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OF THE DERIVATIVE
why the calculus works—
and why it doesn't
by Miles Mathis
A note on my calculus papers, 2006
For briefer, less technical analyses of
contemporary calculus, you may now go here, here and here.
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.
The nest of historical errors I will roust out here is not just a nest of semantics, metaphysics, or failed definitions or methods. It is also an error in finding solutions. I have now used my corrections to theory to show that various proofs are wrong. Furthermore, my better understanding of the calculus has allowed me to show that the calculus is misused in simple physical problems, getting the wrong answer.
By re-defining the derivative I will also undercut the basic assumptions of all current topologies, including symplectic topology—which depends on the traditional definition in its use of points in phase space. Likewise, linear and vector algebra and the tensor calculus will be affected foundationally by my re-definition, since the current mathematics will be shown to be inaccurate representations of the various spaces or fields they hope to express. All representations of vector spaces, whether they are abstract or physical, real or complex, composed of whatever combination of scalars, vectors, quaternions, or tensors will be influenced, since I will show that all mathematical spaces based on Euclid, Newton, Cauchy, and the current definition of the point, line, and derivative are necessarily at least one dimension away from physical space. That is to say that the variables or functions in all current mathematics are interacting in spaces that are mathematical spaces, and these mathematical spaces (all of them) do not represent physical space.
This is not a philosophical contention on my part. My thesis is not that there is some metaphysical disconnection between mathematics and reality. My thesis, proved mathematically below, is that the historical and currently accepted definitions of mathematical points, lines, and derivatives are all false for the same basic reason, and that this falsifies every mathematical space. I correct the definitions, however, which allows for a correction of calculus, topology, linear and vector algebra, and the tensor (among many other things). In this way the problem is solved once and for all, and there need be no talk of metaphysics, formalisms or other esoterica.
In fact, I solve the problem by the simplest method possible, without recourse to any of the mathematical systems I critique. I will not require any math beyond elementary number analysis, basic geometry and simple logic. I do so pointedly, since the fundamental nature of the problem, and its status as the oldest standing problem in mathematics, has made it clearly unresponsive to other more abstract analysis. The problem has not only defied solution; it has defied detection. Therefore an analysis of the foundation must be done at ground level: any use of higher mathematics would be begging the question. This has the added benefit of making this paper comprehensible to any patient reader. Anyone who has ever taken calculus (even those who may have failed it) will be able to follow my arguments. Professional mathematicians may find this annoying for various reasons, but they are asked to be gracious. For they too may find that a different analysis at a different pace in a different “language” will yield new and useful mathematical results.
The end product of my proof will be a re-derivation of the core equation of the differential calculus, by a method that uses no infinite series and no limit concept. I will not re-derive the integral in this paper, but the new algorithm I provide here makes it easy to do so, and no one will be in doubt that the entire calculus has been re-established on firmer ground.
It may also be of interest to many that my method allows me to show, in the simplest possible manner, why umbral calculus has always worked. Much formal work has been done on the umbral calculus since 1970; but, although the various equations and techniques of the umbral calculus have been connected and extended, they have never yet been fully grounded. My re-invention and re-interpretation of the Calculus of Finite Differences allows me to show—by lifting a single curtain—why subscripts act exactly like exponents in many situations.
Finally, and perhaps most importantly, my reinvention and re-interpretation of the Calculus of Finite Differences allows me to solve many of the point-particle problems of QED without renormalization. I will show that the equations of QED required renormalization only because they had first been de-normalized by the current maths, all of which are based upon what I call the Infinite Calculus. The current interpretation of calculus allows for the calculation of instantaneous velocities and accelerations, and this is caused both by allowing functions to apply to points and by using infinite series to approach points in analyzing the curve. By returning to the Finite Calculus—and by jettisoning the point from applied math—I have pointed the way in this paper to cleaning up QED. By making every variable or function a defined interval, we redefine every field and space, and in doing so dispense with the need for most or all renormalization. We also dispense with the primary raison d'etre of string theory.
Newton’s calculus evolved from charts he made himself from his power series, based on the binomial expansion. The binomial expansion was an infinite series expansion of a complex differential, using a fixed method. In trying to express the curve as an infinite series, he was following the main line of reasoning in the pre-calculus algorithms, all the way back to the ancient Greeks. More recently Descartes and Wallis had attacked the two main problems of the calculus—the tangent to the curve and the area of the quadrature—in an analogous way, and Newton’s method was a direct consequence of his readings of their papers. All these mathematicians were following the example of Archimedes, who had solved many of the problems of the calculus 1900 years earlier with a similar method based on summing or exhausting infinite series. However, Archimedes never derived either of the core equations of the calculus proper, the main one being in this paper, y’ = nxn-1.
This equation was derived by Leibniz and Newton almost simultaneously, if we are to believe their own accounts. Their methods, though slightly different in form, were nearly equivalent in theory, both being based on infinite series and differentials that approached zero. Leibniz tells us himself that the solution to the calculus dawned upon him while studying Pascal’s differential triangle. To solve the problem of the tangent this triangle must be made smaller and smaller.
Both Newton and Leibniz knew the answer to the problem of the tangent before they started, since the problem had been solved long before by Archimedes using the parallelogram of velocities. From this parallelogram came the idea of instantaneous velocity, and the 17th century mathematicians, especially Torricelli and Roberval, certainly took their belief in the instantaneous velocity from the Greeks. The Greeks, starting with the Peripatetics, had assumed that a point on a curve might act like a point in space. It could therefore be imagined to have a velocity. When the calculus was used almost two millennia later by Newton to find an instantaneous velocity—by assigning the derivative to it—he was simply following the example of the Greeks.
However, the Greeks had seemed to understand that their analytical devices were inferior to their synthetic methods, and they were even believed by many later mathematicians (like Wallis and Torricelli) to have concealed these devices. Whether or not this is true, it is certain that the Greeks never systematized any methods based on infinite series, infinitesimals, or limits. As this paper proves, they were right not to. The assumption that the point on the curve may be treated as a point in space is not correct, and the application of any infinite series to a curve is thereby an impossibility. Properly derived and analyzed, the derivative equation cannot yield an instantaneous velocity, since the curve always presupposes a subinterval that cannot approach zero; a subinterval that is, ultimately, always one.
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.
The most elementary concept that I must analyze here is the concept of the point. In the Dover edition of Euclid’s Elements, we are told, “a point is that which has no part.” The Dover edition supplies notes on every possible variation of this definition, both ancient and modern, but it fails to answer the question that is central to my paper here: that being, "Does the definition apply to a mathematical point or a physical point” Or, to be even more blunt and vivid, “Are we talking about a point in space, or are we talking about a point on a piece of paper?” This question has never been asked much less answered (until now).
Most will see no point to my question, I know. How is a point on a piece of paper not the same as a point in space? A point on a piece of paper is physical—paper and ink are physical things. So what can I possibly mean?
Let me first be clear on what I do not mean. Some readers will be familiar with the historical arguments on the point, and I must be clear that my question is a completely new one. The historical question, as argued for more than two millennia now, concerns the difference between a monad and a point. According to the ancient definitions a monad was indivisible, but a point was indivisible and had position. The natural question was "position where?" The only answer was thought to be "in space, or in the real world." A thing can have position nowhere else. A point is therefore an indivisible position in the physical world. A monad is a generalized "any-point", or the idea of a point. A point is a specific monad, or the position of a specific monad.
But my distinction between a mathematical point and a physical point is not this historical distinction between a monad and a point. I am not concerned with classifications or with existence. It does not matter to me here, when distinguishing between a physical point and mathematical point, whether one, both, or neither are ideas or objects. What is important is that they are not equivalent. A point in a diagram is neither a physical point nor a monad. A point in a diagram is a specific point; it has (or represents) a definite position. So it is not a monad. But a diagrammed or mathematical point is an abstraction of a physical point; it is not the physical point itself. Its position is different, for one thing. More importantly, whether idea or object, it is one level removed from the physical point, as I will show in some detail below.
