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The Labyrinth of the Continuum: Writings on the Continuum Problem 1672-86
This book gathers together for the first time an important body of texts written between 1672 and 1686 by the great German philosopher and polymath Gottfried Leibniz. These writings, most of them previously untranslated, represent Leibniz's sustained attempt on a problem whose solution was crucial to the development of his thought, that of the composition of the continuum. The volume begins with excerpts from Leibniz's Paris writings, in which he tackles such problems as whether the infinite division of matter entails "perfect points," whether matter and space can be regarded as true wholes, whether motion is truly continuous, and the nature of body and substance. Comprising the second section is Pacidius Philalethi, Leibniz's brilliant dialogue of late 1676 on the problem of the continuity of motion. In the selections of the final section, from his Hanover writings of 1677–1686, Leibniz abandons his earlier transcreationism and atomism in favor of the theory of corporeal substance, where the reality of body and motion is founded in substantial form or force. Leibniz's texts (one in French, the rest in Latin) are presented with facing-page English translations, together with an introduction, notes, appendixes containing related excerpts from earlier works by Leibniz and his predecessors, and a valuable glossary detailing important terms and their translations.
- GenresPhilosophy
576 pages, Hardcover
First published January 1, 1686
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93 people want to read
About the author
Gottfried Wilhelm Leibniz
1,350 books551 followersGerman philosopher and mathematician Baron Gottfried Wilhelm von Leibniz or Leibnitz invented differential and integral calculus independently of Isaac Newton and proposed an optimist metaphysical theory that included the notion that we live in "the best of all possible worlds."
Gottfried Wilhelm von Leibniz, a polymath, occupies a prominent place in the history. Most scholars think that Leibniz developed and published ever widely used notation. Only in the 20th century, his law of continuity and transcendental homogeneity found implementation in means of nonstandard analysis. He of the most prolific in the field of mechanical calculators. He worked on adding automatic multiplication and division to calculator of Blaise Pascal, meanwhile first described a pinwheel in 1685, and used it in the first mass-produced mechanical arithmometer. He also refined the binary number system, the foundation of virtually all digital computers.
Leibniz most concluded that God ably created our universe in a restricted sense, Voltaire often lampooned the idea. Leibniz alongside the great René Descartes and Baruch Spinoza advocated 17th-century rationalism. Applying reason of first principles or prior definitions, rather than empirical evidence, produced conclusions in the scholastic tradition, and the work of Leibniz anticipated modern analytic logic.
Leibniz made major contributions to technology, and anticipated that which surfaced much later in probability, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on politics, law, ethics, theology, history, and philology. Various learned journals, tens of thousands of letters, and unpublished manuscripts scattered contributions of Leibniz to this vast array of subjects. He wrote in several languages but primarily Latin and French. No one completely gathered the writings of Leibniz.
Gottfried Wilhelm von Leibniz, a polymath, occupies a prominent place in the history. Most scholars think that Leibniz developed and published ever widely used notation. Only in the 20th century, his law of continuity and transcendental homogeneity found implementation in means of nonstandard analysis. He of the most prolific in the field of mechanical calculators. He worked on adding automatic multiplication and division to calculator of Blaise Pascal, meanwhile first described a pinwheel in 1685, and used it in the first mass-produced mechanical arithmometer. He also refined the binary number system, the foundation of virtually all digital computers.
Leibniz most concluded that God ably created our universe in a restricted sense, Voltaire often lampooned the idea. Leibniz alongside the great René Descartes and Baruch Spinoza advocated 17th-century rationalism. Applying reason of first principles or prior definitions, rather than empirical evidence, produced conclusions in the scholastic tradition, and the work of Leibniz anticipated modern analytic logic.
Leibniz made major contributions to technology, and anticipated that which surfaced much later in probability, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on politics, law, ethics, theology, history, and philology. Various learned journals, tens of thousands of letters, and unpublished manuscripts scattered contributions of Leibniz to this vast array of subjects. He wrote in several languages but primarily Latin and French. No one completely gathered the writings of Leibniz.
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June 3, 2022
Research into the nature of the continuum has a venerable history going back to the pre-Socratics because regularity of the operation of the laws of nature along with the principle of continuity imply it arises naturally as an idealization of the world of experience. One is immediately confronted with the conundrum as to the existence of indivisibles. Aristotle’s resolution – allowing division to proceed indefinitely but denying that it could ever be actually completed – proves insufficient in the long run since it rules out the limit procedures of the calculus. Speculation continued all through the Middle Ages (see our review of Norman Kretzmann’s Infinity and Continuity in Ancient and Medieval Thought) but the continuum problem becomes particularly topical in the early modern period owing to its relevance to change in the physical world, the quantification of which was to be the keynote of the seventeenth-century scientific revolution.
