1. History & Philosophy of Calculus, Session 6 CONCEPTUAL ISSUES IN THE DEVEOPMENT OF THE CALCULUS

2.  Piecemeal technical breakthroughs over the centuries  Especially algebra & its logical power  Shifted the focus to new techniques & what 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  Infinitesimal numbers  Fermat’s e – not zero? As small as you like?  Are the techniques legitimate?  Theme – Calculus gains independence from geometry  Through new concepts OVERVIEW

3.  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?  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,” UNDERSTANDING TANGENTS

4. CALCULATING THE DERIVATIVE

5.  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  Infinitesimal number?  Wallis introduces  1/∞ - smaller than every positive fraction 1/n but not zero  Fictitious numbers? (like imaginary number √-1 or i or j)  Newton will initially in early work use o in this way  de l'Hôpital “one can take as equal two quantities differing by an infinitely small quantity.”  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' FERMAT’S E

6.  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 ‘Dichtomy’)  Use of infinitely many ‘indivisibles’ or ‘infinitesimals’ to carve up area  Larger 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) – ‘indivisibles’  Surfaces are treated as if composed of parallel lines  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 AREAS

7. 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. JOHN BELL ON INDIVISIBLES

8.  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 calculuate 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’? QUESTIONS FOR CAVALIERI ET. AL.

9.  EF is greater than FG  If we draw in all the lines parallel to EF and FG we have covered the area  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 PARADOX

10.  The shorter 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! TORRICELLI’S SOLUTION

11.  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. DEMOCRITUS’S CYLINDER

12.  Infinitesimals and Indivisibles appear central to interpreting techniques of calculus  But they lack clarity  Concepts that are not mathematical being used to interpret mathematics?  And appear in some cases to lead to contradiction and paradox SUMMARY

13. History & Philosophy of Calculus, Session 6 ATOMISM

14.  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

15.  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

16.  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

17.  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

18.  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 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

19. History & Philosophy of Calculus, Session FORCE

20.  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 FORCE AS MODERN CONCEPT

21. CONCLUSION

22.  Summary  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? SUMMARY & NEXT TIME