List of paradoxes
Encyclopedia
This is a list of paradox
es, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and have their own article.
Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis.
.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and have their own article.
Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis.
Logic
- Barbershop paradoxBarbershop paradoxThe Barbershop Paradox was proposed by Lewis Carroll in a three-page essay entitled "A Logical Paradox" which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses to illustrate the paradox...
: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. - What the Tortoise Said to AchillesWhat the Tortoise Said to Achilles"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief dialogue which problematises the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race...
"Whatever Logic is good enough to tell me is worth writing down...," also known as Carroll's paradox, not to be confused with the physical paradox of the same name. - Crocodile DilemmaCrocodile DilemmaThe Crocodile Dilemma is an unsolvable problem in logic. The premise states that a crocodile who has stolen a child promises the father that his son will be returned if and only if he can correctly predict whether or not the crocodile will return the child....
: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned? - Catch-22 (logic)Catch-22 (logic)A Catch-22, coined by Joseph Heller in his novel Catch-22, is a logical paradox arising from a situation in which an individual needs something that can only be acquired with an action that will lead him to that very situation he is already in; therefore, the acquisition of this thing becomes...
: A situation where someone is in need of something that can only be had by not being in need of it. - Drinker paradoxDrinker paradoxThe drinker paradox is a theorem of classical predicate logic that states: There is someone in the pub such that, if he is drinking, everyone in the pub is drinking. The actual theorem is\exists x.\ [D \rightarrow \forall y.\ D]...
: In any pub there is a customer who, if they drink, everybody in the pub drinks. - Paradox of entailment: Inconsistent premises always make an argument valid.
- Horse paradox: All horses are the same color.
- Lottery paradoxLottery paradoxHenry E. Kyburg, Jr.'s lottery paradox arises from considering a fair 1000 ticket lottery that has exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept that some ticket will win. Suppose that an event is very likely only if the...
: There is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not reasonable to believe that no lottery ticket will win. - Raven paradoxRaven paradoxThe Raven paradox, also known as Hempel's paradox or Hempel's ravens is a paradox proposed by the German logician Carl Gustav Hempel in the 1940s to illustrate a problem where inductive logic violates intuition...
(or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black. - Unexpected hanging paradoxUnexpected hanging paradoxThe unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox about a person's expectations about the timing of a future event The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or...
: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The surprise examination and Bottle Imp paradoxThe Bottle ImpThe Bottle Imp is a short story by the Scottish author Robert Louis Stevenson usually found in the short story collection Island Nights' Entertainments...
use similar logic.
Self-reference
These paradoxes have in common a contradiction arising from self-referenceSelf-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
.
- Barber paradoxBarber paradoxThe Barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible.- The Paradox...
: A male barber shaves all and only those men who do not shave themselves. Does he shave himself? (Russell's popularization of his set theoretic paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
.) - Berry paradoxBerry paradoxThe Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words". Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G...
: The phrase "the first number not nameable in under eleven words" appears to name it in nine words. - Paradox of the CourtParadox of the CourtThe Paradox of the Court, also known as the counterdilemma of Euathlus, is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he wins his...
: A law student agrees to pay his teacher after winning his first case. The teacher then sues the student (who has not yet won a case) for payment. - Curry's paradoxCurry's paradoxCurry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules...
: "If this sentence is true, then Santa Claus exists." - Epimenides paradoxEpimenides paradoxThe Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos , There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:...
: A Cretan says: "All Cretans are liars". This paradox works in mainly the same way as the Liar paradoxLiar paradoxIn philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
. - Exception paradoxException paradoxException paradox: if every rule has an exception , then there must be an exception to the rule that every rule has an exception....
: "If there is an exception to every rule, then every rule must have at least one exception; the exception to this one being that it has no exception." "There's always an exception to the rule, except to the exception of the rule—which is, in of itself, an accepted exception of the rule." - Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterological word? (Another close relative of Russell's paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
.) - Kleene–Rosser paradox: By formulating an equivalent to Richard's paradoxRichard's paradoxIn logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics...
, untyped lambda calculus is shown to be inconsistent. - Liar paradoxLiar paradoxIn philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question no?" And "I'm lying."- Card paradoxCard paradoxThe card paradox is a non-self-referential variant of the liar paradox constructed by Philip Jourdain. It is also known as the postcard paradox, Jourdain paradox or Jourdain's paradox.- The paradox :...
: "The next statement is true. The previous statement is false." A variant of the liar paradox that does not use self-reference. - The Pinocchio paradoxPinocchio paradoxThe Pinocchio paradox arises when Pinocchio says: "My nose grows now", and is a version of the liar paradox. In philosophy and logic, the liar paradox consists of the statement "This sentence is false." Any attempts to assign a classical binary truth value to this statement leads to a...
: What would happen if PinocchioPinocchioThe Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...
said "My nose will be growing"? - Quine's paradoxQuine's ParadoxQuine's paradox is a paradox concerning truth values, attributed to Willard Van Orman Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals...
: "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals. - Yablo's paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are false. Uses neither self-reference nor circular reference.
- Card paradox
- Lynch's ParadoxSelf-awarenessSelf-awareness is the capacity for introspection and the ability to reconcile oneself as an individual separate from the environment and other individuals...
: A machine sufficiently "intelligent" to describe its composition and functionality completely is not necessarily self-aware. - Opposite DayOpposite DayOpposite Day, also known as Opposites Day is a word game where speech is modified so that meaning is inverted. Once Opposite Day is declared, statements mean the opposite of what they usually mean. Usually, a person would say, "After this phrase is over, it will be officially opposite day," and...
: "It is opposite day today." - PetroniusPetroniusGaius Petronius Arbiter was a Roman courtier during the reign of Nero. He is generally believed to be the author of the Satyricon, a satirical novel believed to have been written during the Neronian age.-Life:...
's paradox: "Moderation in all things, including moderation" (unsourced quotation sometimes attributed to Petronius). - Richard's paradoxRichard's paradoxIn logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics...
: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory. - Russell's paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
: Does the set of all those sets that do not contain themselves contain itself? - Socratic paradoxSocratic paradoxThe phrase Socratic paradox can refer to two separate things.The more common usage refers to an object or idea whose very existence, or acknowledgment, is a paradox. Its name is derived from a quote of Socrates from the Republic, where he says, "I know nothing at all." The question that arises is...
: "I know that I know nothing at all."
Vagueness
- Bonini's paradoxBonini's paradoxBonini's Paradox, named after Stanford business professor Charles Bonini, explains the difficulty in constructing models or simulations that fully capture the workings of complex systems .-Statements:...
: models or simulations that explain the workings of complex systems are seemingly impossible to construct: As a model of a complex system becomes more complete, it becomes less understandable; for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents. - Code-talker paradoxCode-Talker ParadoxThe code-talker paradox is an issue in linguistics that brings into question some fundamental ideas of the nature of languages. Specifically, the fact that language can both facilitate and block communication....
: how can a language both enable communication and block communication? - Ship of TheseusShip of TheseusThe Ship of Theseus, also known as Theseus' paradox, or various variants, notably grandfather's axe and Trigger's Broom is a paradox that raises the question of whether an object which has had all its component parts replaced remains fundamentally the same object.The paradox is most notably...
(a.k.a. George Washington's axe or Grandfather's old axe): It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with. - Sorites paradoxSorites paradoxThe sorites paradox is a paradox that arises from vague predicates. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand, from which grains are individually removed...
