CONCEPTUAL ISSUES IN
THE DEVELOPMENT OF
THE CALCULUS
History &
Philosophy of
Calculus,
Session 6
OVERVIEW
 Piecemeal technical breakthroughs over the centuries
 Especially algebra & its logical power
 Shifted the focus to new techniques & what marvels could be accomplished
 Rather than what the implications or underpinnings were ...
 Boyer’s paradox – ‘inappropriate interpretations stimulated the
calculus even though they were ultimately excluded’
 Instantaneous velocity versus Aristotle
 Infinitesimal & Indivisible magnitudes
 Tangents to curves as small linear pieces of curve
 Areas composed of e.g. very small rectangles or even of lines
 Infinitesimals – numbers or variables?
 Fermat’s e – not zero? As small as you like?
 Can be manipulated algebraically (nonzero), but then discarded (zero)
 Are the techniques legitimate?
 Weierstrass – the infinitesimal is the mathematical equivalent of phlogiston
 Theme – Calculus gains independence from geometr y
 Through new concepts
 Mechanics & motion
CALCULATING THE DERIVATIVE
• Here x is horizontal and f(x) gives us the height at each point x.
• The horizontal change is h. The vertical change is f(x + h) – f(x).
• The gradient, therefore, is
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ
UNDERSTANDING TANGENTS
 Classical – tangent is a line touching curve at one point
 Can be constructed in geometry
 Early calculus:
 Torricelli – instantaneous direction of point in motion
 Barrow – ‘to every instant there corresponds some degree of velocity
which the moving body possesses’
 Contrary to Aristotle – do we then open up the problems we saw with
Zeno’s Arrow?
 Aristotle resolved paradox by banning any talk of instants as meaningless
 We need some magnitude however small ...
 Infinitesimal magnitude? (i.e. not a number)
 A linelet (Barrow)?
 A line as small as you like ... Vanishingly small ...
 de l'Hôpital “a curved line may be regarded as being made up of infinitely
small straight line segments,”
FERMAT’S E
 Error – ‘e’ - as new kind of number
 It is not zero so one can divide by it without problems
 Obeys laws of number
 But it is so small that it can be disregarded in resulting equations or ratios
 de l'Hôpital “one can take as equal two quantities differing by an infinitely small
quantity.”
 Infinitesimal number?
 Wallis introduces
 1/∞ - smaller than every positive fraction 1/n but not zero – a very small magnitude
 Eliminated by multiplying by ∞
 Fictitious numbers? Ficti or falsi (like imaginary number √-1 or i or j)
 Newton will initially in early work use o in this way
 Confusion:
 Indeterminate constant– solution to equation
 Continuous variables & functions
 Boyer:
'infinitesimals were uncritically introduced into the analysis, to become firmly
intrenched [sic] as the basis of the subject for about two centuries before giving
way, as the fundamental concept of calculus, to the rigorously defined notion of
the derivative'
AREAS
 Integration appears to depend on two problematic ideas
 The completion of an infinite sum
 (how do we complete an infinite series of steps? Cf Zeno’s ‘Dichotomy’)
 Use of infinitely many ‘indivisibles’ to carve up area
 Area is calculated by dividing into very small elements of area or even
lines
 Different mathematicians have different approaches but no method
seems to be rigorously established
 Kepler – cones are made up of infinitely thin discs
 Stevin – quadrature with very small parallelograms
 Wallis – infinitely small rectangles
 Cavalieri (from week 4) – ‘Geometry of Indivisibles’ (1635)
 Surfaces are treated as if composed of parallel lines (no thickness)
 But with circle – lines are treated like little triangles
 Solids as if composed of planes
 Indivisibles are of lower dimensionality
 But cannot work with points – no meaning can be attached to a ratio of two points
 Barrier to understanding tangents as derivatives
JOHN BELL ON INDIVISIBLES
extended entities such as lines, surfaces, and volumes prove a
much richer source of “indivisibles”.
... In the case of a straight line, such indivisibles would,
plausibly, be points; in the case of a circle, straight lines; and
in the case of a cylinder divided by sections parallel to its
base, circles.
In each case the indivisible in question is infinitesimal in the
sense of possessing one fewer dimension than its generating
figure. In the 16th and 17th centuries indivisibles in this
sense were used in the calculation of areas and volumes of
curvilinear figures, a surface being thought of as the sum of
linear indivisibles and a volume as the sum of planar
indivisibles.
LEIBNIZ PREVIEW
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
HOW LEIBNIZ ADAPTED THE TRIANGLE
As the upper triangle
shrinks to zero by
decreasing the
‘abscissa’:
1. The secant
becomes
equivalent to
subtracting one
ordinate (y2) from
another (y1);
2. The area under
the graph
becomes
equivalent to
adding one
ordinate to
another.
Inverse relationship
QUESTIONS &
PARADOXES
QUESTIONS FOR CAVALIERI ET. AL.
 How can the sum of infinitesimals or indivisibles add up to a
finite quantity?
 If parallel lines have no width how can we use them to calculate area
of plane?
 How can a sum of planes with no thickness produce a volume?
