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- Herausgeber: John Wiley & Sons
- Kategorie: Geisteswissenschaft
- Sprache: Englisch
- Veröffentlichungsjahr: 2018
This book, which studies the links between Mathematics and Philosophy, highlights a reversal. Initially, the (Greek) philosophers were also mathematicians (geometers). Their vision of the world stemmed from their research in this field (rational and irrational numbers, problem of duplicating the cube, trisection of the angle...). Subsequently, mathematicians freed themselves from philosophy (with Analysis, differential Calculus, Algebra, Topology, etc.), but their researches continued to inspire philosophers (Descartes, Leibniz, Hegel, Husserl, etc.). However, from a certain level of complexity, the mathematicians themselves became philosophers (a movement that begins with Wronsky and Clifford, and continues until Grothendieck).
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Table of Contents
Cover
Introduction
PART: 1 The Contribution of Mathematician–Philosophers
Introduction to Part 1
1 Irrational Quantities
1.1. The appearance of irrationals or the end of the Pythagorean dream
1.2. The first philosophical impact
1.3. Consequences of the discovery of irrationals
1.4. Possible solutions
1.5. A famous example: the golden number
1.6. Plato and the dichotomic processes
1.7. The Platonic generalization of ancient Pythagoreanism
1.8. Epistemological consequences: the evolution of reason
2 All About the Doubling of the Cube
2.1. History of the question of doubling a cube
2.2. The non-rationality of the solution
2.3. The theory proposed by Hippocrates of Chios
2.4. A philosophical application: platonic cosmology
2.5. The problem and its solutions
2.6. The trisection of an angle
2.7. Impossible problems and badly formulated problems
2.8. The modern demonstration
3 Quadratures, Trigonometry and Transcendance
3.1.
π
– the mysterious number
3.2. The error of the “squarers”
3.3. The explicit computation of
π
3.4. Trigonometric considerations
3.5. The paradoxical philosophy of Nicholas of Cusa
3.6. What came next and the conclusion to the history of
π
PART: 2 Mathematics Becomes More Powerful
Introduction to Part 2
4 Exploring
Mathesis
in the 17th Century
4.1. The innovations of Cartesian mathematics
4.2. The “plan” for Descartes’
Geometry
4.3. Studying the classification of curves
4.4. Legitimate constructions
4.5. Scientific consequences of Cartesian definitions
4.6. Metaphysical consequences of Cartesian mathematics
5 The Question of Infinitesimals
5.1. Antiquity – the prehistory of the infinite
5.2. The birth of the infinitesimal calculus
6 Complexes, Logarithms and Exponentials
6.1. The road to complex numbers
6.2. Logarithms and exponentials
6.3. De Moivre’s and Euler’s formulas
6.4. Consequences on Hegelian philosophy
6.5. Euler’s formula
6.6. Euler, Diderot and the existence of God
6.7. The approximation of functions
6.8. Wronski’s philosophy and mathematics
6.9. Historical positivism and spiritual metaphysics
6.10. The physical interest of complex numbers
6.11. Consequences on Bergsonian philosophy
PART: 3 Significant Advances
Introduction to Part 3
7 Chance, Probability and Metaphysics
7.1. Calculating probability: a brief history
7.2. Pascal’s “wager”
7.3. Social applications, from Condorcet to Musil
7.4. Chance, coincidences and omniscience
8 The Geometric Revolution
8.1. The limits of the Euclidean demonstrative ideal
8.2. Contesting Euclidean geometry
8.3. Bolyai’s and Lobatchevsky geometries
8.4. Riemann’s elliptical geometry
8.5. Bachelard and the philosophy of “non”
8.6. The unification of Geometry by Beltrami and Klein
8.7. Hilbert’s axiomatization
8.8. The reception of non-Euclidean geometries
8.9. A distant impact: Finsler’s philosophy
9 Fundamental Sets and Structures
9.1. Controversies surrounding the infinitely large
9.2. The concept of “the power of a set”
9.3. The development of set theory
9.4. The epistemological route and others
9.5. Analytical philosophy and its masters
9.6. Husserl with Gödel?
9.7. Appendix: Gödel’s ontological proof
PART: 4 The Advent of Mathematician-Philosophers
Introduction to Part 4
10 The Rise of Algebra
10.1. Boolean algebra and its consequences
10.2. The birth of general algebra
10.3. Group theory
10.4. Linear algebra and non-commutative algebra
10.5. Clifford: a philosopher-mathematician
11 Topology and Differential Geometry
11.1. Topology
11.2. Models of differential geometry
11.3. Some philosophical consequences
12 Mathematical Research and Philosophy
12.1. The different domains
12.2. The development of classical mathematics
12.3. Number theory and algebra
12.4. Geometry and algebraic topology
12.5. Category and sheaves: tools that help in globalization
12.6. Grothendieck’s unitary vision
Conclusion
Bibliography
Index
End User License Agreement
List of Tables
3 Quadratures, Trigonometry and Transcendance
Table 3.1.