(also known as the paradox of the heap): One grain of sand is not a heap. If you don't have a heap, then adding only one grain of sand won't give you a heap. Then no number of grains of sand will make a heap. Similarly, one hair can't make the difference between being bald and not being bald. But then if you remove one hair at a time, you will never become bald. Also similar, one dollar will not make you rich, so if you keep this up, one dollar at a time, you will never become rich, no matter how much you obtain.
Mathematics
- Cramer's paradoxCramer's paradoxIn mathematics, Cramer's paradox is the statement that the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.Cramer's paradox is the result of two theorems: Bézout's theorem In mathematics, Cramer's...
: the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve. - The infinite sum of alternating integers 1 − 2 + 3 − 4 + · · · can be said to equal
, which is not an integer. - Elevator paradoxElevator paradoxThe elevator paradox is a paradox first noted by Marvin Stern and George Gamow, physicists who had offices on different floors of a multi-story building...
: Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement. - Interesting number paradoxInteresting number paradoxThe interesting number paradox is a semi-humorous paradox that arises from attempting to classify natural numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting...
: The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact. - Nontransitive diceNontransitive diceA set of nontransitive dice is a set of dice for which the relation "is more likely to roll a higher number" is not transitive. See also intransitivity....
: You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A. - Russell's paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
: Does the set of all those sets that do not contain themselves contain itself?
Statistics
- Accuracy paradoxAccuracy paradoxThe accuracy paradox for predictive analytics states that predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy...
: predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy. - Berkson's paradoxBerkson's paradoxBerkson's paradox or Berkson's fallacy is a result in conditional probability and statistics which is counter-intuitive for some people, and hence a veridical paradox. It is a complicating factor arising in statistical tests of proportions...
: a complicating factor arising in statistical tests of proportions. - Freedman's paradoxFreedman's paradoxIn statistical analysis, Freedman's paradox, named after David Freedman, describes a problem in model selection whereby predictor variables with no explanatory power can appear artificially important. Freedman demonstrated that this is a common occurrence when the number of variables is similar to...
describes a problem in model selectionModel selectionModel selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered...
where predictor variables with no explanatory power can appear artificially important - Friendship paradoxFriendship paradoxThe friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average. It can be explained as a form of sampling bias in which people with greater numbers of friends have an increased likelihood of...
: For almost everyone, their friends have more friends than they do. - Inspection paradox: Why one will wait longer for a bus than one should.
- Lindley's paradoxLindley's paradoxLindley's paradox is a counterintuitive situation in statistics in which the Bayesian and frequentist approaches to a hypothesis testing problem give opposite results for certain choices of the prior distribution...
: Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results. - Low birth weight paradoxLow birth weight paradoxThe low birth weight paradox is an apparently paradoxical observation relating to the birth weights and mortality of children born to tobacco smoking mothers. Low birth weight children born to smoking mothers have a lower infant mortality rate than the low birth weight children of non-smokers...
: Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. (A special case of Simpson's paradoxSimpson's paradoxIn probability and statistics, Simpson's paradox is a paradox in which a correlation present in different groups is reversed when the groups are combined. This result is often encountered in social-science and medical-science statistics, and it occurs when frequencydata are hastily given causal...
.) - Will Rogers phenomenonWill Rogers phenomenonThe Will Rogers phenomenon is obtained when moving an element from one set to another set raises the average values of both sets. It is based on the following quote, attributed to comedian Will Rogers:...
: The mathematical concept of an averageAverageIn mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
, whether defined as the meanMeanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
or medianMedianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
, leads to apparently paradoxical results — for example, it is possible that moving an entry from an encyclopediaEncyclopediaAn encyclopedia is a type of reference work, a compendium holding a summary of information from either all branches of knowledge or a particular branch of knowledge....
to a dictionaryDictionaryA dictionary is a collection of words in one or more specific languages, often listed alphabetically, with usage information, definitions, etymologies, phonetics, pronunciations, and other information; or a book of words in one language with their equivalents in another, also known as a lexicon...
would increase the average entry length on both books.
Probability
- Bertrand's box paradoxBertrand's box paradoxBertrand's box paradox is a classic paradox of elementary probability theory. It was first posed by Joseph Bertrand in his Calcul des probabilités, published in 1889.There are three boxes:# a box containing two gold coins,...
: A paradox of conditional probability closely related to the Boy or Girl paradox. - Bertrand's paradoxBertrand's paradox (probability)The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités as an example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not...
: Different common-sense definitions of randomness give quite different results. - Birthday paradoxBirthday paradoxIn probability theory, the birthday problem or birthday paradox pertains to the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 366. However, 99%...
: What is the chance that two people in a room have the same birthday? - Borel's paradoxBorel's paradoxIn probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero...
: Conditional probability density functions are not invariant under coordinate transformations. - Boy or Girl paradox: A two-child family has at least one boy. What is the probability that it has a girl?
- False positive paradoxFalse positive paradoxThe false positive paradox is a statistical result where false positive tests are more probable than true positive tests, occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the false positive rate...
: A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny. - Monty Hall problem: An unintuitive consequence of conditional probability.
- Necktie ParadoxNecktie paradoxThe necktie paradox is a puzzle or paradox within the subjectivistic interpretation of probability theory. It is a variation of the two-envelope paradox....
: A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox. - Proebsting's paradoxProebsting's paradoxIn probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first...
: The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin. - Simpson's paradoxSimpson's paradoxIn probability and statistics, Simpson's paradox is a paradox in which a correlation present in different groups is reversed when the groups are combined. This result is often encountered in social-science and medical-science statistics, and it occurs when frequencydata are hastily given causal...
: An association in sub-populations may be reversed in the population. It appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis. - Sleeping Beauty problemSleeping Beauty problemThe Sleeping Beauty problem is a puzzle in probability theory and formal epistemology in which an ideally rational epistemic agent is to be wakened once or twice according to the toss of a coin, and asked her degree of belief for the coin having come up heads....
: A probability problem that can be correctly answered as one half or one third depending on how the question is approached. - Three cards problem: When pulling a random card, how do you determine the color of the underside?
- Three Prisoners problemThree Prisoners ProblemThe Three Prisoners problem appeared in Martin Gardner's Mathematical Games column in Scientific American in 1959. It is mathematically equivalent to the Monty Hall problem with car and goat replaced with freedom and execution respectively, and also equivalent to, and presumably based on,...
: A variation of the Monty Hall problem. - Two-envelope paradox: You are given two indistinguishable envelopes and you are told one contains twice as much money as the other. You may open one envelope, examine its contents, and then, without opening the other, choose which envelope to take.
Infinity and infinitesimals
- Burali-Forti paradoxBurali-Forti paradoxIn set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
: If the ordinal numberOrdinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s formed a set, it would be an ordinal number that is smaller than itself. - Cantor's paradoxCantor's paradoxIn set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
: There is no greatest cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
. - Galileo's paradoxGalileo's paradoxGalileo's paradox is a demonstration of one of the surprising properties of infinite sets.In his final scientific work, the Two New Sciences, Galileo Galilei made two apparently contradictory statements about the positive whole numbers...
: Though most numbers are not squares, there are no more numbers than squares. (See also Cantor's diagonal argumentCantor's diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
) - Hilbert's paradox of the Grand HotelHilbert's paradox of the Grand HotelHilbert's paradox of the Grand Hotel is a mathematical veridical paradox about infinite sets presented by German mathematician David Hilbert .-The paradox:...
