1. History & Philosophy of Calculus, Session 7 NEWTON & LEIBNIZ

2.  Summary  Calculus seems to demand some concept of the infinitesimal  Magnitude  New number  Or both?  But infinitesimals lack clarity  Concepts that are not mathematical being used to interpret mathematics?  Paradoxes are unresolved  If we can’t trust indivisibles or infinitesimals to solve the area of a rectangle, why should we for curves?  Is geometry the true depiction of space? Is it adequate to new sciences of motion? OVERVIEW

3. CALCULATING THE DERIVATIVE

4. History & Philosophy of Calculus, Session 7 ATOMISM

5.  Problem of how to interpret nascent calculus is overlaid by inheritance from Greek Thought  Status of Geometry and its relation to Physical World  Ideal truth of Euclidean demonstrations – proof!  But does it only deal with an ideal realm of mathematical objects?  How does it deal with motion?  Atomism  All physical things are composed of atoms (‘uncuttables’ or ‘indivisibles’)  Objects are compounds of small set of atoms that combine in various ways  Repeated division will hit a terminus with atoms – no infinite divisibility and therefore no continuum as defined by Aristotle  Atoms move through void – empty space which is infinite ANCIENT GREEK THOUGHT

6.  Contingency of cosmos  Atoms arranged in compounds – blind chance  Early multiverse theories  With infinite space and a small number of basic building blocks, parallel worlds will appear identical to this one in all respects  No divine plan or creator – nihilistic?  Void is empty  Versus monotheistic ideas of divine omnipresence  Versus ideas of spiritual presence in spatial world  Return to atomism in Sixteenth & Seventeenth Century Europe  Concerns Church & Jesuits  Discussions of nature of continuum & its composition  1606, 1608, 1613 & 1615  False doctrine – that the continuum is composed of finite number of indivisibles  1632 – campaign launched against infinitesimal ATHEISM OF ATOMISTS

7. Bans several positions on doctrinal grounds including  25. The Continuum and the intensity of qualities are composed of indivisibles.  26. Inflatable points are given, from which the continuum is composed.  30. Infinity in multitude and magnitude can be enclosed between two unities or two points. [contra Torricelli & Cavalieri]  31.Tiny vacuums are interspersed in the continuum, few or many, large or small, depending on its rarity or density. 1651 JESUIT EDICT

8.  Averroes:  A line as a line can be divided indefinitely. But such a division is impossible if the line is taken as made on earth.  Proclus’s Commentary on Euclid  Atomists and Epicureans are ‘those who alone criticise the principles of geometry’  In early modern period, Thomas Hobbes sought to refound geometry on new set of principles  Points with width and breadth  Lines with width and breadth  Early attempt at ‘mechanics’?  Concerned with bodies in motion in world ATOMISM & EUCLIDEAN GEOMETRY

9.  Avicenna: Consider a square and one of its diagonals. If atoms are sizeless, then from every sizeless atom on the diagonal a straight line can be drawn at right angles until it joins a sizeless atom on one of the two sides. When all such lines have been drawn, they will be parallel and no gaps will lie between them. Thus to each atom on the diagonal there corresponds exactly one atom one one of the two sides, and vice-versa. So there must be the same number of atoms along the diagonal of a square as along the two adjoining sides. In that case the absurd conclusion is reached that the route along the diagonal should be no quicker than the route along the two sides. ARGUMENTS AGAINST ATOMISTS

10.  It seems probable to me, that God in the Beginning form’d Matter in solid, massy, hard, impenetrable, moveable Particles, of such Sizes and Figures, and with such other Properties, and in such proportion to Space, as most conduced to the End for which He formed them. Opticks ... had we the proof of but one experiment that any undivided particle, in breaking a hard and solid body, suffered a division, we might conclude by virtue of this rule that the undivided as well as the divided particles may be divided and actually separated to infinity. Principia NEWTON AS ATOMIST

11. History & Philosophy of Calculus, Session 7 THE INVENTION OF THE CALCULUS

12.  Newton and Leibniz in 17 th Century invented the algorithmic procedures underlying the calculus  Established reciprocal relation between differentiation & integration & their general applicability  Fluxionary calculus of Newton  John Bell – Newton is the ‘first person to give a generally applicable procedure for determining an instantaneous rate of change and to invert this in the case of problems involving summations ‘  Differential calculus of Leibniz (Leibniz’s notation is used today)  But struggles over how to interpret its proofs and method  Newton invented his methods in the ‘Plague Year’ of 1665-66 but publication was slower  De analysi per aequationes numero terminorum infinitas; written 1666, published 1711  Methodus fluxionum et serierum infinitarum; written 1671, published 1736  De quadratura curvarum published 1704 as appendix to Opticks. BACKGROUND

13.  Dynamics: Continua and Variable Quantities are generated by (‘indisputable fact of’) motion in time  I don’t here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the rotation of their Legs, Time by a continual flux, and so in the rest. Opening sentences of de quadratura curvarum (cf. Cavalieri)  “Genita” & “moments”  Genita are variable quantities that are in motion or flux  ‘velocities’  Moments are tiny increments or change in genita  Three interpretations of fundamental concepts of his calculus  Equivalent results – but differences in rigour? NEWTON & MOTION

