The following brief article is the Introductory Chapter to a work in progress, entitled:

CLASSICAL MECHANICS FROM FIRST PRINCIPLES (and without Calculus)
which examines 17^th century mechanics as a would-be coherent system of ideas about the physical world. 

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“A basic description of the Scientific Revolution is to say that it represented a successful revolt of the mathematicians against the authority of the philosophers, and of both against the authority of the theologians” — so writes David Wootton in his extremely well documented book, The Invention of Science. This is certainly as good a one sentence description as any of the most important movement in history at least since the Agricultural Revolution.
        David Wootton’s one-liner thus requires a little qualification. The 16^th and 17^th century Western thinkers who inaugurated the ‘Scientific Revolution’ viewed themselves as ‘philosophers’ but  ‘natural philosophers’ — in contrast to the philosophers who concerned themselves with ‘supra-natural’ things. The term ‘science’ or ‘scienzia’ already existed, of course, but it meant  ‘knowledge’ in the widest sense; what we today consider to be ‘physics’ was known as ‘natural science’ and centuries elapsed  before the phrase lost its ‘natural’ part and just became ‘science’. The term ‘scientist’ itself is surprisingly recent. According to Ross,  who wrote a book on the subject, Whewell coined the term as a ‘gender-neutral word’ to replace ‘man of science’, by analogy with ‘artist’, ‘economist’ &c., when reviewing a science book by Mary Somerville in… 1834 !
        But the ‘philosophers’ against whom the Renaissance thinkers revolted were the medieval Scholastics who, in almost all cases, were the same persons as the theologians: there was very little ‘philosophy’ in the Middle Ages that was uncontaminated by theology. And, far from being a hindrance, the re-discovery of the ancient Greek philosophers was a powerful stimulus to 16^th and 17^th century ‘progressive’ thought, mainly because it showed that there had once existed a form of philosophy that concentrated on the things of this world, not the next, and sought explanations of this world phenomena in exclusively this-world terms. The work of the strictly materialist Greek philosophers such as Democritus had unfortunately been lost but at least enquiring minds of the 16^th century learned that such people had existed and they were able to access scraps via Aristotle and, in the case of Epicurus, via his eloquent Roman follower, Lucretius. Galileo viewed himself as a modern emulator of Plato which is why he employed the elegant, if somewhat mannered, ‘Dialogue’ literary form for his defence of the heliocentric theory in his seminal Dialogue Concerning the Two World Systems (1632). Hobbes went so far as to say that Galileo was the greatest philosopher who had ever lived.

        So, according to Wootton, the scientific revolution was all about the mathematicus displacing the philosophus and both of them combining to downstage the theologus. But what about the ‘third man’ of the ‘Scientific Revolution’, the mechanicus? Surely he deserves a mention. In this respect, the most ‘advanced’ thinker of the 16^th and 17^th centuries was not Galileo but the surprisingly early, scientifically speaking, Leonardo da Vinci (1452-1519). For Leonardo, a practising military and civil engineer not a university man, went so far as to sign himself ‘Leonardo Vinci disscepolo della sperientia’ (Leonardo Vinci, disciple of experience) and openly preached the superiority of the ‘method of experience’ over theory, claiming that if “the test of experience is absent from such exercises of the mind…nothing is certain”. Unlike Galileo, Descartes or anyone else before the late 18^th century, Leonardo seems to have been a thorough-going mechanist who had no truck whatsoever with souls or disembodied minds or even vital forces as evinced by some of the more caustic comments in his Notebooks such as “Men are merely passages for food”. But, regrettably, Leonardo kept his more important scientific insights to himself. For, to judge by the diagrams and records of experiments in his Notebooks, Leonardo (re)-discovered the Parallelogram of Forces and even a clear case of the Law of Falling Bodies. But, unlike Galileo, he did not realize the immense potential of these discoveries, especially the latter; in this respect, he was perhaps not enough of a ‘philosophus’. 
        But there can be no doubt that Wootton has captured the long term trend of the West’s cultural evolution. For today, at the beginning of the 21^st century, we have reached a stage when the mathematicus has largely displaced both the philosophus and the mechanicus, at least in the two most prestigious sciences, atomic physics and cosmology. In an era when science has never had more obvious effects on people’s lives, especially in medicine, most professional physicists have abandoned Western science’s traditional concern with objective reality. The leading theoretical physicist of my generation, Stephen Hawking, famously said in a Halley Lecture, “If you take a positivist position, as I do, questions about reality do not have any meaning. All one can ask is whether [for example] imaginary time is useful in formulating certain mathematical models that describe what we observe.” Tegmark goes even further, making the startling claim that “all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time.” Such thinkers, and there are plenty of them in academia, completely reject the age-old notion that the job of the mathematicus is to provide a ‘model’ of the (independently existing) physical world: on the contrary, for them, mathematics is the reality.
