Leibniz and the Invention of Mathematical Transcendence

TABLE OF CONTENTS
LEIBNIZ AND THE INVENTION OF MATHEMATICAL TRANSCENDENCE. THE ADVENTURES OF AN IMPOSSIBLE INVENTORY
	THE DISCOVERY OF THE TRANSCENDENCE
	TRANSCENDENCE AND SYMBOLISM
	TRANSCENDENCE AND GEOMETRY
	LOOKING FOR AN INVENTORY
	SYMBOLICAL INVENTORY?
	GEOMETRICAL INVENTORY?
	ON HIERARCHIES IN TRANSCENDENCE. EXPLORATIONS BY REPRODUCTION
	RECEPTIONS OF THE TRANSCENDENCE
FIRST PART. DISCOVERING TRANSCENDENCE
	CHAPTER I. ON THE GERMINATIONS OF THE CONCEPT OF ‘TRANSCENDENCE’
		1673: “A TRANSCENDENT CURVE SQUARING THE CIRCLE …”
		1674: ABOUT ‘SECRET’ GEOMETRY
		1675: THE LETTER TO OLDENBURG
		1678: THE “PERFECTION OF TRANSCENDENT CALCULUS”
	CHAPTER II. SQUARING THE CIRCLE
		II-A THE ARITHMETICAL QUADRATURE OF THE CIRCLE (1673), OR THE VERY FIRST MATHEMATICAL GLORY OF LEIBNIZ
			The quadrature of the hyperbola in Mercator
			Quadratrices and symbolic substitutions: the mathematical ideas of Leibniz
			Leibniz on “the exact proportion …”
			Some reflections on the integration of rational fractions
		II-B LEIBNIZ AND THE IMPOSSIBILITY OF THE ANALYTICAL QUADRATURE OF THE CIRCLE OF GREGORY
			The true quadrature of the circle and hyperbola, by James Gregory
			The geometric–harmonic mean of Gregory
			Leibniz and the “convergence” of adjacent sequences (1676)
			Leibniz’s “means by composition”
			A mathematical appendix: the geometric-harmonic mean (G-H)
	CHAPTER III THE POWER OF SYMBOLISM: EXPONENTIALS WITH LETTERS
		III-A EPISTOLA PRIOR, OR LEIBNIZ’S DISCOVERY OF SYMBOLIC FORMS WITHOUT A SUBSTANCE (JUNE 1676)
			Fractional exponents
			Symbolism without interpretation?
			On the consistency (?) of Newton’s exponential
			‘Permanence-Ramification’. A scheme
		III-B THE EPISTOLA POSTERIOR (OCTOBER 1676)
			Irrational Exponents
			Letters in Exponents
		III-C DESCARTES, LEIBNIZ AND THE IDEALIZED IMAGE OF THE EXPONENTIAL
			Gradus indefinitus
			Descartes and Newton transcended by Leibniz
		CONCLUSION OF THE FIRST PART: THE TRANSCENDENT IDENTIFIED WITH THE NON-CARTESIAN FIELD
SECOND PART. THE SEARCH FOR AN INVENTORY
	CHAPTER IV FROM INFINITELY SMALL ELEMENTS TO THE EXPONENTIAL UTOPIA
		IV-A TSCHIRNHAUS AND THE INVENTORIES OF 1679–1684
		IV-B DE BEAUNE, DESCARTES, LEIBNIZ, AND THE INFINITELY SMALL ELEMENTS
			When a curve is no longer considered as “a set of points” (in modern terminology)
			Descartes and De Beaune’s problem
			Leibniz and De Beaune’s problem
		IV-C ON EXPONENTIAL SYMBOLISMS
		IV-C1 ON THE RESOLUTION OF THE EXPONENTIAL EQUATIONS. THE “ADMIRABLE EXAMPLE”
			The admirable example
			The impossibility of effective resolutions
		IV-C2 EXPONENTIAL EXPRESSIONS AND THE DIALECTICS OF INDETERMINACY – THE STATUS OF THE LETTER
			The symbolism of quantities “arbitrary, but however fixed”
			When the unknown enters the exponent
			Towards letteralized exponentials
		IV-C3 TOWARDS A HIERARCHY IN TRANSCENDENCE: THE INTERSCENDENT EXPONENTIALS
			When the degree of the exponential “falls between” two integers
			A hierarchy in transcendence?
