Paradox

The impossible cube is a paradoxical object.

A paradox (from the Latin paradoxa, 'the opposite of common opinion') or antilogy is an idea logically contradictory or opposed to what is considered true to the general opinion. A paradox is also considered a proposition that appears to be false or that violates common sense, but does not entail a logical contradiction, as opposed to a sophistry that only appears to be true. be valid reasoning. Some paradoxes are apparently valid reasoning, which start from apparently true premises, but which lead to contradictions or situations contrary to common sense. In rhetoric, it is a figure of thought that consists of using expressions or sentences that imply contradiction. Paradoxes are thought-provoking and often used by philosophers to reveal the complexity of reality. The paradox also makes it possible to demonstrate the limitations of human understanding; the identification of paradoxes based on concepts that at first glance seem simple and reasonable has driven important advances in science, philosophy and mathematics.

Introduction

The term derives from the Latin form paradoxum, which is a loan from the Greek παράδοξον (paradoxon) 'unexpected, incredible, singular', etymologically formed by the preposition para-, which means "next to" or "a part of" plus the root doxon 'opinion, good judgment'.

Examples such as the liar paradox and other similar ones were already studied since antiquity in Greece, and in the Middle Ages they were known as insolubilia. The liar paradox is one of the first cases of self-referential paradox. In fact, among the recurring themes in the paradoxes are direct and indirect self-reference, infinity, circular definitions and confusion of levels of reasoning, although not all paradoxes are self-referential.

In moral philosophy a paradox plays a particularly important role in debates about ethics. For example, the ethical admonition: "love your neighbor" it stands not only in contrast, but also in contradiction, with a neighbor who tries to murder you: if successful, then one would not be able to love him. However, attacking or suppressing the abusive neighbor would not generally be considered loving. This can be called an ethical dilemma. Another example is the conflict between the mandate not to steal and the personal responsibility to feed the family, which under certain circumstances (wars, revolutions, natural disasters) could not be maintained without stealing.

Not all paradoxes are created equal. For example, the birthday paradox may be better defined as a surprise than a logical contradiction, while the resolution of the Curry paradox is still a major topic of debate.

Types of paradoxes

Not all paradoxes fit exactly into a single category. Some examples of paradoxes are:

According to their veracity and the conditions that form them

Some paradoxes only appear to be, since what they state is actually true or false, others contradict themselves, so they are considered true paradoxes, while others depend on their interpretation to be paradoxical or not, such as:

Veridic paradoxes

These are results that perhaps appear to be absurd despite their veracity being demonstrable. To this category belong most of the mathematical paradoxes.

• Birthday Paradox: What is the likelihood that two people at a meeting will meet years the same day?
• Galileo Paradox: Although not all numbers are perfect squares, there are no more numbers than perfect squares.
• Infinity hotel paradox: a hotel of infinite rooms can accept more guests, even if it is full.
• Paradox of the spherical band: it is not a paradox in strict sense, but it clashes with our common sense because it has a solution that seems impossible.

Antinomies

They are paradoxes that reach a self-contradictory result, correctly applying accepted modes of reasoning. They show failure in a previously accepted mode of reason, axiom, or definition. For example, the Grelling-Nelson Paradox points to genuine problems in our understanding of the ideas of truth and description. Many of them are specific cases, or adaptations, of the important Russell Paradox.

• Russell's Paradox: Is there a set of all sets that don't contain themselves?
• Curry Paradox: "If I am not mistaken, the world will end in ten days."
• Liar's paradox: "This prayer is false."
• Paradox of Grelling-Nelson: Is the word "heterological", which means "that it does not describe itself", heterological?
• Berry Paradox: "The least positive integer that cannot be defined with less than fifteen words."
• Paradox of interesting numbers: All integers present some specific interesting property, and therefore the set of non-interesting numbers is empty.

Definitional antinomies

These paradoxes are based on ambiguous definitions, without which they do not reach a contradiction. This type of paradox constitutes a literary resource, in whose use the English writer G. K. Chesterton, who was called the "prince of paradoxes", has stood out. Using the multiple meanings of the words, he sought to mark contrasts that would call attention to an issue that is not commonly considered. These paradoxes, as in his book & # 34; The paradoxes of Mr. Pond & # 34; (1936), are resolved in the course of the stories by clarifying a meaning or adding some key information.

• Paradox deaf: At what time does a lot stop being, when sand grains are removed?
• Teseo Paradox: When all parts of a boat have been replaced, is it still the same boat?
• Boixnet Paradox: I think, then I exist, but when I do not think, do I not exist?
• Examples of Paradox in Chesterton: "He was a very desirable foreigner, and despite that, he was not deported." "Once I met two men who were so completely agreed that, logically, one killed the other."

Conditional Paradoxes

They are only paradoxical if certain assumptions are made. Some of them show that these assumptions are false or incomplete.

• The egg or the hen: The old dilemma about what was first, the egg or the hen?
• Newcomb Paradox: How to play against an omniscient opponent.
• St.Petersburg Paradox: People will only risk a small amount to get an infinite value reward.
• Paradox of the journey in time: What would happen if you travel in time and kill your grandfather before I meet your grandmother?
• Paradox of the serpent: If a snake starts eating its tail, it ends up eating absolutely all of its body, where would the serpent be, inside its stomach, which in turn would be inside it?

According to the area of knowledge to which they belong

All paradoxes are considered to be related to logic, which was once considered a part of philosophy, but has now been formalized and included as an important part of mathematics. Despite this, many paradoxes have helped to understand and advance in some specific areas of knowledge.

Paradoxes in mathematics

• Banach-Tarski Paradox
• Frege Paradox

Paradoxes in probability and statistics

• Birthday paradox: What is the likelihood that two people at a meeting will meet years the same day?
• Simpson Paradox: By adding data, we can find misleading relationships.
• Arrow Paradox: You cannot have all the advantages of an ideal voting system at the same time.
• Monty Hall problem: And behind door number two... (Why is probability not intuitive?)
• St.Petersburg Paradox: How it is not worth risking a lot to win an infinite award.
• Phenomenon Will Rogers: On the mathematical concept of the media, it deals with the median or median of two sets when one of its values is exchanged among them, giving rise to an apparently paradoxical result.
• Paradox of the two envelopes: One of the envelopes contains twice as much money as the other. No matter which of the two envelopes is in my power, the probability always indicates that it is favorable to change it for the remaining envelope.
• The dilemma of the 100 prisoners and 100 cajones: the prisoners (death convicted) to survive, all must find their number in one of the 100 drawers, but each can only open 50 drawers. If one fails in the search, none survives.

Paradoxes in logic

Although all paradoxes are considered to be related to logic, there are some that directly affect its traditional bases and postulates.

The most important paradoxes directly related to the area of logic are antinomies, such as Russell's paradox, which show the inconsistency of traditional mathematics. Despite this, there are paradoxes that do not contradict themselves and that have helped advance concepts such as proof and truth.

• Paradox of the present king of France: Is it true an affirmation of something that does not exist?
• Hempel Raven Paradox: A red apple increases the likelihood that all crows are black.
• Infinite budgetary return: "Every name that designates an object can in turn become the object of a new name that designates its meaning."

Paradoxes about infinity

The mathematical concept of infinity, being counterintuitive, has generated many paradoxes since it was formulated. It is important to highlight that these cases show a paradox but not in the sense of a logical contradiction, but in the sense that they show a result that is counterintuitive, but demonstrably true.

• Galileo Paradox: Although not all numbers are square numbers, there are no more numbers than square numbers.
• Infinity hotel paradox: A hotel of infinite rooms can accept more guests, even if it is full.
• Cantor Set: How to remove elements from a set and keep the same size.
• Horn of Gabriel or Torricelli Trumpet: How can an infinite surface be needed to contain a finite volume?
• Paradoxes of Zenon: Through the concept of division into infinity, Zenon tried to demonstrate that the movement cannot exist, thus confirming the philosophy of his master, Parménides. The best known are the "dicotomy" and the paradox of " Achilles and Turtle".

Paradoxes in geometry

• Optical illusions
• The Fibonacci series
• Disposition of leaves in a stem
• Authentic Division
• logarithmic spiral
• Interior or exterior?
• Hooper Paradox
• Königsberg Bridge Problem
• Klein bottle
• Banda de Möbius
• Problem of the four colors

Paradoxes in physics

Richard Feynman in his Lectures on Physics, clarifies that in Physics paradoxes do not really exist, but in physical paradoxes there is always a misinterpretation of one or both of the reasonings that make up the paradox. This is not necessarily valid in other disciplines where actual paradoxes may exist.