        It needs to be said loud and clear that there is a difference between the process of validation and development of a mathematical and a scientific conjecture. The former, according to the current ‘formalist/Platonic’ view of mathematics, requires nothing more than logical consistency and, to get the ball rolling,  a handful of ‘postulates’ that have been deliberately drained of any connection to sensory experience. A scientist, on the other hand, was, at least until very recently, expected to come up with statements about the physical world which could be tested, or which, alternatively, provided a more satisfactory explanation of well-known and uncontested facts. There remains a certain overlap of pure mathematics and experimental science but this overlap and/or ‘mutual aid’ has become more and more exiguous as the years progress. In the era of Tegmark Platonism and the Multiverse, one might be forgiven for thinking that the mathematicus has become just a little too dominant.
        The decisive Western scientific breakthrough kicked off in 16^th century Italy but, by the second half of the17^th century, the action had shifted to the emerging Protestant sea-powers, Holland and England. Why did this breakthrough come about in the first place? Chiefly, I would argue, because, for a brief golden period, mathematics, experiment/invention AND philosophy converged and cross-fertilized each other. Not only were Galileo, Torricelli, Descartes, Leibnitz, Huyghens, Newton, Hooke and a host of lesser names all, to say the least, capable pure mathematicians, but they were also experimenters who weren’t afraid to get their hands dirty — even Newton, at the time an eccentric Cambridge recluse, originally came to the notice of the Royal Academy for a reflecting telescope he designed and made himself. But third and last, all these names were ‘big thinkers’ committed to the discovery of the ‘truth’ about the world they lived in: they did not view themselves as intellectual gamesters.
        In reaction to the fogs of 19^th century Idealism, especially German, 20^th century thought went to the opposite extreme and increasingly lost interest in ‘objective reality’ and even, with the advent of ‘deconstructivism’, in truth itself. There were ‘good’, or at least understandable, reasons for this volte face, also very bad ones.  The ‘good’ reasons were that subatomic physics, especially Quantum Mechanics, defied attempts to model it realistically. The bad reasons were an outright preference for the abstract over the concrete combined with a sort of smirking superiority with respect to the hoi polloi who still bothered about such outmoded things as ‘meaning’ and ‘deeper reality’. The typical advice to the physics student was, “Don’t try to make sense of it all, just follow the mathematics”. Or there was the ultimate ‘cop out’ that thankfully one does not hear quite so much these days, namely that “It is not the business of science to deal with the why, only the how.” At the same time, since no one but a scientist/mathematician was/is considered to be competent to deal with the why, the result was/is that discussing the why was/is prohibited to practically everyone. This is not even reductionism, it is intellectual nihilism. As one shrewd contemporary observer notes, “The leading intellectuals of our time, instead of being inspired by the search for the great questions ….. have instead become entranced by how to avoid such questions.” And complicated mathematics has proved to be a mastercard in this sterile game, since it is a subject that many lay people find baffling while the initiates immediately close ranks if they are ever called out by whistle blowers to justify their strange ‘beliefs’, or whether they actually have any. 
        Contrary to the present cultural consensus, I consider that ideas matter and that ‘great ideas’ — though they are produced by individual minds, not Hegelian ‘world-forces’ — drive history at least as much as novel technologies or economic forces. Admittedly, most philosophy is a waste of time but that is because there is a good deal more bad philosophy than good: the subject lends itself to intellectual flatulence and casuistry — but so does pure mathematics lend itself to triviality and complication for complication’s sake when it is removed from the invigorating contact with sensory experience. Mathematics was made for man, not man for mathematics. As Stanislaw Ulam put it, “In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs.” Such an attitude is understandable in a society debased by consumerism and trivial entertainment but it is not exactly an attitude to be encouraged in the would-be scientist or serious enquirer after the truth.
        In the past, the ‘great idea’ typically came first, the mathematical embodiment much later. The ‘new science’ got going because Galileo focused attention on ‘change of motion’ or ‘acceleration’ rather than motion per se as in Aristotle. The shift at first sight does not sound particularly revolutionary but it led directly to the key notion of the ‘inertial system’ which proved to be a gift that kept on giving since it took us directly to Newton’s first two Laws of Motion and, two hundred years later, to Special Relativity. But this ‘big idea’ was only given mathematical flesh and bone much later, partly by Newton and Leibnitz but even more decisively by Euler in the 18^th century. Similarly, the great physical idea of the early 19^th century, without which one suspects neither electro-magnetic theory nor General Relativity could have developed, was that of the ‘field’, but its inspired inventor, Faraday, though a first-rate experimenter, was so weak in mathematics we would hardly regard him as numerate today. As we know, the world had to wait for Maxwell and Riemann to provide the mathematical scaffolding. And, if we want to go further back, atomism, the greatest idea of the Greeks, began as a speculative (but well reasoned and plausible) theory that had to wait two and a half thousand years before it gave rise to do-able physics and chemistry. But it was there all the time like a dormant spore waiting for the moment when conditions were finally ripe.
