions and Discoveries (2002) – Essay by G. Stolyarov II

Ancient Chinese Inventions and Discoveries (2002) – Essay by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 18, 2014
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Note from the Author: This essay was originally written in 2002 and published in three parts on Associated Content (subsequently, Yahoo! Voices) in 2007.  The essay earned over 10,900 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
***
~ G. Stolyarov II, July 18, 2014
***

The civilization of ancient China produced a wide array of innovations in science and technology which preceded the rest of the world by centuries and sometimes by millennia. This essay examines some of these remarkable inventions and discoveries.  Chinese inventors developed numerous mechanical implements, engineering advances, and new substances such as gunpowder, which took centuries to spread to or be replicated in other parts of the world. Furthermore, this essay explores the reasons for ancient China’s lack of systematic progress or an industrial revolution despite the presence there of numerous inventive thinkers.

Mathematics

Beginning in the 14th Century BC, the Chinese developed a decimal, or base ten system of recording numbers. This is one of the earliest such systems known.

In the first century AD, Chinese scholars compiled a volume of mathematics, Jin Zhang Suanshu,(Arithmetic in Nine Chapters). Mathematician Zu Chongzhi (429-500) calculated the first 12 digits of the value of pi, while his son, Zu Gengzhi, updated the Jin Zhang Suanshu and determined the correct formula for the volume of a sphere, V= (pi/4)d^3, where d is the diameter.

Paper

Paper was invented by Cai Lun, a scientist at the Imperial Court in 105 AD. It was produced from bamboo and hemp fibers dissolved in water situated in a mold. When the water was drained and the mixture dried, the first genuine design of paper appeared. The Chinese also developed a precursor to the first paper currency in the world, printed in the Ninth Century AD in order to lighten the load carried by tax collectors.

Cast Iron

The first methods for developing raw iron into workable material with the capacity to be crafted into weapons and ornaments were developed in the 4th Century BC, when the Chinese became able to lower iron’s melting temperature by adding phosphorus to the heated metal.

In the 2nd Century BC, this technology served to bring about the manufacture of steel by mixing wrought and cast iron at high temperatures or draining the carbon component from cast iron.

Chain Pump

Invented in China during the 1st Century BC, A chain pump consists of a chain attached to itself at the ends, which carries along it pallets of raw materials, such as water or sand, which are elevated to impressive heights up to about four meters.

Agricultural Technology

The Chinese were the first civilization in the world to plant crops in rows, beginning in the 6th Century BC, in order to obtain rapid crop growth without the crops’ mutual interference. Chinese farmers accomplished this 2200 years before any other culture.

Beginning in the 3rd Century BC, horses were utilized in China to haul loads on farms using an upgraded harness with a collar and chest strap (known a trace harness or horse collar) , which reduced the attachment’s burden on the animal and permitted a single horse to move a ton and a half of material.

The 3rd Century BC also saw the advent of the moldboard plow, or kuan, the design of which included a sharp center for digging into the ground and gradually-sloped wings at the side in order to discard excess soil and ease the friction on the plow.

The wheelbarrow was invented in the 1st Century BC and enabled Chinese farmers to transport massive loads over vast distances with ease.

Gunpowder

Gunpowder was invented in China during the 8th Century AD as a mixture of charcoal, sulfur, and saltpeter and used primarily for fireworks. The fireworks were launched from rockets made of hollowed bamboo tubes. These rockets were lighted through use of matches, invented in the 6th Century AD, carved of pinewood and coated with sulfur. Other civilizations borrowed this aspect and discovered its military utility.

In 1150, fireworks were elaborated as a result of the first multi-staged rockets, where several smaller tubes were attacked to main meter-tall stick, which were ignited in mid-air after the main rocket’s energy became depleted.

Natural-Gas Drilling

During the 1st Century BC, the Chinese discovered methods to drill some 1.5 kilometers into the Earth’s surface. A derrick was constructed, followed by a small shaft that extended until the Earth’s layer of hard rock was reached. Then a cast iron drill would be lowered with bamboo cables, after which the process would often consume years before any actual fuels were located.

Mechanical Clock

Invented in the 8th Century AD, the mechanical clock rapidly spread to other regions of the world. Chinese designs were crucial to inspiring European clock inventors such as Pope Sylvester II. The Chinese mechanical clock was powered by falling water or mercury, which then transmitted the energy to a chain-drive.

Segmental Arch Bridge

A segmental arch, invented by engineer Li Ch’un in the 7th Century AD, consists of only a small fragment of a circle instead of earlier semicircular arches. Ch’un constructed his first bridge over the Chiao Shui River in 610, which was notably lighter, more durable, and more material-efficient than earlier bridges. It is still in frequent use today.

Belt-Drive

A belt-drive (or driving belt) was attached around wheels to ensure smooth transition of mechanical energy between them. Invented in China during the 1st Century BC, the belt-drive was applied extensively to silk manufacture and spinning machines.

The belt-drive made possible the invention of the spinning wheel in 1270, since it provided sufficient cover and attachment for a rimless construction such as a spinning wheel, where a network of threads replaces the rim.

Printing Press

Movable character blocks were invented by Bi Sheng in 1045. A method for arranging and printing pages in mass quantities was devised. However, this was not efficient when applied to the Chinese language, which possesses over 5000 characters, and thus could not spur on the same printing revolution as that which occurred in Europe.

Magnetic Compass

The first magnetic compass was invented in China during the 3rd Century AD, utilizing a piece of magnetite (an ore of iron) which was heated and aligned in a North/South position, afterward being contained in a bowl of water where it floated on a piece of reed. This was integral to early 2nd millennium Chinese explorations in the Indian Ocean.