Leibniz, as one of the principal founders of the calculus, matters therefore to any student of intellectual history. Thus, we are fortunate to have in The Yale Leibniz, the Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686 complete coverage of his early views on the problem of the continuum. This edition has been outfitted with a fine introduction by Richard T.W. Arthur. Along with the Latin texts and English translation, there are appendices on Leibniz’ predecessors and extensive notes to the texts, plus a Latin-English glossary.
What we want to cover in this review: to set the context in the seventeenth century itself and to discuss its relevance to later developments. To begin, let us quote Leibniz himself:
Only Geometry can provide a thread for the Labyrinth of the Composition of the Continuum, of maximum and minimum, and the unassignable and the infinite, and no one will arrive at a truly solid metaphysics who has not passed through that labyrinth. [p. xxiii, Nam filum Labyrintho de Compositione Continui, deque maximo et minimo, ac indesignabili atque infinito, non nisi Geometria praebere potest, ad Metaphysicam vero solidam nemo veniet, nisi qui illac transiverit]
Thus, in his mind’s eye, the problem of the continuum is not isolated to mathematics but is pertinent to all of metaphysics, which for him includes physics as a subordinate part. As Arthur points out, the continuum turns out to be central to Leibniz’ mature monodological metaphysics. Let us try to articulate why Leibniz himself maintains this, against the opinion of many modern experts on his system of thought! For what is at issue is the composition not just of the real line but of anything putatively continuous. In physics: cohesion, atoms and the void. In metaphysics, it implicates the doctrines of substance and of how mind informs substance, primary matter, the Cartesian plenum and individuation. Leibniz’ basic position is that omnia esse plena [p. xlviii]: everything is constituted by a plenum. He advances an ontology of perfect fluids and perfect solids, everything else we see must be constituted from these. It is interesting to note, in this connection, that we are not so far from Descartes’ physics in the Principia Philosophiæ of 1644. For Descartes wants to represent all physical bodies as reducible to extension, i.e., nothing more than the space coterminous with the figure that encloses them. Now, the early Leibniz evidently thinks along similar lines but imposes a further condition, that of fluidity resp. solidity, which merely constrains the spatial interrelations among the component parts, or minima, making up the body. Everyone in the physics world knows that Leibniz is famous for having corrected Descartes on this score, for what Descartes misses (on the level of his official theory at any rate) is the dynamical aspect, namely, that in addition to extension there is also density, for the outcome of a collision between billiard balls is determined not just by their geometrical figure but also by the respective momenta they bear, which are proportional to their respective masses. But it will not be until 1695 that we get a statement reflecting the mature Leibniz’ views, in his Specimen Dynamicum. Therefore, it will be of interest to the reader of these early papers to ponder the young Leibniz’ intellectual trajectory out of Cartesianism.
What more does the early Leibniz bring to the table? A conviction of the true infinite or immensum different from the unbounded [p. liv]. Hence, what is implicit in Descartes’ physics comes to the fore, namely, that body, space and substance are to be regarded as infinite aggregates (he believes in an actual infinite of existing things if not in the existence of an infinite number). Accordingly there will be an infinite regress in bulk matter, reflecting its organization on smaller and smaller scales – remember, for Leibniz, not being an atomist, there is no smallest scale! Here is where the physics gets to be very interesting, for in order to understand the behavior of bulk matter on smaller and smaller scales Leibniz, taking his cue from Hobbes, posits what he calls endeavors, leading to a concept of the differential [p. xxxi]. For perhaps the first time, we find a theory of physics that accounts for macroscopic phenomena as emerging from the behavior of matter on microscopic scales, in principle describable by means of a differential equation. Here, Leibniz’ work is pregnant. For once he has formulated a differential and integral calculus, the great strides to be accomplished in the classical mechanics of the succeeding century become possible – for instance, in Eulerian hydrodynamics. Incidentally, it may be noted that Newton’s corpuscular theory of fluids, which occupies an extensive section in the Principia, is faulty and does not anyway apply to most real fluids in the everyday regime. Indeed, moreover, Newton never writes down his second law of motion in the form of the differential equation so familiar to every physics student: mẍ = F. To develop his ideas in this direction and to contrive a realistic theory of fluids was the task of the eighteenth century.
What perspectives can we gain from Leibniz’ early writings (1672-1686), then? Leibniz’ signature monads would be incomprehensible if one sought to picture them apart from the context afforded by these early writings. For the function a monad performs is to organize into a coherent whole all the endeavors which, as we have seen, characterize bulk matter on the microscopic scale. Thus, in a sense, one implements a change of basis. As an analogy from condensed-matter physics: start from the description of a crystal as a lattice of atoms, then introduce the concept of a phonon, which represents a certain pattern of oscillation of the atoms about their equilibrium positions, at a certain frequency. If all we had was a picture of innumerable atoms jostling one another, condensed matter would be very hard to understand but becomes eminently comprehensible when viewed in terms of phonons – its transport properties etc.; see any textbook on physics of the solid state. Leibniz, of course, would not stop there; for him a phonon would figure as a fairly low-level monad. Especially in organic bodies, there are higher-order monads which control the collective behavior of lower-order monads.
To close, a précis on the later career of the continuum problem in order to show why it still ought to merit our concern. In the original formulation of the calculus the conception of the continuum as enlarged by infinitesimals played a crucial role. Cauchy-Weierstrassian epsilontics – the way we learn the calculus these days, at least at the college level – allows one to evade the labyrinth by handling only finite quantities; polyadic logic takes the place of infinitesimals, for better or for worse. But the seventeenth-century founders employ informal infinitesimal-type reasoning freely. Granted, their practice in this respect is understandable in that infinitesimals call for a lower level of abstraction than that of Fregean predicate calculus.
From our vantage point, what is really exciting is the rise of differential geometry in the nineteenth century. Again, one must appreciate that according to standards then in force, mathematics is not to be sharply distinguished from physics. Differential geometry establishes a tie between the behavior of space in the infinitely small and global properties. In Riemann’s natural philosophy one has to revise existing concepts whenever confronted with inexplicable observations – what Einstein picks up on in his general theory of relativity. For the latter’s genial idea of curvature as a measure of the dynamical properties of the Riemannian metric shows in a wonderful way how, in space viewed as a continuum, the infinitesimal scale becomes relevant to the phenomena of physics on the macroscopic scale, thus realizing in part Leibniz’ vision.
What role could the continuum play in the ongoing search for a theory of quantum gravity? Discretization seems to be the default option but this merely evades the issue. For decades, mathematical physics has been preoccupied by algebraic to the neglect of spatial structure, or the continuum. If anyone were ever to do justice to the contrarian views of P.A.M. Dirac, however, renewed reflection on it would be necessary in order to put oneself in a position to face non-renormalizability of quantum fields squarely. In this enterprise it should prove invaluable to revisit Leibniz’ writings. Four stars: the somewhat rambling pieces reproduced here represent his Nachlaß and were never intended for publication; Leibniz himself never honed his thoughts into sufficiently systematic form to be presentable in a monograph, leaving us to carry out the task for him.
Leibniz, as one of the principal founders of the calculus, matters therefore to any student of intellectual history. Thus, we are fortunate to have in The Yale Leibniz, the Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686 complete coverage of his early views on the problem of the continuum. This edition has been outfitted with a fine introduction by Richard T.W. Arthur. Along with the Latin texts and English translation, there are appendices on Leibniz’ predecessors and extensive notes to the texts, plus a Latin-English glossary.
What we want to cover in this review: to set the context in the seventeenth century itself and to discuss its relevance to later developments. To begin, let us quote Leibniz himself:
Only Geometry can provide a thread for the Labyrinth of the Composition of the Continuum, of maximum and minimum, and the unassignable and the infinite, and no one will arrive at a truly solid metaphysics who has not passed through that labyrinth. [p. xxiii, Nam filum Labyrintho de Compositione Continui, deque maximo et minimo, ac indesignabili atque infinito, non nisi Geometria praebere potest, ad Metaphysicam vero solidam nemo veniet, nisi qui illac transiverit]
Thus, in his mind’s eye, the problem of the continuum is not isolated to mathematics but is pertinent to all of metaphysics, which for him includes physics as a subordinate part. As Arthur points out, the continuum turns out to be central to Leibniz’ mature monodological metaphysics. Let us try to articulate why Leibniz himself maintains this, against the opinion of many modern experts on his system of thought! For what is at issue is the composition not just of the real line but of anything putatively continuous. In physics: cohesion, atoms and the void. In metaphysics, it implicates the doctrines of substance and of how mind informs substance, primary matter, the Cartesian plenum and individuation. Leibniz’ basic position is that omnia esse plena [p. xlviii]: everything is constituted by a plenum. He advances an ontology of perfect fluids and perfect solids, everything else we see must be constituted from these. It is interesting to note, in this connection, that we are not so far from Descartes’ physics in the Principia Philosophiæ of 1644. For Descartes wants to represent all physical bodies as reducible to extension, i.e., nothing more than the space coterminous with the figure that encloses them. Now, the early Leibniz evidently thinks along similar lines but imposes a further condition, that of fluidity resp. solidity, which merely constrains the spatial interrelations among the component parts, or minima, making up the body. Everyone in the physics world knows that Leibniz is famous for having corrected Descartes on this score, for what Descartes misses (on the level of his official theory at any rate) is the dynamical aspect, namely, that in addition to extension there is also density, for the outcome of a collision between billiard balls is determined not just by their geometrical figure but also by the respective momenta they bear, which are proportional to their respective masses. But it will not be until 1695 that we get a statement reflecting the mature Leibniz’ views, in his Specimen Dynamicum. Therefore, it will be of interest to the reader of these early papers to ponder the young Leibniz’ intellectual trajectory out of Cartesianism.
What more does the early Leibniz bring to the table? A conviction of the true infinite or immensum different from the unbounded [p. liv]. Hence, what is implicit in Descartes’ physics comes to the fore, namely, that body, space and substance are to be regarded as infinite aggregates (he believes in an actual infinite of existing things if not in the existence of an infinite number). Accordingly there will be an infinite regress in bulk matter, reflecting its organization on smaller and smaller scales – remember, for Leibniz, not being an atomist, there is no smallest scale! Here is where the physics gets to be very interesting, for in order to understand the behavior of bulk matter on smaller and smaller scales Leibniz, taking his cue from Hobbes, posits what he calls endeavors, leading to a concept of the differential [p. xxxi]. For perhaps the first time, we find a theory of physics that accounts for macroscopic phenomena as emerging from the behavior of matter on microscopic scales, in principle describable by means of a differential equation. Here, Leibniz’ work is pregnant. For once he has formulated a differential and integral calculus, the great strides to be accomplished in the classical mechanics of the succeeding century become possible – for instance, in Eulerian hydrodynamics. Incidentally, it may be noted that Newton’s corpuscular theory of fluids, which occupies an extensive section in the Principia, is faulty and does not anyway apply to most real fluids in the everyday regime. Indeed, moreover, Newton never writes down his second law of motion in the form of the differential equation so familiar to every physics student: mẍ = F. To develop his ideas in this direction and to contrive a realistic theory of fluids was the task of the eighteenth century.
What perspectives can we gain from Leibniz’ early writings (1672-1686), then? Leibniz’ signature monads would be incomprehensible if one sought to picture them apart from the context afforded by these early writings. For the function a monad performs is to organize into a coherent whole all the endeavors which, as we have seen, characterize bulk matter on the microscopic scale. Thus, in a sense, one implements a change of basis. As an analogy from condensed-matter physics: start from the description of a crystal as a lattice of atoms, then introduce the concept of a phonon, which represents a certain pattern of oscillation of the atoms about their equilibrium positions, at a certain frequency. If all we had was a picture of innumerable atoms jostling one another, condensed matter would be very hard to understand but becomes eminently comprehensible when viewed in terms of phonons – its transport properties etc.; see any textbook on physics of the solid state. Leibniz, of course, would not stop there; for him a phonon would figure as a fairly low-level monad. Especially in organic bodies, there are higher-order monads which control the collective behavior of lower-order monads.
To close, a précis on the later career of the continuum problem in order to show why it still ought to merit our concern. In the original formulation of the calculus the conception of the continuum as enlarged by infinitesimals played a crucial role. Cauchy-Weierstrassian epsilontics – the way we learn the calculus these days, at least at the college level – allows one to evade the labyrinth by handling only finite quantities; polyadic logic takes the place of infinitesimals, for better or for worse. But the seventeenth-century founders employ informal infinitesimal-type reasoning freely. Granted, their practice in this respect is understandable in that infinitesimals call for a lower level of abstraction than that of Fregean predicate calculus.
From our vantage point, what is really exciting is the rise of differential geometry in the nineteenth century. Again, one must appreciate that according to standards then in force, mathematics is not to be sharply distinguished from physics. Differential geometry establishes a tie between the behavior of space in the infinitely small and global properties. In Riemann’s natural philosophy one has to revise existing concepts whenever confronted with inexplicable observations – what Einstein picks up on in his general theory of relativity. For the latter’s genial idea of curvature as a measure of the dynamical properties of the Riemannian metric shows in a wonderful way how, in space viewed as a continuum, the infinitesimal scale becomes relevant to the phenomena of physics on the macroscopic scale, thus realizing in part Leibniz’ vision.
What role could the continuum play in the ongoing search for a theory of quantum gravity? Discretization seems to be the default option but this merely evades the issue. For decades, mathematical physics has been preoccupied by algebraic to the neglect of spatial structure, or the continuum. If anyone were ever to do justice to the contrarian views of P.A.M. Dirac, however, renewed reflection on it would be necessary in order to put oneself in a position to face non-renormalizability of quantum fields squarely. In this enterprise it should prove invaluable to revisit Leibniz’ writings. Four stars: the somewhat rambling pieces reproduced here represent his Nachlaß and were never intended for publication; Leibniz himself never honed his thoughts into sufficiently systematic form to be presentable in a monograph, leaving us to carry out the task for him.
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July 22, 2014When I was 17, I claimed I was "wandering the continuous maze of existence."
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