1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Sphere eversion
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In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. Remarkably, it is smoothly and continuously to turn a sphere inside out in this way without cutting or tearing it or creating any crease. More precisely, let f, S2 → R3 be the standard embedding, then there is a homotopy of immersions f t, S2 → R3 such that ƒ0 = ƒ. An existence proof for crease-free sphere eversion was first created by Stephen Smale and it is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including Arnold S. Shapiro, on the other hand, it is much easier to prove that such a turning exists and that is what Smale did. Smales graduate adviser Raoul Bott at first told Smale that the result was obviously wrong and his reasoning was that the degree of the Gauss map must be preserved in such turning—in particular it follows that there is no such turning of S1 in R2. But the degree of the Gauss map for the embeddings f, the degree of the Gauss map of all immersions of S2 in R3 is 1, so there is no obstacle. Smales original proof was indirect, he identified classes of immersions of spheres with a group of the Stiefel manifold. Since the homotopy group that corresponds to immersions of S2 in R3 vanishes, the standard embedding, in principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do. There are several ways of producing explicit examples and beautiful mathematical visualization, Half-way models and this is the original method, first done by Shapiro and Phillips via Boys surface, later refined by many others. The original half-way model homotopies were constructed by hand, and worked topologically, a more recent and definitive graphics refinement is minimax eversions, which is a variational method, and consist of special homotopies. Thurstons corrugations, this is a method and generic, it takes a homotopy. This is illustrated in the computer-graphics animation Outside In and this provides conceptual understanding for the process, revealed as arising from the concrete structure of the 3-dimensional projective plane and the underlying geometry of the Hopf fibration. In spirit, this is closer to the originally suggested by Shapiro. This is partially illustrated in a Povray computer-graphics animation, again found by searching YouTube. Sphere Whitney–Graustein theorem Iain R. Aitchison The Holiverse, holistic eversion of the 2-sphere in R^3, etnyre Review of h-principles and flexibility in geometry, MR1982875. Levy, Silvio, A brief history of sphere eversions, Making waves, Wellesley, MA, Smale, Stephen, A classification of immersions of the two-sphere, Transactions of the American Mathematical Society,90, 281–290, doi,10. The holiverse sphere eversion The deNeve/Hills sphere eversion

3.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times

4.
Parrondo's paradox
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Parrondos paradox, a paradox in game theory, has been described as, A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996, however, the paradox disappears when rigorously analyzed. Consider an example in which there are two points A and B having the same altitude, as shown in Figure 1, in the first case, we have a flat profile connecting them. Here, if we leave some round marbles in the middle that move back and forth in a fashion, they will roll around randomly. Now consider the case where we have a saw-tooth-like region between them. Here also, the marbles will roll towards either ends with equal probability, now if we tilt the whole profile towards the right, as shown in Figure 2, it is quite clear that both these cases will become biased towards B. Now consider the game in which we alternate the two profiles while judiciously choosing the time between alternating from one profile to the other, when we leave a few marbles on the first profile at point E, they distribute themselves on the plane showing preferential movements towards point B. Then we again apply the first profile and repeat the steps, if no marbles cross point C before the first marble crosses point D, we must apply the second profile shortly before the first marble crosses point D, to start over. It easily follows that eventually we will have marbles at point A, hence for a problem defined with having marbles at point A being a win and having marbles at point B a loss, we clearly win by playing two losing games. A second example of Parrondos paradox is drawn from the field of gambling, consider playing two games, Game A and Game B with the following rules. For convenience, define C t to be our capital at time t, winning a game earns us $1 and losing requires us to surrender $1. It follows that C t +1 = C t +1 if we win at step t and C t +1 = C t −1 if we lose at step t. In Game A, we toss a coin, Coin 1. If ϵ >0, this is clearly a losing game in the long run, in Game B, we first determine if our capital is a multiple of some integer M. If it is, we toss a coin, Coin 2. If it is not, we toss another biased coin, Coin 3, the role of modulo M provides the periodicity as in the ratchet teeth. It is clear that by playing Game A, we will almost surely lose in the long run, harmer and Abbott show via simulation that if M =3 and ϵ =0.005, Game B is an almost surely losing game as well. In fact, Game B is a Markov chain, and an analysis of its transition matrix shows that the steady state probability of using coin 2 is 0.3836

5.
Potato paradox
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The potato paradox is a mathematical calculation that has a counter-intuitive result. The paradox involves dehydrating potatoes by a minuscule amount. The paradox has been described as, You have 100 lbs of potatoes and you let them dehydrate until theyre 98 percent water. How much do they weigh now, the Universal Book of Mathematics states the problem as follows, Fred brings home 100 lbs of potatoes, which consist of 99 percent water. He then leaves them outside overnight so that they consist of 98 percent water, the surprising answer is 50 lbs. In Quines classification of paradoxes, the paradox is a veridical paradox. One explanation begins by saying that initially the non-water weight is 1 pound, then one asks,1 pound is 2% of how many pounds. In order for that percentage to be twice as big, the weight must be half as big. 100 lbs of potatoes, 99% water, means that theres 99 lbs of water, if the water decreases to 98%, then the solids account for 2% of the weight. The 2,98 ratio reduces to 1,49, since the solids still weigh 1 lb, the water must weigh 49 lbs. After the evaporating of the water, the total quantity, x, contains 1 lbs pure potatoes. The equation becomes,1 + x = x resulting in x =50 lbs, the weight of water in the fresh potatoes is 0.99 ⋅100. If x is the weight of water lost from the potatoes when they dehydrate then 0.98 is the weight of water in the dehydrated potatoes. For example, if the potatoes were originally 99. 999% water, reducing the percentage to 99. 998% still requires halving the weight

