The Infinitesimal Calculus
I regret not spending any time Wednesday discussing the first of the selections from Child, "The elements of the new calculus for differences and sums...." This is apparently only a draft (I am trying to track down the source--Child is hopeless as an editor), but it matches fairly closely Bos' account of Leibniz's first steps. I find it revealing of the style of his mathematical reasoning (the combination of traditional geometry and infinite series, the reliance on the imagination when it comes to the infinitely small). Below I raise a couple of questions about which I am still not clear, but mainly I just thought it might be useful to work step-by-step through the arguments.
1. Leibniz ties the calculus very closely to his work on series. On p. 137 he states that "to find the differences of series is to find tangents." This becomes clearer when we turn to "The fundamental principle of the calculus," appearing at the end of the document (pp. 142f). Here he defines dx and ∫x as variables that take as values successive terms in a difference or sum series, respectively. But they also can be understood, in effect, as functions that take as arguments successive terms in a given series and deliver as values the difference or sum of those terms. On either interpretation, they are inverse operations: ∫ dx = d ∫x = x.
One of the neat things about this is that dx is not initially connected to the idea of the infinitely small. But it acquires that connection when Leibniz attempts to state rules for taking the differences of products and quotients. In the former case dxy is interpreted as the difference between two successive xy's--one designated as xy, the other as (x + dx)(y + dy). This gives us:
1. Leibniz ties the calculus very closely to his work on series. On p. 137 he states that "to find the differences of series is to find tangents." This becomes clearer when we turn to "The fundamental principle of the calculus," appearing at the end of the document (pp. 142f). Here he defines dx and ∫x as variables that take as values successive terms in a difference or sum series, respectively. But they also can be understood, in effect, as functions that take as arguments successive terms in a given series and deliver as values the difference or sum of those terms. On either interpretation, they are inverse operations: ∫ dx = d ∫x = x.
One of the neat things about this is that dx is not initially connected to the idea of the infinitely small. But it acquires that connection when Leibniz attempts to state rules for taking the differences of products and quotients. In the former case dxy is interpreted as the difference between two successive xy's--one designated as xy, the other as (x + dx)(y + dy). This gives us:
dxy = ((x + dx)(y + dy)) - xy = x dy + y dx + dxdy.
Then Leibniz declares that since "the quantity dx dy is infinitely small in comparison with the rest, for it is supposed that dx and dy are infinitely small (because the lines are understood to be continuously increasing or decreasing by very small increments throughout the series of terms)," we can drop the last term, giving us a version of the familiar product rule:
dxy = x dy + y dx.
The same move is made in the derivation of the quotient rule, with x dx being dropped "as being infinitely small in comparison with xx."
What Leibniz does here is reasonable given that the differences and sums he wants to calculate are those associated with a continuous series (as opposed to a discrete series of natural numbers). All of this suggests, however, that pace Leibniz himself, we are able to reason in a coherent way about the infinitely small. He knows enough to know when it is permissible to omit such a term as negligible and when it is not. Question: can this sort of reasoning be re-expressed in way that is consistent with the syncategorematic interpretation of the infinite?
2. Jump back to p. 137. It helps to rotate the figure 90 degrees anti-clockwise (as Michael did), giving us AB as the x axis and the BC's as ordinates perpendicular to it. The problem is to find a general expression for the tangent of the curve at any point C. Leibniz begins by constructing two difference series. The first gives successive differences of the abscissae, dx:
1C1D = A2B - A1B
2C2D = A3B - A2B
3C3D = A4B - A3B
etc.
The second gives successive differences of the ordinates, dy:
1D2C = 2B2C - 1B1C
2D3C = 3B3C - 2B2C
3D4C = 4B4C - 3B3C
etc.
Then comes the critical move:
3. Leibniz interprets the relation (of equality?) between these finite and infinitesimal ratios as a general result that can be applied to find the tangent of any curve for which we can determine dy. This is shown with the example of a rectangular hyperbola, whose equation is y = aa/x (turn the diagram at the top of p. 138 upside down and think of it as the upper left quadrant of a Cartesian coordinate system). The quotient rule (established on p. 143) gives us:
dy = - (aa/xx) dx.
5. A final thought: there seem to be two distinct kinds of cases in which Leibniz supports his reasoning about the continuum by appeal to the "principle of continuity": 1) in the physical world, things are properly discontinuous (they are divided into contiguous discrete parts ad infinitum), yet we represent them as if they were continuous; 2) in the case of truly continuous objects such as mathematical continua, our reasoning about them (at least as Leibniz conceives it) assumes that we can neglect the infinitesimal difference between a continuous curve and its infinite polygonal approximation. Is this in effect to say that there is no difference between the two, i.e., they are identical? Or is this just a simplifying assumption we make because it makes things work out "smoothly"? This is the main topic for next week's meeting on the principle of continuity. (See, e.g., what Leibniz says in the Reply to Nieuwentijt at the bottom of p. 151.)
What Leibniz does here is reasonable given that the differences and sums he wants to calculate are those associated with a continuous series (as opposed to a discrete series of natural numbers). All of this suggests, however, that pace Leibniz himself, we are able to reason in a coherent way about the infinitely small. He knows enough to know when it is permissible to omit such a term as negligible and when it is not. Question: can this sort of reasoning be re-expressed in way that is consistent with the syncategorematic interpretation of the infinite?
2. Jump back to p. 137. It helps to rotate the figure 90 degrees anti-clockwise (as Michael did), giving us AB as the x axis and the BC's as ordinates perpendicular to it. The problem is to find a general expression for the tangent of the curve at any point C. Leibniz begins by constructing two difference series. The first gives successive differences of the abscissae, dx:
etc.
The second gives successive differences of the ordinates, dy:
etc.
Then comes the critical move:
"If now these dx and dy are taken to be infinitely small, or the two points on the curve are understood to be at a distance apart that is less than any given length... then it is plain that the straight line joining these two points, 2C1C say, (which is an elements of the curve or a side of the infinite-angled polygon that stands for the curve)..."
With this imaginative leap, we form the hypotenuse of the characteristic triangle 1D1C2C, whose sides are infinitesimal values of dx and dy. The parenthetical comment is interesting, for in it Leibniz seems to allow that the straight line joining the two points is not properly a part of CC, whose curvature is continuous, but a side of an infinite-angled polygon that models CC. So, one question to ask here is: in the case of a mathematical figure (as opposed to a physical motion), what is the relation between a continuous curve and a representation of it in terms of an infinity of infinitesimal straight lines: identity or approximation? And, does Leibniz have a satisfactory way of re-phrasing this kind of (loose?) talk of infinity?
Continuing with the argument: extend 2C1C until it intersects the axis at 1T. This gives the tangent to the curve at 1C, as well as the hypotenuse of the triangle 1B1T1C. Since this triangle is similar to the characteristic triangle (not proved here), we have a pair of proportionalities between their sides. If t is the base of the larger triangle and y its adjacent side, then t : y :: dx : dy.3. Leibniz interprets the relation (of equality?) between these finite and infinitesimal ratios as a general result that can be applied to find the tangent of any curve for which we can determine dy. This is shown with the example of a rectangular hyperbola, whose equation is y = aa/x (turn the diagram at the top of p. 138 upside down and think of it as the upper left quadrant of a Cartesian coordinate system). The quotient rule (established on p. 143) gives us:
Then comes a series of substitutions. Substitute for dy in dx : dy, cancel the dx's, and multiply both sides by -1, gives -xx : aa. Rearranging this and substituting y for aa/x (from our original equation), gives -x : y. From this, finally, Leibniz concludes t = -y. Is this a mistake? Should this be t = -x, derived from the pair of proportionalities -x : y and t : y?
4. Leibniz then goes on to illustrate the method of quadrature using sum series (Michael discussed this at length). Here I would raise the same question I raised above about approximation, based on passages like this;
"Also I represent the area of a figure by the sum of all the rectangles contained by the ordinates and the differences of the abscissae, i.e. by the sum 1B1D + 2B2D + 3B3D + etc. For the narrow triangles 1C1D2C, 2C2D3C, etc., since they are infinitely small compared with the said rectangles, may be omitted without risk; and thus I represent in my calculus the area of the figure by ∫y dx, or the sum of the rectangles contained by each y and the dx that corresponds to it...."
4. Leibniz then goes on to illustrate the method of quadrature using sum series (Michael discussed this at length). Here I would raise the same question I raised above about approximation, based on passages like this;
"Also I represent the area of a figure by the sum of all the rectangles contained by the ordinates and the differences of the abscissae, i.e. by the sum 1B1D + 2B2D + 3B3D + etc. For the narrow triangles 1C1D2C, 2C2D3C, etc., since they are infinitely small compared with the said rectangles, may be omitted without risk; and thus I represent in my calculus the area of the figure by ∫y dx, or the sum of the rectangles contained by each y and the dx that corresponds to it...."
5. A final thought: there seem to be two distinct kinds of cases in which Leibniz supports his reasoning about the continuum by appeal to the "principle of continuity": 1) in the physical world, things are properly discontinuous (they are divided into contiguous discrete parts ad infinitum), yet we represent them as if they were continuous; 2) in the case of truly continuous objects such as mathematical continua, our reasoning about them (at least as Leibniz conceives it) assumes that we can neglect the infinitesimal difference between a continuous curve and its infinite polygonal approximation. Is this in effect to say that there is no difference between the two, i.e., they are identical? Or is this just a simplifying assumption we make because it makes things work out "smoothly"? This is the main topic for next week's meeting on the principle of continuity. (See, e.g., what Leibniz says in the Reply to Nieuwentijt at the bottom of p. 151.)
posted by Don at 1:11 PM
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