return to homepage A REDEFINITION First draft begun December 2002. First finished draft May 2003 (in my files). That draft submitted to American Mathematical Society August 2003. This is the extended draft of 2004, updated several times since then.
In this paper I will prove that the invention of the calculus using infinite series and its subsequent interpretation using limits were both errors in analyzing the given problems. In fact, as I will show, they were both based on the same conceptual error: that of applying diminishing differentials to a mathematical curve (a curve as drawn on a graph). In this way I will bypass and ultimately falsify both standard and nonstandard analysis.
To prove this I must first provide the groundwork for my theory by analyzing at some length a number of simple concepts that have not received much attention in mathematical circles in recent years. Some of these concepts have not been discussed for centuries, perhaps because they are no longer considered sufficiently abstract or esoteric. One of these concepts is the cardinal number. Another is the cardinal (or natural) number line. A third is the assignment of variables to a curve. A fourth is the act of drawing a curve, and assigning an equation to it. Were these concepts still taught in school, they would be taught very early on, since they are quite elementary. As it is, they have mostly been taken for granted—one might say they have not been deemed worthy of serious consideration since the fall of Athens. Perhaps even then they were not taken seriously, since the Greeks also failed to understand the curve—as their use of an instantaneous velocity makes clear. This is of utmost importance for several reasons. The most critical reason is that once you draw a graph, you must assign variables to the axes. Let us say you assign the axes the variables x and y, as is most common. Now, you must define your variables. What do they mean? In physics, such a variable can mean either a distance or a point. What do your variables mean? No doubt you will answer, "my variables are points." You will say that x stands for a point x-distance from the origin. You will go on to say that distances are specified in mathematics by Δx (or some such notation) and that if x were a Δx you would have labeled it as such. I know that this has been the interpretation for all of history. But it turns out that it is wrong. You build a graph so that you can assign numbers to your variables at each point on the graph. But the very act of assigning a number to a variable makes it a distance. You cannot assign a counting number to a point. I know that this will seem metaphysical at first to many people. It will seem like philosophical mush. But if you consider the situation for a moment, I think you will see that it is no more than common sense. There is nothing at all esoteric about it. Let us say that at a certain point on the graph, y = 5. What does that mean? You will say it means that y is at the point 5 on the graph. But I will repeat, what does that mean? If y is a point, then 5 can’t belong to it. What is it about y that has the characteristic “5”? Nothing. A point can have no magnitudes. The number belongs to the graph. The “5” is counting the little boxes. Those boxes are not attributes of y, they are attributes of the graph. You might answer, “That is just pettifoggery. I maintain that what I meant is clear: y is at the fifth box, that is all. It doesn’t need an explanation.” But the number “5” is not an ordinal. By saying “y is at the fifth box” you imply that 5 is an ordinal. We have always assumed that the numbers in these equations are cardinal numbers numbers [I use “cardinal” here in the traditional sense of cardinal versus ordinal. This is not to be confused with Cantor’s use of the term cardinal]. The equations could hardly work if we defined the variables as ordinals. The numbers come from the number line, and the number line is made up of cardinals. The equation y = x² @ x = 4, doesn’t read “the sixteenth thing equals the fourth thing squared.” It reads “sixteen things equals four things squared.” Four points don’t equal anything. You can’t add points, just like you can’t add ordinals. The fifth thing plus the fourth thing is not the ninth thing. It is just two things with no magnitudes. The truth is that variables in mathematical equations graphed on axes are cardinal numbers. Furthermore, they are delta variables, by every possible implication. That is, x should be labeled Δx. The equation should read Δy = Δx². All the variables are distances. They are distances from the origin. x = 5 means that the point on the curve is fives little boxes from the origin. That is a distance. It is also a differential: x = (5 – 0).Think of it this way. Each axis is a ruler. The numbers on a ruler are distances. They are distances from the end of the ruler. Go to the number “1” on a ruler. Now, what does that tell us? What informational content does that number have? Is it telling us that the line on the ruler is in the first place? No, of course not. It is telling us that that line at the number “1” is one inch from the end of the ruler. We are being told a distance. You may say, “Well, but even if it is a distance, your number “5” still applies to the boxes, not to the variable. So your argument fails, too.” No, it doesn't. Let’s look at the two possible variable assignments: x = five little boxes or Δx = five little boxes The first variable assignment is absurd. How can a point equal five little boxes? A point has no magnitude. But the second variable assignment makes perfect sense. It is a logical statement. Change in x equals five little boxes. A distance is five little boxes in length. If we are physicists, we can then make those boxes meters or seconds or whatever we like. If we are mathematicians, those boxes are just integers. You can see that this changes everything, regarding a rate of change problem. If each variable is a delta variable, then each point on a curve is defined by two delta variables. The point on the curve does not represent a physical point. Neither variable is a point in space, and the point on the graph is also not a point in space. This is bound to affect applying the calculus to problems in physics. But it also affects the mathematical derivation. Notice that you cannot find the slope or the velocity at some point (x, y) by analyzing the curve equation or the curve on the graph, since neither one has a point x on it or in it. I have shown that the whole idea is foreign to the preparation of a graph. No point on the graph stands for a point in space or an instant in time. No point on any possible graph can stand for a point in space or an instant in time. A point graphed on two axes stands for two distances from the origin. To graph a line in space, you would need one axis. To graph a point in space, you would need zero axes. You cannot graph a point in space. Likewise, you cannot graph an instant in time. Therefore, all the machinations of calculus, all the dx's and dy's and limits, are not applicable. You cannot let x go to zero on a graph, because that would mean you were really taking Δx to zero, which is either meaningless or pointless. It either means you are taking Δx to the origin, which is pointless; or it means you are taking Δx to the point x, which is meaningless (point x does not exist on the graph--you are postulating