Dear Prasanna

You can see discursion on page

http://mathoverflow.net/questions/33280/how-would-calculus-be-possible-in-a-finitist-axiom-system

Sincerelly Yours Yuri

The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.

The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.

You may refer to work of Madhava and Nilakantha.

and refer to article as below

http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433

Calculus created in India 250 years before Newton: study

CBC News Posted: Aug 14, 2007 12:55 PM ET Last Updated: Aug 14, 2007 2:25 PM ET

Researchers in England may have finally settled the centuries-old debate over who gets credit for the creation of calculus.

For years, English scientist Isaac Newton and German philosopher Gottfried Leibniz both claimed credit for inventing the mathematical system sometime around the end of the seventeenth century.

Now, a team from the universities of Manchester and Exeter says it knows where the true credit lies — and it's with someone else completely.

The "Kerala school," a little-known group of scholars and mathematicians in fourteenth century India, identified the "infinite series" — one of the basic components of calculus — around 1350.

Dr. George Gheverghese Joseph, a member of the research team, says the findings should not diminish Newton or Leibniz, but rather exalt the non-European thinkers whose contributions are often ignored.

"The beginnings of modern mathsis usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten," he said. "The brilliance of Newton's work at the end of the seventeenth century stands undiminished — especially when it came to the algorithms of calculus.

"But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus — infinite series."

He argues that imperialist attitudes are to blame for suppressing the true story behind the discovery of calculus.

"There were many reasons why the contribution of the Kerala school has not been acknowledged," he said. "A prime reason is neglect of scientific ideas emanating from the Non-European world, a legacy of European colonialism and beyond."

However, he concedes there are other factors also in play.

"There is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written," he admits.

Joseph made the discovery while conducting research for the as-yet unpublished third edition of his best-selling book The Crest of the Peacock: the Non-European Roots of Mathematics.

Dear authors please understand my question. I would exactly like to know how can I start teaching calculus in a systematic way for , say a high school student with the subject developed axiomatically. i.e. starting from the introduction of the primitives, then stating the axioms and then proceeding to theorems. I suppose the fundamental axioms of calculus would be stated in terms of  limits of functions or sequences. I actually need their complete list. Please don't give links to totally irrelevant pages. 

Then kindly refer to standard textbooks.

Then kindly refer to standard textbooks.

Then kindly refer to standard textbooks.

Then kindly refer to standard textbooks.

Dear Prasanna. One thing is the fundamental theorem of Calculus and another thing is what a professor should teach on Calculus. If you would like to know what to teach to your students on Calculus, then the best thing is to consult one of the several books that exist on this matter. In summary you have to teach functions, limits, continuity, derivatives of several orders, partial derivatives, and defined and undefined integrals, among others, including their theorems and corollaries.

Dear Jorge, I suppose theorems are statements of truth which are derived from even more basic set of statements called axioms. The axioms are statements assumed to be true. The mathematician doesn't care whether they are true in the real world. All that a mathematician does is to say the implications of these statements for an infinite set of primitives . For instance, see this link which gives a presentation on axioms of Boolean algebra based on which the entire set of Boolean theorems are derived.

http://academic.eng.au.edu/~ce3704/02_2up.pdf

In the field of calculus. I know that understanding the concept of limit should precede that of understanding derivatives. But during the process of obtaining the derivatives of many common functions we have to use more basic truth statements in the form of limits of sequences or limits of functions. (for example to find  the derivative of the exponential function we use a fundamental limit statement ). My question is how many such limit relations exist. I suppose it should be a finite number as in the case of Boolean algebra.( maybe a set of ten statements for diffential calculus?). If such a finite set of statements exist, what are they.

Axiom vs Theorem

An axiom is a statement that is considered to be true, based on logic; however, it cannot be proven or demonstrated because it is simply considered as self-evident. Basically, anything declared to be true and accepted, but does not have any proof or has some practical way of proving it, is an axiom. It is also sometimes referred to as a postulate, or an assumption.

An axiom’s basis for its truth is often disregarded. It simply is, and there is no need to deliberate any further. However, lots of axioms are still challenged by various minds, and only time will tell if they are crackpots or geniuses.

Axioms can be categorized as logical or non-logical. Logical axioms are universally accepted and valid statements, while non-logical axioms are usually logical expressions used in building mathematical theories.

It is much easier to distinguish an axiom in mathematics. An axiom is often a statement assumed to be true for the sake of expressing a logical sequence. They are the principal building blocks of proving statements. Axioms serve as the starting point of other mathematical statements. These statements, which are derived from axioms, are called theorems.

A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives. Theorems are often proven through rigorous mathematical and logical reasoning, and the process towards the proof will, of course, involve one or more axioms and other statements, which are already accepted to be true.

Theorems are often expressed to be derived, and these derivations are considered to be the proof of the expression. The two components of the theorem’s proof are called the hypothesis and the conclusion. It should be noted that theorems are more often challenged than axioms, because they are subject to more interpretations, and various derivation methods.

It is not difficult to consider some theorems as axioms, since there are other statements that are intuitively assumed to be true. However, they are more appropriately considered as theorems, due to the fact that they can be derived via principles of deduction.

Summary:

1. An axiom is a statement that is assumed to be true without any proof, while a theory is subject to be proven before it is considered to be true or false.

2. An axiom is often self-evident, while a theory will often need other statements, such as other theories and axioms, to become valid.

3. Theorems are naturally challenged more than axioms.

4. Basically, theorems are derived from axioms and a set of logical connectives.

5. Axioms are the basic building blocks of logical or mathematical statements, as they serve as the starting points of theorems.

6. Axioms can be categorized as logical or non-logical.

7. The two components of the theorem’s proof are called the hypothesis and the conclusion.

 Of course, the different concepts associated to Calculus are used to define other concepts and definitions. For example, to define the concept of derivatives you have to use the concepts of functions and limits. Calculus can be considered as a building, you should start constructing the building from the soil, and you build the different floors one in the top of the other. However, to go to the fifth floor you have to pass the first, second, third and fourth floors.

The number of theorems and corollaries that should be used in your course of Calculus will depend how far the professor wishes or is obliges to go. If you wish to teach Calculus to future economists, you should not need to go too deep into the materia, because they do not need it. However, if you are teaching Calculus to future mathematicians, then you must go deep into the materia. In the first case you can avoid several theorems and corollaries. In the second to have to used many theorems and corollaries

Dear jorge,

thanks for the elaboration. Still it doesn't provide an answer. What I am asking is not " how many of the theorems of calculus should be taught?". They may virtually be infinite in number so there is no point in planning to teach  all the theorems of calculus. What I am asking is " is it at least possible to teach all the axioms of calculus?", Since I believe the number of axioms will be finite.

Dear Prasanna. There is a finite number of theorems, corollaries and axioms on Calculus and can be teach in a course of two years, depending how far the professor wish to go and the program of the course. I gave lessons on Calculus when I was professor in my country.

Dear Prasanna, This is also my problem. I think, theory of analysis is not ready yet. There is not a list of axioms. (Somebody think, that all algebra, topology, geometry need to built analysis. Other people think that this question is not interesting.) Of course we do not teach axiomatic system of analysis (or calculus), but the teacher should it know. Unfortunately axiom system of analysis does not exist. It should be written.