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A paradox (Greek, ‘against expectation’), in mathematics, is a pair of mutually contradictory statements, or apparently contradictory statements, which are both deductions from statements which are accepted as true. Paradoxes have cropped up several times in the history of mathematics, and have usually led to major developments as they have been reconciled. There is one interesting exception to this ‘rule’: the Greeks\' discovery of the (to them, paradoxical) fact that there were irrational numbers so unsettled them, that they virtually gave up the study of numbers.
One field that paradoxes have played a major part in developing is set theory. This is principally because this field was never really given a formal basis until the members of the school of logicism wished to give it the status of the most basic part of mathematics. The paradoxes they examined were of two kinds. The first kind was caused because the new set theory developed by Georg Cantor (1845 - 1918) (and particularly his assertions about infinity) ran counter to the prejudices of other mathematicians. They would say that no set could be equinumerous with a proper subset (that is, a subset which is not the whole thing) of itself, but that Cantor had shown that the set of natural numbers is equinumerous with the set of even numbers. The problem was not with set theory itself, but in the fact that these mathematicians had assumed that a result true of finite sets would also be true of infinite sets.
The second type of paradox was more serious in its implications, particularly as set theory went on to become the basis of mathematics. If there were truly contradictions inherent in set theory, then there were contradictions inherent in the most basic structure of mathematics; and in logic, a contradiction means that any result can be proved to be true, a most unsatisfactory state of affairs. There were several different versions of essentially the same paradox, which was to do with the nature of mathematical property.
The best-known version is named after , Bertrand Russell (1872 - 1970). Suppose you consider the set A of all sets which are not members of themselves. (For example, the empty set has no members at all, so is not a member of itself.) Is A a member of A or not? If A is a member of A, then it is not a member of A, by the definition of A; and if A is not a member of A, then it is a member of A, also by the definition of A. This is clearly contradictory, but surely A must either be a member of A or not? The solution to this paradox is to recognize that A is not actually a legitimate set, but what is known as a proper class. Classes have sets as members, but do not themselves have to be sets, though all sets are classes. A class is defined by a formula of symbolic logic, and contains all the sets which satisfy that formula. A set is basically a class which can be shown to be a set using the axioms of set theory.
Another similar paradox is to do with numbers. It is clear that the names for numbers in English become longer (in general) as the numbers get larger. So there will be numbers that cannot be described in English in fewer than 250 words. As that is the case, we can define a particular number n to be ‘the first number which cannot be described in English in fewer than 250 words’. The problem is that we now have a description of this number in fewer than 250 words, so that we have a contradiction. The key to this paradox lies in the words ‘in English’. We have here an indication that English is not the language in which mathematics should be performed—and neither is any other human language. Properties which are to be investigated mathematically should be capable of being expressed in symbolic form, and this one is not. So the nature of properties to be investigated mathematically is limited by paradox to logical sentences. SMcL
Further reading E.P. Northrop, Riddles in Mathematics. |
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