: If a hotel with infinitely many rooms is full, it can still take in more guests. - Pigeonhole Principle: E.g. there must be at least two people in London with the same number of hairs on their heads
- Russell's paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
: Does the set of all those sets that do not contain themselves contain itself? - Skolem's paradoxSkolem's paradoxIn mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as...
: Countably infinite models of set theory contain uncountably infinite sets. - SupertaskSupertaskIn philosophy, a supertask is a quantifiably infinite number of operations that occur sequentially within a finite interval of time. Supertasks are called "hypertasks" when the number of operations becomes innumerably infinite. The term supertask was coined by the philosopher James F...
s can result in paradoxes such as the Ross-Littlewood paradox and Benardete's paradox. - Zeno's paradoxesZeno's paradoxesZeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...
: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on." (This is also a physical paradox.)
Geometry and topology
- Banach–Tarski paradoxBanach–Tarski paradoxThe Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
: Cut a ball into a finite number of pieces, re-assemble the pieces to get two balls, both of equal size to the first. The von Neumann paradoxVon Neumann paradoxIn mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the...
is a two-dimensional analogue.- Paradoxical setParadoxical setIn set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is a partitioning of the set into exactly two subsets, along with an appropriate group of functions that operate on some universe , such that each partition can be mapped back onto...
: A set that can be partitioned into two sets, each of which is equivalent to the original.
- Paradoxical set
- Coastline paradoxCoastline paradoxThe coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. It was first observed by Lewis Fry Richardson....
: the perimeter of a landmass is in general ill-defined. - Gabriel's HornGabriel's HornGabriel's Horn is a geometric figure which has infinite surface area but encloses a 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, with the finite...
or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot setMandelbrot setThe Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
and various other fractalFractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottleKlein bottleIn mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
. - Hausdorff paradoxHausdorff paradoxIn mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S2, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent...
: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself. - Missing square puzzleMissing square puzzleThe missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures. It depicts two arrangements of shapes, each of which apparently forms a 13×5 right-angled triangle, but one of which has a 1×1 hole in it.-Solution:The key to the puzzle...
: Two similar-looking figures appear to have different areas while built from the same pieces. - Nikodym setNikodym setIn mathematics, a Nikodym set is the seemingly paradoxical result of a construction in measure theory. A Nikodym set in the unit square S in the Euclidean plane E2 is a subset N of S such that* the area In mathematics, a Nikodym set is the seemingly paradoxical result of a construction in measure...
: A set contained in and with the same Lebesgue measureLebesgue measureIn measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
as the unit squareUnit squareIn mathematics, a unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at , , , and .-In the real plane:...
, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point. - Smale's paradoxSmale's paradoxIn differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion...
: A sphere can, topologically, be turned inside out.
Decision theory
- Abilene paradoxAbilene paradoxThe Abilene paradox is a paradox in which a group of people collectively decide on a course of action that is counter to the preferences of any of the individuals in the group. It involves a common breakdown of group communication in which each member mistakenly believes that their own preferences...
: People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do. - Apportionment paradoxApportionment paradoxAn apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense.To apportion is to divide into parts according to some rule, the rule typically being one of proportion...
: Some systems of apportioning representation can have unintuitive results due to rounding- Alabama paradox: Increasing the total number of seats might shrink one block's seats.
- New states paradox: Adding a new state or voting block might increase the number of votes of another.
- Population paradox: A fast-growing state can lose votes to a slow-growing state.
- Arrow's paradoxArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
: Given more than two choices, no system can have all the attributes of an ideal voting systemVoting systemA voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
at once. - Buridan's assBuridan's assBuridan's ass is an illustration of a paradox in philosophy in the conception of free will.It refers to a hypothetical situation wherein an ass is placed precisely midway between a stack of hay and a pail of water...
: How can a rationalRationalityIn philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...
choice be made between two outcomes of equal valueValue theoryValue theory encompasses a range of approaches to understanding how, why and to what degree people should value things; whether the thing is a person, idea, object, or anything else. This investigation began in ancient philosophy, where it is called axiology or ethics. Early philosophical...
? - Chainstore paradoxChainstore paradoxChainstore paradox is a concept that purports to refute standard game theory reasoning.-The chain store game:A monopolist has branches in 20 towns. He faces 20 potential competitors, one in each town, who will be able to choose IN or OUT. They do so in sequential order and one at a time...
: Even those who know better play the so-called chain store game in an irrational manner. - Fenno's paradoxFenno's ParadoxFenno's Paradox is the belief that people generally disapprove of the United States Congress as a whole, but support the Congressmen from their own Congressional district. It is named after Richard Fenno who discussed this in his 1978 book Home Style: House Members in Their Districts....
: The belief that people generally disapprove of the United States Congress as a whole, but support the Congressman from their own Congressional district. - Green paradoxGreen paradoxThe Green Paradox is a phrase coined by German economist Hans-Werner Sinn to describe the fact that an environmental policy that becomes greener with the passage of time acts like an announced expropriation for the owners of fossil fuel resources, inducing them to anticipate resource extraction and...
: Policies intending to reduce future CO2 emissions may lead to increased emissions in the present. - Inventor's paradoxInventor's paradoxThe inventor's paradox is a phenomenon that occurs in seeking a solution to a given problem. Instead of solving a specific type of problem, which would seem intuitively easier, it is easier to solve a more general problem, which covers the specifics of the sought after solution...
: It is easier to solve a more general problem that covers the specifics of the sought-after solution. - Kavka's toxin puzzleKavka's toxin puzzleKavka's toxin puzzle is a thought experiment about the possibility of forming an intention to perform an act which, following from reason, is an action one would not actually perform. It was presented by moral and political philosopher Gregory S...
: Can one intend to drink the non-deadly toxin, if the intention is the only thing needed to get the reward? - Morton's forkMorton's ForkA Morton's Fork is a choice between two equally unpleasant alternatives , or two lines of reasoning that lead to the same unpleasant conclusion...
: Choosing between unpalatable alternatives. - Navigation paradoxNavigation paradoxThe Navigation paradox states that increased navigational precision may result in increased collision risk. In the case of ships and aircraft, the advent of Global Positioning System navigation has enabled craft to follow navigational paths with such greater precision , that, without better...
: Increased navigational precision may result in increased collision risk. - Newcomb's paradoxNewcomb's paradoxNewcomb's paradox, also referred to as Newcomb's problem, is a thought experiment involving a game between two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox is disputed....
: How do you play a game against an omniscient opponent? - Paradox of hedonismParadox of hedonismThe paradox of hedonism, also called the pleasure paradox, is the idea in the study of ethics which points out that pleasure and happiness are strange phenomena that do not obey normal principles....
: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness. - Paradox of toleranceParadox of toleranceThe tolerance paradox arises from a problem that a tolerant person might be antagonistic toward intolerance, hence intolerant of it. The tolerant individual would then be by definition intolerant of intolerance...
: Should one tolerate intolerance; if intolerance would destroy the possibility of tolerance? - Paradox of votingParadox of votingThe paradox of voting, also called Downs paradox, is that for a rational, self-interested voter the costs of voting will normally exceed the expected benefits. Because the chance of exercising the pivotal vote The paradox of voting, also called Downs paradox, is that for a rational,...
: Also known as the Downs paradox. For a rational, self-interested voter the costs of voting will normally exceed the expected benefits, so why do people keep voting? - Parrondo's paradoxParrondo's paradoxParrondo's paradox, a paradox in game theory, has been described as: A losing strategy that wins. It is named after its creator, Spanish physicist Juan Parrondo, who discovered the paradox in 1996...
