The official New Calculus page resides at: http://johngabrie1.wix.com/newcalculus
A typical dialogue between myself and other academics:
Academic: For every epsilon greater than zero, there exists a delta greater than zero,
such that for all x close to c, 0<xc<delta implies f(x)L<epsilon.
Gabriel: How do you find the limit L when you do not know its value?
Academic: You can approximate L by using some small h and observing what happens
to the finite difference ratio.
Gabriel: L is never an exact value. How can you be sure it will always satisfy f(x)L<epsilon?
Academic: You can determine L exactly through the first principles method, that is, using
the limit definition: f'(x) = (lim as h approaches 0) { f(x+h)  f(x) } / h
Gabriel: The first principles method requires that L exists because it is a limit definition. By
forming the symbolic difference quotient, one automatically assumes that L exists.
The first principles method assumes the limit exists; it does not prove a limit exists,
but merely provides a flawed guide or method of how to find the limit. A fast way
to see this, is to recognize that the limit (lim) appears on the right hand side of the
first principles method. f'(x) is an Lvalue (an assigned value). Although defined as:
f'(x) = (lim as h approaches 0) { f(x+h)  f(x) } / h , academics have misinterpreted
the definition as: (lim as h approaches 0) { f(x+h)  f(x) } / h = f'(x) = L
Still incorrect, but it should not surprise one that these misguided academics got it backward, they
have been using flawed methods and vague understanding since Newton and Leibniz.
In one part of the method you treat h different from 0 and in another part you treat h
as you would 0. Does h undergo a change in its nature  perhaps due to certain
quantum fluctuations? (joke) The limit definition states h cannot be zero, yet deriving from the first
principles method cannot work unless h is in fact zero.
Academic: (Confounded...)
Does any of the previous circular logic make sense to you?
The following applet demonstrates the stark differences between Cauchy's juryrigged calculus and the rigorous formulation of the New Calculus. Observe how the limit L or f'(2) becomes indeterminate as the finite difference ratio in Cauchy's Kludge approaches the form 0/0 and implodes. Also note how the gradient is always correct in the New Calculus, but never correct using standard calculus. Be sure to drag the slider in the following applet so that dst = 0. Naturally, to ignorant academics, it's more appealing to use a limit definition that never represents a tangent line gradient, even at the most critical stage.
Cauchy's Kludge

The next applet demonstrates exactly what Cauchy thought about the derivative. Observe that the limit is some value you have to imagine, because it is not attainable in the classic calculus by any finite number of steps in a welldefined algorithm (first principles method). Not too surprising, because poor Isaac Newton and Gottfried Leibniz had no idea exactly why their method worked, except that it worked.
Academic: So what's wrong with the idea in the following applet then (Cauchy's Epic Failure)?
Gabriel: There is this illdefined concept of a limit which one has to imagine. How close does one have to get to the point of tangency before deciding what the limit is?
Academic: Well, you can use the first principles method to find the limit. Just set h=0 after you simplify the quotient.
Gabriel: That's exactly the problem. Cauchy's limit definition states that h can never be zero, so how can you use the "first principles" method that does exactly the opposite of what is stated in the limit definition?
Academic: It produces the right answer.
Gabriel: So by using an illdefined concept in an illogical method, confirms you have a jury rigged definition (also known as a kludge). Do you think a kludge has any part in rigorous mathematics?
Academic: The modern definition of derivative is algebraic, not geometric. We do not rely on the tangent line any longer as the limit definition has replaced it.
Gabriel: Does that mean that Newton's root approximation method does not rely on the tangent line then?! Please explain how you would guess the next value of a root without the tangent line?
Academic: Confounded.
Observe that Cauchy's kludge is a marriage of Cauchy's illdefined limit idea and the mean value theorem. Stated in words, Cauchy's definition says exactly:
A derivative exists at x, if and only if, the mean value theorem holds for every point in the interval containing x, except perhaps at x.
It's easy to see how the previous definition fails in the case of the cubic, that is, f(x)=x^3.
An equivalent statement is:
A derivative exists at x, if and only if, a derivative exists for every point in the interval containing x, except perhaps at x.
This statement is an irony, since the definition is meant to define a derivative at the point x! This epic failure of Cauchy to notice the circularity in his definitions and reasoning has resulted in a flawed calculus that has stagnated the last 150 years.
About the applet Cauchy's Epic Failure:
The green secant line and tangent line are parallel. The purple point indicates the coordinates at which a tangent line exists as the green point is moved along the curve and clearly demonstrates the mean value theorem.
Academic: So what's wrong with the idea in the following applet then (Cauchy's Epic Failure)?
Gabriel: There is this illdefined concept of a limit which one has to imagine. How close does one have to get to the point of tangency before deciding what the limit is?
Academic: Well, you can use the first principles method to find the limit. Just set h=0 after you simplify the quotient.
Gabriel: That's exactly the problem. Cauchy's limit definition states that h can never be zero, so how can you use the "first principles" method that does exactly the opposite of what is stated in the limit definition?
Academic: It produces the right answer.
Gabriel: So by using an illdefined concept in an illogical method, confirms you have a jury rigged definition (also known as a kludge). Do you think a kludge has any part in rigorous mathematics?
Academic: The modern definition of derivative is algebraic, not geometric. We do not rely on the tangent line any longer as the limit definition has replaced it.
Gabriel: Does that mean that Newton's root approximation method does not rely on the tangent line then?! Please explain how you would guess the next value of a root without the tangent line?
Academic: Confounded.
Observe that Cauchy's kludge is a marriage of Cauchy's illdefined limit idea and the mean value theorem. Stated in words, Cauchy's definition says exactly:
A derivative exists at x, if and only if, the mean value theorem holds for every point in the interval containing x, except perhaps at x.
It's easy to see how the previous definition fails in the case of the cubic, that is, f(x)=x^3.
An equivalent statement is:
A derivative exists at x, if and only if, a derivative exists for every point in the interval containing x, except perhaps at x.
This statement is an irony, since the definition is meant to define a derivative at the point x! This epic failure of Cauchy to notice the circularity in his definitions and reasoning has resulted in a flawed calculus that has stagnated the last 150 years.
About the applet Cauchy's Epic Failure:
The green secant line and tangent line are parallel. The purple point indicates the coordinates at which a tangent line exists as the green point is moved along the curve and clearly demonstrates the mean value theorem.
Cauchy's Epic Failure
Created by John Gabriel 
Academics will argue that the derivative is produced on completion of the limit (whatever this nonsense means  usually finding the limit through use of Cauchy's flawed first principles method). The only problem however, is that on completion of the limit, h must be 0 and two illegal operations have occurred: 1) division by h to simply the difference quotient and 2) setting h=0 after simplifying the difference quotient. Academics cannot have it both ways: either h is not zero or it is zero. Neither scenario works as I have demonstrated beyond any shadow of doubt. Even in terms of a Cauchy sequence, the derivative must be that value on completion of the limit (finding the limit through visual observation), that is, when h=0.
Riemann's Kludge:
The Riemann Integral suffers from the same illdefined problems as the Cauchy derivative, that is, the Riemann sum is rational until the limit completes("a jump to infinity" ?) and the infinitely many rectangular areas become zero, but the limit is the area! The illdefined limit definition, is in the end equivalent to a dog chasing its tail. One cannot argue the limit is that number which arises as the partition sizes approach zero, because while the partitions are approaching zero, the infinite sum is always an approximation. Professing mathematicians claim the infinite sum is the limit(number) when the partition size is zero, but the partition size can never be zero. And then of course the limit may not be a number after all, but instead it is an incommensurable magnitude. See article that proves Riemann integral is in fact a product of two averages: /uploads/5/6/7/4/5674177/riemannfaux.pdf
Academic: The partitions never actually become zero. It is the partial sum that each time approaches a certain "number". This number is known as the "limit".
Gabriel: So you recognize this "number" by the radix representation of the partial sum each time the partition becomes smaller?
Academic: Yes. It's common in numeric integration. We needn't do this if we can find an antederivative (fundamental theorem of calculus).
Gabriel: The radix representation is always a rational number. By observation, you assign this representation a name, that is, pi or square root of 2, etc. But we know that magnitudes such as pi are not rational numbers. Therefore, the "limit" you refer to cannot be this number in the case of pi. Either the limit is a number or it is not. Which is it?
Academic: (Confounded...)
