Part 4: Footnotes

[1]. Velocity is just the speed along a defined straight line. A change in velocity can therefore be a change in speed in the same direction or the same speed in another direction (or a different speed and a different direction). For the scientifically inclined velocity is a vector quantity and speed a scalar; the less scientifically inclined should read the first law as saying that ‘A body will remain at rest or move with a constant speed in a straight line unless acted upon by a force’. 2]. Doubts about the independance of the first law are sometimes posted onto physics forums in the internet. They also occasionally appear in text-books written for other disciplines by non-physicists (see for example Gerhard Schurz’s ‘Einführung in die Wissenschaftstheorie’, 2008 WBG Darmstadt, p.177). Because there are excellent internet articles explaining the fallacy for physicists my approach here is addressed more towards students of philosophy or the history of science. [3]. The full title was “Philosophiae Naturalis Principia Mathematica”, or “the Mathematical Principles of Natural Philosophy”. The words “Philosophiae” and “Principia” were both underscored in the original publication to emphasise that Newton regarded his book as an answer to Descartes’ system of mechanics published in 1644 under the title “Principia philosophiae”. Although strongly disagreeing with Descartes’ natural philosophy, Newton does incorporate one of Descartes’ laws of motion rewording it slightly to become Newton’s first law. [4]. What we now refer to as “the calculus” was developed independently by Leibniz and by Newton. In the seventeenth century the educated public, whom Newton wished to address with his publication, although converse with Latin was not familiar with this new mathematical invention and could better understand arguments when presented in terms of geometry. [5]. In the “Principia” Newton describes this important asymmetry so: “True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed on the body.” (The Principia, Translated by Andrew Motte, Prometheus Books, 1955, p.16). Shortly before this sentence Newton had defined relative motion as motion in an accelerating frame of reference whilst true motion was in a non-accelerating frame. [6]. I say ‘of course’ but the supporter of the redundancy theory believes only in Newton’s second law and must consequently tell us what force is acting on the station to make it accelerate away! [7]. The concept of the inertial framework was first formulated by the German physicist, Ludwig Lange , (1863-1936) in 1885 and thus, like Euler’s equation, it does not appear in the “Principia”. In practical terms inertial frameworks are at rest with, or moving with constant velocity relative to, the average proper motion of the fixed stars. One framework cannot be differentiated from another because the laws of mechanics are the same in each. [8]. This does not mean that Newton’s laws cannot be used in non-inertial frameworks. They can be employed but fictitious, non-Newtonian, forces (such as the Coriolis and centrifugal forces) have to be invented to account for the additional accelerations. [9]. Westfall, Richard S., Never at Rest: A Biography of Issac Newton (Cambridge University Press, 1980), Chapter 10 ‘Principia’ . [10]. For a discussion of this see ‘Leibniz and the Plurality of Space-Time Frameworks’, in Rescher, Nicholas., Leibniz’s Metaphysics of Nature – a group of essays (D. Reidel Publishing Company, Dornrecht, Holland, 1981) 84-100. [11]. These two geniuses at the beginning of the Age of Enlightenment, Newton and Leibniz, supported unconventional theological, even mystical, convictions. Newton decided that the diatonic intervals in music were mirrored in colours of the spectrum. This meant that there must be seven; a belief reinforced by that number’s significance in Scripture (see for example http://www.biblewheel.com/Topics/Seven_Meaning.php). Leibniz, who was born in a Germany savaged and destroyed by the Thirty-Year War, believed that we must be living in God’s finest creation, ‘the Best of all Possible Worlds’. Leibniz’s optimism was mercilessly pilloried by Voltaire in ‘Candide’. Schopenhauer, with typical pessimistic hyperbole, dismissed it rather as ‘the worst of all possible worlds’. However Leibniz’s criterion for ‘The Best’ was not anthropocentric and not defined purely by human suffering. Rather what was ‘The Best’ was the optimisation of variety over order [see ‘Leibniz on Creation and the Evaluation of Possible Worlds’, in Rescher, Nicholas., Leibniz’s Metaphysics of Nature – a group of essays (D. Reidel Publishing Company, Dornrecht, Holland, 1981) 1-19]. Leibniz realised, nevertheless, that what resulted – the theodicy problem: the existence of ‘evil’ in God’s chosen world – had to be addressed and put forward explanations in his writings. [12]. One of the manifestations of the ‘Principle of Sufficient Reason’, nothing can be as it is without a sufficient reason or cause why it is so and not otherwise, was the foundation of physics on causal explanations. Even today it accounts for the ‘God does not play dice’ objection to the measurement problem in an otherwise deterministic quantum theory. But it seemed in the seventeenth and eighteenth centuries to lead to an endless chain of events, to infinite time, even to the denial of time, as well as to determinism and fatalism, consequences which Spinoza was prepared to accept but which were too radical for the theistic philosophers of the era. Their belief in an ultimate Being which/who created the world allowed them to terminate this endless chain but only at the cost of other problems. The objection why this ultimate Being would have chosen that particular moment for the creation of the world rather than any other is, in essence, the same as Leibniz’s objection to Newton why an ultimate Being would select a particular framework rather than any other. Eventually the inability of the rationalists to justify theism was to become a central issue in the ‘Pantheism Controversy’ in Germany and in the rise of pietistic protestantism (Beiser, Frederick C., The Fate of Reason Harvard University Press, 1987, Chapters 2 and 3). [13]. These arguments may be found in the Leibniz-Clarke correspondence, a series of letters exchanged between Gottfried Wilhelm Leibniz and Samuel Clarke (who was Newton’s parish vicar and neighbour and with whom Newton is generally assumed to have consulted closely in the formulation of Clarke’s letters) starting in November 1715 and ending in October 1716 due to the death of Leibniz. They are published in numerous editions, for a list see: Gjertsen, Derek The Newton Handbook (London, Routledge and Kegan Paul, 1986), 301. For those interested in pursuing this debate further N.L. Wilson’s article might prove a useful primer (‘Individual Identity, Space, and Time in the Leibniz-Clarke Correspondence’, in ‘The Philosophy of Leibniz and the Modern World’, ed. Ivor Leclerc (Nashville USA: Vanderbilt University Press, 1973) p. 189-206). It illustrates how tortured the arguments were as both sides tried to understand the implications of the concepts they were advocating. In my explanation of Leibniz’s objection to Newton using the ‘Principle of Sufficient Reason’ I have paraphrased a section of Leibniz’s letter to Clarke, quoted in N. L. Wilson’s article on page 202, into the language of inertial frames. [14]. Accounts of Descartes’ theories of motion may be read, for example, in Scott, J.F., The Scientific Work of René Descartes (Taylor and Francis Ltd, London, 1976), Chapter 10 and in Daniel Garber’s contribution entitled Descartes’ physics in The Cambridge Companion to Descartes edited by John Cottingham (Cambridge University Press, 1992). [15]. The ‘Special Theory of Relativity’, (1905), dealing with non-gravitational motion, employs inertial frames. Leibniz, because of his opposition to the Newtonian concepts of absolute space and time, might appear to anticipate Einstein. But any resemblance is superficial being founded only on confusing Leibniz’s ‘relationalism’ with Einstein’s ‘relativitism’. The ‘Logical Empiricists’ because of their objection to metaphysics were also prone to this mistake (see for example: Reichenbach, Hans, The Philosophy of Space and Time 1958, Dover Publications, New York, p.210). The difference between ‘relationism’ and ‘relativitism’ is comprehensively explained in John Earman’s book ‘World Enough and Space-Time: Absolute and Relational Theories of Space and Time’, 1989 MIT Press.

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