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Philosophiæ Naturalis Principia Mathematica
Title page of 'Principia', first edition (1687)
Original title	Philosophiæ Naturalis Principia Mathematica

Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687.^[1][2] After annotating and correcting his personal copy of the first edition,^[3] Newton also published two further editions, in 1713 and 1726.^[4] The Principia states Newton's laws of motion, forming the foundation of classical mechanics, also Newton's law of universal gravitation, and a derivation of Kepler's laws of planetary motion (which Kepler first obtained empirically). The Principia is "justly regarded as the most important work in the history of science".^[5]

The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of mathematical Principles of natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses."^[6] A more recent assessment has been that while acceptance of Newton's theories was not immediate, by the end of a century after publication in 1687, "no one could deny that" (out of the 'Principia') "a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally."^[7]

In formulating his physical theories, Newton developed and used mathematical methods now included in the field of calculus. But the language of calculus as we know it was largely absent from the Principia; Newton gave many of his proofs in a geometric form of infinitesimal calculus, based on limits of ratios of vanishing small geometric quantities.^[8] In a revised conclusion to the Principia (see General Scholium), Newton used his expression that became famous, Hypotheses non fingo ("I contrive no hypotheses"^[9]).

Contents[link]

Expressed aim and topics covered[link]

In the preface of the Principia, Newton wrote^[10]

[...] Rational Mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated [...] And therefore we offer this work as mathematical principles of philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena [...]

The 'Principia' deals primarily with massive bodies in motion, initially under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed. It attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles. It explores difficult problems of motions perturbed by multiple attractive forces. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites. It shows how astronomical observations prove the inverse square law of gravitation (to an accuracy that was high by the standards of Newton's time); offers estimates of relative masses for the known giant planets and for the Earth and the Sun; defines the very slow motion of the Sun relative to the solar-system barycenter; shows how the theory of gravity can account for irregularities in the motion of the Moon; identifies the oblateness of the figure of the Earth; accounts approximately for marine tides including phenomena of spring and neap tides by the perturbing (and varying) gravitational attractions of the Sun and Moon on the Earth's waters; explains the precession of the equinoxes as an effect of the gravitational attraction of the Moon on the Earth's equatorial bulge; and gives theoretical basis for numerous phenomena about comets and their elongated, near-parabolic orbits.

It was perhaps the force of the 'Principia', which revealed so many different things about the natural world with such economy, that caused this method to become synonymous with physics, even as it is practiced almost three and a half centuries after its beginning. Today the two methodological aspects that Newton outlined could be called analysis and synthesis.

The opening sections of the 'Principia' contain, in revised and extended form, nearly all of the content of Newton's 1684 tract 'De motu...' (see article De motu corporum in gyrum which summarises the topics and indicates where they reappear in the 'Principia').

The 'Principia'^[11] begins with 'Definitions' and 'Axioms or Laws of Motion'^[12] and continues in three books:

Book 1, De motu corporum[link]

Book 1, subtitled De motu corporum (On the motion of bodies) concerns motion in the absence of any resisting medium. It opens (Section I) with a mathematical exposition of "the method of first and last ratios," a geometrical form of infinitesimal calculus.^[8]

Section II (Propositions 1-10) establishes relationships between centripetal forces and the law of areas now known as Kepler's second law (Proposition 1-3), and relates circular velocity and radius of path-curvature to radial force^[13] (Proposition 4), and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form.

Sections III to VI (Propositions 11-31) establish properties of motion in paths of eccentric conic-section form including ellipses, and their relation with inverse-square central forces directed to a focus, and include Newton's theorem about ovals (lemma 28).

Section IX includes Newton's demonstration (Propositions 43-45) that in an eccentric orbit under centripetal force where the apse may move, a steady nonmoving orientation of the line of apses is an indicator of an inverse-square law of force.

Book 1 contains some proofs with little connection to real-world dynamics. But there are also sections with far-reaching application to the solar system and universe:-

Section XI (Propositions 57-69) deals with the "motion of bodies drawn to one another by centripetal forces." This section is of primary interest for its application to the solar system, and includes Proposition 66 along with its 22 Corollaries: here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which later gained name and fame (among other reasons, for its great difficulty) as the three-body problem.

Section XII (Propositions 70-84) deals with the attractive forces of spherical bodies. This section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation. [3]The conclusion made by Isaac Newton, stating that the change of movement of some bodies is caused by the impact of other bodies - is wrong and does not correspond to the facts.^{[citation needed]}

Book 2[link]

Part of the contents originally planned for the first book was divided out into a second book, which largely concerns motion through resisting mediums. Just as Newton examined consequences of different conceivable laws of attraction in Book 1, here he examines different conceivable laws of resistance; thus Section 1 discusses resistance in direct proportion to velocity, and Section 2 goes on to examine the implications of resistance in proportion to the square of velocity. Book 2 also discusses (in Section 5) hydrostatics and the properties of compressible fluids. The effects of air resistance on pendulums are studied in Section 6, along with Newton's account of experiments that he carried out, to try to find out some characteristics of air resistance in reality by observing the motions of pendulums under different conditions. Newton compares the resistance offered by a medium against motions of bodies of different shape, attempts to derive the speed of sound, and gives accounts of experimental tests of the result.

Less of Book 2 has stood the test of time than of Books 1 and 3, and it has been said that Book 2 was largely written on purpose to refute a theory of Descartes which had some wide acceptance before Newton's work (and for some time after). According to this Cartesian theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them.^[14] Newton wrote at the end of Book 2 (in the Scholium to proposition 53) his conclusion that the hypothesis of vortices was completely at odds with the astronomical phenomena, and served not so much to explain as to confuse them.

Book 3, De mundi systemate[link]

Book 3, subtitled De mundi systemate (On the system of the world) is an exposition of many consequences of universal gravitation, especially its consequences for astronomy. It builds upon the propositions of the previous books, and applies them with further specificity than in Book 1 to the motions observed in the solar system. Here (introduced by Proposition 22, and continuing in Propositions 25-35) are developed several of the features and irregularities of the orbital motion of the Moon (see Lunar theory -- Newton), especially the variation. Newton lists the astronomical observations on which he relies (in 'The Phaenomena'), and establishes in a stepwise manner that the inverse square law of mutual gravitation applies to solar system bodies, starting with the satellites of Jupiter and going on by stages to show that the law is of universal application. He also gives starting at Lemma 4 and Proposition 40) the theory of the motions of comets (for which much data came from John Flamsteed and from Edmond Halley), and accounts for the tides, attempting quantitative estimates of the contributions of the Sun and Moon to the tidal motions; and offers the first theory of the precession of the equinoxes. Book 3 also considers the harmonic oscillator in three dimensions, and motion in arbitrary force laws.

In Book 3 Newton also made clear his heliocentric view of the solar system, modified in a somewhat modern way, since already in the mid-1680s he recognized the "deviation of the Sun" from the centre of gravity of the solar system.^[15] For Newton, "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World" (Proposition 12, corollary), and that this centre "either is at rest, or moves uniformly forward in a right line" (Proposition 11 & preceding Hypothesis). (Newton rejected the second alternative after adopting the position that "the centre of the system of the world is immoveable", which "is acknowledg'd by all, while some contend that the Earth, others, that the Sun is fix'd in that centre", Proposition 11.) Newton estimated the mass ratios Sun:Jupiter and Sun:Saturn (Proposition 8, Corollary 2), and pointed out that these put the centre of the Sun usually a little way off the common center of gravity, but only a little, the distance at most "would scarcely amount to one diameter of the Sun" (Proposition 12).^[16]

The sequence of definitions used in setting up dynamics in the Principia is recognisable in many textbooks today. Newton first set out the definition of mass⁶

The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity. This quantity I designate by the name of body or of mass.

This was then used to define the "quantity of motion" (today called momentum), and the principle of inertia in which mass replaces the previous Cartesian notion of intrinsic force. This then set the stage for the introduction of forces through the change in momentum of a body. Curiously, for today's readers, the exposition looks dimensionally incorrect, since Newton does not introduce the dimension of time in rates of changes of quantities.

He defined space and time "not as they are well known to all". Instead, he defined "true" time and space as "absolute" and explained:

Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. [...] instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them.

To some modern readers it can appear that some dynamical quantities recognized today were used in the 'Principia' but not named. The mathematical aspects of the first two books were so clearly consistent that they were easily accepted; for example, Locke asked Huygens whether he could trust the mathematical proofs, and was assured about their correctness.

However, the concept of an attractive force acting at a distance received a cooler response. In his notes, Newton wrote that the inverse square law arose naturally due to the structure of matter. However, he retracted this sentence in the published version, where he stated that the motion of planets is consistent with an inverse square law, but refused to speculate on the origin of the law. Huygens and Leibniz noted that the law was incompatible with the notion of the aether. From a Cartesian point of view, therefore, this was a faulty theory. Newton's defence has been adopted since by many famous physicists—he pointed out that the mathematical form of the theory had to be correct since it explained the data, and he refused to speculate further on the basic nature of gravity. The sheer number of phenomena that could be organised by the theory was so impressive that younger "philosophers" soon adopted the methods and language of the Principia.