        So, where is all this leading? Apart from its usefulness, so-called ‘classical’ —  by which I mean 15^th – 20^th century Western — mechanics is an impressive system of (natural) philosophic ideas, a lens with which to view reality, a ‘paradigm’ in Kuhn’s sense. As such it has an integrity that more recent ensembles of scientific ideas lack. Moreover, since I can see the principles of ‘leverage’ and ‘inertia’ being put into practice all around me all the time, I actually ‘believe’ it, or a good deal of it, while I am not so sure that I do believe Quantum Mechanics or General Relativity. “Believe is a strong word”, as one of the main characters says in the brilliant Canadian TV series, The Crossing. While stopping well short of ‘blind faith’, ‘belief’ in something means rather more than ‘reluctant acceptance because I can’t actually put my finger on what’s wrong with it’. Quantum Mechanics is troublesome because it doesn’t make sense, and, instinctively, I distrust and dislike things that don’t make sense. However, there can be no doubt that Quantum Mechanics ‘works’, and I suppose that I, along with the rest of the population, depend on it to some extent since QM is built into a lot of clever devices, medical or otherwise, on which modern society depends, transistors, lasers, electron tunnelling microscopes &c. &c.
        As for General Relativity, it offends because, in the current fashionable Block Universe format, it completely excludes free will and indeed the passage of time itself: anything that can happen has, in a certain sense, ‘already happened’. Well, maybe. (Also, the mathematics is repellent.) But, in the case of GR, I can dismiss it as irrelevant to my life since the only application of GR in modern society I know of is the GPS system and I don’t drive, never have, and, apart from that, I don’t much like the potential for secret surveillance from Big Brother up in the skies. Humanity could perfectly well thrive without GR, maybe not so well without QM.
        What you get today in elementary physics or mechanics textbooks is the result of a long, complex evolutionary thought process initiated by the 16^th and 17^th philosophes and mathematically given canonical form by brilliant 18^th century applied mathematicians such as the Bernoullis and Euler. But mechanics as a subject, and certainly as a means of viewing the world, lost its way when people like Lagrange deliberately turned it into an arid  piece of pure mathematics. The result was that practising mechanics, the people who inaugurated the Age of Steam, Newcomen, Savery, Trevithick, Watt and the like, didn’t ‘know mechanics’ while Lagrange actually boasted of having written a long treatise La Mécanique Analytique without a single diagram (or in some editions, just one). During the 19^th century Calculus was ‘tidied up’ logically by German and French professional mathematicians but in such a way that it ceased to have much relevance to mundane reality and was for that reason widely ignored by British engineers who nonetheless managed to outperform their far more learned French rivals. Also, the impact of the steam engine, to which initially classical mechanics contributed very little, brought to the forefront the question of ‘energy’ which has, since then, blotted out more or less all other considerations. In my view, analyses on the basis of ‘energy’ should not be used when doing ‘classical’ mechanics: both the term and the concept are quite foreign to Newton though we do find faint anticipations of the concept in Leibnitz. Newton and the 17^th century dealt in ‘force’, not ‘energy’ and, in point of fact, the great 19^th century pioneer of the ‘conservation of energy’ principle, Mayer, still spoke of ‘force’, not energy as such. ‘Energy’ is, when all is said and done, something of a wraith for it is potential only, while force is something whose dramatic effects, e.g. in collisions, we are all familiar with. Scientifically speaking, Energy is Work that could be done but in actual fact is not being done, it is two places removed from actuality, whereas ‘force’ is only one place removed. (‘What, then, is ‘actuality’?’ said Pontius Pilate and did not wait for an answer. But to this I do have an answer, “ ‘Actuality’ is ‘what happens and what makes what happens happen ”, i.e. events and forces.)
        17^th century classical mechanics was not, of course, a completely water tight system of ideas, but inasmuch as it fails to entirely convince a modern reader (such as myself), this is not for the usual reasons given in textbooks, namely that Newton didn’t know Vector Calculus and Leibnitz hadn’t heard of Hilbert Space. I have found it instructive to evaluate the system according to its own standards and see what can be done with it using only processes and ideas intelligible to the 17^th century participants and all the while keeping to the basically realistic approach that was typical of the times. Treatments today go straight to the (19^th century) mathematics, but I want to consider classical mechanics in the first instance as a coherent system of ideas about physical reality which will eventually be fleshed out by some ‘elementary’ (i.e. non calculus) mathematics.                                                Sebastian Hayes   16/04/23