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A
REDEFINITION OF THE DERIVATIVE why
the calculus works— and why it doesn't
by
Miles Mathis
A
note on my calculus papers, 2006
For a simplification
or gloss of this paper, you may go here
For briefer,
less technical analyses of contemporary calculus, you may now
go here or 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.
Introduction
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
redefining 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
redefinition, 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 rederivation of the core equation
of the differential calculus, by a method that uses no infinite
series and no limit concept. I will not rederive 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 reestablished 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
reinvention and reinterpretation 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
reinterpretation of the Calculus of Finite Differences allows me
to solve many of the pointparticle problems of QED without
renormalization. I will show that the equations of QED required
renormalization only because they had first been denormalized 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 precalculus
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’ = nx^{n1}.
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.
Part
One The Groundwork
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 "anypoint", 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 counterintuitive. 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
twodimensional yardstsick. This analysis may seem selfevident,
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 zerodimensional
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 xdistance 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 graphyou are
postulating making the graph disappear, which would also make the
curve disappear).
In its own way, the historical
derivation of the derivative sometimes understands and admits
this. Readers of my papers like to send me to the epsilon/delta
definition, as an explanation of the limit concept. The
epsilon/delta definition is just this: For all ε>0
there is a δ>0 such that whenever x 
x_{0}<δ then f(x) 
y_{0}<ε. What I want to
point out is that x  x_{0} 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 nearzero 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 reallife 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 threevariable 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 threevariable 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 taxis. 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 twovariable 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 reallife 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.
Part
Two Historical Interlude & 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
uptodate. 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 [(x0) and (y0)].
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
nonstarter. 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}
 x^{2} = 2xδx + δx^{2} 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
coordinates of another anypoint on the curve—this
anypoint being some distance further along the curve than the
first anypoint. But a closer examination will show that the
second curve equation is not the same as the first. The anypoint
expressed by the second equation is not on the curve y = x^{2}.
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 anypoint on a curve from an anypoint 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 + δx^{2}
to the right side. This might have been justified by some
argument if it gave us two anypoints on the same curve, but it
doesn’t. We have completed an illegal operation for no
apparent reason.
Now we subtract the first anypoint from the second anypoint.
What do we get? Well, we should get a third anypoint. What is
the coordinate of this third anypoint? It is impossible to say,
since we got rid of the variable y. A coordinate 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 changeiny 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 anypoint
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.]
Part
Three 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 90^{o} 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 = Δx_{f}
 Δx_{i} Where Δ x_{f} is the
final cardinal number and Δx_{i} 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 seconddegree 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 xaxis itself has a rate of change of
one, and the y or taxis 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 firstdegree 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 x_{1} and x_{2} arbitrarily, say x_{1}
= .0000000001 and x_{2} = .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
seconddegree rate of change (or 2RoC), and so on.
Part
Four The Algorithm
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... Δx^{2} = 1, 4, 9, 16, 25, 36,
49, 64, 81... Δx^{3} = 1, 8, 27, 64, 125, 216,
343... Δx^{4} = 1, 16, 81, 256, 625, 1296... Δx^{5}
= 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 ΔΔx^{2} = 1, 3, 5,
7, 9, 11, 13, 15, 17, 19 ΔΔx^{3} = 1, 7,
19, 37, 61, 91, 127 ΔΔx^{4} = 1, 15, 65,
175, 369, 671 ΔΔx^{5} = 1, 31, 211, 781,
2101, 4651, 9031 ΔΔΔx = 0, 0, 0, 0, 0, 0, 0
ΔΔΔx^{2} = 2, 2, 2, 2, 2, 2, 2, 2,
2, 2 ΔΔΔx^{3} = 6, 12, 18, 24, 30,
36, 42 ΔΔΔx^{4} = 14, 50, 110, 194,
302 ΔΔΔx^{5} = 30, 180, 570, 1320,
2550, 4380 ΔΔΔΔx^{3} = 6, 6,
6, 6, 6, 6, 6, 6 ΔΔΔΔx^{4} =
36, 60, 84, 108 ΔΔΔΔx^{5} =
150, 390, 750, 1230, 1830 ΔΔΔΔΔx^{4}
= 24, 24, 24, 24 ΔΔΔΔΔx^{5}
= 240, 360, 480, 600 ΔΔΔΔΔΔx^{5}
= 120, 120, 120 from this, one can predict that ΔΔΔΔΔΔΔx^{6}
= 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 seconddegree 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 ΔΔΔx^{2} = 2,
2, 2, 2, 2, 2, 2 ΔΔΔΔx^{3} =
6, 6, 6, 6, 6, 6, 6 ΔΔΔΔΔx^{4}
= 24, 24, 24, 24 ΔΔΔΔΔΔx^{5}
= 120, 120, 120 ΔΔΔΔΔΔΔx^{6}
= 720, 720, 720
Do you see it? 2ΔΔx =
ΔΔΔx² 3ΔΔΔx^{2}
= ΔΔΔΔx^{3} 4ΔΔΔΔx^{3}
= ΔΔΔΔΔx^{4} 5ΔΔΔΔΔx^{4}
= ΔΔΔΔΔΔx^{5}
6ΔΔΔΔΔΔx^{5} =
ΔΔΔΔΔΔΔx^{6} 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
nearzero 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 rederive 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 preexisting 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’ = nx^{n1} 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 xdirection (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
curveequation 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 seconddegree change. A length is a
static change in x. A distance is a movement from one x to the
other.
