Real Numbers - some history

Greek beginnings building on sand back to basics

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.

Greek Beginnings

Number in Classical Greek Mathematics
The social division in classical Greece, between slaves and citizens, supported a division of practical computation and mathematical theory. Though Greek mathematicians understood as numbers only what we now call "the natural numbers", they dealt also with whole number ratios, and with geometric magnitudes, corresponding to what we now call rational and real numbers. None of these systems were treated as we would treat them today, but even geometric magnitudes were treated in Greece with greater rigour than at any subsequent period until the real number system was placed on a firm foundation in the 19th century.
The Method of Exhaustion
Before any of the number systems had been established to modern standards Eudoxus developed the Method of Exhaustion. This was used extensively in Book XII of Euclid's Elements to demonstrate results about areas and volumes which would today be done using real numbers and the integral calculus. Euclid's work in this area sustained a standard of rigour which surpassed that of Newton and Leibniz when they later invented the calculus. This standard was not subsequently matched until the rigourisation of analysis in the 19th century.

Building on Sand

Creativity and Application without Rigour
After the conquest of Greece by Phillip II of Macedonia, and the creation of Alexandria by his son, the centre of mathematical development moved to that cosmopolitan city and the character of mathematics began to change.

The 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.

The conquest of Alexandria by Rome marks the beginning of a period of decline culminating in the destruction of Alexandria by the Mohammedans in A.D.640.

Mathematics then reached its lowest ebb, sustained only by Arabic and Hindu activity.

European Mathematics from the Renaissance
Mathematics gradually re-emerges in Europe as a tool for science and engineering.

Under 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.

Back to Basics

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has lead to so few of the so-called paradoxes.
Abel, 1826
Rigour in Analysis
Untrammelled creativity engenders chaos and confusion. The solution of hard mathematical problems depends upon intuitions founded in a clear understanding of basic concepts. By the 19th century further progress in analysis was increasingly dependent on improving the foundations on which this field was based.

Bolzano, Cauchy, Abel, Dirichlet and later Weierstrass set about putting the house in order.

Central to this was the notion of limit, both differentiation and integration could be defined rigorously using limits. In this way the calculus was freed from a dependence on geometric or other physical intuitions.

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 Last
In the latter part of the 19th century attention turned to irrational numbers.

Real 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.

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