The Idea of Principles in Early Modern Thought

Interdisciplinary Perspectives

Peter R. Anstey, Peter R. Anstey

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781315452678
• Edition: 1
• Topic: Filosofia
• Subtopic: Storia e teoria della filosofia

This collection presents the first sustained examination of the nature and status of the idea of principles in early modern thought. Principles are almost ubiquitous in the seventeenth and eighteenth centuries: the term appears in famous book titles, such as Newton's Principia; the notion plays a central role in the thought of many leading philosophers, such as Leibniz's Principle of Sufficient Reason; and many of the great discoveries of the period, such as the Law of Gravitational Attraction, were described as principles.

Ranging from mathematics and law to chemistry, from natural and moral philosophy to natural theology, and covering some of the leading thinkers of the period, this volume presents ten compelling new essays that illustrate the centrality and importance of the idea of principles in early modern thought. It contains chapters by leading scholars in the field, including the Leibniz scholar Daniel Garber and the historian of chemistry William R. Newman, as well as exciting, emerging scholars, such as the Newton scholar Kirsten Walsh and a leading expert on experimental philosophy, Alberto Vanzo. The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives charts the terrain of one of the period's central concepts for the first time, and opens up new lines for further research.

1 Early Modern Mathematical Principles and Symmetry Arguments

James Franklin

Mathematics is the home ground of principles. Since Euclid, mathematics has been the model of a body of knowledge organized as a deductive structure based on self-evident axioms. The prestige of that model was highest in early modern times, lying after the vast extension of the realm of mathematics in the Scientific Revolution but before the discovery of non-Euclidean geometries and the foundational crises of the late nineteenth century. When the Jesuit missionaries chose Euclid as the first book to be translated into Chinese (Engelfriet 1998) and when Spinoza offered to exhibit ethics as a system demonstrated more geometrico from definitions and simple self-evident axioms, they paid tribute to the place of the Euclidean model at the summit of intellectual achievement.

Seventeenth-century thinkers who applied mathematics to physics—Stevin, Galileo, Descartes, Pascal, Huygens, Barrow, Newton, and Leibniz, to name only the most prominent—were captivated by a model of applied mathematics, apparently realized in Euclid’s Optics and Archimedes’s mechanics, according to which pure thought could establish principles for empirical reality. Their ambition stood at the opposite extreme from the Baconian style of “experimental philosophy,” which recommended generalization from carefully collected facts (Domski 2013).

Their hopes were not fully realized because of such awkward facts as the contingency of the constant of gravitational attraction. But their vast optimism concerning the power of mathematical reasoning and the possibilities of understanding reality through pure symmetry arguments proved astonishingly successful and created modern physics.

The Background: Aristotle’s Posterior Analytics and Euclid’s Elements

The model of mathematics as a deductive structure of propositions proved from self-evident truths was backed by the two greatest possible authorities, Aristotle and Euclid, and was in any case open to inspection if one followed the proofs in Euclid oneself. Virtually all thinkers accepted it without question.

The essentials of the model were laid down in Aristotle’s Posterior Analytics. A true or fully developed science should demonstrate its truths by syllogistic deduction from self-evident first principles, explaining why the truths of the science must be as they are. The first principles should be simple enough to be evident to the pure light of reason, or nous (Latin intellectus), a divinely granted faculty of the soul capable of grasping necessities.

If to modern ears that seems a promise too good to be true, for early modern thinkers it was confirmed by the existence and obvious success of Euclidean geometry, a science that seemed to conform exactly to Aristotle’s model (McKirahan 1992: Ch. 12). Euclid’s Elements begins with twenty-three definitions, such as

1. 1. A point is that which has no part …
2. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another …

These are meant to be not arbitrary stipulations but contentful truths about real geometrical objects. Though Euclid omits any philosophical commentary, modern writers rightly emphasize that to appreciate the definition of a circle, for example, one needs an act of insight to understand that something uniformly round does, in fact, consist of all the points equidistant from a centre (Lonergan 1997: Ch. 1). There follow five postulates or axioms, such as

1. 1. To draw a straight line from any point to any point …
2. 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The way in which the Fifth Postulate is posed is such as to make it appear as self-evident as possible, despite the worries, expressed since ancient times, as to whether anything involving lines “produced indefinitely” could be truly self-evident. Indeed, behind the scenes there were considerable doubts among the best mathematicians as to whether Euclid’s logic was watertight (De Risi 2016), but from the perspective of outsiders the Euclidean structure was monolithic and impregnable.

There follow a large number of propositions or theorems derived logically from the postulates, beginning with

I.1 To construct an equilateral triangle on a given finite straight line.

These truths were taken to be necessary truths about the real space we live in, even though Aristotle, to some extent, adopts the Platonist idea that the geometer’s lines and circles are assumed perfect and hence are idealizations of the real shapes of rulers and wheels. Any doubts about the principles of geometry and about its direct applicability to the physical world were suppressed in the seventeenth century. Even as contra-suggestible a mind as Hobbes, initially doubtful about Euclid, was convinced by the proofs. Virtually the only doubter was the Chevalier de Méré, a “man of the world” without intellectual standing. When he wrote to Pascal with some puzzles about probability which he took to show that mathematics was self-contradictory, Pascal wrote to Fermat that de Méré’s incompetence in mathematics was clear from his belief that space was atomic (contrary to the infinite divisibility of space in Euclid) (Pascal 1964–1970, 2: 1142; embarrassingly, it turned out that whether space is continuous or atomic is not provable; see Franklin 1994).

To appreciate the early modern understanding of mathematical principles, it is essential to put to one side certain more recent philosophical views which make it appear implausible. Right or wrong, certain contemporary received ideas about mathematics and its relation to reality impede a clear view of early modern assumptions. We discuss briefly the two main ones, which both assume a divorce between mathematics and reality that was not part of, or even one of the possibilities considered by, the early modern view of mathematics.

First, the legacy of four hundred years of experimental science is an assumption that substantial scientific knowledge is contingent and must be established by experiment; hence, the role of mathematics is as a “theoretical juice extractor,” a collection of methods to codify the generalizations arising from experiment and to enable predictions from them. Mathematics itself is not seen as directly about reality. As Einstein put it, “[a]s far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality” (Einstein 1954: 233; argued against in Franklin 2014: Ch. 5).

The second and related contemporary obstacle to understanding early modern views on mathematics is the oscillation in post-Fregean philosophy of mathematics between nominalism and Platonism. Nominalist philosophies that take mathematics to be not about anything but a language of science, or a collection of tautologies, or manipulation of formal symbols, or an investigation of what follows from arbitrary axioms, stand against Platonism, which regards mathematics as about “abstract objects” such as sets and numbers (or idealized geometrical objects like perfect circles). (The dichotomy explained and criticized in Franklin 2014: Ch. 7; a textbook on mathematical proof from an Aristotelian point of view is Franklin and Daoud 2011.)

Nothing could be further from the early modern conception of mathematics, which was neither nominalist nor Platonist but Aristotelian (or “moderate”) realist—mathematics was said to be the “science of quantity,” with quantity understood as a property of extended physical things (e.g., Barrow 1734: 10–15; further in Barrow 1860 [1664]: lectiones 4–8; Gillette 2009; Encyclopaedia Britannica 1771: article ‘Mathematics’, vol. 3: 30–31; Mancosu 1996: 16, 35–37, 56, 88; Jesseph 1993: Ch. 1. Gassendi and Hobbes do have nominalist tendencies: Sepkoski 2005, Pycior 1987; Dear 1995 puts a constructivist interpretation on the topic; eighteenth-century continuations in Franklin 2006). Quantity, one of the basic Aristotelian categories, comes in discrete (studied by arithmetic) and continuous (studied by geometry). (For the bridging of this gap in the sixteenth and seventeenth centuries, see Malet 2006; Neal 2002a.) As Barrow writes,

[i]t is plain the Mathematics is conversant about two Things especially, viz Quantity strictly taken, and Quotity; or if you please, Magnitude and Multitude. By others they are called Continued and Discontinued Quantity …

(Barrow 1734: 10)

There are also subordinate or mixed sciences, such as music (subordinate to arithmetic) and optics and astronomy (subordinate to geometry). Barrow says,

But because both Magnitude and Multitude may be considered in a double respect; viz. either as they are mentally separated, or abstracted from all Matter, material Circumstances, and Accidents; … or as they inhere in some particular Subject, and are found conjoined with certain other physical Qualities, Actions and Circumstances: Hence arises the Division of Mathematics into Pure or Abstract, and Mixed or Concrete.

Being concrete does not impede the certainty of the mixed mathematical sciences. They are again conceived as sciences of real but necessary aspects of the world. The search for self-evident principles for the subordinate sciences is the most interesting aspect of the topic and we will return to it.

The Aristotelian-Euclidean background to seventeenth-century mathematics means that mathematics does not fit at all into the old picture of the Scientific Revolution as a revolt against scholastic obfuscation in favour of empiricism and experimentalism. That model, even if oversimplified, does make some sense as applied to the natural sciences, especially chemistry, biology and the more empirical parts of physics. Mathematics, including applied mathematics, is otherwise. As Derek Whiteside put it after his extensive investigation of early British mathematics,

I have neglected a prevailing fashion which sees mathematics as a mere handmaiden to the sciences, and the 17th century scientific achievement as a revolution in which scientific thought was freed from the largely sterile dominance of scholastic authority under a universal guiding principle of the primacy of theory induced from observed instances in phenomena.

(Whiteside 1961: 180)

The story of mathematics is more like an extension of scholasticism than a retreat from it. Where the old scholastics had been excessively modest about the possibilities of reducing the contingent physical world to quantitative order and demonstration, the mathematicians showed that it could be done by doing it. Early modern applied mathematics is the pursuit of the scholastic vision by other means.

Mathematical Developments in the Seventeenth Century

Before the Scientific Revolution was a revolution in science, it was a mathematical revolution. Copernicus’s astronomy was purely geometrical, an attempt to rearrange the geometry of the heavens without any considerations of the causes of celestial motions. The same is largely true of the successful parts of Kepler’s astronomy, though he did add some speculations about magnetic attractions as a possible cause of orbits. John Dee’s preface to the first English Euclid, of 1570, describes a culture of some thirty useful mathematical sciences like perspective, navigation, astrology, and statics (Euclid [1570]: Billingsley’s Preface). The important mathematical achievements around 1600 of Viète, Stevin, and Harriot in fields like algebra, statics, navigation, and logarithms preceded the first new experimental results of Galileo and Bacon (Franklin 2000).

Then the seventeenth century itself saw a flowering of mathematical genius—much more so than in the two adjacent centuries. Galileo, Descartes, Fermat, Pascal, Huygens, Newton, and Leibniz were mathematicians of the first rank; Kepler, Cavalieri, Barrow, Wallis, Jacob Bernoulli, and others, important contributors. All were trained in Euclid’s model of how to do mathematics. All had, so to speak, a Euclid baton in their knapsack; they harboured the ambition of becoming the new Euclid of their fields. In many but not all cases, as we will see, that involved explicitly laying down definitions and axioms for new fields and deriving theorems from them.

What they hoped for is expressed in a confident, even impudent, exchange about theory and experiment from Galileo’s Dialogue Concerning the Two Chief World Systems:

• SIMPLICIO: So you have not made a hun...