The following applet demonstrates how ridiculous is the idea of summing an infinite number of rectangle areas. Riemann's Kludge suffers from exactly the same illdefined problems as Cauchy's Kludge. The New Calculus definition of integral as a product of two averages is rigorous and easy to understand. Bernhard Riemann was probably a real nice guy, but some of his ideas were just plain wrong. Fix your gaze firmly on the Area and try to imagine what the limit will be, because once you get to "infinity", the sum implodes due to the partition size of zero. The limit is the "number" that describes the area, but if you don't know the limit, then you have to observe visually what happens as you approach infinity. But, you may say, "we can use the fundamental theorem of calculus", to which I respond: "What if the function does not have a closed primitive form or is described by a transcendental series?" In this case, you cannot find the exact limit, and if you are astute, you will concede the idea of summing an infinite number of rectangle areas is clearly flawed.
The flaws of Riemann's misguided thinking are summarized as follows:
1) One is informed that the sum is that "number" (unknown at this time) which is approached
as n gets closer to infinity.
2) However, as n gets closer to infinity, the sum is always an approximation.
3) One must at some stage reach a stopping condition.
In order to reach a stopping condition, one must guess the value of the sum! Can you tell why? Because one must be able to show that the difference between this number guessed (say L) and any sum (say S) close to L can be made as small as one wishes, that is, SL < epsilon. Guess you say?! "Yes", I respond, "and there's the rub. In most cases, you will not be able to do any better than some approximation, because no irrational number has a finite representation of any sort". The only exceptions are wellknown incommensurable magnitudes such as pi, e, square root of 2, etc that have a recognizable approximation. If L is not a rational number, then one can't even show that the difference SL can be made as small as one wishes! Because one can't even begin to guess L! Unless the "irrational numbers" are already known (such as concepts defined in other ways, for example pi, e, square root 2, etc which are recognized by abbreviated radix representations), the limit, which supposedly represents the number, is an ethereal concept whose value cannot ever be known, because n never reaches infinity. An academic retort might be that one can assign each of these ethereal limits a unique name. Well, in that case, there are infinitely many such "numbers" that can be placed into a onetoone correspondence with a unique name?! This is absurd to say the least, and also in complete contradiction to the spirit of the deified mathematician Georg Cantor, whose only claim to fame is a worthless concept that states a set is countable if it can be placed into a onetoone correspondence with the set of natural numbers, that is, a bijection. Of course Cantor was a misguided fool, but that is another issue.
So once again, we have the illdefined limit concept  similar to a dog chasing its tail. The same rot that we observed used in the definition of the derivative by Cauchy.
The integral is welldefined in the New Calculus  there is no direct computation of infinite sums, use of limits or infinitesimals. In fact, the new calculus standard integral is defined as follows in terms of welldefined averages:
Area = Average length of infinitely many vertical lines
x
Average length of infinitely many horizontal lines (the interval width)
An astute reader says: But your Average lengths of infinitely many lines require an infinite sum!
My response to the reader: Correct! However, my method does not compute an infinite sum directly. In the case where a primitive function exists, the average can be found because of a telescoping series (used in proof of Mean Value Theorem). In all other cases, it is always an approximation, just like Riemann's faux definition of integral. But unlike Riemann's faux definition, my definition is a welldefined product of two averages, that are understood to be approximations.
Riemann's Kludge
Created by John Gabriel 
Instantaneous Rates:
At some time in the twentieth century, an ignorant mathematics professor had a scatterbrained idea about instantaneous rates. I don't know who (Joseph La Grange from the18th Century may have been the first to get the idea) coined this term, but frankly it makes little difference, because it is easy to see that it makes no sense. Ever since the Newton/Berkeley fiasco, academics have been trying to make sense of the derivative. Any mix of the expressions average rate and instantaneous rate has a mindaltering effect on the way modern academics reach nonerudite conclusions. In fact, what they imagine to be an instantaneous rate is in fact a rate corresponding to many average rates over a given interval. Now that the New Calculus is here, and the derivative and integral are welldefined for the first time in history, there is no longer a need for redundant terms such as instantaneous rate. Instantaneous rate must be replaced by the phrase "rate at time t" or "rate at a given point".
Academic: We think about the derivative as a rate of change.
Gabriel: The derivative at a given time t is a rate that is expressed as a ratio of differences (each differential is exactly a welldefined difference, not infinitesimal or any other nonsense), so why call it an instantaneous rate?
Academic: We call it an instantaneous rate because it represents the ratio of two differentials.
Gabriel: The derivative is not generally a rate of any kind unless a time differential is involved. Therefore it makes no sense to call a derivative an instantaneous rate. However, even if time is one of the differentials, the derivative at a given time t is a rate, so why call it an instantaneous rate? What else can it be but the rate at the time t?
Academic: Confounded.
The phrase instantaneous rate is not only a redundancy, but it makes no sense even when time differentials are involved. Furthermore, to think of the derivative as the change in one differential with respect to another differential is fallacious. The derivative has nothing to do with change unless time is involved. The derivative is a ratio of finite differences. In fact, the derivatives for a given differentiable function have always existed, so that nothing is changing or has for that matter, ever changed. That the comparison of two differentials is called a rate, does not mean it is related to change. To wit, most derivatives do not contain time differentials. As the most common example, consider the thousands of functions that are everywhere differentiable and contain no time differential. The differentials are finite differences, not changes that take place in given variables. There is a significant semantic difference (excuse the pun) between change and difference.
The New Method:
Although I first called my new idea the Secant Theorem (because it has been proved), it is actually also an axiom. One might equally well have called it the SecantTangent Axiom. Ask yourself why it is that academics have been so ignorant and incompetent all these centuries.
The idea of limit was born from the inability of mathematicians to realize what I have finally accomplished in the New Calculus  a task not completed by any one else in history.
What better method exists to find the slope of a tangent than by means of a parallel secant? The New Calculus is founded on this idea. This beautiful axiom/theorem makes it possible to develop the entire calculus without the use of limits or other illdefined concepts such as infinitesimals. If you are astute, then you will notice that the introductory graphic on this page summarizes all calculus. Well, you might have to study a bit...
The next applet demonstrates the secant theorem. Slide the point m or c (by using slider tangentpoint) and observe how the gradient is always correct in the New Calculus. The derivative f'(c)=E(c)+Q(m,n) is also displayed. Check its values manually to satisfy yourself the secant method is true. Remember m must be less than or equal to c and n must be greater than or equal to c. m and n are the distances on either side of point c. Moving point m to the right of c will yield unpredictable results. Note that f'(c)=E(c) where E(c) is an expression for the derivative in terms of c. Q(0,0) is always equal to 0 in the secant method. Hold down CTRL; press and release + while the applet is not in focus in order to view the whole applet; then click on the applet to give it focus.
If at any time, the values of m or n are undefined, this is due to the limitations of the GeoGebra software. Observe that you can find an (m;n) distance pair for any distance m in the interval [cm;c] regardless of whether cm>nc or cm<nc. When the software fails to work, you can determine the (m,n) pair from the auxiliary equation, that is, Q(m,n)=0. One of the powerful features of the New Calculus is its potential use in computer graphics. If the creators of Geogebra knew the New Calculus, the following applet would probably not suffer from undefined values.
Secant Theorem

Differentiability:
The concept of differentiability is poorly defined in standard calculus, with incorrect results at points of inflection. A function is not differentiable at inflection or saddle points. This is caused by a change in concavity. If a tangent line could be constructed at a point of inflection, then there would be no change in concavity. A tangent line (of finite length) by definition cannot cross a function's path. It intersects a path in exactly one point, extends to both sides of the point and crosses it nowhere (*). If a "tangent line" does not extend to both sides of a point, then by this false definition, it is possible to have an infinite number of tangent lines at the same point provided one end of the tangent lies on the path.
A serious misconception is that one can use differentiability to draw conclusions about smoothness. Smoothness must have already been established or assumed in order for differentiation to be possible, that is, in order for any function to be differentiable over a given interval, it must first be known to be smooth over the same interval. Stating that a function is smooth over an interval if it is differentiable over the same interval, is like saying: "A function is smooth over an interval if it is smooth over the interval". The standard calculus is replete with many such circular definitions. A good example here is the work of Maurice Frechet, yet another incompetent French mathematician.
A final comment on the subject of differentiability: it seems quite odd that one should even care that a tangent line to a function exists at any given point, unless there are questions about the function behaving strangely at that point (it appears to be discontinuous or not smooth). After all, it's impossible to check every point for differentiability in a given interval using the flawed calculus. However, the New Calculus is designed to check for differentiability over an entire finite interval using only one point in that interval and a parallel secant (provided it exists)!