Rules of Reasoning in Philosophy[link]

Perhaps to reduce the risk of public misunderstanding, Newton included at the beginning of Book 3 (in the second (1713) and third (1726) editions) a section entitled "Rules of Reasoning in Philosophy." In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them. The four Rules of the 1726 edition run as follows (omitting some explanatory comments that follow each):

Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes.

Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

This section of Rules for philosophy is followed by a listing of 'Phenomena', in which are listed a number of mainly astronomical observations, that Newton used as the basis for inferences later on, as if adopting a consensus set of facts from the astronomers of his time.

Both the 'Rules' and the 'Phenomena' evolved from one edition of the 'Principia' to the next. Rule 4 made its appearance in the third (1726) edition; Rules 1-3 were present as 'Rules' in the second (1713) edition, and predecessors of them were also present in the first edition of 1687, but there they had a different heading: they were not given as 'Rules', but rather in the first (1687) edition the predecessors of the three later 'Rules', and of most of the later 'Phenomena', were all lumped together under a single heading 'Hypotheses' (in which the third item was the predecessor of a heavy revision that gave the later Rule 3).

From this textual evolution, it appears that Newton wanted by the later headings 'Rules' and 'Phenomena' to clarify for his readers his view of the roles to be played by these various statements.

In the third (1726) edition of the Principia, Newton explains each rule in an alternative way and/or gives an example to back up what the rule is claiming. The first rule is explained as a philosophers' principle of economy. The second rule states that if one cause is assigned to a natural effect, then the same cause so far as possible must be assigned to natural effects of the same kind: for example respiration in humans and in animals, fires in the home and in the Sun, or the reflection of light whether it occurs terrestrially or from the planets. An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalization of observational results, with a caution against making up fancies contrary to experiments, and use of the rules to illustrate the observation of gravity and space.

Isaac Newton’s statement of the four rules revolutionized the investigation of phenomena. With these rules, Newton could in principle begin to address all of the world’s present unsolved mysteries. He was able to use his new analytical method to replace that of Aristotle, and he was able to use his method to tweak and update Galileo’s experimental method. The re-creation of Galileo’s method has never been significantly changed and in its substance, scientists use it today.

General Scholium[link]

The General Scholium is a concluding essay added to the second edition, 1713 (and amended in the third edition, 1726).^[17]

Here Newton used what became his famous expression 'Hypotheses non fingo', "I frame no hypotheses",^[9] in response to criticisms of the first edition of the 'Principia'. ('Fingo' is sometimes nowadays translated 'feign' rather than the traditional 'frame'.) Newton's gravitational attraction, an invisible force able to act over vast distances, had led to criticism that he had introduced "occult agencies" into science.^[18] Newton firmly rejected such criticisms and wrote that it was enough that the phenomena implied gravitational attraction, as they did; but the phenomena did not so far indicate the cause of this gravity, and it was both unnecessary and improper to frame hypotheses of things not implied by the phenomena: such hypotheses "have no place in experimental philosophy", in contrast to the proper way in which "particular propositions are inferr'd from the phenomena and afterwards rendered general by induction".^[19]

Newton also underlined his criticism of the vortex theory of planetary motions, of Descartes, pointing to its incompatibility with the highly eccentric orbits of comets, which carry them "through all parts of the heavens indifferently".

Newton also gave theological argument. From the system of the world, he inferred the existence of a Lord God, along lines similar to what is sometimes called the argument from intelligent or purposive design. It has been suggested that Newton gave "an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity",^[20][21] but the General Scholium appears to say nothing specifically about these matters.

Writing and publication[link]

Halley and Newton's initial stimulus[link]

In January 1684, Halley, Wren and Hooke had a conversation in which Hooke claimed to not only have derived the inverse-square law, but also all the laws of planetary motion. Wren was unconvinced, Hooke did not produce the claimed derivation although the others gave him time to do it, and Halley, who could derive the inverse-square law for the restricted circular case (by substituting Kepler's relation into Huygens' formula for the centrifugal force) but failed to derive the relation generally, resolved to ask Newton.^[22]

Halley's visits to Newton in 1684 thus resulted from Halley's debates about planetary motion with Wren and Hooke, and they seem to have provided Newton with the incentive and spur to develop and write what became Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). Halley was at that time a Fellow and Council member of the Royal Society in London, (positions that in 1686 he resigned in order to become the Society's paid Clerk).^[23] Halley's visit to Newton in Cambridge in 1684 probably occurred in August.^[24] When Halley asked Newton's opinion on the problem of planetary motions discussed earlier that year between Halley, Hooke and Wren,^[25] Newton surprised Halley by saying that he had already made the derivations some time ago; but that he could not find the papers. (Matching accounts of this meeting come from Halley and Abraham De Moivre to whom Newton confided.) Halley then had to wait for Newton to 'find' the results, but in November 1684 Newton sent Halley an amplified version of whatever previous work Newton had done on the subject. This took the form of a 9-page manuscript, "De motu corporum in gyrum" ("Of the motion of bodies in an orbit"): the title is shown on some surviving copies, although the (lost) original may have been without title.

Newton's tract 'De motu...', which he sent to Halley in late 1684, derived what are now known as the three laws of Kepler, assuming an inverse square law of force, and generalized the result to conic sections. It also extended the methodology by adding the solution of a problem on the motion of a body through a resisting medium. The contents of 'De motu...' so excited Halley by their mathematical and physical originality and far-reaching implications for astronomical theory, that he immediately went to visit Newton again, in November 1684, to ask Newton to let the Royal Society have more of such work.^[26] The results of their meetings clearly helped to stimulate Newton with the enthusiasm needed to take his investigations of mathematical problems much further in this area of physical science, and he did so in a period of highly concentrated work that lasted at least until mid-1686.^[27]

Newton's single-minded attention to his work generally, and to his project during this time, is shown by later reminiscences from his secretary and copyist of the period, Humphrey Newton. His account tells of Isaac Newton's absorption in his studies, how he sometimes forgot his food, or his sleep, or the state of his clothes, and how when he took a walk in his garden he would sometimes rush back to his room with some new thought, not even waiting to sit before beginning to write it down.^[28] Other evidence also shows Newton's absorption in the Principia: Newton for years kept up a regular programme of chemical or alchemical experiments, and he normally kept dated notes of them, but for a period from May 1684 to April 1686, Newton's chemical notebooks have no entries at all.^[29] So it seems that Newton abandoned pursuits to which he was normally dedicated, and did very little else for well over a year and a half, but concentrated on developing and writing what became his great work.

Newton's own first edition copy of his Principia, with handwritten corrections for the second edition.

Titlepage and frontispiece of the third edition, London, 1726 (John Rylands Library)

The first of the three constituent books was sent to Halley for the printer in spring 1686, and the other two books somewhat later. The complete work, published by Halley at his own financial risk,^[30] appeared in July 1687. (Newton had also communicated 'De motu...' to Flamsteed, and during the period of composition he exchanged a few letters with Flamsteed about observational data on the planets, eventually acknowledging Flamsteed's contributions in the published version of the 'Principia' of 1687.)

Preliminary version[link]

The process of writing that first edition of the Principia went through several stages and drafts: some parts of the preliminary materials still survive, others are lost except for fragments and cross-references in other documents.^[31]

Surviving preliminary materials show that Newton (up to some time in 1685) conceived his book as a two-volume work: The first volume was to be 'De motu corporum, Liber primus', with contents that later appeared (in extended form) as Book 1 of the 'Principia'.

A fair-copy draft of Newton's planned second volume 'De motu corporum, Liber secundus' still survives, and its completion has been dated to about the summer of 1685. What it covers is the application of the results of 'Liber primus' to the earth, the moon, the tides, the solar system, and the universe: in this respect it has much the same purpose as the final Book 3 of the 'Principia', but it is written much less formally and is more easily read.

It is not known just why Newton changed his mind so radically about the final form of what had been a readable narrative in 'De motu corporum, Liber secundus' of 1685, but he largely started afresh in a new, tighter, and less accessible mathematical style, eventually to produce Book 3 of the 'Principia' as we know it. Newton frankly admitted that this change of style was deliberate when he wrote, in the introduction to Book 3, that he had (first) composed this book "in a popular method, that it might be read by many", but to "prevent the disputes" by readers who could not "lay aside the[ir] prejudices", he had "reduced" it "into the form of propositions (in the mathematical way) which should be read by those only, who had first made themselves masters of the principles established in the preceding books".^[32] The final Book 3 also contained in addition some further important quantitative results arrived at by Newton in the meantime, especially about the theory of the motions of comets, and some of the perturbations of the motions of the Moon.

The result was numbered Book 3 of the 'Principia' rather than Book 2, because in the meantime, drafts of 'Liber primus' had expanded and Newton had divided it into two books. The new and final Book 2 was concerned largely with the motions of bodies through resisting mediums.

