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Calculus
Simplified
by
Miles Mathis
A
note on my calculus papers, 2006
Several years
ago I wrote a long paper on the foundation of
the calculus. The paper was not really dense or difficult—as
these things usually go—since I made a concentrated effort to
keep both the language and the math fairly simple. But because I
was tackling a large number of problems that had accumulated over
hundreds of years, and because calculus is considered a bit scary
to start with, the paper was still hard to absorb. I found it
necessary to talk a lot about history and theory and to bring up
very old and outdated ideas, like those of Archimedes and Euclid.
This ended up confusing most of my readers, I think, and very few
made it through to the end.
For this reason I have now returned to the subject, hoping to
further shorten and simplify my findings. What I plan to do here
is try to sell my idea to a hypothetical reader. I will imagine I
am talking to a high school student just entering first-semester
calculus. I will explain to him or her why my explanation is
necessary, why it is better, and why he or she should prefer to
take a course based on my explanation rather than a course based
on current theory. In doing this, I will show that current
notation and the current method of teaching calculus is a
gigantic mess. In a hundred years, all educated people will look
back and wonder how calculus could exist, and be taught, in such
a confusing manner. They will wonder how such basic math, so
easily understood, could have remained in a halfway state for so
many centuries. The current notation and derivation for the
equations of calculus will look to them like the leeches that
doctors used to put on patients, as an all-round cure, or like
the holes they drilled in the head to cure headache. Many
students have felt that learning calculus is like having holes
drilled in their heads, and I will show that they were right to
feel that way.
What some of you students have no doubt
already felt is that the further along in math you get, the more
math starts to seem like a trick. When you first start out, math
is pretty easy, since it makes sense. You don’t just learn an
equation. No, you learn an equation and you learn why the
equation makes sense. You don’t just acquire a fact, you
acquire understanding. For example, when you learn addition, you
don’t just learn how to use a plus sign. You also learn why the
sign works. You are shown the two apples and the one apple, and
then you put them together to get three apples. You see the
apples and you go, “Aha, now I see!” Addition makes sense to
you. It doesn’t just work. You fully understand why it
works.
Geometry is also
understood by most students, since geometry is a physical math.
You have pictures you can look at and line segments you can
measure and so on, so it never feels like some kind of magic. If
your trig teacher was a good teacher, you may have felt this way
about trig as well. The sine and cosine stuff seems a bit
abstract at first, but sooner or later, by looking at triangles
and circles, it may dawn on you that everything makes absolute
sense.
Algebra is the next
step, and many people get lost there. But if you can get your
head around the idea of a variable, you are halfway home.
But
when we get to calculus, everyone gets swamped. Notice that I did
not say, “almost everyone.” No, I said everyone. Even the
biggest nerd with the thickest glasses who gets A’s on every
paper is completely confused. Those who do well in their first
calculus courses are the ones that just memorize the equations
and don’t ask any questions. One reason for this is that with
calculus you will be given some new signs, and these signs will
not really make sense in the old ways. You will be given an arrow
pointing at zero, and this little arrow and zero will be
underneath variables or next to big squiggly lines. This arrow
and zero are supposed to mean, “let the variable or function
approach zero,” but your teacher probably won’t have time to
really make you understand what a function is or why anyone
wanted it to approach zero in the first place. Your teacher would
answer such a question by saying, “Well, we just let it go
toward zero and then see what happens. What happens is that we
get a solution. We want a solution, don’t we? If going to zero
gives us a solution, then we are done. You can’t ask questions
in math beyond that.”
Well,
if you teacher says that to you, you can tell your teacher he or
she is wrong. Math is not just memorizing equations, it is
understanding equations. All math, no matter how difficult, is
capable of being understood in the same way that 2+2=4 can be
understood; and if your teacher cannot explain it to you, then he
or she does not understand it.
What is happening with calculus is that you are taking your first
step into a new kind of math and science. It is a kind of
faith-based math. Almost everything you will learn from now on is
math of this sort. You will not have time to understand it,
therefore you must accept it and move on. Unless you plan to
become a professor of the history of math, you will not have time
to get to the roots of the thing and really make sense of it in
your head. What no high school or college student is supposed to
know is that even the history-of-math professors don’t
understand calculus. No one understands or ever understood
calculus, not Einstein, not Cauchy, not Cantor, not Russell, not
Bohr, not Feynman, no one. Not even Leibniz or Newton understood
it. That is a big statement, I know, but I have already proved it
and I will prove it again below. The short proof is to point out
that if they had really understood it, they would have corrected
it like I am about to. If any of these people had understood
calculus, they would have reconstructed the whole thing so that
you could understand it, too. There is no reason to teach you a
math that can’t be explained simply. There is no conspiracy.
You are taught calculus as a big mystery simply because, until
now, it was a big mystery.
