Paradox
's self-flowing flask fills itself in this diagram, but perpetual motion machines don't exist.]]A paradox is an apparently true statement that seems to lead to an illogical contradiction, or to a situation that contradicts common intuition. Put simply, a paradox is 'the opposite of what one thinks to be true.' The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics.
The etymology of paradox can be traced back to texts appearing at the dawn of the renaissance, a period of accelerated scientific thought in Europe and Asia sometime after the year 1500 AD. The first forms of the word appeared as the late Latin word paradoxum, but is also found in Greek texts as paradoxon (however, Latin is predominantly derived from the Greek alphabet--furthermore, English is derived from Roman Latin, but with the addition of the letters "J", "U" and "W"). The word is composed of the prefix para- which means "contrary to", "altered" or "opposite of", and conjoined with the noun suffix doxa, meaning "opinion." Compare orthodox and heterodox.
In moral philosophy, paradox plays a particularly central role in debates on ethics. For instance, an ethical admonition to "love thy neighbor" is not just in contrast with, but in contradiction to an armed neighbor actively trying to kill you: if he or she succeeds, then, you will not be able to love them. But to preemptively attack them or restrain them is not usually understood as very loving. This might be termed an ethical dilemma. Another example is the conflict between an injunction not to steal and one to care for a family that you cannot afford to feed without stolen money.
It should be noted that many paradoxes rely on an essential assumption: that language (be it spoken, visual, or mathematic) accurately models the reality it is describing. In quantum physics, many paradoxical behaviors can be observed (the Heisenberg uncertainty principle, for instance) and some have attributed these paradoxes to inherent limitations of language and scientific models. Alfred Korzybski, who founded the study of General Semantics, sums up this concept quite simply by having stated that, "The map is not the territory." A common example of the limitations of language are the forms of the word "to be." "Being" is not clearly defined (the area of philosophical study called ontology has yet to produce a concrete meaning) and thus if a statement includes being as an essential element, it may be subject to paradox.
Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning.
Not all paradoxes are equal. For example, the Birthday paradox is more of a surprise than a paradox, while the resolution of Curry's paradox is still a matter of contention.
W. V. Quine (1962) distinguished three classes of paradox:
Not all paradoxes fit neatly into one category. Some paradoxes include:
These are unintuitive results of correct logical reasoning.
These are incorrect results of subtly false reasoning.
Paradoxes that show flaws in accepted reasoning, axioms, or definitions. Note that many of these are special cases, or adaptations, of Russell's paradox.
These paradoxes rest simply on an ambiguous definition.
These are paradoxes only if certain special assumptions are made.
Some of these show that those assumptions are false or incomplete, others are other types of paradoxes.
Quine, W. V. (1962) "Paradox". Scientific American, April 1962, pp. 84–96.Types of paradoxes
List of paradoxes
Veridical paradoxes
Mathematical/Logical
Psychological/Philosophical
Physical
Falsidical paradoxes
Antinomies
Antinomies of definition
Conditional paradoxes
Other paradoxes
References