Scientific theories are perpetually changing. This is not due to mere chance but might well be expected, according to our characterization of empicrical science.
perhaps this is why, as a rule, only branches of science-and these only temporarily-ever acquire the form of an elaborate and logically well-constructed system of theories. in spite of this, a tentative system can usually be quite well surveyed as a whole, with all its important consequences. this is very necessary; for a severe test of a system presupposes that it is at the time sufficiently definite and final in form to make it impossible for new assumptions to be smuggled in. in other words, the system must be formulated sufficiently clearly and definitely to make every new assumption easily recognizable for what it is: a modification and therefore a revision of the system.
this, I believe, is the reason why the form of a rigorous system is aimed at. it is the form of a so-called 'axiomatized system'-the form which Hilbert, for example, was able to give to certain branches of theoretical physics. the attempt is made to collect all the assumption which are needed, but no more, to form the apex system. they are usually called the 'axioms' (or 'postulates', or, 'primitive propositions'; no claim to truth is implied in the term 'axiom' as here used). the axioms are choosen in such a way that all the other statements belonging to the theoretical system can be derived from the axioms by purely logical or mathematical transformations.
a theoretical system amy be said to be axiomatized if a set of statements, the axioms, has been formulated which satisfies the following four fundamental requirements. (a) the system of axioms must be free from contradiction (whether self-contradiction or mutual contradiction). this is equivalent to the demand that not every arbitrarily chosen statement is deducible from it. (b) the system must be independent, i.e. it must not contain any axiom deducible from the remaining axioms. (in other words, a statement is to be called an axiom only if it is not deducible within the rest of the system.) these two conditions concern the axiom system to the bulk of the theory, the axioms should be (c) sufficient for deduction of all statements belonging to the theory which is to be axiomatized, and (d) necessary, for the same purpose; which means that they should contain no superfluous assumptions.
perhaps this is why, as a rule, only branches of science-and these only temporarily-ever acquire the form of an elaborate and logically well-constructed system of theories. in spite of this, a tentative system can usually be quite well surveyed as a whole, with all its important consequences. this is very necessary; for a severe test of a system presupposes that it is at the time sufficiently definite and final in form to make it impossible for new assumptions to be smuggled in. in other words, the system must be formulated sufficiently clearly and definitely to make every new assumption easily recognizable for what it is: a modification and therefore a revision of the system.
this, I believe, is the reason why the form of a rigorous system is aimed at. it is the form of a so-called 'axiomatized system'-the form which Hilbert, for example, was able to give to certain branches of theoretical physics. the attempt is made to collect all the assumption which are needed, but no more, to form the apex system. they are usually called the 'axioms' (or 'postulates', or, 'primitive propositions'; no claim to truth is implied in the term 'axiom' as here used). the axioms are choosen in such a way that all the other statements belonging to the theoretical system can be derived from the axioms by purely logical or mathematical transformations.
a theoretical system amy be said to be axiomatized if a set of statements, the axioms, has been formulated which satisfies the following four fundamental requirements. (a) the system of axioms must be free from contradiction (whether self-contradiction or mutual contradiction). this is equivalent to the demand that not every arbitrarily chosen statement is deducible from it. (b) the system must be independent, i.e. it must not contain any axiom deducible from the remaining axioms. (in other words, a statement is to be called an axiom only if it is not deducible within the rest of the system.) these two conditions concern the axiom system to the bulk of the theory, the axioms should be (c) sufficient for deduction of all statements belonging to the theory which is to be axiomatized, and (d) necessary, for the same purpose; which means that they should contain no superfluous assumptions.
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