making the graph disappear, which would also make the curve disappear). In its own way, the historical derivation of the derivative sometimes understands and admits this. Readers of my papers like to send me to the epsilon/delta definition, as an explanation of the limit concept. The epsilon/delta definition is just this: For all ε>0 there is a δ>0 such that whenever |x - x|<_{0}δ then |f(x) - y. What I want to point out is that |_{0}|<εx - x| is not a point, it is a differential. The epsilon/delta definition is sometimes simplified as "Whatever number you can choose, I can choose a smaller one." Which might be modified as "You can choose a point as near to zero as you like; but I can choose a point even nearer." But this is not what the formal epsilon/delta proof states, as you see. The formal proof defines both epsilon and delta as differentials. In physics or applied math, that would be a length. Stated in words, the formal epsilon/delta definition would say, "Whatever _{0}length you choose, I can choose a shorter one." Epsilon/delta is dealing with lengths, not points. If you define your numbers or variables or functions as lengths, as here, then you cannot later claim to find solutions at points. If you are taking differentials or lengths to limits, then all your equations and solutions must be based on lengths. You cannot take a length to a limit and then find a number that applies to a point. Currently, the calculus uses a proof of the derivative that takes lengths to a limit, as with epsilon/delta. But if you take lengths to a limit, then your solution must also be a length. If you take differentials to a limit, your solution must be a differential. Which is all to say that the calculus contains no points. It contains differentials only. That is why it is called the differential calculus. All variables and functions in equations are differentials and all solutions are differentials. The only possible point in calculus is at zero, and if that limit is reached then your solution is zero. You cannot find numerical solutions at zero, since the only number at zero is zero. If this is all true, how is it possible to solve a calculus problem? The calculus has to do with instants and instantaneous things and infinitesimals and limits and near-zero quantities, right? No, the calculus initially had to do with solving areas under curves and tangents to curves, as I said above. I have shown that a curve on a graph has no instants or points on it, therefore if we are going to solve without leaving the graph, we will have to solve without instants or infinitesimals or limits. It is also worth noting that finding an instantaneous velocity appears to be impossible. A curve on a graph has no instantaneous velocities on it anywhere—therefore it would be foolish to pursue them mathematically by analyzing a curve on a graph. To sum up: You cannot analyze a curve on a graph to find an instantaneous value, since there are no instantaneous values on the graph. You cannot analyze a curve off a graph to find an instantaneous value, since a curve off a graph has a different shape than the same curve on the graph. It is a different curve. The given curve equation will not apply to it. Some will say here, “There is a simple third alternative to the two in this summation. Take a curve off the graph, a physical curve—like that bug crawling or your brother in his car—and assign the curve equation to it directly. Do not import some curve equation from a graph. Just get the right curve equation to start with.” First of all, I hope it is clear that we can’t use the car as a real-life curve equation, since it is not curving. How about the bug? Again, three variables where we need two. Won’t work. To my dissenters I say, find me a physical curve that has two variables and I will use the calculus to analyze it with you without a graph. They simply cannot do it. It is logically impossible. One of the dissenters may see a way out: “Take the bug’s curve and apply an equation to it, with three variables, x, y and t. The t variable is not a constant, but its rate of change is a constant. Time always goes at the same rate! Therefore we can cancel it and we are back to the calculus. What is happening is that the calculus curve is just a simplification of this curve on three axes.” To this I answer, yes, we can use three axes, but I don’t see how you are going to apply variables to the curve without putting it on the graph. Calculus is worked upon the curve equation. You must have a curve equation in order to find a derivative. To discover a curve equation that applies to a given curve, you must graph it and plot it. The dissenter says, “No, no. Let us say we have the equation first. We are given a three-variable curve equation, and we just draw it on the wall, like the bug did. Nothing mysterious about that.” I answer, where is the t axis, in that case? How are you or the bug drawing the t component on the wall? You are not drawing it, you are ignoring it. In that case the given curve equation does not apply to the curve you have drawn on the wall, it applies to some three-variable curve on three axes. The dissenter says, “Maybe, but the curvature is the same anyway, since t is not changing.” I say, is the curve the same? You may have to plot some “points” on a three dimensional graph to see it, but the curve is not the same. Plot any curve, or even a straight line on an (x, y) graph. Now push that graph along a t axis. The slope of the straight line decreases, as does the curvature of any curve. Even a circle is stretched. This has to affect the calculus. If you change the curve you change the areas under the curve and the slope of the tangent at each point. The dissenter answers, “It does not matter, since we are getting rid of the t-axis. We are going to just ignore that. What we are interested in is just the relationship of x to y, or y to x. It is called a function, my friend. If it is a simplification or abstraction, so what? That is what mathematics is.” To that I can only answer, fine, but you still haven’t explained two things. 1) If you are talking of functions, you are back on the two-variable graph, and your curve looks the way it does only there. To build that graph you must assign an origin to the movement of your little bug, in which case your two variables become delta variables. In which case you have no points or instants to work on. The calculus is useless. 2) Even if you somehow find values for your curve, they will not apply to the bug, since his curve is not your curve. His acceleration is determined by his movement in the continuum x, y, t. You have analyzed his movement in the continuum x, y which is not equivalent. The dissenter will say, “Whatever. Apply my curve to your brother’s car, if you want. It does not matter what his tire tracks look like. What matters is the curve given by the curve equation. An x, t graph will then be an abstraction of his motion, and the values generated by the calculus on that graph will be perfectly applicable to him.” I answer that we are back to square one. You either apply the calculus to the real-life curve, where there are points in space, or you apply it to the curve on the graph, where there are not. In real life, where there are points, there is no curvature. On the graph, where there is curvature, there are no points. If my dissenter does not see this as a problem, he is seriously deluded.
