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In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says

In modern times mathematics was recreated and vastly expanded on a foundation of number concepts rather than geometrical ones.

Why such recreation happened ? For getting rid of the limitations of geometrical methods ?

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    $\begingroup$ Because you can't construct some numbers with geometry but you need them those numbers for various reason. Eg $\pi$, $e$. $\endgroup$
    – user312097
    May 28, 2016 at 11:19
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    $\begingroup$ @ritwiksinha I'd say we can "construct" or define $\;\pi\;$ with geometry: it is simply the ratio of any circle's perimeter to twice its radius. Did you have perhaps some other meaning of "construction" in mind? $\endgroup$
    – DonAntonio
    May 28, 2016 at 11:21
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    $\begingroup$ Algebra: the development of algebra (folowing arabic sources) during the Renaissance and Scientific Revolution produced tools for "symbolic manipulation" much more flexible and powerful that ancient Greek "geometrical" techniques of proof. $\endgroup$
    – Mauro ALLEGRANZA
    May 28, 2016 at 11:26
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    $\begingroup$ @Joanpemo: Perhaps the "construction" refers to construction using Euclidean tools (ruler and compass). If $x$ is a positive real number then it is not difficult to prove that starting from a line segment of unit length it is possible to construct a line segment of length $x$ by using ruler and compass if and only if $x$ can be expressed as a finite combinations of rational numbers together with operations $+,-,\times,/,\sqrt{}$. $\endgroup$
    – Paramanand Singh ♦
    May 28, 2016 at 11:57
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    $\begingroup$ @iMath: Thank you. :) In general it's a good idea to make questions as self-contained as is feasible, since web links break over time. Separately, I (for one) wasn't going to click a google books link to find the source of the quote. $\endgroup$
    – Andrew D. Hwang
    May 28, 2016 at 12:20

1 Answer 1

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Proofs using geometrical intuition and Euclid axioms were almost the norm in ancient Greek mathematics. However side by side the idea of numbers was also developing for obvious practical needs (i.e. counting and measuring). With advent of algebra it became obvious that methods based on algebraic manipulation of numbers were far more powerful than the non-obvious geometrical arguments based on Euclidean axioms and this was the main reason for shift of focus from geometry to arithmetic/algebra.

Use of algebra became so much so prevalent that it became fashionable to study geometry using numbers (co-ordinate geometry). However as many mathematicians during 1800-1900 realized the idea of replacing geometrical concepts with numbers was not easy. The concept of a straight line as a smooth continuum of points was very difficult to map with the existing theory of numbers (which basically was a theory of numbers accessible via algebraic techniques i.e. the numbers considered were algebraic). Only with the development of a proper theory of real numbers by Cantor, Dedekind, Weierstrass it was possible to map the points of a line with real numbers and the set of real numbers could then be viewed as a sort of arithmetical continuum.

All this development by the way had a side effect. The charm of Euclid's Elements is no more available in standard high school curriculum. Only the bare minimum material based on Euclid axioms is covered in school syllabus and students get to learn analytic geometry and tools of calculus to deal with general curves. The beautiful theory of conics by Apollonius of Perga is a remarkable work based on Euclid axioms and sadly very few students are aware of it. I did study some of it from the book Apollonius, Treatise on Conic Sections by T. L. Heath and wrote some posts on conics.

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    $\begingroup$ +1 for the side effects. I think both methodologies should be taught. because i think, euclidean geometry is much more beautiful than analytical geometry with calculus. $\endgroup$
    – user312097
    May 28, 2016 at 13:05
  • $\begingroup$ agree with your views @ritwiksinha $\endgroup$
    – Paramanand Singh ♦
    May 28, 2016 at 13:35
  • $\begingroup$ Why the downvote?? $\endgroup$
    – Paramanand Singh ♦
    May 28, 2016 at 18:45
  • $\begingroup$ @ParamanandSingh Great ! you helped me again ! Where have you learned these history of math? $\endgroup$
    – iMath
    May 29, 2016 at 13:02
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    $\begingroup$ @iMath: check my profile on MSE. After school years (meaning till age 17) I studied most of the stuff online. $\endgroup$
    – Paramanand Singh ♦
    May 29, 2016 at 13:38

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