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Originally, the science of number and quantity. But with the birth of numerous more qualitative formalisms, (e.g., logic, propositional calculi, set theory), with the emergence of the unifying idea of a mathematical structure, with the advent of the axiomatic method emphasising inference, proof and the descriptions of complex systems in terms of simple axioms, and, finally, with self-reflective efforts such as meta-mathematics, mathematics has become the autonomous (see autonomy) science of formal constructions. Emphasising its formal character and its applicability to all conceivable worlds, mathematics has been likened to a language whose semantics is supplied by other sciences or by particular applications. Although all constructions are inventions of the human mind, cannot be found in nature and have no necessary connection with the world outside mathematics, they nevertheless arise in conjunction with solving certain kinds of problems: (1) real world problems, (e.g., geometry evolved in efforts of measuring the earth, game theory grew out of concerns for social conflict resolution, statistics from the need to test hypotheses on large numbers of observations, recursive function theory from the desire for efficient algorithms), (2) intellectual curiosity and playfulness, (e.g., markov chain theory stems from interest in poetry, probability theory from games of chance, the four-color problem, symmetry and much of topology (see the Mobiusband) from interest in artistic expression), and (3) interest in the powers and limitations of mathematics and the mind, (e.g., Goedel's incompleteness theorem from the inherent undecidability or incompleteness of systems, the theory of logical types from disturbing paradoxes, the differential and integral calculi from efforts to transcend the smallest distinctions practically possible). However, it is a characteristic of mathematics that the problems giving rise to its constructions are soon forgotten and the constructions develop a life of their own, checked only by such validity criteria as internal consistency, decidability and completeness. Empirical data from an existing world do not threaten the products of mathematics. (Krippendorff)
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