But the 'Liber secundus' of 1685 can still be read today. Even after it was superseded by Book 3 of the Principia, it survived complete, in more than one manuscript. After Newton's death in 1727, the relatively accessible character of its writing encouraged the publication of an English translation in 1728 (by persons still unknown, not authorised by Newton's heirs). It appeared under the English title A Treatise of the System of the World. This had some amendments relative to Newton's manuscript of 1685, mostly to remove cross-references that used obsolete numbering to cite the propositions of an early draft of Book 1 of the Principia. Newton's heirs shortly afterwards published the Latin version in their possession, also in 1728, under the (new) title De Mundi Systemate, amended to update cross-references, citations and diagrams to those of the later editions of the Principia, making it look superficially as if it had been written by Newton after the Principia, rather than before.^[33] The System of the World was sufficiently popular to stimulate two revisions (with similar changes as in the Latin printing), a second edition (1731), and a 'corrected' reprint of the second edition (1740).

Halley's role as publisher[link]

The text of the first of the three books of the Principia was presented to the Royal Society at the close of April, 1686. Hooke made some priority claims (but failed to substantiate them), causing some delay. When Hooke's claim was made known to Newton, who hated disputes, Newton threatened to withdraw and suppress Book 3 altogether, but Halley, showing considerable diplomatic skills, tactfully persuaded Newton to withdraw his threat and let it go forward to publication. Samuel Pepys, as President, gave his imprimatur on 30 June 1686, licensing the book for publication. The Society had just spent its book budget on a History of Fishes,^[34] and the cost of publication was borne by Edmund Halley (who was also then acting as publisher of the Philosophical Transactions of the Royal Society): the book appeared in summer 1687.^[35]

Historical context[link]

Beginnings of the scientific revolution[link]

Nicolaus Copernicus had firmly moved the Earth away from the center of the universe with the heliocentric theory for which he presented evidence in his book De revolutionibus orbium coelestium (On the revolutions of the heavenly spheres) published in 1543. The structure was completed when Johannes Kepler wrote the book Astronomia nova (A new astronomy) in 1609, setting out the evidence that planets move in elliptical orbits with the sun at one focus, and that planets do not move with constant speed along this orbit. Rather, their speed varies so that the line joining the centres of the sun and a planet sweeps out equal areas in equal times. To these two laws he added a third a decade later, in his book Harmonices Mundi (Harmonies of the world). This law sets out a proportionality between the third power of the characteristic distance of a planet from the sun and the square of the length of its year.

The foundation of modern dynamics was set out in Galileo's book Dialogo sopra i due massimi sistemi del mondo (Dialogue on the two main world systems) where the notion of inertia was implicit and used. In addition, Galileo's experiments with inclined planes had yielded precise mathematical relations between elapsed time and acceleration, velocity or distance for uniform and uniformly accelerated motion of bodies.

Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Another mistake was his treatment of circular motion, but this was more fruitful in that it led others to identify circular motion as a problem raised by the principle of inertia. Christiaan Huygens solved this problem in the 1650s and published it much later as a book.

Newton's role[link]

Newton had studied these books, or, in some cases, secondary sources based on them, and taken notes entitled Quaestiones quaedam philosophicae (Questions about philosophy) during his days as an undergraduate. During this period (1664–1666) he created the basis of calculus, and performed the first experiments in the optics of colour. At this time, his proof that white light was a combination of primary colours (found via prismatics) replaced the prevailing theory of colours and received an overwhelmingly favourable response, and occasioned bitter disputes with Robert Hooke and others, which forced him to sharpen his ideas to the point where he already composed sections of his later book Opticks by the 1670s in response. Work on calculus is shown in various papers and letters, including two to Leibniz. He became a fellow of the Royal Society and the second Lucasian Professor of Mathematics (succeeding Isaac Barrow) at Trinity College, Cambridge.

Newton's early work on motion[link]

In the 1660s Newton studied the motion of colliding bodies, and deduced that the centre of mass of two colliding bodies remains in uniform motion. Surviving manuscripts of the 1660s also show Newton's interest in planetary motion and that by 1669 he had shown, for a circular case of planetary motion, that the force he called 'endeavour to recede' (now called centrifugal force) had an inverse-square relation with distance from the center.^[36] After his 1679-1680 correspondence with Hooke, described below, Newton adopted the language of inward or centripetal force. According to Newton scholar J Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The difference between the centrifugal and centripetal points of view, though a significant change of perspective, did not change the analysis.^[37] Newton also clearly expressed the concept of linear inertia in the 1660s: for this Newton was indebted to Descartes' work published 1644.^[38]

Controversy with Hooke[link]

Hooke published his ideas about gravitation in the 1660s and again in 1674 (see Robert Hooke - Gravitation). He argued for an attracting principle of gravitation in Micrographia of 1665, in a 1666 Royal Society lecture "On gravity", and again in 1674, when he published his ideas about the "System of the World" in somewhat developed form, as an addition to "An Attempt to Prove the Motion of the Earth from Observations".^[39] Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, along with a principle of linear inertia. Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.^[40] Hooke also did not provide accompanying evidence or mathematical demonstration. On these two aspects, Hooke stated in 1674: "Now what these several degrees [of gravitational attraction] are I have not yet experimentally verified" (indicating that he did not yet know what law the gravitation might follow); and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e., "prosecuting this Inquiry").^[39]

In November 1679, Hooke began an exchange of letters with Newton (of which the full text is now published.^[41]). Hooke told Newton that Hooke had been appointed to manage the Royal Society's correspondence,^[42] and wished to hear from members about their researches, or their views about the researches of others; and as if to whet Newton's interest, he asked what Newton thought about various matters, giving a whole list, mentioning "compounding the celestial motions of the planets of a direct motion by the tangent and an attractive motion towards the central body", and "my hypothesis of the lawes or causes of springinesse", and then a new hypothesis from Paris about planetary motions (which Hooke described at length), and then efforts to carry out or improve national surveys, the difference of latitude between London and Cambridge, and other items. Newton's reply offered "a fansy of my own" about a terrestrial experiment (not a proposal about celestial motions) which might detect the Earth's motion, by the use of a body first suspended in air and then dropped to let it fall. The main point was to indicate how Newton thought the falling body could experimentally reveal the Earth's motion by its direction of deviation from the vertical, but he went on hypothetically to consider how its motion could continue if the solid Earth had not been in the way (on a spiral path to the centre). Hooke disagreed with Newton's idea of how the body would continue to move.^[43] A short further correspondence developed, and towards the end of it Hooke, writing on 6 January 1679|80 to Newton, communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance."^[44] (Hooke's inference about the velocity was actually incorrect.^[45])

In 1686, when the first book of Newton's 'Principia' was presented to the Royal Society, Hooke claimed that Newton had obtained from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated therby" was wholly Newton's.^[41]

A recent assessment about the early history of the inverse square law is that "by the late 1660s," the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons".^[46] Newton himself had shown in the 1660s that for planetary motion under a circular assumption, force in the radial direction had an inverse-square relation with distance from the center.^[36] Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea, giving reasons including the citation of prior work by others before Hooke.^[41] Newton also firmly claimed that even if it had happened that he had first heard of the inverse square proportion from Hooke, which it had not, he would still have some rights to it in view of his mathematical developments and demonstrations, which enabled observations to be relied on as evidence of its accuracy, while Hooke, without mathematical demonstrations and evidence in favour of the supposition, could only guess (according to Newton) that it was approximately valid "at great distances from the center".^[41]

The background described above shows there was basis for Newton to deny deriving the inverse square law from Hooke. On the other hand, Newton did accept and acknowledge, in all editions of the 'Principia', that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1.^[47] Newton also acknowledged to Halley that his correspondence with Hooke in 1679-80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ...".^[41]) Newton's reawakening interest in astronomy received further stimulus by the appearance of a comet in the winter of 1680/1681, on which he corresponded with John Flamsteed.^[48]

In 1759, decades after the deaths of both Newton and Hooke, Alexis Clairaut, mathematical astronomer eminent in his own right in the field of gravitational studies, made his assessment after reviewing what Hooke had published on gravitation. "One must not think that this idea ... of Hooke diminishes Newton's glory", Clairaut wrote; "The example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".^[49][50]

Location of copies[link]

A page from the Principia

Several national rare-book collections contain original copies of Newton's Principia Mathematica, including:

• The Martin Bodmer Library[4] keeps a copy of the original edition that was owned by Leibniz. In it, we can see handwritten notes by Leibniz, in particular concerning the controversy of who discovered calculus (although he published it later, Newton argued that he developed it earlier).
• The library of Trinity College, Cambridge, has Newton's own copy of the first edition, with handwritten notes for the second edition.[5]
• The Whipple Museum of the History of Science in Cambridge has a first-edition copy which had belonged to Robert Hooke.
• The Pepys Library in Magdalene College, Cambridge, has Samuel Pepys' copy of the third edition.
• Fisher Library in the University of Sydney has a first-edition copy, annotated by a mathematician of uncertain identity and corresponding notes from Newton himself.
• The University College London library holds a copy in 'Strong Room E' of its Rare Books collection.
• The University of Wisconsin - Madison, Memorial Library at Special Collections
• The Harry Ransom Center at The University of Texas in Austin holds two first edition copies, one with manuscript additions and corrections.
• The Earl Gregg Swem Library at the College of William & Mary has a first edition copy of the Principia [6]
• The Frederick E. Brasch Collection of Newton and Newtoniana in Stanford University also has a first edition of the Principia.[7]
• A first edition is also located in the archives of the library at the Georgia Institute of Technology. The Georgia Tech library is also home to a second and third edition.
• A first edition forms part of the Crawford Collection, housed at the Royal Observatory, Edinburgh. The collection also holds a third edition copy.
• The Uppsala University Library owns a first edition copy, which was stolen in the 1960s and returned to the library in 2009. [8]
• The University of Michigan Special Collections Library owns several early printings, including the first (1687), second (1713), second revised (1714), unnumbered (1723), and third (1726) editions of the Principia.
• The Royal Society in London holds John Flamsteed's first edition copy, and also the manuscript of the first edition. The manuscript is complete containing all three books but does not contain the figures and illustrations for the first edition.
• The Burns Library at Boston College contains a 1723 copy published between the second and third editions.
• The George C. Gordon Library at the Worcester Polytechnic Institute holds a third edition copy. [9]
• The Gunnerus Library at the Norwegian University of Science and Technology in Trondheim holds a first edition copy of the Principia.
• Haverford College Quaker & Special Collections owns a first edition of the Principia.
• The Fellows Library at Winchester College owns a first edition of the Principia.
• The Fellows' Library at Jesus College, Oxford, owns a copy of the first edition.
• The Old Library of Magdalen College, Oxford owns a first edition copy.
• The Library of New College, Oxford owns a first edition copy.
• The Library of Somerville College, Oxford owns a second edition copy.
• The Southwest Research Institute in Texas owns a third edition copy dated 1726CE.

A facsimile edition (based on the 3rd edition of 1726 but with variant readings from earlier editions and important annotations) was published in 1972 by Alexandre Koyré and I. Bernard Cohen.^[4]

Later editions[link]

Two later editions were published by Newton:

Second edition, 1713[link]

Newton had been urged to make a new edition of the 'Principia' since the early 1690s, partly because copies of the first edition had already become very rare and expensive within a few years after 1687.^[51] Newton referred to his plans for a second edition in correspondence with Flamsteed in November 1694:^[52] Newton also maintained annotated copies of the first edition specially bound up with interleaves on which he could note his revisions; two of these copies still survive:^[53] but he had not completed the revisions by 1708, and of two would-be editors, Newton had almost severed connections with one, Fatio de Duillier, and the other, David Gregory seems not to have met with Newton's approval and was also terminally ill, dying later in 1708. Nevertheless, reasons were accumulating not to put off the new edition any longer.^[54] Richard Bentley, master of Trinity College, persuaded Newton to allow him to undertake a second edition, and in June 1708 Bentley wrote to Newton with a specimen print of the first sheet, at the same time expressing the (unfulfilled) hope that Newton had made progress towards finishing the revisions.^[55] It seems that Bentley then realised that the editorship was technically too difficult for him, and with Newton's consent he appointed Roger Cotes, Plumian professor of astronomy at Trinity, to undertake the editorship for him as a kind of deputy (but Bentley still made the publishing arrangements and had the financial responsibility and profit). The correspondence of 1709-1713 shows Cotes reporting to two masters, Bentley and Newton, and managing (and often correcting) a large and important set of revisions to which Newton sometimes could not give his full attention.^[56] Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz,^[57] and by troubles at the Mint,^[58] Cotes was able to announce publication to Newton on 30 June 1713.^[59] Bentley sent Newton only six presentation copies; Cotes was unpaid; Newton omitted any acknowledgement to Cotes.

Among those who gave Newton corrections for the Second Edition were: Firmin Abauzit, Roger Cotes and David Gregory. However, Newton omitted acknowledgements to some because of the priority disputes. John Flamsteed, the Astronomer Royal, suffered this especially.

Third edition, 1726[link]

The third edition was published 25 March 1726, under the stewardship of Henry Pemberton, M.D., a man of the greatest skill in these matters ...; Pemberton later said that this recognition was worth more to him than the two hundred guinea award from Newton.^[60]

Annotated and other editions[link]

In 1739-42 two French priests, Pères Thomas LeSeur and François Jacquier (of the 'Minim' order, but sometimes erroneously identified as Jesuits) produced with the assistance of J-L Calandrini an extensively annotated version of the 'Principia' in the 3rd edition of 1726. Sometimes this is referred to as the 'Jesuit edition': it was much used, and reprinted more than once in Scotland during the 19th century.^[61]

Gabrielle Emilie le Tonnelier de Breteuil, marquise du Chatelet also made a translation of Newton's Principia into French. Unlike LeSeur and Jacquier's edition, hers was a complete translation of Newton's three books and their prefaces. She also included a Commentary section where she fused the three books into a much clearer and easier to understand summary. She included an analytical section where she applied the new mathematics of calculus to Newton's most controversial theories. Previously, geometry was the standard mathematics used to analyze theories. Du Chatelet's translation is the only complete one to have been done in French and hers remains the standard French translation to this day. See "Translating Newton's 'Principia': The Marquise du Châtelet's Revisions and Additions for a French Audience." Author(s): Judith P. Zinsser Source: Notes and Records of the Royal Society of London, Vol. 55, No. 2 (May, 2001), pp. 227-245.

English translations[link]

Two full English translations of Newton's 'Principia' have appeared, both based on Newton's 3rd edition of 1726.

The first, from 1729, by Andrew Motte,^[2] was described by Newton scholar I. Bernard Cohen (in 1968) as "still of enormous value in conveying to us the sense of Newton's words in their own time, and it is generally faithful to the original: clear, and well written".^[62] The 1729 version was the basis for several republications, often incorporating revisions, among them a widely used modernized English version of 1934, which appeared under the editorial name of Florian Cajori (though completed and published only some years after his death). Cohen pointed out ways in which the 18th-century terminology and punctuation of the 1729 translation might be confusing to modern readers, but he also made severe criticisms of the 1934 modernized English version, and showed that the revisions had been made without regard to the original, also demonstrating gross errors "that provided the final impetus to our decision to produce a wholly new translation".^[63]

The second full English translation, into modern English, is the work that resulted from this decision by collaborating translators I. Bernard Cohen and Anne Whitman; it was published in 1999 with a guide by way of introduction.^[64]

William H. Donahue has published a translation of the work's central argument, published in 1996, along with expansion of included proofs and ample commentary.^[65] The book was developed as a textbook for classes at St. John's College in Annapolis and the aim of this translation is to be faithful to the Latin text.^[66]

See also[link]

	Book: Isaac Newton
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• Atomism
• Elements of the Philosophy of Newton
• Previous writings by Newton, including Quaestiones quadem philosophicae, De motu corporum in gyrum

References[link]

Further reading[link]

• Alexandre Koyré, Newtonian studies (London: Chapman and Hall, 1965).
• I. Bernard Cohen, Introduction to Newton's Principia (Harvard University Press, 1971).
• Richard S. Westfall, Force in Newton’s physics; the science of dynamics in the seventeenth century (New York: American Elsevier, 1971).
• S. Chandrasekhar, Newton’s Principia for the common reader (New York: Oxford University Press, 1995).
• Guicciardini, N., 2005, "Philosophia Naturalis..." in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 59-87.
• Andrew Janiak, Newton as Philosopher (Cambridge University Press, 2008).
• François De Gandt, Force and geometry in Newton’s Principia trans. Curtis Wilson (Princeton, NJ: Princeton University Press, c1995).
• Steffen Ducheyne, The main Business of Natural Philosophy: Isaac Newton’s Natural-Philosophical Methodology (Dordrecht e.a.: Springer, 2012).
• John Herivel, The background to Newton’s Principia; a study of Newton’s dynamical researches in the years 1664-84 (Oxford, Clarendon Press, 1965).
• Brian Ellis, "The Origin and Nature of Newton's Laws of Motion" in Beyond the Edge of Certainty, ed. R. G. Colodny. (Pittsburgh: University Pittsburgh Press, 1965), 29-68.
• E.A. Burtt, Metaphysical Foundations of Modern Science (Garden City, NY: Doubleday and Company, 1954).

External links[link]

Latin versions[link]

Wikisource has original text related to this article: Philosophiae Naturalis Principia Mathematica

• Cambridge University, Cambridge Digital Library High resolution digitised version of Newton's own copy of the first edition, interleaved with blank pages for his annotations and corrections.
• 1687: Newton's 'Principia', first edition (1687, in Latin). High-resolution presentation of the Gunnerus Library's copy.
• 1687: Newton's 'Principia', first edition (1687, in Latin).
• Principia (in Latin, annotated). 1833 Glasgow reprint (volume 1) with Books 1 & 2 of the Latin edition annotated by Leseur, Jacquier and Calandrini 1739-42 (described above).
• Project Gutenberg
• Archive.org

English translations[link]

Wikisource has original text related to this article: The Mathematical Principles of Natural Philosophy (American edition, 1846)

• Andrew Motte, 1729, first English translation of third edition (1726)
  • WikiSource, Partial
  • Google books, vol.1 with Book 1.
  • Google books, vol.2 with Books 2 and 3. (Book 3 starts at p.200.) (Google's metadata wrongly labels this vol.1).
  • Partial HTML
• Robert Thorpe 1802 translation
• N. W. Chittenden, ed., 1846 "American Edition" a partly modernized English version, largely the Motte translation of 1729.
  • Wikisource
  • Archive.org #1
  • Archive.org #2
• Percival Frost 1863 translation with interpolations Archive.org
• Florian Cajori 1934 modernization of 1729 Motte and 1802 Thorpe translations
• [10], Ian Bruce has made a complete translation of the third edition, with notes, on his website.

