Infinity. Newton, Leibniz & the Calculus

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1 Infinity Newton, Leibniz & the Calculus

2 Aristotle: Past time can t be infinite because there can t be an endless chain of causes (movements) preceding the present. Descartes: Space as extension; the res extensa make up space. Time as movement. Hence no spatial vacuum and no eventless time. To move, objects must somehow squeeze past each other. Newton: Space &time as empty containers that exist independently of matter and motion and instead provide the stage they occupy. Absolute vs Relative Space

3 Cartesian Coordinates

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5 The 1671 Method of Fluxions was written in Latin and never published until after his death (when it was published in English). It speaks freely of indefinitely little quantities of both space and time and uses them in calculations. By the Principia (1687) he already has something like the idea of limit in the form of the ultimate ratio. But the Principia wasn t written in the new language of the calculus, which his contemporaries wouldn t have understood. The infinitesimals persisted in later work, and drew criticism. Newton s Calculus

6 In Newton s terminology a fluent is a thing that changes over time, and its fluxion is the way in which it changes. Example: the height of a thrown ball over time is a fluent; its velocity (how the height changes over time) is its fluxion. These terms haven t passed into mainstream mathematical usage In the Method of Fluxions he is clear that there are two key and complementary problems: Given a fluent, find its fluxion: this is the problem solved by differentiation. Given a fluxion, find its fluent, which will be solved by integration. Fluents & Fluxions

7 65. And God, the author of Nature, was able to carry out this divine and infinitely marvellous artifice because every portion of matter is not only divisible to infinity, as the ancients realised, but is actually sub-divided without end, every part divided into smaller parts, each one of which has some motion of its own rather than having only such motion as it gets from the motion of some larger lump of which I is a part. Without this infinite dividedness it would be impossible for each portion of matter to express the whole universe. 66. And from this we can see that there is a world of creatures of living things and animals, entelechies and souls in the smallest fragment of matter. 67. Every portion of matter can be thought of as a garden full of plants or a pond full of fish. But every branch of the plant, every part of the animal (every drop of its vital fluids, even) is another such garden or pond. 68. And although the earth and air separating the plants in the garden and the water separating the fish in the pond are not themselves plants or fish, they contain other organisms, but usually ones that are too small for us to perceive them. Leibniz s Monadology

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9 70. [When dividing a line, o]ne imagines a final end, a number that is infinite, or infinitely small; but that is all simple fiction. Every number is finite and specific; every line is so likewise, and the infinite or infinitely small signify only magnitudes that one may take as great or as small as one wishes, to show that an error is smaller than that which has been specified, that is to say, that there is no error; or else by the infinitely small is meant the state of a magnitude at its vanishing point or its beginning, conceived after the pattern of magnitudes already actualized. Leibniz on Infinitesimals

10 Natura non facit saltus: Nature does not make leaps. Even apparently instantaneous changes are really continuous. From Aristotle, we know a continuum can t be composed of its infinitesimal parts. But Leibniz held that it also can t be a simple unity precisely because it s divisible. This leads him to conclude that any continuum, along with its infinitesimals, are purely ideal, not real. They re tools we need to think with but don t correspond directly to anything actual. Leibniz is here developing a calculus: a method of reckoning with the infinite, not a metaphysics. Leibniz on Continua

11 Gradient (Roughly Speaking) The gradient of a slope is a measure of how fast height is changing as you walk forwards. How far up have I gone for each unit I ve gone forward? The sign says that, on the upcoming slope, I ll go up 10% of the amount I go forward. E.g. I go up 1m for every 10m I go forward. 1/10 (one tenth) is 10%

12 Instantaneous Change

13 Instantaneous Change

14 The gradient from where I m standing now to the summit is defined as the vertical distance ( rise ) divided by the horizontal distance ( run ). This line is called a secant line. Instantaneous Change

15 But does it make sense to talk about the gradient at the point where I m standing right now? Remember the gradient is how steep the slope is but also how fast the height is changing. Can this latter concept have meaning at a single point? Instantaneous Change

16 Here s one way to try. Instead of looking at the gradient from me to the summit, move the point at the summit closer and closer to me. What happens when it reaches me? Instantaneous Change

17 Instantaneous Change

18 As the point moves closer and closer to me, the slope of the line gets closer and closer to the true gradient where I m standing. It will then be the tangent to the curve roughly speaking, the straight line that just touches it at a single point. Its slope will then be the gradient the rate of change of height at that point. Gradient (More Precisely Speaking)

19 Here s one way to try. Instead of looking at the gradient from me to the summit, move the point at the summit closer and closer to me. What happens when it reaches me? Instantaneous Change

20 We re now looing at a graph of the height of the ball over time (NB the ball is being thrown directly upwards!). The slope of the gradient line can be found in exactly the same way and represents the rate of change of height over time, that is, the velocity of the ball. Instantaneous Change

21 The gradient at a point seems to be the vertical change divided by the horizontal change. But at a point the horizontal change seems to be zero. We can t divide by zero! In fact the vertical change ought to be zero, too. Although it makes geometric sense, the process just discussed looks like mathematical nonsense. Early analysts: The changes aren t zero, but are infinitesimal. Berkeley s reply: These are the ghosts of departed quantities an absurd fiction. We must choose one side of Zeno s paradoxes or the other: either the smallest changes are finite amounts, in which case we can t speak of a gradient at a point or at an instant, or they have zero extension. Oops

22 Long ago, Archimedes had used a clever method to find an approximate value for the area of a given circle. This ancient problem squaring the circle is now known to be unsolvable using the standard methods of Euclid. Archimedes did it by approximating the circle with polygons and arguing correctly that if you add enough edges you can get as close as you like to the circle. Method of Exhaustion

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24 ½ 0.5 ½ + ¼ 0.75 ½ + ¼ + 1/ ½ + ¼ + 1/8 + 1/ ½ + ¼ + 1/8 + 1/16 + 1/ ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/ Similarly, the person in Zeno s dichotomy never completes her journey, but she does get closer and closer to completing it. In particular, if she takes enough steps she ll be as close as we like to having completed it. Recall Leibniz: the infinite or infinitely small signify only magnitudes that one may take as great or as small as one wishes, to show that an error is smaller than that which has been specified Zeno s Dichotomy

25 This suggests the idea of an infinite limit. For example: Limits 1 lim n n = 0 Our intuition here is that 1/n gets smaller and smaller closer and closer to 0 as n gets bigger. We can make 1/n as close to 0 as we like just by making n very big. This does NOT commit us to saying something mathematically silly like 1 = 0 We can countenance a process that goes on as long as we like (the potential infinite) but not an actual infinite quantity.

26 Now we can define the gradient or rate of change at a point or instant, which we call the derivative. Suppose t is some increasing quantity (e.g. time) and f(t) assigns a number (e.g. height) to each value of t. Say t increases by an infinitesimal amount, dt, and f(t) by the infinitesimal df. Then we have: df dt = lim f t + dd f(t) dd 0 dd The thing on the left is the symbol for the rate of change of f with respect to t. The thing on the right is a limit The Derivative

27 It seems as if the limit gives us a way to pass from the finite to the infinite. But there are still philosophical and mathematical problems here. These were really discovered and investigated in the later 18 th and early 19 th century. The result was a much more rigorous and careful set of definitions. The cost is it s also more abstract, more intellectually difficult and so fewer people learn it. Today this is done in terms of topology, which has its own definition of continuity. and the philosophical worries still haven t entirely gone away, especially in connection with the actuality of continua.

28 The continuum any continuum should be representable in terms of numbers. But which numbers? In the nineteenth century the real numbers were redefined as the limits of those sequences that ought to have limits (technically, Cauchy sequences ). For example, we can use Archimedes method as the definition of pi. The rational numbers have gaps (like root 2), the real numbers have no gaps by definition; they are complete (another technical term). The success of physics based on this system has led many to believe that the fine structure of the universe must somehow match this very abstract definition.

29 FURTHER EXAMPLES

30 Two curves are transversal if they touch and no arbitrarily small displacement can separate them. The curve and line at the top are not transversal, since the dotted line shows how to separate them. Those in the bottom image are. Imagine the top curve moving slowly downwards. It starts not touching the line and ends crossing it transversally in two places. By continuity, there must have been a moment when it just touched the line. A Geometric Example

31 In the real world of sense perceptions and physical measurements, no continuous quantity or [ ] relationship is ever perfectly determined. The only physically meaningful properties of a mapping, consequently, are those that remain valid when the map is slightly deformed. Such properties are stable properties [ ] Transversality, a notion that at first appears unintuitively formal, is all we can really experience. -- Guillemin & Pollack, Differential Topology

32 Here s a sequence of numbers: 0.9, 0.99, 0.999, , , , , Write x(n) for the n th number on this list and, evidently, lim x n = 1 n So is a zero followed by an infinite number of nines really equal to 1? In all but the most exotic domains of mathematics, the answer is yes. If you want to say no, you must say by what finite amount the two numbers differ! A Numerical Example

33 APPENDIX Some mathematical details for those that want them

34 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

35 2 F(x) = x

36 The only way to see that this works is by calculating a derivative, so with apologies Suppose f(t) = t 2 : dd dt = lim f t + dd f(t) dd 0 dd Definition of the derivative. = lim dd 0 (t + dd) 2 t 2 dd = lim dd 0 t dd t + (dd) 2 t 2 dd = lim dd 0 2 dd t + (dd) 2 dd = lim dd 0 2t + dd = 2t Replace f with what it actually does. Multiply out the bracket. The x 2 and x 2 cancel each other out. The h on top and bottom cancel out. Now we re not dividing by h any more, we can let it go to 0 and see the result.