java physics

Acceleration is the time rate at which a velocity is changing. Because velocity has both magnitude and direction, it is called a vector quantity; acceleration is also a vector quantity and must account for changes in both the magnitude and direction of a velocity. The velocity of a point or an object moving on a straight path can change in magnitude only; on a curved path, it may or may not change in magnitude, but it will always change in direction. This condition means that the acceleration of a point moving on a curved path can never be zero.

If the velocity of a point moving on a straight path is increasing (i.e., if the speed, which is the magnitude of the velocity, is increasing), the acceleration vector will have the same direction as the velocity vector. If the velocity is decreasing (that is, the point or object is decelerating), the acceleration vector will point in the opposite direction. The average acceleration during a time interval is equal to the total change in the velocity during the interval divided by the time interval. The acceleration at any instant is equal to the limit of the ratio of the velocity change to the length of the time interval, as the time interval approaches zero.

When a point moves on a curved path, the component of the acceleration that results from the change in the direction of the velocity vector is perpendicular to the velocity vector and is directed inward, to the concave side of the path; its magnitude is given by the square of the velocity divided by the radius of curvature r of the path: v²/r. The change in the magnitude of v may be represented by another vector (that is, a second component of the acceleration) collinear with v and in the same direction if v is increasing and the opposite direction if v is decreasing. If velocity is stated in meters per second, acceleration will be stated in meters per second per second.

Excerpt from the Encyclopedia Britannica without permission.

Excerpt from the Encyclopedia Britannica without permission.

Metaphysics is the science that seeks to define what is ultimately real as opposed to what is merely apparent.

The contrast between appearance and reality, however, is by no means peculiar to metaphysics. In everyday life people distinguish between the real size of the Sun and its apparent size, or again between the real color of an object (when seen in standard conditions) and its apparent color (nonstandard conditions). A cloud appears to consist of some white, fleecy substance, although in reality it is a concentration of drops of water. In general, men are often (though not invariably) inclined to allow that the scientist knows the real constitution of things as opposed to the surface aspects with which ordinary men are familiar. It will not suffice to define metaphysics as knowledge of reality as opposed to appearance; scientists, too, claim to know reality as opposed to appearance, and there is a general tendency to concede their claim.

It seems that there are at least two components in the metaphysical conception of reality. One characteristic, which has already been illustrated by Plato, is that reality is genuine as opposed to deceptive. The ultimate realities that the metaphysician seeks to know are precisely things as they are--simple and not variegated, exempt from change and therefore stable objects of knowledge. Plato's own assumption of this position perhaps reflects certain confusions about the knowability of things that change; one should not, however, on that ground exclude this aspect of the concept of reality from metaphysical thought in general. Ultimate reality, whatever else it is, is genuine as opposed to sham.

Second, and perhaps most important, reality for the metaphysician is intelligible as opposed to opaque. Appearances are not only deceptive and derivative, they also make no sense when taken at their own level. To arrive at what is ultimately real is to produce an account of the facts that does them full justice. The assumption is, of course, that one cannot explain things satisfactorily if one remains within the world of common sense, or even if one advances from that world to embrace the concepts of science. One or the other of these levels of explanation may suffice to produce a sort of local sense that is enough for practical purposes or that forms an adequate basis on which to make predictions. Practical reliability of this kind, however, is very different from theoretical satisfaction; the task of the metaphysician is to challenge all assumptions and finally arrive at an account of the nature of things that is fully coherent and fully thought-out.

Excerpt from the Encyclopedia Britannica without permission.

Calculus is the branch of mathematical analysis concerned with the rates of change of continuous functions as their arguments change. Two men are now credited with discovering calculus, Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany. For almost a century, development of the subject was inhibited by a bitter controversy over priority between supporters of Newton and those of Leibniz.

A basic concept of calculus is "limit," an idea applied by the early Greeks in geometry. Archimedes inscribed equilateral polygons in a circle. Upon increasing the number of sides, the areas of the polygons (which he could calculate) approach the area of the circle as a limit. Using this result together with a similar idea involving circumscribed polygons, he was able to find the area of the circle as r², in which r is the radius of the circle and (pi) is a constant that has a value between 3 1/7 and 3 10/71.

The area of an irregularly shaped plate also can be found by subdividing it into rectangles of equal width. If the number of rectangles is made larger and larger, the sum of their areas (found by multiplying base by height) approaches the required area as a limit. The same procedure can be used to find volumes of spheres, cones, and other solid objects. The beauty and importance of calculus is that it provides a systematic way for the exact calculation of many areas, volumes, and other quantities that were beyond the methods of the early Greeks.

Newton's discovery of calculus, legend says, may very well have been inspired by an apple falling from a tree. As an apple falls, it moves faster and faster; that is, it has not only a velocity but an acceleration. Newton expressed this mathematically by supposing that at any stage of its motion the apple drops a small additional distance s (delta s) during a brief additional time interval t (delta t). Then the velocity is very nearly equal to the distance s divided by the time t--i.e., s/t. The exact velocity v would be the limit of s/t as t gets closer and closer to zero or, as we say, approaches zero. That is,

The quantity ds/dt is called the derivative of s with respect to t, or the rate of change of s with respect to t. It is possible to think of ds and dt as numbers whose ratio ds/dt is equal to v; ds is called the differential of s, and dt the differential of t.