The historical question has concerned one sort of abstraction—from the specific to the general. My question concerns a completely different sort of abstraction—the representation of one specific thing by another specific thing. The Dover edition calls its question ontological. My question is operational. A mathematical point represents a physical point, but it is not equivalent to a physical point since the operation of diagramming creates fields that are not directly transmutable into physical fields.
Applied mathematics must be applied to something. Mathematics is abstract, but applied mathematics cannot be fully abstract or it would be applicable to nothing. Applied mathematics applies to diagrams, or their equivalent. It cannot apply directly to the physical world. And this is why I call a diagrammed point a mathematical point. Applied geometry and algebra are applied to mathematical points, which are diagrammed points.
A point on a piece of paper is a diagram, or the beginning of a diagram. It is a representation of a physical point, not the point itself. When we apply mathematics, we do so by assigning numbers to points or lengths (or velocities, etc.). Physics is applied mathematics. It is meant to apply to the physical world. But the mathematical numbers may not be applied to physical points directly. Mathematics is an abstraction, as everybody knows, and part of what makes it abstract even when it is applied to physics is that the numbers are assigned to diagrams. These diagrams are abstractions. A Cartesian graph is one such abstraction. The graph represents space, but it is not space itself. A drawing of a circle or a square or a vector or any other physical representation is also an abstraction. The vector represents a velocity, it is not the velocity itself. A circle may represent an orbit, but it is not the orbit itself, and so on. But it not just that the drawing is simplified or scaled up or down that makes it abstract. The basic abstraction is due to the fact that the math applied to the problem, whatever it is, applies to the diagram, not to the space. The numbers are assigned to points on the piece of paper or in the Cartesian graph, not to points in space.
All this is true even when there is no paper or pixel diagram used to solve the problem. Whenever math is applied to physics, there is some spatial representation somewhere, even if it is just floating lines in someone’s head. The numbers are applied to these mental diagrams in one way or another, since numbers cannot logically be applied directly to physical spaces.
The easiest way to prove the inequivalence of the physical point and the mathematical point is to show that you cannot assign a number to a physical point. We assign numbers to mathematical points all the time. This assignment is the primary operational fact of applied mathematics. Therefore, if you cannot assign a number to a physical point, then a physical point cannot be equivalent to a mathematical point.
Pick a physical point. I will assume you can do this, although metaphysicians would say that this is impossible. They would give some variation of Kant’s argument that whatever point you choose is already a mental construct in your head, not the point itself. You will have chosen a phenomenon, but a physical point is a noumenon. But I am not interested in metaphysics here; I am interested in a precise definition, one that has the mathematically required content to do the job. A definition of “point” that does not tell us whether we are dealing with a physical point or a mathematical point cannot fully do its job, and it will lead to purely mathematical problems.
So, you have picked a point. I am not even going to be rigorous and make you worry about whether that point is truly dimensionless or indivisible, since, again, that is just quibbling as far as this paper is concerned. Let us say you have picked the corner of a table as your point. The only thing I am going to disallow you to do is to think of that point in relation to an origin. You may not put the corner of your table into a graph, not even in your head. The point you have chosen is just what it is, a physical point in space. There are no axes or origins in your room or your world. OK, now try to assign a number to that point. If you are stubborn you can do it. You can assign the number 5 to it, say, just to vex me. But now try to give that number some mathematical meaning. What about the corner of that table is “5”? Clearly, nothing. If you say it is 5 inches from the center of the table or from your foot, then you have assigned an origin. The center of the table or your foot becomes the origin. I have disallowed any origins, since origins are mathematical abstractions, not physical things.
The only way to assign a number to your point is to assign the origin to another point, and to set up axes, so that your room becomes a diagram, either in your head or on a piece of paper. But then the number 5 applies to the diagram, not to the corner of the table.
What does this prove? Euclid’s geometry is a form of mathematics. I don’t think anyone will argue that geometry is not mathematics. Geometry becomes useful only when we can begin to assign numbers to points, and thereby find lengths, velocities and accelerations. If we assign numbers to points, then those points must be mathematical points. They are not physical points. Euclid’s definitions apply to points on pieces of paper, to diagrams. They do not apply directly to physical points.
I am not going to argue that you cannot translate your mathematical findings to the physical world. That would be nihilistic and idiotic, not to say counter-intuitive. But I am going to argue here that you must take proper care in doing so. You must differentiate between mathematical points and physical points, because if you do not you will misunderstand all higher math. You will misinterpret the calculus, to begin with, and this will throw off all your other maths, including topology, linear and vector algebra, and the tensor calculus.
To show how all this applies to the calculus, I will start with a close analysis of the curve. Let us say we are given a curve, but are not given the corresponding curve equation. To find this equation, we must import the curve into a graph. This is the traditional way to “measure” it, using axes and an origin and all the tools we are familiar with. Each axis acts as a sort of ruler, and the graph as a whole may be thought of simply as a two-dimensional yardstsick. This analysis may seem self-evident, but already I have enumerated several concepts that deserve special attention. Firstly, the curve is defined by the graph. When we discover a curve equation by our measurement of the curve, the equation will depend entirely on the graph. That is, the graph generates the equation. Secondly, if we use a Cartesian graph, with two perpendicular axes, then we have two and only two variables. Which means that we have two and only two dimensions. Thirdly, every point on the graph will likewise have two dimensions. Let me repeat: every POINT on the graph will have TWO DIMENSIONS (let that sink in for a while). Using the most common variables, it will have an x dimension and a y dimension. This means that any equation with two variables implies two dimensions, which implies two dimensions at every point on the graph and every point on the curve.
If you are mathematician who is chafing under all this "philosophy talk"—or anyone else who is the least bit lost among all these words, for whatever reason—let me explain very directly why I have bolded the words above. For it might be called the central mathematical assertion of this whole paper: the primary thesis of my analysis. A point on a graph has two dimensions. But of course a physical point does not have two dimensions. A mathematician who defined a point as a quantity having two dimensions would be an oddity, to say the least. No one in history has proposed that a point has two dimensions. A point is generally understood to have no dimensions. And yet we have no qualms calling a point on a graph a point. This imprecision in terminology has caused terrible problems historically, and it is one of these problems that I am unwinding here. The historical and current proof of the derivative both treat a point on a graph and on a curve as a zero-dimensional variable. It is not a zero dimension variable; it is a two dimensional variable. A point in space can have no dimensions, but a point on a piece of paper can have as many dimensions as we want to give it. However, we must keep track of those dimensions at all times. We cannot be sloppy in our language or our assignments. The proof of the calculus has been imprecise in its language and assignments.
Let me clarify this with an example. Say a bug crawls by on the wall. You mark off its trail. Now you try to apply a curve equation to the trail, without axes. Say the curve just happens to match a curve you are familiar with. Say it looks just like the curve y = x². OK, try to assign variables to the bug’s motion. You can’t do it. The reason why you can’t is that a curve drawn on the wall, whether by a bug or by Michelangelo, needs three axes to define it. You need x, y and t. The curve may look the same, but it is not the same. A curve on the wall and a curve on the graph are two different things.
As a second example, say your little brother jumps into his new car and peals off down the street. You run out after him and look at the black marks trailing off behind his car. He’s accelerating still, so you should see those marks curve, right? A curve describes an acceleration, right? Not necessarily. The car is going in only one direction, so you can plot x against t. There is no third variable y. But it still doesn’t mark off a curve. The car is going in a straight line. Mystifying.
The curve for an equation looks like it does only on a graph. Its curve is dependent on the graph. That is, its rate of change is defined by the graph. All those illustrations and diagrams you have seen in books with curves drawn without graphs are incomplete. Years ago—nobody knows how many years—books stopped drawing the lines, since they got in the way. Even Descartes, who invented the lines, probably let them evaporate as an artistic nuisance. And so we have ended up forgetting that every mathematical curve implies its own graph.
A physical curve and a mathematical curve are not equivalent. They are not mathematically equivalent.
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 - x0|<δ then |f(x) - y0|<ε. What I want to point out is that |x - x0| 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 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.
a Critique of
The Current Derivation
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.1 He showed the error in assigning the point to the “unit” or the number one; the point must be assigned to its analogous magnitude, which was zero. He proved that the point was indivisible precisely because it was zero. This correction to both geometry and arithmetic pointed Stevin in the direction of my solution here, but he never realized the operational import of the diagram in geometry. In refining the concepts of number and point, he did not see that both the Greeks and the moderns were in possession of two separate concepts of the point: the point in space and the point in diagrammatica.
John Wallis came even nearer this recognition. Following Stevin, he wrote extensively of the importance of the point as analogue to the nought. He also did very important work on the calculus, being perhaps the greatest influence on Newton. He was therefore in the best position historically to have discovered the disjunction of the two concepts of point. Unfortunately he continued to follow the strong current of the 17th century, which was dominated by the infinite series and the infinitesimal. After his student Newton created the current form of the calculus, mathematicians were no longer interested in the rigorous definitions of the Greeks. The increasing abstraction of mathematics made the ontological niceties of the ancients seem quaint, if not passé. The mathematical current since the 18th century has been strongly progressive. Many new fields have arisen, and studying foundations has not been in vogue. It therefore became less and less likely that anyone would notice the conceptual errors at the roots of the calculus. Mathematical outsiders like Bishop Berkeley in the early 18th century failed to find the basic errors (he found the effects but not the causes), and the successes of the new mathematics made further argument unpopular.
I have so far critiqued the ability of the calculus to find instantaneous values. But we must remember that Newton invented it for that very purpose. In De Methodis, he proposes two problems to be solved. 1) “Given a length of the space continuously, to find the speed of motion at any time.” 2) “Given the speed of motion continuously, to find the length of space described at any time.” Obviously, the first is solved by what we now call differentiation and the second by integration. Over the last 350 years, the foundation of the calculus has evolved somewhat, but the questions it proposes to solve and the solutions have not. That is, we still think that these two questions make sense, and that it is sensible that we have found an answer for them
Question 1 concerns finding an instantaneous velocity, which is a velocity over a zero time interval. This is done all the time, up to this day. Question 2 is the mathematical inverse of question 1. Given the velocity, find the distance traveled over a zero time interval. This is no longer done, since the absurdity of it is clear. On the graph, or even in real life, a zero time interval is equal to a zero distance. There can be no distance traveled over a zero time interval, even less over a zero distance, and most people seem to understand this. Rather than take this as a problem, though, mathematicians and physicists have buried it. It is not even paraded about as a glorious paradox, like the paradoxes of Einstein. No, it is left in the closet, if it is remembered to exist al all.
As should already be clear from my exposition of the curve equation, Newton’s two problems are not in proper mathematical or logical form, and are thereby insoluble. This implies that any method that provides a solution must also be in improper form. If you find a method for deriving a number that does not exist, then your method is faulty. A method that yields an instantaneous velocity must be a suspect method. An equation derived from this method cannot be trusted until it is given a logical foundation. There is no distance over a zero distance; and, equally, there is no velocity over a zero interval.
Bishop Berkeley commented on the illogical qualities of Newton’s proofs soon after they were published (The Analyst, 1734). Ironically, Berkeley’s critiques of Newton mirrored Newton’s own critiques of Leibniz’s method. Newton said of Leibniz, “We have no idea of infinitely little quantities & therefore I introduced fluxions into my method that it might proceed by finite quantities as much as possible.” And, “The summing up of indivisibles to compose an area or solid was never yet admitted into Geometry.”2
This “using finite quantities as much as possible” is very nearly an admission of failure. Berkeley called Newton’s fluxions “ghosts of departed quantities” that were sometimes tiny increments, sometimes zeros. He complained that Newton’s method proceeded by a compensation of errors, and he was far from alone in this analysis. Many mathematicians of the time took Berkeley’s criticisms seriously. Later mathematicians who were much less vehement in their criticism, including Euler, Lagrange and Carnot, made use of the idea of a compensation of errors in attempting to correct the foundation of the calculus. So it would be unfair to dismiss Berkeley simply because he has ended up on the wrong side of history. However, Berkeley could not explain why the derived equation worked, and the usefulness of the equation ultimately outweighed any qualms that philosophers might have. Had Berkeley been able to derive the equation by clearly more logical means, his comments would undoubtedly have been treated with more respect by history. As it is, we have reached a time when quoting philosophers, and especially philosophers who were also bishops, is far from being a convincing method, and I will not do more of it. Physicists and mathematicians weaned on the witticisms of Richard Feynman are unlikely to find Berkeley’s witticisms quite up-to-date.
I will take this opportunity to point out, however, that my critique of Newton is of a categorically different kind than that of Berkeley, and of all philosophers who have complained of infinities in derivations. I have not so far critiqued the calculus on philosophical grounds, nor will I. The infinite series has its place in mathematics, as does the limit. My argument is not that one cannot conceive of infinities, infinitesimals, or the like. My argument has been and will continue to be that the curve, whether it is a physical concept or a mathematical abstraction, cannot logically admit of the application of an infinite series, in the way of the calculus. In glossing the modern reaction to Berkeley’s views, Carl Boyer said, “Since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than plausibility in the light of sense perception or intuition.”3 I agree, and I stress that my main point already advanced is that there is no inner consistency in letting a differential [f(x + i) – f(x)] approach a point when that point is already expressed by two differentials [(x-0) and (y-0)].
Boyer gives the opinion of the mathematical majority when he defends the instantaneous velocity in this way: “[Berkeley’s] argument is of course absolutely valid as showing that the instantaneous velocity has no physical reality, but this is no reason why, if properly defined or taken or taken as an undefined notion, it should not be admitted as a mathematical abstraction.”4 My answer to this is that physics has treated the instantaneous velocity as a physical reality ever since Newton did so. Beyond that, it has been accepted by mathematicians as an undefined notion, not as a properly defined notion, as Boyer seems to admit. He would not have needed to include the proviso “or taken as an undefined notion” if all notions were required to be properly defined before they were accepted as “mathematical abstractions.” The notion of instantaneous velocity cannot be properly defined mathematically since it is derived from an equation that cannot be properly defined mathematically. Unless Boyer wants to argue that all heuristics should be accepted as good mathematics (which position contemporary physics has accepted, and contemporary mathematics is closing in on), his argument is a non-starter.
Many mathematicians and physicists will maintain that the foundation of the calculus has been a closed question since Cauchy in the 1820’s, and that my entire thesis can therefore only appear Quixotic. However, as recently as the 1960’s Abraham Robinson was still trying to solve perceived problems in the foundation of the calculus. His nonstandard analysis was invented for just this purpose, and it generated quite a bit of attention in the world of math. The mathematical majority has not accepted it, but its existence is proof of widespread unease. Even at the highest levels (one might say especially at the highest levels) there continue to be unanswered questions about the calculus. My thesis answers these questions by showing the flaws underlying both standard and nonstandard analysis.
Newton’s original problems should have been stated like this: 1) Given a distance that varies over any number of equal intervals, find the velocity over any proposed interval. 2) Given a variable velocity over an interval, find the distance traveled over any proposed subinterval. These are the questions that the calculus really solves, as I will prove below. The numbers generated by the calculus apply to subintervals, not to instants or points. Newton’s use of infinite series, like the power series, misled him to believe that curves drawn on graphs could be expressed as infinite series of (vanishing) differentials. All the other founders of the calculus made the same mistake. But, due to the way that the curve is generated, it cannot be so expressed. Each point on the graph already stands for a pair of differentials; therefore it is both pointless and meaningless to let a proposed differential approach a point on the graph.
To show precisely what I mean, let us now look to the current derivation of the derivative equation. Take a functional equation, for example
y = x²
Increase it by δy and δx to obtain
y + δy = (x + δx) 2
subtract the first equation from the second:
δy = (x + δx)2 - x2
= 2xδx + δx2
divide by δx
δy /δx = 2x + δx
Let δx go to zero (only on the right side, of course)
δy / δx = 2x
y' = 2x
Most will expect that my only criticism is that δx should not go to zero on the left side, since that would imply to ratio going to infinity. But that is not my primary criticism at all. My primary criticism is this:
In the first equation, the variables stand for either “all possible points on the curve” or “any possible point on the curve.” The equation is true for all points and any point. Let us take the latter definition, since the former doesn’t allow us any room to play. So, in the first equation, we are at “any point on the curve”. In the second equation, are we still at any point on the same curve? Some will think that (y + δy) and (x + δx) are the co-ordinates of another any-point on the curve—this any-point being some distance further along the curve than the first any-point. But a closer examination will show that the second curve equation is not the same as the first. The any-point expressed by the second equation is not on the curve y = x2. In fact, it must be exactly δy off that first curve. Since this is true, we must ask why we would want to subtract the first equation from the second equation. Why do we want to subtract an any-point on a curve from an any-point off that curve?
Furthermore, in going from equation 1 to equation 2, we have added different amounts to each side. This is not normally allowed. Notice that we have added δy to the left side and 2xδx + δx2 to the right side. This might have been justified by some argument if it gave us two any-points on the same curve, but it doesn’t. We have completed an illegal operation for no apparent reason.
Now we subtract the first any-point from the second any-point. What do we get? Well, we should get a third any-point. What is the co-ordinate of this third any-point? It is impossible to say, since we got rid of the variable y. A co-ordinate is in the form (x,y) but we just subtracted away y. You must see that δy is not the same as y, so who knows if we are off the curve or on it. Since we subtracted a point on the first curve from a point off that curve, we would be very lucky to have landed back on the first curve, I think. But it doesn’t matter, since we are subtracting points from points. Subtracting points from points is illegal. If you want to get a length or a differential you must subtract a length from a length or a differential from a differential. Subtracting a point from a point will only give you some sort of zero—another point. But we want δy to stand for a length or differential in the third equation, so that we can divide it by δx. As the derivation now stands, δy must be a point in the third equation.
Yes, δy is now a point. It is not a change-in-y in the sense that the calculus wants it to be. It is no longer the difference in two points on the curve. It is not a differential! Nor is it an increment or interval of any kind. It is not a length, it is a point. What can it possibly mean for an any-point to approach zero? The truth is it doesn’t mean anything. A point can’t approach a zero length since a point is already a zero length.
Look at the second equation again. The variable y stands for a point, but the variable δy stands for a length or an interval. But if y is a point in the second equation, then δy must be a point in the third equation. This makes dividing by δx in the next step a logical and mathematical impossibility. You cannot divide a point by any quantity whatsoever, since a point is indivisible by definition. The final step—letting δx go to zero—cannot be defended whether you are taking only taking the denominator on the left side to zero or whether you are taking the whole fraction toward zero (which has been the claim of most). The ratio δy/δx was already compromised in the previous step. The problem is not that the denominator is zero; the problem is that the numerator is a point. The numerator is zero.
To my knowledge the calculus derivation has never been critiqued in this way. From Berkeley on the main criticism concerned explaining why the ratio δy/δx was not precisely zero, and why letting δx go to zero did not make the fraction go to infinity. Newton tried to explain it by the use of prime and ultimate ratios, and Cauchy is believed to have solved it by having the ratio approach a limit. But according to my analysis the ratio already had a numerator of zero in the previous step, so that taking it to a limit is moot.
Nonstandard analysis has no answer to this either. Abraham Robinson’s “rigorously” defining the infinitesimal has done nothing to solve my critique here. Adding new terminology does not clarify the problem, since it is beside the point whether one part of these equations is called “standard” or “nonstandard.” If δy is a point on the curve in the third equation, then it is no longer an infinitesimal. At that point it doesn’t matter what we call it, how we define it, or how we axiomatize our logic. It isn’t a distance and cannot yield what we want it to yield, not with infinitesimals, limits, diminishing series or anything else.
[I have gotten several emails over the years from angry mathematicians, saying or implying that my mentioning this derivation is some sort of strawman. They tell me they don't prove the derivative that way and then launch into some longwinded torture of both mediums (math and the English language) to show me how to do it. Unfortunately, this derivation above is much more than a strawman. It is the way I was taught the calculus in high school in the early 80's, and it is posted all over the internet to this day. If it is a strawman, it is the mainstream's own strawman, and they had best stop propping it up. These mathematicians are just angry I am using their own equations against them. They reference Newton and Leibniz when it suits them, but when someone else references them, it is a strawman. These mathematicians are slippery than eels. If you mention one derivation, they misdirect you into another, claiming that one has been superceded. If you then destroy the new one, they find a third one to hide behind. And you won't ever finding them address the main points of your papers. For instance, I have never had a single mathematician respond to the central points of this paper. They ignore those and look for tangential arguments they can waste my time with indefinitely. This, by itself, is a sign of the times.]
The Rest of the Groundwork
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.
I have already shown that the variables in a curve equation are cardinal numbers, and as such they must be understood as delta variables. In mathematical terms, they are differentials; in physical terms, they are lengths or distances. This is because a curve is defined by a graph and a graph is defined by axes. The numbers on these axes signify distances from zero or differentials: (x – 0) or (y – 0). In the same way the cardinal number line is also a compendium of distances or differentials. In fact, each axis on a graph may be thought of as a separate cardinal number line. The Cartesian graph is then just two number lines set zero to zero at a 90o angle.
This being true, a subtraction of one number from another—when these numbers are taken from Cartesian graphs or from the cardinal number line—is the subtraction of one distance from another distance, or one differential from another. Written out in full, it would look like this:
ΔΔx = Δxf - Δxi
Where Δ xf is the final cardinal number and Δxi is the initial cardinal number. This is of course rigorous in the extreme, and may seem pointless. But be patient, for we are rediscovering things that were best not forgotten. This equation shows that a cardinal number stands for a change from zero, and that the difference of two cardinal numbers is the change of a change. All we have done is subtract one number from another and we already have a second-degree change.
Following this strict method, we find that any integer subtracted from the next is equal to 1, which must be written ΔΔx = 1. On a graph each little box is 1 box wide, which makes the differential from one box to the next 1. To go from one end of a box to the other, you have gone 1. This distance may be a physical distance or an abstract distance, but in either case it is the change of a change and must be understood as ΔΔx = 1.
Someone might interrupt at this point to say, "You just have one more delta at each point than common usage. Why not simplify and get back to common usage by canceling a delta in all places?” We cannot do that because then we would have no standard representation for a point. If we let a naked variable stand for a cardinal number, which I have shown is not a point, then we have nothing to let stand for a point. To clear up the problem like I believe is necessary, we must let x and y and t stand for points or instants or ordinals, and only point or instants or ordinals. We must not conflate ordinals and cardinals, and we must not conflate points with distances. We must remain scrupulous in our assignments.
Next, it might be argued that we can put any numbers into curve equations and make them work, not just integers. True, but the lines of the graph are commonly integers. Each box is one box wide, not ½ a box or e box or π box. This is important because the lines define the graph and the graph defines the curve. It means that the x-axis itself has a rate of change of one, and the y or t-axis also. The number line itself has a rate of change of one, by definition. None of my number theory here would work if it did not.
For instance, the sequence 1, 1, 1, 1, 1, 1.... describes a point. If you remain at one you don’t move. A point has no RoC (rate of change). Its change is zero, therefore its RoC is zero. The sequence of cardinal integers 1, 2, 3, 4, 5…. describes motion, in the sense that you are at a different number as you go down the sequence. First you are at 1, then at 2. You have moved, in an abstract sense. Since you change 1 number each time, your RoC is steady. You have a constant RoC of 1. A length is a first-degree change of x. Every value of Δx we have on a graph or in an equation is a change of this sort. If x is a point in space or an ordinal number, and Δx is a cardinal number, then ΔΔx is a RoC.
I must also stress that the cardinal number line has a RoC of 1 no matter what numbers you are looking at. Rationals, irrationals, whatever. Some may argue that the number line has a RoC of 1 only if you are talking about the integers. In that case it has a sort of “cadence,” as it has been suggested to me. Others have said that the number line must have a RoC of zero, even by my way of thinking, since it has an infinite number of points, or numbers. There are an infinite number of points from zero to 1, even. Therefore, if you “hop” from one to the other, in either a physical or an abstract way, then it will take you forever to get from zero to one. But that is simply not true. As it turns out, in this problem, operationally, the possible values for Δx have a RoC of 1, no matter which ones you choose. If you choose numbers from the number line to start with (and how could you not) then you cannot ever separate those numbers from the number line. They are always connected to it, by definition and operation. The number line always “moves” at a RoC of 1, so the gap between any numbers you get for x and y from any equation will also move with a RoC of 1.
If this is not clear, let us take the case where I let you choose values for x1 and x2 arbitrarily, say x1 = .0000000001 and x2 = .0000000002. If you disagree with my theory, you might say, "My gap is only .0000000001. Therefore my RoC must be much slower than one. A sequence of gaps of .0000000001 would be very very slow indeed." But it wouldn’t be slow. It would have a RoC of 1. You must assume that your .0000000001 and .0000000002 are on the number line. If so, then your gap is ten billion times smaller than the gap from zero to 1. Therefore, if you relate your gap to the number line—in order to measure it—then the number line, galloping by, would traverse your gap ten billion times faster than the gap from zero to one. The truth is that your tiny gap would have a tiny RoC only if it were its own yardstick. But in that case, the basic unit of the yardstick would no longer be 1. It would be .0000000001. A yardstick, or number line, whose basic unit is defined as 1, must have a RoC of 1, at all points, by definition.
From all this you can see that I have defined rate of change so that it is not strictly equivalent to velocity. A velocity is a ratio, but it is one that has already been established. A rate of change, by my usage here, is a ratio waiting to be calculated. It is a numerator waiting for a denominator. I have called one delta a change and two deltas a rate of change. Three deltas would be a second-degree rate of change (or 2RoC), and so on.
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.
Δx = 1, 2, 3, 4, 5, 6, 7, 8, 9...
Δ2x = 2, 4, 6, 8, 10, 12, 14, 16, 18...
Δx2 = 1, 4, 9, 16, 25, 36, 49, 64, 81...
Δx3 = 1, 8, 27, 64, 125, 216, 343...
Δx4 = 1, 16, 81, 256, 625, 1296...
Δx5 = 1, 32, 243, 1024, 3125, 7776, 16807
ΔΔx = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
ΔΔ2x = 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
ΔΔx2 = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
ΔΔx3 = 1, 7, 19, 37, 61, 91, 127
ΔΔx4 = 1, 15, 65, 175, 369, 671
ΔΔx5 = 1, 31, 211, 781, 2101, 4651, 9031
ΔΔΔx = 0, 0, 0, 0, 0, 0, 0
ΔΔΔx2 = 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
ΔΔΔx3 = 6, 12, 18, 24, 30, 36, 42
ΔΔΔx4 = 14, 50, 110, 194, 302
ΔΔΔx5 = 30, 180, 570, 1320, 2550, 4380
ΔΔΔΔx3 = 6, 6, 6, 6, 6, 6, 6, 6
ΔΔΔΔx4 = 36, 60, 84, 108
ΔΔΔΔx5 = 150, 390, 750, 1230, 1830
ΔΔΔΔΔx4 = 24, 24, 24, 24
ΔΔΔΔΔx5 = 240, 360, 480, 600
ΔΔΔΔΔΔx5 = 120, 120, 120
from this, one can predict that
ΔΔΔΔΔΔΔx6 = 720, 720, 720
And so on
This is what you call simple number analysis. It is a table of differentials. The first line is a list of the potential integer lengths of an object. It is also a list of the cardinal integers, as you can see. It is also a list of the possible values for the number of boxes we could count in our graph. It is therefore both physical and abstract, so that it may be applied in any sense one wants. Line 2 lists the potential lengths or box values of the variable Δ2x. Line 3 lists the possible box values for Δx². Line seven begins the second-degree differentials. It lists the differentials of line 1, as you see. To find differentials, I simply subtract each number from the next. Line eight lists the differentials of line 2, and so on. Line 14 lists the differentials of line 9. I think you can follow the logic of the rest.
Now let's pull out the important lines and relist them in order:
ΔΔx = 1, 1, 1, 1, 1, 1, 1
ΔΔΔx2 = 2, 2, 2, 2, 2, 2, 2
ΔΔΔΔx3 = 6, 6, 6, 6, 6, 6, 6
ΔΔΔΔΔx4 = 24, 24, 24, 24
ΔΔΔΔΔΔx5 = 120, 120, 120
ΔΔΔΔΔΔΔx6 = 720, 720, 720
Do you see it?
2ΔΔx = ΔΔΔx²
3ΔΔΔx2 = ΔΔΔΔx3
4ΔΔΔΔx3 = ΔΔΔΔΔx4
5ΔΔΔΔΔx4 = ΔΔΔΔΔΔx5
6ΔΔΔΔΔΔx5 = ΔΔΔΔΔΔΔx6
and so on.
Voila. We have the current derivative equation, just from a table. All I have to do now is explain what it means. Instead of looking where the differentials approach zero, as the calculus did, I have looked for a place where the differentials are constant—as in the second little table. I have had to look farther and farther up in the rate of change table each time to find it, but it is always there. The calculus solves up from a near-zero differential. I solve down from a constant differential. Their differential is never fully defined or explained (despite their claims); mine will be in the paragraphs that follow.
But before we proceed, I will stop for a moment to point out that our final numbers are all factorials. Each line may be expressed as n factorial, as in 6 = 3 factorial, 24 = 4 factorial, 720 = 6 factorial, and so on. A full analysis of this would lead us back into Pascal and Euler and the current complicated expressions of the calculus, which I am simplifying here. But some readers may find this fact meaningful or suggestive.
I will explain in great detail below what is being expressed as I re-derive the derivative equation; but let me first gloss the important aspects of this chart. The chart is generated by basic number theory, as I have already said. That means that it is true for any and all variables. It is an analysis of the number line, and the relationship of integers and all exponents of integers. Therefore we can use the information in the chart to give us more information about any curve equation. The information in the chart is defined by the number line itself. Meaning that it is true by definition. In that way it may be thought of as a cache of pre-existing information or tautological equalities. As you can see, the chart needs no proof, since it is simply a list of givens. It is a direct result of exponential notation, and I have done nothing more than list values.
Lagrange claimed that the Taylor series was the secret engine behind the calculus, but this chart is the secret engine behind both the Taylor series and the calculus. I personally don’t believe that the Greeks were concealing any algorithms or other devices, but if they were this is the algorithm they were likely concealing. I don’t believe Archimedes was aware of this chart, for if he had been he would not have continued to pursue his solutions with infinite series.
The calculus works only because the equations of the calculus work. The equation y’ = nxn-1 and the other equations of the calculus are the primary operational facts of the mathematics, not the proofs of Newton or Leibniz or Cauchy. Newton’s and Leibniz’s most important recognition was that these generalized equations were the most needful things, and that they must be achieved by whatever means necessary. The means available to them in the late 17th century was a proof using infinitesimals. A slightly finessed proof yielded results that far outweighed any philosophical cavils, and this proof has stood ever since. But what the calculus is really doing when it claims to look at diminishing differentials and limits is take information from this chart. This chart and the number relations it clearly reveals are the foundations of the equations of the calculus, not infinite series or limits.
To put it in even balder terms, the equalities listed above may be used to solve curve equations. By “solve” I mean that the equalities listed in this chart are substituted into curve equations in order to give us information we could not otherwise get. Rate of change problems are thereby solved by a simple substitution, rather than by a complex proof involving infinities and limits. A curve equation tells us that one variable is changing at a rate equal to the rate that another variable (to some exponent) is changing. The chart above tells us the same thing, but in it the same variable is on both sides of the equation. So obviously all we have to do is substitute in the correct way and we have solved our equation. We have taken information from the chart and put it into the curve equation, yielding new information. It is really that simple. The only questions to ask are, "What information does the chart really contain?" And, "What information does it yield after substitution into a curve equation?"
I have defined Δx as a linear distance from zero on the graph, in the x-direction (if the word "distance" has too much physical baggage for you, you may substitute "change from zero"). ΔΔx is then the change of Δx, and so on. Since ΔΔx/ΔΔt is a velocity, ΔΔΔx is sort of constant acceleration, waiting to be calculated (given a ΔΔt). In that sense, ΔΔΔΔx is a variable acceleration waiting to be calculated. ΔΔΔΔΔx is a change of a variable acceleration, and ΔΔΔΔΔΔx is a change of a change of a variable acceleration. Some may ask, "Do these kinds of accelerations really exist? They boggle the mind. How can things be changing so fast?" High exponent variables tell us that we are dealing with these kinds of accelerations, whether they exist in physical situations or not. The fact is that complex accelerations do exist in real life, but this is not the place to discuss it. Most people can imagine a variable acceleration, but get lost beyond that. Obviously, in strictly mathematical situations, changes can go on changing to infinity.
I said in the previous paragraph that velocity is ΔΔx/ΔΔt. By my notation it must be. Current notation has one less delta at each point than I do. Current notation assumes that curve-equation variables are naked variables: x, t. I assume they are delta variables, Δx, Δt. But I agree with current theory that velocity is a change of these variables. Therefore velocity must be ΔΔx/ΔΔt.
You will say, "Then you are implying that velocity is not distance over time. You are saying by your notation that velocity is change in distance over change in time." Precisely. Look at it this way: say I am sitting at the number 3 on a big ruler. I have shown that the number three is telling the world that I am three inches from the end. It is giving a distance. Now, can I use that distance to calculate a velocity? How?—I just said I was sitting there. I am not moving. There is no velocity involved, so it would be ridiculous to calculate one. To calculate a velocity, we must have a velocity, in which case I must move from one number mark on the ruler to another one. In which case we have a change in distance, you see.
You may answer, “What if you were at the origin to begin with? Then the distance and the change in distance are the same thing.” They would be the same number, yes. But mathematically the calculation would still involve a subtraction, if you were writing out the whole thing. It would always be implied that ΔΔx = Δx(final) - Δx(initial) = Δx(final) - 0. Your final number would be the same number, and the magnitude would be the same, but conceptually it is not the same. Δx and ΔΔx are both measured in meters, say, but they are not the same conceptually.
One way to clear up part of this confusion is to differentiate between length and distance. In physics, they are often used interchangeably. In our rate of change problems, we may create more clarity by assigning one word exclusively to one situation, and the other word to the other situation. Let us assign length to Δx and distance to ΔΔx. A cardinal number represents a length from zero. It is the extension between two static points, but no movement is implied. One would certainly have to move to go from one to the other, but a length implies no time variable, no change in time. A length can exist in the absence of time. A distance, however, cannot. A distance implies the presence of another variable, even if that variable is not a physical variable like time. For instance, to actually travel from one point to another requires time. Distance implies movement, or it implies a second-degree change. A length is a static change in x. A distance is a movement from one x to the other.
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
*Why can we cancel deltas here? That is a very important question. Is a delta a variable? Is every delta equal to every other delta? The answer is that a delta is not a variable; and that every delta does not equal every other delta. Therefore the rules of cancellation are a bit tricky. A delta is not a free-standing mathematical symbol. You will never see it by itself. It is connected to the variable it precedes. A variable and all its deltas must therefore be taken as one variable. This would seem to imply that cancelling deltas is forbidden. However a closer analysis shows that in some cases it is allowed. A variable and all its deltas stand for an interval, or a differential. At a particular point on the graph, that would be a particular interval. But in a general equation, that stands for all possible intervals of the variable. As you can see from my table, some delta variables have the same interval value at all points. Most don’t. High exponent variables with few deltas have high rates of change. However, all the lines in the table are dependent on the first line. Notice that each line could be read as, " If Δx = 1,2,3&c , then this line is true." You can see that you put those values for Δx into every other line, in order to get that line. Each line of the table is just reworking the first line. Line three is what happens when you square line one, for instance. So that the underlying variable Δx is the same for every line on the table. Therefore, if you set up equalities between one line and another, the rates of change are relatable to each other. They are all rates of change of Δx. That is why you can cancel deltas here.
Δt = Δx3
Then the derivative is
Δt' = 3Δx2
From my chart we can see that
3ΔΔΔx2 = ΔΔΔΔx3
So, 3Δx2 = ΔΔx3
[Deltas may be cancelled across these particular equalities]*
And, Δt' = 3Δx2 = ΔΔx3
Δt = Δx3
Therefore, Δt' = ΔΔt
The derivative is just the rate of change of our dependent variable Δt. But I repeat, it is the rate of change of a length or period. It is not the rate of change of a point or instant. A point on the graph stands for a value for Δt, not a point in space. The derivative is a rate of change of a length (or a time period).
This all goes to say that if x is on both sides of the equation, you can cancel deltas. Otherwise you cannot.
Now let's do that again without using what we already know from the calculus. Let's prove the derivative equation logically just from the chart without making any assumptions that the historical equation is correct. Again, we are given the curve equation and a curve on a graph.
Δt = Δx3
We then look at my second little chart to find Δx3 . We see that the differential is constant (6) when the variable is changing at this rate: ΔΔΔΔx3. You will say, "Wait, explain that. Why did you go there on the chart? Why do we care where the differential is constant?" We care because when the differential is constant, the curve is no longer curving over that interval. If the curve is no longer curving, then we have a straight line. That straight line is our tangent. That is what we are seeking.
Now let's show what 2ΔΔx = ΔΔΔx2 means. The equation is telling us "two times the rate of change of x is equal to the 2RoC of x2." This is somewhat like saying "twice the velocity of x is equal to the acceleration of x2." These equalities are just number equalities. They do not imply spatial relationships. For instance, if I say, “My velocity is equal to your acceleration,” I am not saying anything about our speeds. I am not saying that we are moving in the same way or covering the same ground. I am simply noticing a number equality. The number I calculate for my velocity just happens to be the number you are calculating for your acceleration. It is a number relation. This number relation is the basis for the calculus. The table above is just a list of some slightly more complex number relations. But they are not very complex, obviously, since all we had to do is subtract one number from the next.
Next let's look again at our given equation, Δt = Δx3
What exactly is that equation telling us? Since the graph gives us the curve—defines it, visualizes, everything—we should go there to find out. If we want to draw the curve, what is the first thing we do? We put numbers in for Δx and see what we get for Δt, right? What numbers do we put in for Δx? The integers, of course. You can see that if we put integers in, then Δx is changing at the rate of one. We put in 1 first, and then 2, and so on. So Δx is changing at a rate of one. As I proved above, we don't have to put in integers. Even if we put in fractions or decimals, Δx will be changing at the rate of one. It just won't be so easy to plot the curve. If Δx is changing at the rate of one, then Δt will be changing at the rate of Δx3. That is all the equation is telling us.
Now that we are clear on what everything stands for, we are ready to solve.
