1. History & Philosophy of Calculus, Session 7
Newton & Leibniz
History & Philosophy of Calculus, Session 7

2. Overview Summary of last week
Calculus seems to demand some rigorous conceptualisation of the infinitesimal
Magnitude
New number
Or both?
But infinitesimals lack clarity
How do they relate to indivisibles? (geometric & physical?)
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?

3. Atomism
History & Philosophy of Calculus, Session 6

4. Ancient Greek thought
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 – atoms as building blocks
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
Early theories of parallel worlds
Mid 17th century: Torricelli & Pascal conducted experiments with mercury to prove possibility of void or vacuum
Cf indivisible vs infinite divisibility of Aristotle (infinitesimal?)

5. Atheism of atomists Contingency of cosmos Void is empty
Atoms arranged in compounds – blind chance
No soul – on death or atoms disperse
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

6. 1651 Jesuit Edict
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.

7. Atomism & Euclidean geometry
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 re-found 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 – if motion is to be in our conclusions then it must be in our axioms

8. Arguments against Atomists
Avicenna:
Consider a square and one of its diagonals. If atoms are sizeless, then, Nazzam contends, 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 on 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.
Cf our paradoxes from before – is atomism the idea that links infinitesimals & indivisibles?

9. Newton as Atomist
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

10. Leibniz’s Monads “The New System” 1695
§3 “At first, when I had freed myself from the yoke of ·the schools, and thus of· Aristotle, I was in favour of ·an approach to physics based on· atoms and empty space…”
“I had to bring in what might be called a real and living point, an atom of substance [monad] that is a complete being only because it contains some kind of form or activity.”
Primary forces – actuality & activity
Unities – devoid of parts
What substantial things are made of
Contra Newton, a monad’s force is always internal & expressive
Each monad is a microcosm of entire universe
‘A perpetual living mirror of the universe’ (“Monadology” 1714)
Cf Newton – vis insita (inertia) & vis impressa (external force)
Euler: Force is a property of matter that means that one body can change the state of another

11. Leibniz’s 3 kinds of point
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
Generative - Cf. Newton’s fluxions & genita
mathematical points, or positions in space (do not have extension as in Euclid);
Mathematical points cannot form differentials – no magnitude
Differentials are always infinitely small lines (same dimensionality)
and physical points
have extension or magnitude as in atomism;
Russell identifies with “an infinitesimal extension of the kind used in the Infinitesimal Calculus.”
“A [mathematical] point is not a certain part of matter, nor would an infinite number of [mathematical] points make an extension.”

12. New System, §11
“Atoms of Substance [monads] might be called metaphysical points: they are related to mathematical points, which are their points of view for expressing the universe, but they are not themselves mathematical points because they have something alive about them, and a kind of perception.
“When a bodily substance is contracted ·far enough·, all its organs together make what to us is only a physical point. So physical points only seem to be indivisible. Mathematical points really are indivisible, but they are not things.
“It is only metaphysical or substantial points (constituted by forms or souls) that are both exact and real, and without them there wouldn’t be any things at all, because without true unities there would be no multiplicity—·without true ones there would be no manies.”

13. The invention of the calculus
History & Philosophy of Calculus, Session 7

14. background
Newton and Leibniz in 17th 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 but publication was slower (private circulation)
De analysi per aequationes numero terminorum infinitas (‘On Analysis’); written 1666, published 1711
Methodus fluxionum et serierum infinitarum (The method of fluxions); written 1671, published 1736
De quadratura curvarum (‘On the quadrature of curves’) published 1706 as appendix to Opticks. Written c
In interim, Principia Mathematica appeared in 1687.

15. Newton & Motion
From Isaac Barrow: grounding of geometry on motion (causal)
Geometrical Lectures (1670 – edited by Newton)
Hobbes de Corpore (1655): “science is the demonstration of the causes and demonstrations of things” – if that’s not present in the axioms, then it won’t be in conclusions! Euclidean geometry needs rethinking!
Roberval: “the direction of the motion of a point which describes a curved line is the tangent to the curved line at each position of that point.” (1835/1693)
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 et al)
Fermat & the rethinking of the equation of the line. Roberval is similar to method proposed by Torricelli in 1644.