Other links[link]

• In Search of Principia, regarding online editions

NCB Naturalis
the Netherlands Centre for Biodiversity
Mammoth skeleton on display in the section about prehistoric animals
Location	Leiden, Netherlands
Type	Natural history museum
Collection size	37 million objects[1]
Visitor figures	271,000 (2011 est.)[2] Ranked 11th nationally (2011)
Director	Edwin van Huis[3]
President	Cees Veerman[3]
Public transit access	Leiden Centraal[4]
Website	www.naturalis.nl

NCB Naturalis, the Netherlands Centre for Biodiversity (Dutch: NCB Naturalis, Nederlands Centrum voor Biodiversiteit) is a national museum of natural history and a research center on biodiversity in Leiden, Netherlands.

History[link]

Naturalis originated from the merger of the Rijksmuseum van Natuurlijke Historie (abbreviated RMNH) and the Rijksmuseum van Geologie en Mineralogie (abbreviated RGM) in 1984. In 1986 it was decided that the museum had to become a public museum and a new building was built. The new building cost about €60 million, making it the second most expensive museum building in the Netherlands.^{[citation needed]}

Collection[link]

The tower of Naturalis, in which almost the entire collection is housed

The current museum is known for the numerous objects in its collections. It has approximately 10 million zoological and geological specimens.

• 5,250,000 insects
• 2,290,000 other invertebrates
• 1,000,000 vertebrates
• 1,160,000 fossils
• 440,000 rocks and minerals
• 2,000 gemstones
• 10,000 objects in an educational collection

The collections will be fused with the collections of the Zoölogisch Museum Amsterdam and expanded to approximately 37 million specimens. The fusion started in 2011 and is scheduled to be competed in 2012.

The collections are stored in a 60-meter-high tower, a landmark in Leiden, opened in April 1998.

Exhibitions[link]

Besides its temporary exhibition the museums has several permanent exhibitions:

• Nature Theater (Animals, plants, fungi, one-celled organisms, bacteria, stones, and minerals: an impression of nature in all its various forms.)
• Primeval Parade (A parade of fossils shows the history of the earth and the development of life.)
• Earth (Games and signs inform the visitor about the Earth's complexities.)
• Life (It displays how plants and animals live and survive on earth. )
• Earth Inside (For children and their parents to discover in a playful way how nature works.)
• Biotechnology (Games and movies show the visitor how essential DNA is to all life processes.)
• Treasure Chamber (Special security and storage conditions protect the precious gemstones, including a collection that once belonged to the Dutch King William I, and the mounted skins of animals that became extinct over the past few hundred years.)

References[link]

External links[link]

Wikimedia Commons has media related to: Naturalis Leiden

• Naturalis, official website
• Naturalis extinct birds, 3D presentation

Coordinates: 52°09′53″N 4°28′24″E / 52.16472°N 4.47333°E / 52.16472; 4.47333

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The title page of the shortened version of the Principia Mathematica to *56.

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.

PM, as it is often abbreviated, is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for PM was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. These were avoided in PM by building an elaborate system of types: a set of elements is of a different type than is each of its elements (a set is not an element; one element is not the set) and one cannot speak of the "set of all sets" and similar constructs, which would lead to paradoxes (see Russell's paradox).

PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."

PM is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy since Aristotle's Organon.^[1] The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.^[2]

Scope of foundations laid[link]

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.

A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

[edit] The construction of the theory of PM

As noted in the criticism of the theory by Kurt Gödel (below), unlike a Formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that almost immediately in the theory, interpretations (in the sense of model theory) are presented in terms of truth-values for the behavior of the symbols "⊢" (assertion of truth), "~" (logical not), and "V" (logical inclusive OR).

Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw (pure) Formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of "values", a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the "interpretation" of what the symbols commonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)". But this is not a pure Formalist theory.

The contemporary construction of a formal theory[link]

The following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows:

1. Symbols used: This set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols "→" (implies, IF-THEN, "⊃"), "&" (and), "V" (or), "¬" (not), "∀" (for all), "∃" (there exists); predicate symbol "=" (equals); function symbols "+" (arithmetic addition), "∙" (arithmetic multiplication), "'" (successor); individual symbol "0" (zero); variables "a", "b", "c", etc.; and parentheses "(" and ")".^[3]
2. Symbol strings: The theory will build "strings" of these symbols by concatenation (juxtaposition).^[4]
3. Formation rules: The theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable strings or "well-formed formulas" (wffs).^[5] This includes a rule for "substitution".^[6] of strings for the symbols called "variables" (as opposed to the other symbol-types).
4. Transformation rule(s): The axioms that specify the behaviors of the symbols and symbol sequences.
5. Rule of inference, detachment, modus ponens : The rule that allows the theory to "detach" a "conclusion" from the "premises" that led up to it, and thereafter to discard the "premises" (symbols to the left of the line │, or symbols above the line if horizontal). If this were not the case, then substitution would result in longer and longer strings that have to be carried forward. Indeed, after the application of modus ponens, nothing is left but the conclusion, the rest disappears forever.

Contemporary theories often specify as their first axiom the classical or modus ponens or "the rule of detachment":

A, A ⊃ B │ B

The symbol "│" is usually written as a horizontal line, here "⊃" means "implies". The symbols A and B are "stand-ins" for strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in a manner similar to IF-THEN but with a difference: given symbol string IF A and A implies B THEN B (and retain only B for further use). But observe that the symbols have no "interpretation" (e.g., no "truth table" or "truth values" or "truth functions") and modus ponens proceeds mechanistically, by grammar alone.

[edit] The logicistic construction of the theory of PM

The reader will observe both significant similarities, and similar differences, to a contemporary formal theory. Kleene states that "this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intended to be believed, or at least to be accepted as plausible hypotheses concerning the world".^[7] Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, PM introduces the notion of "truth-values", i.e., truth and falsity in the real-world sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory (PM 1962:4-36):

• 1. Variables.
• 2. Uses of various letters.
• 3. The fundamental functions of propositions: "the Contradictory Function" symbolized by "~" and the "Logical Sum or Disjunctive Function" symbolized by "∨" being taken as primitive and logical implication defined (the following example also used to illustrate 9. Definition below) as

p ⊃ q .=. ~ p ∨ q Df. (PM 1962:11)

and logical product defined as

p . q .=. ~(~p ∨ ~q) Df. (PM 1962:12)

(See more about the confusing "dots" used as both a grammatical device and as to symbolize logical conjunction (logical AND) at the section on notation.)

• 4. Equivalence: Logical equivalence, not arithmetic equivalence: "≡" given as a demonstration of how the symbols are used, i.e., "Thus ' p ≡ q ' stands for '( p ⊃ q ) . ( q ⊃ p )'." (PM 1962:7). Notice that to discuss a notation PM identifies a "meta"-notation with "[space] ... [space]":^[8]

Logical equivalence appears again as a definition:

p ≡ q .=. ( p ⊃ q ) . ( q ⊃ p. ) (PM 1962:12),

Notice the appearance of parentheses. This grammatical usage is not specified and appears sporadically; parentheses do play an important role in symbol strings, however, e.g., the notation "(x)" for the contemporary "∀x".

• 5. Truth-values: "The 'Truth-value' of a proposition is truth if it is true, and "falsehood if it is false" (this phrase is due to Frege) (PM 1962:7).
• 6. Assertion-sign: "'⊦'. p may be read 'it is true that' ... thus '⊦: p .⊃. q ' means 'it is true that p implies q ', whereas '⊦. p .⊃⊦. q ' means ' p is true; therefore q is true'. The first of these does not necessarily involve the truth either of p or of q, while the second involves the truth of both" (PM 1962:92).
• 7. Inference: PM 's version of modus ponens. "[If] '⊦. p ' and '⊦ (p ⊃ q)' have occurred, then '⊦ . q ' will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of '⊦. p ' [in other words, the symbols on the left disappear or can be erased]" (PM 1962:9).
• 8. The Use of Dots: See the section on notation.
• 9. Definitions: These use the "=" sign with "Df" at the right end. See the section on notation.
• 10. Summary of preceding statements: brief discussion of the primitive ideas "~ p" and "p ∨ q" and "⊦" prefixed to a proposition.
• 11. Primitive propositions: the axioms or postulates. This was significantly modified in the 2nd edition.
• 12. Propositional functions: The notion of "proposition" was significantly modified in the 2nd edition, including the introduction of "atomic" propositions linked by logical signs to form "molecular" propositions, and the use of substitution of molecular propositions into atomic or molecular propositions to create new expressions.
• 13. The range of values and total variation.
• 14. Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned in the 2nd edition. In particular, the distinction between the concepts defined in sections 15. Definition and the real variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.
• 17. Formal implication and formal equivalence.
• 18. Identity: See the section on notation. The symbol "=" indicates "predicate" or arithmetic equality.
• 19. Classes and relations.
• 20. Various descriptive functions of relations.
• 21. Plural descriptive functions.
• 22. Unit classes.