The first approximations of π
7 Chance, Probability and Metaphysics
Table 7.1.
Matrix of gains
12 Mathematical Research and Philosophy
Table 12.1.
2010 Mathematics Subject Classification (top levels)
List of Illustrations
Introduction
Figure I.1.
The YBC 7289 tablet (source: Yale Babylonian Collection)
1 Irrational Quantities
Figure 1.1.
Meno’s square
Figure 1.2.
Plato’s line, from the Republic
Figure 1.3.
The pentacle of the Pythagoreans
2 All About the Doubling of the Cube
Figure 2.1.
The procedure used by Hippocrates of Chios
Figure 2.2.
The “mechanical” solution proposed by the academy
Figure 2.3.
Archytas’ solution
Figure 2.4.
Archytas’ curve
Figure 2.5.
Archytas’ curve in 3D-space
Figure 2.6.
Menaechmus’ first solution
Figure 2.7.
Menaechmus’ second solution
Figure 2.8.
Hippias’ quadratrix and its generator mechanism
Figure 2.9.
The division of an angle into any number of parts
Figure 2.10.
The conchoids of Nicomedes. For a color version of this figure, see www.iste.co.uk/parrochia/philosophy.zip
Figure 2.11.
Morley’s theorem. For a color version of this figure, see www.iste.co.uk/parrochia/philosophy.zip
3 Quadratures, Trigonometry and Transcendance
Figure 3.1.
Squaring the lune
Figure 3.2.
Computing π
Figure 3.3.
The coincidence of extremities according to Nicholas of Cusa
Figure 3.4.
The “passions” of the maximal and infinite line
4 Exploring
Mathesis
in the 17th Century
Figure 4.1.
The construction of curves
5 The Question of Infinitesimals
Figure 5.1.
The tangents problem
Figure 5.2.
The problem of Spinoza’s (Letter XII to Louis Meyer)
6 Complexes, Logarithms and Exponentials
Figure 6.1.
Argand’s plane
Figure 6.2.
The luminous vibration
8 The Geometric Revolution
Figure 8.1.
Right parallels and left parallels
Figure 8.2.
The parallelism angle
Figure 8.3.
Specific configuration
9 Fundamental Sets and Structures
Figure 9.1.
The “method of the diagonal”
10 The Rise of Algebra
Figure 10.1.
Clifford parallelism on a sphere and on a torus
Guide
Cover
Table of Contents
Begin Reading
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Series Editor
Nikolaos Limnios
Mathematics and Philosophy
Daniel Parrochia
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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© ISTE Ltd 2018
The rights of Daniel Parrochia to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018938064
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-209-0
Introduction
Philosophy is not descended from heaven. It does not follow a completely autonomous line of thought or a mode of speculation that is unknown to this world. Experience has shown us that the problems, concepts and theories of philosophy are born out of a certain economic and political context, in close conjunction with sources of knowledge that fall within positive learning and practices. It is within these sites that philosophy normally discovers the inductive elements for its thinking. This is where, as they say, it finds life. A little historical context, therefore, often makes it possible to reconstitute these elements that may sometimes leap off the surface of a text but always inform its internal working. All we have to do is identify them. Thus, metaphysics, from Plato to Husserl and beyond, has largely benefited from advances made in an essential field of knowledge: mathematics. Any progress and revolution in this discipline has always provided philosophy with not only schools of thought, but also tools and instruments of thinking.
This is why we will study here the link between philosophy and the discipline of mathematics, which is today an immense reservoir of extremely refined structures with multiple interconnections. We will examine the vicissitudes of this relationship through history. But the central question will be that of the knowledge that today can be drawn from this discipline, which has lately become so powerful and complex that it often and in large part soars out of reach of the knowledge and understanding of the philosopher. How can contemporary mathematics serve today’s philosophy? This is the real question that this book explores, being neither entirely an history of philosophy, nor an history of the sciences, and even less so that of epistemology.