Now, when I say that math after calculus is faith-based, I am
offending a lot of important people. Mathematicians are very
proud of their field, as you would expect, and they don’t want
some cowboy coming in and comparing it to religion. But I am not
just saying things to be novel or to get attention. I can give
you famous examples of how math has become faith-based. Many of
you will have heard of Richard Feynman, and not just because I
mentioned him ten sentences ago. He is probably the most famous
physicist after Einstein, and he got a lot of attention in the
second half of the 20th century—as one of the fathers of QED,
among other things. One of his most quoted quotes is, “Shut up
and calculate!” Meaning, “Don’t ask questions. Don’t try
to understand it. Accept that the equation works and memorize it.
The equation works because it matches experiment. There is no
understanding beyond that.”
All of quantum dynamics is based on this same idea, which started
with Heisenberg and Bohr back in the early 1900’s. “The
physics and math are not understandable, in the normal way, so
don’t ask stupid questions like that any more.” This last
sentence is basically the short form of what is called the
Copenhagen Interpretation of quantum dynamics. The Copenhagen
Interpretation applies to just about everything now, not just
QED. It also applies to Relativity, in which the paradoxes must
simply be accepted, whether they make sense or not. And you might
say that it also applies to calculus. Historically, your
professors have accepted the Copenhagen Interpretation of
calculus, and this interpretation states that students’
questions cannot be answered. You will be taught to understand
calculus like your teacher understands it, and if your teacher is
very smart he understands it like Newton understood it. He will
have memorized Newton’s or Cauchy’s derivation and will be
able to put it on the blackboard for you. But this derivation
will not make sense like 2+2=4 makes sense, and so you will still
be confused. If you continue to ask questions, you will be read
the Copenhagen Interpretation, or some variation of it. You will
be told to shut up and calculate.
The first semester of
calculus you will learn differential calculus. The amazing thing
is that you will probably make it to the end of the semester
without ever being told what a differential is. Most
mathematicians learn that differential calculus is about solving
certain sorts of problems using a derivative, and later courses
called “differential equations” are about solving more
difficult problems in the same basic way. But most never think
about what a differential is, outside of calculus. I didn’t
ever think about what a differential was until later, and I am
not alone. I know this because when I tell people that my new
calculus is based on a constant differential instead of a
diminishing differential, they look at me like I just started
speaking Japanese with a Dutch accent. For them, a differential
is a calculus term, and in calculus the differentials are always
getting smaller. So talking about a differential that does not
get smaller is like talking about a politician that does not lie.
It fails to register.
A
differential is one number subtracted from another number: (2-1)
is a differential. So is (x-y). A “differential” is just a
fancier term for a “difference”. A differential is written as
two terms and a minus sign, but as a whole, a differential stands
for one number. The differential (2-1) is obviously just 1, for
example. So you can see that a differential is a useful
expansion. It is one number written in a longer form. You can
write any number as a differential. The number five can be
written as (8-3), or in a multitude of other ways. We may want to
write a single number as a differential because it allows us to
define that differential as some useful physical parameter. For
instance, a differential is most often a length. Say you have a
ruler. Go to the 2-inch mark. Now go to the 1-inch mark. What is
the difference between the two marks? It is one inch, which is a
length. (2-1) may be a length. (x-y) may also be a length. In
pure math, we have no lengths, of course, but in math applied to
physics, a differential is very often a length.
The problem is that modern mathematicians do not like to teach
you math by drawing you pictures. They do not like to help you
understand concepts by having you imagine rulers or lengths or
other physical things. They want you to get used to the idea of
math as completely pure. They tell you that it is for your own
good. They make you feel like physical ideas are equivalent to
pacifiers: you must grow up and get rid of them. But the real
reason is that, starting with calculus, they can no longer draw
you meaningful pictures. They are not able to make you
understand, so they tell you to shut up and calculate. It is kind
of like the wave/particle duality, another famous concept you
have probably already heard of. Light is supposed to act like a
particle sometimes and like a wave at other times. No one has
been able to draw a picture of light that makes sense of this, so
we are told that it cannot be done. But in another one of my
papers I have drawn a picture of light that makes sense of this,
and in this paper I will show you a pretty little graph that
makes perfect sense of the calculus. You will be able to look at
the graph with your own eyes and you will see where the numbers
are coming from, and you will say, “Aha, I understand. That was
easy!”
There is basically only one equation that you
learn in your first semester of calculus. All the other equations
are just variations and expansions of the one equation. This one
equation is also the basic equation of what you will learn next
semester in integral calculus. All you have to do is turn it
upside down, in a way. This equation is
y’
= nxn-1
This
is the magic equation. What you won’t be told is that this
magic equation was not invented by either Newton or Leibniz. All
they did is invent two similar derivations of it. Both of them
knew the equation worked, and they wanted to put a foundation
under it. They wanted to understand where it came from and why it
worked. But they failed and everyone else since has failed. The
reason they failed is that the equation was used historically to
find tangents to curves, and everyone all the way back to the
ancient Greeks had tried to solve this problem by using a
magnifying glass. What I mean by that is that for millennia, the
accepted way to approach the problem and the math was to try to
straighten out the curve at a point. If you could straighten out
the curve at that point you would have the tangent at that point.
