Development of Mathematical Thought 2024-2025, University of Padua

Our Programme

A history of the concept of function, from Descartes to Riemann. (16 hours) . A history of the concept of curvature, from Euler to Levi-Civita, through Gauss and Riemann. (16 hours)

A History of the Concept of Function

1. Analysis without functions: some preliminary on Greek mathematics, Pappus, Descartes and La géométrie (I-II)
2. The birth of the calculus: Newton and Leibiniz
3. A brand new discipline: the role of Euler (IV)
4. The controversy on what an arbitrary function is: D'Alembert, Euler, Daniel Bernoulli (V)
5. Fourier's work on trigonometric series produces unexpected results and new problems. (VI)
6. Cauchy's Cours d'Analyse and the rigorization of analysis. (VII)
7. Riemann's Habilitationsschrift: Riemann integral and the introduction of pathological functions (VIII)

Characteristics of Greek Mathematics

The origins of analysis can be traced back to Greek mathematics. This is true in a twofold sense. Firstly, Greek mathematicians addressed and solved, at least partly, problems that we would now place within the realm of analysis: the calculation of areas and volumes of figures, the determination of tangent and normal lines to given (elementary) curves (conic sections, spirals, etc.). Secondly, the word analysis (ovadvouc) was employed by Greek mathematicians to denote a particular method of discovery or problem solving that, working step by step backward from what is sought as if it had already been achieved, eventually arrives at what is known. However, Greek mathematics is characterized by a radical rejection of limit pro- cedures of any kind. It devised elegant and rigorous techniques that allowed the solution of a large number of problems without resorting to infinitesimal con- cepts: the determination of the area of the circle and the parabolic sector, the volume and surface of a sphere, the volume of a pyramid, the construction of tangent and normal lines to conic sections, are examples in which such techniques proved their effectiveness.

Greek Mathematical Corpus

The study of the Greek mathematical corpus in general, and more specifically those parts that are connected to the historical development of analysis, is made difficult by the fact that primary sources, both for what concern content and language, are quite distant to our mathematical knowledge. To simplify and generalize a little too much, Greek mathematics is characterized by an almost total lack of symbolic language and a tendency to reason and solve problems using notions and concepts that are predominantly geometric in nature. For example, the theory of proportions, which constitutes one of the highest achievements of ancient science, was formulated with respect to unspecified magnitudes that we can interpret as line segments, planar regions, angles, etc., and the determination of the area of given planar regions or the volume of portions of space was never expressed in terms of assigning a number or a formula to calculate areas and volumes but always in terms of ratios or more elementary geometric constructions. Naturally, the concept of number does appear in the Elements and other works of the Greek corpus (think of Diophantus). However, we must always keep in mind that the Greek notion encompasses a much more limited scope than modern numerical fields (rational, real). Please, remember that the first axiomatic and fully rigorous treatment of real numbers was provided by Dedekind in 1872 for the first time.

Book VII of the Elements

To see this, we can turn to Book VII of the Elements. Properly speaking, only positive integers greater than 2 are considered to be numbers. Indeed, according to Def. VII.2, a number is "a multiple composed of units"; the unit is not a number, since the definition of number requires the presence of more than one unit. The notion of multiple was linked to the possibility of being completely measured by another number; more precisely, Euclid expressed this idea by stating that a number is a part of another (larger) number when the first completely measures the second; conversely, a number is parts of another when it does not completely measure it, meaning that it is a proper fraction of the first. It is important to note that rational numbers are not conceived as autonomous entities. However, it should be pointed out that in Def. VII.21, the idea of equality between two ratios of integers is only formalized in an indirect way.