We will not study science, its methods and laws, its evaluation or its status in the field of knowledge. We will simply ask how this science can still be of use to philosophers today in building a new vision of the world, and what this might be. A reader who is a philosopher will, therefore, certainly be asked to invert their thinking and reject their usual methods. Rather than placing scientific knowledge entre parentheses and embarking on a quest for a hypothetical other knowledge, assumed to be more remarkable, more native or more radical (the method called the “phenomenological method”), we prefer suspending judgment, using the epoché (reduction) method for phenomenology itself and sticking to the only effective knowledge that truly makes up reason (or, at any rate, a considerable part of reason): mathematical knowledge. This knowledge contains within itself the most remarkable developments and transformations not only of thought, but also of the world. This knowledge, by itself, has the capacity of constructing, in a methodical and reflective manner, the basic conceptual architecture needed to create worldviews. It would seem that philosophers have long forgotten this elementary humility that consists of beginning only with which is proven, instead of developing, through a blind adherence to empiricism, theories and dogma that lasted only a season, failing the test of time, their weaknesses revealed over the course of history.
In doing this, we follow in the footsteps of thinkers who are more or less forgotten today, but who kept repeating exactly what we say here. Gaston Milhaud, for example, had already noted this remarkable influence. In the opening lesson of a course taught at Montpellier in 1908–1909, which was then published in the Revue Philosophique and reprinted in one of his books [MIL 11, pp. 21–22], we find the following text:
“My intention is to bind myself to certain essential characteristics of mathematical thought and, above all, to study the repercussions it has had on the concepts and doctrines of philosophers and even on the most general tendencies of the human mind.
How can we doubt that these repercussions have been significant when history shows us mathematical speculations and philosophical reflections often united in the same mind; when so often, from the Pythagoreans to thinkers such as Descartes, Leibniz, Kant and Renouvier (to speak only of the dead), some fundamental doctrines, at least, have been based on the idea of mathematics; when on all sides and in all times we see the seeds of not only critical views, but even systems that weigh in on the most difficult and obscure metaphysical problems and which reveal especially, through the justifications offered by the authors, a sort of vertigo born out of the manipulation of or just coming into contact with the speculations of geometricians? The excitation in a thinker’s mind, far from being an accident in the history of ideas, appears to us as a continuous and almost universal fact”.
A few years later, in 1912, Léon Brunschvicg published Les étapes de la philosophie mathématique (Stages in Mathematical Philosophy), a book in which, as Jean-Toussaint Desanti noted in his preface to the 1981 reprint, it clearly appears that mathematics informed philosophy1. In this book, hailed by Borel as “one of the most powerful attempts by any philosopher to assimilate a discipline as vast as mathematical science”, we can already see, as Desanti recalls, that “the slow emergence of forms of mathematical intelligibility provided the reader with a grid through which to interpret the history of different philosophies”. [BRU 81, p. VII]. The fact remains, of course, that these two effects were secondary to Brunschvicg’s chief project: to give an account of mathematical discourse itself in its operational kernels, where the forms of construction of intelligible objects take place and where the activity of judgment (which he found so important) chiefly manifests itself, along with the dynamism inherent to the human intellect.
Admittedly, today mathematics is no longer accepted as truth in itself. Shaken to its foundations and now seen as being multivarious, if not uncertain2, it has seen its relevance diminish further of late. Knowing that 95% of truths are not demonstrable within our current systems and that the more complex a formula the more random it is3, we may well wonder as to the philosophical interest of the discipline. And so, Brunschvig’s concluding remark, according to which, “the free and fertile work of thought dates back to the time when mathematics gave man the true norm for the truth” [BRU 81, p. 577], may well make us smile. His Spinozian inspiration seems quite passé now and the lazy philosopher will delight in stepping into the breach.
Nonetheless, not even recent masters – Jules Vuillemin, Gilles-Gaston Granger, Roshdi Rashed – who dedicated a large part of their work to mathematical thought and its philosophical consequences, have gone down this path. If they are often close, it is in the sense that their work generally looks at measuring the influence, or even truly the impact, of mathematics on philosophy4. We will thus content ourselves with modestly following in their path. This book will thus undoubtedly follow a counter current. However, it joins certain observations made by contemporary mathematicians in the wake of Bachelard. “The truth is that science enriches and renews philosophy more than the other way around”5, as Jean-Paul Delahaye wrote in the early 2000s [DEL 00, p. 95]. In addition, we do not seek to lay out a pointless culture. We only aim to communicate the essential. That is, in the teacher’s experience, what is most easily lost or forgotten. The majority of this book will thus redemonstrate that philosophical reason, while it has undoubtedly been subject to multiple inflexions over its history, can only be constructed by looking at the corresponding advances made in science, and especially the discipline that contains the major victories of the sciences: mathematics. From the Pythagoreans to the post-modern philosophers, nothing of any importance has ever been conceived of without this near-constant reference.