		IV-C4 THE EXPONENTIAL UTOPIA
		IV-C5 LETTERALIZED EXPONENTIALS ARE SOLUBLE IN THE DIFFERENTIAL CALCULUS
			An exchange of correspondence with Huygens (1690)
			The response to Nieuwentijt (1695)
	CHAPTER V GEOMETRY AND TRANSCENDENCE: THE CURVES
		V-A THE UNIVERSES OF THE GEOMETRICAL DISCOURSES OF DESCARTES AND LEIBNIZ
			Descartes and the ‘geometrical’ curves
			Leibniz and the ‘algebraical’ curves
		V-B LEIBNIZ’S CRITIQUE OF DESCARTES’ GEOMETRY‘ OF THE STRAIGHT LINE’ AND THE PARAMETERIZED CURVES.
			Descartes’ practice of the construction of curves, ‘linear, pointwise’
			Leibniz and the parameterized curves
		V-C EVOLUTES, ENVELOPES AND THE GENERATION OF TRANSCENDENT CURVES.
			The evolute of an algebraical curve is – usually – a transcendent curve
			Quadratures and evolutes are processes that are at the same time reciprocal and similar
			‘On the art of the discovery in general’: Leibniz and the need for reasoned inventories
			Leibniz and the envelope of a family of curves
			Variables or constants? Exchanging interpretations
		V-D TRANSCENDENT CURVES AND ‘POINTWISE’ CONSTRUCTIONS
			The equilibrium position of an inextensible heavy thread
			Some elements of hyperbolic trigonometry
			The catenary
			“On the curve formed by a thread …”.
			The geometric-harmonic construction and the letter to Huygens of July 1691
			The parameter of the figure is a transcendent quantity
			The catenary, starting from the logarithmic
			The logarithmic, starting from the catenary
			“The construction of Leibniz is the most geometrical possible …”
			To find points “as much as we want?” Or, “any point?”
			Among the transcendent curves, the ‘percurrent’ curves are a minima
			Descartes and the ‘pointwise’ constructions
			The mathematician as a constructor or as a prescriber?
		V-E PERCURRENCE AND TRANSCENDENCE, LEIBNIZ AND JOHN BERNOULLI – THE STORY OF A MISUNDERSTANDING
			The catenary, regarded as an avatar of the strategy of a geometrical inventory
			Leibniz and Bernoulli – the birth of a misunderstanding
			Bernoulli and the percurrent calculus
			A middle way between algebraicity and transcendence?
			The end of an ambiguity – the ‘hypo-transcendence’ of the exponentials
			On a possible hierarchy in percurrence
		V-F MATHEMATICAL COMPLEMENTS – ENVELOPES AND EVOLUTES
			Determination of the envelope of a family of curves
			Evolutes and involutes
			Curves without equations – physical constructions
	CHAPTER VI QUADRATURES, THE INVERSE METHOD OF TANGENTS, AND TRANSCENDENCE
		QUADRATURES AS A ‘BLIND’ MODE OF GENERATING TRANSCENDENCE
			The typology of quadratures in Leibniz
			The inverse method of tangents
			Curves subject to a condition and differential equations
			Arc lengths and rectifications
			On ‘constants of integration’
	CHAPTER VII LEIBNIZ AND TRANSCENDENT NUMBERS
		A VALUE, WHICH ‘IS NOT A UNIQUE NUMBER’
		THE NUMBER IS “HOMOGENEOUS TO THE UNIT”
		THE TRANSCENDENT NUMBERS IN DE ORTU
		TRANSCENDENT NUMBERS IN THE NOUVEAUX ESSAIS
		TRANSCENDENT NUMBERS IN INITIA RERUM MATHEMATICARUM METAPHYSICA
		DIRECT EXPRESSION OF A NUMBER AND EXPRESSIONS WHICH CONTAIN IT
THIRD PART. TOWARDS AN “ANALYSIS OF THE TRANSCENDENT”
	INTRODUCTION
	CHAPTER VIII ON HARMONY IN LEIBNIZ
		VIII-A UNITAS IN VARIETATE
			Harmony is all the greater as it is revealed in a greater diversity
			Harmony and the identity of indiscernibles
			Harmony in the mathematics of Leibniz
			Symbolism and social communication
			Harmony via reciprocity, via homogeneity, via symmetry
		VIII-B THE TWO TRIANGLES AND THE LETTER TO L’HOSPITAL OF DECEMBER 1694
		VIII-C PASCAL AND LEIBNIZ. THE FIRST CONSTRUCTION OF THE TRIANGLES (BY ELEMENTS)