Just as velocity is the rate of change, or derivative, of the distance with respect to time, so the acceleration is the rate of change, or derivative, of the velocity with respect to time. Therefore a, the acceleration, would be

where v is the increase in velocity that occurs during the interval t. Since a is the derivative of v and v is the derivative of s, a is called the second derivative of s:

To find derivatives of s with respect to t, the dependence of s on t must be known; in other words, s must be expressed as a function of t. Usually this functional dependence is stated as a formula relating s and t. That part of calculus dealing with derivatives is called differential calculus.

Given s as a function of t, the derivative (that is, v) of s can be found. Conversely, if v is known it is possible to work backward to get s. This process of finding what is called the anti-derivative of v is begun by rewriting the equation v = ds/dt as ds = vdt. The quantity s is here regarded as the anti-differential of ds, denoted by a special symbol called an integral sign:

The last equation specifies s the integral of v with respect to t. That part of calculus dealing with integrals is called integral calculus. Applications of integral calculus involve finding the limit of a sum of many small quantities, such as the rectangular slices of an irregular plane figure.

Excerpt from the Encyclopedia Britannica without permission.

Chaos is where apparently random or unpredictable behaviour in systems governed by deterministic laws. A more accurate term, "deterministic chaos," suggests a paradox because it connects two notions that are familiar and commonly regarded as incompatible. The first is that of randomness or unpredictability, as in the trajectory of a molecule in a gas or in the voting choice of a particular individual from out of a population. In conventional analyses, randomness was considered more apparent than real, arising from ignorance of the many causes at work. In other words, it was commonly believed that the world is unpredictable because it is complicated. The second notion is that of deterministic motion, as that of a pendulum or a planet, which has been accepted since the time of Isaac Newton as exemplifying the success of science in rendering predictable that which is initially complex.

In recent decades, however, a diversity of systems have been studied that behave unpredictably despite their seeming simplicity and the fact that the forces involved are governed by well-understood physical laws. The common element in these systems is a very high degree of sensitivity to initial conditions and to the way in which they are set in motion. For example, the meteorologist Edward Lorenz discovered that a simple model of heat convection possesses intrinsic unpredictability, a circumstance he called the "butterfly effect," suggesting that the mere flapping of a butterfly's wing can change the weather. A more homely example is the pinball machine: the ball's movements are precisely governed by laws of gravitational rolling and elastic collisions--both fully understood--yet the final outcome is unpredictable.

In classical mechanics the behaviour of a dynamical system can be described geometrically as motion on an "attractor." The mathematics of classical mechanics effectively recognized three types of attractor: single points (characterizing steady states), closed loops (periodic cycles), and tori (combinations of several cycles). In the 1960s a new class of "strange attractors" was discovered by the American mathematician Stephen Smale. On strange attractors the dynamics is chaotic. Later it was recognized that strange attractors have detailed structure on all scales of magnification; a direct result of this recognition was the development of the concept of the fractal (q.v.; a class of complex geometric shapes that commonly exhibit the property of self-similarity), which led in turn to remarkable developments in computer graphics.

Applications of the mathematics of chaos are highly diverse, including the study of turbulent flow of fluids, irregularities in heartbeat, population dynamics, chemical reactions, plasma physics, and the motion of groups and clusters of stars.

Excerpt from the Encyclopedia Britannica without permission.

Ellipses:

An ellipse is a closed curve, the intersection of a right circular cone and a plane that is not parallel to the base, the axis, or an element of the cone. It may be defined as the path of a point moving in a plane so that the ratio of its distances from a fixed point (the focus) and a fixed straight line (the directrix) is a constant less than one. Any such path has this same property with respect to a second fixed point and a second fixed line, and ellipses often are regarded as having two foci and two directrixes. The ratio of distances, called the eccentricity, is the discriminant (q.v.; of a general equation that represents all the conic sections [see conic section]). Another definition of an ellipse is that it is the locus of points for which the sum of their distances from two fixed points (the foci) is constant. The smaller the distance between the foci, the smaller is the eccentricity and the more closely the ellipse resembles a circle.

A straight line drawn through the foci and extended to the curve in either direction is the major diameter (or major axis) of the ellipse. Perpendicular to the major axis through the centre, at the point on the major axis equidistant from the foci, is the minor axis. A line drawn through either focus parallel to the minor axis is a latus rectum (literally, "straight side").

The path of a heavenly body moving around another in a closed orbit in accordance with Newton's gravitational law is an ellipse (see Kepler's laws of planetary motion). In the solar system one focus of such a path about the Sun is the Sun itself.

For an ellipse the centre of which is at the origin and the axes of which are coincident with the x and y axes, the equation is x²/a² + y²/b² = 1. The length of the major diameter is 2a; the length of the minor diameter is 2b. If c is taken as the distance from the origin to the focus, then c² = a² - b², and the foci of the curve may be located when the major and minor diameters are known. The problem of finding an exact expression for the perimeter of an ellipse led to the development of elliptic functions, an important topic in mathematics and physics.

Excerpt from the Encyclopedia Britannica without permission.

Euclid (fl. c. 300 BC, Alexandria), the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements.