Confusion about the meaning of tangent line.
Modern mathematicians (read as: mostly incompetent) will tell you that tangent lines are defined in terms of derivatives and then add that the motivation of the derivative definition is the tangent line. The derivative definition is based entirely on the fact that the given secant line finite difference approaches the gradient of the tangent line as discussed earlier.
(*) Webster's dictionary defines a tangent line as follows:
a : meeting a curve or surface in a single point if a sufficiently small interval is considered
First Known Use: 1594
One might try to argue that an endpoint of the sufficiently small interval considered, is the same as the point of tangency. This argument is quickly refuted, that is, in such a scenario, one can construct infinitely many tangents with the same endpoint, but this is obviously misguided.
To think of a function f being differentiable at a point c in standard calculus is in fact incorrect, because in order to be differentiable at c, implies that f is continuous and smooth on the interval or subinterval containing c, that is, f is differentiable over the interval containing c, not only at c. Therefore, it is ignorant to state that if the Cauchy limit can be found, then a function is differentiable at a given point, to wit, the cubic (x^3) is not differentiable at the origin due to the fact that a finite tangent line with defined gradient cannot be constructed at the origin. In fact, it is just absurd to talk about a function being differentiable at a point. More importantly, a function f is differentiable over an interval, if and only if, one finite tangent line with defined gradient can be constructed at every point in the interval. In other words, the question to ask is: "Is the function differentiable over a given interval?" and not "Is a function differentiable at a given point?" Moreover, one assumes that a function is differentiable on a given interval and then finds the gradient of a tangent line at a point in the interval to confirm this is true or not. By calculating a derivative, one always assumes a function is differentiable. Therefore it is incorrect to think that one proves differentiability, rather one confirms or rejects that a function is differentiable over a given interval.
In the New Calculus, provided a single tangent with a defined gradient can be constructed at any point in a given interval, then it follows the function is differentiable over the finite interval.
Cauchy did not realize that in order to form the difference quotient used in his kludgy limit definition, the function must be continuous and smooth, that is, differentiable over the interval for which the secant line gradients (difference quotients) are formed. It is absurd to think that one can find tangent line gradients to functions that are not differentiable over a given interval! Yet this is exactly the implication of Cauchy's kludgy limit. The unique finite tangent lines with defined gradients are the reason a function is differentiable over a given interval. Ignorant academics did not realize this important fact for centuries until I made them aware. Calculus is about natural averages which are only possible with continuous and smooth functions.
Certain academics will shun the new calculus because they are lazy and do not like to exercise what little brains they have. A typical academic might claim the new calculus is difficult. For one who has learned the wrong methods all one's life and is resistant to change, this is true. However, I have found that students learn the New Calculus far easier than the flawed standard calculus. Their general mathematics ability also improves tenfold because the new calculus requires sound reasoning and thought processes which are rigorous.
The new calculus is not only the first rigorous formulation of calculus, but also the clearest exposition of calculus ever. One can also say it is the only sound formulation, given Cauchy's limit based calculus is fake.
Newton and Leibniz were both credited for inventing calculus independently of each other. Although both these individuals were fine academics, the truth is neither of them invented calculus. Furthermore, neither individual was able to formulate a sound definition of either the integral or derivative. Moreover, neither academic was able to state the mean value theorem which is the fundamental theorem of calculus.
The Real Beginning:
The first three propositions mentioned in the section called Quadrature of the Parabola (The Works of Archimedes), were stated without proof. Archimedes claims these were proved in the Elements of the Conics, presumably a work by Euclid and Aristarchus, that is thought to be lost forever. Calculus began with these propositions. The geometric objects called curves and tangent lines were the source of the modern concepts such as continuity, smoothness and differentiability. These concepts are not welldefined in standard calculus but they are welldefined and easy to understand in the new calculus.
Proposition 1 (Quadrature of the parabola) states that if a straight line from a point V on a chord of a parabola is constructed parallel to the axis (or the axis itself) and meets a tangent line at some point P, then V is the midpoint of the chord. Had Archimedes thought of the tangent line concept in the same incorrect way as modern academics, the first three propositions would never have been thought of or published.
Ideas and concepts exist independently of the human mind. Those who chance to think of them are but flash moments in the history of time.
The Secant Theorem.
Given a function f, that has exactly one tangent line (with defined gradient) at each point in an interval [cm,c+n] containing some point c, the gradient of the tangent line to the function at the point c is given by:
However, we use the following notation once it is understood that c is a point, while m and n are distances on either side of c such that cm < c < c+n. c_x means the x coordinate of the point c.
m and n are related distances corresponding to the endpoints [cm,f(cm)] and [c+m,f(c+m] of a parallel secant to the tangent line at c.
For any function f with a tangent line (at x=c) having gradient k, the ordinate difference of the secant endpoints is always k(m+n) because of the gradient ratio. This implies the ordinate difference is always divisible by (m+n). Provided f is continuous and smooth over any interval (cm,c+n), there are infinitely many secant ordinate pairs f(c+n) and f(cm), such that any secant gradient ratio [f(c+n)f(cm)]/(m+n) produces k.
To see how the New Calculus works dynamically, click on the slider called dst in the applet that follows. Changing the value of dst (by dragging the black dot on the slider in either direction) repositions the red line which is parallel to the blue tangent line. You will notice that there is always a relationship between m, n and c. The tangent line is drawn at the point (c,0). You can click on the function f(x) in the Objects pane, and change it to see how the secant method works with other functions. If for some reason when you change the function, nothing seems to happen, your best bet is to reload/refresh the page and try again. There are some bugs in the Geogebra applet API. Experiment by changing f(x) and moving the sliders dst and c to see the effect on the secant and tangent line. Please bear in mind that if a discrepancy occurs between the software and the algebra, it could be that the limitations of the graphing software are inadvertently misleading.
Illustration of the Secant Method in the New Calculus

Finding the relationship between m and n is a feature of the New Calculus. There is no equivalent in Newton's flawed calculus.
Differences between the standard and new calculus:
If f(x) = x^2 (x squared), then the general derivative is given by f ' (x) = 2x and a numeric derivative is given by f ' (c) = 2c, where c is the x coordinate of the tangent point.
The standard definition (with limit), that is, [f(x+h)f(x)] / h as h approaches 0, never represents the tangent line gradient until cancellation has taken place (even in this case it represents only the general derivative, never the numeric derivative), that is, k(h)/(h), where h/h is replaced by 1, h is replaced by 0 in any terms still containing h, and k is the gradient.
Newton's idea of considering a denominator in a finite difference ratio that decreases to an infinitesimal value is incorrect. In fact, it is incorrect to substitute zero for h (old calculus) or m+n (new calculus) in the terms of the finite difference quotient that contain them, until after cancellation, for otherwise the ratio does not represent any meaningful gradient. In the New Calculus, one of the terms which is the gradient, does not contain m or n. The remaining terms always sum to zero. By taking the limit as per standard calculus, one is concluding that the sequence of finite differences:
{ [f(x1)f(c)]/[x1c] ; [f(x2)f(c)]/[x2c] ; [f(x3)f(c)]/[x3c] ; ... k }
converges to some real number k, which is the tangent line gradient. Although this is generally true, the sequence depends on the validity of the limit as a welldefined concept, which of course is not the case. Furthermore, k derived in this way, is always a numeric derivative, never a general derivative. In fact the value of k is always approximate, unless k is a rational number.
Bishop Berkeley was correct in being skeptical. The denominator of a finite difference ratio which describes the consequent of a gradient can never be zero. Setting h=0 in the polynomial resulting from the difference quotient (that is, after cancellation) produces correct results, although the method is juryrigged, because as previously explained, the derivative produced is a general derivative as opposed to a numeric derivative, that is, f ' (x) for some value of x
On the other hand, the new calculus definition always represents the tangent line gradient (given appropriate values of m and n for each parallel secant line in the interval considered) and does not rely on the illdefined notion of limit  this in contrast to the standard definition which erroneously refers to a nonparallel secant line gradient until after the difference ratio is reduced by cancellation. Furthermore, it is entirely correct in the New Calculus to set m=n=0 in the polynomial resulting from the difference quotient after cancellation.
The new calculus does not suffer from absurd results such as 0/0 in the finite difference ratio, the illdefined concept of infinitesimals or any other confusion that arises naturally from Cauchy's illdefined derivative.