Other Noteworthy Advances

The Chinese were the first to develop a kite in the 4th Century BC. Craftsmen like Kungshu P’an possessed mastery to the extent of developing kites that stayed afloat for three days. These kites had military applications as well, carrying messages to isolated troop formations on the battlefield.

Commissioned by the imperial government in 132 AD, mathematician and cartographer Chang Heng devised the first seismograph, which allowed fairly accurate forecasts of earthquakes, leading to more efficient economic planning.

The Yellow Emperor’s Manual of Corporeal Medicine, composed in the 2nd Century BC, contains an advanced treatise on the circulation of blood. This was published fifteen centuries before William Harvey developed a work of comparable caliber in the West.

Why the Ancient Chinese Failed to Achieve Routine Technological Progress

Despite numerous ingenious technological innovations throughout its history, China failed to develop an industrial revolution and a routine theory like the Scientific Method to render inventions and discoveries systematic and not merely the spontaneous products of ingenious minds.

Ancient China was a generally stagnant society which, despite the presence of numerous brilliant thinkers, failed to achieve any regular technological progress until the late 19th century. So dramatic was this stagnation that it led Victor Hugo to compare China to “a fetus in a jar.” While it witnessed numerous promising technological developments in their embryonic stages, ancient China failed to harness these developments into a consistent advance.

The reason for this unfortunate lack of progress was above all institutional. Although the earlier Han and Tang dynasties among others were receptive to advancements and scientific practice, the Ming, following the defeat of the Mongols, isolated China from the remainder of the world and focused on civil stability to a greater extent than technological progress.

The heavily Confucian paradigm of the era from 1368 to 1911 focused more on adaptation to nature and the arts rather than the sciences. Scholars were trained in extensive law memorization rather than further studies of the external world. This caused China to lag behind the West, and contact with the Occident was required to re-establish its rich technological tradition.

Sources

1997 World Book Encyclopedia: Vol. 3 C-Ch. World Book Inc. Chicago. 1997.

Franklin Institute Online. “China: Ancient Arts and Sciences.” Available March 31, 2002: http://sln.fi.edu/tfi/info/current/china.html

Latourette, Kenneth S. A Short History of the Far East. The Macmillan Company. New York. 1964.

Reischauer, Edwin O. Fairbank, John K. East Asia: The Great Tradition. Houghton Mifflin Company. Boston. 1960.

Schurmann, Franz. Schell, Orville. Imperial China. Random House Inc. New York. 1967.

Think Quest Library of Entries. “Ancient Chinese Technology.” Available March 31, 2002: http://www.thinkquest.org/library/lib/site_sum_outside.html?tname=23062&url=23062/frameset.html.

Wagner, Donald B. “Liu Hui and Zu Gengzhi on the Volume of a Sphere.” Available March 31, 2002: http://www.staff.hum.ku.dk/dbwagner/SPHERE/SPHERE.html.

Ideas in Mathematics and Probability: Conditional Probabilities and Bayes’ Theorem (2007) – Article by G. Stolyarov II

Ideas in Mathematics and Probability: Conditional Probabilities and Bayes’ Theorem (2007) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 18, 2014
******************************
Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 2,100 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
***
~ G. Stolyarov II, July 18, 2014
***
When analyzing dependent events, the concept of conditional probability becomes a useful tool. The conditional probability of A given B is the probability that event A will occur, given that event B has occurred. In mathematical notation, the probability of A given B is expressed as P(A|B).
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Bayes’ Theorem enables us to determine the probability of both of two dependent events occurring when we know the conditional probability of one of the events occurring, given that the other has occurred. Bayes’ Theorem states the probability of A and B occurring is equal to the product of the probability of B and the conditional probability of A given B or the product of the probability of A and the conditional probability of B given A:
P(A and B) = P(B)* P(A|B) = P(A)*P(B|A).

This theorem works for both independent and dependent events, but for independent events, the result is equivalent to what is given by the multiplication rule: P(A and B) = P(B)*P(A). Why is this the case? When two events are independent, the occurrence of one has no effect on whether the other will occur, so the probability of the event taking place should be equal to the conditional probability of that event given that the other event has taken place. So for independent events A and B: P(A) = P(A|B) and P(B) = P(B|A). If one ever wishes to determine whether two events are independent, it is possible to do so by computing their individual probabilities and their conditional probabilities and seeing if the former equal the latter.

The following sample problem can illustrate the kinds of probability questions that Bayes’ Theorem can be used to answer. This particular problem is of my own invention, but the first actuarial exam (Exam P) has been known to have other problems of this sort, which are virtually identical in format.

Problem: A company has four kinds of machines: A, B, C, and D. The probabilities that a machine of a given type will fail on a certain day are: 0.02 for A, 0.03 for B, 0.05 for C, and 0.15 for D. 10% of a company’s machines are of type A, 25% are of type B, 30% are of type C, and 35% are of type D. Given that a machine has failed on a certain day, what is the probability of the machine being of type B?