: It is possible to play two losing games alternately to eventually win. - Prevention paradoxPrevention paradoxThe prevention paradox was first formally described in 1981 by the epidemiologist Geoffrey Rose. The prevention paradox describes the seemingly contradictory situation where the majority of cases of a disease come from a population at low or moderate risk of that disease, and only a minority of...
: For one person to benefit, many people have to change their behavior — even though they receive no benefit, or even suffer, from the change. - Prisoner's dilemmaPrisoner's dilemmaThe prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
: Two people might not cooperate even if it is in both their best interests to do so. - Relevance paradoxRelevance ParadoxThe relevance paradox describes an attempt to gather information relevant to a decision, which fails because the elimination of information perceived as distracting or unnecessary and thus detrimental to making an optimal decision, also excludes information that is actually crucial.-Definition:In...
: Sometimes relevant information is not sought out because its relevance only becomes clear after the information is available. - Voting paradoxVoting paradoxThe voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
: Also known as Condorcet's paradox and paradox of voting. A group of separately rational individuals may have preferences that are irrational in the aggregate.
Physics
- Cool tropics paradoxCool tropics paradoxThe cool tropics paradox refers to an apparent difference between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the colder temperatures which proxies suggested were present...
: A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the colder temperatures that proxies suggest were present. - The holographic principleHolographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
: The amount of information that can be stored in a given volume is not proportional to the volume but to the area that bounds that volume. - Irresistible force paradoxIrresistible force paradoxThe Irresistible force paradox, also the unstoppable force paradox, is a classic paradox formulated as "What happens when an unstoppable force meets an immovable object?" This paradox is a form of the omnipotence paradox, which is a simple demonstration that challenges omnipotence:...
: What would happen if an unstoppable force hit an immovable object?
Astrophysics
- Algol paradoxAlgol paradoxIn stellar astronomy, the Algol paradox is an apparently paradoxical situation when elements of a binary star seem to evolve in discord with the established theories of stellar evolution...
: In some binaries the partners seem to have different ages, even though they're thought to have formed at the same time. - Faint young Sun paradoxFaint young Sun paradoxThe faint young Sun paradox or problem describes the apparent contradiction between observations of liquid water early in the Earth's history and the astrophysical expectation that the Sun's output would be only 70% as intense during that epoch as it is during the modern epoch. The issue was raised...
: The apparent contradiction between observations of liquid water early in the Earth's history and the astrophysical expectation that the output of the young sun would have been insufficient to melt ice on earth. - The GZK paradox: High-energy cosmic rayCosmic rayCosmic rays are energetic charged subatomic particles, originating from outer space. They may produce secondary particles that penetrate the Earth's atmosphere and surface. The term ray is historical as cosmic rays were thought to be electromagnetic radiation...
s have been observed that seem to violate the Greisen-Zatsepin-Kuzmin limitGreisen-Zatsepin-Kuzmin limitThe Greisen–Zatsepin–Kuzmin limit is a theoretical upper limit on the energy of cosmic rays coming from "distant" sources. The limit is 5×1019 eV, or about 8 joules. The limit is set by slowing-interactions of cosmic ray protons with the microwave background radiation over long distances...
, which is a consequence of special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.
Classical mechanics
- Archer's paradoxArcher's paradoxThe term archer's paradox was coined by Robert P. Elmer in the 1930s. The paradox refers to the phenomenon that in order to strike the center of the target, the arrow must be pointed slightly to the side of the target...
: An archer must, in order to hit his target, not aim directly at it, but slightly to the side. - Archimedes paradoxArchimedes paradoxThe Archimedes paradox, named after Archimedes of Syracuse, states that an object can float in a quantity of water that has less volume than the object itself, if its average density is less than that of water....
: A massive battleship can float in a few litres of water. - Aristotle's wheel paradoxAristotle's wheel paradoxAristotle's wheel paradox is a paradox from the Greek work Mechanica traditionally attributed to Aristotle. There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution...
: Rolling joined concentric wheels seem to trace the same distance with their circumferences, even though the circumferences are different. - Carroll's paradoxCarroll's paradoxIn physics, Carroll's paradox arises when considering the motion of a falling rigid rod that is specially constrained. Considered one way, the angular momentum stays constant; considered in a different way, it changes. It is named after Michael M...
: The angular momentum of a stick should be zero, but is not. - D'Alembert's paradoxD'Alembert's paradoxIn fluid dynamics, d'Alembert's paradox is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to the fluid...
: Flow of an inviscidViscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
fluid produces no net force on a solid body. - Denny's paradoxDenny's paradoxIn the study of animal locomotion on the surface layer of water, Denny's paradox refers to the apparent impossibility of surface-dwelling animals such as the water strider generating enough propulsive force to move...
: Surface-dwelling arthropodArthropodAn arthropod is an invertebrate animal having an exoskeleton , a segmented body, and jointed appendages. Arthropods are members of the phylum Arthropoda , and include the insects, arachnids, crustaceans, and others...
s (such as the water striderWater striderGerridae is a family of true bugs in the order Hemiptera, commonly known as water striders, water bugs, magic bugs, pond skaters, skaters, skimmers, water scooters, water skaters, water skeeters, water skimmers, water skippers, water spiders, or Jesus bugs...
) should not be able to propel themselves horizontally. - Elevator paradoxElevator paradox (physics)The elevator paradox relates to a hydrometer placed on an "elevator" or vertical conveyor that, by moving to different elevations, changes the atmospheric pressure. In this classic demonstration, the floating hydrometer remains at an equilibrium position. Essentially, a hydrometer measures...
: Even though hydrometers are used to measure fluid density, a hydrometer will not indicate changes of fluid density caused by changing atmospheric pressure. - Feynman sprinklerFeynman sprinklerA Feynman sprinkler, also referred to as a Feynman inverse sprinkler or as a reverse sprinkler, is a sprinkler-like device which is submerged in a tank and made to suck in the surrounding fluid...
: Which way does a sprinkler rotate when submerged in a tank and made to suck in the surrounding fluid? - Painlevé paradoxPainlevé paradoxThe Painlevé paradox is a well known example by Paul Painlevé in rigid-body dynamics that showed that rigid-body dynamics with both contact friction and Coulomb friction is inconsistent. This is due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in...
: Rigid-body dynamics with contact and friction is inconsistent. - Tea leaf paradoxTea leaf paradoxThe tea leaf paradox describes a phenomenon where tea leaves in a cup of tea migrate to the center and bottom of the cup after being stirred rather than being forced to the edges of the cup, as would be expected from a spiral centrifugal force...
: When a cup of tea is stirred, the leaves assemble in the center, even though centrifugal force pushes them outward.
Cosmology
- Bentley's paradoxBentley's paradoxBentley's paradox is a cosmological paradox pointing to a problem occurring when Newton's theory of the gravitation is applied to cosmology: "According to Newton, each star in the universe ought to be attracted towards every other star. They should not remain motionless, at a constant distance from...
: In a Newtonian universe, gravitation should pull all matter into a single point. - Fermi paradoxFermi paradoxThe Fermi paradox is the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilizations and the lack of evidence for, or contact with, such civilizations....
: If there are, as probability would suggest, many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious? - Heat death paradox: Since the universe is not infinitely old, it cannot be infinite in extent.
- Olbers' paradoxOlbers' paradoxIn astrophysics and physical cosmology, Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. The argument is also...
: Why is the night sky black if there is an infinity of stars?