One could say that standard calculus is fake because it is based on definitions that are illformed. If you understood the contents of the Cauchy Kludge pdf, then no doubt you will agree that Cauchy's definition is a perfect example of juryrigging.
Simplifying Explanation of Cauchy's wrong ideas:
The main problem with Cauchy's definition is that h is infinitesimal (greater than zero but less than every other magnitude?) before cancellation (*) and zero after cancellation  neither of which are permitted for a numeric derivative. 'h' cannot be infinitesimal before cancellation as it represents the horizontal component for the gradient of a nonparallel secant line and h cannot be zero after cancellation because then the horizontal component of the gradient is zero and thus the finite difference ratio is meaningless. Standard calculus performs bogus arithmetic operations twice:
1. Assumes h is not zero and performs division by h.
2. Assumes h is very small (meaningless nonsense) and discards terms in h effectively treating h as if it were 0.
Moreover, the Cauchy definition relies on the empirical approach of Newton which is an indeterminate process directed at finding a numeric derivative rather than a logical method in finding a general derivative. For example, it does not matter how small the denominator of the ratio is made, because no infinitely small denominator exists that can be the horizontal component of the tangent line gradient. This means that although f '(x) represents the general derivative (after cancellation and h=0), it never represents the derivative when x=x in Newton's empirical approach! In fact, Cauchy's definition is incorrect if it is interpreted as valid for both a numeric and general derivative. Newton was stumped by this, which I think is one of the reasons he refrained from publishing his ideas sooner. The following graphic will help clarify these ideas:
The numeric derivative presumes prior knowledge of limits. The general derivative is only possible if the red secant lines are parallel to the green tangent line. If not, then one cannot set h=0 to find the general derivative because the difference quotient is that for a nonparallel secant. By setting h=0 in Cauchy's definition after the quotient is reduced (diagram on right in previous illustration), one arrives at f '(c), where x<c<c+h, and f ' (c) is not equal to f ' (x). This is discussed in Cauchy's kludge.
In the previous illustration, the diagram on the left shows how h can never be 0. To make sense of the difference quotient, one must use parallel secants (as in the diagram on the right), in which case h can be 0, and subsequently a general derivative can be found. However, the general derivative is not the same as Cauchy's numeric derivative. In my new calculus, one arrives at the same derivative (numeric or general) every time.
Can you see how Cauchy's definition is juryrigged? You may have to study these facts several times before your understanding becomes clear because Cauchy's error is subtle. The diagram on the right illustrates why the kludge works for general derivatives, that is, f(x+h) and f(x) in the left diagram correspond to f(c+h) and f(x) in the right diagram respectively. These facts do not affect the use of Cauchy's definition, except for pedagogical purposes and numerous incorrect theorems. The last couple of centuries have shown that students and mathematics professors never acquire a clear understanding of the derivative (where limits are not required at all).
These inconsistencies are removed in my New Calculus. As a result, differentials are welldefined in the New Calculus. dy/dx always means the same thing in the New Calculus, whereas it has a serious identity crisis in the standard calculus, that is, it can be interpreted as a limit or as a rational expression depending on context. Cauchy would have been correct if he defined the derivative as follows from the diagram on the right:
f ' (c) = [ f(c+h)  f(cx) ] : (c+hx)
Cauchy's first error was in his conception of the limit. His second error was using the limit in an attempt to define the derivative, where it results in his juryrigged definition. If there were any justification for using limits in calculus, it would probably be in respect to integration, however, even in this regard, limits are not required. Using my systematic approach in finding antederivatives, it is now possible to find any antederivative of any function that is continuous and smooth.
Leibniz's definition is really not much better than Newton's, even though it appears to be geometric and somewhat tidier. Leibniz would have defined the derivative as follows had he understood calculus as well as I :
df:dx = [ f(x+n)  f(xm) ] : (m+n) [LD]
The differentials df and dx are exactly equal symbolically (or proportional if numeric) to
[ f(x+n)  f(xm) ] and (m+n) respectively where [LD] represents the gradient of a secant line parallel to the tangent line whose gradient is being determined. In the previous form [LD], df/dx is a symbolic fraction. df/dx becomes an exact fraction when the symbols (function placeholders and variables) are replaced with numbers. The difference between a symbolic and exact fraction is that a symbolic fraction's value is determined according to a given difference ratio whereas an exact fraction has known values. However, this difference is superficial as the fractions are used exactly the same way in algebra.
(*) Cancellation is the process of forming the separate individual quotients by considering the quotient of each term in the numerator with the denominator. The astute reader will notice that this process is assumed to be complete (according to Cauchy), before the finite difference is reduced later (through cancellation) in order to find the general derivative. Therefore, Cauchy's definition is not only fake, but is also fraudulent in terms of the simple properties of arithmetic.
Back to the Beginning.
If you read and understood the previous files containing some general and actual examples, then I have no doubt you will want to know more. What happened between the time of Archimedes and Newton? It is a wellknown fact that the foundations of integral calculus were laid by Archimedes. He used the method of exhaustion to approximate incommensurable magnitudes such as Pi and to calculate irregular areas. The mathematical objects that Archimedes used were the rational numbers.
Over 2,000 years later, these are still the same objects we use today. There is some new terminology (real number) and also some illdefined concepts (limit, complex number). 'Real number' is terminology used to describe any magnitude that is incommensurable in addition to welldefined magnitudes that are known as rational numbers. Contrary to the opinion of most academics, a real number has yet to be welldefined. In fact, most academics don't know the difference between number and magnitude.
A mathematician is like an artist: the objects arising from concepts in a mathematician's mind, are only as appealing as they are welldefined.
Zero enters the number club.
Many centuries passed after rational numbers had been introduced. Eastern mathematicians were researching the concept of zero magnitude that was initially rejected by the Greeks. At first, zero failed almost every requirement needed to qualify as a number. The main requirements were: (i) the mathematical object must be a magnitude (ii) the object must be measurable by the magnitude of an object different to itself, presumably the larger measured by the smaller as Euclid states in his masterpiece, The Elements.
In order for an object to qualify as a magnitude, it was considered imperative that such an object could be instantiated, if not physically, then at the very least it must be welldefined in one's imagination, which implies its physical instantiation is irrelevant. If what one imagines is welldefined, then that which is being imagined is as good as real!
Zero was not immediately accepted, for it could not even satisfy the first requirement, that is, it is not a magnitude. Although the first requirement was eventually waived, zero still failed to satisfy the second requirement, for no other object or magnitude can measure zero, except zero itself.
Zero manifested great potential in serving as a placeholder in radix systems and also in denoting equality between arithmetical operations, as opposed to the unit which denotes equality between numbers. For example, comparing p with p, that is, taking the difference (p  p), results in equality where p is some number, while a  b = c/d results in equality between the expressions a  b and c/d, that is, (a  b)  c/d = 0. It was eventually realized that granting zero membership had certain potential benefits. Provided there was a way to resolve this conflict, zero could be quite useful as a number.
What are the irrational numbers?
Given any integers c and d and some real number k, one can form averages k/c and 1/d. If no c and d exist such that k/c=1/d, then k is said to be irrational. In other words, no proper fraction p/q exists (where p and q are integers), such that p/q equals the fractional part of k, given that k is irrational.
This definition states that k is irrational if it cannot be written as c/d. The definition states what is not a rational number, but says nothing about what constitutes an irrational number. Consider the following analogy:
Humans are not human because they don't have tails. They are human because they think and reason. Chimpanzees do not have tails, yet they are not human. A quality or trait or characteristic, must indicate a feature of the object or idea being defined.
Would it have made any sense to say, Primates without tails are human ? Obviously not. The analogy here is that primates are magnitudes where humans are rational and chimpanzees are irrational.
A property of being welldefined is evident by defining a concept through an attribute it possesses, not by an attribute it lacks. This property is one of the core characteristics of an object that is welldefined. To state that irrational numbers are numbers that are not rational, does not say much, except that such magnitudes cannot be described by any known rational number. So, exactly what is an irrational number?
Unlike a rational number, an irrational number is an incommensurable magnitude that can neither be defined, nor represented in terms of comparison. It is possible only to represent incommensurable magnitudes through averages that are mere approximations. However, these same approximations never define incommensurable magnitudes which came to be known over time as irrational numbers.
In fact, irrational 'numbers' are not numbers; rather these are incommensurable magnitudes, which are now represented and used as approximations through sums of averages.