We are given Δt = Δx3
We find from the table 3ΔΔΔx2 = ΔΔΔΔx3
We simplify 3Δx2 = ΔΔx3
We seek ΔΔt
We notice ΔΔt = ΔΔx3 since we can always add a delta to both sides*
We substitute ΔΔt = 3Δx2
ΔΔt = Δt'
So Δt' = 3Δx2
Now I explain the steps thoroughly. The final equation reads, in full: "When the rate of change of the length Δx is one, the rate of change of the length (or period, in this case) Δt is 3Δx2." The first part of that sentence is implied from my previous explanations, but it is good for us to see it written out here, in its proper place. For it tells us that when we are finding the derivative, we are finding the rate of change of the first variable (the primed variable) when the other variable is changing at the rate of one. Therefore, we are not letting either variable approach a limit or go to zero. To repeat, ΔΔx is not going to zero. It is the number one.
That is why you can let it evaporate in the denominator of the current calculus proof. In the current proof the fraction Δy/Δx (this would be ΔΔy/ΔΔx by my notation) is taken to a limit, in which case Δx is taken to zero, we are told. But somehow the fraction does not go to infinity, it goes to Δy. The historical explanation has never been satisfactory. I have shown that it is simply because the denominator is one. A denominator of one can always be ignored.
*We were allowed to add deltas to both sides of the equation in this case because we were adding the same deltas. Deltas aren’t always equivalent, but we can multiply both sides by deltas that are equivalent. What is happening is that we have an equality to start with. We then give the same rate of change to both sides: so the equality is maintained.
You may now ask, "OK, but how did you know to seek ΔΔt? You have shown above that the current proof seeks that, but you were supposed to be solving without taking any assumptions from the current proof or use of the calculus. Why did you seek it? What does it stand for in your interpretation? What is happening on the graph or in real life that explains ΔΔt?"
Good question. By answering that I can pretty much finish off this proof. I have shown that by the very way the equation and the graph are set up, we can show that it must be true that ΔΔx = 1. Given that, what are we seeking? The tangent to the curve on the graph. The tangent to the curve on the graph is a straight line intersecting the curve at (Δx, Δt). Each tangent will hit the curve at only one (Δx, Δt), otherwise it wouldn't be the tangent and the curve wouldn't be a differentiable curve. Since the tangent is a straight line, its slope will be ΔΔt/ΔΔx. So we need an equation that gives us a ΔΔt/ΔΔx for every value of Δt and Δx on our curve. Nothing could be simpler. We know ΔΔx = 1, so we just seek ΔΔt.
ΔΔt/ΔΔx = ΔΔt/1 = ΔΔt
ΔΔt is the slope of the tangent at every point on the curve on the graph.
If Δt = Δx3
Then ΔΔt = 3Δx2
Application to Physics
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)'
I made t the dependent variable initially, but this was an arbitrary choice on my part. If I had made x the dependent variable, then we would have had
(Δx)' = ΔΔx/ΔΔt
So the derivative looks like a velocity.
But the velocity at the point on the graph is not the velocity at a point in space, therefore the slope of the tangent does not apply to the instantaneous velocity. It is the velocity during a period of time of acceleration, not the velocity at an instant. You will say, “Yes, but by your own method we may continue to cancel deltas, in which case we will get
ΔΔt/ΔΔx = Δt/Δx = t/x.
If the Δt's are equal then the t's are equal, and so on."
No they're not. Notice that the equation x/t doesn't even describe a velocity. It is a point over an instant. That is not a velocity. It is not even a meaningful fraction. As I have shown, t in that case is really an ordinal number. You cannot have an ordinal as a denominator in a fraction. It is absurd. In reducing that last fraction, you are saying that 5 meters/5 seconds would equal the fifth meter mark over the fifth second tick. But the fifth meter mark is equivalent to the first meter mark and the hundredth meter mark. And the fifth tick is the same as every other tick. Therefore, I could say that 5 meters/5 seconds = 5th mark/5th tick = 100th mark/ 7th tick. Gobbledygook.
Furthermore, your method of cancellation is not allowed. I cancelled deltas across equalities, under strictly analyzed circumstances (x was on both sides of the equation); you are canceling across a fraction. You are simplifying a fraction by canceling a delta in the numerator and denominator. This is not the same as canceling a term on both sides of an equation. Obviously, ΔΔt/ΔΔx cannot equal Δt/Δx, since the derivative is not the same as the values at the point on the graph. The slope of a curve is not just Δy/Δx. A delta does not stand for a number or a variable, therefore it does not cancel in the same ways. It sometimes cancels across an equality, as I have shown. But the delta does not cancel in the fraction ΔΔt/ΔΔx, because Δt and Δx are not changing at the same rate. If they changed at the same rate, then we would have no acceleration. The deltas are therefore not equivalent in value and cannot be cancelled.
You will answer, “OK, fine. But if the velocity you have found is not an instantaneous velocity, it must be the velocity over some interval. You have just shown that is not the velocity of the interval Δxfinal - Δxinitial. That only applies if the curve is a straight line. So what interval is it?"
It is the velocity over the nth interval of ΔΔx, where ΔΔx = 1. [If t were the independent variable, then the interval would be ΔΔt.] Again, ΔΔt/ΔΔx is the velocity equation, according to our given equation. Therefore the velocity at a given point on the graph (Δxn, Δtn) is the velocity over the nth interval ΔΔx. Very straightforward. The velocity equation tells us that itself: the denominator is the interval. Each interval ΔΔx is one, but the velocity over those intervals is not constant, since we have an acceleration. The velocity we find is the velocity over a particular subinterval of Δx. The subinterval of Δx is ΔΔx. The velocity may be written this way:
Δt' / ΔΔx
We have not gone to a limit or to zero; we have gone to a subinterval—the interval directly below the length and the period. What do I mean by this? I mean that our basic intervals or differentials are Δx and Δt. But if we have a curve equation, we have an acceleration or its mathematical equivalent. If we have an acceleration, then while we are measuring distance and period, something is moving underneath us. We have a change of a change. A rate of change. Our basic intervals are undergoing intervals of change. Not that hard to imagine. It happens all the time. While I am walking in the airport (measuring off the ground with my feet and my watch) I step onto a moving sidewalk. The ground has changed over a subinterval. It changes over only one subinterval, so I feel acceleration only over this subinterval. Once I achieve the speed of the sidewalk, my change stops, the subinterval ends, and I am at a new constant velocity. The subinterval is not an instant, it is the time(beginning of change) to the time(end of change). But in constant acceleration, I would be stepping onto faster sidewalks during each subsequent subinterval, and I would continue to accelerate.
All this means that the subinterval is not an instant. It is a definite period of time or distance, and this time or distance is given by the equation and the graph. As I have exhaustively shown, the subinterval in any graph where the box length is one and the independent variable is Δx is simply ΔΔx = 1. If we assign the box length to the meter, then ΔΔx = 1m. If we find the velocity "at a point," then we must assign that velocity to the interval preceding that point. Not an infinitesimal interval, but the interval 1 meter. If we then assign that velocity to a real object at a point in space, an object we have been plotting with our graph and our curve, then the velocity of the object must also be assigned to the preceding one-meter interval.
You will say, “But a real object does not accelerate by fits and starts. Nor does the curve on the graph. We should be able to find the velocity at any fractional point, in space or on the graph.”
Yes, you can, but the value you achieve will apply to the interval, not the instant. You can find the velocity at the value Δx = 5m or Δx = 9.000512m or at any other value, but any velocity will apply to the metric interval preceding the value.
You will say, "Good God, we need to be more precise than that. Can't I make that interval smaller somehow?”
Of course: just assign your box length to a smaller magnitude. If you let each box equal an angstrom, then the interval preceding your velocity is also an angstrom. However, notice that you cannot arbitrarily assign magnitude. That is, if you are actually measuring your object to the precision of angstroms, fine. You can mirror that precision on your graph. But if you are not being that precise in your operation of measurement, then you can’t assign a very small magnitude to your box length just because you want to be closer to an instant or a point. Your graph is a representation of your operation of measurement. You cannot misrepresent that operation without cheating. It would be like using more significant digits than you have a right to.