Life and work. Of Euclid's life it is known only that he taught at and founded a school at Alexandria in the time of Ptolemy I Soter, who reigned from 323 to 285/283 BC. Medieval translators and editors often confused him with the philosopher Eucleides of Megara, a contemporary of Plato about a century before, and therefore called him Megarensis. Writing in the 5th century AD, the Greek philosopher Proclus told the story of Euclid's reply to Ptolemy, who asked whether there was any shorter way in geometry than that of the Elements--"There is no royal road to geometry." Another anecdote relates that a student, probably in Alexandria, after learning the very first proposition in geometry, wanted to know what he would get by learning these things, whereupon Euclid called his slave and said, "Give him threepence since he must needs make gain by what he learns."

Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (5th century BC), not to be confused with the physician Hippocrates of Cos (flourished 400 BC). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle. The older elements were at once superseded by Euclid's and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own. He evidently altered the arrangement of the books, redistributed propositions among them and invented new proofs if the new order made the earlier proofs inapplicable. Thus, while Book X was mainly the work of the Pythagorean Theaetetus (flourished 369 BC), the proofs of several theorems in this book had to be changed in order to adapt them to the new definition of proportion developed by Eudoxus (q.v.). According to Proclus, Euclid incorporated into his work many discoveries of Eudoxus and Theaetetus. Most probably Books V and XII are the work of Eudoxus, X and XIII of Theaetetus. Book V expounds the very influential theory of proportion that is applicable to commensurable and incommensurable magnitudes alike (those whose ratios can be expressed as the quotient of two integers and those that cannot). The main theorems of Book XII state that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These theorems are certainly the work of Eudoxus, who proved them with his "method of exhaustion," by which he continuously subdivided a known magnitude until it approached the properties of an unknown. Book X deals with irrationals of different classes. Apart from some new proofs and additions, the contents of Book X are the work of Theaetetus; so is most of Book XIII, in which are described the five regular solids, earlier identified by the Pythagoreans. Euclid seems to have incorporated a finished treatise of Theaetetus on the regular solids into his Elements. Book VII, dealing with the foundations of arithmetic, is a self-consistent treatise, written most probably before 400 BC.

Excerpt from the Encyclopedia Britannica without permission.

Here is a list of everyday fallacies take from Peter A. Angeles Dictionary of Philosophy-- published by Barnes and Noble, copyright 1981.

Fallacy, classification of informal. Informal fallacies may be classified in a variety of ways. Three general categories: (a) Material fallacies have to do with the facts (the matter, the content) of the argument in question. Two subcategories of material fallacies are: (1) fallacies of evidence, which refer to arguments that do not provide the required factual support (ground, evidence) for their conclusions, and (2) fallacies of irrelevance (or relevance) which refer to arguments that have supporting statements that are irrelevant to the conclusion being asserted and therefore cannot establish the truth of that conclusion. (b) Linguistic fallacies have to do with defects in arguments such as ambiguity (in which careless shifts of meanings or linguistic imprecisions lead to erroneous conclusions), vagueness, incorrect use of words, lack of clarity, linguistic inconsistencies, circularities. (c) Fallacies of irrelevant emotional appeal have to do with affecting behavior (responses, attitudes). That is, arguments are presented in such a way as to appeal to one's prejudices, biases, loyalty, dedication,fear, guilt, and so on. They persuade, cajole, threaten, or confuse in order to win assent to an argument.

Fallacy, types of informal. Sometimes semi-formal or quasi-formal fallacies. The following is a list of 40 informal fallacies which is by no means eshaustive. No attempt has been made to subsume them under general categories such as Fallacies, Classification of Informal [which I will also include].

1. Black-and-white fallacy. Arguing (a) with the use of sharp ("black-and-white") distinctions despite any factual or theoretical support for them, or (b) by classifying any middle point between the extremes ("black-and-white") as one of the extremes. Examples: "If he is an atheist then he is a decent person." "He is either a conservative or a liberal." "He must not be peace-loving, since he participated in picketing the American embassy."

2. Fallacy of argumentum ad baculum (argument from power or force.) The Latin means "an argument according to the stick." "argument by means of the rod," "argument using force." Arguing to support the acceptance of an argument by a threat, or use of force. Reasoning is replaced by force, which results in the termination of logical argumentation, and elicits other kinds of behavior (such as fear, anger, reciprocal use of force, etc.).

3. Fallacy of argumentum ad hominem (argument against the man) [a personal favorite of mine]. The Latin means "argument to the man." (a) Arguing against, or rejecting a person's views by attacking or abusing his personality, character, motives, intentions, qualifications, etc. as opposed to providing evidence why the views are incorrect. Example: "What John said should not be believed because he was a Nazi sympathizer." [Well, there goes Heidegger.]

4. Fallacy of argumentum ad ignorantiam (argument from ignorance). The Latin means "argument to ignorance." (a) Arguing that something is true because no one has proved it to be false, or (b) arguing that something is false because no one has proved it to be true. Examples: (a) Spirits exist since no one has as yet proved that there are not any. (b) Spirits do not exist since no one has as yet proved their existence. Also called the appeal to ignorance: the lack of evidence (proof) for something is used to support its truth.

5. Fallacy of argumentum ad misericordiam (argument to pity). Arguing by appeal to pity in order to have some point accepted. Example: "I've got to have at least a B in this course, Professor Angeles. If I don't I won't stand a chance for medical school, and this is my last semester at the university." Also called the appeal to pity.

