Computer Software in Science and Mathematics

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Computation offers a new means of describing and investigating scientific and mathematical systems. Simulation by computer may be the only way to predict how certain complicated systems evolve by Stephen Wolfram

Scientific laws give algorithms, or procedures for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced.

Executing a computer program is much like performing an experiment. Unlike the physical objects in a conventional experiment, however, the objects in a computer experiment are not bound by the laws of nature. Instead they follow the laws embodied in the computer program, which can be of any consistent form. Computation thus extends the realm of experimental science: it allows experiments to be performed in a hypothetical universe. Computation also extends theoretical science. Scientific laws have conventionally been constructed in terms of a particular set of mathematical functions and constructs, and they have often been developed as much for their mathematical simplicity as for their capacity to model the salient features of a phenomenon. A scientific law specified by an algorithm, however, can have any consistent form. The study of many complex systems, which have resisted analysis by traditional mathematical methods, is consequently being made possible through computer experiments and computer models. Computation is emerging as a major new approach to the science, supplementing the long-standing methodologies of theory and experiment.

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The Main Principles of Axiomatic Methods

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The axiomatic method consists simply in making a complete collection of the basic concepts as well as the basic facts from which all concepts and theorems of a science can be derived by definition and deduction respectively. If this is possible, then the scientific theory in question is said to be definite according to Husseri. Such is the case for the theory of space. Of course, from the axioms of geometry we cannot possible deduce the law of gravitation. Similarly the axioms of geometry fail to disclose whether Zurich is farther from Hamburg than Paris. Though the question deals with a geometrical relation, the relation is one between individually exhibited locations. Thus precisely speaking, what are supposed to be deducible from the axioms are the pertinent general true prepositions.

An axiom system must under all circumstances be free from contradictions, in which case it is called consistent; that is to say, it must be certain that logical inference will never lead from the axioms to a proposition «a» whiles some other proof will yield the opposite proposition «a». If the axioms reflect the truth regarding some field of objects, then, indeed, there can be no doubt as to their consistency. But the facts do not always answer our questions as unmistakably as might be desirable, a scientific theory rarely provides a faithful edition of the data but is almost invariably a bold construction.

Not indispensable but desirable is the independence of the individual axioms of an axioms system. It should contain no superfluous components, no statements which are already demonstrable on the basis of the other axioms. The question of independence is closely connected with that of consistence, for the proposition «a» is independent of a given set of axioms if and only if the proposition «a» is consistent with them.

The dependence of a proposition «a» on other propositions A (an axiom system) is established as soon as a concrete proof of «a» on the basis of A is given. In order to establish the independence on the other hand, it is required to make sure that no combination of inferences, however intricate, is capable of yielding the proposition «a». There are some methods at one’s disposal of reaching this goal; by what has been said above, each of them qualifies also for proving the consistency of an axiom system.

(1) The first method is based on the following principle: if «a» contains a new original concept, not defined in terms of those occurring in A, then «a» cannot be a consequence of A. For example: a ship is 250 feet long and 60 feet wide; how old is its captain? Only in the most trivial cases does this simple idea accomplish our objective.

(2) The construction of a model. Objects and relations are exhibited which, upon suitable naming, satisfy all of the propositions A, and yet fail to satisfy «a». This method has been the most successful so far invented.

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