## What is the New Calculus?

History:

The New Calculus (NC) was conceived in the early part of the second half of the twentieth century. It is the first and only rigorous formulation of calculus in human history. I say in "human" history, because it's quite possible advanced alien civilisations might have discovered it a long time ago. Based only on well-formed concepts, the new calculus does not use any ill-formed concepts such as limit, infinity or infinitesimals.

Due to persecution from mainstream academia, this is the sixth time this site has been rebuilt. In fact, Weebly once took down the site due to false reports of spamming from Google Inc. Weebly subsequently apologised, but I had already created a new site on Wix. As of 9/17/2016, the owners of Wix succumbed to pressure from mainstream academics to take the site offline. It is unbelievable that in this day and age, such Nazi tactics are used by the supposedly enlightened, educated and open-minded academic mainstream community. These efforts were driven by Prof. Gilbert Strang (aka Port563 on sci.math) of MIT and Prof. Jack Huizenga, a Harvard alumnus. Both these academics did not like the comments I made about them. You can read these comments here and here. Neither of the previous two links are on Weebly.

The single variable New Calculus can be learned and mastered in just 2-4 weeks with the only prerequisite being high school mathematics. The multivariable New Calculus can be mastered in 8 weeks. However, the New Calculus contains a new kind of mathematics, one that includes tangent objects. While the single NC is easily extended to the multivariable NC in the same way as mainstream calculus, tangent objects provide a far more efficient way of working with multivariable calculus, including differential equations and partial differential equations.

The purpose of this site is to expose the flaws of mainstream calculus and elaborate on the beauty and light of the New Calculus which contains many features that are not possible using the flawed mainstream formulation:

1. The Auxiliary equation is the first and most powerful feature that a student learns. Not possible in the flawed formulation, this feature has been used by thousands of STEM professionals worldwide in fields as varied as technology, computer aided design, statistics and last but not least, education. Some important uses of the auxiliary equation include:

a. Non-linear regression

b. Finite difference estimation

c. Solution without iteration

and much more. There is no similar or equivalent possible in the flawed mainstream formulation.

2. Systematic integration is possible for any function in the New Calculus.

3. The Gabriel polynomial (single and multivariable forms) far surpasses the Taylor analog in mainstream calculus. It contains a varied number of fixed terms and is a closed form always.

4. Solution of differential equations and use of superior numeric integration techniques.

The entire single variable New Calculus is summarised in the leading graphic on this page.

Resources:

Articles.

Interactive and dynamic applets.

The 10 lesson NC course.

The 9 lesson NC Applet course.

YouTube Videos.

High School Students:

You can learn single variable calculus in just a couple of hours by downloading New Calculus for grade 8 students.

Here is a short excerpt:

We can prove that if f(x) is a function with tangent line equation t(x)=kx+b and a parallel secant line equation

s(x)=[{f(x+n)-f(x-m)}/(m+n)] x + p, then f'(x)={f(x+n)-f(x-m)}/(m+n).

Proof:

Let t(x)=kx+b be the equation of the tangent line to the function f(x).

Then a parallel secant line is given by s(x)=[{f(x+n)-f(x-m)}/(m+n)] x + p.

So, k={f(x+n)-f(x-m)}/(m+n) because the secant lines are all parallel to the tangent line.

But the required derivative f'(x) of f(x), is given by the slope of the tangent line t(x).

Therefore f'(x)={f(x+n)-f(x-m)}/(m+n).

Q.E.D.

Also, m+n is a factor of the expression f(x+n)-f(x-m).

Proof:

From k(m+n)=f(x+n)-f(x-m), it follows that m+n divides the LHS exactly. But since m+n divides the left hand side exactly, it follows that m+n must also divide the RHS exactly. Hence, m+n is a factor of the expression f(x+n)-f(x-m). This means that if we divide f(x+n)-f(x-m) by m+n, the expression so obtained must be equal to k. This is only possible if the sum of all the terms in m and n are 0.

Q.E.D.

The previous two proofs hold for any function f.

It is easy to derive the new calculus integral definition using the new derivative definition.

