Paradox

Etymology

Étymologiquement, paradox (παράδοξος) means “opposite with the common direction” (contrary with orthodoxe in conformity with the opinions). In the beginning, a paradox is an idea which goes against the common direction. The concept, more restrictive, of contradiction, which is the everyday usage today of the term, appeared only later.

History

The oldest trace of paradox is reported in the Bible:

“Somebody of them, their own prophet, said: “Crétois are always lying, of malicious animals, the lazy bellies. ””
the epistle with Tite, chapter 1, verset12, Paul de Tarse.

This prophet, who lived at seventh century BC, would be Épiménide ''' Crétois '''. However, this first formulation of the Paradoxe of the liar appeared paradoxical only well later.

It is then necessary to mention the paradoxical thought of CAD De Jing founder of the Taoïsme at sixth century BC.

The first paradoxes, clearly stated like such, appear in Greek antiquity and are the work of Zénon d' Elée (for example: Paradox of Achilles or paradox of the arrow). They were then apprehended in manner purely Rhétorique. However they precede and announce physical sciences and mathematics.

Thereafter, the paradox will be a motive fluid of science in becoming. Thus, each discipline balbultiante will generate its paradoxes

* the Paradoxe of egg and the hen in Biologie,

* the Paradoxe of Condorcet, the Paradoxe of Alabama and especially the Paradoxe of Saint-Pétersbourg, in Decision theory,

* the paradox known as of the sky on fire (called later Paradoxe of Olbers) in Cosmologie,…

The Theory of probability, elaborate as of the 17th century, is more particularly fertile in paradoxes: Paradox of Borel, Paradox of the birthdays,…

The birth of modern physics, at the beginning of the 20th century, involved the appearance of many paradoxes. In Quantum physics, the Paradox EPR, and that of the Cat of Schrödinger, reflect in obviousness, the conceptual opposition between this physics and traditional physics; but also, the difficulty (even impossibility) with " interpréter" quantum physics. In parallel the Theory of relativity also generated its batch of paradoxes: Paradox of Selleri, Paradox of Ehrenfest, Paradox of the twins, Paradox of the train.

Lastly, in mathematics, certain theorems among most recent (Paradox of Banach-Tarski, Paradox of Skolem) run up against the intuition, and thus are wrongly described as paradox.

Epistemology

“ the good sense, no matter what it does, cannot miss being let surprise on the occasion. The goal of science is to save this surprise to him and to create mental processes which will have to be in close agreement with the process of the outside world, in order to avoid, in any case, unforeseen the ”
(Bertrand Russell).

The paradox plays a driving role in sciences, because it pushes with the fine analysis, and from there with a better thorough formalization and in the search of a better coherence. It has moreover one effect if justifying and mobilizing (although not federator) that one meets it like basic element of scientific constructions (the Paradoxe of Olbers and the Paradoxe of Saint-Pétersbourg, for example).

The Paradox of Hempel, in particular, plays an epistemological part extremely: that of a warning statement concerning sciences of nature. It states indeed that a theory cannot be completely validated by a simple accumulation of observations. In other words, it draws the attention to the dangers of the logical induction when its implicit is not clearly defined.

In mathematics, the diagonal Argument at the base of the Paradoxe of Russell is founder of the Logique mathematics which modified in-depth, the apprehension of the Mathématiques. The paradox of Bertrand Russell has a particular status: that of mathematical Theorem (within a erroneous theory). It was thus, not vector of progress, but with the source of a renouncement: the characterization of the Whole . Moreover, this paradox (and the paradox in general) reveal that the nonsense of a proposal is not intuitively obvious.

The paradoxical form can be used to express a major truth: " The first will be the derniers". " Happy pauvres".

Some types of paradox

It is necessary for all to keep in mind that the paradox is business of interprétation' S . However, without seeking to categorize, one can state some mechanisms of creation and resolution of paradoxes:

the erroneous premise

To find a erroneous Prémisse is the means simplest to build or solve a paradox. The Paradox EPR, for example, was knowingly written in order to determine the erroneous premise (a Postulat in fact).

the diagonal argument

“… I will continue in this feel questioned until I know something of some, or at least, in the absence of another thing, that I know as certain that precisely there are nothing of some. ”
Second Meditations metaphysics, Descartes.

The best means of apprehending the diagonal Argument is still the study of the paradoxes built by employing it. The most concise example is I lie (in this moment). As for any paradox of this type, one arrives at the conclusion that If it is true then they is false… and conversely.

The paradoxes of this category are based on the Auto-référence. In a more worked out way, by defining an object, an entity or a state; then one shows that the definite object involves, from his definition even, a nonsense. See for example: Paradox of the liar, Paradox of the barber, Paradox '' heterologic '' of Grelling, paradoxes of Berry, or: Paradox of the pope (After having declared: " I am not infaillible" , would a pope be always infallible? )

Except the Paradox of Russell, mathematical logic rejects this asserting kind of reflexive definitions which they cannot be formalized. They proceed of the amalgam between " thought " and " méta-thought " , or more prosaically, between definition and definite object .

the semantic or contextual amalgam

It is a very subtle process. It consists in, without that appearing, employing a word, or a turning of sentence, in two different directions or two contexts (angle or point of view) different. One carries out then an amalgam (a confusion) and one obtains a nonsense.

The paradoxes built on the model of the Syllogisme are characteristic of this process. A slip of direction or Contexte takes place between the two Prémisse S. Then the conclusion creates the amalgam, which results in aberrations. The fraud thus lies in the invalid use of the syllogism.

