1. What is the source of the practical value of math?
There was practical value also in traditional math.
For example the practical value of arithmetic for commerce.
But this practical value originated from efficient techniques of calculation, NOT from metaphysical proof.
Peano's axioms or Bertrand Russell's 378 page proof of 1+1=2 adds NOTHING to the practical value of arithmetic for commerce.
Indian arithmetic algorithms were efficient techniques of calculation compared to inefficient Greek and Roman arithmetic done on an abacus.
The computer is an efficient technique of calculation and is widely used today for various practical purposes including calculating the trajectory of spacecraft.
2. Any practical calculation involves approximation.
For example, the calculation of
involves
a non-terminating algorithm.
The practical process is to truncate or round the infinite series and accept the value as an approximation. Practically, no interest in and no possibility of knowing ALL infinite digits in the expansion of
Hence, the sulba sutra refers to
as savisesa (or “with a remainder”).Similarly, a computer cannot do the metaphysics of infinity. To represent one place in the binary expansion of a number, a computer needs one bit. A computer has finite memory, howsoever large. Therefore, a computer can never represent any “irrational” number such as
.In fact a computer cannot also represent an infinity of integers, therefore a computer can never do integer arithmetic exactly.
But a computer is today used to do most practical tasks related to mathematics and its applications to commerce, engineering, and science. Therefore, the metaphysics of infinity used in axiomatic math is not a source of practical value in most cases of applications to commerce, engineering and science.
Note that I am NOT saying that we go back to indigenous tradition and reject everything modern. The computer is not part of traditional arithmetic, but we accept the computer as an efficient technique of calculation. But the philosophy of math it uses (math as approximation) is similar to that of traditional math.
3. Pedagogy: Can axiomatic math (which teaches avoidance of the empirical) be taught at the school level?
NO. Teaching at school level CAN ONLY be done by referring to the empirical.
Therefore, teaching formal math in schools lands us in the absurd and confusing situation of teaching avoidance of the empirical by referring to the empirical.
For example and invisible point is defined by means of a dot, but people are told the dot is not a point.
It is this absurd attempt to teach avoidance of empirical through the empirical which leads to the confusion in math teaching.
For example, the mixing of synthetic and metric concepts.
The mixing of empirical and metaphysical concepts (“straight line is obtained by extending a line segment indefinitely in both directions”), etc.
Math is difficult because of this intrinsic confusion in teaching axiomatic math by empirical means.
This confusion arises from the desire to thrust axiomatic math on children, an impossible project.
Who benefits from this project?
The majority of children drop math after class X. They learn nothing from this attempt to teach axiomatic math. The difficulty puts them off math for ever.
They are taught ignorance. For example they are taught about formal real numbers, but never told exactly what a formal real number is.
Ignorant people are forced to grope in the dark. This groping is guided in wrong directions with the help of all sorts of other false stories that are instilled along with the teaching of axiomatic math: such as false stories about “Euclid”.
The real interest is that of political domination by the church.
(Why is axiomatic math taught to teach reasoning and logic. Logic could be taught differently as in the Nyaya sutra. We imitate this church practice without asking for the relation of the “reason” being taught to the church theology of reason.) As explained earlier, the real aim is not to teach reasoning but to teach avoidance of the empirical.
Teaching axiomatic math only teaches children to distrust commonsense and accept Western authority. This is a project which delivers political advantage of intellectual dominance to the West. By imitating it our interests suffer. There is no practical advantage for the student, in this.
So, the conclusion is that we should teach empirical geometry. We have two choices compass box geometry vs string geometry. Today we will review these choices.
What is your role?
Many of you felt that these revelations make it difficult for you to continue teaching from the confused school text.
You felt that this should be made widely known.
But how?
The easiest way to spread something widely is through the education system.
The education system is fixed by the government.
Therefore, the best way of spreading this is to persuade the government to change the curriculum..
This can be done. The government is not against us, but it needs solid evidence to change the system.
The aim of this project is to compile this solid evidence.
Your role is to try out the new methods of teaching traditional geometry with small groups of students and report the results.