Solution: First, let us designate the event of a machine’s failure with the letter F. Thus, from the given information in the problem, P(A) = 0.10, P(B) = 0.25, P(C) = 0.3, and P(D) = 0.35. P(F|A) = 0.02, P(F|B) = 0.03, P(F|C) = 0.05, and P(F|D) = 0.15. We want to find P(B|F). By Bayes’ Theorem, P(B and F) = P(F)* P(B|F). We can transform this to

P(B|F) = P(B and F)/P(F). To solve this, we must determine P(B and F). By another application of Bayes’ Theorem, P(B and F) = P(B)* P(F|B) = 0.25*0.03 = 0.0075. Furthermore,

P(F) = P(A and F) + P(B and F) + P(C and F) + P(D and F)

P(F) = P(A)*P(F|A) + P(B)* P(F|B) + P(C)* P(F|C) + P(D)* P(F|D)

P(F) = 0.10*0.02 + 0.25*0.03 + 0.3*0.05 + 0.35*0.15 = 0.077So P(B|F) = P(B and F)/P(F) = 0.0075/0.077 = 15/154 or about 0.0974025974. Thus, if a machine has failed on a certain day, the probability that it is of type B is 15/154.

Ideas in Mathematics and Probability: Independent Events and Dependent Events (2007) – Article by G. Stolyarov II

Ideas in Mathematics and Probability: Independent Events and Dependent Events (2007) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 18, 2014
******************************
Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 3,300 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
***
~ G. Stolyarov II, July 18, 2014
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This essay discusses independent and dependent events and their role in probability theory and analyses.
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Let us consider two events, A and B. If the probability that event A occurs has no effect on the probability that event B occurs, then A and B are independent events. A classic example of independent events is two tosses of the same fair coin. If a coin lands heads once, this has no influence on whether it will land heads again. The probability of landing heads on any given toss of the fair coin is ½.

It is a common error to presume that once a coin has landed heads for a number of times, this increases its probability of landing tails the next time it is tossed. If each toss is an independent event, this cannot be the case. Even if the coin has landed heads for 1000 consecutive times previously, its probability of landing heads the next time it is tossed is ½.

With two dependent events, on the other hand, the outcome of the first event affects the probability of the second. A classic example of such events would be drawing cards from a deck without replacement. A standard 52-card deck contains 4 aces. On the first draw, the probability of choosing an ace is 4/52 or 1/13. However, the probability of choosing an ace on the second draw will depend on whether an ace was selected on the first draw.

If an ace was selected on the first draw, there are 51 cards left to choose from, 3 of which are aces. So the probability of selecting an ace on the second draw is 3/51. But if an ace was not selected on the first draw, there are 4 aces left among 51 cards, so the probability of selecting an ace on the second draw is 4/51. Clearly, then, multiple drawings of cards from a deck without replacement are dependent events.

With any number of independent events, it is possible to use the multiplication rule to know the probability of some number of these events occurring. For example, if A, B, and C are independent events, and P(A) — the probability of A — is 1/3, P(B) is 3/5, and P(C) is 4/11, then the probability that A and B will occur is P(A)*P(B) = (1/3)(3/5) = 1/5. The probability that A, B, and C will occur is P(A)*P(B)*P(C) = (1/3)(3/5)(4/11) = 4/55.

It is important to only use the multiplication rule for independent events. With dependent events, the computation of probabilities for multiple events is not so straightforward and depends on the specific situation and dependence relationship among events. But further explorations of the world of probability theory will acquaint one with methods of analyzing probabilities of multiple dependent events as well.

Collectivism is Ancient; Freedom, Reason, and Progress Are New (2010) – Article by G. Stolyarov II

Collectivism is Ancient; Freedom, Reason, and Progress Are New (2010) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
Originally Published April 23, 2010
as Part of Issue CCXLV of The Rational Argumentator
Republished July 18, 2014
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Note from the Author: This essay was originally published as part of Issue CCXLIV of The Rational Argumentator on April 23, 2010, using the Yahoo! Voices publishing platform. Because of the imminent closure of Yahoo! Voices, the essay is now being made directly available on The Rational Argumentator. The arguments in it continue to be relevant to discussions regarding reason, individualism, and liberty, and therefore it is fitting for this publication to provide these arguments a fresh presence.
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~ G. Stolyarov II, July 18, 2014
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Irrational, illiberal collectivism had its beginnings along with the beginnings of the human species. How else could it be the case that the overwhelming majority of the history of our species took place with virtually no progress whatsoever? Indeed, even the advent of basic agriculture and the written word occurred quite late in our history, considering that humans virtually identical in body and mind to our contemporaries appeared circa 50000 B.C.E., whereas the beginnings of agriculture occurred circa 10000 B.C.E., and writing emerged even later. How could this have been the case? Surely, with the proper freedom-respecting, individualistic mindsets and institutions, our remote ancestors could have accomplished noticeable progress every generation. Instead, about 80% of human history passed without any progress whatsoever, and another 19% passed with minimal progress and centuries where previous progress had been reversed and nearly eliminated (e.g., the Dark Ages and the 14th Century in Europe, and the era of Mongol conquests in Russia, the Middle East, and the Far East). And yet superbly intelligent, capable people existed in every generation, and would, if placed in our time or the recent past, have become great innovators.

The sensible explanation of these otherwise perplexing facts is that absolutely stifling mindsets afflicted the majority of human societies during the majority of history. Although they left no written records, most Paleolithic hunter-gatherer societies can be safely assumed to have held ultra-tribalist, collectivist views of the world – in addition to a persistently animistic, superstitious view of the inanimate world and a violently intense xenophobia. Moreover, in a small nomadic tribe, an “us versus them” attitude would have been quite easy and tempting to adopt; one relied on one’s fellow tribesmen to protect one against aggression by other humans, wild animals, and myriad miscellaneous perils. Departure from the norms and societal structures of the tribe, through either material or intellectual innovation, would likely have resulted in ostracism from the tribe or worse.