The ancient Greeks had the novel idea of looking at smaller and
smaller segments of the curve, closer and closer to the point in
question. The smaller the segment, the less it curved. Rather
than use a real curve and a real magnifying glass, the Greeks
just imagined the segment shrinking down. This is where we come
to the diminishing differential. Remember that I said the
differential was a length. Well, the Greeks assigned that
differential to the length of the segment, and then imagined it
getting smaller and smaller.
Two thousand years later, nothing had changed. Newton and Leibniz
were still thinking the same way. Instead of saying the segment
was “getting smaller” they said it was “approaching zero”.
That is why we now use the little arrow and the zero. Newton even
made tables, kind of like I will make below. He made tables of
diminishing differentials and was able to pull the magic equation
from these tables.
The problem
is that he and everyone else has used the wrong tables. You can
pull the magic equation from a huge number of possible tables,
and in each case the equation will be true and in each case the
table will “prove” or support the equation. But in only one
table will it be clear why the equation is true. Only one table
will be simple enough and direct enough to show a 16-year-old
where the magic equation comes from. Only one table will cause
everyone to gasp and say, “Aha, now I understand.” Newton and
Leibniz never discovered that table, and no one since has
discovered it. All their tables were too complex by far. Their
tables required you to make very complex operations on the
numbers or variables or functions. In fact, these operations were
so complex that even Newton and Leibniz got lost in them. As I
will show after I unveil my table, Newton and Leibniz were forced
to perform operations on their variables that were actually
false. Getting the magic equation from a table of diminishing
differentials is so complex and difficult that no one has ever
been able to do it without making a hash of it. It can be done,
but it isn’t worth doing. If you can pull the magic equation
from a simple table of integers, why try to pull it from a
complex table of functions with strange and confusing scripts?
Why teach calculus as a big hazy mystery, invoking infinite
series or approaches to 0’s or infinitesimals, when you can
teach it at a level that is no more complex than 1+1=2?
So
here is the lesson. I will teach you differential calculus in one
day, in one paper. If you have reached this level of math, the
only thing that should look strange to you in the magic equation
is the y’. You know what an exponent is, and you should know
that you can write an exponent as (n-1) if you want to. That is
just an expansion of a single number into a differential, as I
taught you above. If n=2, for instance, then the exponent just
equals 1, in that case. Beyond that, “n” is just another
variable. It could be “z” or “a” or anything else. That
variable just generalizes the equation for us, so that it applies
to all possible exponents.
All
that is just simple algebra. But you don’t normally have primed
variables in high school algebra. What does the prime signify?
That prime is telling you that y is a different sort of variable
than x. When you apply this magic equation to physics, x is
usually a distance and y is a velocity. A variable could also be
an acceleration, or it could be a point, or it could be just
about anything. But we need a way to remind ourselves that some
variables are one kind of parameter and some variables are
another. So we use primes or double primes and so on.
This is important, because it means that mathematically, a
velocity is not a distance, and an acceleration is not a
velocity. They have to be kept separate. A calculus equation
takes you from one sort of variable to another sort. You cannot
have a distance on both sides of the magic equation, or a
velocity on both sides. If x is a distance, y’ cannot be a
distance, too.
Some people
will try to convince you later that calculus can be completely
divorced from physics, or from the real world. They will stress
that calculus is pure math, and that you don’t need to think of
distances or velocities or physical parameters. But if this were
true, we wouldn’t need to keep our variables separate. We
wouldn’t need to keep track of primed variables, or later
double-primed variables and so on. Variables in calculus don’t
just stand for numbers, they stand for different sorts of
numbers, as you see. In pure math, there are not different sorts
of numbers, beyond ordinal and cardinal, or rational and
irrational, or things like that. In pure math, a counting integer
is a counting integer and that is all there is to it. But in
calculus, our variables are counting different things and we have
to keep track of this. That is what the primes are for.
What, you may ask, is the difference between a length and a
velocity? Well, I think you can probably answer that without the
calculus, and probably without much help from me. To measure a
length you don’t need a watch. To measure velocity, you do.
Velocity has a “t” in the denominator, which makes it a rate
of change. A rate is just a ratio, and a ratio is just one number
over another number, with a slash in between. Basically, you hold
one variable steady and see how the other variable changes
relative to it. With velocity, you hold time steady (all the
ticks are the same length) and see how distance changes during
that time. You put the variable you know more about (it is
steady) in the denominator and the variable you are seeking
information about (you are measuring it) in the numerator. Or,
you put the defined variable in the denominator (time is defined
as steady) and the undefined variable in the numerator (distance
is not known until it is measured).