Electromagnetism
- Faraday paradoxFaraday paradoxThe Faraday paradox is an experiment that illustrates Michael Faraday's law of electromagnetic induction. Faraday deduced this law in 1831, after inventing the first electromagnetic generator or dynamo, but was never satisfied with his own explanation of the paradox.-The equipment:The experiment...
: An apparent violation of Faraday's law of electromagnetic induction.
Quantum mechanics
- Bell's theoremBell's theoremIn theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
: Why do measured quantum particles not satisfy mathematical probability theory? - Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
? - Extinction paradox: In the small wavelength limit, the total scattering cross section of an impenetrable sphere is twice its geometrical cross-sectional area (which is the value obtained in classical mechanics).
- Hardy's paradoxHardy's paradoxHardy's paradox is a thought experiment in quantum mechanics devised by Lucien Hardy in which a particle and its antiparticle may interact without annihilating each other...
: How can we make inferences about past events that we haven't observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to? - Klein paradoxKlein paradoxIn 1929, physicist Oskar Klein obtained a surprising result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping...
: When the potential of a potential barrier becomes similar to the mass of the impinging particle, it becomes transparent. - The Mott problemMott problemIn quantum mechanics, the Mott problem is a paradox that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics...
: spherically symmetric wave functions, when observed, produce linear particle tracks. - Quantum LC circuit paradox: Energies stored on capacitance and inductance are not equal to the ground state energy of the quantum oscillator.
- Quantum pseudo-telepathyQuantum pseudo-telepathyQuantum pseudo-telepathy is a phenomenon in quantum game theory resulting in anomalously high success rates in coordination games between separated players. These high success rates would require communication between the players in a purely classical world; however, the game is set up such that...
: Two players who can not communicate accomplish tasks that seemingly require direct contact. - Schrödinger's cat paradox: A quantum paradox — Is the cat alive or dead before we look?
Relativity
- Bell's spaceship paradoxBell's spaceship paradoxBell's spaceship paradox is a thought experiment in special relativity involving accelerated spaceships and strings. The results of this thought experiment are for many people paradoxical. While J. S. Bell's 1976 version of the paradox is the most widely known, it was first designed by E. Dewan and M...
: concerning relativityTheory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
. - Black hole information paradoxBlack hole information paradoxThe black hole information paradox results from the combination of quantum mechanics and general relativity. It suggests that physical information could disappear in a black hole, allowing many physical states to evolve into the same state...
: Black holes violate a commonly assumed tenet of science — that information cannot be destroyed. - Ehrenfest paradoxEhrenfest paradoxThe Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry...
: On the kinematics of a rigid, rotating disk. - Ladder paradoxLadder paradoxThe ladder paradox is a thought experiment in special relativity. It involves a ladder travelling horizontally and undergoing a length contraction, the result of which being that it can fit into a much smaller garage...
: A classic relativity problem. - Mocanu's velocity composition paradox: a paradox in special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. - Supplee's paradoxSupplee's paradoxIn relativistic physics, Supplee's paradox arises when considering the buoyant force exerted on a relativistic bullet immersed in a fluid subject to an ambient gravitational field...
: the buoyancyBuoyancyIn physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...
of a relativisticTheory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
object (such as a bulletBulletA bullet is a projectile propelled by a firearm, sling, or air gun. Bullets do not normally contain explosives, but damage the intended target by impact and penetration...
) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest. - Trouton-Noble or Right-angle lever paradox. Does a torqueTorqueTorque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
arise in static systems when changing frames? - Twin paradoxTwin paradoxIn physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
: The theory of relativityTheory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
predicts that a person making a round trip will return younger than his or her identical twin who stayed at home.
Thermodynamics
- Gibbs paradoxGibbs paradoxIn statistical mechanics, a semi-classical derivation of the entropy that doesn't take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive...
: In an ideal gasIdeal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, is entropyEntropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
an extensive variable? - Loschmidt's paradoxLoschmidt's paradoxLoschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics...
: Why is there an inevitable increase in entropy when the laws of physicsPhysical lawA physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...
are invariant under time reversalT-symmetryT Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...
? The time reversal symmetry of physical laws appears to contradict the second law of thermodynamics. - Maxwell's DemonMaxwell's demonIn the philosophy of thermal and statistical physics, Maxwell's demon is a thought experiment created by the Scottish physicist James Clerk Maxwell to "show that the Second Law of Thermodynamics has only a statistical certainty." It demonstrates Maxwell's point by hypothetically describing how to...
: The second law of thermodynamicsSecond law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
seems to be violated by a cleverly operated trapdoor. - Mpemba paradoxMpemba effectThe Mpemba effect is the observation that warmer water sometimes freezes faster than colder water. Although the observation has been verified, there is no single scientific explanation for the effect.-Historical observations:...
: Hot water can, under certain conditions, freeze faster than cold water, even though it must pass the lower temperature on the way to freezing.
Biology
- Paradox of enrichmentParadox of enrichmentThe paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971. He described an effect in six predator-prey models wherein increasing the food available to the prey caused the predator's population to destabilize...
: Increasing the food available to an ecosystem may lead to instability, and even to extinction. - French paradoxFrench paradoxThe French Paradox is the observation that French people suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats...
: the observation that the French suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats. - Glucose paradoxGlucose paradoxGlucose paradox is a theory formulated by biochemist Dr. J. D. McGarry and his research associates. It is a theory that demonstrates the importance of lactic acid in carbohydrate metabolism...
: The large amount of glycogen in the liver cannot be explained by its small glucose absorption. - Gray's ParadoxGray's ParadoxGray's Paradox is a paradox posed in 1936 by British zoologist Sir James Gray. The paradox was to figure out how dolphins can obtain such high speeds and accelerations with what appears to be a small muscle mass. Gray made an estimate of the power a dolphin could exert based on its physiology, and...
: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations. - Hispanic ParadoxHispanic ParadoxThe Hispanic Paradox, or Latino Paradox, also known as the "Epidemiologic Paradox," refers to the epidemiological finding that Hispanic and Latino Americans tend to have health outcomes that paradoxically are comparable to, or in some cases better than, those of their U.S. white counterparts, even...
: The finding that Hispanics in the U.S. tend to have substantially better health than the average population in spite of what their aggregate socio-economic indicators predict. - Lombard's ParadoxLombard's ParadoxLombard's Paradox describes a paradoxical muscular contraction in humans. When rising to stand from a sitting or squatting position, both the hamstrings and quadriceps contract at the same time, despite their being antagonists to each other....
: When rising to stand from a sitting or squatting position, both the hamstrings and quadriceps contract at the same time, despite their being antagonists to each other. - Mexican paradoxMexican paradoxThe Mexican paradox is the observation that the Mexican people exhibit a surprisingly low incidence of low birth mass, contrary to what would be expected from their socioeconomic status...
: Mexican children tend to have higher birth weights than can be expected from their socio-economic status. - Paradox of the pesticidesParadox of the pesticidesThe paradox of the pesticides is a paradox that states that by applying pesticide to a pest, one may in fact increase its abundance. This happens when the pesticide upsets natural predator-prey dynamics in the ecosystem....
: Applying pesticide to a pest may increase the pest's abundance. - Paradox of the planktonParadox of the planktonIn aquatic biology, the paradox of the plankton is the name given to the situation where a limited range of resources supports a much wider range of planktonic organisms...
: Why are there so many different species of phytoplankton, even though competition for the same resources tends to reduce the number of species? - PetoRichard PetoSir Richard Peto FRS is Professor of Medical Statistics and Epidemiology at the University of Oxford.He attended Taunton's School in Southampton and subsequently studied Natural Sciences at Cambridge University....