6. Fallacy of argumentum ad personam (appeal to personal interest). Arguing by appealing to the personal likes (preferences, prejudices, predispositions, etc.) of others in order to have an argument accepted.

7. Fallacy of argumentum as populum (argument to the people). Also the appeal to the gallery, appeal to the majority, appeal to what is popular, appeal to popular prejudice, appeal to the multitude, appeal to the mob instinct [appeal to the stupid, stinking masses]. Arguing in order to arouse an emotional, popular acceptance of an idea without resorting to logical justification of the idea. An appeal is made to such things as biases, prejudices, feelings, enthusiasms, attitudes of the multitude in order to evoke assent rather than to rationally support the idea.

8. Fallacy of argumentum ad verecundiam (argument to authority or to veneration) [another of my personal favorites]. (a) appealing to authority (including customs, traditions, institutions, etc.) in order to gain acceptance of a point at issue and/or (b) appealing to the feelings of reverence or respect we have of those in authority, or who are famous. Example: "I believe that the statement 'YOu cannot legislate morality' is true, because President Eisenhower said it."

9. Fallacy of accent. Sometimes clasified as ambiguity of accent. Arguing to conclusions from undue emphasis (accent, tone) upon certain words or statements. Classified as a fallacy of ambiguity whenever this anphasis creates an ambiguity or AMPHIBOLY in the words or statements used in an argument. Example: "The queen cannot but be praised." [also "We are free iff we could have done otherwise."-- as this statement is used by incompatibilists about free-will and determinism.]

10. Fallacy of accident. Also called by its Latin name a dicto simpliciter asd dictum secundum quid. (a) Applying a general rule or principle to a particular instance whose circumstances by "accident" do not allow the proper application of that generalization. Example: "It is a general truth that no one should lie. Therefore, no one should lie if a murderer at the point of a knife asks you for information you know would lead to a further murder." (b) The error in arumentation of applying a general statement to a situation to which it cannot, and was not necessarily intended to, be applied.

11. Fallacy of ambiguity. An argument that has at least one ambiguous word or statement from which a misleading or wrong conclusion is drawn.

12. Fallacy of amphiboly. Arguing to conclusions from statements that themselves are amphibolous-- ambiguous because of their syntax (grammatical construction). Sometimes classified as a fallacy of ambiguity.

13. Fallacy of begging the question. (a) Arriving at a conclusion from statements that themselves are questionable and hae to be proved but are assumed true. Example: The universe has a beginning. Every thing that has a beginning has a beginner. Therefore, the universe has a beginner called God. This assumes (begs the question) that the universe does indeed have a beginning and also that all things that have a beginning have a beginner. (b) Assuming the conclusion ar part of the conclusion in the premises of an argument. Sometimes called circular reasoning, vicious circularity, vicious circle fallacy [Continental Philosophy-- sorry, I just couldn't resist]. Example: "Everything has a cause. The universe is a thing. Therefore, the universe is a thing that has a cause." (c) Arguing in a circle. One statement is supported by reference to another statement which is itself supported by reference to the first statement [such as a coherentist account of knowledge/truth]. Example: "Aristocracy is the best form of government because the best form of government if that which has strong aristocratic leadership."

14. Fallacy of complex question (or loaded question). (a) Asking questions for which either a yes or no answer will incriminate the respondent. The desired answer is already tacitly assumed in the question and no qualification of the simple answer is allowed. Example: "Have you discontinued the use of opiates?" (b) Asking questions that are based on unstated attitudes or questionable (or unjustified) assumptions. These questions are often asked rhetorically of the respondent in such a way as to elicit an agreement with those attitudes or assumptions from others. Example: "How long are you going to put up with this brutality?"

15. Fallacy of composition. Arguing (a) that what is true of each part of a whole is also (necessarily) true of the whole itself, or (b) what is true of some parts is also (necessarily) true of the whole itself. Example: "Each member (or some members) of the team is married, therefore the team also has (must have) a wife." [A less silly example-- you promise me that you will come to Portland tomorrow, you also promise someone else that you will go to Detroit tomorrow. Now, you ought to be in Portland tomorrow, and you ought to be in Detroit tomorrow (because you ought to keep your promises). However, it does not follow that you ought to be in both Portland and Detroit tomorrow (because ought implies can).] Inferring that a collection has a certain characteristic merely on the basis that its parts have them erroneously proceeds from regarding the collection DISTRIBUTIVELY to regarding it COLLECTIVELY.

16. Fallacy of consensus gentium. Arguing that an idea is true on the basis (a) that the majority of people believe it and/or (b) that it has been universally held by all men at all times. Example: "God exists because all cultures hae had some concept of a God."

17. Fallacy of converse accident. Sometimes converse fallcy of accident. Also called by its Latin name a dicto secumdum quid ad dictum simpliciter. The error of generalizing from atypical or exceptional instances. Example: "A shot of warm brandy each night helps older people relax and sleep better. People in general ought to drink warm brandy to relieve their tension and sleep better."

18. Fallacy of division. Arguing that what is true of a whole is (a) also (necessarily) true of its parts and/or (b) also true of some of its parts. Example: "The community of Pacific Palisades is extremely wealthy. Therefore, every person living there is (must be) extremely wealthy (or therefor Adma, who lives there, must be extremely wealthy." Inferring that the parts of a collection have certain characteristic merely on the basis that their collection has them erroneously proceeds from regarding the collection collectively to regarding it distributively.

