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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    $\begingroup$ This book was essentially written by Courant in 1927 (first edition), so "modetn mathematics" means 'early 20century mathematics'. This was indeed the time of 'arithmetization' of foundations. $\endgroup$ – Alexandre Eremenko May 29 '16 at 16:11
  • $\begingroup$ Suggestion; read Isaac Barrow, The Geometrical lectures (original Latin edition: 1670 - J.M.Child edition: 1916) the first para of Lecture IX, page 101-on, regarding tangents, and compare with current "differential" proof. $\endgroup$ – Mauro ALLEGRANZA May 29 '16 at 16:57
  • $\begingroup$ And consider a letter from Tschirnhaus dated 1675 that, after careful study of Barrow's Lectures, commenting on Lectio IV, §16-17 states explicitly that he cannot see anything essentially new in Leibniz's symbolism with its "monstrae characteres" in comparison with Barrow's text that appears to be so much more readily intelligible. $\endgroup$ – Mauro ALLEGRANZA May 29 '16 at 17:20

The "reasons" behind the history of the calculus are many.

But a possible source of the "project" of

founding mathematics on number concepts rather than geometrical ones

may be searched in the development of algebraic symbolism and tools in the 16th and 17th Centuries.

As an example, consider Leibniz's own words in his Historia et origo calculi differentialis (ca.1714), translated into:

it certainly never entered the mind of anyone else before Leibniz [the memoir is written by Leibniz in third person] to institute the notation peculiar to the new calculus by which the imagination is freed from a perpetual reference to diagrams [emphasis added], as was made by Vieta and Descartes in their ordinary or Apollonian geometry [i.e. conic sections]; moreover, the more advanced parts pertaining to Archimedean geometry, and to lines which were called "mechanical" [i.e. transcendent curves] by Descartes, were excluded by the latter in his calculus.

But now by the calculus of Leibniz the whole of geometry is subjected to analytical computation [emphasis added], and those transcendent lines that Descartes called mechanical are also reduced to equations chosen to suit them, by considering the differences $dx, ddx$, etc., and the sums that are the inverses of these differences, as functions of the $x$'s; and this, by merely introducing the calculus, whereas before this no other functions were admissible but [...] powers and roots.

Consider Euclid's Elements (c.300 BC): Book V deals with magnitudes:

Prop.V.1: If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.

The proposition is illustrated with segments.

In 1638 Galileo Galilei, in the Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze, still illustrates his theorems on motion with ratio between segments.

Then we have the invention of the calculus and the language began to change.

The first calculus textbook from Guillaume de l'Hôpital:

Definition I. Those quantities [quantités] are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change.

With Leonhard Euler:

Caput I: De Functionibus in Genere. Quantitas constans est quantitas determinata, perpetuo eumdem valorem servans.

Ejusmodi quantitates sunt numeri cujusvis generis [emphasis added], quippe qui eumdem, quem semel obtinuerunt, valorem constanter conservant [...].

Finally we may consider Augustin-Louis Cauchy's Cours d'Analyse (1821), Preliminaries:

First of all, we will indicate what idea will be appropriate to attach to the two words number and quantity. We always take the meaning of numbers in the sense that is used in arithmetic, where numbers arise from the absolute measure of magnitudes, and we will only apply the term quantities to real positive or negative quantities, that is to say to numbers preceded by the signs $+$ or $−$. Furthermore, we regard these quantities as intended to express increase and decrease, so that a given magnitude will simply be represented by a number if we only mean to compare it to another magnitude of the same type taken as a unit, and by the same number preceded by the sign $+$ or the sign $−$, if we consider it as being capable of increasing or decreasing a given magnitude of the same kind.

Conclusion: whith the development of the main new branch of mathematics: analysis, the concepts of number and function, rather than geometrical concepts, became the fundamental ones.

  • $\begingroup$ what is the advantage of lay the foundation of mathematics on number concepts rather than geometrical ones? $\endgroup$ – iMath May 31 '16 at 9:59
  • $\begingroup$ @iMath - Personally, I do not know... but was so for many centuries, at least not in terms of "modern" forundations, but at least in terms of methods. But the same is today with set-concept: its "foundational" role is more in term of language and methods of proof that in terms of "basic ontology". $\endgroup$ – Mauro ALLEGRANZA May 31 '16 at 10:26
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    $\begingroup$ The advantage is that a number of foundational questions was largely clarified. For example that Euclid's Vth book was basically laying foundations of the theory of real numbers (or rather, of the passage from arithmetic to real numbers). The theory of "magnitudes" is much more difficult to understand and explain in geometrical terms than in algebraic ones. A very interesting reading is the historical part on the book of Bourbaki regarding General Topology chap.5. $\endgroup$ – Nicola Ciccoli Jun 2 '16 at 8:43

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