What are real numbers?
Real numbers came into existence as a result of irrational "numbers". Before incommensurable magnitudes (aka irrational numbers) were discovered by the ancient Greeks, all magnitudes were considered to be rational numbers, that is, all magnitudes were thought to be measurable. Therefore, incommensurable magnitudes, were simply renamed irrational numbers and lumped together with the rational numbers to form the new real numbers.
Given these facts, it is immediately evident that an irrational number is illdefined. Now since real numbers are defined to be the union of a set which includes the set of rational numbers and the set of irrational numbers, it follows the real numbers also, are not welldefined.
It would thus be in order to say that when a real number is not rational, the same number represents some incommensurable magnitude, for example Pi, the square root of two or any other incommensurable magnitude.
Academics after the ancient Greeks (with the exception of John Gabriel) have failed to realize that a magnitude becomes a number once it is completely measurable.
The Limit.
It did not take long for mathematicians to become dissatisfied with their inability to welldefine incommensurable magnitudes. Rather than concede these incommensurable magnitudes cannot be represented by numbers, those misguided 'mathematicians' devised a new concept called the limit which they would use to redefine, not only all real numbers, but also attempt to provide rigour to the many problematic definitions in calculus. They failed, and what is taught in today's lecture rooms is the nonsense created by them.
Limits are not required in calculus or any branch of mathematics. If set theory (not all set theory is unsound) can be called mathematics, perhaps this is the only place where limits might be studied. Indeed, the arrival of the theory of limits is a major setback in the progress of mathematics. The theory is flawed for many reasons, but the most important reason is that limit theory was developed from the theory of sets, which is fundamentally in error.
The popular notion regarding the concept of limit being at the core of calculus is entirely false. In fact, as stated earlier, the inventors of calculus knew nothing about limits. The limit itself is poorly defined and takes on several different meanings, depending on context. For example, a limit can denote some value of an expression through cancellation of certain factors in the expression (*). A limit can also denote the upper bound of an infinite sum as in the case of a convergent sequence (also known as Cauchy sequence).
The most popular use of limits is for determination of gradients where they are not required at all. A gradient of a curve at a point is the gradient of the tangent line at that point, provided the tangent line exists and has a defined slope. The only geometric object that has a gradient or slope is a straight line. By gradient of a curve at a point, the implied meaning is the gradient of the tangent line to a curve at that point.
(*) This notion is extended to expressions where very small magnitudes (illdefined nonsense) are discarded and the result is called a limit.
How did we get the idea of limit?
The great mathematician Archimedes was the first to formalize the ideas that eventually led to the unfortunate adoption of the limit by modern academics, and consequently also the theory that arose from this idea. The Archimedean Property states:
If x is any magnitude, then there exists a well defined natural number N such that x < N.
Note that I use the word magnitude  because a real number is not welldefined. In fact, Real Analysis, which was created from the nonsense of Georg Cantor’s misguided ideas, defines real numbers using the limit concept (Dedekind cuts or Cauchy sequences). Arbitrarily small distances between rational numbers are used to demonstrate convergence which leads to the notion of a limit. That which is being defined is used in its own definition! Refer to my article on magnitude and number (section called Debunking Real Analysis Myths).
The Archimedean property is misunderstood by most professors of mathematics. The property was not intended to be used as a prelude for the concept of a least upper bound of a set which is bounded above. Its purpose was to establish a method for measuring incommensurable magnitudes by means of approximation. For example, Archimedes knew that his method of exhaustion would never result in a value greater than some rational number that is larger than Pi. In fact, pi is the reason Archimedes arrived at the Archimedean property. Archimedes knew after he discovered the property, that the perimeter of a polygon inscribed in a circle, would always be some rational number less than pi and that the perimeter of a polygon circumscribing the same circle would always be some rational number greater than pi. This is the sole reason for the discovery of the Archimedean property revealed for the first time in history on this page.
The property is stated more accurately as follows:
If x is any magnitude, then there exists a commensurable magnitude N such that x<N.
What every academic (except John Gabriel) failed to understand is that the following is also true:
If x is any magnitude, then there exists a commensurable magnitude N such that x>N.
These last definitions use commensurable magnitude because natural numbers are an abstraction derived from ratios of commensurable magnitudes  a fact that every professor of mathematics does not know, unless such a professor learned it from me. What does the Archimedean property mean? The property simply states that given any commensurable or incommensurable magnitude, a commensurable magnitude exists which is greater or smaller. I suppose one can write the revised Archimedean property as follows:
If x is any magnitude, then there exist commensurable magnitudes M and N such that M > x > N.
The Archimedean property is best illustrated geometrically through the triangle inequality which states that the sum of any two sides of a triangle are greater than the remaining side. One can therefore let the magnitude be a side of the triangle and construct the other two sides such that they are commensurable, and thus the property is proved.
Note that Archimedes knew nothing about infinitesimal/s. In fact, Archimedes rejected the idea of real number as "understood" by ignorant modern academics. He knew that irrational numbers do not exist and hence neither do real numbers exist. Archimedes understood that pi is an incommensurable magnitude. It could not be a number because it is impossible to completely measure the magnitude known as pi.
Anal retentive academics were not satisfied with Archimedes' ideas. They preferred to obfuscate matters by introducing nonissues that are unrelated to Archimedes' original ideas. What started out as simple and elegant ideas morphed into:
(i) a least upper bound concept for a given set called a limit.
(ii) the name given to a result of simplifying a given expression
through cancellation of its factors, is called a limit.
(iii) zero which assumes the role of a limit when the denominator
of a given fraction is assumed to increase or decrease without bound.
(iv) the name given to a result of simplifying a given expression by
disregarding terms whose values are decided by an indeterminately
large denominator. This is an extension of (iii).
The categorization of these various aspects based on this illdefined concept of limit evolved into the ubiquitous calculus limit as it is known today.
Epsilonics (EpsilonDelta) proofs.
Unfortunately, an entire semester is usually wasted learning epsilonic proofs. The fact that many students drop out of math courses because they think they do not have what it takes, is good reason to be alarmed. To those students who are contemplating dropping out, my advice to you is to simply tough it out. Your difficulty in understanding is mainly due to the fact this theory is illdefined. And then you have lecturers who are absolutely incompetent, which compounds the problem. If only they understood some mathematics... To help you understand what the nonnoteworthy fuss is about, I have compiled a document called limits.pdf (link follows). Use this document in conjunction with the GeoGebra applet to understand the meaning. Whatever you do, don't quit your math studies because of this unimportant topic in Real Analysis. I expect that when my New Calculus eventually replaces standard calculus, this nonsense will be a thing of the past.
Stating a theory using numbers and symbols does not make it more rigorous. In most cases, unless the theory can be properly worded and more importantly welldefined, it usually ends up containing many flaws. Prior to learning standard calculus, students are required to spend much fruitless time trying to prove limits exist, when all they are doing is stating a limit exists using epsilonics. Finding a delta and epsilon, is analogous to saying that a triangle is a triangle because it has three angles. In fact it is insufficient to find a delta and epsilon  one has to find a relation between the two. Again, it is insufficient because of the reasons explained in limits.pdf.
The following applet calculates the exact epsilondelta region for a given planar function f(x), which you can change by clicking on f(x) in the Free Objects region and entering your own function. The anchor button (rightmost button on toolbar) allows you to move the graph (move graphics view mode), to rescale the axes and to zoom in and out. To rescale an axis, click on any part of it whilst in move graphics view mode and whilst holding your mouse button down, drag in either direction (northsouth or eastwest) until you are satisfied and then release the mouse button. You may have to move the graph back into view (using anchor button) after rescaling the axes. To move the sliders you must be in pointer mode (leftmost button on toolbar). Move the sliders called delta and c to observe the effect on the limiting region.
In some cases it might appear as if there is no graph, but what this means is that you have to resize and recentre the graph. Rescaling the axes helps to see more granular details when the limiting region is very small.
Dynamic Illustration of EpsilonDelta Limit theory on your choice of function.
Purpose of this applet: To demonstrate the uselessness of epsilonics in limit theory. 
Calculus without the use of limits.
Calculus is possible because of the properties of natural averages. Calculus is about natural averages, not limits. The New Calculus does not use limits. The following pdf provides a proof of the average value theorem and the fundamental theorem of calculus without the use of limits.
Newton and Leibniz would have loved to know this information. Both these academics were challenged to provide rigorous definitions of their calculus. Both failed.