's paradox: Humans gets cancer with high frequency, while larger mammals, like whales, don’t. If cancer is essentially a negative outcome lottery at the cell level, and larger organisms have more cells, and thus more potentially cancerous cell divisions, you would expect larger organisms to be more predisposed to cancer. - Pulsus paradoxusPulsus paradoxusIn medicine, a pulsus paradoxus , also paradoxic pulse or paradoxical pulse, is defined as an exaggeration of the normal variation during the inspiratory phase of respiration, in which the blood pressure declines as one inhales and increases as one exhales...
: Sometimes it is possible to hear, with a stethoscope, heartbeats that can't be felt at the wrist. - Sherman paradox: An anomalous pattern of inheritance in the fragile X syndrome.
- Temporal paradox (paleontology)Temporal paradox (paleontology)The temporal paradox, or time problem is a controversial issue in the evolutionary relationships of birds. It was described by paleornithologist Alan Feduccia.-Objection to consensus:The concept of a "temporal paradox" is based on the following facts...
: When did the ancestors of birds live?
Chemistry
- Faraday paradox (electrochemistry)Faraday paradox (electrochemistry)The Faraday paradox was a once inexplicable aspect of the reaction between nitric acid and steel. Around 1830, the English scientist Michael Faraday found that diluted nitric acid would attack steel, but concentrated nitric acid would not. The attempt to explain this discovery led to advances in...
: Diluted nitric acid will corrode steel, while concentrated nitric acid doesn't. - Levinthal paradoxLevinthal paradoxLevinthal's paradox is a thought experiment, also constituting a self-reference in the theory of protein folding. In 1969, Cyrus Levinthal noted that, because of the very large number of degrees of freedom in an unfolded polypeptide chain, the molecule has an astronomical number of possible...
: The length of time that it takes for a protein chain to find its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations. - SAR paradox: Exceptions to the principle that a small change in a moleculeMoleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
causes a small change in its chemical behaviour are frequently profound.
Time
- Grandfather paradoxGrandfather paradoxThe grandfather paradox is a proposed paradox of time travel first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent . The paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler's...
: You travel back in timeTime travelTime travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...
and kill your grandfather before he conceives one of your parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your grandfather. - Bootstrap paradoxBootstrap paradoxThe bootstrap paradox is a paradox of time travel in which information or objects can exist without having been created. After information or an object is sent back in time, it is recovered in the present and becomes the very object/information that was initially brought back in time in the first...
: Can a time traveler send himself information with no outside source? - Predestination paradoxPredestination paradoxA predestination paradox is a paradox of time travel that is often used as a convention in science fiction. It exists when a time traveller is caught in a loop of events that "predestines" or "predates" them to travel back in time...
: A man travels back in time to discover the cause of a famous fire. While in the building where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would inspire him, years later, to travel back in time. The bootstrap paradoxBootstrap paradoxThe bootstrap paradox is a paradox of time travel in which information or objects can exist without having been created. After information or an object is sent back in time, it is recovered in the present and becomes the very object/information that was initially brought back in time in the first...
is closely tied to this, in which, as a result of time travel, information or objects appear to have no beginning. - Temporal paradoxTemporal paradoxTemporal paradox is a theoretical paradoxical situation that happens because of time travel. A time traveler goes to the past, and does something that would prevent him from time travel in the first place...
: What happens when a time traveler does things in the past that prevent him from doing them in the first place?
Philosophy
- Paradox of analysisParadox of analysisThe Paradox of Analysis is a paradox that concerns how an analysis can be both correct and informative. Although the problem takes its origin from the conflict in Plato's Meno, the term "Paradox of Analysis" actually came from philosopher G. E. Moore's work in 1952.-The Paradox:A conceptual...
: It seems that no conceptual analysis can both meet the requirement of correctness and of informativeness. - Buridan's bridgeBuridan's bridgeBuridan's Bridge is described by Jean Buridan, one of the most famous and influential philosophers of the Late Middle Ages, in his book Sophismata...
: Will Plato throw Socrates into the water or not? - Chicken or the egg: Which came first, the chicken or the egg?
- Fitch's paradoxFitch's paradox of knowabilityFitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that any truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that any truth is known...
: If all truths are knowable, then all truths must in fact be known. - Paradox of free willArgument from free willThe argument from free will contends that omniscience and free will are incompatible, and that any conception of God that incorporates both properties is therefore inherently contradictory. The argument may focus on the incoherence of people having free will, or else God himself having free will...
: If God knew how we will decide when he created us, how can there be free will? - Goodman's paradox: Why can induction be used to confirm that things are "green", but not to confirm that things are "grue"?
- Paradox of hedonismParadox of hedonismThe paradox of hedonism, also called the pleasure paradox, is the idea in the study of ethics which points out that pleasure and happiness are strange phenomena that do not obey normal principles....
: In seeking happiness, one does not find happiness. - Hutton's ParadoxDream argumentThe dream argument is the postulation that the act of dreaming provides preliminary evidence that the senses we trust to distinguish reality from illusion should not be fully trusted, and therefore any state that is dependent on our senses should at the very least be carefully examined and...
: If asking oneself "Am I dreaming?" in a dream proves that one is, what does it prove in waking life? - Liberal paradoxLiberal paradoxThe liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his impossibility theorem, which showed that within a system of menu-independent social choice, it is impossible to have both a commitment to "Minimal Liberty", which was defined as the...
: "Minimal Liberty" is incompatible with Pareto optimality. - Mere addition paradoxMere addition paradoxThe mere addition paradox is a problem in ethics, identified by Derek Parfit, and appearing in his book, Reasons and Persons . The paradox identifies apparent inconsistency between three seemingly true beliefs about population ethics by arguing that utilitarianism leads to an apparent overpopulated...
: Also known as Parfit's paradox: Is a large population living a barely tolerable life better than a small, happy population? - Moore's paradoxMoore's paradoxMoore's paradox concerns the putative absurdity involved in asserting a first-person present-tense sentence such as 'It's raining but I don't believe that it is raining' or 'It's raining but I believe that it is not raining'. The first author to note this apparent absurdity was G.E. Moore...
: "It's raining, but I don't believe that it is." - Newcomb's paradoxNewcomb's paradoxNewcomb's paradox, also referred to as Newcomb's problem, is a thought experiment involving a game between two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox is disputed....
: A paradoxical game between two players, one of whom can predict the actions of the other. - Paradox of nihilismParadox of nihilism- Meaning :According to Hegarty, the paradox of nihilism is "that the absence of meaning seems to be some sort of meaning".- Truth :Luhmann construes the paradox as stating "that consequently, only the untrue could be the truth"...
: Several distinct paradoxes share this name. - Omnipotence paradoxOmnipotence paradoxThe omnipotence paradox is a family of semantic paradoxes which address two issues: Is an omnipotent entity logically possible? and What do we mean by 'omnipotence'?. The paradox states that if a being can perform any action, then it should be able to create a task it is unable to perform, and...
: Can an omnipotent being create a rock too heavy for itself to lift? - Preface paradoxPreface paradoxThe preface paradox, or the paradox of the preface, was introduced by David Makinson in 1965. Similar to the lottery paradox, it presents an argument according to which it can be rational to accept mutually incompatible beliefs...