Implementing philosophy today assumes an awareness of this creative trajectory. Once this is done, there are, of course, still some evident problems: if we believe in our schema, then should today’s philosophy follow the same inspiration as the philosophies of earlier ages? Is it possible for today’s philosophy to escape the biases that burden ancient systematic thinkers without denying its own nature? What definitive form must philosophy take today? These are but a few of the many questions that surround this reflection, which is, in our view, constantly inspired by mathematics. History has shown us that the true philosophers have not always been those who stirred up radical ideas, political criticisms or those short-sighted moralists who, today, many consider great philosophers. This is chiefly due to their lack of knowledge of science as well as the echo-chamber created by the media around the most insignificant things, which pushes the media itself to discuss nothing but this phenomenon. However, the existence of real facts and strong movements, generally ignored by the media buzz, leads us to think that things of true importance are happening elsewhere. Philosophy, with all due respect to Voltaire, used to be something quite different. And, for those who are serious, this remains an undertaking that goes well beyond what we find today in journals and magazines.
A note on the notations used here: When we speak of the mathematics of antiquity, the Middle Ages or the Classical Age – in brief, the mathematics of the past! – we will use present-day notations to ensure clarity. However, it must be understood that the symbols that we will use to designate the usual arithmetic operations have only existed in their current usage for about three centuries [BRU 00, p. 57]. It was at the beginning of the 17th Century, for example, that the “plus” sign, (+) (a deformation of the “and” sign (&)) and the “minus” sign (–) began to be widely used. These symbols are likely to have appeared in Italy in 1480; however, at that time it was more common to write “piu” and “meno”, with “piu” often being shortened to “pp”. In the 16th Century (1545, to be exact), a certain Michael Stiffel (1487–1567) denoted multiplication by a capital M. Then, in 1591, the algebraist François Viète (1540–1603), a specialist in codes who used to transcribe Henri IV’s secret messages, replaced this sign by “in”. The present-day use of the cross (×) was only introduced in 1632, by William Oughtred (1574–1660), a clergyman with a passion for mathematics. The notation for the period (.) owes itself to Leibniz (1646–1716), who used it for the first time in 1698. He also generalized the use of the “equal to” sign (=). This was used by Robert Recorde (1510–1558) in 1557 but was later often written as the Latin word (œqualitur) or, as used by Descartes and many of his contemporaries, was abridged to a backward “alpha”. While the notation for the square root appeared on Babylonian tablets dating back to 1800 or 1600 B.C. (see Figure I.1), its representation in the form we know today, dates back no earlier than the 17th Century. Its use in earlier mathematics is, thus, only a simplification and has no historical value.
Finally, this book is not a history of mathematics, but rather a study of the impact of mathematical ideas on representations in Western Philosophy over time, with the aim of highlighting teachings that we can use today.
Figure I.1.The YBC 7289 tablet (source: Yale Babylonian Collection)
1
J.-T. Desanti, preface to L. Brunschvicg [BRU 81, p. VI].
2
See the title of the book by Kline [KLI 93].
3
Toward the end of
Chapter 7
, we will be able to return to this and comment on the results obtained notably from the work of Gregory Chaitin.
4
See, for example, Rashed [RAS 91]. G.-G. Granger has sometimes highlighted the reverse, as is the case with Leibniz, where the philosophical principle of continuity determines different aspects of his mathematics. But this, in his own words, is an exceptional phenomenon [GRA 86].
5
Further on in this book (pp. 95–104), the author lists different important philosophical consequences of the progress of Kolmogorov’s theory of complexity and, notably, the definition of the randomness of a string as algorithmic “incompressibility”, which resulted in: (1) a new understanding of Gödel’s theorems of incompleteness; (2) an objective conception of physical entropy (Zurek); (3) Chaitin’s Omega number and the assurance of coherence in the theory of measurement; (4) a new understanding of scientific induction, of Bayes’ rule and Occam’s razor; (5) the distinction between random complexity and organized complexity (Bennett). A final epistemologically non-negligible consequence is the famous law propounded by Kreinovich and Longpré, according to which if a mathematical result is potentially useful, then it is not possible for it to have a complex proof. It would seem to result from this that which is complex is potentially useless, a result which many long-winded philosophers would do well to contemplate (see [KRE 00] and [LI 97]).