			Construction ‘by elements’ of Pascal’s arithmetical triangle
			Construction ‘by elements’of Leibniz’s harmonic triangle
		VIII-D HUYGENS, LEIBNIZ AND THE SUM OF RECIPROCAL TRIANGULAR NUMBERS (FIRST METHOD)
			The problem of Huygens
			Leibniz’s first method
			The sums of reciprocal figurate numbers. Analysis of Leibniz’s first method
			Harmonic progressions
		VIII-E) A MATHEMATICAL COMPLEMENT: THE DETERMINATION OF THE ELEMENTS OF THE HARMONIC TRIANGLE
	CHAPTER IX FROM THE HARMONIC TRIANGLE TO THE CALCULUS OF TRANSCENDENTS
		IX-A THE SECOND CONSTRUCTION OF THE TRIANGLES (BY LINES)
			The Leibnizian logical pattern
			The Leibnizian pattern: presentation and justifications
			Proofs of these results
			On convergence. The second method of Leibniz for the reciprocal triangular numbers
			Higher-order differentials
			Descartes and Leibniz, numbers versus functions
		IX-B THE ‘SUM OF ALL THE DIFFERENCES’, FROM LEIBNIZ TO LAMBERT AND LEBESGUE
			The ‘sum of all the differences’ in a contemporary environment
			Lambert, continued fractions, and the ‘sum of all the differences’
		IX-C ON THE HARMONY OF THE HARMONIC TRIANGLE
			Leibniz and Mengoli
			Leibniz and the harmony of the triangle
		IX-D FROM RECIPROCITY TO THE “NEW CALCULATION OF THE TRANSCENDENTS”. THE “CONSIDERATIONS (…)” OF 1694. THE PATTERN.
			A new calculation with eight pairwise reciprocal operations
			From Descartes to Leibniz, the rational functions
			Leibniz and the partial fraction expansion of the rational fractions
			Leibniz and the primitives of the rational fractions
			‘Ordinary’ Analysis versus ‘New’ Calculation of the Transcendents
			A truly general calculation
			Transcendent expressions are exactly those that are impossible to explicate
			The advent of the transcendence and the detachment from geometry as the only guarantee of the truth
			Harmony is restored: transcendence is no longer an obstacle to the calculation but an opening towards a new mathematics
			The Leibnizian scheme: a calculation with reciprocity and iterations
		IX-E MATHEMATICAL COMPLEMENTS. ON THE PROPERTIES OF THE HARMONIC TRIANGLE
	CHAPTER X TRANSCENDENCE AND IMMANENCE. SOME TERMINOLOGICAL MARKS BEFORE AND AFTER LEIBNIZ
		X-A ON NICHOLAS OF CUSA AND ON THE ORIGIN OF THE TERM ‘TRANSCENDENT’ IN LEIBNIZ
		X-B TRANSCENDENCE AND IMMANENCE. MATHEMATICS AND PHILOSOPHY
			The use of the word by Leibniz
			Transcendence: some philosophical definitions
			On immanence
			The immanence of the Cartesian algebra, as opposed to the transcendence of the Leibnizian infinitesimal Calculus
FOURTH PART. THE RECEPTION OF THE TRANSCENDENCE
	INTRODUCTION TO THE FOURTH PART
	CHAPTER XI THE RECEPTION OF THE TRANSCENDENCE BY THE CONTEMPORARIES OF LEIBNIZ
		XI-A THE RECEPTION BY TSCHIRNHAUS (1678–1682)
			“Transcendentes, ut vocas …”
			Tschirnhaus and the tangents to an arbitrary transcendent curve
		XI-B THE RECEPTION BY CRAIG (1685 AND 1693)
			John Craig and the impossibility of the quadrature of the transcendent figures
		XI-C THE RECEPTION BY STURM (1689)
			John-Christopher Sturm and the ‘transcendent degree’
		XI-D L’HOSPITAL, DIFFERENTIAL CALCULUS AND TRANSCENDENT CURVES
			The ‘Analyse des infiniment petits’ (1696)
			The organization of the book
			L’Hôpital and the tangents to the transcendent curves
			The glory of Leibniz
		XI-E JOHN BERNOULLI. ABOUT THE ORGANISATION OF THE TRANSCENDENT COMPLEXITY (1695–1730)
			Bernoulli, Huygens and Leibniz, between hypotranscendence and hypertranscendence
			Bernoulli and the “degrees of transcendence”
			On the comparative transcendence orders for the quadratures
			Only the first order subsists in fine
			Bernoulli and the repetition of the transcendent creation