* the paradox " good-market/cher" is a semantic example of amalgam.

* the Paradoxe of the Gruyere is an example of contextual amalgam.

the absence of demarcation

In any general information, such a paradox (it is rather necessary to speak about thorny question) is built on the opposition of two proposals. One puts forward whereas there does not exist demarcation between the validity of the one and other. More formally, one regards an axis (time, a quantity or an unspecified size) appearing as a Continuum, and a Prédicat of kind one of the proposals is its assertion of this predicate, in a point of the axis, the other its refutation in second points. Where is then, on the segment thus defined, the limit of veracity of the predicate?

More coarsely: if a thing is true here, and distorts there, where is the limit?

The most representative example is the paradox of bearded the: Where is the border (in term of many hairs or length of the hairs) between the bearded one and the beardless one? The questions of this quality are innumerable. Inter alia, many problems metaphysics, ethics, or legislatures (concerning the responsibility, heart, limits of life, etc), can be stated by this process.

The situation appears really paradoxical only if or postulated the inexistence of the limit is shown.

For example, let us consider the question of egg and hen and that of the birth of the heart in the center of the controversy on the Théorie of the evolution. Do these questions become paradoxes only if it is admitted that any hen is born from an egg, any egg (of hen) leaves a hen or that if a being has a heart, then its parent has of it a , or if all the men go down from a first man, and if it is admitted that the first man was created by God, the nature of this first man cannot differ from that of God then comes the paradox if God is a man, then which created God? .

Such paradoxes “exist” in reality for all the physical phenomena which one cannot approach without modifying them (for example the principle of uncertainty of Heisenberg: beyond a certain limit, it becomes impossible to check which of two assumptions is real, and one must be based on two contradictory assumptions, possibly probabilistic, therefore in a continuous reality between truth and the forgery and without no limit between the two possibilities. To be able to explain the other phenomena, one must start from a postulate (a principle) on which builds and is explained all the remainder while noting by the experiment that the imagined model functions accurately. But for that it is necessary to admit at the beginning a paradoxical postulate which one cannot explain nor to justify who implies that all the reasoning that one will make while trying to explain a phenomenon will be based on an unverifiable assumption.

Other paradoxes exist since mathematics, while being based only at the beginning on the Cartesian principle where can exist only truth or the forgery (without any other medium), makes it possible to show by applying only this reasoning which an assertion can be at the same time true and false (or more exactly than it is impossible to show with this reasoning than it is true nor to show that it is false). That creates another model or Cartesian principle is not applicable for all to explain, and that should be also included the assertions indécidables, and which it is impossible to know what is reality also far that one pushes the reasoning to try to check it.

That is also summarized in another so paradoxical sentence: “I am certain that the doubt exists and that I doubt. ” or in his form even shorter “I doubt all. ” Another way of perceiving the problem is the apparently paradoxical assertion “I do not know large thing, but that of which I am sure, it is that I do not know anything. ” which can be included/understood as paradoxical only if it is not admitted that “Nothing” can be something of “tangible” with a probability infinitesimal (but not null) vis-a-vis an infinite reality which remains inaccessible; so that this explanation makes it possible to lead a reasoning explaining in a tangible world, the existence should be admitted also nontangible, unexplainable world because unjustifiable in the reality which one can perceive.

The physical phenomena go still even further since it is possible to give a quantified measurement (according to our own system of measurement) nonnull to the probabilistic limit of our uncertainty, and that uncertainty obtained and limited is inherent in any phenomenon observed: even if one seems to perceive a “real” phenomenon like the only possible one (and explainable), the contrary phenomenon also exists and can be observed then also simultaneously with a nonnull and measurable probability. in other words that at the same time truth and the forgery exist simultaneously according to our model of Cartesian reasoning.

One can then include/understand this reality observed dual only if ourself, measuring instrument, and the phenomena observed belong to same and single all, inseparable, without no origin in time or the space or any other projection of what one perceives of reality, which calls into question the principle even of causality on which tries to be based our usual reasoning in vain trying all to explain while dividing the various cases artificially to be explained.

the underhand reasoning

This kind of paradox is built like a demonstration concealing an error cunningly dissimulated; it is thus a Sophisme. It then acts more than one exercise intended to trap the student or to test his vigilance. For example, the Paradox of the three coins, the Paradox of the two accounts - checks.

Canted logic

Certain situations are regarded as paradoxical because they concern a canted logic. Historical examples:

Guy of Maupassant hated the Eiffel Tower and yet, it went up there as often as possible, while explaining to its astonished interlocutors: " It is the only place from where I do not see it any more! "
Whereas Gestapo comes to stop it with his/her partner, Tristan Bernard however posts his serenity: " Up to now we lived in fear; now, we will live in the hope! "
Solicited by an admiror for an autograph, Sacha Guitry written with its hand: " Forgive me, but I never give autographe." And it signs!
During prohibition, sobriety was not the principal quality of W.C Fields which came on scene with a supposed thermos flask to contain grapefruit juice. One day when his/her friends had made him a joke, he exclaimed: “But which thus put grapefruit juice in my grapefruit juice? ”

Appendices

logical Concepts
List of paradoxes
Paradoxes and double constraint
School of Systemic Palo-Viola
Communication
systemic Approach
Vicious circle
Sophism
impossible Aphorism
Object
Paradoxes of Zénon

External bonds

paradox|paradox

Roles of the paradoxes in the evolution of mathematics, Stephan Genard.

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