All this can also be applied to velocity and acceleration. The
magic equation can be applied to velocity and acceleration, too.
If x is a velocity, then y’ is an acceleration. This is because
acceleration is the rate of change of the velocity. Acceleration
is v/t. So you can see that y’ is always the rate of change of
x. Or, y’ is always x/t. This is another reason that calculus
can’t really be divorced completely from physics. Time is a
physical thing. A pure mathematician can say, “Well, we can say
that y’ is always x/z, where z is not time but just a pure
variable.” But in that case, x/z is still a rate of change. You
can refuse to call “z” a time variable, but you still have
the concept of change. A pure number changing still implies time
passing, since nothing can change without time passing.
Mathematicians want “change” without “time”, but change
is time. If a mathematician can imagine or propose change without
time, then he is cleverer than the gods by half, since he has
just separated a word from its definition.
At any rate, I
think you are already in a better position to understand the
calculus than any math student in history. Whether you like that
little diversion into time and change is really beside the point,
since even if you believe in pure math it doesn’t effect my
argument.
All the famous
mathematicians in history have studied the curve in order to
study rate of change. To develop the calculus, they have taken
some length of some curve and then let that length diminish. They
have studied the diminishing differential, the differential
approaching zero. This approach to zero gives them an infinite
series of differentials, and they apply a method to the series in
order to understand its regression.
But it is much more useful to notice that curves always concern
exponents. Curves are all about exponents, and so is the
calculus. So what I did is study integers and exponents, in the
simplest situations. I started by letting z equal some point. If
I let a variable stand for a point, then I have to have a
different sort of variable stand for a length, so that I don’t
confuse a point and a length. The normal way to do this is to let
a length be Δz (read “change in z”). I want lengths instead
of points, since points cannot be differentials. Lengths can. You
cannot think of a point as (x-y). But if x and y are both points,
then (x-y) will be a length, you see.
In the first line of my table, I list the possible integer values
of Δz. You can see that this is just a list of the integers, of
course. Next I list some integer values for other exponents of
Δz. This is also straightforward. At line 7, I begin to look at
the differentials of the previous six lines. In line 7, I am
studying line 1, and I am just subtracting each number from the
next. Another way of saying it is that I am looking at the rate
of change along line 1. Line 9 lists the differentials of line 3.
Line 14 lists the differentials of line 9. I think you can follow
my logic on this, so meet me down below.
1
Δz
1, 2, 3, 4, 5, 6, 7, 8, 9….
2
Δ2z
2, 4, 6, 8, 10, 12, 14, 16, 18….
3
Δz2
1, 4, 9, 16, 25, 36, 49 64,
81
4
Δz3
1, 8, 27, 64, 125, 216, 343
5
Δz4
1,
16, 81, 256, 625, 1296
6
Δz5
1,
32, 243, 1024, 3125, 7776, 16807
7
ΔΔz
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
8
ΔΔ2z
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
9
ΔΔz2
1, 3, 5, 7, 9, 11, 13, 15, 17,
19
10
ΔΔz3
1,
7, 19, 37, 61, 91, 127
11
ΔΔz4
1,
15, 65, 175, 369, 671
12
ΔΔz5
1,
31, 211, 781, 2101, 4651, 9031
13
ΔΔΔz
0, 0, 0, 0, 0, 0, 0
14
ΔΔΔz2
2,
2, 2, 2, 2, 2, 2, 2, 2, 2
15
ΔΔΔz3
6,
12, 18, 24, 30, 36, 42
16
ΔΔΔz4
14,
50, 110, 194, 302
17
ΔΔΔz5
30, 180, 570, 1320, 2550,
4380
18
ΔΔΔΔz3
6,
6, 6, 6, 6, 6, 6, 6
19
ΔΔΔΔz4
36,
60, 84, 108
20
ΔΔΔΔz5
150,
390, 750, 1230, 1830
21
ΔΔΔΔΔz4
24, 24, 24, 24
22
ΔΔΔΔΔz5
240, 360, 480, 600
23
ΔΔΔΔΔΔz5
120,
120, 120
from this, one can
predict that
24
ΔΔΔΔΔΔΔz6
720, 720, 720
And
so on.
Again, 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, and a length
is a differential. It is also a list of the integers, as I said.
After that it is easy to follow my method. It is easy until you
get to line 24, where I say, “One can predict that. . . .” Do
you see how I came to that conclusion? I did it by pulling out
the lines where the differential became constant.
7
ΔΔz 1, 1, 1, 1, 1, 1, 1
14
ΔΔΔz2
2,
2, 2, 2, 2, 2, 2
18 ΔΔΔΔz3
6,
6, 6, 6, 6, 6, 6
21 ΔΔΔΔΔz4
24,
24, 24, 24
23 ΔΔΔΔΔΔz5
120,
120, 120
24 ΔΔΔΔΔΔΔz6
720,
720, 720
Do you see it now?