		XI-F ON THE PHILOSOPHY OF SIMPLICITY. DESCARTES VERSUS NEWTON
			The ‘geometrically irrational’ curves in Newton (1687)
			From Descartes (1637) to Newton (1707). Symbolical simplicity versus geometrical simplicity
	CHAPTER XII THE TRANSCENDENCE BONE OF CONTENTION BETWEEN HUYGENS AND LEIBNIZ (1690)
		XII-A THE CONTROVERSY
			Leibniz and Huygens
			The controversy of 1690 between Huygens and Leibniz
			A lesson on method from Leibniz to Huygens: an equivalence between power series and exponentials
			Huygens’ conversion
	THE TRANSCENDENCE, BONE OF CONTENTION BETWEEN HUYGENS AND LEIBNIZ (1690)
		XII-B A MATHEMATICAL STUDY
			XII-B1 Huygens’ geometrical problem
			XII-B2 The Cartesian equation of Huygens’ curve (H)
			XII-B3 The equation of Huygens’ curve (H) in polar coordinates
			XII-B4 On calculations of the sub-tangent
			The “normal vector” method
			The sub-tangent to Huygens’ cubic curve
			XII-B5 Leibniz’s ‘supertranscendent’ curve
			XII-B6 The controversy (fourth and fifth letters)
			XII-B7 Huygens’ method for the tangents to algebraic curves
	CHAPTER XIII TRANSCENDENCE: THE WORD AND THE CONCEPTS, FROM EULER AND LAMBERT TO HILBERT
		XIII-A EULER AND TRANSCENDENCE
			Euler and the classification of functions:algebraic or transcendent
			Euler, algebraic curves and transcendent curves
			An extensive overview of transcendent functions
			Euler, algebraic quantities and transcendent quantities
			On quantities that are not expressible by radicals
			A hierarchy of transcendent numbers?
			Transcendent numbers have a very extensive domain
			Euler: proofs of irrationality and conjectures of transcendence
		XIII-B LAMBERT AND TRANSCENDENT NUMBERS
			Introduction to Mémoire: on “proof by simplicity”
			Conclusions of Mémoire: a classification of numbers
			On the supremacy of symbolism and issues of ‘transcendent’ terminology
		XIII-C RECEPTION OF THE ENCYCLOPÉDIE (1784–1789)
		XIII-D ON THE TRANSCENDENCE OF NUMBERS: THE WORD AND THE CONCEPTS, FROM LEGENDRE TO HILBERT
			From Legendre to Liouville
			Hermite and Lindemann: “La transcendance est fille de l’irrationalité”42
			Hilbert’s “seventh problem”: Euler-Hilbert conjecture
	CHAPTER XIV COMTE AND THE PHILOSOPHY OF THE TRANSCENDENT ANALYSIS
		The glorification of the transcendent analysis
		On the philosophy of mathematics
		About equations: formation versus resolution
		Leibniz, Newton, Lagrange: the trio of interpreters
		For a reasoned history
		Leibniz, the authentic creator of the transcendence. “The loftiest idea ever yet attained by the human mind”
		Newton: an effort to rationalize
		Lagrange, and the promotion of “abstraction”
		Differential calculus and geometry in two and three dimensions
		Differential calculus and integral calculus
		On the comparison of the interpretations
		The contributions of D’Alembert and Lagrange. On partial derivative equations
		The calculus of variations, and the ‘hypertranscendent’ analysis
		On the philosophy of the hypertranscendence. Singular and multiple. Variables and functions.
	CHAPTER XV AFTER LEIBNIZ: SOME MODERN EPISTEMOLOGICAL ASPECTS OF THE TRANSCENDENCE
		ON THE CONCEPTS OF LEIBNIZ TODAY
			Algebraic functions, transcendent functions. An initial approach
			Algebraic curves, transcendent curves. Modern definitions
			On the supremacy of the symbolism (again)
			On the ontology of the plane curves
			Curves implicitly defined
			On Huygens’ cubic curve
			Curves defined parametrically
			The cycloid as an example
			On the dialectic of the duality
			Algebraic numbers, transcendent numbers
			Leibniz, as the “founder of discursivity” of mathematical transcendence
REFERENCES
	LEIBNIZ
	DESCARTES
	GENERAL REFERENCES