6.
Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic

7.
Missing square puzzle
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It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it, the key to the puzzle is the fact that neither of the 13×5 triangles is truly a triangle, because what appears to be the hypotenuse is bent. In other words, the hypotenuse does not maintain a consistent slope, a true 13×5 triangle cannot be created from the given component parts. The four figures total 32 units of area, the apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S =13 ×52 =32.5 units. However, the triangle has a ratio of 5,2, while the red triangle has the ratio 8,3. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the figure occupies 33. The amount of bending is approximately 1/28th of a unit, which is difficult to see on the diagram of the puzzle, overlaying the hypotenuses from both figures results in a very thin parallelogram with an area of exactly one grid square, so the missing area. According to Martin Gardner, this puzzle was invented by a New York City amateur magician, Paul Curry. However, the principle of a dissection paradox has been known since the start of the 16th century, the integer dimensions of the parts of the puzzle are successive Fibonacci numbers, which leads to the exact unit area in the thin parallelogram. Many other geometric dissection puzzles are based on a few properties of the Fibonacci sequence. In the smaller rearrangement, each needs to overlap the triangle by an area of half a unit for its top/bottom edge to align with a grid line. Mitsunobu Matsuyamas Paradox uses four congruent quadrilaterals and a small square, when the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new square is a little smaller than the original one. If θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2θ −1, for θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0. 8%. Einstellung effect Missing dollar riddle A printable Missing Square variant with a video demonstration, currys Paradox, How Is It Possible. At cut-the-knot Triangles and Paradoxes at archimedes-lab

8.
Zeno's paradoxes
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It is usually assumed, based on Platos Parmenides, that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides view. Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point, some of Zenos nine surviving paradoxes are essentially equivalent to one another. Aristotle offered a refutation of some of them, three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below. Zenos arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction and they are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zenos paradoxes are simply mathematical problems, some philosophers, however, say that Zenos paradoxes and their variations remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear, Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zenos teacher Parmenides was the first to introduce the Achilles and the tortoise paradox. But in a passage, Laertius attributes the origin of the paradox to Zeno. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. – as recounted by Aristotle, Physics VI,9, 239b15 In the paradox of Achilles, Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at constant speed, then after some finite time, Achilles will have run 100 meters. During this time, the tortoise has run a shorter distance. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go, therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. That which is in locomotion must arrive at the stage before it arrives at the goal. – as recounted by Aristotle, Physics VI,9. Before he can get there, he must get halfway there, before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth, before an eighth, one-sixteenth, the resulting sequence can be represented as, This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a problem in that it contains no first distance to run, for any possible first distance could be divided in half. Hence, the trip cannot even begin, the paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion is not actually divisible and this argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts

9.
Gabriel's Horn
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Gabriels horn is a geometric figure which has infinite surface area but finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, the properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century. Gabriels horn is formed by taking the graph of x ↦1 x, with the domain x >1, mathematically, the volume approaches π as a approaches infinity. Using the limit notation of calculus, lim a → ∞ V = lim a → ∞ π = π ⋅ lim a → ∞ = π, the surface area formula above gives a lower bound for the area as 2π times the natural logarithm of a. There is no upper bound for the logarithm of a as a approaches infinity. That means, in case, that the horn has an infinite surface area. That is to say, lim a → ∞ A ≥ lim a → ∞2 π ln = ∞, actually, while the section lying in the xy-plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the sum of sections, is finite. A perhaps more convincing approach is to treat the horn as a stack of disks with diminishing radii, as their shape is identical, one is tempted to calculate just the sum of radii, which produces the harmonic series that goes to infinity. A more careful consideration shows that one should calculate the sum of their squares, every disk has a radius r = 1/x and an area πr2 or π/x2. The series 1/x diverges but 1/x2 converges, in general, for any real ε >0, 1/x1+ε converges. The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei. However, to coat the surface of the horn with a constant thickness of paint, no matter how thin. Of course, in reality, paint is not infinitely divisible and we talk therefore of an ideal paint in a world where limits do smoothly tend to zero well below atomic and quantum sizes, the world of the continuous space of mathematics. Write S for the solid of revolution of the graph y = f about the x-axis, if the surface area of S is finite, then so is the volume. Therefore, there exists a t0 such that the supremum sup is finite, hence, M = sup must be finite since f is a continuous function, which implies that f is bounded on the interval [1, ∞). Gabriels Horn by John Snyder, the Wolfram Demonstrations Project,2007, gabriels Horn, An Understanding of a Solid with Finite Volume and Infinite Surface Area by Jean S. Joseph

10.
Cramer's paradox
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It is named after the Swiss mathematician Gabriel Cramer. This paradox is the result of an understanding or a misapplication of two theorems, Bézouts theorem. The resolution of the paradox is that in degenerate cases n /2 points are not enough to determine a curve uniquely. The paradox was first published by Maclaurin, Cramer and Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. It has become known as Cramers paradox after featuring in his 1750 book Introduction à lanalyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement. At about the time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points. The result was publicized by James Stirling and explained by Julius Plücker, for first order curves the paradox does not occur, because n =1 so n2 =1 < n /2 =2. In general two distinct lines L1 and L2 intersect at a single point P unless the lines are of equal gradient, in which case they do not intersect at all. A single point is not sufficient to define a line, through the point P there pass not only the two lines but an infinite number of other lines as well. In a letter to Euler, Cramer pointed out that the cubic curves x3 − x =0, hence 9 points are not sufficient to uniquely determine a cubic curve, at least in degenerate cases such as these. Ed Sandifer Cramer’s Paradox Cramers Paradox at MathPages

11.
0.999...
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics