Is Newton's first law redundant?

. . . . . .   its justification and significance. An explanation for students of philosophy and the history of science.

Part 1:    What the problem seems to be

Isaac Newton gave us three laws of motion. They form the foundations of classical mechanics; they are the basis of scientific achievements ranging from explaining the motions of the planets, through engineering the dynamics of automobiles and planes, to mastering moon-landings. But do we need all of his three laws?

Expressed succinctly Newton's laws are:

  1. A body will remain at rest or move with a constant velocity1 unless acted upon by a force.
  2. Force is equal to mass multiplied by acceleration.
  3. To every action there is an equal and opposite reaction.

These laws have been handed down generation to generation. But for some non-professional physicists they raise doubts2. For these sceptics Newton's first law of motion does not seem to have the same status as the other two. The second and third laws appear to be totally independent of each other. But Newton's first law, for this minority, is seen to be just a consequence of the second law. Could it be that the first law is redundant? Did Newton give us three laws when two would have been adequate? It seems unlikely that this colossus of science would make such a mistake but the argument supporting this "redundancy hypothesis" is so convincing that doubts arise.

The "redundancy hypothesis" argument is easy to understand. Two premises lead to a deductive conclusion. The first premise comes directly from Newton's second law which asserts that "force is equal to mass multiplied by acceleration". The second premise is the definition of acceleration. It is that an acceleration means a change in velocity. (The acceleration, of course, can be negative meaning that the velocity is decreasing.) No acceleration means no change in velocity

So the two premises are:

     Premise 1: If there is no force then there is no acceleration
     Premise 2: If there is no acceleration then there is no change in velocity

From these a conclusion may be deduced:

     Conclusion: If there is no force then there is no change in velocity

The conclusion is that a change in velocity can only occur if a force is present. But this is just what Newton's first law states. In other words, it appears that the first law can be derived from the second, and if it is nothing more than a consequence of the second, then the first law is redundant.

The argumentation presented above uses the logic of categorical statements but some might be more familiar with the mathematical exposition. So let us come to the same conclusion using the familiar mathematical formula for Newton's second law which was published, some thirty years after the death of Newton, by the Swiss-born mathematician, Leonhard Euler (1707-1783):

        F = m . d2x/dt2

(where F represents the applied force, m is the inertial mass and d2x/dt2 is the acceleration).

If we make the force, F, equal to zero, then

      d2x/dt2    =    dv/dt    =    0      (where v is the velocity)

which on integration gives:    v = constant.

In otherwords if there is no force then there is no change in velocity. And this again is just what the first law states.

Thus both a logical argument and a mathematical formula seem to arrive at the same conclusion: Newton's first law can be derived from his second. If this is true then the first law is redundant.


Part 2:    Where the fallacy lies

The arguments above are like a good conjuring trick; they are impressive but built around a deception. The truth is that Newton's first law limits the scope of the second law: it is an independent rule. The key to unmasking the deception lies in understanding what Newton meant by "force".

Newton published his laws of motion on July 5, 1687 in a work which is commonly referred to as "The Principia"3. It was written in Latin but even in translation the book is rarely read today mainly because the arguments are presented in terms of geometry rather than the now more familiar calculus4. It is because it is not read that misunderstandings arise. At the beginning of the "Principia" Newton carefully defined the concepts he would use in his laws. Eight definitions and a lengthy discussion (which is entitled "Scholium") precede his definition of the three laws and these in turn are followed by six corollaries. All are essential reading for an understanding of Newton's concept of force.

For Newton, force is intimately connected with the frame of reference (or co-ordinate system) in which acceleration is measured. This leads to an important asymmetry: a force will cause an acceleration but an acceleration might not necessarily be caused by a force. An object can appear to accelerate when, in reality, it is the reference frame which is accelerating5. For example, if I am seated in a train compartment then this is my reference frame. When the train leaves the station, then from my reference system, it is the train station that is accelerating away although, of course, no force is acting on the station6.

If the reference frame is accelerating then a body otherwise at rest will appear to be accelerating away. Only in a framework which is stationary or moving with a constant velocity will a body remain at rest or move with a constant velocity unless acted upon by a Newtonian force. In all other frameworks a body will accelerate even when no force is present.

But the above phrase "a body [will] remain at rest or move with a constant velocity unless acted upon by a Newtonian force" is exactly Newton's first law. Therefore the first law defines the frames of reference in which Newton's concept of force is valid. They are frames of reference in which a body remains at rest or moves with a constant velocity unless acted upon by a Newtonian force. Such reference frames are called inertial frames of reference7. All of Newton's three laws involve his concept of force so all three laws are only properly defined within inertial frames of reference8.