: The author of a book may be justified in believing that all his statements in the book are correct, at the same time believing that at least one of them is incorrect. - Problem of evilProblem of evilIn the philosophy of religion, the problem of evil is the question of how to explain evil if there exists a deity that is omnibenevolent, omnipotent, and omniscient . Some philosophers have claimed that the existences of such a god and of evil are logically incompatible or unlikely...
(EpicureanEpicureanismEpicureanism is a system of philosophy based upon the teachings of Epicurus, founded around 307 BC. Epicurus was an atomic materialist, following in the steps of Democritus. His materialism led him to a general attack on superstition and divine intervention. Following Aristippus—about whom...
paradox): The existence of evil seems to be incompatible with the existence of an omnipotent, omniscient, and morally perfect God. - Zeno's paradoxesZeno's paradoxesZeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...
: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on..." (This is also a paradox of the infinite)
Mysticism
- TzimtzumTzimtzumTzimtzum is a term used in the kabbalistic teaching of Isaac Luria, explaining his concept that God began the process of creation by "contracting" his infinite light in order to allow for a "conceptual space" in which a finite and seemingly independent world could exist...
: In Kabbalah, how to reconcile self-awareness of finite Creation with Infinite Divine source, as an emanatedEmanationismEmanationism is an idea in the cosmology or cosmogony of certain religious or philosophical systems. Emanation, from the Latin emanare meaning "to flow from" or "to pour forth or out of", is the mode by which all things are derived from the First Reality, or Principle...
causal chain would seemingly nullify existence. Luria's initial withdrawal of God in Hasidic panentheismPanentheismPanentheism is a belief system which posits that God exists, interpenetrates every part of nature and timelessly extends beyond it...
involves simultaneous illusionismAcosmismAcosmism, in contrast to pantheism, denies the reality of the universe, seeing it as ultimately illusory, , and only the infinite unmanifest Absolute as real....
of Creation (Upper Unity) and self-aware existence (Lower Unity), God encompassing logical opposites.
Economics
- Allais paradoxAllais paradoxThe Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...
: A change in a possible outcome that is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory. - Arrow information paradoxArrow Information ParadoxThe Arrow information paradox, named after Kenneth Arrow, is a problem that companies face when managing intellectual property across their boundaries. This happens when they seek external technologies for their business or external markets for their own technologies.The paradox is that the...
: To sell information you need to give it away before the sale. - Bertrand paradoxBertrand paradox (economics)In economics and commerce, the Bertrand paradox—named after its creator, Joseph Bertrand—describes a situation in which two players reach a state of Nash equilibrium where both firms charge a price equal to marginal cost. The paradox is that in reality, it usually takes a large number of firms to...
: Two players reaching a state of Nash equilibriumNash equilibriumIn game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
both find themselves with no profits. - Braess's paradox: Adding extra capacity to a network can reduce overall performance.
- Demographic-economic paradoxDemographic-economic paradoxThe demographic-economic paradox is the inverse correlation found between wealth and fertility within and between nations. The higher the degree of education and GDP per capita of a human population, subpopulation or social stratum, the fewer children are born in any industrialized country...
: nations or subpopulations with higher GDP per capita are observed to have fewer children, even though a richer population can support more children. - Diamond-water paradoxParadox of valueThe paradox of value is the apparent contradiction that, although water is on the whole more useful, in terms of survival, than diamonds, diamonds command a higher price in the market. The philosopher Adam Smith is often considered to be the classic presenter of this paradox...
(or paradox of value) Water is more useful than diamonds, yet is a lot cheaper. - Downs–Thomson paradox: Increasing road capacity at the expense of investments in public transport can make overall congestion on the road worse.
- Easterlin paradoxEasterlin paradoxThe Easterlin Paradox is a key concept in happiness economics. It is named for economist and USC Professor Richard Easterlin who discussed the factors contributing to happiness in the 1974 paper "Does Economic Growth Improve the Human Lot? Some Empirical Evidence." Easterlin found that within a...
: For countries with income sufficient to meet basic needs, the reported level of happiness does not correlate with national income per person. - Edgeworth paradoxEdgeworth paradoxIn economics, the Edgeworth paradox describes a situation in which two players cannot reach a state of equilibrium with pure strategies, i.e. each charging a stable price....
: With capacity constraints, there may not be an equilibrium. - Ellsberg paradoxEllsberg paradoxThe Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices violate the expected utility hypothesis.An alternate viewpoint is that expected utility theory does not properly describe actual human choices...
: People exhibit ambiguity aversionAmbiguity aversionIn decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...
(as distinct from risk aversionRisk aversionRisk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
), in contradiction with expected utility theory. - European paradoxEuropean paradox"European paradox" refers to the perceived failure of European countries to translate scientific advances into marketable innovations. The term was coined in a European Commission Green Paper in 1995.- References :- External links :* * * , , May 2007...
: The perceived failure of European countries to translate scientific advances into marketable innovations. - Gibson's paradoxGibson's paradoxGibson's Paradox is the observation that the rate of interest and the general level of prices are positively correlated. It is named for British economist Alfred Herbert Gibson who noted the correlation in a 1923 article for Banker's Magazine....
: Why were interest rates and prices correlated? - Giffen paradoxGiffen goodIn economics and consumer theory, a Giffen good is one which people paradoxically consume more of as the price rises, violating the law of demand. In normal situations, as the price of a good rises, the substitution effect causes consumers to purchase less of it and more of substitute goods...
: Increasing the price of bread makes poor people eat more of it. - Icarus paradoxIcarus ParadoxThe Icarus paradox is a neologism coined by Danny Miller, and popularized by his 1990 book by the same name, for the observed phenomenon of businesses that fail abruptly after a period of apparent success....
: Some businesses bring about their own downfall through their own successes. - Jevons paradoxJevons paradoxIn economics, the Jevons paradox is the proposition that technological progress that increases the efficiency with which a resource is used, tends to increase the rate of consumption of that resource...
: Increases in efficiency lead to even larger increases in demand. - Leontief paradoxLeontief paradoxLeontief's paradox in economics is that the country with the world's highest capital-per worker has a lower capital/labor ratio in exports than in imports....
: Some countries export labor-intensive commodities and import capital-intensive commodities, in contradiction with Heckscher–Ohlin theory. - Lucas paradoxLucas paradoxIn economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker....
: Capital is not flowing from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker, and therefore higher returns to capital. - Mandeville's paradoxMandeville's paradoxMandeville's paradox is named after Bernard Mandeville, who shows that actions which may be qualified as vicious with regard to individuals have benefits for society as a whole. This is already clear from the subtitle of his most famous work, The Fable of The Bees: ‘Private Vices, Publick Benefits’...
: Actions that may be vicious to individuals may benefit society as a whole. - Metzler paradoxMetzler paradoxIn economics, the Metzler paradox is the theoretical possibility that the imposition of a tariff on imports may reduce the relative internal price of that good. It was proposed by Lloyd Metzler in 1949 upon examination of tariffs within the Heckscher–Ohlin model...
: The imposition of a tariff on imports may reduce the relative internal price of that good. - Paradox of thriftParadox of thriftThe paradox of thrift is a paradox of economics, popularized by John Maynard Keynes, though it had been stated as early as 1714 in The Fable of the Bees, and similar sentiments date to antiquity...
: If everyone saves more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population. - Paradox of toilParadox of toilThe paradox of toil is the economic hypothesis that employment will continue to shrink when "the short-term nominal interest rate is zero and there aredeflationary pressures and output contraction"...