 Boyer:
 ‘Cavalieri failed to provide a convincing answer to this question,
sidestepping it with the response that, while the indivisibles are
correctly regarded as lacking thickness or breadth, nevertheless one
“could substitute for them small elements of area and volume in the
manner of Archimedes”.’
 Simply the ancient ‘method of exhaustion’?
DEMOCRITUS’S CYLINDER
If a cone is cut by sections parallel to its base, are we to say
that the sections are equal or unequal?
If we suppose that they are unequal, they will make the
surface of the cone rough and indented by a series of steps.
If the surfaces are equal, then the sections will be equal and
the cone will become a cylinder, being composed of equal,
instead of unequal, circles. This is a paradox.
GALILEO’S 1 ST PARADOX
Every radius determines a
point on each of the
concentric circles.
Therefore there are as
many points on the
circumference of the
bigger circle as on the
smaller.
Indivisibles are collections
of all the points, no?
Does this mean the
circumferences are of
equal length?
Cavalieri – the continuum
is not composed of
indivisibles and …
TORRICELLI’S PARADOX
 EF is greater than FG
 If we draw in all the lines parallel to EF and FG we have
covered the area of the rectangle
 In each case the line parallel to EF is greater than the
corresponding line parallel to FG
 Therefore triangle ABC is greater in area than triangle ACD
TORRICELLI’S SOLUTION
 The shor ter lines parallel to FG are wider than the lines parallel to EF!
 The indivisibles are not equal to each other in width
 The ratio of the rectangles sides determines the width of the indivisibles!
 You must compare like ratios with like: find the equivalent length to EF from AC
to CD (parallel to AD) and so on
 Cavelieri – “oblique transit”
 You must “cut” in such a way as to avoid problems of density
 Same problem applied to Galileo’s circles!
GALILEO’S 2 ND PARADOX (1634 LETTER
TO CAVALIERI)
By Pythagoras’s Theorem, the area
between the circumferences of the
red and green circles must equal π
* AE2 : the area of the purple circle.
(Red circle has centre E and radius
DE – which must equal red line AF )
As AE can be reduced to
infinitesimal – what does that mean
for white area inside the red circle,
given dimensionality issue?
When A coincides with E, does area
of point equal the area of the
circumference? Are all
circumferences of all circles the
same size?
Cavalieri: they are all zero! But they
are not indivisibles.
SUMMARY
 Infinitesimals and Indivisibles appear central to interpreting techniques
of calculus
 They stimulated new discoveries & insights
 But they lack clarity
 Concepts that are not mathematical being used to interpret mathematics?
 How do we deal with the problem of dimensionality?
 e.g. “the last indivisible” in Galileo’s paradox
 Cavalieri later concedes that the first and last indivisible should be included in the aggregates
to calculate area
 This idea of ‘first and last’ indivisible will have an echo in Newton’s interpretations of the
calculus
 And appear in some cases to lead to contradiction and paradox
 Is the method reliable as proof?
 Taquet in 1651:
 “To be sure, geometers admit that a line can be generated by the flowing of a point
[ex fluxi puncti] and a surface by a line … . But it is quite different to say that a
quantity is produced by the flowing of an indivisible, and to say that it is composed
by indivisibles. The former is an indisputable proposition. The latter is so opposed
to geometry … that it must be destroyed by geometry itself. Even if the inventions
are beautiful … ”
 Leibniz – points are not indivisibles: only infinitely small lines can be used as
indivisibles, because they are still divisible ?!?!?
 Metaphysics appear s ineliminable …
ATOMISM
History &
Philosophy
of
Calculus,
Session 6
ANCIENT GREEK THOUGHT
 Problem of how to interpret nascent calculus is overlaid by
inheritance from Greek Thought
 Status of Geometr y 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 17 th century: Torricelli & Pascal conducted experiments with mercury to
prove possibility of void or vacuum
 Robert Boyle & the air-pump
ATHEISM OF ATOMISTS
 Contingency of cosmos
 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
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.
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
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.
FORCE
History &
Philosophy
of
Calculus,
Session
FORCE AS MODERN CONCEPT
 Ancient & Medieval Philosophy dominated by
 Form and matter
 Latitude of forms
 Hot /cold
 At rest / in motion
 Substance and attributes
 Forces acting on bodies
 Gravity & acceleration
 Velocitas & velocitatio
 Galileo’s inclined plane experiments vs thought experiments
 Curve of moving body produced by effects of different forces
 Vectors & tangents – representing motion
 Euler 1760
 Force is a property of matter that means that one body can change the state of
another
 New inclination towards physics
 Meaning authority of geometry & Euclid might be circumvented
CONCLUSION
SUMMARY & NEXT TIME
 Summar y
 Calculus seems to demand some concept of the infinitesimal
 Magnitude
 New number
 Or both?
 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?
 Next time: how did Newton & Leibniz understand their formalisation of
the calculus?
 Did they overcome the problems outlined here around infinitesimal numbers &
magnitudes?
 How did they think through the implications of their technical innovations in
relation to metaphysical questions?