2ΔΔz = ΔΔΔz2
3ΔΔΔz2
= ΔΔΔΔz3
4ΔΔΔΔz3
= ΔΔΔΔΔz4
5ΔΔΔΔΔz4
= ΔΔΔΔΔΔz5
6ΔΔΔΔΔΔz5
= ΔΔΔΔΔΔΔz6
All these equations
are equivalent to the magic equation, y’ = nxn-1.
In any of those equations, all we have to do is let x equal the
right side and y’ equal the left side. No matter what exponents
we use, the equation will always resolve into our magic equation.
If I know anything about teenagers, I will expect this
reaction: “Well, sir, that may be a great simplification of
Newton, for all we know, but it is not exactly 1+1=2.” Fair
enough. It may take a bit of sorting through. But I assure you
that compared to the derivation you will learn in school, my
table is a miracle of simplicity and transparence. Not only that,
but I will continue to simplify and explain. Since in those last
equations we have z on both sides, we can cancel a lot of those
deltas and get down to this:
2z = Δz2
3z2
= Δz3
4z3
= Δz4
5z4
= Δz5
6z5
= Δz6
Now,
if we reverse it, we can read that first equation as, “the rate
of change of z squared is two times z.” That is information
that we just got from a table, and that table just listed
numbers. Simple differentials. One number subtracted from the
next.
This is useful to us
because it is precisely what we were looking for when we wanted
to learn calculus. We use the calculus to tell us what the rate
of change is for any given variable and exponent. Given an x, we
seek a y’, where y’ is the rate of change of x. And that is
what we just found. Currently, calculus calls y’ the
derivative, but that is just fancy terminology that does not
really mean anything. It just confuses people for no reason. The
fact is, y’ is a rate of change, and it is better to remember
that at all times.
You may still have one very important
question. You will say, “I see where the numbers are coming
from, but what does it mean?
Why are we selecting the lines in the table where the numbers are
constant?” We are going to those lines, because in those lines
we have flattened out the curve. If the numbers are all the same,
then we are dealing with a straight line. A constant differential
describes a straight line instead of a curve. We have dug down to
that level of change that is constant, beneath all our other
changes. As you can see, in the equations with a lot of deltas,
we have a change of a change of a change. . . . We just keep
going down to sub-changes until we find one that is constant.
That one will be the tangent to the curve. If we want to find the
rate of change of the exponent 6, for instance, we only have to
dig down 7 sub-changes. We don’t have to approach zero at all.
In a way we have done the same
thing that the Greeks were doing and that Newton was doing. We
have flattened out the curve. But we did not use a magnifying
glass to do it. We did not go to a point, or get smaller and
smaller. We went to sub-changes, which are a bit smaller, but
they aren’t anywhere near zero. In fact, to get to zero, you
would have to have an infinite number of deltas, or sub-changes.
And this means that your exponent would have to be infinity
itself. Calculus never deals with infinite exponents, so there is
never any conceivable reason to go to zero. We don’t need to
concern ourselves with points at all. Nor do we need to talk of
infinitesimals or limits. We
don't have an infinite series, and we don't have any vanishing
terms. We
have a definite and limited series, one that is 7 terms long with
the exponent 6 and only 3 terms long with the exponent 2.
I
hope you can see that the magic equation is just a generalization
of all the constant differential equations we pulled from the
table. To “invent” the calculus, we don’t have to derive
the magic equation at all. All we have to do is generalize a
bunch of specific equations that are given us by the table. By
that I mean that the magic equation is just an equation that
applies to all similar situations, whereas the specific equations
only apply to specific situations (as when the exponent is 2 or
3, for example). By using the further variable “n”, we are
able to apply the equation to all exponents. Like this:
nzn-1
= Δzn
And we don’t have to prove or derive the table either. The
table is true by definition. Given the definition of integer and
exponent, the table follows. The table is axiomatic number
analysis of the simplest kind. In this way I have shown that the
basic equation of differential calculus falls out of simple
number relationships like an apple falls from a tree.
Even pure mathematicians can have nothing to say against my
table, since it has no necessary physical content. I call my
initial differentials lengths, but that is to suit myself. You
can subtract all the physical content out of my table and it is
still the same table and still completely valid.
We don’t need to consider any infinite series, we don’t need
to analyze differentials approaching zero in any strange way, we
don’t need to think about infinitesimals, we don’t need to
concern ourselves with functions, we don’t need to learn weird
notations with arrows pointing to zeros underneath functions, and
we don’t need to notate functions with parentheses and little
“f’s”, as in f(x). But the most important thing we can
ditch is the current derivation of the magic equation, since we
have no need of it. I will show you that this is important,
because the current derivation is gobblydegook.