: If everyone tries to work during times of recession, lower wages will reduce prices, leading to more deflationary expectations, leading to further thrift, reducing demand and thereby reducing employment. - Productivity paradoxProductivity paradoxThe productivity paradox was analyzed and popularized in a widely-cited article by Erik Brynjolfsson, which noted the apparent contradiction between the remarkable advances in computer power and the relatively slow growth of productivity at the level of the whole economy, individual firms and many...
(also known as Solow computer paradox): Worker productivity may go down, despite technological improvements. - Scitovsky paradoxScitovsky paradoxThe Scitovsky paradox is a theory which states that in welfare economics there is no increase in social welfare by a return to the original part of the losers...
: Using the Kaldor–Hicks criterion, an allocation A may be more efficient than allocation B, while at the same time B is more efficient than A. - Service recovery paradox: Successfully fixing a problem with a defective product may lead to higher consumer satisfaction than in the case where no problem occurred at all.
- St. Petersburg paradoxSt. Petersburg paradoxIn economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
: People will only offer a modest fee for a reward of infinite expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
.
Perception
- Tritone paradoxTritone paradoxThe tritone paradox is an auditory illusion in which a sequentially played pair of Shepard tones separated by an interval of a tritone, or half octave, is heard as ascending by some people and as descending by others. Different populations tend to favor one of a limited set of different spots...
: An auditory illusion in which a sequentially played pair of Shepard tones is heard as ascending by some people and as descending by others.
Politics
- Stability-instability paradoxStability-instability paradoxThe stability-instability paradox is an international relations theory regarding the effect of nuclear weapons and mutually assured destruction. It states that when two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor...
: When two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases.
History
- Georg Wilhelm Friedrich HegelGeorg Wilhelm Friedrich HegelGeorg Wilhelm Friedrich Hegel was a German philosopher, one of the creators of German Idealism. His historicist and idealist account of reality as a whole revolutionized European philosophy and was an important precursor to Continental philosophy and Marxism.Hegel developed a comprehensive...
: We learn from history that we do not learn from history. (paraphrased)
See also
- Absence paradoxAbsence paradoxThe absence paradox while named a paradox, is more precisely an informal fallacy and humorous misuse of language which results in the conclusion that "No one is ever present." The statement of the argument is some formulation of the following:...
: No one is ever "here". - Bracketing paradoxBracketing paradoxIn linguistic morphology, the term bracketing paradox refers to morphologically complex words which apparently have more than one incompatible analysis, or bracketing, simultaneously....
: Is a "nuclear physicist" a physicist who had an accidentDemon coreThe Demon core was the nickname given to a subcritical mass of plutonium that accidentally went briefly critical in two separate accidents at the Los Alamos laboratory in 1945 and 1946. Each incident resulted in the acute radiation poisoning and subsequent death of a scientist...
, or is it just someone who studied nuclear physics? - Buttered cat paradoxButtered cat paradoxThe buttered cat paradox is a paradox based on the tongue-in-cheek combination of two adages:* Cats always land on their feet.* Buttered toast always lands buttered side down....
: Humorous example of a paradox from contradicting folk tales. - Excusable negligenceExcusable negligenceExcusable negligence is a paradoxical phrase, since if the failure to exercise reasonable care under the circumstances is excusable, there is no negligence. 38 Am J1st Negl § 12...
: If a behavior is excusable, it is not negligence. - Godel's incompleteness theoremsGödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
- Ignore all rulesIgnore all rules"Ignore all rules" is a rule to set rules aside.The stance of ignoring all rules is itself a rule, constituting a paradox. A scholar on Immanuel Kant's view of genius states that this critical stance is accordingly transcended by the autonomy of genius: "Genius demonstrates its autonomy not by...
: To obey this rule, it is necessary to ignore it. - Impossible objectImpossible objectAn impossible object is a type of optical illusion consisting of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object although it is not actually possible for such an object to exist An impossible...
: A type of optical illusion. - Intentionally blank pageIntentionally blank pageAn intentionally blank page is a page that is devoid of content, and may be unexpected. Such pages may serve purposes ranging from place-holding to space-filling and content separation...
: Many documents contain pages on which the text "This page is intentionally left blank" is printed, thereby making the page not blank. - Invalid proofInvalid proofIn mathematics, certain kinds of mistakes in proof, calculation, or derivation are often exhibited, and sometimes collected, as illustrations of the concept of mathematical fallacy...
: An apparently correct mathematical derivation that leads to an obvious contradiction. - Little WymondleyLittle WymondleyLittle Wymondley is a village situated between Hitchin and Stevenage in Hertfordshire. Paradoxically, it is larger than its near neighbour Great Wymondley...
is much larger than nearby Great WymondleyGreat WymondleyGreat Wymondley is a village situated near Hitchin in Hertfordshire. It is dominated by its small Norman church with a large apsidal chancel and 15th century nave roof. There are also earthworks of a former motte-and-bailey castle, and Delamere House, an elegant Elizabethan building...
. - Logical fallacy: A misconception resulting from incorrect reasoning in argumentation.
- Moral paradox: A situation in which moral imperatives clash without clear resolution.
- Moravec's paradoxMoravec's paradoxMoravec's paradox is the discovery by artificial intelligence and robotics researchers that, contrary to traditional assumptions, high-level reasoning requires very little computation, but low-level sensorimotor skills require enormous computational resources. The principle was articulated by Hans...
: Logical thought is hard for humans and easy for computers, but picking a screw from a box of screws is an unsolved problem. - Movement paradoxMovement paradoxA movement paradox is a grammatical phenomenon which, particularly according to proponents of lexical functional grammar, presents some problems for a transformational approach to syntax...
: In transformational linguistics, there are pairs of sentences in which the sentence without movement is ungrammatical while the sentence with movement is not. - Observer's paradoxObserver's paradoxIn the social sciences, the observer's paradox refers to phenomena where the observation of an event or experiment is influenced by the presence of the observer/investigator.-In linguistics:...
: The outcome of an event or experiment is influenced by the presence of the observer. - Paradox gunParadox gunA Paradox gun is a firearm made by Holland & Holland with the last two to three inches of the muzzle rifled and the rest smooth, intended to be used as both a rifle and shotgun....
: A gun that has characteristics of both (smoothbore) shotguns and rifles. - Paradoxical laughterParadoxical laughterParadoxical laughter is an exaggerated expression of humour which is unwarranted by external events. It may be uncontrollable laughter which may be recognised as inappropriate by the person involved...
: Inappropriate laughter, often recognized as such by the laughing person. - The Paradox of Anti-SemitismThe Paradox of Anti-SemitismThe Paradox of Anti-Semitism is a book written by American Reform Rabbi Dan Cohn-Sherbok and published in 2006. The book primarily asserts that anti-Semitism has served as a useful external reinforcement for Jewish culture in the diaspora, and that the lack of such external persecutions and...
: A book arguing that the lack of external persecutions and antagonisms results in the dissolution of Jewish identity, a theory that resonates in works of Dershowitz and Sartre. - Proof that 0.999... equals 1
- PuzzlePuzzleA puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...
- Stapp's Ironical ParadoxStapp's Ironical ParadoxStapp's Ironical Paradox was formulated by Colonel John Paul Stapp and says "The universal aptitude for ineptitude makes any human accomplishment an incredible miracle."-References:...
: "The universal aptitude for ineptitude makes any human accomplishment an incredible miracle." - Status paradoxStatus paradoxA status paradox can be one of several paradoxes involving "status", in the meaning of either medical state or social status.- Status paradox of migration :...
: Several paradoxes involve the concept of medical or social status.