19. Fallacy of equivocation. An argument in which a word is used with one meaning in one part of the argument and with another meaning in another part. A common example: "The end of a thing is its perfection; death is the end of life; hence, death is the perfection of life." 20. Fallacy of non causa pro causa. the LAtin may be translated as "there is no cause of the sort that has been given as the cause." (a) Believing that something is the cause of an effect when in reality it is not. Example: "My incantations caused it to rain." (b) Arguing so that a statement appears unacceptable because it implies another statement that is false (but in reality does not).

20. Fallacy of post hoc ergo propter hoc. The Latin means "after this therefore the consequence (effect) of this," or "after this therefore because of this." Sometimes simply fallacy of false cause. Concluding that one thing is the cause of another thing because it precedes it in time. A confusion between the concept of succession and that of causation. Example: "A black cat ran across my path. Ten minutes mater I was hit by a truck. Therefore, the cat's running across my path was the cause of my being hit by a truck."

21. Fallacy of hasty generalization. Sometimes fallacy of hasty induction. An error of reasoning whereby a general statement is asserted (inferred) based on (a) limited information or (b) inadequate evidence, or (c) an unrepresentative sampling.

22. Fallacy of ignoratio elenchi (irrelevant conclusion). An argument that is irrelevant; that argues for something other than that which is to be proved and thereby in no way refutes (or supports) the points at issue. Example: A lawyer in defending his alcoholic client who has murdered three people in a drunken spree argues that alcoholism is a terrible disease and attempts should be made to eliminate it. IGNORATIO ELENCHI is sometimes used as a general name for all fallacies that are based on irrelevancy (such as ad baculum, ad hominem, as misericordiam, as populum, ad verecundiam, consensus gentium, etc.)

23. Fallacy of inconsistency. Arguing from inconsistent statements, or to conclusions that are inconsistent with the premises. See fallacy of tu quoque below.

24. Fallacy of irrelevant purpose. Arguing against something on the basis that it has not fulfilled its purpose (although in fact that was not its intended purpose).

26 Fallacy of 'is' to 'ought.' Arguing from premises that have only descriptive statements (is) to a conclusion that contains an ought, or a should.

1. Fallacy of limited (or false) alternatives. The error of insisting without full inquiry or evidence that the alternatives to a course of action have been exhausted and/or are mutually exclusive.

2. Fallacy of many questions. Sometimes fallact of the false question. Asking a question for which a single and simple answer is demanded yet the question (a) requires a series of answers, and/or (b) requires answers to a host of other questions, each of which have to be answered separately. Example: "Have you left school?"

3. Fallacy of misleading context. Arguing by misrepresenting, distorting, omitting or quoting something out of context.

4. Fallacy of prejudice. Arguing from a bias or emotional identification or involvement with an idea (argument, doctrine, institution, etc.).

5. Fallacy of red herring. Ignoring criticism of an argument by changing attention to another subject. Examples: "You believe in abortion, yet you don't believe in the right-to-die-with-dignity bill before the legislature."

6. Fallacy of slanting. Deliberately omitting, deemphasizing, or overemphasizing certain points to the exclusion of others in order to hide evidence that is important and relevant to the conclusion of the argument and that should be taken into accoun of in an argument.

7. Fallacy of special pleading. (a) Accepting an idea or criticism when applied to an opponent's argument but rejecting it when applied to one's own argument. (b) rejecting an idea or criticism when applied to an opponent's argument but accepting it when applied to one's own.

8. Fallacy of the straw man. Presenting an opponent's position in as weak or misrepresented a version as possible so that it can be easily refuted. Example: "Darwinism is in error. It claims that we are all descendents from an apelike creature, from which we evolved according to natural selection. No evidence of such a creature has been found. No adequate and consistent explanation of natural selection has been given. Therefore, evolution according to Darwinism has not taken place."

9. Fallacy of the beard. Arguin (a) that small or minor differences do not (or cannot) make a difference, or are not (or cannot be) significant, or (b) arguing so as to find a definite point at which something can be named. For example, insisting that a few hairs lost here and there do not indicate anything about my impending baldness; or trying to determine how many hairs a person must have before he can be called bald (or not bald).

10. Fallacy of tu quoque (you also). (a) Presenting evidence that a person's actions are not consistent with that for which he is arguing. Example: "John preaches that we should be kind and loving. He doesn't practice it. I've seen him beat up his kids." (b) Showing that a person's views are inconsistent with what he previously believed and therefore (1) he is not to be trusted, and/or (2) his new view is to be rejected. Example: "Judge Egener was against marijuana legislation four years ago when he was running for office. Now he is for it. How can you trust a man who can change his mind on such an important issue? His present position is inconsistent with his earlier view and therefore it should not be accepted." (c) Sometimes related to the Fallacy of two wrongs make a right. Example: The Democrats for years used illegal wiretapping; therefore the Republicans should not be condemned for their use of illegal wiretapping.

11. Fallacy of unqualified source. Using as support in an argument a source of authority that is not qualified to provide evidence.