Primitive ideas[link]

Cf. PM 1962:90-94, for the first edition:

• (1) Elementary propositions.
• (2) Elementary propositions of functions.
• (3) Assertion: introduces the notions of "truth" and "falsity".
• (4) Assertion of a propositional function.
• (5) Negation: "If p is any proposition, the proposition "not-p", or "p is false," will be represented by "~p" ".
• (6) Disjunction: "If p and q are any propositons, the proposition "p or q, i.e., "either p is true or q is true," where the alternatives are to be not mutually exclusive, will be represented by "p ∨ q" ".
• (cf. section B)

Primitive propositions (Pp)[link]

The first edition (see discusion relative to the second edition, below) begins with a definition of the sign "⊃"

✸1.01. p ⊃ q .=. ~ p ∨ q. Df.

✸1.1. Anything implied by a true elementary proposition is true. Pp modus ponens

(✸1.11 was abandoned in the second edition.)

✸1.2. ⊦: p ∨ p .⊃. p. Pp principle of tautology

✸1.3. ⊦: q .⊃. p ∨ q. Pp principle of addition

✸1.4. ⊦: p ∨ q .⊃. q ∨ p. Pp principle of permutation

✸1.5. ⊦: p ∨ ( q ∨ r ) .⊃. q ∨ ( p ∨ r ). Pp associative principle

✸1.6. ⊦:. q ⊃ r .⊃: p ∨ q .⊃. p ∨ r. Pp principle of summation

✸1.7. If p is an elementary proposition, ~p is an elementary proposition. Pp

✸1.71. If p and q are elementary propositions, p ∨ q is an elementary proposition. Pp

✸1.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨ ψp is an elementary proposition. Pp

Together with the "Introduction to the Second Edition", the second edition's Appendix A abandons the entire section ✸9. This includes six primitive propositions ✸9 through ✸9.15 together with the Axioms of reducibility.

The revised theory is made difficult by the introduction of the Sheffer stroke ("|") to symbolize "incompatibility" (i.e., if both elementary propositions p and q are true, their "stroke" p | q is false), the contemporary logical NAND (not-AND). In the revised theory, the Introduction presents the notion of "atomic proposition", a "datum" that "belongs to the philosophical part of logic". These have no parts that are propositions and do not contain the notions "all" or "some". For example: "this is red", or "this is earlier than that". Such things can exist ad finitum, i.e., even an "infinite eunumeration" of them to replace "generality" (i.e., the notion of "for all").^[9] PM then "advance[s] to molecular propositions" that are all linked by "the stroke". Definitions give equivalences for "~", "∨", "⊃", and ".".

The new introduction defines "elementary propositions" as atomic and molecular positions together. It then replaces all the primitive propositions ✸1.2 to ✸1.72 with a single primitive proposition framed in terms of the stroke:

"If p, q, r are elementary propositions, given p and p|(q|r), we can infer r. This is a primitive proposition."

The new introduction keeps the notation for "there exists" (now recast as "sometimes true") and "for all" (recast as "always true"). Appendix A strengths the notion of "matrix" or "predicative function" (a "primitive idea", PM 1962:164) and presents four new Primitive propositions as ✸8.1–✸8.13.

✸88. Multiplicative axiom

✸102. Axiom of infinity

[edit] Notation used in PM

One author^[1] observes that "The notation in that work has been superseded by the subsequent development of logic during the 20th century, to the extent that the beginner has trouble reading PM at all"; while much of the symbolic content can be converted to modern notation, the original notation itself is "a subject of scholarly dispute", and some notation "embod[y] substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism".^[10]

Kurt Gödel was harshly critical of the notation:

"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in ✸1–✸21 of Principia [i.e., sections ✸1–✸5 (propositional logic), ✸8–14 (predicate logic with identity/equality), ✸20(introduction to set theory), and ✸21 (introduction to relations theory)]) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs".^[11]

This is reflected in the example below of the symbols "p", "q", "r" and "⊃" that can be formed into the string "p ⊃ q ⊃ r". PM requires a definition of what this symbol-string means in terms of other symbols; in contemporary treatments the "formation rules" (syntactical rules leading to "well formed formulas") would have prevented the formation of this string.

Source of the notation: Chapter I "Preliminary Explanations of Ideas and Notations" begins with the source of the notation:

"The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico [i.e., Peano 1889]. His use of dots as brackets is adopted, and so are many of his symbols" (PM 1927:4).^[12]

PM adopts the assertion sign "⊦" from Frege's 1879 Begriffsschrift:^[13]

"(I)t may be read 'it is true that'"^[14]

Thus to assert a proposition p PM writes:

"⊦. p." (PM 1927:92)

(Observe that, as in the original, the left dot is square and of greater size than the period on the right.)

An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)[link]

PM 's dots^[15] are used in a manner similar to parentheses. Later in section ✸14, brackets "[ ]" appear, and in sections ✸20 and following, braces "{ }" appear. Whether these symbols have specific meanings or are just for visual clarification is unclear. More than one dot indicates the "depth" of the parentheses, e.g., ".", ":" or ":.", "::", etc. Unfortunately for contemporary readers, the single dot (but also ":", ":.", "::", etc.) is used to symbolize "logical product" (contemporary logical AND often symbolized by "&" or "∧").

Logical implication is represented by Peano's "Ɔ" simplified to "⊃", logical negation is symbolized by an elongated tilde, i.e., "~" (contemporary "~" or "¬"), the logical OR by "v". The symbol "=" together with "Df" is used to indicate "is defined as", whereas in sections ✸13 and following, "=" is defined as (mathematically) "identical with", i.e., contemporary mathematical "equality" (cf. discussion in section ✸13). Logical equivalence is represented by "≡" (contemporary "if and only if"); "elementary" propositional functions are written in the customary way, e.g., "f(p)", but later the function sign appears directly before the variable without parenthesis e.g., "φx", "χx", etc.

Example, PM introduces the definition of "logical product" as follows:

✸3.01. p . q .=. ~(~p v ~q) Df.

where "p . q" is the logical product of p and q.

✸3.02. p ⊃ q ⊃ r .=. p ⊃ q . q ⊃ r Df.

This definition serves merely to abbreviate proofs.

Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitive translation can be given. However, because of criticisms such as that of Kurt Gödel below, the best contemporary treatments will be very precise with respect to the "formation rules" (the syntax) of the formulas.

The first formula might be converted into modern symbolism as follows:^[16]

(p & q) =_df (~(~p v ~q))

alternately

(p & q) =_df (¬(¬p v ¬q))

alternately

(p ∧ q) =_df (¬(¬p v ¬q))

etc.

The second formula might be converted as follows:

(p → q → r) =_df (p → q) & (q → r)

But note that this is not (logically) equivalent to (p → (q → r)) nor to ((p → q) → r), and these two are not logically equivalent either. The fact that such an ambiguous formula as p ⊃ q ⊃ r might appear as a result of the application of the formalism of PM reflects the harsh criticism of Kurt Gödel.

An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)[link]

These sections concern what is now known as Predicate logic, and Predicate logic with identity (equality).

• NB: As a result of criticism and advances, the second edition of PM (1927) replaces ✸9 with a new ✸8 (Appendix A). This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'.^[17] To add to the complexity of the treatment, ✸8 introduces the notion of substituting a "matrix", and the Sheffer stroke:

• Matrix: In contemporary usage, PM 's matrix is (at least for propositional functions), a truth table, i.e., all truth-values of a propositional or predicate function.
• Sheffer stroke: Is the contemporary logical NAND (NOT-AND), i.e., "incompatibility", meaning:

"Given two propositions p and q, then ' p | q ' means "proposition p is incompatible with proposition q, i.e., if both propositions p and q evaluate as false, then p | q evaluates as true." After section ✸8 the Sheffer stroke sees no usage.

Section ✸10: The existential and universal "operators": PM adds "(x)" to represent the contemporary symbolism "for all x " i.e., " ∀x", and it uses a backwards serifed E to represent "there exists an x", i.e., "(Ǝx)", i.e., the contemporary "∃x". The typical notation would be similar to the following:

"(x) . φx" means "for all values of variable x, function φ evaluates to true"

"(Ǝx) . φx" means "for some value of variable x, function φ evaluates to true"

Sections ✸10, ✸11, ✸12: Properties of a variable extended to all individuals: section ✸10 introduces the notion of "a property" of a "variable". PM gives the example: φ is a function that indicates "is a Greek", and ψ indicates "is a man", and χ indicates "is a mortal" these functions then apply to a variable x. PM can now write, and evaluate:

(x) . ψx

The notation above means "for all x, x is a man". Given a collection of individuals, one can evaluate the above formula for truth or falsity. For example, given the restricted collection of individuals { Socrates, Plato, Russell, Zeus } the above evaluates to "true" if we allow for Zeus to be a man. But it fails for:

(x) . φx

because Russell is not Greek. And it fails for

(x) . χx

because Zeus is not a mortal.