What is relatively new in human history – dating back to ancient Greece – is early true liberal, pro-freedom thinking; I still believe that we are in the early stages of the development of liberal thought, considering how illiberal the majority of human societies today are and how the majority of human progress (and, indeed, of human sanity altogether) can be attributed to only a handful of forward-thinking individuals. Free the human mind just a little, give even a few reasonably intelligent people just a small amount of material and intellectual space to decide how to live and to think – and you get all that human civilization has accomplished thus far. Free humans completely, and astonishing accomplishments would be possible, even from the “average” person.

Of course, the reverse is possible, too: such a severe degeneration of human thinking and institutions as to produce a relapse into barbarism. This would be the worst, most tragic outcome to befall mankind.

Particular, Principled, Context-Specific Justice (2010) – Article by G. Stolyarov II

Particular, Principled, Context-Specific Justice (2010) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
Originally Published April 11, 2010
as Part of Issue CCXLIV of The Rational Argumentator
Republished July 18, 2014
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Note from the Author: This essay was originally published as part of Issue CCXLIV of The Rational Argumentator on April 11, 2010, using the Yahoo! Voices publishing platform. Because of the imminent closure of Yahoo! Voices, the essay is now being made directly available on The Rational Argumentator. The arguments in it continue to be relevant to discussions regarding justice, natural law, and a merit-based society, and therefore it is fitting for this publication to provide these arguments a fresh presence.
~ G. Stolyarov II, July 18, 2014
***

Here, I will briefly outline the fundamental features of a new approach to justice that departs radically from the egalitarian view typical of our era. A departure from egalitarianism may appear to some to be reactionary – with the alternative being a reversion to the older, class-based systems of justice, where different individuals were afforded different treatments on the basis of membership in rather arbitrarily defined groups. However, the approach of particular, principled, context-specific justice is in fact highly progressive in that it rejects the collectivism and suffering of innocents inherent in both class-based and egalitarian systems of justice. If we use an analogy to medical evolution, class-based justice could be compared to the pre-scientific treatments of bleeding and leeches; egalitarian justice could be compared to a mass-marketed pill that helps some people, but not in all ways, and also causes substantial adverse side effects in others; particular and context-specific justice is like an army of tiny nano-machines, repairing specific instances of bodily damage cell by cell without damaging healthy tissues. What nano-medicine promises to accomplish for the principle of health, particular and context-specific justice can accomplish in advancing the principle of merit.

The best way of encapsulating particular, principled, context-specific justice is to say that justice should not be blind. Indeed, justice should see as much as possible about the situation which is being judged and use all relevant information to arrive at a remedy specifically tailored to that situation. Any simplification of this principle – including the invocation of group- or class-based stereotypes, inflexible norms, and binding precedents – leads a departure from the just outcome.

It is a necessary component of justice that no innocent person should be harmed by its application – and that no guilty person should be harmed by it beyond the extent specifically warranted by his guilt. To hold otherwise is to embrace not justice, but pseudo-pragmatic trade-offs, where the suffering of some innocents is weighed against the perceived greater or lesser suffering of other innocents. To enforce such trade-offs is not within the legitimate power of any human being, nor is it necessitated by the natures of things or genuine practicality.

Unfortunately, “justice” as conceived by many of our contemporaries – egalitarian justice, or, phrased less generously, one-size-fits-all justice – necessitates the making of trade-offs that harm innocent people in virtually every case. Egalitarian justice is based on the premise that all persons must be treated in the same manner, irrespective of their individual qualities, context, and the consequences of a particular treatment. The uniform treatment is intended to produce the “greatest good for the greatest number” – but it often results in the lowering of the manner in which people are actually treated to a mediocre level, or even to the level of the lowest common denominator. Egalitarian justice typically imposes mandates or prohibitions deemed to improve the position of the “average person” or the majority of people; in reality, such impositions hamstring the above-average individuals while providing only slight, if any, benefits for the others. Indeed, many egalitarians, after the failure of their attempts to elevate the majority through one-size-fits-all measures, resort to insisting that everyone must “share the burden” equally – i.e., suffer by the same amount in situations where, before, no suffering was necessary.

Egalitarian justice is misguided, because it is premised on the idea that justice applies fundamentally to collectives of people, as opposed to individuals – who are the basic units where human perception, thinking, creation, and decision-making are concerned. Egalitarian justice seeks – at least in its best-intentioned variant – to bring about societal improvement by imposing the same rules and treatments upon all of society.

By contrast, reason and morality – natural law – require that every individual be treated in accordance with the merits or demerits of that individual’s own actions. Individuals who act rationally and morally, to the genuine benefit of themselves and others, should be rewarded, and individuals who act detrimentally – by harming others or themselves – should suffer the naturally ensuing adverse consequences of their actions. Individuals who harm only themselves are already punished sufficiently by the harm they inflict; there is no need for an external entity to disproportionately magnify that harm. However, individuals whose actions also adversely affect innocent others will not always be thwarted in time to prevent the harm. Hence arises the need for societal institutions, external to a particular situation where harm to others can be caused, to prevent or remedy such harm. This is the function of justice.

Thus, to have true justice in a particular case, it is clear that the harm to innocent persons in that case must be prevented or remedied – and, just as importantly, no harm must be caused by the process of justice itself. This is impossible to accomplish without a finely targeted approach: one that attempts to fathom the particular situation in all its relevant details, to establish the harm being committed or threatened, and to develop a way of neutralizing that harm which will punish only the guilty, and only in proportion to their guilt. A simplistic rule, conceived to apply to a myriad of diverse cases, apart from the context of these particular cases, is not adequate to this task.