Newton's greatest mathematical accomplishment and supreme work was his discovery of the finite difference interpolation polynomial. In terms of mathematics, all of Newton's remaining works are in my opinion, insignificant. Leibniz on the other hand, almost succeeded in redefining his definite integral correctly. He was on the right track by researching moments, but was unable to complete his task. Leibniz may not have been too surprised to learn from my redefinition of the integral, that all integrals are path or line integrals.
Most modern academics fail to understand that calculus only applies where natural averages are present, that is, continuous and smooth functions. For example, they have been known to apply calculus to conditional functions (aka piecewise functions) where they remove discontinuities inadvertently or purposely, and then try to deduce conclusions about continuity and differentiability. Newton and Leibniz knew nothing about conditional functions.
Finally, a rigorous calculus without the use of limits.
Almost 330 years later, I have presented a rigorous calculus and redefined not only the derivative, but also the definite integral without the use of limits. Newton, Leibniz and Cauchy were mistaken, but their wrong ideas have been corrected in the new calculus. Even differentials are now welldefined in the new calculus. PDE experts normally spend their entire lives mastering DEs. The New Calculus has already changed this (personal research) and when it is adopted, mathematicians will no longer require a lifetime to become an expert in a topic as complex as PDEs (partial differential equations).
Conclusion:
I trust your visit to my site has been informative and also entertaining. These few ideas and secrets I have shared with you, are only a fraction of what is contained in my new calculus. One of the secrets I recently shared is the existence of a cumulative probability function for the normal density function. It might have been impossible or very difficult without the methods of the New Calculus.
Future of the new calculus:
1. The primary purpose of the new calculus is for academics to adopt a sound and rigorous branch of mathematics that one can learn and master in a short time. Standard calculus has proved to be difficult and hard to master in any reasonable period of time, one of the key problems being its use of the limit concept and real analysis.
2. Real calculus reform: there are no ambiguities, paradoxes or contradictions. Concepts are welldefined unlike the standard calculus which is fake.
3. The following key fields have been identified as important research areas in
the new calculus:
Solution of differential and partial differential equations, including the
use of numeric integration techniques.
Solution of area, volume and hypervolume problems using welldefined
concepts such as natural averages.
Spatial representation of curves in ndimensions (useful in computer
graphics).
Accurate curve fitting.
Systematic indefinite integration and differentiation.
New regression analysis (statistics).
Application to abstract algebra and discrete mathematics.
And much more...
4. Due to decreased complexity in the new calculus, research and understanding
is facilitated and thereby enhanced.
I have done substantial research in these aforementioned areas. With regards to numeric differentiation and integration the New Calculus has significant advantages over the nPoint central difference formulas and wellknown quadrature/cubature algorithms. This information is currently withheld pending the publication of What you had to know... I am certain there are many more benefits to be realized by adopting the new calculus which the future is certain to reveal.
(C) John Gabriel, The New Calculus, 2010
All Rights Reserved
Fools in academia:
These foolish academics need to be exposed, not because they are jealous and arrogant (they generally are), but rather because they are ignorant beyond belief. It is a tragedy that today's most influential academics censor and denigrate those, whose ideas they neither understand nor like. While ignoramuses are abundant throughout the earth, it is a crying shame that modern academia is the new Catholic church of knowledge. As the Catholics suppressed those who in the middle ages differed from their ideas, modern academia has established a biased knowledge repository in the form of academic journals.
Modern academia have through the medium of journals formed an elite clique which one might call the academic bourgeoisie. If new knowledge is not printed in one of the recognized journals, then sheepish learners are trained to shun it, just as Catholics of the middle ages were warned of committing blasphemy. In the end, everyone is a loser.
Georg Cantor: Believed he was onto something with regards to countability of sets, in particular infinite sets. Cantor foolishly assumed that the set of real numbers is uncountable for all the wrong reasons, when in fact the set is illdefined for two reasons: it is infinite and its members are not welldefined. If the real numbers can be represented in base 10 (as Cantor assumed), then the set of real numbers is indeed countable because any radix system only represents rational numbers. In fact, the set of real "numbers" is uncountable because real numbers are not welldefined. Indeed, how can one count anything if one does not know what it is? Preposterous, of course! See article on magnitude and number at the top of this page.
Not too long ago, I had some correspondence with a theoretical physicist (who by his own admission stated he is not a good mathematician) who claimed that it is not possible to formulate (construct) the numbers in only a few pages. Without even reading my formulation (see magnitude and number article and also powerpoint document called construction of numbers), he dismissed it as spurious. It is easy to show the development of the number concept up to rational numbers in just a few paragraphs.
Guiseppe Peano/Abraham Fraenkel/Ernst Zermelo: These "mathematicians" formulated the ZF axioms which are nothing more than a juvenile attempt to form what they perceived as the foundations of arithmetic. Built on Cantor's dumb ideas, ZF is an attempt to define numbers without an understanding of ratio or measure.
David Hilbert: An ardent follower of Cantor whose efforts placed Cantorian ideas at the forefront of mathematics  a devastating action that set mathematics off course the last 120 years. Set theory is a failed attempt at understanding numbers in terms of containment rather than measure. Containment disregards the aspect of comparison, whereas measure is defined by it. The New Mathematics (of set theory) espoused these wrong ideas and till this day mathematicians never fully grasp the concept of ratio. In fact, most mathematics professors do not know the difference between a magnitude and number. I had a foolish professor write and tell me that the number of elements in each set can be compared and thus there is measure in set theory! What the nitwit failed to realize, is that to find the number of elements, one must assume prior knowledge of measure, that is, "number".
Kurt Godel: The father of the completeness/incompleteness theorems in logic. A little known fact is that Godel's own theorems disprove themselves. Only academic ignoramuses can be trusted to miss such a simple fact as this.
Bertrand Russell: Discovered paradoxes in set theory (unsurprising because set theory is illdefined) and prepared the stage for flawed modern logic theory. Russell was an overrated logician whose debating skills earned him a place in the history of mathematics. He was a notorious smoker who never failed to appear in any photograph without a tobacco pipe in mouth or hand. Russell's Principia Mathematica is less worth than the paper used to print it.
Abraham Robinson: A twentieth century American Jewish mathematician, who designed nonstandard analysis on Cauchy's kludgy idea from classic analysis, that is, the illdefined concept of infinitesimal. The infinitesimals according to Robinson are a subset of the interval (0,1) with no least upper bound. Ignorant academics claim that an infinitesimal is greater than 0 but less than every positive number. How they arrive at the plural is absurd, that is, there is no way of comparing infinitesimals with each other.
It is impossible to tell where infinitesimals end and the "real" numbers begin since the infinitesimal set has no least upper bound. Not one instance of an infinitesimal number can be demonstrated either in theory or practice. Rather an attempt is made to draw conclusions about more theorems using the same illdefined concepts, unsurprisingly often with incorrect results. It is impossible to compare infinitesimals (measure them). Ironically, Robinson based his theory on the assumption that the real numbers are welldefined (and used the transfer principle to validate his nonstandard theory)  I have proved this to be false, that is, the real numbers are not welldefined. In fact, real numbers do not exist. See my article on magnitude and number.
Robinson's useless theory survives because of the Jewish influence in academia. Wikipedia's Jewish sysops and admins tried to give this unsound theory of Robinson's more credibility by claiming Archimedes used infinitesimals, until they finally understood that Archimedes had no idea about any such nonsense, nor is his method of exhaustion in any way related to Robinson's absurd ideas.
Stephen Hawking: The obscurities of his theories are tantalizing to the community because they sound exotic and alluring. It is my opinion that most theoretical physicists are duller than dish water. In 2012, Stephen Hawking was awarded $3 million dollars for his useless theory on black holes. What Hawking knows about calculus is quite precarious. When Hawking creates theory using a calculus that's questionable and nonestablished "facts" about the universe, what you get is sheer speculation. Just one more example of ignorant academics awarding each other accolades and prizes for theory that's not worth the ink used to publish it. If future historians are honest (somehow doubtful), Hawking will be remembered as a speculating ignoramus. This type of news almost makes me want to take down this site and all others in which I share my knowledge of the New Calculus. While Hawking has been awarded millions, cowardly academics who have acknowledged my work privately, remain silent in public. My horoscope has more chances of being true than a theoretical physicist's theories ever being proved sound  at any rate, such theories will not be confirmed or discarded for centuries. Perhaps I have been too harsh in my criticism of Hawking, but frankly I think he knows his theories are bogus. He does not think it is his problem that so many ignorant academics and people consider his theory to be doctrine. After all, it puts more money in his pocket and earns him many awards.
