Greek beginnings | building on sand | back to basics |
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There are two major periods in the historical development of the real number system which we consider here. The first is the period of classical Greek mathematics in which mathematics first emerged as a deductive science. The second is that of the rigourisation of analysis and the formalisation of mathematics which took place mostly in the 19th century. Between these periods mathematics expanded very much in areas which depended on real numbers despite weakness in the understanding of real numbers. |

Number in Classical Greek Mathematics |
The Method of Exhaustion |

Creativity and Application without RigourThe social system which had made mathematicians stand aloof from applications was gone and the development of mathematics was stimulated by practical needs. Geometry had to be quantitative, despite difficulties with the concept of number this demanded. Rigour declined as Alexandrian mathematics demonstrated how much can be done with only a tenuous grasp on key underlying concepts. |
NadirMathematics then reached its lowest ebb, sustained only by Arabic and Hindu activity. |
European Mathematics from the RenaissanceUnder this stimulus mathematics developed for four centuries in Europe without a return to Greek standards of rigour or the realisation of a clear understanding of the number systems used in applications of mathematics. This includes the arithmetisation of geometry using Descartes' coordinate system, Galileo and the mathematisation of science, the development by Newton and Leibniz of the differential and integral calculi, and a century of rapid further development of analysis. None of these developments, achieved without much understanding of the numbers they depend upon, would have been possible if classical Greek standards of rigour had prevailed. |

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Abel, 1826 |

Rigour in AnalysisBolzano, Cauchy, Abel, Dirichlet and later Weierstrass set about putting the house in order.
Central to this was the notion of This new rigour presupposed but did not supply a number system in which convergent sequences have limits, and gave the impetus to work on the theory of real numbers. |
Real Numbers at LastReal numbers were defined by Dedekind as certain sets of rationals. The theory of rational and natural numbers were then clarified in turn, ultimately reducing all of these systems to set theory and logic. |