Understand why the standard integral is a product of two arithmetic means using one of the following applets:

Applet 1. The New Calculus uses no ill-formed concepts such as infinite sums, limits or infinitesimals.

Applet 2. This applet explains the new calculus integral in great detail.

The Future.

Rather than rebuild this site from scratch again, I have chosen to use the Google New Calculus Site exclusively for the New Calculus and this site to expose the fallacies in mainstream mathematics. You are invited to join my New Calculus group in order to ask questions and to comment.

The ideas that added zero rigour to calculus.

The video and this applet explain the ideas that added no rigour to calculus, but in fact made it more obscure and complicated.

In the applet, you can click on the check box called "Show Information" to see why it is an ill-formed definition. It fails for many reasons, but the two most important are:

i. There is no valid construction of real numbers.

ii. Circular definitions are invalid.

The 13 fallacies of mainstream mathematics.

What's kind of sad and also kind of funny is how mainstream academics always ask for sources, usually printed or peer reviewed. The greatest mathematicians never had their work reviewed because they had no peers on their level. It is the same with me. In order for my work to be reviewed, my "peers" (I have none) would need to possess my intelligence and not to be infinitely ignorant like Prof. Gilbert Strang from MIT.

There is no such thing as an instantaneous rate of change.

So, courtesy of the internet, I am able to share some of my ideas and educate those aspiring young mathematicians with knowledge that is well formed.

Here are my four essential requirements for any concept to be well defined:

In order to be well defined, a concept

1. Must be reifiable either intangibly or tangibly.

2. Must be defined in terms of attributes which it possesses, not those it lacks.

3. Must not lead to any logical contradictions.

4. Must exist in a perfect Platonic form. What this means is that it exists independently of the human mind or any other mind.

If you can't reify a concept, then it may not exist outside your mind. If a group of mainstream academics get together and claim an infinite sum is possible, even among themselves, they do not think of it the same way. The fallacy that 0.999... and 1 are the same, is a fine example. Some academics think that it is actually possible to sum the series.

Others (such as Rudin) realise that only a limit is possible. Still others believe that it's a good idea to give a sequence a value in terms of its limit even when the limit is not known.

To *reify*, means to produce an instance of the concept so that someone who knows nothing about it, can understand it exactly the same way as another. Even though the 0.999... fallacy has been around for so many decades, ask yourself how it is that so many students and even educators have different views on it, with most forums split almost evenly among those who acknowledge the fallacy and those who don't.

You can reify a concept without someone else being able to understand it for many other reasons; some include intelligence, ignorance, etc. However, I am talking about all things being equal, in which case the concept can be acknowledged as having been reified.

If you can't get past reification, then your concept is no doubt nonsense. Some examples are:

Irrational numbers.

Infinitesimals.

Limits when there is in fact no valid construction of "real" numbers.

Infinity.

Einstein's theories.

Hawking's theories.

Definitions that are self-referential.

If a concept is not defined in terms of attributes it possesses, then you may as well be talking about innumerably many other concepts. You have endless ambiguity. It is the most important second step after reification. It describes the boundaries or limits, the extent of the instantiated object from the concept. I used to think:

"*Mathematicians are like artists, the objects arising from concepts in a mathematician's mind are only as appealing as they are well defined*"

Clearly, they are not even usable if they cannot be well defined. A good example is 0.999... - it has no use and nothing worthwhile can be done with such an idiotic definition, that is, S = Lim S. This is what I call the Eulerian Blunder because it was Euler who stated S = Lim S.

Once a concept has been reified and well defined (there are limitations to being well defined and this is why one needs to have checks for contradictions until the concept becomes axiomatic over a long period of vetting), it has to be vetted. This is done by always verifying that any results stemming from its use do not produce logical contradictions.

Finally, the last point which is sufficient for a well-formed concept, is that it must exist outside of the human mind or any other mind. For example, if aliens think of pi, they will think of it in the only logical way: ratio of the circumference magnitude to the diameter magnitude.

Perfect concepts exist whether life exists or not. That is what the Greeks discovered when they studied geometry. The concepts of geometry are all without any exception perfect concepts (Platonic).

What is pseudo mathematics?

The 13 fallacies that form the foundation of mythmatics (mainstream mathematics):

1. Infinity is not a well-formed concept.

2. There is no such thing as an infinite set. Proof by Prof. W. Mueckenheim using mainstream theory that a contradiction exists in set theory. Mueckenheim does a great job of showing how infinite set theory invalidates itself. The video is an eye opener.

3. Non-terminating radix representation cannot be used to represent "real" numbers. Much of the source for such confusion goes back all the way to Newton and the Dutch engineer Simon Stevin. It can be shown that multiplication is not distributed over infinite series. Newton experienced some really dull moments in his life and thoughts.

4. There are no irrational numbers. Hence no real numbers also.

5. An infinite sum is not possible. It is impossible to sum an infinite series according to mainstream academics. The mean value theorem is the reason calculus works and the fundamental theorem (which does not have two parts) is derived directly from it. The single variable New Calculus can be explained in under 25 minutes.

6. 1/3 =/= 0.333... Newton never used infinite series. Most of his misguided ideas came from manipulating partial divisors. This fallacy is easily debunked. In fact, the very idea of limit is flawed. Find out 5 reasons why Cauchy's derivative definition is flawed.

7. 1 =/= 0.999... It's hard to find a more controversial topic than the fallacy that 0.999... is anything but a sequence and that it's a very bad idea to define it as a limit. Irrational numbers do not exist and the worthless (not to mention absolutely useless) idea that 0.999... can be well defined as 1 is easy to dismiss. 0.999... is not well defined as 1 in this universe or any other. An article describing many reasons why it's a bad idea. My debate on SpaceTimeAndTheUniverse debunks each and every fake proof of this fallacy.

8. The integral is not the limit of an infinite sum. The definite integral is a product of arithmetic means and has nothing to do with non-existent infinite sums. The mean value theorem is the reason we can evaluate definite integrals. Riemann's ideas in this regard are completely misguided.

9. Numbers cannot be derived using sets. The von Neumann ordinal approach is a joke. The first major stumbling block is that in order to define rational numbers using set theory, we already need to know how to "count". That's right, you need to be able to compute the cardinality of a given set. Unless you are one of Cantor's delusional followers, cardinal value means *number*, not bijective cardinality myths involving sets whose members are not distinct, that is, the illusion of infinitely many points. Now, do you have any clue what effort went into deriving the machinery of counting numbers which came long after ratios of magnitudes? Of course you don't. Unless you have read my article, you don't have a clue what it means to be a *number*. Thousands of worthless books on numbers and history of numbers have been printed - all of them worthless rot.

After reading that article, ask yourself, does set theory require the natural numbers to be in place? Hint: YES

Does the von Neumann ordinal approach make any sense at all? Hint: NO

Is there any valid construction of irrational number? Hint: NO

Since there is no valid construction of irrational number, can there be any valid mathematical concept for real number? Hint: NO

10. The derivative is not a limit and does not require the limit concept. The nonsense of limit theory is trivially debunked.

11. Natural numbers did not come first. The real story of numbers is very different to anything ever published on thousands of worthless books on the same. The understanding of mainstream academics is based entirely on that mythical object called a *real number line. *It is absurd how they imagine numbers correspond to indistinguishable points on a number line.

12. There are no axioms in real mathematics.

13. Neither Cauchy sequences nor Dedekind cuts are valid constructions of real numbers.

**John Gabriel**

Age: 55 (Photo: 3/7/2017)

Favourite Original Quotes:

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Age: 55 (Photo: 3/7/2017)

Favourite Original Quotes:

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*Whatever I imagine is real because whatever I imagine is well defined*""

*The objects that arise from concepts in a mathematician's mind are only as appealing as they are well defined*""

*All well-formed knowledge is discovered, never invented*""

*Future inventions have always existed, only we cannot realise them all*""

*A number is the measure of a magnitude*"**Contact:**

**john_gabriel AT yahoo.com**