Thanks to Einstein's nonsensical theories, theoretical physics is now a lucrative occupation, especially if you are an academic suffering from some kind of handicap. For those of you that hate mathematics, your next best bet might be a career in theoretical physics. If "physicists" can tell so much from ancient light sources and a flawed calculus, surely there must be some truth in Astrology also.
AMS (American Mathematical Society) and MAA (Mathematical Association of America): Both societies will gladly publish antimathematical nonsense on illdefined calculus limits and nonexistent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by misguided and shortsighted academics, is beyond question.
ACTA (The Royal Swedish Academy of Sciences): A journal which does not bother refereeing papers. The editors glance at new submissions while drinking tea and base their "expert" opinions on what "appears or seems to be".
I could spend my entire life refuting all them or continue working on my new ideas in mathematics. I choose to sound the warning bells and let those who ignore pay the price.
These foolish academics need to be exposed, not because they are jealous and arrogant (they generally are), but rather because they are ignorant beyond belief. It is a tragedy that today's most influential academics censor and denigrate those, whose ideas they neither understand nor like. While ignoramuses are abundant throughout the earth, it is a crying shame that modern academia is the new Catholic church of knowledge. As the Catholics suppressed those who in the middle ages differed from their ideas, modern academia has established a biased knowledge repository in the form of academic journals.
Modern academia have through the medium of journals formed an elite clique which one might call the academic bourgeoisie. If new knowledge is not printed in one of the recognized journals, then sheepish learners are trained to shun it, just as Catholics of the middle ages were warned of committing blasphemy. In the end, everyone is a loser.
Georg Cantor: Believed he was onto something with regards to countability of sets, in particular infinite sets. Cantor foolishly assumed that the set of real numbers is uncountable for all the wrong reasons, when in fact the set is illdefined for two reasons: it is infinite and its members are not welldefined. If the real numbers can be represented in base 10 (as Cantor assumed), then the set of real numbers is indeed countable because any radix system only represents rational numbers. In fact, the set of real "numbers" is uncountable because real numbers are not welldefined. Indeed, how can one count anything if one does not know what it is? Preposterous, of course! See article on magnitude and number at the top of this page.
Not too long ago, I had some correspondence with a theoretical physicist (who by his own admission stated he is not a good mathematician) who claimed that it is not possible to formulate (construct) the numbers in only a few pages. Without even reading my formulation (see magnitude and number article and also powerpoint document called construction of numbers), he dismissed it as spurious. It is easy to show the development of the number concept up to rational numbers in just a few paragraphs.
Guiseppe Peano/Abraham Fraenkel/Ernst Zermelo: These "mathematicians" formulated the ZF axioms which are nothing more than a juvenile attempt to form what they perceived as the foundations of arithmetic. Built on Cantor's dumb ideas, ZF is an attempt to define numbers without an understanding of ratio or measure.
David Hilbert: An ardent follower of Cantor whose efforts placed Cantorian ideas at the forefront of mathematics  a devastating action that set mathematics off course the last 120 years. Set theory is a failed attempt at understanding numbers in terms of containment rather than measure. Containment disregards the aspect of comparison, whereas measure is defined by it. The New Mathematics (of set theory) espoused these wrong ideas and till this day mathematicians never fully grasp the concept of ratio. In fact, most mathematics professors do not know the difference between a magnitude and number. I had a foolish professor write and tell me that the number of elements in each set can be compared and thus there is measure in set theory! What the nitwit failed to realize, is that to find the number of elements, one must assume prior knowledge of measure, that is, "number".
Kurt Godel: The father of the completeness/incompleteness theorems in logic. A little known fact is that Godel's own theorems disprove themselves. Only academic ignoramuses can be trusted to miss such a simple fact as this.
Bertrand Russell: Discovered paradoxes in set theory (unsurprising because set theory is illdefined) and prepared the stage for flawed modern logic theory. Russell was an overrated logician whose debating skills earned him a place in the history of mathematics. He was a notorious smoker who never failed to appear in any photograph without a tobacco pipe in mouth or hand. Russell's Principia Mathematica is less worth than the paper used to print it.
Abraham Robinson: A twentieth century American Jewish mathematician, who designed nonstandard analysis on Cauchy's kludgy idea from classic analysis, that is, the illdefined concept of infinitesimal. The infinitesimals according to Robinson are a subset of the interval (0,1) with no least upper bound. Ignorant academics claim that an infinitesimal is greater than 0 but less than every positive number. How they arrive at the plural is absurd, that is, there is no way of comparing infinitesimals with each other.
It is impossible to tell where infinitesimals end and the "real" numbers begin since the infinitesimal set has no least upper bound. Not one instance of an infinitesimal number can be demonstrated either in theory or practice. Rather an attempt is made to draw conclusions about more theorems using the same illdefined concepts, unsurprisingly often with incorrect results. It is impossible to compare infinitesimals (measure them). Ironically, Robinson based his theory on the assumption that the real numbers are welldefined (and used the transfer principle to validate his nonstandard theory)  I have proved this to be false, that is, the real numbers are not welldefined. In fact, real numbers do not exist. See my article on magnitude and number.
Robinson's useless theory survives because of the Jewish influence in academia. Wikipedia's Jewish sysops and admins tried to give this unsound theory of Robinson's more credibility by claiming Archimedes used infinitesimals, until they finally understood that Archimedes had no idea about any such nonsense, nor is his method of exhaustion in any way related to Robinson's absurd ideas.
Stephen Hawking: The obscurities of his theories are tantalizing to the community because they sound exotic and alluring. It is my opinion that most theoretical physicists are duller than dish water. In 2012, Stephen Hawking was awarded $3 million dollars for his useless theory on black holes. What Hawking knows about calculus is quite precarious. When Hawking creates theory using a calculus that's questionable and nonestablished "facts" about the universe, what you get is sheer speculation. Just one more example of ignorant academics awarding each other accolades and prizes for theory that's not worth the ink used to publish it. If future historians are honest (somehow doubtful), Hawking will be remembered as a speculating ignoramus. This type of news almost makes me want to take down this site and all others in which I share my knowledge of the New Calculus. While Hawking has been awarded millions, cowardly academics who have acknowledged my work privately, remain silent in public. My horoscope has more chances of being true than a theoretical physicist's theories ever being proved sound  at any rate, such theories will not be confirmed or discarded for centuries. Perhaps I have been too harsh in my criticism of Hawking, but frankly I think he knows his theories are bogus. He does not think it is his problem that so many ignorant academics and people consider his theory to be doctrine. After all, it puts more money in his pocket and earns him many awards.
Thanks to Einstein's nonsensical theories, theoretical physics is now a lucrative occupation, especially if you are an academic suffering from some kind of handicap. For those of you that hate mathematics, your next best bet might be a career in theoretical physics. If "physicists" can tell so much from ancient light sources and a flawed calculus, surely there must be some truth in Astrology also.
AMS (American Mathematical Society) and MAA (Mathematical Association of America): Both societies will gladly publish antimathematical nonsense on illdefined calculus limits and nonexistent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by misguided and shortsighted academics, is beyond question.
ACTA (The Royal Swedish Academy of Sciences): A journal which does not bother refereeing papers. The editors glance at new submissions while drinking tea and base their "expert" opinions on what "appears or seems to be".
I could spend my entire life refuting all them or continue working on my new ideas in mathematics. I choose to sound the warning bells and let those who ignore pay the price.
Laughs and Criticism:
The following section contains criticism of articles I sometimes read when I am bored. There is so much nonsense constantly being published by mathematics professors and other academics. The same nonsense is read and digested by naive students worldwide. The results are undesirable.
For a good laugh, visit Crank Chu Carroll's (a selfconfessed Jew and a scoundrel) comical blog at:
http://scientopia.org/blogs/goodmath/2010/02/04/acrankamongcranksdebatingjohngabriel
I kind of feel sorry for Crank Carroll now. I suppose that if I had my fingers broken by my naziinspired school mates, I too would be very defensive and angry with even the slightest challenge to my ideas.
http://scienceblogs.com/goodmath/2009/07/16/veryofftopicwhyiwontbea/
The Cantor Debate with Crank Chu Carroll.