12. Gambler's fallacy. (a) Arguing that since, for example, a penny has fallen tails ten times in a row then it will fall heads the eleventh time or (b) arguing that since, for example, an airline has not had an accident for the past ten years, it is then soon due for an accident. The gambler's fallacy rejects the assumption in probability theory that each event is independent of its previous happening. the chances of an event happening are always the same no matter how many times that event has taken place in the past. Given those events happening over a long enough period of time then their frequency would average out to 1/2. Sometimes referred to as the Monte Carlo fallacy (a generalized form of the gambler's fallacy): The error of assuming that because something has happened less frequently than expected in the past, there is an increased chance that it will happen soon.

13. Genetic fallacy. Arguing that the origin of something is identical with that thing with that from which it originates. Example: 'Consciousness orinates in neural processes. Therefore, consciousness is (nothing but) neural processes. Sometimes referred to as the nothing-but fallacy, or the REDUCTIVE FALLACY. (b) Appraising or explaining something in terms of its origin, or source, or beginnings. (c) Arguing that something is to be rejected because its origins are [unknown] and/or suspicious.

14. Pragmatic fallacy. Arguing that something is true because it has practical effects upon people: it makes them happier, easier to deal with, more moral, loyal, stable. Example: "An immortal life exists because without such a concept men would have nothing to live for. There would be no meaning or purpose in life and everyone would be immoral."

15. Pathetic fallacy. Incorrectly projecting (attributing) human emotions, feeling, intentions, thoughts, traits upon events or ojects which do not possess the capacity for such qualities.

16. Naturalist fallacy (ethics). 1. The fallacy of reducing ethical statements to factual statements, to statements about natural events. 2. The fallacy of deriving (deducing) ethical statements from nonethical statements. [is/ought fallacy]. 3. The fallacy of defining ethical terms in nonethical (descriptive, naturalistic, or factual) terms [ought/is fallacy].

Excerpt from the Encyclopedia Britannica without permission.

Galileo Galilei :

Italian mathematician, astronomer, and physicist, made several significant contributions to modern scientific thought. As the first man to use the telescope to study the skies, he amassed evidence that proved the Earth revolves around the Sun and is not the centre of the universe, as had been believed. His position represented such a radical departure from accepted thought that he was tried by the Inquisition in Rome, ordered to recant, and forced to spend the last eight years of his life under house arrest. He informally stated the principles later embodied in Newton's first two laws of motion. Because of his pioneer work in gravitation and motion and in combining mathematical analysis with experimentation, Galileo often is referred to as the founder of modern mechanics and experimental physics. Perhaps the most far-reaching of his achievements was his reestablishment of mathematical rationalism against Aristotle's logico-verbal approach and his insistence that the "Book of Nature is . . . written in mathematical characters." From this base, he was able to found the modern experimental method.

Galileo was born at Pisa on February 15, 1564, the son of Vincenzo Galilei, a musician. He received his early education at the monastery of Vallombrosa near Florence, where his family had moved in 1574. In 1581 he entered the University of Pisa to study medicine. While in the Pisa cathedral during his first year at the university, Galileo supposedly observed a lamp swinging and found that the lamp always required the same amount of time to complete an oscillation, no matter how large the range of the swing. Later in life Galileo verified this observation experimentally and suggested that the principle of the pendulum might be applied to the regulation of clocks.

Until he supposedly observed the swinging lamp in the cathedral, Galileo had received no instruction in mathematics. Then a geometry lesson he overheard by chance awakened his interest, and he began to study mathematics and science with Ostilio Ricci, a teacher in the Tuscan court. But in 1585, before he had received a degree, he was withdrawn from the university because of lack of funds. Returning to Florence, he lectured at the Florentine academy and in 1586 published an essay describing the hydrostatic balance, the invention of which made his name known throughout Italy. In 1589 a treatise on the centre of gravity in solids won for Galileo the honourable, but not lucrative, post of mathematics lecturer at the University of Pisa.

Galileo then began his research into the theory of motion, first disproving the Aristotelian contention that bodies of different weights fall at different speeds. Because of financial difficulties, Galileo, in 1592, applied for and was awarded the chair of mathematics at Padua, where he was to remain for 18 years and perform the bulk of his most outstanding work. At Padua he continued his research on motion and proved theoretically (about 1604) that falling bodies obey what came to be known as the law of uniformly accelerated motion (in such motion a body speeds up or slows down uniformly with time). Galileo also gave the law of parabolic fall (e.g., a ball thrown into the air follows a parabolic path). The legend that he dropped weights from the leaning tower of Pisa apparently has no basis in fact.

Galileo became convinced early in life of the truth of the Copernican theory (i.e., that the planets revolve about the Sun) but was deterred from avowing his opinions--as shown in his letter of April 4, 1597, to Kepler--because of fear of ridicule. While in Venice in the spring of 1609, Galileo learned of the recent invention of the telescope. After returning to Padua he built a telescope of threefold magnifying power and quickly improved it to a power of 32. Because of the method Galileo devised for checking the curvature of the lenses, his telescopes were the first that could be used for astronomical observation and soon were in demand in all parts of Europe.

As the first person to apply the telescope to a study of the skies, Galileo in late 1609 and early 1610 announced a series of astronomical discoveries. He found that the surface of the Moon was irregular and not smooth, as had been supposed; he observed that the Milky Way system was composed of a collection of stars; he discovered the satellites of Jupiter and named them Sidera Medicea (Medicean Stars) in honour of his former pupil and future employer, Cosimo II, grand duke of Tuscany. He also observed Saturn, spots on the Sun, and the phases of Venus . His first decisive astronomical observations were published in 1610 in Sidereus Nuncius ("The Starry Messenger").

