## onsdag 20 november 2013

### 10 Reasons Why Standard Calculus is Wrong Today

In recent posts I have argued that Calculus in its standard symbolic mathematics form,  in the computer era of our time should be given a new form as constructive mathematics, where symbols are supported by numerical algorithms. Let me here list 10 reasons why the standard text Calculus: A Complete Course by Adams and Essex, is no good today:
1. Real numbers are introduced with geometric reference as points on a number line.
2. Completeness of the set of real numbers is identified as a geometric property of the number line to "have no holes", which as a pretentious triviality can only be mystifying to the student.
3. The derivative is introduced through a limit process with division by infinitely small quantities, which is numerically ill-posed.
4. The integral is introduced with geometric reference to area defined by equality of sup of lower sums and inf of upper sums, which can only be mystifying to the student. The Fundamental Theorem is expressed as differentiation of area, which misses the esssence of the Fundamental Theorem with the integral arising as the solution to a basic initial value problem.
5. The exponential function is introduced as the inverse of the logarithm with the basic differential equation satisfied by the exponential arising as a strange mystifying surprise at the end.
6. Trigonometric functions are defined geometrically and not constructively as solutions to certain basic initial value problems computable by time stepping. This is wrong from mathematical point of view.
7. The central Fixed Point Theorem is not proved in the text, but left as an exercise for students to skip.
8. The proof of existence of solutions to the basic initial value problem is not given, just vaguely hinted at.
9. The presentation of calculus of several variables uses an awkward notation, and is severely limited with very little about the basic field of partial differential equations. Implicit function theorem not proved. Inverse function theorem not even stated.
10. The scope of Calculus today as constructive mathematics designed by human minds performed by computers, is missing.
BodyandSoul contains corrections of these errors of Standard Calculus, and much more...

PS1 The mission of Calculus: A Complete Course is stated in its Preface to be:
• Of course it is not true that computers can contain all of mathematics; so one challenge (of a new edition) is to ensure that no one imagines that they can.
PS2 The authors of Calculus: A Complete Course want to impress the reader by using a "mathematical style" with DEFINITION, THEOREM, PROOF and EXAMPLE displayed in very big letters, but then in reality offer little, since proofs of important results are generally omitted and the proofs actually offered are mostly pretentious trivialities. BodyandSoul (BS) was written in order to replace the emptiness of a standard calculus text. Seeing BS by teachers at Chalmers with experience of BS, being replaced by Calculus: A Complete Course, makes me sad or angry, or both.