Leibniz and the Invention of Mathematical Transcendence

Name: Leibniz and the Invention of Mathematical Transcendence
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Erschienen am 25. Oktober 2018

XIX, 225 Seiten

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Intro
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

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