Although the Venetian senate had granted Galileo a lifetime appointment as professor at Padua because of his findings with the telescope, he left in the summer of 1610 to become "first philosopher and mathematician" to the grand duke of Tuscany, an appointment that enabled him to devote more time to research.

In 1611 Galileo visited Rome and demonstrated his telescope to the most eminent personages at the pontifical court. Encouraged by the flattering reception accorded to him, he ventured, in three letters on the sunspots printed at Rome in 1613 under the title Istoria e dimostrazioni intorno alle macchie solari e loro accidenti . . . , to take up a more definite position on the Copernican theory. Movement of the spots across the face of the Sun, Galileo maintained, proved Copernicus was right and Ptolemy wrong.

His great expository gifts and his choice of Italian, in which he was an acknowledged master of style, made his thoughts popular beyond the confines of the universities and created a powerful movement of opinion. The Aristotelian professors, seeing their vested interests threatened, united against him. They strove to cast suspicion upon him in the eyes of ecclesiastical authorities because of contradictions between the Copernican theory and the Scriptures. They obtained the cooperation of the Dominican preachers, who fulminated from the pulpit against the new impiety of "mathematicians" and secretly denounced Galileo to the Inquisition for blasphemous utterances, which, they said, he had freely invented. Gravely alarmed, Galileo agreed with one of his pupils, B. Castelli, a Benedictine monk, that something should be done to forestall a crisis. He accordingly wrote letters meant for the Grand Duke and for the Roman authorities (letters to Castelli, to the Grand Duchess Dowager, to Monsignor Dini) in which he pointed out the danger, reminding the church of its standing practice of interpreting Scripture allegorically whenever it came into conflict with scientific truth, quoting patristic authorities and warning that it would be "a terrible detriment for the souls if people found themselves convinced by proof of something that it was made then a sin to believe." He even went to Rome in person to beg the authorities to leave the way open for a change. A number of ecclesiastical experts were on his side. Unfortunately, Cardinal Robert Bellarmine, the chief theologian of the church, was unable to appreciate the importance of the new theories and clung to the time-honoured belief that mathematical hypotheses have nothing to do with physical reality. He only saw the danger of a scandal, which might undermine Catholicity in its fight with Protestantism. He accordingly decided that the best thing would be to check the whole issue by having Copernicanism declared "false and erroneous" and the book of Copernicus suspended by the congregation of the Index. The decree came out on March 5, 1616. On the previous February 26, however, as an act of personal consideration, Cardinal Bellarmine had granted an audience to Galileo and informed him of the forthcoming decree, warning him that he must henceforth neither "hold nor defend" the doctrine, although it could still be discussed as a mere "mathematical supposition."

For the next seven years Galileo led a life of studious retirement in his house in Bellosguardo near Florence. At the end of that time (1623), he replied to a pamphlet by Orazio Grassi about the nature of comets; the pamphlet clearly had been aimed at Galileo. His reply, titled Saggiatore . . . ("Assayer . . . "), was a brilliant polemic on physical reality and an exposition of the new scientific method. In it he distinguished between the primary (i.e., measurable) properties of matter and the others (e.g., odour) and wrote his famous pronouncement that the "Book of Nature is . . . written in mathematical characters." The book was dedicated to the new pope, Urban VIII, who as Maffeo Barberini had been a longtime friend and protector of Galileo. Pope Urban received the dedication enthusiastically.

In 1624 Galileo again went to Rome, hoping to obtain a revocation of the decree of 1616. This he did not get, but he obtained permission from the Pope to write about "the systems of the world," both Ptolemaic and Copernican, as long as he discussed them noncommittally and came to the conclusion dictated to him in advance by the pontiff--that is, that man cannot presume to know how the world is really made because could have brought about the same effects in ways unimagined by him, and he must not restrict God's omnipotence. These instructions were confirmed in writing by the head censor, Monsignor Niccol Riccardi.

Galileo returned to Florence and spent the next several years working on his great book Dialogo sopra i due massimi sistemi del mondo, tolemaico e copernicano (Dialogue Concerning the Two Chief World Systems--Ptolemaic and Copernican). As soon as it came out, in the year 1632, with the full and complete imprimatur of the censors, it was greeted with a tumult of applause and cries of praise from every part of the European continent as a literary and philosophical masterpiece.

On the crisis that followed there remain now only inferences. It was pointed out to the Pope that despite its noncommittal title, the work was a compelling and unabashed plea for the Copernican system. The strength of the argument made the prescribed conclusion at the end look anticlimactic and pointless. The Jesuits insisted that it could have worse consequences on the established system of teaching "than Luther and Calvin put together." The Pope, in anger, ordered a prosecution. The author being covered by license, the only legal measures would be to disavow the licensers and prohibit the book. But at that point a document was "discovered" in the file, to the effect that during his audience with Bellarmine on February 26, 1616, Galileo had been specifically enjoined from "teaching or discussing Copernicanism in any way," under the penalties of the Holy Office. His license, it was concluded, had therefore been "extorted" under false pretenses. (The consensus of historians, based on evidence made available when the file was published in 1877, has been that the document had been planted and that Galileo was never so enjoined.) The church authorities, on the strength of the "new" document, were able to prosecute him for "vehement suspicion of heresy." Notwithstanding his pleas of illness and old age, Galileo was compelled to journey to Rome in February 1633 and stand trial. He was treated with special indulgence and not jailed. In a rigorous interrogation on April 12, he steadfastly denied any memory of the 1616 injunction. The commissary general of the Inquisition, obviously sympathizing with him, discreetly outlined for the authorities a way in which he might be let off with a reprimand, but on June 16 the congregation decreed that he must be sentenced. The sentence was read to him on June 21: he was guilty of having "held and taught" the Copernican doctrine and was ordered to recant. Galileo recited a formula in which he "abjured, cursed and detested" his past errors. The sentence carried imprisonment, but this portion of the penalty was immediately commuted by the Pope into house arrest and seclusion on his little estate at Arcetri near Florence, where he returned in December 1633. The sentence of house arrest remained in effect throughout the last eight years of his life.