It may seem at first glance that the attainment of particular justice precludes the application of any principles whatsoever. After all, are principles not themselves general rules that are developed apart from any given particular case? Yet it is not possible to reach a non-arbitrary decision on any matter without having some standards on which to base that decision. And there are indeed standards which are universally applicable to all human beings – derivable from the desirability of human life and flourishing, and from the mechanisms by which such values can be preserved and expanded. Among these standards are the natural rights of all humans: the right to act in the furtherance of one’s life, the right to acquire and keep property by naturally legitimate means, the right to interact with consenting others, and the right to be free from aggression, expropriation, and unwarranted punishment.

Indeed, the very definition of what constitutes an unjust harm is dependent on the principles of natural law. For instance, it is not an unjust harm if a person becomes displaced from a particular field of work because technological advances by others rendered that field of work obsolete. Because the technological advances and their creators did not rob, injure, kill, threaten, or defraud anyone, they are in complete accord with justice. The people displaced from their jobs may be worse off temporarily, but they always have an opportunity to retrain themselves in a society that respects their rights. Moreover, because they did not have the right to hold a particular job in the first place – as such a job was the result of an agreement that requires the continuing consent of two parties – they lost nothing to which they were entitled. On the other hand, it may be salutary from the standpoint of voluntary, private morality for the employers of such displaced individuals to offer to support their re-training or to aid them in finding alternate jobs.

But the universal standards of natural law are not the standards used by egalitarian justice; rather, egalitarianism tends to develop highly concrete criteria that are applied irrespective of whether they satisfy the abstract universal principles of justice. According to the most widespread embodiments of this philosophy, everyone must be subjected to the same minutiae, in an attempt to approximate just outcomes on a society-wide level. By contrast, in true justice, universal principles are not tied to any specific set of objects, procedures, or prescriptions for concrete behaviors. Rather, each principle can only be properly applied by considering the context in which it is relevant. To say, for instance, that honesty is a universal principle does not translate into concrete mandates or prohibitions for every situation; while it may not be justified to lie in most situations, in some – including situations where an aggressor demands the truth so as to inflict harm on its basis – lying may be morally necessary. It is an unfortunate characteristic of the egalitarian thinking of our era that abstract principles often become reified into a laundry list of byzantine particulars, whose “uniform” imposition then becomes seen as synonymous with justice – to the detriment of the very principles of justice that were supposed to be advanced in the first place.

While universal moral principles do not change, there are two important aspects of the world that do change continually: (1) our knowledge and understanding of these principles and (2) the specific concretes of our existence, to which those principles need to be applied. Moral philosophy is, and should be, an ever-evolving discipline, not because there are no truths to be found, but because no one can claim to have found all the truths or to have developed all of the facets of any true idea. At the same time, new discoveries, inventions, and societal changes raise new questions and dilemmas regarding how moral principles ought to be applied. The attempt of egalitarianism to set uniform concrete norms that apply to all people in all cases stands in defiance of the dynamic context in which we live and strive to fathom justice and reality. Egalitarianism, even based on the best effort to integrate the most advanced knowledge and the most rational thinking currently available, freezes justice in time and cuts off the prospects for a variety of innovative approaches that often occur simultaneously with one another within different subsets of any given society.

Because of the complexity of individual circumstances, every concrete norm, applied too broadly, will harm some innocent people. Particular, principled, context-specific justice would avoid this problem by being flexible with respect to concrete norms. For this, the discretion of the entity that dispenses justice is of foremost importance. Without discretion, no deviation from a concrete norm is possible – and, consequently, there is no way to avert innocent suffering. Discretion by a reasonable intelligent person, however, can avoid all of the obvious harms of a given norm – and the most competent and scrupulous dispensers of justice can even structure remedies so as to avoid subtle and indirect harms. Discretion should not be unlimited, and its exercise should be allowed in such a manner as would not extend the authority of the dispenser of justice beyond its intended sphere. Moreover, every care should be taken to prevent such discretion from resulting in draconian outcomes. But the limits imposed upon discretion should never prevent contextually warranted leniency or experimentation with remedies that are more palatable to all parties involved than those suggested by precedent or tradition.

To apply a general principle properly to a given situation, knowledge of the situation is crucial. The difference between true justice and egalitarian justice is akin to the difference between two applications of the principle of healthy eating: one approach makes choices regarding the nutritional value of every particular item of food one encounters, in the context in which one encounters it, while the other approach develops in advance a “diet” that consists of context-independent prohibitions on certain foods and requirements for certain other foods. Following a sub-optimal strategy for healthy eating may still make one healthier on net and, in that case, is not perilous. But this is because the individual is the basic moral unit; actions that benefit an individual on net while causing some discomfort, inconvenience, or inefficiency to that individual are therefore acceptable. But there can be no legitimate consideration of what benefits “society on net” which disregards harms to any individuals that occur in the process. Society is not a moral unit, and harms to its “components” cannot be brushed aside as necessary to advance an ostensibly greater goal.

Of course, for particular, principles-based justice to be applied to any systematic extent, both prevailing legal systems and moral understandings would need to change; the latter change would most likely need to precede the former, at least among the people who can affect the legal systems. Egalitarian justice attempts to treat particular situations independently of context or consequences; such treatment cannot be reconciled with the principles of justice. True justice encounters reality directly and infuses into it improvements – protections for the innocent, punishments for the guilty, and a closer approximation of a society where natural law is obeyed and the principle of merit is reflected.

Read other articles in The Rational Argumentator’s Issue CCXLIV.