I am once
again making a very big claim, but once again I can prove it, in
very simple language. Let’s look at the current derivation of
the magic equation. This derivation is a simplified form of
Newton’s derivation, but conceptually it is exactly the same.
Nothing important has changed in 350 years. This is the
derivation you will be taught this semester. The figure δ stands
for “a very small change”. It is the small-case Greek “d”,
which is called delta. The large-case is Δ, remember, which is a
capital delta. Sometimes the two are used interchangeably, and
you may see the derivation below with Δ instead of δ. You may
even see it with the letter “d”. I will not get into which is
better and why, since in my opinion the question is moot. After
today we can ditch all three.
Anyway, we start by taking any functional equation. “Functional”
just means that y depends upon x in some way. Think of how a
velocity depends on a distance. To measure a velocity you need to
know a distance, so that velocity is a function of distance. But
distance is not a function of velocity, since you can measure a
distance without being concerned at all about velocity. So, we
take any functional equation, say
y = x2
Increase
it by δy and δx to obtain
y + δy = (x + δx)2
subtract
the first equation from the second:
δy = (x + δx)2
- x2
= 2xδx + δx2
divide
by δx
δy /δx = 2x + δx
Let δx go to zero (only on the
right side, of course)
δy / δx = 2x
y’ = 2x
That
is how they currently derive the magic equation. Any teenager, or
any honest person, will look at that series of operations and go,
“What the. . . ?” How can we justify all those seemingly
arbitrary operations? The answer is, we can’t. As it turns out,
precisely none of them are legal. But Newton used them, he was a
very smart guy, and we get the equation we want at the end. So we
still teach that derivation. We haven’t discovered anything
better, so we just keep teaching that.
Let me run through the operations quickly, to show you what is
going on. We only have four operations, so it isn’t that
difficult, really. Historically, only the last operation has
caused people to have major headaches. Newton was called on the
carpet for it soon after he published it, by a clever bishop
named Berkeley. Berkeley didn’t like the fact that δx went to
zero only on the right side. But no one could sort through it one
way or the other and in a few decades everyone just decided to
move on. They accepted the final equation because it worked and
swept the rest under the rug.
But what I will show you is that the derivation is lost long
before the last operation. That last operation is indeed a big
cheat, but mathematicians have put so many coats of pretty paint
on it that it is impossible to make them look at it clearly
anymore. They answer that δx is part of a ratio on the left
side, and because of that it is sort of glued to the δy above
it. They say that δy/δx must be considered as one entity, and
they say that this means it is somehow unaffected by taking δx
to zero on the right side. That is math by wishful thinking, but
what are you going to do?
To
get them to stand up and take notice, I have been forced to show
them the even bigger cheats in the previous steps. Amazingly, no
one in all of history has noticed these bigger cheats, not even
that clever bishop. So let us go through all the steps.
In the first equation, the variables stand for either “all
possible points on the curve” or “any possible point on the
curve.” The equation is true for all points and any point. Let
us take the latter definition, since the former doesn’t allow
us any room to play. So, in the first equation, we are at “any
point on the curve”. In the second equation, are we still at
any point on the same curve? Some will think that (y + δy) and
(x + δx) are the co-ordinates of another any-point on the
curve—this any-point being some distance further along the
curve than the first any-point. But a closer examination will
show that the second curve equation is not the same as the first.
The any-point expressed by the second equation is not on the
curve y = x2.
In fact, it must be exactly δy off
that first curve. Since this
is true, we must ask why we would want to subtract the first
equation from the second equation. Why do we want to subtract an
any-point on a curve from an any-point off that curve?
Furthermore, in going from equation 1 to equation 2, we have
added different amounts to each side. This is not normally
allowed. Notice that we have added δy to the left side and 2xδx
+ δx2
to the right side. This might
have been justified by some argument if it gave us two any-points
on the same curve, but it doesn’t. We have completed an illegal
operation for no apparent reason.
Now we subtract the first any-point from the second any-point.
What do we get? Well, we should get a third any-point. What is
the co-ordinate of this third any-point? It is impossible to say,
since we got rid of the variable y. A co-ordinate is in the form
(x,y) but we just subtracted away y. You must see that δy is not
the same as y, so who knows if we are off the curve or on it.
Since we subtracted a point on the first curve from a point off
that curve, we would be very lucky to have landed back on the
first curve, I think. But it doesn’t matter, since we are
subtracting points from points. Subtracting points from points is
illegal anyway. If you want to get a length or a differential you
must subtract a length from a length or a differential from a
differential. Subtracting a point from a point will only give you
some sort of zero—another point. But we want δy to stand for a
length or differential in the third equation, so that we can
divide it by δx. As the derivation now stands, δy must be a
point in the third equation.
Yes, δy is now a point. It is not a change-in-y in the sense
that the calculus wants it to be. It is no longer the difference
in two points on the curve. It is not a differential! Nor is it
an increment or interval of any kind. It is not a length, it is a
point. What can it possibly mean for an any-point to approach
zero? The truth is it doesn’t mean anything. A point can’t
approach a zero length since a point is already a zero length.