Let us take a short break from this groundwork and return to the history of the calculus for just a moment. Two mathematicians in history came nearest to recognizing the difference between the mathematical point and the physical point. You will think that Descartes must be one, since he invented the graph. But he is not. Although he did much important work in the field, his graph turned out to be the greatest obstruction in history to a true understanding of the problem I have related here. Had he seen the operational significance of all diagrams, he would have discovered something truly basic. But he never analyzed the fields created by diagrams, his or any others. No, the first to flirt with the solution was Simon Stevin, the great Flemish mathematician from the late 16th century. He is the person most responsible for the modern definition of number, having boldly redefined the Greek definitions that had come to the “modern” age via Diophantus and Vieta.
Now let us return to the groundwork. The next stone I must lay concerns rate of change, and the way the concept of change applies to the cardinal number line. Rate of change is a concept that is very difficult to divorce from the physical world. This is because the concept of change is closely related to the concept of time. This is not the place to enter a discussion about time; suffice it to say that rate of change is at its most abstract and most mathematical when we apply it to the number line, rather than to a physical line or a physical space. But the concept of rate of change cannot be left undefined, nor can it be taken for granted. The concept is at the heart of the problem of the calculus, and therefore we must spend some time analyzing it.
With this established I am finally ready to unveil my algorithm. We have a tight definition of a rate of change, we have our variable assignments clearly and unambiguously set, and we have the necessary understanding of the number line and the graph. Using this information we can solve a calculus problem without infinite series or limits. All we need is this beautiful table that I made up just for this purpose. I have scanned the math books of history to see if this table turned up somewhere. I could not find it. It may be buried out there in some library, but if so it is unknown to me. I wish I had had it when I learned calculus in high school. It would have cleared up a lot of things.
Now, let’s see what the current value for the derivative is telling us, according to my chart. If we have a curve equation, say
We have solved the first part of our problem. We have found the derivative without calculus and have assigned its value to the general equation for the slope of the tangent to the curve. Now we must ask whether we can assign this equation to the velocity at all "points on the curve". This is no longer a math question. It is a physics question. The answer appears to be "yes."
ΔΔt/ΔΔx = ΔΔt = (Δt)'
As a final step, let me show that the second derivative is also not found at an instant. There is no such thing as an instantaneous acceleration, any more than there is an instantaneous velocity. What we seek for the acceleration at the point on the graph is this equation:
Δt'' = ΔΔΔt/ΔΔx
The proof is complete. Newton’s analysis was wrong, and so was Leibniz’s. No fluxions are involved, no vanishing values, no infinitesimals, no indivisibles (other than zero itself). Nothing is taken to zero. No denominator goes to zero, no ratio goes to zero. Infinite progressions are not involved. Even Archimedes was wrong. Archimedes invented the problem with his analysis, which looked toward zero 2200 years ago. All were guilty of a misapprehension of the problem, and a misunderstanding of rate of change. Euler and Cauchy were also wrong, since there is no sense in giving a foundation to a falsehood. The concept of the limit is historically an
In subsequent papers, I show how my tables may be converted to find integrals, trig functions, herelogarithms, and so on. I think it is clear that integrals may be found simply by reading up the table rather than down. But there are several implications of this that must be enumerated in full. And the conversion to trig functions and the rest is somewhat more difficult, although not, I hope, esoteric in any sense. All we have to do to convert the above tables to any function is to consider the way that numbers are generated by the various methods, keeping in mind the provisos I have already covered here. ^{1}See for example, Jacob Klein, Greek Mathematical Thought and the Origin of Algebra. ^{2}Newton, Isaac, Mathematical Papers, 8: 597.^{3}Boyer, Carl. B., The History of the Calculus and its Conceptual Development, p. 227.^{4}Ibid.Links: To see how this paper ties into the problems of Quantum Mechanics, see my paper Quantum Mechanics and Idealism. If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |