Education - History and Philosophy of Mathematics - Uncategorized

How to make calculus easy

Calculus is difficult because real numbers are wrongly believed essential to it. Reverting to the way calculus originated makes it easy.

Bertrand Russell arguing with a grocer
Axiomatic proofs (like Russell’s 378 page proof of 1+1=2) add zilch to the practical value of math in a grocer’s shop, but they make math excessively difficult. The purported “superiority” of axiomatic proofs is a church superstition which was politically convenient to the Crusading church in developing its rational theology which used proofs based on reason minus facts (or observations).

Having stolen the calculus from India, Europeans eventually realized that they did not fully understand it. Hence, they invented “real” numbers, long after the purported “discovery” of calculus by Newton and Leibniz. Real numbers add to the difficulty but not to the practical value of calculus.

Real numbers are regarded as essential to define core calculus concepts such as derivative and integral, and to sum its infinite series. But they are not actually defined in the typical fat “Thomas’ calculus” texts, of over 1300 pages. Failure to define core concepts naturally makes calculus difficult.

Further, real numbers have nil practical value, since all current applications of the calculus, such as calculating rocket trajectories, are done numerically on computers which use floating point numbers and cannot use metaphysical (=unreal) real numbers.

Original Indian understanding of calculus

The solution to calculus difficulties, then, is not to cancel the calculus, but to make it easy, by cancelling real numbers. This can be easily achieved by reverting to the original Indian understanding (epistemology) with which calculus originated. That has three key components. (1) Aryabhata’s (5th c.) approach to calculus as concerning the numerical solution of difference/differential equations. (2) Using the “non-Archimedean” arithmetic of Brahmagupta’s (7th c.) polynomials (or unexpressed arithmetic), which algebra Europeans failed to grasp, but which first enabled Nilakantha (15th c.) to correctly sum infinite series. (This “cancels” real numbers, which are Archimedean.) (3) The philosophy of zeroism or sunyavada (which asserts that belief in exactitude is erroneous, since nothing endures exactly for even two instants).

This way of doing calculus makes it very easy. This has been pedagogically demonstrated with 8 groups of students across 5 universities in 3 countries. It also enables students to solve harder, real-life problems not covered in calculus courses at K-12 or beginning undergraduate level.

Church origins of axiomatic proofs

Axiomatic real numbers only provide the metaphysical (or fantasized) exact sum of an infinite series, such as that for the number pi(π). The axiomatic method was brazenly imputed to the “Euclid” book, when that book first arrived in Europe as a Crusading trophy. Hilariously, however, the “Euclid” book has no axiomatic proofs, and “real” numbers are typically used for an axiomatic proof of even its first proposition.

Actually, axiomatic proof, or a method of reasoning which dodges facts, was a church innovation. During the Crusades, the church had a political requirement for a Christian rational theology to counter Islamic rational theology. Hence, the church accepted reason, but cunningly prohibited facts which contradict so many of its dogmas. It declared reasoning (minus facts) from (metaphysical) assumptions a “superior” method of reasoning. The church brazenly read this axiomatic method into the “Euclid” book, to hide its real origins. Europeans, under church hegemony, gullibly believed that for centuries, though the “Euclid” book actually has no axiomatic proofs.

Today, it is a stock Western superstition that axiomatic proof is (epistemically) “superior”, since “infallible”, like the pope. Actually, axiomatic proofs are highly fallible.

The political advantage of making math difficult

But the West won’t abandon the axiomatic method, because the resulting difficulty of math makes most people ignorant of math. Ignorant people must trust authority, and trusted authorities are Western, as colonial education and Wikipedia both teach us. That is, the difficulty of math has the current political advantage (for the West) that it helps Western (or Western-approved) mathematicians to dominate mathematics, needed for science.

However, the difficulty of calculus also means fewer people available for technology development. Thus, the question today is whether, for the sake of its soft power, the West is willing to lose the technology race, and the hard power from its slender technological lead over others, as the California new math framework indicates.

How long can the false history and bad philosophy of math last?

The question is also how long that soft power will last, before the truth is exposed, in this age of information. And whether and for how long the colonised non-West will continue to blindly believe those false myths and superstitions (about the history and philosophy of math) used by the coloniser to teach the fundamental lesson of colonial education: “we are superior, imitate us”.

Despite indoctrination through globalised colonial education, the colonised just might wake up and cross-check that false history and discard that bad philosophy of math.

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