The deception in the "redundancy hypothesis" can now be seen. When we said that the acceleration is zero because the force is zero we were saying that accelerations can only occur because of Newtonian forces. In other word we are limiting our choice of reference frame to only inertial systems. But such inertial frames of reference have properties that are defined by Newton's first law. Thus we did not derive Newton's first law from his second, we built it into our argument when we eliminated any possible acceleration other than those driven by a force.

Let us first see how this deception was smuggled into our mathematical argument based around Euler's equation. By assuming that there can be no acceleration when there is no force, we have defined the "x" in "d2x/dt2" as being measured not in any possible co-ordinate system but only in an inertial frame. In otherwords we were assuming Newton's first law.

Similarly the sleight of hand occurs in the first premise of the logical argument. "If there is no force then there is no acceleration" is logically equivalent to saying "If there is an acceleration then there is a force". But this is only true in an inertial frame. Thus the first premise defines the reference frame as being inertial, which, in turn, is defined by Newton's first law. The argument's conclusion was, therefore, just a rephrasing of this premise.

So the conjuring trick is explained: the magician was able to pull the rabbit out of the hat because he had just, a split-second beforehand, put it there himself !

Part 3:   Why the mistake arises

Newton took great care in formulating his laws. We are told in Richard S. Westfall's comprehensive biography of Isaac Newton9 that the first law, in particular, caused him much effort. Over many months Newton modified his ideas about force, inertia and absolute space causing the statement of the first law to undergo many revisions. In his choice of wording Newton reflected the philosophical climate of the second half of the seventeenth century; a world whose ideas about motion were influenced by the ideas of Galileo, Huygens and Descartes. It was towards his contemporaries that Newton directed his writings and it was towards their search for quantitative laws of motion that the rigour of his definition of force, which the first law embodies, was focused. The ideas expressed in the "Principia" were not initially accepted by everyone. In particular the German mathematician and philosopher, Gottfried Wilhelm Leibniz (1646 - 1716), who was probably Newton's only intellectual peer, was a bitter critic of the Newtonian system.

Leibniz fully understood the implication of Newton's first law. It assumed that space existed in its own right even if matter was not present. (It is in this 'absolute space' that the inertial frameworks exist.) But this was a concept which the philosopher utterly rejected. Leibniz's preference was for space and the motion of bodies to be defined relative to other objects. For him there was no 'absolute space' which was common to his 'possible worlds'. The space in any world was defined through the arangement of matter in it. All such 'possible worlds' would be different and so no universal, absolute space could permeate them all10. Leibniz's knew that his metaphysics could accomodate Newton's second law but not the first.

In the history of science there has rarely, if at all, been such a bitter dispute than that between Newton and Leibniz. There was little love lost between the two. But as a supreme logician Leibniz realised that Newton's first law was not derived logically from his second but was a separate metaphysical claim. So Leibniz attacked Newton's first law from this direction11. Leibniz's objection was based on his 'Principle of Sufficient Reason'12. His argument, expressed in modern terms, is that Newton's laws are valid in any member of the set of inertial frameworks one of which is at absolute rest in the universe (and the others are moving with a constant velocity relative to that stationary framework). But why Leibniz is asking has an omnipotent being chosen that particular framework to be at rest rather than any one of the others? There must be a reason so what was it?13

Why is it that the independance of the first law now seems to trouble some modern readers? The reason, as has already been inferred, is that it is quoted out of context. Simply stating the first law as 'a body will remain at rest or move with a constant velocity unless acted upon by a force' assumes a knowledge of the foundations which Newton had errected at the beginning of the 'Principia' to support this statement. In particular it assumes to know what Newton meant by 'force'.

The 'Principia' is rarely read today but without it Newton's statement of the first law can lead, as we have seen, to a misconception: to the error of the 'redundancy theory'. Furthermore without this back-ground knowledge the first law seems to have a different purpose: that of opposing the theories of Aristotle. In the Aristotelian world the absence of a force on a moving body would eventually cause it to come to rest in its 'natural' place. For Aristotle, unlike Newton or Descartes, motion could only be maintained through the constant presence of a 'moving' force. If the 'Principia' is not read then it seems that Newton by insisting that natural motion will continue without the presence of a force, is just opposing the Aristotelian tradition. But while this rejection of Aristotle might be read into Descartes' formulation of his first law, Newton was saying more14. Newton was claiming that space was absolute. It was a thing in its own right indepedent of whether matter was present of not. Furthermore it occupied the whole universe. Future physicists such as Ernst Mach and Albert Einstein were to refine these ideas but they, like Leibniz, had no problem in seeing the first law as saying something more than the second15.

If we are not inclined to read the 'Principia' how might the first law be better formulated? Using modern terminology such a re-phrasing might be: 'When measured within an inertial frame of reference a body will remain at rest or move with a constant velocity unless acted upon by a force'. This formulation compels us to consider what we mean by 'force'.

It compels us to understand that not every acceleration results from a universal force

and this is the essential idea of the first law.

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 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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