Ideas in Mathematics and Probability: The Uniform Distribution (2007) – Article by G. Stolyarov II

Ideas in Mathematics and Probability: The Uniform Distribution (2007) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 17, 2014
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Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 4,800 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
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~ G. Stolyarov II, July 17, 2014
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The uniform distribution is alternately known as the de Moivre distribution, in honor of the French mathematician Abraham de Moivre (1667-1754) who introduced it to probability theory. The fundamental assumption behind the uniform distribution is that none of the possible outcomes is more or less likely than any other. The uniform distribution applies to continuous random variables, i.e., variables that can assume any values within a specified range.***

Let us say that a given random variable X is uniformly distributed over the interval from a to b. That is, the smallest value X can assume is a and the largest value it can assume is b. To determine the probability density function (pdf) of such a random variable, we need only remember that the total area under the graph of the pdf must equal 1. Since the pdf is constant throughout the interval on which X can assume values, the area underneath its graph is that of a rectangle — which can be determined by multiplying its base by its height. But we know the base of the rectangle to be (b-a), the width of the interval over which the random variable is distributed, and its area to be 1. Thus, the height of the rectangle must be 1/(b-a), which is also the probability density function of a uniform random variable over the region from a to b.

What is the mean of a uniformly distributed random variable? It is, conveniently, the halfway point of the interval from a to b, since half of the entire area under the graph of the pdf will be to the right of such a midway point, and half will be to the left. So the mean or mathematical expectation of a uniformly distributed random variable is (b-a)/2.

It is also possible to arrive at a convenient formula for the variance of such a uniform variable. Let us consider the following equation used for determining variance:

Var(X) = E(X2) – E(X)2 , where X is our uniformly distributed random variable.

We already know that E(X) = (b-a)/2, so E(X)2 must equal (b-a)2/4. To find E(X2), we can use the definition of such an expectation as the definite integral of x2*f(x) evaluated from b to a, where f(x) is the pdf of our random variable. We already know that f(x) = 1/(b-a); so E(X2) is equal to the integral of x2/(b-a), or x3/3(b-a), evaluated from b to a, which becomes (b-a)3/3(b-a), or (b-a)2/3.

Thus, Var(X) = E(X2) – E(X)2 = (b-a)2/3 – (b-a)2/4 = (b-a)2/12, which is the variance for any uniformly distributed random variable.

Ideas in Mathematics and Probability: Covariance of Random Variables (2007) – Article by G. Stolyarov II

Ideas in Mathematics and Probability: Covariance of Random Variables (2007) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 17, 2014
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Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 5,200 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
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~ G. Stolyarov II, July 17, 2014
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Analyzing the variances of dependent variables and the sums of those variances is an essential aspect of statistics and actuarial science. The concept of covariance is an indispensable tool for such analysis.
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Let us assume that there are two random variables, X and Y. We can call the mathematical expectations of each of these variables E(X) and E(Y) respectively, and their variances Var(X) and Var(Y) respectively. What do we do when we want to find the variance of the sum of the random variables, X+Y? If X and Y are independent variables, this is easy to determine; in that case, simple addition accomplishes the task: Var(X+Y) = Var(X) + Var(Y).

But what if X and Y are dependent? Then the variance of the sum most often does not simply equal sum of the variances. Instead, the idea of covariance must be applied to the analysis. We shall denote the covariance of X and Y as Cov(X, Y).

Two crucial formulas are needed in order to deal effectively with the covariance concept:

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)

Cov(X, Y) = E(XY) – E(X)E(Y)

We note that these formulas work for both independent and dependent variables. For independent variables, Var(X+Y) = Var(X) + Var(Y), so Cov(X, Y) = 0. Similarly, for independent variables, E(XY) = E(X)E(Y), so Cov(X, Y) = 0.

This leads us to the general insight that the covariance of independent variables is equal to zero. Indeed, this makes conceptual sense as well. The covariance of two variables is a tool that tells us how much of an effect the variation in one of the variables has on the other variable. If two variables are independent, what happens to one has no effect on the other, so the variables’ covariance must be zero.

Covariances can be positive or negative, and the sign of the covariance can give useful information about the kind of relationship that exists between the random variables in question. If the covariance is positive, then there exists a direct relationship between two random variables; an increase in the values of one tends to also increase the values of the other. If the covariance is negative, then there exists an inverse relationship between two random variables; an increase in the values of one tends to decrease the values of the other, and vice versa.

In some problems involving covariance, it is possible to work from even the most basic information to determine the solution. When given random variables X and Y, if one can compute E(X), E(Y), E(X2), E(Y2), and E(XY), one will have all the data necessary to solve for Cov(X, Y) and Var(X+Y). From the way each random variable is defined, one can derive the mathematical expectations above and use them to arrive at the covariance and the variance of the sums for the two variables.

Concepts in Probability Theory: Mathematical Expectation (2007) – Article by G. Stolyarov II

Concepts in Probability Theory: Mathematical Expectation (2007) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 17, 2014
******************************
Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 10,000 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
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~ G. Stolyarov II, July 17, 2014
***

The idea of expectation is crucial to probability theory and its applications. As one who has successfully passed actuarial Exam P on Probability, I would like to educate the general public about this interesting and useful mathematical concept.

The idea of expectation relies on some set of possible outcomes, each of which has a known probability and a known, quantifiable payoff — which can be positive or negative. Let us presume that we are playing a game called X with possible outcomes A, B, and C on a given turn. Each of these outcomes has a known probability P(A), P(B), and P(C) respectively. Each of the outcomes is associated with set payoffs a, b, and c, respectively. How much can one expect to win on an average turn of playing this game?

This is where the concept of expectation comes in. There is a P(A) probability of getting payoff a, a P(B) probability of getting payoff b, and a P(C) probability of getting payoff c. The expectation for a given turn of game X, E(X) is equal to the sum of the products of the probabilities for each given event and the payoffs for that event. So, in this case,

E(X) = a*P(A) + b*P(B) + c*P(C).

Now let us substitute some numbers to see how this concept could be applied. Let us say that event A has a probability of 0.45 of occurring, and if A occurs, you win $50. B has probability of 0.15 of occurring, and if B occurs, you lose $5. C has a probability of 0.4 of occurring, and if C occurs, you lose $60. Should you play this game? Let us find out.

E(X) = a*P(A) + b*P(B) + c*P(C). Substituting the values given above, we find that E(X) = 50*0.45 + (-5)(0.15) + (-60)(0.40) = -2.25. So, on an average turn of the game, you can be expected to lose about $2.25.

Note that this corresponds to neither of the three possible outcomes A, B, and C. But it does inform you of the kinds of results that you will approach if you play this game for a large number of turns. The Law of Large Numbers implies that the more times you play such a game, the more likely your average payoff per turn is to approach the expected value E(X). So if you play the game for 5 turns, you can be expected to lose 5*2.25 = $11.25, but you will likely experience some deviation from this in the real world. Yet if you play the game for 100 turns, you can be expected to lose 100*2.25 = $225, and your real-world outcome will most likely be quite close to this expected value.

In its more general form for some random variable X, the expectation of X or E(X) can be phrased as the sum of the products of all the possible outcomes x and their probabilities p(x). In mathematical notation, E(X) = sigma(x*p(x)) for all values of x. You can apply this formula to any discrete random variable, i. e., a random variable which assumes only a finite set of particular values.

For a continuous random variable Y, the mathematical expectation is equal to the integral of y*f(y) over the region on which the variable is defined. The function f(y) is called the probability density function of Y; its height over a given domain on a graph can be an indication the likelihood of the random variable assuming values over that domain.

Epsilon-N Proof of a Limit of a Sequence: Lim[2n/(3n+2)] = 2/3 – Article by G. Stolyarov II

Epsilon-N Proof of a Limit of a Sequence: Lim[2n/(3n+2)] = 2/3 – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 12, 2014
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Note from the Author: This proof was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.  The article earned over 17,250 page views on Associated Content/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.  ***
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~ G. Stolyarov II, July 12, 2014
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This is the first in a series of formal mathematical proofs I intend to present in order to assist anybody who has ventured into the challenging but fascinating world of advanced calculus and real analysis. I start with a fairly basic proof: the limit of the nth term of a sequence as n becomes increasingly large. This is an epsilon-N proof, which uses the following definition: lim(n approaches ∞)xn= L iff for each real number ε>0, there exists a positive integer N(ε) such that if n ≥ N(ε), then │xn-L│< ε, i.e., L- ε < xn< L+ ε.

The epsilon-N proof has two component steps; first, we assume that our ε>0 is given and work backward to transform the inequality │xn-L│< ε in order to find an appropriate value of N(ε) that corresponds to a given value of ε. Then, using the value of N(ε) we found, we work forward to show that if n ≥ N(ε), then │xn-L│< ε.

The proof also uses the Archimedean Property, which states that the set of positive integers is not bounded above. There are actually four equivalent conditions that are known as the Archimedean Property:

1. If a,b are in R, a>0, b>0, then there is a positive integer n such that na>b

2. The set of positive integers is not bounded above.

3. For each real number x, there exists an integer n such that n ≤x < n+1

4. For each positive real number x, there exists a positive integer n such that 1/n ≤ x

Prove: lim(n approaches ∞)[2n/(3n+2)] = 2/3

Proof: Let ε>0 be given. Find N(ε)є Z+ such that if N(ε) < n, then │2n/(3n+2)-2/3│< ε.

Working backward to transform the inequality: │2n/(3n+2)-2/3│< ε

│6n/[3(3n+2)]-2(3n+2)/[3(3n+2)]│< ε

│[6n-2(3n+2)]/[3(3n+2)]│< ε

│[6n-6n-4]/[3(3n+2)]│< ε

│-4/[3(3n+2)]│< ε

Since ε>0, (3n+2)>0, the above inequality is the same as

4/[3(3n+2)] < ε

4/(3ε) < (3n+2)

4/(3ε)- 2 < 3n

4/(9ε)- 2/3 < n

Now I work forward to prove the original result:

Let N(ε)є Z+ э 4/(9ε)- 2/3 < N(ε).

Since 4/(9ε)- 2/3 is a real number, by the Archimedean Property it must be the case that some integer exists which is greater than that real number-since the set of positive integers is not bounded above. Here, we call that integer N(ε).

For all n>N, if 4/(9ε)- 2/3 < N < n, then:

4/(3ε)- 2 < 3n

4/(3ε) < (3n+2)

4/[3(3n+2)] < ε

│-4/[3(3n+2)]│< ε

│6n/[3(3n+2)]-2(3n+2)/[3(3n+2)]│< ε

│2n/(3n+2)-2/3│< ε

I have demonstrated the above inequality, which is sufficient to demonstrate that lim(n approaches ∞)[2n/(3n+2)] = 2/3.

I have hence proved what was desired. Another way to express that the desired objective has been obtained (which I shall use in future proofs of this sort) is the abbreviation

“Q. E. D.” of the Latin “Quod Erat Demonstraterum,” which means “that which was to be demonstrated.”