Look at the second equation
again. The variable y stands for a point, but the variable δy
stands for a length or an interval. But if y is a point in the
second equation, then δy must be a point in the third equation.
This makes dividing by δx in the next step a logical and
mathematical impossibility. You cannot divide a point by any
quantity whatsoever, since a point is indivisible by definition.
The final step—letting δx go to zero—cannot be defended
whether you are taking only taking the denominator on the left
side to zero or whether you are taking the whole fraction toward
zero (which has been the claim of most). The ratio δy/δx was
already compromised in the previous step. The problem is not that
the denominator is zero; the problem is that the numerator is a
point. The
numerator is zero.
My new method drives right around this mess by
dispensing with points altogether. You can see that the big
problem in the current derivation is in trying to subtract one
point from another. But you cannot subtract one point from
another, since each point acts like a zero. Every point has zero
extension in every direction. If you subtract zero from zero you
can only get zero.
You will
say that I subtracted one point from another above (x-y) and got
a length, but that is only because I treated each variable as a
length to start with. Each “point” on a ruler or curve is
actually a length from zero, or from the end of the ruler. Go to
the “point” 5 on the ruler. Is that number 5 really a point?
No, it is a length. The number 5 is telling you that you are five
inches from the end of the ruler. The number 5 belongs to the
length, not the point. Which means that the variable x, that may
stand for 5 or any other number on the ruler, actually stands for
a length, not a point. This is true for curves as well as
straight lines or rulers. Every curve is like a curved ruler, so
that all the numbers at “points” on the curve are actually
lengths.
You may say, “Well,
don’t current mathematicians know that? Doesn’t the calculus
take that into account? Can’t you just go back into the
derivation above and say that y is a length from zero instead of
a point, which means that in the third equation δy is a length,
which means that the derivation is saved?” Unfortunately, no.
You can’t say any of those things, since none of them are true.
The calculus currently believes that y’ is an instantaneous
velocity, which is a velocity at a point and at an instant. You
will be taught that the point y is really a point in space, with
no time extension or length. Mathematicians believe that the
calculus curve is made up of spatial points, and physicists of
all kinds believe it, too. That is why my criticism is so
important, and why it cannot be squirmed out of. The variable y
is not a length in the first equation of the derivation, and this
forces δy to be a point in the third equation.
A differential stands for a length only if the two terms in the
differential are already lengths. They must both have extension.
Five inches minus four inches is one inch. Everything in that
sentence is a length. But the fifth-inch mark minus the
fourth-inch mark is not the one inch-mark, nor is it the length
one inch. A point minus a point is a meaningless operation. It is
like 0 – 0.
This is the
reason I was careful to build my table only with lengths. I don’t
use points. This is because I discovered that you can’t assign
numbers to points. If you can’t assign numbers to points, then
you can’t assign variables or functions to points. When I was
building my table above, I kind of blew past this fact, since I
didn’t want to confuse you with too much theory. My table is
all lengths, but I didn’t really tell you why it had to be like
that. Now, however, I think you are ready to notice that points
can’t really enter equations or tables at all. Only ordinal
numbers can be applied to points. These are ordinal numbers: 1st,
2nd, 3rd. The fifth point, the eighth point, and so on. But math
equations apply to cardinal or counting numbers, 1, 2, 3. You
can’t apply a counting number to a point. As I showed with the
ruler, any time you apply a counting number to a “point” on
the ruler, that number attaches to the length, not the point. The
number 5 means five inches, and that is a length from zero or
from the end of the ruler. It is the same with all lines and
curves. And this applies to pure math as well as to applied math.
Even if your lines and curves are abstract, everything I say here
still applies in full force. The only difference is that you no
longer call differentials lengths; you call them intervals or
differentials or something.
The students will now say,
“Can’t you go back yourself and redefine all the points as
lengths, in the existing derivation? Can’t you fix it somehow?”
The answer is no. I can’t. I
have showed you that Newton cheated on all four steps, not just
the last one. You can’t “derive” his last equation from his
first by applying a series of mathematical operations to them
like this, and what is more you don’t need to. I have showed
with my table that you don’t need to derive the magic equation
since it just drops out of the definition of exponent fully
formed. The equation is axiomatic. What I mean by that is that it
really is precisely like the equation 1+1=2. You don’t need to
derive the equation 1+1=2, or prove it. You can just pull it from
a table of apples or oranges and generalize it. It is
definitional. It is part of the definition of number and
equality. In the same way, the magic equation is a direct
definitional outcome of number, equality, and exponent. Build a
simple table and the equation drops out of it without any work at
all.