Part
Five The Derivation
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 Δt
= Δx^{3} Then the derivative is Δt' =
3Δx^{2} From my chart we can see that 3ΔΔΔx^{2}
= ΔΔΔΔx3 So, 3Δx^{2} =
ΔΔx^{3}
[Deltas may be cancelled across these particular
equalities]* And, Δt' = 3Δx^{2} = ΔΔx^{3}
Δt = Δx^{3} 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).
*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 freestanding 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.
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 = Δx^{3} We then look
at my second little chart to find Δx^{3} . We see
that the differential is constant (6) when the variable is
changing at this rate: ΔΔΔΔx^{3}.
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 = ΔΔΔx^{2}
means. The equation is telling us "two times the rate of
change of x is equal to the 2RoC of x^{2}." This is
somewhat like saying "twice the velocity of x is equal to
the acceleration of x^{2}." 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 = Δx^{3}
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 Δx^{3}. 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 = Δx^{3} We
find from the table 3ΔΔΔx^{2} =
ΔΔΔΔx^{3} We simplify 3Δx^{2}
= ΔΔx^{3} We seek ΔΔt We
notice ΔΔt = ΔΔx^{3} since we can
always add a delta to both sides* We substitute ΔΔt
= 3Δx^{2} ΔΔt = Δt' So
Δt' = 3Δx^{2}
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 3Δx^{2}." 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 = Δx^{3}
Then ΔΔt = 3Δx^{2}
Part
Six 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 (Δx_{n},
Δt_{n}) 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 onemeter 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.
Part
Seven 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 = Δx^{3} We find from the table
3ΔΔΔx^{2} = ΔΔΔΔx^{3}
We simplify 3ΔΔx^{2} = ΔΔΔx3 We
seek Δt'' or ΔΔΔt We notice ΔΔΔt
= ΔΔΔx^{3} since we can add the same
deltas to both sides We substitute 3ΔΔx^{2}
= ΔΔΔt Back to the table 2ΔΔx =
ΔΔΔx^{2} Simplify 2Δx = ΔΔx^{2}
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.
Summation
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 reanalysis 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 oversigned 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 = v^{2}/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
upsidedown, 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 R^{2}
represents a onedimensional subspace. But I have just shown that
a line in R^{2}
represents a velocity, which is not a onedimensional subspace. I
proved in section 1 above that a point in R^{2}
was already a twodimensional entity, so a line must be a
threedimensional subspace. In R^{3}
a line represents an acceleration. In R^{4}
a line represents a cition (Δa). Since velocity is a
threedimensional 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 R^{n}
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.
Addendum
In subsequent
papers, I show how my tables may be converted to find integrals,
trig functions,
herelogarithms, and so on. I think it is
clear that integrals may be found simply by reading up the table
rather than down. But there are several implications of this that
must be enumerated in full. And the conversion to trig functions
and the rest is somewhat more difficult, although not, I hope,
esoteric in any sense. All we have to do to convert the above
tables to any function is to consider the way that numbers are
generated by the various methods, keeping in mind the provisos I
have already covered here.
^{1}See
for example, Jacob Klein, Greek
Mathematical Thought and the Origin of Algebra.
^{2}Newton,
Isaac, Mathematical
Papers, 8:
597. ^{3}Boyer,
Carl. B., The
History of the Calculus and its Conceptual Development,
p. 227. ^{4}Ibid.
Links:
To see how this paper ties into the problems of Quantum
Mechanics, see my paper Quantum Mechanics
and Idealism.
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