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Quote this page as follows:
The New Calculus (c) John Gabriel 2010 [http://thenewcalculus.weebly.com]
New Calculus at a glance (Powerpoint Presentation):
/uploads/5/6/7/4/5674177/the_new_calculus_-_presentation.ppt
A short comparison between the New Calculus and Newton's Flawed calculus:
/uploads/5/6/7/4/5674177/comparing_the_new_calculus_to_newton.pdf
For Educators:
You might be skeptical that the New Calculus can be learned in just two weeks. For this reason, I have compiled the following lesson plans that can be used by high school students/graduates or in first year university calculus courses. A student can quickly grasp and understand both differentiation and integration in only two lessons of 60 minutes each:
The New Calculus derivative: /uploads/5/6/7/4/5674177/calculus-derivative-lessonplan.pdf
The New Calculus integral: /uploads/5/6/7/4/5674177/calculus-integral-lessonplan.pdf
A quick introduction to the New Calculus (for mathematicians):
The following documents are simple and a teaching approach is used:
/uploads/5/6/7/4/5674177/newcalculus-abstract-part-1.pdf
/uploads/5/6/7/4/5674177/newcalculus-abstract-part-2.pdf
About John Gabriel and the New Calculus:
John Gabriel is the first and only mathematician in history to resolve the problem of rigour in calculus. The New Calculus is designed on well-defined concepts. Based on sound analytic geometry, the New Calculus can be learned and mastered in a short time. There are no ill-defined concepts such as limits or infinitesimals. The ideas of the New Calculus were known to Gabriel before the year 2000. His first electronic articles on the subject began to appear in 2002. Several sites (no longer live) had been established before this site. His unpublished book What you had to know in mathematics but your educators could not tell you is a work that started over 30 years ago, in the form of short papers but has now grown close to 2000 pages of which 800 pages are about the New Calculus. You are very fortunate to read what the greatest mathematicians longed to know - on this page. As a mathematician or an educator, you have a responsibility to teach truth and to encourage independent thinking.
The New Calculus is not just a better calculus, it is a different kind of calculus. Almost the same in terms of results, only more expedient, and without ill-defined concepts. Euclid's elements is an enduring masterpiece because he attempted to well-define all the concepts he used. Nikola Tesla was a prolific inventor because whatever he imagined, was well-defined. Well-defined concepts are not only easy to learn, but inevitably lead to more discovery, and are accessible to every kind of learner. There is an unbridgeable gulf between the New Calculus and Newton's flawed formulation in terms of potential. Ill-defined concepts are not only a problem in calculus, but in all mathematics.
From this, it is understandable that most learners hate mathematics - because the average human brain is conditioned to steer away from concepts, situations, etc that fail to make sense. Subject an individual to ill-defined concepts long enough, and you will notice that at first there is dislike, then gradually fear and discomfort, followed by intense loathing. The question is not whether every student can learn mathematics, rather the question is: Can educators be retrained to teach mathematics using well-defined concepts?
I am, what learning theorists might call an abstract learner (INTJ Myers Briggs personality for what it's worth), and even though I hated the theory of limits, I was able to master it and understand its flaws. Limit theory has no place in calculus - certainly not in differential or integral calculus. Furthermore, set theory which is the foundation of limits, is seriously flawed. The New Math of the 1960s accomplished exactly the opposite of what its proponents intended. Can you guess why? (Hint: Georg Cantor)
Personal: The information you find on my sites is only the proverbial tip of the ice-berg. I have not shared the most interesting discoveries with my readers for obvious reasons (think revenue). You will experience many instances of enlightenment as you read this site. There are many more concepts that you as a mathematician never understood, even though you may have learned to use theorems based on these same concepts. The reason is that these concepts are not well-defined. In my articles and web pages, one will see the phrase well-defined many times. The machinery of Cauchy's calculus is clunky and rusted, but more importantly, it is seriously flawed. It's just a matter of time before honest mathematicians will realize that my New Calculus is far superior. I am hoping it will happen in my lifetime, but am very doubtful because of the arrogance and obnoxiousness of modern academia, which is rather sad. These same foolish academics cling to the useless and incorrect ideas of Georg Cantor and his bipolar brainchild - set theory. The real numbers are uncountable (countability is a worthless Cantorian idea) because real numbers do not exist. What modern academics think of as a real number, is an ill-defined concept. Every academic I have met does not know the difference between a magnitude and a number. Einstein, the purported "great mind" of the twentieth century had no idea about the significance of a number. A magnitude becomes a number when it is possible to measure it (the magnitude) completely. Approximations of incommensurable magnitudes are not numbers representing these incommensurable magnitudes.
Modern academics would rather die than concede the New Calculus is the first rigorous formulation of calculus in history. To acknowledge that I am correct, would be to admit their incompetence and that of mathematicians the last few hundred years.
To read many more interesting mathematics articles by John Gabriel, search the web with this page's URL or simply continue reading this page. The following article is one of many. Even if you experience difficulty understanding its contents, don't let this discourage you from reading the others. It often takes more than one reading for most people to understand problems, where ill-defined concepts are involved. /uploads/5/6/7/4/5674177/proof_that_0.999_not_equal_1.pdf A little secret I recently shared is the fact that 1/3 is not equal to 0.333... in decimal. You can see the proof of this on pages 33 through 37 of the previous pdf. Although it was unnecessary to know this fact in order to determine that 0.333... or 0.xxx... is always an ill-defined concept. There are many interesting facts that I have not shared for obvious reasons. The disproof of the previous asinine idea is yet another nail in Georg Cantor's coffin, who thought erroneously that all real numbers can be represented using the decimal radix system. If you are lazy and don't like reading, here is a presentation that debunks the 3 most common fake proofs used: /uploads/5/6/7/4/5674177/debunking_0.9991.ppt
It amazes me how many people without any understanding so readily subscribe to this ridiculous idea that 0.999...=1. The only reason I have included it on my web page, is to perhaps get them interested in something far bigger and more important: The New Calculus. Of course it's completely untrue that 0.999...=1 and it does not matter at all, but I hope that while you are here, you will take some time to learn about the New Calculus.
Any magnitude that cannot be measured completely in terms of a unit is not a number. The only true numbers are rational numbers. "Irrational numbers" are a figment of the dysfunctional/irrational mind.
A number is the abstract object that describes the complete measure of a magnitude.
I have received a lot of correspondence from readers asking me to debunk all the "proofs" in favour of the equality of 0.999... and 1. I don't know what they all are. Some are so silly that I won't bother. However, if you find any convincing "proofs" that I think are worth debunking, I'll be glad to update the above article. I have included only the refutation of those supposed "proofs" that are convincing through the methods used. Your first task as a mathematics educator is to understand the difference between a magnitude and a number. For this, I recommend you read my article on magnitude and number (link provided further on this page). One of the main reasons this silly debate exists today is that until I came along, no one had formalized the difference between magnitude and number. Euclid tried but failed. I am the first mathematician in history to construct the number concept from scratch in a logical way. Had others been able to do this before me, the dumb idea that 0.999... = 1 would not exist today. I am engaged in a titanic battle with academic fools in high places. It pains me to say that some of the most ignorant and dumbest people I have ever met work in education. They (with their PhDs) have made stupidity an art form. Alas, the light and beauty of Ancient Greek mathematics will be snuffed out in exchange for the deceit of Cantor's dark and poisonous ideas. Be careful what you wish for.
What others say.
The New Calculus is far superior and easier to use in every discipline: from applied mathematics (differential and partial differential equations that are intrinsically difficult using the flawed formulation of calculus), mechanical engineering, physics, computer graphics (3-D visualization and simulation), mathematical modelling, mathematical statistics, real-time processes, and the list just goes on.
One of the most important applications of the New Calculus is in Education and future Mathematical Research. It cannot be emphasized enough how critical well-defined concepts are to learning and future progress. About 60% of my research is based in this direction.
Although I receive quite a bit of positive feedback, the most surprising comes from the least expected sources. The following screenshot is from an amateur mathematician (member of ResearchGate) who found the New Calculus extremely useful in his work on Quadratic Bezier curves (very important in computer graphics). In fact I have done some substantial research in this field also (computer graphics) which will be shared if What you had to know in mathematics but your educators could not tell you, is ever published. It is my opinion that independent thinking mathematicians and aspiring amateurs who are not trapped in group-think, will be at the forefront of new developments in mathematics and science.
Audio introduction by John Gabriel follows (25 seconds):
Companion Sites:
New Calculus Course: http://whatyouhadtoknowbutyoureducators.weebly.com/the-new-calculus-course.html
Educators: http://whatyouhadtoknowbutyoureducators.weebly.com
German language: http://hanspeterguettinger.weebly.com
Wordpress Blog: http://mathphile.wordpress.com
Google Profile: https://plus.google.com/u/0/115441392355463289173/posts/p/pub
Welfare of humanity: http://godisimaginary.com
New Calculus on a single page:
For a quick glance at the ideas behind the New Calculus, the following graphic may be useful: /uploads/5/6/7/4/5674177/spnc.jpg From the graphic (spnc.jpg) which shows only the curve f(x), one can understand that the area between the curve f ' (x) (not shown in the graphic) and the x-axis is given by the product of the rectangular width and height corresponding to the same. The height is equivalent to the average value of f ' (x). The width is simply the length of the interval, that is, m+n. Observe that the value of the summation index k, does not influence the average value which is the same for any integer k>0. The identity SMA (Sub-interval Mean Average) is new in calculus. There are no ill-defined concepts such as sums of infinitely many rectangles or cubes in the calculation of areas and volumes. Leibniz's wrong ideas led to these incorrect definitions in standard calculus. A standard integral is not an infinite sum, it is always the product of two averages. See applet further on this page called Riemann's Kludge.
The first rigorous mathematical axioms and how we got numbers:
Learn how we really got numbers and the first rigorous mathematical axioms - a process that has taken almost 6,000 years. /uploads/5/6/7/4/5674177/magnitude_and_number.pdf The following articles may help to understand the concept of number, arithmetic and calculus:
/uploads/5/6/7/4/5674177/mathematical_axioms.pdf
/uploads/5/6/7/4/5674177/construction-of-numbers.ppt
Copyright Notice:
The New Calculus is the sole intellectual property of John Gabriel. No articles, books or any other publications in electronic form or otherwise, are permitted without prior written consent from John Gabriel. In the event of his death, permission to publish any information on the New Calculus or What you had to know in mathematics but your educators could not tell you, must be obtained from his beneficiary, Amanda Irene Duminy.
What is wrong with the standard calculus?
Standard Calculus evolved from the flawed ideas of Newton and Leibniz that Cauchy eventually enhanced. Until the New Calculus was developed, a rigorous formulation of calculus was unknown or unrealized (to or by humans of course). Rather than admit they had no answers, academics conjured up theories based on half truths and in many cases, self-deception. For almost 300 years academics were unable to resolve the problem of finding the gradient of a tangent line to a curve using a rigorous method. Instead they chose to base the entire theory of calculus on the ill-defined notion of limit (/uploads/5/6/7/4/5674177/magnitude_and_number.pdf)
When one repeats an untruth sufficient times, one begins to believe it. I have lost count of how many times I have read or heard the untrue phrases "The calculus was placed on a firm foundation." or "The calculus was placed on a rigorous footing." or "The calculus was made more rigorous." If I were able to sue all the authors of the publications containing these false statements, no doubt I would be rich. While many ignoramuses have imbibed these untruths in ignorance, the progress of calculus has suffered from zero growth for almost 150 years. Concepts such as tangent, differential, area, rate and also number are not well-understood by those very mathematics professors who teach the same. In most cases, these academics have no clue what is a number, thanks to set theory, Cauchy's limits and the introduction of the subject called Real Analysis. There are many PhDs of mathematics who would not pass a comprehensive examination on Euclid's elements, never mind advanced calculus.