16. “Genita” & “moments”
Genita are variable quantities that are in motion or flux
‘velocities’
Their flow or flux appears to create “stock” (quantity) of higher dimension – movement of point creates line (with length); sweep of line creates surface (with area) etc.
Moments are tiny increments or change in genita
Three interpretations of fundamental concepts of Newton’s calculus
Equivalent results – but differences in rigour?
Evanescence & nascence – versus part and whole conceptualisation
(Equation as genetic principle governing the motion of the point and line, rather than description)
But no explanation of ‘instantaneous velocity’

17. Equations & Curves – further observations
Equation captures motion (the tangent) that generates curve (Barrow & Newton)
"This means that one is no longer dealing with continuous wholes as wholes which are bounded and limited and in this way given before their parts. Instead they are treated as generated wholes which may be potentially infinite but which are given by the algebraically expressed law constraining and determining their generation.“ Mary Tiles, The Philosophy of Set Theory, p. 75
Equation is characterization of the whole curve (a linear continuum) which is not built up from information about points
Hobbes & notion of conatus (‘striving’) – an intensive magnitude or principle

18. Newton’s interpretations of calculus
History & Philosophy of Calculus, Session 7

19. First interpretation De Analysi – employs infinitesimal quantities
‘Moment’ is treated as equivalent 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 are removed
“terms multiplied by ‘o’ will be nothing in respect to the rest”
Newton did not explain what entitled him to do this
Boyer – [in this essay] ‘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 technical manipulation of the calculus
in rebus mathematicis errores quam minimi non sunt contemnendi –
In mathematical matters errors are not to be ignored, no matter how small Principia Mathematica
Example page of calculations?

20. Second interpretation
Method of Fluxions
Quantity (stock) generated by motion – fluent
Rate of generation or ‘instantaneous velocity’ – fluxion (cf ‘genita’)
'augments of fluents generated in equal but very small particles of time - the first ratio of nascent augments' Opticks 1704
Traditionally – the rate of change – fluxion - would be considered to be a property of the fluent (cf ‘latitude of form’)
Newton – fluxions (genita) generate the fluent as quantity or magnitude
From derivative we can calculate position at any time by integration
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’
Stock / flow is my terminology – analogy. Fluxion generates fluents (quantities); fluxion is the rate of generation of fluents. Evanescent divisible quantities.

21. Third interpretation
Prime & Ultimate ratios & sums in de quadratura curvarum
Or ‘First’ and ‘last’ ratios and sums
Idea appears in fragmentary and unclear form in Principia
Stress on ratio rather than two evanescent quantities forming the ratio
viz. ‘slope’ calculation: “ultimate ratio of arc, chord & tangent is equality”
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
Inflection? – are we talking about technical matter or metaphysics?
Hard to shed conceptual underpinning in infinitesimal though Newton claims it does not need infinitesimally small
Emphasis on ratio rather than nascent and evanescent divisible quantities which are captured by ratio (again description or principle?). Shift between technique and metaphysics.

22. Changes between editions of Principia
Lemma II, Book II in 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' (infinitesimal magnitudes?)
Lemma II, Book II in Second edition of Principia (1713)
'Finite particles are not moments, but the very quantities generated by the moments.‘ (finite, but small, numbers?)
Recap
Fluent – variable quantity generated by motion
Fluxions / genita are rates of change of nascent or evanescent quantities
‘velocities’
Moments are evanescent variables used to express fluxions at any instant
2nd edition of Principia appears to have elements of all three interpretations contained within. Reference to Leibniz excised – ‘co-inventor’. Immediately before Scholium.

23. Scholium to Lemma XI in Book 1 of Principia
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...
Stress here on evanescence vs part/whole.

24. Scholium to Lemma XI in Book 1 of Principia (cont.)
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.

25. Berkeley’s criticisms
Are Moments Infinitesimals?
If by a Moment you mean more than the very initial Limit, it must be either a finite Quantity or an Infinitesimal. But all finite Quantities are expressly excluded from the Notion of a Moment. Therefore the Moment must be an Infinitesimal. And indeed, though much Artifice hath been employ'd to escape or avoid the admission of Quantities infinitely small, yet it seems ineffectual. For ought I see, you can admit no Quantity as a Medium between a finite Quantity and nothing, without admitting Infinitesimals. An Increment generated in a finite Particle of Time, is itself a finite Particle; and cannot therefore be a Moment. You must therefore take an Infinitesimal Part of Time wherein to generate your Moment.
... it requires a marvellous sharpness of Discernment, to be able to distinguish between evanescent Increments and infinitesimal Differences. The Analyst, XI & XVII
‘ghosts of departed quantities’ – fluxionary calculus
It must, indeed, be acknowledged, that he used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities? The Analyst, XXXV