A few years ago I wrote a Knol disproving Cantor's diagonal argument. Firstly, let me say that the real numbers are uncountable because they don't exist, that is, they are not welldefined. Secondly, I wanted to prove that if the real numbers are represented in decimal, I could list all of them in a tree using a sound algorithm and locate the start node of any real number. In fact, this is what Cantor was trying to do. Cantor knew nothing about a mapping or inverse pairing function  these came much later and were named after him. It is possible to find a mapping and an inverse pairing function for the rational numbers because they are welldefined. My primary goal was to disprove the false Diagonal Argument that it was not possible to list all the real numbers, if they can be represented as decimals. Also note that when Cantor came up with his idea, a bijection was not known. The first known use of bijection was in 1963 (according to Webster), not too long after WWII (and the Holocaust). Of course this does not mean it was unknown under another name or the idea was unknown, but it is highly unlikely Cantor knew anything about bijections. The bijection came much later, as did the pairing and inverse functions for rational numbers. Were Jewish mathematicians scrambling to defend Cantor's wrong ideas?
Crank Chu Carroll was immediately hung up on many things at the start of the debate. He began by trying to differentiate between representation and enumeration. I never got round to explaining my enumeration (part of the mapping algorithm) because Chu Carroll knew he lost the argument and continued to insult me. Aside from this objection, Carroll would just not accept that 1/3 (or 0.333....) was present in my tree. He was comfortable that 1/3 = 0.333... (with endless 3s following) but would not accept that I could do the same thing in my decimal tree traversal. He became obsessed and infuriated that I was able to do this in my tree traversals. For someone who has a PhD in computer science, Carroll knows very little about traversals. Any astute mathematician will see that this is a contradiction, for if one cannot represent 0.333... in my tree, then 0.333... cannot represent 1/3 in decimal. Last I looked at the above link, it has been heavily edited: the comments where Chu Carroll raised the subject of his Jewish ethnicity are completely removed. The comments where I corrected his wrong ideas regarding tree traversals are also removed. Many of the comments are either modified or deleted to suit Carroll's agenda.
One of the funny comments Carroll made was "No, you can't take 0.333... to infinity. You just can't!". And in the same breath, all the idiots who were agreeing with him in the forum, would say that 0.333... is equal to 1/3, which is only possible if the limit is considered. Well, if ignorant mathematicians can use the illdefined limit in their mathematics, why can't I also use it in my mapping algorithm? Answer: It refutes the dumb ideas of Cantor regarding countable infinite sets. To read about how I have completely debunked the fallacy that fractions can be represented using nonterminating notation, find my debate on Space Time and the universe: http://www.spacetimeandtheuniverse.com/math/66610999reallyequal1a.html
Also read a disproof of the existence of real numbers here:
http://www.spacetimeandtheuniverse.com/math/45070999equalone317.html
And for more laughs, another Jewcontrolled (?) site:
http://forums.xkcd.com/viewtopic.php?f=17&t=51224
By all means, read what others say about me, but remember to form your own opinion after you read what I have to say about myself. I have found that one can have all the knowledge in the universe but is unable to form an opinion. Being able to remember many facts is not a sign of intelligence but that is a different topic.
Some of the false criticism leveled against me by the "erudite" mathematics community:
 lunatic
 psychotic
 crank
 kook
 narcissist
 sufferer of BPD (borderline personality disorder)
 schizophrenic
 nutcase
 exhibitor of the Dunning Krueger effect
 Your choice of mental disorder/syndrome/dysfunctional effect here...
I have to say that inside my mind there is an immeasurable hatred for these scoundrels who have leveled baseless accusations against me and in the process destroyed or stymied my chances of success. I hate them with a perfect hatred and sincerely hope that every worthless dog who has leveled these accusations against me, suffers a terrible life and an even worse death. They have no excuse and ignorance is not an excuse. In fact, words cannot even begin to describe the loathing I feel for these academics  many of whom are pedophiles and perverts. Perhaps I am not such a nice person, because I cannot find it in me to forgive and forget. Why should I... Hatred is not illogical. It is every bit as logical as love or kindness or gentleness or any other human emotion. Anyone who claims not to hate is a liar.
Many of my accusers overlook the fact that I am not good at expressing myself in writing. Not everyone is a natural born writer or speaker. I find it extremely difficult to write especially seeing that I loathe writing (*). There are many other factors that influence my writing, including foreign language and culture differences. I speak/read/write several languages and have lived in many parts of the world. I often think about how I will be understood or perceived. Many times I am horrified at what I once wrote. I can't change it soon enough, but by the time the realization has sunk in, it may already have been read and the damage done. Unlike my accusers, I am not afraid to admit I might have been wrong or to apologize. In my opinion, this makes me a better human than they all.
My confidence is often interpreted as arrogance. I understand mathematics better than anyone I have ever known or any of the mathematical authors I have studied (including Newton, Leibniz and Cauchy).
One of the reasons I have created this page is for the sake of young learners. I hope to make it easier for them to learn calculus. I also encourage students to study the old flawed methods of calculus. Graduates can also learn much from studying the New Calculus  it can help to consolidate and dispel some of the wrong concepts they are taught at university.
So, try to focus on the facts rather than the inferences, which might pop out at you while reading this web page. If you try to psychoanalyze me from my writing, you will be completely misled.
Well, if you read this far, you are entitled to know a little more about the greatest mathematician ever:
I once had an acquaintance who would tell me that one should never call oneself the greatest, but let others say this of him. What this acquaintance with his limited intelligence (IQ of 139 according to him) failed to realize, is that the opinion of others is completely worthless to those who are the greatest. Needless to say, he is/was a bumbling religious fool. Thus, if others call me the greatest, I would be very worried, because the world is full of idiots.
Throughout my life I have always kept an open mind. While I do not claim to know everything, I do claim to know what I know, better than anyone else. Of course I have doubted myself. Not once, but many times. Over and over again, I have questioned my logic and conclusions. In fact, I never cease to question my conclusions. Knowledge of any subject is like a model. My approach to knowledge, is that if a flaw is found, the entire model, or at the very least those flawed parts of it, must be revised, discarded or replaced completely. Never have I closed the door on any knowledge and by so doing, considered it beyond further investigation. I openly acknowledge my imperfections and admit I could be wrong. What I find most irksome, are ignorant academics (and they are in no short supply!) who will argue a subject they do not and have not ever understood.
Other passionate views:
I am a supporter of assisted suicide. It is my hope that assisted suicide will be legalized in the United States, as it has been in Switzerland and other more advanced civilizations. In my opinion, terminal illness or lack thereof, should not be a consideration if an individual chooses to end his life. Even healthy individuals should have access to assisted suicide. It's too bad one does not have a choice regarding entry into this earthly reality, but it's very sad and unfortunate that one cannot choose a time to exit peacefully. The most powerful mindcontrol drug (religion) is protected by the constitution of the United States, but the most basic right of choice to die is illegal in most states.
I once had an acquaintance who would tell me that one should never call oneself the greatest, but let others say this of him. What this acquaintance with his limited intelligence (IQ of 139 according to him) failed to realize, is that the opinion of others is completely worthless to those who are the greatest. Needless to say, he is/was a bumbling religious fool. Thus, if others call me the greatest, I would be very worried, because the world is full of idiots.
Throughout my life I have always kept an open mind. While I do not claim to know everything, I do claim to know what I know, better than anyone else. Of course I have doubted myself. Not once, but many times. Over and over again, I have questioned my logic and conclusions. In fact, I never cease to question my conclusions. Knowledge of any subject is like a model. My approach to knowledge, is that if a flaw is found, the entire model, or at the very least those flawed parts of it, must be revised, discarded or replaced completely. Never have I closed the door on any knowledge and by so doing, considered it beyond further investigation. I openly acknowledge my imperfections and admit I could be wrong. What I find most irksome, are ignorant academics (and they are in no short supply!) who will argue a subject they do not and have not ever understood.
Other passionate views:
I am a supporter of assisted suicide. It is my hope that assisted suicide will be legalized in the United States, as it has been in Switzerland and other more advanced civilizations. In my opinion, terminal illness or lack thereof, should not be a consideration if an individual chooses to end his life. Even healthy individuals should have access to assisted suicide. It's too bad one does not have a choice regarding entry into this earthly reality, but it's very sad and unfortunate that one cannot choose a time to exit peacefully. The most powerful mindcontrol drug (religion) is protected by the constitution of the United States, but the most basic right of choice to die is illegal in most states.