This means that in physics, the precision of your measurement of your given variables completely determines the precision of your velocity. This is logically just how it should be. We should not be able to find the velocity at an instant or a point, when we cannot measure an instant or a point. An instantaneous velocity would have an infinite precision. We have a margin of error in all measurement of length and time, since we cannot achieve absolute accuracy. But heretofore we expected to find instantaneous velocities and accelerations, which would imply absolute accuracy.
The Second Derivative—Acceleration
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
Acceleration is traditionally Δv/Δt. By current notation, that is (ΔΔx /Δt)/ Δt. By my notation of extra deltas, that would be [Δ(ΔΔx)/ΔΔt] / ΔΔt . My variables have been upside down this whole paper, meaning I have been finding slope and velocity as t/x instead of x/t. So flip that last equation
[Δ(ΔΔt)/ΔΔx] / ΔΔx
As we have found over and over, ΔΔx = 1, therefore that equation reduces to ΔΔΔt. For the acceleration we seek ΔΔΔt. The denominator is one, as you can plainly see, which means we are still seeking ΔΔΔt over a subinterval of one, not an interval diminishing to zero or to a limit.
We are given Δt = Δx3
We find from the table 3ΔΔΔx2 = ΔΔΔΔx3
We simplify 3ΔΔx2 = ΔΔΔx3
We seek Δt'' or ΔΔΔt
We notice ΔΔΔt = ΔΔΔx3 since we can add the same deltas to both sides
We substitute 3ΔΔx2 = ΔΔΔt
Back to the table 2ΔΔx = ΔΔΔx2
Simplify 2Δx = ΔΔx2
Substitute once more 6Δx = ΔΔΔt
At Δx = 5, ΔΔΔt = 30
The subinterval for the acceleration is the same as the subinterval for velocity. This subinterval is 1.
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 ad hoc invention regarding the calculus: one which may now be jettisoned. My redefinition of the derivative as simply the rate of change of the dependent variable demands a re-analysis of almost all higher math.*
The entire mess was built on one great error: all these mathematicians thought that the point on the graph or on the mathematical curve represented a point in space or a physical point. There was therefore no way, they thought, to find a subinterval or a differential without going to zero. But the subinterval is just the number one, as I have shown. That was the first given of the graph, and of the number line. The differential ΔΔx = 1 defines the entire graph, and every curve on it. That constant differential is the denominator of every possible derivative—first, second or last. The derivative is not the limit as Δx approaches zero of Δf(x)/Δx. It is the value Δf(x)/1.
And this is precisely why the Umbral Calculus works. The current interpretation and formalism of the Calculus of Finite Differences is so complex and over-signed that it is difficult to tell what is going on. But my simple explanation of it above shows the groundwork clearly, even to those who are not experts in this subfield. Once you limit the Calculus of Finite Differences to the integers, build a simple table, and refuse to countenance things like forward differences and backward differences (which are just baggage), the clouds begin to dissipate. You give the constant differential 1 to the table, not arbitrarily, but because the number line itself has a constant differential of 1. We have defined the number 1 as the constant differential of the world and of every possible space. Mathematicians seem apt to forget it, but it is so. Every time we apply numbers to a problem, we have automatically defined our basic differential as 1. What this means, operationally, is that in many problems, exponents begin to act like subscripts, or the reverse. To see what I mean, go back to the table above. Because the integer 1 defines the table and the constant differentials on it, the exponents could be written as subscripts without any change to the math.
Once we have defined our basic differential as 1, we cannot help but mirror much of the math of subscripts, since subscripts are of course based on the differential 1. Unless you are very iconoclastic, your subscript changes 1 each time, which means your subscript has a constant differential of 1. So does the Calculus of Finite Differences, when it is used to replace the Infinite Calculus and derive the derivative equation like I have done here. Therefore it can be no mystery when other subscripted equations—if they are explicitly or implicitly based on a differential of 1—are differentiable.
Beyond this, by redefining the problem completely, I have been able to prove that instantaneous values are a myth. They do not exist on the curve or on the graph. Furthermore, they imply absolute accuracy in finding velocities and accelerations, when the variables these motions are made of—distance and time—are not, and cannot be, absolutely accurate. Instantaneous values do not exist even as undefined mathematical concepts in the calculus, since they were arrived at by assigning diminishing differentials to points that were not points. You cannot postulate the existence of a limit at a “point” that is already defined by two differentials, (x - 0) and (y - 0).
I achieved all this with an algorithm that is simple and easy to understand. Calculus may now be taught without any mystification. No difficult proofs are required; nothing must be taken on faith. Every step of my derivation is capable of being explained in terms of basic number theory, and any high school student will see the logic in substituting values from the chart into curve equations.
[As proof that the calculus does not go to a limit, an infinitesimal, or approach zero, you may consult my second paper on Newton's orbital equation a = v2/r. There, I use the equation on the Moon, showing that the acceleration of the Moon due to the Earth is not an instantaneous acceleration. In other words, it does not take place at an instant or over an infinitesimal time. I actually calculate the real time that passes during the given acceleration, showing in a specific problem that the calculus goes to a subinterval, not a limit or infinitesimal. That subinterval is both finite and calculable in any physical problem. In other words, I find the subinterval that acts as 1 in a real problem. I find the value of the baseline differential.]
[In a new paper, I prove my contention here that calculus is fundamentally misunderstood to this day by analyzing a textbook solution of variable acceleration. I show that the first integral is used where the second derivative should be used, proving that scientists don't comprehend the basic manipulations of the calculus. Furthermore, I show that calculus is taught upside-down, by defining the derivative in reverse.]
*For example, my correction to the calculus changes the definition of the gradient, which changes the definition of the Lagrangian, which changes the definition of the Hamiltonian. Indeed, every mathematical field is affected by my redefinition of the derivative. I have shown that all mathematical fields are representations of intervals, not physical points. It is impossible to graph or represent a physical point on any mathematical field, Cartesian or otherwise. The gradient is therefore the rate of change over a definite interval, not the rate of change at a point.
Symplectic topology also relies upon the assumptions I have overturned in this paper. If points on a Cartesian graph are not points in real space, then quantum mechanical states are not points in a symplectic phase space. Hilbert space also crumbles, since the mathematical formalism cannot apply to the fields in question. Specifically, the sequence of elements, whatever they are, does not converge into the vector space. Therefore the mathematical space is not equivalent to the real space, and the one cannot fully predict the other. This means that the “uncertainty” of quantum mechanics is due (at least in part) to the math and not to the conceptual framework. That is to say, the various difficulties of quantum physics are primarily problems of a misdefined Hilbert space and a misused mathematics (vector algebra), and not problems of probabilities or philosophy.
In fact, all topologies are affected by this paper. Elementary topology makes the same mistake as the calculus in assuming that a line in R2 represents a one-dimensional subspace. But I have just shown that a line in R2 represents a velocity, which is not a one-dimensional subspace. I proved in section 1 above that a point in R2 was already a two-dimensional entity, so a line must be a three-dimensional subspace. In R3 a line represents an acceleration. In R4 a line represents a cition (Δa). Since velocity is a three-dimensional quantity—requiring the dimensions y and t, for instance, plus a change (a change always implies an extra dimension)—it follows that a line in Rn represents an (n + 1)—dimensional subspace. This means that all linear and vector algebras must be reassessed. Tensors are put on a different footing as well, and that is a generous assessment. Not one mathematical assumption that relies on the traditional assumptions of differential calculus, topology, linear algebra, or measure theory is untouched by this paper.
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.
1See for example, Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.
2Newton, Isaac, Mathematical Papers, 8: 597.
3Boyer, Carl. B., The History of the Calculus and its Conceptual Development, p. 227.
To see how this paper ties into the problems of Quantum Mechanics, see my paper Quantum Mechanics and Idealism.
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