Although confined to his estate, Galileo's prodigious mental activity continued undiminished to the last. In 1634 he completed Discorsi e dimostrazioni mathematiche intorno a due nuove scienze attenenti alla meccanica (Dialogue Concerning Two New Sciences), in which he recapitulated the results of his early experiments and his mature meditations on the principles of mechanics. This, in many respects his most valuable work, was printed by Louis Elzevirs at Leiden in 1638. His last telescopic discovery--that of the Moon's diurnal and monthly librations (wobbling from side to side)--was made in 1637, only a few months before he became blind. But the fire of his genius was not even yet extinct. He continued his scientific correspondence with unbroken interest and undiminished acumen; he thought out the application of the pendulum to the regulation of clockwork, which the Dutch scientist Christiaan Huygens put into practice in 1656; he was engaged in dictating to his disciples, Vincenzo Viviani and Evangelista Torricelli, his latest ideas on the theory of impact when he was seized with the slow fever that resulted in his death at Arcetri on January 8, 1642.

The direct services of permanent value that Galileo rendered to astronomy are virtually summed up in his telescopic discoveries. His name is justly associated with a vast extension of the bounds of the visible universe, and his telescopic observations are a standing monument of his ability. Within two years after their discovery, he had constructed approximately accurate tables of the revolutions of Jupiter's satellites and proposed their frequent eclipses as a means of determining longitudes on land and at sea. The idea, though ingenious, has been found of little use at sea. His observations on sunspots are noteworthy for their accuracy and for the deductions he drew from them with regard to the rotation of the Sun and the revolution of the Earth.

A puzzling circumstance is Galileo's neglect of Kepler's laws, which were discovered during his lifetime. But then he believed strongly that orbits should be circular (not elliptical, as Kepler discovered) in order to keep the fabric of the cosmos in its perfect order. This preconception prevented him from giving a full formulation of the inertial law, which he himself discovered, although it usually is attributed to the French mathematician Ren Descartes. Galileo believed that the inertial path of a body around the Earth must be circular. Lacking the idea of Newtonian gravitation, he hoped this would allow him to explain the path of the planets as circular inertial orbits around the Sun.

The idea of a universal force of gravitation seems to have hovered on the borders of this great man's mind, but he refused to entertain it because, like Descartes, he considered it an "occult" quality. More valid instances of the anticipation of modern discoveries may be found in his prevision that a small annual parallax would eventually be found for some of the fixed stars and that extra-Saturnian planets would at some future time be ascertained to exist and in his conviction that light travels with a measurable although extremely great velocity. Although Galileo discovered, in 1610, a means of adapting his telescope to the examination of minute objects, he did not become acquainted with the compound microscope until 1624, when he saw one in Rome and, with characteristic ingenuity, immediately introduced several improvements into its construction.

A most substantial part of his work consisted undoubtedly of his contributions toward the establishment of mechanics as a science. Some valuable but isolated facts and theorems had previously been discovered and proved, but it was Galileo who first clearly grasped the idea of force as a mechanical agent. Although he did not formulate the interdependence of motion and force into laws, his writings on dynamics are everywhere suggestive of those laws, and his solutions of dynamical problems involve their recognition. In this branch of science he paved the way for the English physicist and mathematician Isaac Newton later in the century. The extraordinary advances made by him were due to his application of mathematical analysis to physical problems.

Galileo was the first man who perceived that mathematics and physics, previously kept in separate compartments, were going to join forces. He was thus able to unify celestial and terrestrial phenomena into one theory, destroying the traditional division between the world above and the world below the Moon. The method that was peculiarly his consisted in the combination of experiment with calculation--in the transformation of the concrete into the abstract and the assiduous comparison of results. He created the modern idea of experiment, which he called cimento ("ordeal"). This method was applied to check theoretical deductions in the investigation of the laws of falling bodies, of equilibrium and motion on an inclined plane, and of the motion of a projectile. The latter, together with his definition of momentum and other parts of his work, implied a knowledge of the laws of motion as later stated by Newton. In his Discorso intorno alle cose che stanno in su l'acqua ("Discourse on Things That Float"), published in 1612, he used the principle of virtual velocities to demonstrate the more elementary theorems of hydrostatics, deducing the equilibrium of fluid in a siphon, and worked out the conditions for the flotation of solid bodies in a liquid. He also constructed, in 1607, an elementary form of air thermometer.

Excerpt from the Encyclopedia Britannica without permission.