26. Leibniz’s differential calculus
History & Philosophy of Calculus, Session 7

27. Pascal’s Triangle Pascal: triangle ADC is congruent with triangle HGI
where D cuts GI in half and AC does likewise for HG;
And HG is tangent to the circle centred on A.
Boyer: “If Pascal had at this point only been more interested in arithmetic considerations … he might have anticipated the important concept of a limit of quotient.”
Leibniz: “Pascal sometimes seemed to have had a bandage over his eyes.”

28. How leibniz adapted the triangle
As the upper triangle shrinks to zero by decreasing the ‘abscissa’:
The secant becomes equivalent to subtracting one ordinate (y2) from another (y1);
The area under the graph becomes equivalent to adding one ordinate to another.
Inverse relationship
Every curve is reconceived as infinitary polygon.

29. Pascal’s Triangle & the differential calculus
Leibniz reads Pascal in And realises:
The tangent is determined by the ratio of the differences between changes in y (ordinate) and changes in x (abscissa)
AND
That the quadrature is determined by the sum of the ordinates as infinitely thin rectangles where each base is the infinitesimal abscissa.
AND that the two processes are inverse.
Stress on changes in x and changes in y – differential calculus

30. Differential calculus
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’ Letter to Grandi (cf. Newton)

31. AcadÉmie des sciences controversy
L’Hôpital Analyse des infiniment petits (1696)
Paris Academie dispute over ‘new methods’
Rolle – main critic – advanced arguments very similar to those of Berkeley’s Analyst 30 years later
Infinitary methods are not rigorous!
“It is not enough for conclusions in geometry to be true, they must be reached well.” (Abbé Gouye)
Status of infinitesimals – no demonstration of their existence
Archimedes’s Axiom is breached
x + dx = x, so dx = 0?
Inconsistent: sometimes non-zero sometimes as zero (cf. Fermat)
Non-zero when being manipulated algebraicaly
As zero when being excluded from results
L’Hôpital initiated by Johann I Bernoulli who learnt it from Leibniz. Law of excluded midddle.

32. Leibniz’s response
Leibniz to Pinson 1701: “… in place of the infinite or of the infinitely small, one can take the quantities as great or as small as one needs so that the error be less than the given error.”
dx is a variable not a quantity! Dynamic universe not stasis of geometry
“I did not believe at all in the existence of truly infinite magnitudes or truly infinitesimal magnitudes.” Leibniz in 1715 recalling the controversy
Can treat differentials as ‘well-founded fictitious entities’ like imaginary numbers
Retreat from infinitesimal? Intimation of limits?
Disappointed, French infinitesimalists made appeal to Newton’s Lemma XI of Newton’s Principia
Wanted a ‘true metaphysics’ of infinitesimals

33. Resolving the differential mathematically
Confusion: 1712 Royal Society report into priority claims for invention of the calculus
Thought Differential Calculus of Leibniz & Newton’s method of fluxions were the same
Developments: 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 – infinitesimal with property that its powers are all equal to zero – ‘nilsquare’ property
Infinitesimal is different to zero but its powers are not (cf. Elimination of powers of ‘e’ and ‘o’)
Clarification of algebraic manipulation of new numbers – viz NSA

34. Arguments over infinitesimals
John Bell The Continuous and the Infinitesimal in Mathematics and Philosophy 2005

35. Options for interpreting differentials
dx ≈ 0 (dx is ‘indistinguishable from’ zero)
neither dx = 0 nor dx ≠ 0
dx ≠ 0, but dx2 = 0 (Nieuwentijdt)
dx → 0 (dx ‘becomes vanishingly small’ or is ‘evanescent’ – a variable quantity tending to zero)

36. conclusion
History & Philosophy of Calculus, Session 7

37. Conclusion & summary
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
2nd interpretation appears to have more metaphysical inflection
Leibniz – emphasis on algorithmic processes (‘rules for calculating’ and ‘notation’) 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