The Ratio of Integers

Numbers are in proportion when the first is the same multiple, or the same part, or the same parts of the second that the third is of the fourth This definition can be symbolically expressed (and thus through an anachronism) as follows: a, b, c, d are in proportion when there exist integers m, n such that: a = m - b n ◆ c= m n ◆ ; It is clear that in the case where n = 1, the first and third are "the same multiple" of the second and fourth, respectively; if m = 1, the first and third are " the same part" of the second and fourth, respectively; finally, if n = 1 Am /1, the first and third are "the same parts" of the second and fourth, respectively. Elements, Definition VII, 21 Αριθμοί άνάλογόν είσιν, όταν ο πρώτος του δευτέρου και ο τρίτος τού τετάρτου ίσάκις ή πολλαπλάσιος ή τό αύτό μέρος ή τα αύτά μέρη ώσιν.

Euclid's Elements, 3rd Century BC

• A work composed of 13 Books
• Book I. After giving three sets of principles (definitions, postulates, axioms), which serve as a general introduction to the entire work, it presents the theory of congruence of triangles, the theory of perpendiculars, the theory of parallels, and the theory of equivalence of polygons. Book I revolves around two fundamental theorems: prop. 32 (the sum of the internal angles of a triangle equals two right angles) and prop. 47-48 (Pythagoras' theorem and its inverse), with which it concludes.
• Book II. It revisits and completes some procedures already started in the previous book, reaching the quadrature of any polygon, i.e., the construction of a square equivalent to a given polygon.
• Book III is dedicated to the theory of circle.
• Book IV provides the constructions of regular polygons inscribed and circumscribed (triangle, square, hexagon, pentagon, and pentadecagon).
• Book V contains the general theory of magnitudes and proportions.

Euclid's Elements Continued

• Book VI contains the geometric applications of the theory of proportions: it studies the properties of similar polygons (third segment, fourth proportional, golden ratio of a segment). It ends with the generalization of the quadrature problems addressed in the second book: one polygon is transformed into another equivalent to a given shape.
• Books VII, VIII, IX: these are the arithmetic books of the Elements, where arithmetic is understood in the sense of number theory: they exclusively deal with integers and their properties (proportions between integers, greatest common divisor and least common multiple, prime factorization of integers, powers, geometric progression). The properties are always studied in general, without giving any numerical examples.
• Book X is the longest and most complex among the books of the Elements. It meticulously studies quadratic irrationals, i.e., "irrational numbers" obtained through repeated root extractions.
• Books XI, XII, XIII cover the principles of solid geometry. The method of exhaustion is applied to determine certain plane areas and the volume of a pyramid. It concludes with the study of the five regular polyhedra (Platonic solids): tetrahedron, cube, octahedron, icosahedron, dodecahedron.

The Theory of Proportions

The treatment in Book V, intended as an expansion of the theory of arithme- tic ratios, was largely motivated by the need to account for the existence of incommensurable magnitudes, such as the side and diagonal of a given square. The discovery of the existence of incommensurable magnitudes, which occurred within the Pythagorean school, led to a foundational crisis that undermined the Pythagorean conception to the effect that the world can be described in terms of integer numbers only. There are two fundamental definitions in the theory. The first, Def. V.4: Magni- tudes are said to have a ratio to each other when they can, if multiplied, exceed each other. In modern terms, Euclid requires that the magnitudes he is dealing with are Archimedean, meaning that given magnitudes A, B, there exists a multiple of one that exceeds the other: mA > B. An example of non-Archimedean magnitudes, also present in the Elements, is the class of rectilinear angles and horn angles. Indeed, as shown in III.16, there is no multiple of a tangent angle (the plane area between the circumference and the tangent to the circumference at a point) that exceeds a rectilinear angle.

Definition of Proportion

A much more complicated definition is provided by Def. V.5, which introduces the notion of having the same ratio and indirectly, of proportion. In symbolic language, it states that four magnitudes A, B, C, D are in proportion if, when A and C are multiplied by any integer m and B and D by any integer n, for any choice of m and n, the following holds: i) mA < nB implies mC < nD; ii) mA = nB implies mC = nD; iii) mA > nB implies mC > nD.