14. History & Philosophy of Calculus, Session 7 NEWTON’S INTERPRETATIONS OF CALCULUS

15. De Analysi – employs infinitesimal quantities  ‘Moment’ is treated asequivalent to Fermat’s ‘e’  X is replaced by x+o in equations  Where ‘o’ is a letter not a zero – but a momentary change  At conclusion all results containing ‘o’ or its powers were removed  ‘terms multiplied by ‘o’ will be nothing in respect to the rest’  Newton did not explain what entitled him to do this  Boyer – ‘Newton facilitated the operations [of calculus] but did not clarify them’  Later, Newton abandons and tries to eliminate all traces of the infinitesimal in the calculus FIRST INTERPRETATION

16. Method of Fluxions  Quantity generated by motion – fluent (cf ‘genita’)  Rate of generation or ‘velocity’ – fluxion  'augments of fluents generated in equal but very small particles of time - the first ratio of nascent augments' Opticks 1704  These quantities or rates are expressed by ratios of finite magnitudes  (but do we need infinitesimals to make sense of these ratios?)  Newton emphasises ‘nascent’ and ‘evanescent’ quantities  Where ratios ‘retain the character of that which is appearing or disappearing’  First edition of Principia (1687)  'Moments, as soon as they are of finite magnitude, cease to be moments. To be given finite bounds is in some measure contradictory to their continuous increase or decrease'  Second edition of Principia (1713)  ' Finite particles are not moments, but the very quantities generated by the moments.‘ SECOND INTERPRETATION

17.  Prime & Ultimate ratios & sums in de quadratura curvarum  ‘First’ and ‘last’ ratios  Idea appears in fragmentary and unclear form in Principia  Stress on ratio rather than two quantities forming the ratio (viz. ‘slope’ calculation’)  Hard to shed conceptual underpinning in infinitesimal though Newton claims it does not need infinitesimally small  Anticipates the limit concept of Nineteenth century  Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal. Lemma 1 in Section 1 of Book I Principia THIRD INTERPRETATION

18. For demonstrations are shorter by the method of indivisibles; but because the hypothesis of indivisibles seems somewhat harsh[durior 'problematic'], and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following Propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios, and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with greater safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curved lines for right ones, I would not be understood to mean indivisibles, but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios... SCHOLIUM TO LEMMA XI IN BOOK 1 OF PRINCIPIA

19. by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be.... ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.... Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end. SCHOLIUM TO LEMMA XI IN BOOK 1 OF PRINCIPIA (CONT.)

20. History & Philosophy of Calculus, Session 7 LEIBNIZ’S DIFFERENTIAL CALCULUS

21.  Nova Methodus 1684 & De Geometri Recondita 1686  Fundamental concept is the ‘differential’  dy/dx – where dy & dx are ‘differentials’ – variables ranging over differences  Where it is taken to be Velocity – the tangent is an ‘infinitely small line’  Integral – sum of infinitely narrow rectangles  Use of both infinitesimal number & infinitesimal magnitude  Infinitesimal number  nothing but quantities that can be as small as one wants  governed by laws of number but are smaller than any number  Calculating rules were adapted from rules for ordinary numbers (iteration)  Infinitesimal magnitude  We have to keep in mind that to find a tangent means to draw a line that connects two points of a curve at an infinitely small distance, or the continued side of a polygon with an infinite number of angles, which for us takes the place of a curve.  ‘a relative zero’ – ‘as an evanescent quantity which yet retains the character of that which is disappearing’ (cf. Newton) DIFFERENTIAL CALCULUS

22.  metaphysical points, or “monads”, from which actual entities such as bodies are compounded;  monad as formal atom - their nature consists in force - feeling and appetite (akin to souls)  atoms that are not of matter  mathematical points, or positions in space (do not have extension as in Euclid);  and physical points, (have extension as in atomism)  Russell identifies with “an infinitesimal extension of the kind used in the Infinitesimal Calculus.”  “A point is not a certain part of matter, nor would an infinite number of points make an extension.” LEIBNIZ’S 3 KINDS OF POINT

23.  Confusion  Differential Calculus of Leibniz & Newton’s method of fluxions were the same  1712 Royal Society report into priority claims for invention of the calculus confused the issue  New concept of number  More rigorous conception or ratios or ‘rational numbers’ (a/b)  Ratio as itself a number rather than a relation of two numbers  Infinitesimal & Zero  Traditionally – 0 is the only number incapable of being multiplied repeatedly to equal any given number  Nieuwentijdt 1694-96 – infinitesimal with property that powers are all equal to zero – ‘nilsquare’ property  Infinitesimal is different to zero but its power are not (cf. Elimination of powers of ‘e’ and ‘o’) RESOLVING THE DIFFERENTIAL MATHEMATICALLY

24. John Bell The Continuous and the Infinitesimal in Mathematics and Philosophy 2005 ARGUMENTS OVER INFINITESIMALS

25. 1.dx ≈ 0 (dx is ‘indistinguishable from’ zero) 2.neither dx = 0 nor dx ≠ 0 3.dx 2 = 0 (Nieuwentijdt) 4.dx → 0 (dx ‘becomes vanishingly small’ is ‘evanescent’ – a variable quantity tending to zero) OPTIONS FOR INTERPRETING DIFFERENTIALS

26. History & Philosophy of Calculus, Session 7 CONCLUSION

27.  Newton – less metaphysically inclined than Leibniz  Three interpretations do not rest on metaphysical notions besides a strong basis in motion in time  Mathematical foundation more important  Leibniz – emphasis on algorithmic processes rather than dynamics or mechanics  But has metaphysical cosmology and ontology into which differential interpretation fits  Eighteenth Century does not resolve differences of interpretation  Rival interpretations of calculus and infinitesimal CONCLUSION & SUMMARY