Equipped with this notation PM can create formulas to express the following: "If all Greeks are men and if all men are mortals then all Greeks are mortals". (PM 1962:138)

(x) . φx ⊃ ψx :(x). ψx ⊃ χx :⊃: (x) . φx ⊃ χx

Another example: the formula:

✸10.01. (Ǝx). φx . = . ~(x) . ~φx Df.

means "The symbols representing the assertion 'There exists at least one x that satisfies function φ' is defined by the symbols representing the assertion 'It's not true that, given all values of x, there are no values of x satisfying φ'".

The symbolisms ⊃_x and "≡_x" appear at ✸10.02 and ✸10.03. Both are abbreviations for universality (i.e., for all) that bind the variable x to the logical operator. Contemporary notation would have simply used parentheses outside of the equality ("=") sign:

✸10.02 φx ⊃_x ψx .=. (x). φx ⊃ ψx Df

Contemporary notation: ∀x(φ(x) → ψ(x)) (or a variant)

✸10.03 φx ≡_x ψx .=. (x). φx ⊃ ψx Df

Contemporary notation: ∀x(φ(x) ↔ ψ(x)) (or a variant)

PM attributes the first symbolism to Peano.

Section ✸11 applies this symbolism to two variables. Thus the following notations: ⊃_x, ⊃_y, ⊃_{x, y} could all appear in a single formula.

Section ✸12 reintroduces the notion of "matrix" (contemporary truth table), the notion of logical types, and in particular the notions of first-order and second-order functions and propositions.

New symbolism "φ ! x" represents any value of a first-order function. If a circumflex "＾" is placed over a variable, then this is an "individual" value of y, meaning that "ŷ" indicates "individuals" (e.g., a row in a truth table); this distinction is necessary because of the matrix/extensional nature of propositional functions.

Now equipped with the matrix notion, PM can assert its controversial axiom of reducibility: a function of one or two variables (two being sufficient for PM 's use) where all its values are given (i.e., in its matrix) is (logically) equivalent ("≡") to some "predicative" function of the same variables. The one-variable definition is given below as an illustration of the notation (PM 1962:166-167):

✸12.1 ⊢: (Ǝ f): φx .≡_x. f ! x Pp;

Pp is a "Primitive proposition" ("Propositions assumed without proof" (PM 1962:12, i.e., contemporary "axioms"), adding to the 7 defined in section ✸1 (starting with ✸1.1 modus ponens). These are to be distinguished from the "primitive ideas" that include the assertion sign "⊢", negation "~", logical OR "V", the notions of "elementary proposition" and "elementary propositional function"; these are as close as PM comes to rules of notational formation, i.e., syntax.

This means: "We assert the truth of the following: There exists a function f with the property that: given all values of x, their evaluations in function φ (i.e., resulting their matrix) is logically equivalent to some f evaluated at those same values of x. (and vice versa, hence logical equivalence)". In other words: given a matrix determined by property φ applied to variable x, there exists a function f that, when applied to the x is logically equivalent to the matrix. Or: every matrix φx can be represented by a function f applied to x, and vice versa.

✸13: The identity operator "=" : This is a definition that uses the sign in two different ways, as noted by the quote from PM:

✸13.01. x = y .=: (φ): φ ! x . ⊃ . φ ! y Df

means:

"This definition states that x and y are to be called identical when every predicative function satisfied by x is also satisfied by y ... Note that the second sign of equality in the above definition is combined with "Df", and thus is not really the same symbol as the sign of equality which is defined."

The not-equals sign "≠" makes its appearance as a definition at ✸13.02.

✸14: Descriptions:

"A description is a phrase of the form "the term y which satisfies φŷ, where φŷ is some function satisfied by one and only one argument."^[18]

From this PM employes two new symbols, a forward "E" and an inverted iota "ɿ". Here is an example:

✸14.02. E ! ( ɿy) (φy) .=: ( Ǝb):φy . ≡_y . y = b Df.

This has the meaning:

"The y satisfying φŷ exists," which holds when, and only when φŷ is satisfied by one value of y and by no other value." (PM 1967:173-174)

Introduction to the notation of the theory of classes and relations[link]

The text leaps from section ✸14 directly to the foundational sections ✸20 GENERAL THEORY OF CLASSES and ✸21 GENERAL THEORY OF RELATIONS. "Relations" are what known in contemporary set theory as ordered pairs. Sections ✸20 and ✸22 introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" (PM 1962:188); "⊂" (✸22.01) signifies "is contained in", "is a subset of"; "∩" (✸22.02) signifies the intersection (logical product) of classes (sets); "∪" (✸22.03) signifies the union (logical sum) of classes (sets); "–" (✸22.03) signifies negation of a class (set); "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse.

Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.) (PM 1962:188):

x ε α

"The use of single letter in place of symbols such as ẑ(φz) or ẑ(φ ! z) is practicallly almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' x ε α' will mean ' x is a member of the class α'". (PM 1962:188)

α ∪ –α = V

The union of a set and its inverse is the universal (completed) set.^[19]

α ∩ –α = Λ

The intersection of a set and its inverse is the null (empty) set.

When applied to relations in section ✸23 CALCULUS OF RELATIONS, the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example: "⊍", "∸".^[20]

The notion, and notation, of "a class" (set): In the first edition PM asserts that no new primitive ideas are necessary to define what is meant by "a class", and only two new "primitive propositions" called the axioms of reducibility for classes and relations respectively (PM 1962:25).^[21] But before this notion can be defined, PM feels it necessary to create a peculiar notation "ẑ(φz)" that it calls a "fictitious object". (PM 1962:188)

⊢: x ε ẑ(φz) .≡. (φx)

"i.e., ' x is a member of the class determined by (φẑ)' is [logically] equivalent to ' x satisfies (φẑ),' or to '(φx) is true.'". (PM 1962:25)

At least PM can tell the reader how these fictitious objects behave, because "A class is wholly determinate when its membership is known, that is, there cannot be two different classses having he same membership" (PM 1962:26). This is symbolized by the following equality (similar to ✸13.01 above:

ẑ(φz) = ẑ(ψz) . ≡ : (x): φx .≡. ψx

"This last is the distinguishing characteristic of classes, and justifies us in treating ẑ(ψz) as the class determined by [the function] ψẑ." (PM 1962:188)

Perhaps the above can be made clearer by the discussion of classes in Introduction to the 2nd Edition, which disposes of the Axiom of Reducibility and replaces it with the notion: "All functions of functions are extensional" (PM 1962:xxxix), i.e.,

φx ≡_x ψx .⊃. (x): ƒ(φẑ) ≡ ƒ(ψẑ) (PM 1962:xxxix)

This has the reasonable meaning that "IF for all values of x the truth-values of the functions φ and ψ of x are [logically] equivalent, THEN the function ƒ of a given φẑ and ƒ of ψẑ are [logically] equivalent." PM asserts this is "obvious":

"This is obvious, since φ can only occur in ƒ(φẑ) by the substitution of values of φ for p, q, r, ... in a [logical-] function, and, if φx ≡ ψx, the substitution of φx for p in a [logical-] function gives the same truth-value to the truth-function as the substitution of ψx. Consequently there is no longer any reason to distinguish between functions classes, for we have, in virtue of the above,

φx ≡_x ψx .⊃. (x). φẑ = . ψẑ".

Observe the change to the equality "=" sign on the right. PM goes on to state that will continue to hang onto the notation "ẑ(φz)", but this is merely equivalent to φẑ, and this is a class. (all quotes: PM 1962:xxxix).

Consistency and criticisms[link]

According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.

Beyond the status of the axioms as logical truths, the questions remained:

• whether a contradiction could be derived from the Principia's axioms (the question of inconsistency), and
• whether there exists a mathematical statement which could neither be proven nor disproven in the system (the question of completeness).

Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)

Gödel 1930, 1931[link]

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms (such as those of Principia Mathematica) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

Gödel's incompleteness theorems cast unexpected light on these two related questions.

Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, for every sufficiently powerful logical system (such as Principia), there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.

Gödel's second incompleteness theorem (1931) shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).

Wittgenstein 1919, 1939[link]

By the second edition of PM, Russell had removed his axiom of reducibility to a new axiom (although he does not state it as such). Gödel 1944:126 describes it this way: "This change is connected with the new axiom that functions can occur in propositions only "through their values", i.e., extensionally . . . [this is] quite unobjectionable even from the constructive standpoint . . . provided that quantifiers are always restricted to definite orders". This change from a quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i.e. functions of functions: "We can decide that mathematics is to confine itself to functions of functions which obey the above assumption" (PM 2nd Edition p. 401, Appendix C).