Conciseness on Actuarial Essay Exams (2010) – Article by G. Stolyarov II

Conciseness on Actuarial Essay Exams (2010) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
July 11, 2014
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This essay, originally written and published on Associated Content/Yahoo! Voices in 2010, has helped many actuarial candidates to prepare for essay exams. I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time. 

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~ G. Stolyarov II, July 11, 2014
***

Upper-level actuarial exams are in essay format, requiring both conceptual discussions and extensive calculations to answer 30-50 questions within a 4-hour timeframe. Even for highly knowledgeable candidates, the sheer time constraints of the exam render it difficult to respond both thoroughly and within the allotted time. Thus, conciseness, without compromising the communication of understanding, becomes a priority.

The following ideas for condensing actuarial exam responses were derived from reviewing past sample answers released by the Casualty Actuarial Society. By understanding which answers received full credit while employing certain shortcuts of presentation, I was able to arrive at ideas that, when used in combination, may save candidates tens of minutes on the exam. This time can be devoted to reviewing one’s answers or to answering more questions than would otherwise be possible. While, as an outsider to the grading process, I can offer no guarantees, I plan to personally use these approaches to the extent they are relevant.

If other actuarial candidates have additional ideas to facilitate concise, effective exam answers, I welcome their input.

1. Common Abbreviations

Many insurance concepts have generally known abbreviations that do not need to be defined unless an explicit definition is requested. On most questions, it would be safe, for instance, to assume that the grader will know what ALAE, ULAE, IBNR, IBNER, PDLD, GAAP, SAP, and terms of similarly common usage stand for.

There are also commonly used general abbreviations, such as “&” for “and”, “b/c” for “because”, “w.r.t” for “with respect to”.

2. Uncommon Abbreviations

It is also possible to define uncommon (even self-invented) abbreviations once, and use them thereafter. For instance, one could refer to “the Bornhuetter-Ferguson method (B-F)” and then subsequently state that “B-F assumes…” or “according to B-F…”.

As long as the grader understands what the abbreviations mean in the context of one’s answer, full credit should be possible.

Here is a non-exhaustive list of abbreviations that may be useful for the 2010 CAS Exam 6 in particular:

B-F: Bornhuetter-Ferguson method
B-S: Berquist-Sherman method
Cat.: Catastrophe
CL: Chain ladder
C-N: Conger-Nolibos generalized approach
Co-part.: Co-participation
Cov.: Coverage
Dev.: Developed or Development (depending on context)
G-B: Gunnar Benktander method
GL: General liability
Inc.: Incurred
Lim.: Limit
M-A: Mango-Allen adjustment
O/S: Outstanding
QS: Quota share
S-B: Stanard-Bühlmann method (“CC” for “Cape Cod method” can also be used).
SS: Surplus share (definitely define that one before using!)
U/W: Underwriting
WC: Workers’ compensation
XOL: Excess-of-loss

3. Shortcuts for Repetitive Calculations

It is possible to save time in cases where one must perform multiple calculations using the same basic formula or approach. Instead of displaying every single calculation, one could simply display (1) the formula used, (2) a sample calculation, and (3) the final results of all the other calculations.

As a non-insurance illustration, suppose you were faced with the following problem:

Find the hypotenuses of the right triangles with the following legs:
(3, 4)
(8, 15)
(9, 40)
(20, 21)

The long way to answer would be to display all four calculations. A shorter way would be the following:
Formula: c = √(b2 + a2)
Sample: √(32 + 42) = 5
Answers: 5, 17, 41, 29

The only possible drawback to this approach is that, if one makes a mistake in a calculation other than the sample calculation, the specific nature of the mistake will not be visible to the grader. It is possible that the grader will simply assume a mechanical error and therefore be lenient in giving partial credit, because the formula and sample calculation demonstrate an understanding of the ideas involved. However, it is impossible to offer any guarantees here.

4. Alternatives to Complete Sentences

While, in academic settings, answering in complete sentences is a requirement for most exams and assignments, the sheer time pressure of an actuarial essay exam renders this approach sub-optimal. A review of past exam answers that have received full credit suggests that graders do not remove points from responses that convey a candidate’s knowledge of the tested content but are written in sentence fragments.

Instead of writing in complete sentences, there are many possible alternative ways of answering, depending on the question. For instance, a question asking the candidate to compare and contrast certain aspects of Method X and Method Y might be answered as follows:

Method X: (List features of method)
Method Y: (List features of method, preferably using language parallel to what was used for Method X.)

Using a bulleted or numbered list to answer some questions may not only save time but may make it easier for the grader to identify the substance of the answer.

Chains of causation or implication may be expressed via an “→” symbol (e.g., “Writing new business → acquisition expense recognized immediately, premiums earned over time → decline in policyholders’ surplus → need for surplus relief.”

It is also acceptable to omit certain articles and to omit stating the premise of the question in the answer’s first sentence, as long as the meaning is clear. Furthermore, some instances of expressions like “that”, “then”, and “in order” may be omitted without compromising the answer’s intent.

As an illustration, I present two ways of answering my Problem S6-9-3(b): “What effect should be removed in order to evaluate development patterns correctly (Statement of Principles, p. 16)?”

Complete-sentence answer (my original): “The effect of discounting should be removed in order to evaluate development patterns correctly. If a reserve is established as a present value of future costs, then upward development may occur simply as a result of paying claims, and this may send a misleading signal.”

Condensed answer: “Effect of discounting should be removed. If reserve is set as present value of future costs, upward development may occur simply as result of paying claims → misleading signal may result.”

Again, I welcome input on these ideas and other ideas for facilitating conciseness on actuarial essay exams.