• Bell Paradox: Plant a classic special relativity problem.
• Paradox of Olbers: Why, if there are infinite stars, the sky is black? Olberts calculated that the lightness of the sky would correspond to a temperature of the order of 5,500 °C, which, in fact, is not observed. It is now known that the luminosity calculated by Olberts does not become such by the important red corrimiento of the most remote light sources, which the most accepted theory attributes to the removal of galaxies or expansion of the universe. In addition, the finite age of the universe is opposed, its remarkable changes during its history and that the number of galaxies is not infinite. The paradox comes from a time when galaxies were not known and tended to believe that the universe was infinite and static, so it was also plausible that there were infinite stars.
• Maxwell Paradox or Maxwell Demon: An apparent classical paradox of thermodynamics.
• Paradox of twins: When one of the brothers returns from a journey at speeds close to those of light, he discovers that he is much younger than his brother.
• Einstein-Podolsky-Rosen paradox: A paradox about the nature of quantum mechanics proposed by these three physicists.
• Fermi Paradox: If the Universe were populated by technologically advanced civilizations, where are they?
• Young's experiment. A quantum paradox in its electron version to electron. In Young's experiment, electrons can be passed through a double slit one to one in a corpuscular way, as if they were particles, however obtaining a figure of interference.
• Schrödinger Paradox: The paradox par excellence of quantum mechanics.
• D'Alembert Paradox: Related to body resistance to viscous and non-viscous fluids, in Fluid Mechanics.
• Klein's paradox: Predicts the non-conservation of the wave amplitude of a particle. It appears when attempting to apply relativistic quantum mechanics without the concept of quantum field theory.

Paradoxes in economics

• Abilene Paradox: A group of people often make decisions against their own interests.
• Saving Paradox: If everyone tries to save during a recession, the aggregate demand will fall and the total savings of the population will be lower, this paradox is similar to Kalecki's paradox.
• Allais Paradox: In some kind of bets, even though people prefer certainty to uncertainty, if the problem is posed differently, they will prefer the uncertainty they rejected earlier.
• Bertrand Paradox: Two players who achieve the same balance of Nash are each without any benefit.
• Bird paradox in the bush: Why do people avoid risk?
• Paradox of value (or paradox of diamond and water): Why is water cheaper than diamonds, while humans need water, not diamonds, to survive?
• Edgeworth Paradox: With capacity constraints, there can be no balance.
• Ellsberg Paradox: In some kind of bets, even though it is logically equivalent, people bet on something that against something, that is, they get more useful by betting in favor.
• Gibson Paradox: Why are interest rates and positively correlated prices?
• Giffen Paradox: Can it be that the poor eat more bread even if their price rises?
• Jevons Paradox: An increase in efficiency leads to increased demand.
• Kalecki Paradox of Costs: A generalized decline in wages (cost reduction) and fixed prices away instead of increasing profits reduce sales by a drop in aggregate demand.
• Leontief Paradox: Contrary to Heckscher-Ohlin's theory, some countries import goods that are intensive in factors that abound relatively in that country and export goods that are intensive in factors that are relatively scarce in that country, for example the United States.
• Parrondo Paradox: It is possible to play in two games that cause losses alternatively to end up winning.
• St. Petersburg Paradox: How it is not worth risking a lot to win an infinite award
• Paradox of the voter: The more people participate in a election by vote, the less the benefit of going to vote, being every less decisive voter.

Other paradoxes

• Moore's paradox: a paradox in epistemology and in the philosophy of language, which examines the apparent absurdity in phrases like It's raining, but I don't think it's raining.
• Paradox of tolerance

Paradoxes and abstraction

It is essential to correctly use the abstraction capacities of the mind to achieve an adequate understanding of the aforementioned paradoxes. As such, their goal is not to get the individual to come up with fabulous, imaginative ideas for their resolution. Within the general scope of people without scientific or philosophical pretensions, an adequate interpretation of paradoxes and their explanations contributes to the development of analysis and processing of abstract information.