This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as "All 'x' are blue" now has to list all of the 'x' that satisfy (are true in) the proposition, listing them in a possibly infinite conjunction: e.g. x₁ V x₂ V . . . V x_n V . . .. Ironically, this change came about as the result of criticism from Wittgenstein in his 1919 Tractatus Logico-Philosophicus. As described by Russell in the Preface to the 2nd edition of PM:

"There is another course, recommended by Wittgenstein† (†Tractatus Logico-Philosophicus, *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. . . . [Working through the consequences] it appears that everything in Vol. I remains true . . . the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2ⁿ > n breaks down unless n is finite." (PM 2nd edition reprinted 1962:xiv, also cf new Appendix C).

In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in PM Second Edition.

Wittgenstein in his Lectures on the Foundations of Mathematics, Cambridge 1939 criticised Principia on various grounds, such as:

• It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g., that Principia did not characterize numbers or addition correctly), not as evidence of an error in everyday counting.
• The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g., billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental and hence questionable methods such as induction). So again Principia depends on everyday techniques, not vice versa.

Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.

Gödel 1944[link]

In his 1944 Russell's mathematical logic, Gödel offers a "critical but sympathetic discussion of the logicistic order of ideas"^[22]:

"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in *1-*21 of Principia) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs . . . The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens . . . it is chiefly the rule of substitution which would have to be proved" (Gödel 1944:124)^[23]

Quotations[link]

• "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful.")

See also[link]

• Axiomatic set theory
• Begriffsschrift
• Boolean algebra (logic)
• Information Processing Language

Footnotes[link]

References[link]

Primary:

• Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.
• Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume One. Merchant Books. ISBN 978-1-60386-182-3. 
• Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume Two. Merchant Books. ISBN 978-1-60386-183-0. 
• Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume Three. Merchant Books. ISBN 978-1-60386-184-7. 

Secondary:

• Stephen Kleene 1952 Introduction to Meta-Mathematics, 6th Reprint, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9.
  • Stephen Cole Kleene; Michael Beeson (March 2009). Introduction to Metamathematics (Paperback ed.). Ishi Press. ISBN 978-0-923891-57-2. 
• Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870-1940, Princeton University Press, Princeton N.J., ISBN 0-691-05857-1 (alk. paper).
• Ludwig Wittgenstein 2009 Major Works: Selected Philosophical Writings, HarperrCollins, NY, NY, ISBN 978-0-06-155024-9. In particular:

Tractatus Logico-Philosophicus (Vienna 1918, original publication in German).

• Jean van Heijenoort editor 1967 From Frege to Gödel: A Source book in Mathematical Logic, 1879-1931, 3rd printing, Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 (pbk.)

External links[link]

• Stanford Encyclopedia of Philosophy:
  • Principia Mathematica -- by A. D. Irvine.
  • The Notation in Principia Mathematica -- by Bernard Linsky.
• Principia Mathematica online (University of Michigan Historical Math Collection):
  • Volume I
  • Volume II
  • Volume III
• Proposition ✸54.43 in a more modern notation (Metamath)

Sir Isaac Newton
Godfrey Kneller's 1689 portrait of Isaac Newton (age 46)
Born	(1642-12-25)25 December 1642 [NS: 4 January 1643][1] Woolsthorpe-by-Colsterworth Lincolnshire, England
Died	20 March 1727(1727-03-20) (aged 84) [NS: 31 March 1727][1] Kensington, Middlesex, England
Residence	England
Nationality	English
Fields	Physics, mathematics, astronomy, natural philosophy, alchemy, Christian theology
Institutions	University of Cambridge Royal Society Royal Mint
Alma mater	Trinity College, Cambridge
Academic advisors	Isaac Barrow[2] Benjamin Pulleyn[3][4]
Notable students	Roger Cotes William Whiston
Known for	Newtonian mechanics Universal gravitation Infinitesimal calculus Optics Binomial series Newton's method Philosophiæ Naturalis Principia Mathematica
Influences	Henry More[5] Polish Brethren[6]
Influenced	Nicolas Fatio de Duillier John Keill
Signature
Notes His mother was Hannah Ayscough. His half-niece was Catherine Barton.

Sir Isaac Newton PRS MP (25 December 1642 – 20 March 1727 [NS: 4 January 1643 – 31 March 1727])^[1] was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."^[7] His monograph Philosophiæ Naturalis Principia Mathematica, published in 1687, lays the foundations for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws, by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the Scientific Revolution.

The Principia is generally considered to be one of the most important scientific books ever written, due, independently, to the specific physical laws the work successfully described, and for the style of the work, which assisted in setting standards for scientific publication down to the present time. Newton built the first practical reflecting telescope^[8] and developed a theory of colour based on the observation that a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. In mathematics, Newton shares the credit with Gottfried Leibniz for the development of differential and integral calculus. He also demonstrated the generalised binomial theorem, developed Newton's method for approximating the roots of a function, and contributed to the study of power series. Newton's work on infinite series was inspired by Simon Stevin's decimals.^[9] Newton, although an unorthodox Christian, was deeply religious, and wrote more on Biblical hermeneutics and occult studies than on science and mathematics. Newton secretly rejected Trinitarianism, and feared being accused of refusing holy orders.^[10]

Life[link]

Early life[link]

Isaac Newton was born on what is retroactively considered 4 January 1643 [OS: 25 December 1642]^[1] at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. At the time of Newton's birth, England had not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, 25 December 1642. Newton was born three months after the death of his father, a prosperous farmer also named Isaac Newton. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug (≈ 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young Isaac disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house over them."^[11] Although it was claimed that he was once engaged,^[12] Newton never married.

Newton in a 1702 portrait by Godfrey Kneller

Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)

From the age of about twelve until he was seventeen, Newton was educated at The King's School, Grantham (where his alleged signature can still be seen upon a library window sill).[1]^{[citation needed]} He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, where his mother, widowed by now for a second time, attempted to make a farmer of him. He hated farming.^[13] Henry Stokes, master at the King's School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student.^[14] The Cambridge psychologist Simon Baron-Cohen considers it "fairly certain" that Newton had Asperger syndrome.^[15]

In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role.^[16] At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,^[17] Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus,^[18] optics and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity.^[19] Fellows were required to become ordained priests, something Newton desired to avoid due to his unorthodox views. Luckily for Newton, there was no specific deadline for ordination and it could be postponed indefinitely. The problem became more severe later when Newton was elected for the prestigious Lucasian Chair. For such a significant appointment, ordaining normally could not be dodged. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II (see "Middle years" section below).

Middle years[link]

Mathematics[link]

Newton's work has been said "to distinctly advance every branch of mathematics then studied".^[20] His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton's mathematical papers.^[21] The author of the manuscript De analysi per aequationes numero terminorum infinitas, sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as:^[22]

Mr Newton, a fellow of our College, and very young ... but of an extraordinary genius and proficiency in these things.

Newton later became involved in a dispute with Leibniz over priority in the development of infinitesimal calculus. Most modern historians believe that Newton and Leibniz developed infinitesimal calculus independently, although with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in 1684. (Leibniz's notation and "differential Method", nowadays recognised as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) Such a suggestion, however, fails to notice the content of calculus which critics of Newton's time and modern times have pointed out in Book 1 of Newton's Principia itself (published 1687) and in its forerunner manuscripts, such as De motu corporum in gyrum ("On the motion of bodies in orbit"), of 1684. The Principia is not written in the language of calculus either as we know it or as Newton's (later) 'dot' notation would write it. But his work extensively uses an infinitesimal calculus in geometric form, based on limiting values of the ratios of vanishing small quantities: in the Principia itself Newton gave demonstration of this under the name of 'the method of first and last ratios'^[23] and explained why he put his expositions in this form,^[24] remarking also that 'hereby the same thing is performed as by the method of indivisibles'.

Because of this, the Principia has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times^[25] and "lequel est presque tout de ce calcul" ('nearly all of it is of this calculus') in Newton's time.^[26] His use of methods involving "one or more orders of the infinitesimally small" is present in his De motu corporum in gyrum of 1684^[27] and in his papers on motion "during the two decades preceding 1684".^[28]

Newton had been reluctant to publish his calculus because he feared controversy and criticism.^[29] He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's Principia, and corresponded with Leibniz.^[30] In 1693 the relationship between Duillier and Newton deteriorated, and the book was never completed.

Starting in 1699, other members of the Royal Society (of which Newton was a member) accused Leibniz of plagiarism, and the dispute broke out in full force in 1711. The Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later found that Newton himself wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter's death in 1716.^[31]

Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula), and was the first to use power series with confidence and to revert power series.

He was appointed Lucasian Professor of Mathematics in 1669 on Barrow's recommendation. In that day, any fellow of Cambridge or Oxford was required to become an ordained Anglican priest. However, the terms of the Lucasian professorship required that the holder not be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the ordination requirement, and Charles II, whose permission was needed, accepted this argument. Thus a conflict between Newton's religious views and Anglican orthodoxy was averted.^[32]

Optics[link]

A replica of Newton's second Reflecting telescope that he presented to the Royal Society in 1672^[33]