Aside from being false, this belief that calculus was made rigorous has generated much flawed theory which should never have been thought of, never mind printed. For example, the idea of an infinitesimal is a myth because it is an ill-defined concept. Whatever one imagines is real, if and only if what one imagines is well-defined.
Epsilonics theory was an attempt to avoid the use of infinitesimal ideas. One important reason epsilonics fails, is that it is based on the concept of arbitrarily small distances (meaningless nonsense) where there is no clear boundary or distinction between given magnitudes (aka real numbers) and so called infinitesimal "numbers". Any academic who supports infinitesimal theory, is unfit to be called a mathematician.
Newton was a primarily a scientist and then a mathematician. The finite difference ratios used in his experiments were by no means well-defined mathematical objects, even in his mind. In his flagship publication (The Analyst) he does not reveal all the details, and his explanations are clearly dubious. Newton was a master at approximations, to the extent that his entire contribution in mathematics would amount to very little without these. In fact, Newton's approach to finding the gradient of a tangent line was an approximation! The flawed calculus was built on this approximation.
The mathematician Cauchy, was the first to attempt to fix Newton's and Leibniz's wrong ideas. He tried to do this by adding the limit concept to Newton's finite difference quotient, but a new problem arose: Cauchy's limit concept is ill-defined. The arrival of Cauchy's limit was hailed as the Holy Grail of calculus. Little did those foolish academics in Cauchy's time know that nothing had been made more rigorous or placed on a firmer foundation. Rather, the flawed machinery had been completed, which subsequent mathematicians used in their research.
Weierstrass cut the ribbon to Cauchy's Calculus of limits by heralding epsilonics theory which in fact is also a failed theory addressing the conspicuous infinitesimal thorn in Newton's calculus.
Debunking Modern Academic ideas about the tangent line:
The modern "understanding" is that a function is differentiable at a given point if the limit of a secant line finite difference ratio (gradient) exists at that point, that is to say, the left hand limit is equal to the right hand limit. Let me begin by making it clear that the finite difference ratios in the flawed calculus, always represent the gradient of a secant line, as a point is approached by a function from both directions. The limit stopping or terminating condition, occurs when the value of the secant line gradient is indistinguishable from either side of the point. Now ask yourself, what happens to these secant lines as they converge on the point.
Indeed, they intersect the function in exactly one point, and cross it nowhere. Now ask yourself what kind of geometric object has this property. If you thought of anything else besides the planar tangent object (tangent line), you should probably try another profession. To claim that tangent objects are no longer relevant in calculus, confirms that your thinking is clearly flawed. The limit can never replace tangent objects. Contradictions exist in the flawed calculus because of the efforts to discard geometry (tangent objects). In fact, set theory has been the most powerful assault on geometry. The limit concept relies on geometry for its validity, but has its buttresses in set theory. The foundation of mathematics is geometry, not set theory.
In planar calculus, the tangent object is the tangent line. In three-dimensional calculus, the tangent object is a disc with finite radius. In 4-dimensional calculus, the tangent object is a sphere with finite radius. Tangent objects cannot be visualized for dimensions higher than 4. The flawed calculus uses the same ill-defined machinery to study properties of functions across all dimensions, that is, the ill-defined limit. Although the New Calculus can be extended to multi-variable calculus in exactly the same fashion as flawed calculus(example on this page), it has new tools (a new kind of mathematics) to deal with the same problems far more efficiently and without contradictions. Rather than study well-defined tangent objects, ignorant academics settled for Cauchy's kludge.
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<|x-c|<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 L-value (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 jury-rigged 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 most 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 well-defined 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 ill-defined 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 ill-defined 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 ill-defined 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.
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.
And this 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 ill-defined 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 ill-defined 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 ill-defined 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.
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.
And this 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 ill-defined 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 ill-defined 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 ante-derivative (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 ill-defined 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, |S-L| < 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 well-known 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 |S-L| 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 one-to-one 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 one-to-one correspondence with the set of natural numbers, that is, a bi-jection. Of course Cantor was a misguided fool, but that is another issue.
So once again, we have the ill-defined 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 well-defined 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 well-defined 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 well-defined 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 scatter-brained 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 mind-altering effect on the way modern academics reach non-erudite 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 well-defined 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 well-defined 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 Secant-Tangent 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 ill-defined 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 [c-m;c] regardless of whether c-m>n-c or c-m<n-c. 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
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For more information on why epsilon-delta theory is flawed, be sure to read: /uploads/5/6/7/4/5674177/limits.pdf The flawed theory of epsilonics is discussed in more detail further on in this page.
Cauchy's Kludge (/uploads/5/6/7/4/5674177/cauchykludge.pdf) demonstrates the errors of Cauchy's definition in about 6 pages. One of many contradictions and errors that arise from the Kludge is a theorem that states polynomial functions are generally everywhere differentiable. One can read about the fallacy regarding the differentiable cubic /uploads/5/6/7/4/5674177/cubicfallacy.pdf at the origin. The New Calculus has a rigorous method for determining inflection points: in the case of a point where a tangent line cannot be constructed, the method in the new calculus shows the existence of an inflection point by proving that an (m,n) pair other than (0,0) does not exist. For certain special functions such as the hyperbola, one uses the (0,0) pair in conjunction with the auxiliary equation in order to find a relationship between m, n and the point at which the tangent line is constructed. The flawed calculus recipe for finding inflection points fails in several cases.
For the single variable New Calculus, a tangent line at any point P is constructed by knowing the relationship between the abscissa of P and the distances m and n from the abscissa to the endpoints of a parallel secant. It is my correct opinion that tangent lines were invented by Ancient Greeks to measure smoothness of curves when continuity is given, that is, a curve is smooth if exactly one tangent line can be constructed at every point except perhaps a point of inflection. If more than one tangent line can be constructed at the same point, then the curve is no longer smooth at that point. Many modern academics do not know what is a tangent line because their definitions are ill-defined. Bad mathematics resulting from ill-defined concepts does not end with the tangent line definition. Consider that until my New Calculus, academics were completely in the dark about the nature of differentials. I now enlighten all those who are interested: /uploads/5/6/7/4/5674177/dydx_compared.pdf
Even though Cauchy's fake calculus generally "works", consider that it has given rise to more false and unsound theory (infinitesimals, limits, real analysis, etc) and obstructed the progress of mathematics in this regard.
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. The file called cubicfallacy.pdf on this page explains much more. Read the following article to see what it means for a function to be differentiable at a given point in single variable calculus (not the same in new multi-variable calculus): /uploads/5/6/7/4/5674177/differentiability.pdf
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 sub-interval 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.
Introduction:
I decided to call my reformulation of this important branch of mathematics: The New Calculus. In fact, it is not so much a new calculus, as much as it is a sound, well-defined and easy-to-learn calculus, without the use of limit theory or real analysis, of which neither existed in Newton's period. The secant method is at the core of single variable differentiation, just as the tangent disc method is at the core of 3 dimensional differentiation (*) in the new calculus.
Tangent objects exist for higher dimensions even though these can only be visualized up to and including 4 dimensions. Calculus, in one of its aspects, is the subject that describes how to calculate attributes of tangent objects (such as gradient, normal vector, etc), to functions that are both continuous and smooth. In another aspect, it is the branch of mathematics that through the use of natural averages, describes distances, tangent objects, areas, volumes and hyper-volumes in different dimensions. Calculus developed into a very complex subject when real analysis became widely used in education.
(*) A new kind of mathematics vastly different from standard multi-variable calculus but far simpler and easier to learn. Most mathematicians like to work from a set of axioms. For a simple, no nonsense list of well-defined mathematical axioms, the following will be very useful:
/uploads/5/6/7/4/5674177/mathematical_axioms.pdf
This web page is about introducing the New Calculus which will change the way mathematicians do business. Although single variable calculus is discussed, there is one example of how it can be extended to multi-variable calculus (/uploads/5/6/7/4/5674177/partial_derivatives.pdf).
The New Calculus will change the way students perceive calculus, which has nothing to do with limits, but due to Cauchy's wrong ideas, is now mistakably associated with limit theory.
If this is your first visit, you will encounter new knowledge. Whether you are a renowned mathematics professor or a high school graduate, prepare to be astounded.
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 following document explains: /uploads/5/6/7/4/5674177/mvt-indivisibles.pdf
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 well-defined in standard calculus but they are well-defined 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 [c-m,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 c-m < c < c+n. c_x means the x coordinate of the point c.
m and n are related distances corresponding to the endpoints [c-m,f(c-m)] 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 (c-m,c+n), there are infinitely many secant ordinate pairs f(c+n) and f(c-m), such that any secant gradient ratio [f(c+n)-f(c-m)]/(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
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Finding the relationship between m and n is not always an easy task and in certain cases, may require some ingenuity. It is always possible to determine a general derivative (explained shortly) given a primitive function that is differentiable.
Finding an auxiliary equation:
To find a relationship between m and n by means of an auxiliary equation is described further here: /uploads/5/6/7/4/5674177/appetizer.pdf
In order to convince oneself the secant theorem is true, form the quotient described by f ' (c) for a given differentiable function and simplify the result. Next, one must find a relationship between m and n. This can be done by equating the sum of all the terms in m and n [denoted by Q(m,n)] to zero. Choose a suitable m or n and then find the corresponding m or n. Finally, substitute the values of m and n into the finite difference directly to obtain the gradient f '(c). Any (m,n) pair can be substituted into the finite difference quotient except (m,n)=(0,0) because the (0,0) pair does not belong to any secant.
Note that an expression such as [sin(c+n)-sin(c-m)]/(m+n) is not an expression whose terms all contain m and/or n. Thus, equating [sin(c+n)-sin(c-m)]/(m+n) to zero would yield incorrect results. However, (m+n) is a factor of every term found in the ordinate difference formed by f(c+n)-f(c-m). It can be demonstrated that m+n is a factor of every term in the ordinate difference sin(c+n)-sin(c-m). So, what do you think the relationship between m and n might be in the case of the derivative where f(x)=sin(x) ?
All differentiable functions possess the special relationship where the sum of the terms in m and n equals to 0 after cancellation (known in the new calculus as the distance pair (0;0) ) in the difference quotient as will be explained shortly. Once again, it must be remembered that although we can disregard terms in m and n, this does not matter in the case of a straight line where m and n can take any values. A straight line is the only geometric object whose gradient does not depend on secant ordinates because straight lines do not have tangents. In fact, it can be proved that the values of m and n have no effect on the secant line gradients or tangent line gradient.
For a given interval (m;n), all the parallel secants are part of a tangent space for the given (c, f(c)) which is the point of tangency. Each parallel secant has its own unique (m;n) pair. The tangent line owns the (0;0) pair. A special relationship exists between m, n and c for any continuous and smooth function. If only the pair (0;0) is possible, then a point of inflection exists.
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)]/[x1-c] ; [f(x2)-f(c)]/[x2-c] ; [f(x3)-f(c)]/[x3-c] ; ... 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 well-defined 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 jury-rigged, 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 ill-defined notion of limit - this in contrast to the standard definition which erroneously refers to a non-parallel 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 ill-defined concept of infinitesimals or any other confusion that arises naturally from Cauchy's ill-defined derivative.
One could say that standard calculus is fake because it is based on definitions that are ill-formed. 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 jury-rigging.
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. As explained in the pdf called Debunking wrong ideas about the derivative /uploads/5/6/7/4/5674177/debunking_wrong_ideas_about_derivative.pdf,
h cannot be infinitesimal before cancellation as it represents the horizontal component for the gradient of a non-parallel 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 non-parallel 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 jury-rigged? 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 well-defined 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(c-x) ] : (c+h-x)
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 jury-rigged 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 ante-derivatives (/uploads/5/6/7/4/5674177/indefinite_integral_-_systematic_method.pdf ) it may now be possible to find any ante-derivative, although not a trivial process.
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(x-m) ] : (m+n) [LD]
The following links explain more:
/uploads/5/6/7/4/5674177/meaning_of_dydx.pdf
/uploads/5/6/7/4/5674177/what_does_leibniz_notation_mean_exactly.pdf
/uploads/5/6/7/4/5674177/dydx_related_rates_example.pdf
The differentials df and dx are exactly equal symbolically (or proportional if numeric) to
[ f(x+n) - f(x-m) ] 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.
The following file called newcalculus.pdf contains a few general examples. Divisibility_identities.pdf contains a proof that supports the ideas in newcalculus.pdf.
| newcalculus.pdf |
| divisibility_identities.pdf |
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 well-known 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 ill-defined concepts (limit, complex number). 'Real number' is terminology used to describe any magnitude that is incommensurable in addition to well-defined magnitudes that are known as rational numbers. Contrary to the opinion of most academics, a real number has yet to be well-defined. In fact, most academics don't know the difference between number and magnitude. Following article explains: /uploads/5/6/7/4/5674177/magnitude_and_number.pdf
A mathematician is like an artist: the objects arising from concepts in a mathematician's mind, are only as appealing as they are well-defined.
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 well-defined in one's imagination, which implies its physical instantiation is irrelevant. If what one imagines is well-defined, 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 well-defined 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 well-defined. 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 ill-defined. 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 well-defined.
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 well-define 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 (ill-defined 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 well-defined. 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 non-issues 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 ill-defined concept of limit evolved into the ubiquitous calculus limit as it is known today.
Epsilonics (Epsilon-Delta) 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 ill-defined. 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 non-noteworthy 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 well-defined, 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. Whichever way one chooses to look at this epsilon-delta theory, it is ill-defined as the following article explains: /uploads/5/6/7/4/5674177/limits.pdf
The following applet calculates the exact epsilon-delta 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 re-scale the axes and to zoom in and out. To re-scale 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 (north-south or east-west) until you are satisfied and then release the mouse button. You may have to move the graph back into view (using anchor button) after re-scaling 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 re-size and re-centre the graph. Re-scaling the axes helps to see more granular details when the limiting region is very small.
Dynamic Illustration of Epsilon-Delta 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.
| proof_avtftc.pdf |
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 piece-wise 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.
To help you learn all about Newton's interpolation polynomial (and much more...), I have compiled the following publications:
How we got calculus: /uploads/5/6/7/4/5674177/howwegotcalculus.pdf
Newton's methods: /uploads/5/6/7/4/5674177/newton-methods.pdf
How we got calculus is a tour de force of Newton's most important work which I think is his interpolation polynomial. Only standard calculus is used in How we got calculus. Taylor's theorem can be derived from Newton's interpolation polynomial by using only the new calculus. Start with formula [T1] on page 15 of How we got calculus. Observe that my new calculus version of the Taylor polynomial is an equality (Gabriel polynomial), unlike Taylor's polynomial which is always an approximation. The main idea behind Taylor's theorem is ease of calculation even though convergence is a disadvantage in terms of acquiring sufficient accuracy.
Newton's methods were based largely on approximation because he lacked certain knowledge, that is, he probably did not know of the mean value theorem and certainly did not know of my New Calculus. The mean value theorem is hardly understood by most mathematics professors. There are many GeoGebra applets supposedly demonstrating the workings of the mean value theorem, but all of them are superficial and miss the most important aspect which is captured in the following GeoGebra applet: /uploads/5/6/7/4/5674177/mvtvisual.ggb
Roger Cotes is a mathematician who is not well-known. Yet the impact of Cotes' work has been significant. Without it there would have been very little progress in numeric integration and therefore the solving of differential equations. The following document shows how Cotes arrived at the general formula: /uploads/5/6/7/4/5674177/cotesintegration.pdf
A proof of L'Hopital's Rule using the New Calculus.
I am often asked how I would explain L'Hopital's rule without the use of limits. The following file contains an elegant proof that is less than one page. /uploads/5/6/7/4/5674177/lhopital.pdf
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 well-defined 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 well-defined 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 hyper-volume problems using well-defined
concepts such as natural averages.
Spatial representation of curves in n-dimensions (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 n-Point central difference formulas and well-known 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 ill-defined for two reasons: it is infinite and its members are not well-defined. 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 well-defined. 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 power-point document called construction of numbers), he dismissed it as spurious. To show the world how ignorant this academic is, I have demonstrated (further down) on this web page 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 ill-defined) 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 non-standard analysis on Cauchy's kludgy idea from classic analysis, that is, the ill-defined 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 ill-defined 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 well-defined (and used the transfer principle to validate his non-standard theory) - I have proved this to be false, that is, the real numbers are not well-defined. 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 non-established "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 anti-mathematical nonsense on ill-defined calculus limits and non-existent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by misguided and short-sighted 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.
To contact me: john underscore gabriel at Ya hooooo dot com
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 ill-defined for two reasons: it is infinite and its members are not well-defined. 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 well-defined. 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 power-point document called construction of numbers), he dismissed it as spurious. To show the world how ignorant this academic is, I have demonstrated (further down) on this web page 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 ill-defined) 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 non-standard analysis on Cauchy's kludgy idea from classic analysis, that is, the ill-defined 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 ill-defined 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 well-defined (and used the transfer principle to validate his non-standard theory) - I have proved this to be false, that is, the real numbers are not well-defined. 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 non-established "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 anti-mathematical nonsense on ill-defined calculus limits and non-existent infinitesimals. Unfortunately most sheepish journals follow their lead. That these societies are run by misguided and short-sighted 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.
To contact me: john underscore gabriel at Ya hooooo dot com
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.
A few common fallacies:
/uploads/5/6/7/4/5674177/fallacies.pdf
Debunking the Riemann Integral:
/uploads/5/6/7/4/5674177/riemannfaux.pdf
Mathematics professors who do not understand the concept of tangent line:
/uploads/5/6/7/4/5674177/knisleyrefute.pdf
The same professor (Knisley) has some coherent thoughts regarding calculus:
http://faculty.etsu.edu/knisleyj/calculus/facing.htm
For a good laugh, visit Crank Chu Carroll's (a self-confessed Jew and a scoundrel) comical blog at:
http://scientopia.org/blogs/goodmath/2010/02/04/a-crank-among-cranks-debating-john-gabriel
I kind of feel sorry for Crank Carroll now. I suppose that if I had my fingers broken by my nazi-inspired 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/very-off-topic-why-i-wont-be-a/
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 well-defined. 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 well-defined. 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 ill-defined 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.
And for more laughs, another Jew-controlled (?) 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 (border-line 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.
(*) If you have visited this page, you will know that it has changed many times. Even as I write this, I am not fooled about not having to change the page contents at some future time. I am especially amused when academic ignoramuses attribute the title crank to me. If I am a crank, what are they? Anyone who disagrees with what they think are "well-established" views, is by their definition, a crank. In the United States, it is permissible (legal) to call someone a crank. For example, take a look at Mark Chu Carroll: an imbecile, who knows nothing about anything, but can mouth off his stupidity on his blogs without any fear of being sued. This vile reptile (Chu Carroll) is protected by a ruling passed by Chief Judge Posner, in the following court case: https://bulk.resource.org/courts.gov/c/F3/75/75.F3d.307.95-2282.html
One might say, "Why don't you wait a while before you publish?". My response is that I will probably end up waiting indefinitely because I always find something I don't like about my writing.
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).
On occasion I have shared some personal information. Probably not a wise thing to do. The reason is that I want my readers to know a little more of me, aside from the mathematics. Sharing information was also an attempt to be more approachable, but evidently not too successful. Oh well, at least I tried, which is more than what many others might do.
Often I think that these last few paragraphs (under Laughs and Criticism) do not belong on this web page. I ask myself many times what to include and omit. It would be nice if there were a science language which lacked inference, so that when communicating in this language, one would not have to worry about hurting or offending others, even though the fault usually lies with those who take offence. Unfortunately, there is no such language. If it were possible to describe everything in terms of mathematical symbols, then mathematics would be a language in its own right. However, it is common knowledge that mathematics is not a language.
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.
Educating the uneducated:
The following exercises are appropriate for professors of mathematics and mathematics teachers because most of them do not know what is a number.
Study the following publications:
/uploads/5/6/7/4/5674177/magnitude_and_number.pdf
1. Explain why a real number is ill-defined in standard calculus by considering that the decimal representation (or any radix representation in base 10) of an "irrational" number is always rational.
Observe that a limit may itself be a number or not, then argue that since "real" numbers that include "irrational" numbers are defined in terms of limits (equivalent Cauchy sequences), it follows that "real" numbers also are not well-defined. Submit solution in not more than 3 pages.
The convergent sequence [pi] = (3; 3.1; 3.14; .... ) is bounded above (*) by the magnitude known as pi. Yet pi is NOT a number (pi is a ratio of two magnitudes). Therefore it is incorrect to state that the sequence [pi] has a limit pi, because the ill-defined concept of "limit" has a dual nature: sometimes it is a number and sometimes an incommensurable magnitude. In any event, neither pi nor any other incommensurable number (known as irrational numbers to modern academics) can be well-defined using Cauchy sequences.
(*) While the concept of being bounded above by a rational number makes perfect sense, it is ridiculous to think of an incommensurable magnitude being an upper bound because its complete dimensions are unknown, that is, it is an ill-defined concept.
Study the following publication:
http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/proof_that_0.999_not_equal_1.pdf
2. Prove that radix systems can only be used to represent rational numbers.
However, qualitative measurement can be inaccurate. For example, if a mathematician were able to discern that two magnitudes are not equal, it would still be impossible to determine how much larger or smaller the one is from the other, that is, a difference is not well-defined.
To define difference, the brilliant Ancient Greeks began by formalizing the process of comparison, that is, they formed ratios of magnitudes. For example:
magnitude (object A) : magnitude (object B)
which means literally:
The magnitude of A compared with the magnitude of B qualitatively, that is, one of two outcomes only: magnitude(A) = magnitude(B) or magnitude(A) not equal to magnitude(B).
Note that the magnitudes are unknown till this stage. They are only being compared qualitatively (by visual observation).
The incredible breakthrough occurred when they discovered the abstraction of a unit, from which the natural numbers were born. A unit is defined very simply as the comparison of any magnitude with itself or another equal magnitude, that is,
magnitude (X) : magnitude (X) = UNIT (**)
(**) This great accomplishment led to quantitative measurement. Now, it was possible to tell the difference between two magnitudes being compared (provided both are measurable in whole units), and a known symbol could be associated (having a known magnitude value) with both magnitude(A) and magnitude(B), that is, both magnitudes are now comparable quantitatively as natural numbers.
Note that the magnitudes are known as natural numbers at this stage. They are being compared quantitatively (by finding the difference in terms of a natural number).
Recap: We started with an unknown magnitude (X) and arrived at a quantitative measurement of X in terms of natural numbers. A natural number is a ratio X : N where X is a ratio of measurable magnitudes and N is a unit.
Now, this great knowledge was insufficient where incommensurable magnitudes (pi, sqrt(2), e, etc) are concerned. That is, incommensurable magnitudes can be represented only by approximations through use of fractions (or radix systems).
I am not unaware of how dull mathematics professors and teachers can be, so I suggest you find a quiet room and study these things long and diligently.
These few paragraphs are meant only to whet your appetite. For the full story of the intricate thought processes of the Ancient Greeks and much more on the development and history of numbers, calculus and mathematics in general, you will have to wait for the publication of the greatest unpublished work in mathematics:
What you had to know in mathematics, but your educators could not tell you.
This great work is the correction and completion of Euclid's Elements, The Works of Archimedes and Apollonius' Conics. It incorporates the New Calculus. Contained therein is new knowledge of averages, natural averages, area, volume and hyper volume, which the Ancient Greeks understood but were unable to formalize.
Reader FAQs:
May I write articles about the New Calculus?
The short answer to this question is: NO. The current circumstances are such that the academic community as a whole, needs to publicly acknowledge the New Calculus is the first rigorous formulation. Until this happens, I cannot grant permission to anyone wishing to write about the New Calculus.
May I teach the New Calculus in class?
Yes, on one condition: You must use the correct attribution:
"The New Calculus was discovered by John Gabriel in 2002 and published online in 2010."
It would also not hurt to refer your students to my websites.
Can I write a review?
Unfortunately, I can't stop you from writing a review. However, be warned, if you misrepresent information, you may be sued. Truth is, you probably won't be sued because I don't have money to hire a lawyer, but that could change at any time, and you could be in trouble if the statute of limitations has not run out.
To define difference, the brilliant Ancient Greeks began by formalizing the process of comparison, that is, they formed ratios of magnitudes. For example:
magnitude (object A) : magnitude (object B)
which means literally:
The magnitude of A compared with the magnitude of B qualitatively, that is, one of two outcomes only: magnitude(A) = magnitude(B) or magnitude(A) not equal to magnitude(B).
Note that the magnitudes are unknown till this stage. They are only being compared qualitatively (by visual observation).
The incredible breakthrough occurred when they discovered the abstraction of a unit, from which the natural numbers were born. A unit is defined very simply as the comparison of any magnitude with itself or another equal magnitude, that is,
magnitude (X) : magnitude (X) = UNIT (**)
(**) This great accomplishment led to quantitative measurement. Now, it was possible to tell the difference between two magnitudes being compared (provided both are measurable in whole units), and a known symbol could be associated (having a known magnitude value) with both magnitude(A) and magnitude(B), that is, both magnitudes are now comparable quantitatively as natural numbers.
Note that the magnitudes are known as natural numbers at this stage. They are being compared quantitatively (by finding the difference in terms of a natural number).
Recap: We started with an unknown magnitude (X) and arrived at a quantitative measurement of X in terms of natural numbers. A natural number is a ratio X : N where X is a ratio of measurable magnitudes and N is a unit.
Now, this great knowledge was insufficient where incommensurable magnitudes (pi, sqrt(2), e, etc) are concerned. That is, incommensurable magnitudes can be represented only by approximations through use of fractions (or radix systems).
I am not unaware of how dull mathematics professors and teachers can be, so I suggest you find a quiet room and study these things long and diligently.
These few paragraphs are meant only to whet your appetite. For the full story of the intricate thought processes of the Ancient Greeks and much more on the development and history of numbers, calculus and mathematics in general, you will have to wait for the publication of the greatest unpublished work in mathematics:
What you had to know in mathematics, but your educators could not tell you.
This great work is the correction and completion of Euclid's Elements, The Works of Archimedes and Apollonius' Conics. It incorporates the New Calculus. Contained therein is new knowledge of averages, natural averages, area, volume and hyper volume, which the Ancient Greeks understood but were unable to formalize.
Reader FAQs:
May I write articles about the New Calculus?
The short answer to this question is: NO. The current circumstances are such that the academic community as a whole, needs to publicly acknowledge the New Calculus is the first rigorous formulation. Until this happens, I cannot grant permission to anyone wishing to write about the New Calculus.
May I teach the New Calculus in class?
Yes, on one condition: You must use the correct attribution:
"The New Calculus was discovered by John Gabriel in 2002 and published online in 2010."
It would also not hurt to refer your students to my websites.
Can I write a review?
Unfortunately, I can't stop you from writing a review. However, be warned, if you misrepresent information, you may be sued. Truth is, you probably won't be sued because I don't have money to hire a lawyer, but that could change at any time, and you could be in trouble if the statute of limitations has not run out.
Above photograph has been edited to prevent positive identification using facial recognition software.
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.
My life's journey thus far has been very "interesting". I am certain my days are numbered (as are everyone else's) and whether I am recognized or not, will mean nothing once I am gone. True knowledge is always discovered, never invented. There is never anything new - neither ideas, nor inventions. Future inventions and ideas have always existed - we just cannot think of them all, because we exist a finite time in our reality. What one imagines is as good as real, if and only if it is well-defined.
So what where you hoping to realize by reading this far? I do not have all the answers - neither to the meaning of life(if there is any), nor to mathematics. The New Calculus is only "new" because I was the first known to discover it in the history of man. However, the ideas and knowledge of the New Calculus have always existed, just as all other knowledge has always existed. In this sense, attributing greatness really means nothing more than acknowledging the individual who first thought of useful knowledge in a given time frame of history.
Ethnic roots: Cyprus and Lesvos are the islands of my recent ancestors. Before them, your guess is as good as any.
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 mind-control 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. An irony?
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.
My life's journey thus far has been very "interesting". I am certain my days are numbered (as are everyone else's) and whether I am recognized or not, will mean nothing once I am gone. True knowledge is always discovered, never invented. There is never anything new - neither ideas, nor inventions. Future inventions and ideas have always existed - we just cannot think of them all, because we exist a finite time in our reality. What one imagines is as good as real, if and only if it is well-defined.
So what where you hoping to realize by reading this far? I do not have all the answers - neither to the meaning of life(if there is any), nor to mathematics. The New Calculus is only "new" because I was the first known to discover it in the history of man. However, the ideas and knowledge of the New Calculus have always existed, just as all other knowledge has always existed. In this sense, attributing greatness really means nothing more than acknowledging the individual who first thought of useful knowledge in a given time frame of history.
Ethnic roots: Cyprus and Lesvos are the islands of my recent ancestors. Before them, your guess is as good as any.
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 mind-control 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. An irony?
Hall of Infamy: Slander and Disrepute Infamers:
The following slanderers/murderers have at one or other time attributed slander and disrepute to my name.
Mark Chu Carroll (self-confessed Jew, slanderer)
Kurt Deligne (self-confessed homosexual, slanderer)
All it takes to destroy a man's reputation is a few slanderers. Once a reputation is destroyed, it's not possible to accomplish success or continue earning a living.
This day, 16th February 2013, I swear that I will take the most prized knowledge and ideas, which I have not shared with the world to the grave with me. I hope that my accusers and slanderers suffer a long and terrible life, followed by a slow, agonizing death. Thanks to the efforts of slanderers, I have become a pariah amongst the mathematics community. I have been compared to the likes of Gene Ray of Time Cube - a delusional individual, whom I knew nothing about until recently (Feb 2013). Such comparisons are irresponsible and destructive, and amount to slander and disrepute.
How to commit murder and not be prosecuted:
In the United States, it is legal to call someone a crank. You have the right by law to call anyone you like or don't like, a crank. It does not matter if you are correct or not. It does not matter if you know your subject well enough or not. It does not even matter if you are educated. In fact, you do not even have to prove that the person you are defaming is a crank! You may call anyone you like a crank.
Disrepute and slander are the most effective tools of murder. One is never charged and never pays for one's heinous crimes. There are no court appearances, no jail time, no execution. It is a fact that once a reputation is destroyed or severely blemished, then it is effectively the same as murder. The reasons are not hard to see. In our modern age, people frequently are attracted to the most negative reports of others they hear about. Often individuals accept such reports to be true at face value. With the advanced media of the internet and television, it takes only a few seconds to destroy a good man's reputation. Given the varying degrees of intelligence, many people will often not bother verifying negative reports. For them, where there's smoke, there must be fire.
Whatever you do, don't call anyone a crank. Even if a person is wrong, no one has any right to call any person a crank. We are not all equal in terms of intelligence. History has shown over and over again how the greatest minds were often perceived to be cranks. If you have nothing good to say, just don't say anything. I am the first to admit, that even I have on many occasions used profanity out of anger. Just as profanity is damaging and wrong, so is slander and disrepute even more deadly than profanity.
Hardly anyone takes profanity seriously, but slander and disrepute are cankerous. Once the seed of doubt has been sown, it grows almost like an avalanche. An avalanche is unstoppable in nature. Slander and disrepute are also unstoppable, but unlike an avalanche, slander and disrepute are preventable.
A slanderer is a murderer with a licence to kill.
The following slanderers/murderers have at one or other time attributed slander and disrepute to my name.
Mark Chu Carroll (self-confessed Jew, slanderer)
Kurt Deligne (self-confessed homosexual, slanderer)
All it takes to destroy a man's reputation is a few slanderers. Once a reputation is destroyed, it's not possible to accomplish success or continue earning a living.
This day, 16th February 2013, I swear that I will take the most prized knowledge and ideas, which I have not shared with the world to the grave with me. I hope that my accusers and slanderers suffer a long and terrible life, followed by a slow, agonizing death. Thanks to the efforts of slanderers, I have become a pariah amongst the mathematics community. I have been compared to the likes of Gene Ray of Time Cube - a delusional individual, whom I knew nothing about until recently (Feb 2013). Such comparisons are irresponsible and destructive, and amount to slander and disrepute.
How to commit murder and not be prosecuted:
In the United States, it is legal to call someone a crank. You have the right by law to call anyone you like or don't like, a crank. It does not matter if you are correct or not. It does not matter if you know your subject well enough or not. It does not even matter if you are educated. In fact, you do not even have to prove that the person you are defaming is a crank! You may call anyone you like a crank.
Disrepute and slander are the most effective tools of murder. One is never charged and never pays for one's heinous crimes. There are no court appearances, no jail time, no execution. It is a fact that once a reputation is destroyed or severely blemished, then it is effectively the same as murder. The reasons are not hard to see. In our modern age, people frequently are attracted to the most negative reports of others they hear about. Often individuals accept such reports to be true at face value. With the advanced media of the internet and television, it takes only a few seconds to destroy a good man's reputation. Given the varying degrees of intelligence, many people will often not bother verifying negative reports. For them, where there's smoke, there must be fire.
Whatever you do, don't call anyone a crank. Even if a person is wrong, no one has any right to call any person a crank. We are not all equal in terms of intelligence. History has shown over and over again how the greatest minds were often perceived to be cranks. If you have nothing good to say, just don't say anything. I am the first to admit, that even I have on many occasions used profanity out of anger. Just as profanity is damaging and wrong, so is slander and disrepute even more deadly than profanity.
Hardly anyone takes profanity seriously, but slander and disrepute are cankerous. Once the seed of doubt has been sown, it grows almost like an avalanche. An avalanche is unstoppable in nature. Slander and disrepute are also unstoppable, but unlike an avalanche, slander and disrepute are preventable.
A slanderer is a murderer with a licence to kill.