If you must have a derivation, the simplest possible
one is this one:
We are given
a functional equation of the general sort
y = xn
and we seek y’, where, by definition
y’ = Δxn
Then we go to our generalized equation from the table, which
is
nxn-1
= Δxn
By substitution, we get
y’ = nxn-1
That’s all we need.
But I will give you one other piece of information that will come
in handy later. Remember how we cancelled all those deltas, to
simplify the first equations coming out of the table? Well, we
did that just to make things look tidier, and to make the
equations look like the current calculus equations. But those
deltas are really always there. You can cancel them if you want
to clean up your math, but when you want to know what is going on
physically, you have to put them back in. What they tell you is
that when you are dealing with big exponents, you are dealing
with very complex accelerations. Once you get past the exponent
two, you aren’t dealing with lengths or velocities anymore. The
variable x to the exponent 6 will have 7 deltas in front of it,
as you can see by going back to the table. That is a very high
degree of acceleration. Three deltas is a velocity. Four is an
acceleration. Five is a variable acceleration. Six is a change of
a variable acceleration. And so on. Most people can’t really
visualize anything beyond a variable acceleration, but high
exponent variables do exist in nature, which means that you can
go on changing changes for quite a while. If you go into physics
or engineering, this knowledge may be useful to you. A lot of
physicists appear to have forgotten that accelerations are often
variable to high degrees. They assume that every acceleration in
nature is a simple acceleration.
In my long paper I
covered a lot of other interesting topics, but I will only
mention one more of them here. I have told you a bit about
quantum mechanics above, so I will give you a clue about the end
of that story, too. QED hit a wall about 20 years ago, and that
is why all the big names are now working on string theory. String
theory is a horrible mess, one that makes the mess of calculus
look like spilled milk. But one of the main reasons it was
invented was to save QED from the point. This problem I have
solved for you about the point is exactly the same one that
cold-cocked QED. All of physics is dependent on calculus and its
offshoots, and using calculus with points in the equations has
ended up driving everyone a little mad. The only way that
physicists could make the equations of QED keep working is by
performing silly operations on them, like the ones that Newton
performed in his derivation. These operations in QED are called
“renormalization”. That is a big word for fudging. The
inventor of renormalization was the same Richard Feynman who I
told you about above. His students are still finding new ways to
renormalize equations that won’t work in normal ways. Mr.
Feynman was a big mess maker, but he did have the honesty to at
least admit it, regarding renormalization. He himself called it
“hocus pocus” and a “dippy process” that was “not
mathematically legitimate.” It would have been nice if Newton
or Leibniz or Cauchy had had the intellectual honesty to say the
same about the calculus derivation.
The reason this should be interesting to you is that my
correction to the calculus solves all the problems of QED at one
blow, although they haven’t figured that out yet.* Just by
reading this paper you are now smarter than all the “geniuses”
fudging giant equations. With your new knowledge, you can go to
college, wade briskly through all the muck, and start putting the
house in order. Your understanding of calculus and the point will
allow you to climb ladders that no one even knew existed. So
please remember me when you get to the top. And don't dump any
more garbage that might land on my head.
Addendum:
Here is an email I got from a
reader, confirming that—at least for some—my method does
indeed make calculus transparent at last:
Miles,
I
just wanted to say thank you for teaching me calculus in a day
(actually it was really only a couple of hours but who's
counting). I came across your work recently and have been
devouring it page by page. I especially love your candor
(frankness, honesty, truthfulness, sincerity, bluntness,
straightforwardness etc.). When I was in high school, I got 80's
and 90's in all my maths and sciences, except for calculus where
I got a 50. I had no idea what was going on. Luckily, I still got
into university where I again got 50 in first year calculus. I
re-did the class and got 95 but that is only because I gave up
trying to understand calculus and just memorized the rules. Even
after graduating from university, and after 30 years working as a
computer scientist doing advanced imaging and robotics, the
calculus still mystified me. That was until a couple of days ago
when I read your pages on calculus. Now I am 100 percent certain
I understand differential calculus.
I'm
sure this will go a long ways in helping me understand physics
which has also mystified me for many years. I have been trying to
redefine (re-divine) physics from the perspective of the fractal
paradigm (see attached paper). As a computer scientist with
expertise in graphics, I have always known that there is no such
thing as a point particle or a continuous curve. All particles
have extent (pixel/voxel size), and all curves are generated
using line segments (MOVETO, LINETO).
I
just wanted to let you know that you did help someone and that
someone out there does care.
Sincerely,
Lori Gardi
*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. My correction to the calculus allows for a fix of all
higher maths, spaces, and theories.
If this paper was useful to you in
any way, please consider donating a dollar (or more) to the SAVE
THE ARTISTS FOUNDATION. This will allow me to continue writing
these "unpublishable" things. Don't be confused by
paying Melisa Smith--that is just one of my many noms de
plume. If you are a Paypal user, there is no fee; so it might
be worth your while to become one. Otherwise they will rob us 33
cents for each transaction.
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