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Axiomatization is the process of taking a body of knowledge, usually but not exclusively in mathematics, and separating out certain sentences to describe it. These sentences are the axioms, and they will have the following useful properties: (1) They should be appear to be intuitively true. (2) Their meaning should be easy to grasp. (3) They should be consistent (that is, no contradictory result should follow from them as a logical consequence). (4) As much as possible of the body of knowledge (known as the theory) should be a logical consequence of them. (5) They should be as few in number as possible (that is, none of them should be a logical consequence of the others). (The reason for saying ‘as much as possible’ rather than ‘all’ in (4) is Gödel\'s incompleteness theorem which makes ‘all’ an unattainable goal in most reasonably powerful systems.) The mathematical sentences of the theory which are logical consequences of the axioms are called ‘theorems’. Properties (4) and (5) together are usually coalesced as the property of ‘productiveness’—a small number of axioms produce a large number of theorems.
Axiomatization has played an increasingly important role in mathematics. The original paradigm of an axiom system is the axioms for geometry of Euclid (295 BCE) (he actually called them ‘postulates’). The search for increasing rigour in mathematics has led to the axiomatization of more and more theories, a process which culminated in the axiomatization of set theory by Ernst Zermelo (1871 - 1953) and , Adolf Abraham Fraenkel (1891 - 1965). It is now common to start with a group of axioms and to see what can be deduced from them, rather than taking an existing body of knowledge and picking a group of axioms for it.
The role of axioms in mathematics has been to make far clearer exactly what is being said, particularly since the introduction of Gottlob Frege\'s symbolic logic. They expose exactly what is being assumed by particular mathematicians—the idea is never to assume something which is not an axiom. The success of axioms in many bodies of mathematics led to (not unreasonable) attempts to axiomatize the whole of mathematics, first in the Principia Mathematica of , Bertrand Russell (1872 - 1970) and , Alfred North Whitehead (1861 - 1947) and then in the Hilbert programme. The former was never finished, and the latter was proved impossible by Gödel. It has also led to attempts to duplicate the idea in other fields. Spinoza (1639 - 1677) attempted to set up a system of philosophy which was axiomatic, and in the 19th century Marx tried to do the same in political theory. The problem that ruined the effectiveness of their work was that both of them failed to realize that unless the axioms are accepted, neither can the consequences of the axioms. The same problem has dogged others who have tried to bring ‘mathematical’ precision to areas other than mathematics.
One of the problems of formalization is that a series of formal proofs in symbolic logic soon becomes tedious; after a few theorems, the reader begins to scream for a less fussy approach. As a result, the tendency in modern mathematics has been to take what is known as a ‘Platonic viewpoint’: to give informal proofs to convince a ‘reasonable’ person which can (theoretically, at least) be backed up with the more formal proof to counter objections. There are clearly disadvantages to this, as one person\'s idea of a convincing argument is different from another\'s. Also, the retreat from formalism makes it possible to hide mistakes (from oneself and others) in imprecise language and to make unstated assumptions. In an incorrect proof, it is usually the word ‘clearly’ that hides the error. SMcL |
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