Our present system of education came to us through the coloniser who came to India to loot, and not to benefit us.
Colonial education facilitated this loot by stabilising colonial rule, and developing a slave mentality of respect for white/Western authority.
Creating a slave mentality as a stabilising force was needed since the British were militarily weak and easily overthrown by the uprising of 1857.
Colonial education (contd)
This education was NOT designed for our benefit as people have naively assumed.
All education reform committees from the Zakir Hussain committee onwards noted the problems with the education system (rote learning etc) but wrongly assumed that these were aberrations rather than part of the design of the missionary education system created by the church.
So, we need to change the education system in a fundamental way.
Colonial education (contd)
We have never actually examined or critically compared that education system with more traditional education or with better alternatives available today. How does it serve our interests. E.g. of the calendar.
We should do a critical comparison now. How does this imported education system serve OUR interests? And how to eliminate notions (like the calendar) which are clearly contrary to our interests.
The myth of Euclid
Our school texts in math tell us a false story of an early Greek called Euclid. Then they tell us to imitate him and do math the way he supposedly did.
In fact (a) there is no evidence that there actually was any Euclid. I have a prize of Rs 2 lakhs for serious evidence about Euclid. No one has been able to claim it.
There is ample counter-evidence that the book was written at another time (over 700 years later, 5th c. CE) by another person for a completely different purpose.
The myth of Euclid (contd)
All Greek manuscripts of the Elements claim it was authored by Theon +4th c. CE, the last librarian of the Library of Alexandria. His daughter completed the book. Since she was from Alexandria, in Africa, she was black.
The relevance for us is that the reference to early Greeks hides the church connection. We keep quarreling about Pythagoras, without seeing the role of the church. (The church regarded the early Greeks as the only “friends of Christians”. )
Mathematics and religion
The book Elements was actually written for religious reasons.
The Egyptians did two types of math: a practical math and a mystery math.
Mathematics and religion (contd)
The early Greeks learnt from Egyptians, but focused on mystery math.
Plato explains in Meno that the function of (mystery) math is to arouse the soul. In Republic he prescribed the teaching of math for the young men of the Republic since arousing the soul makes people virtuous.
The Pythagoreans thought similarly.
Mathematics and religion (contd)
The aim of reason was to draw the mind inwards so that the soul could combine with Nous. Similar to yoga = union of atman with Brahman (not some physical exercise).
This line of thought carried forward by Neoplatonists (like Plotinus and Proclus. We know them today as sufis. Through sufism, this went into Islam.
Mathematics and religion (contd)
Relevance for us is this: This led to the Western superstition that math is ETERNAL KNOWLEDGE, hence EXACT.
Since there was no way to make infinite (non-terminating) algorithms, like square rootws exact, the West converted math to metaphysics.
In India गणित was done for practical purposes, hence accepted as approximate.
The church theology of reason
The only solid fact we have is that the book Elements was used by the church as a textbook for its priests for almost 7 centuries. Therefore, the philosophy attributed to the book had to be theologically correct and serve the church’s political interests.
The church theology of reason (contd)
Few people know that the church developed a theology of reason during the Crusades. This theology of reason was developed in order to persuade Muslims who were (a) too strong to be converted by force, and (b) rejected the Christian scriptures as corrupted. Muslims however accepted reason, as in the aqi-i-kalam or Muslim rational theology. Christian rational theology was an attempt to counter that.
The church theology of reason (contd)
The church political requirement was to persuade Muslims. Since Muslims accepted reason. hence, the church spoke of universal reason. Since the sole aim of the church was persuasion, it spoke of reason as a way to provide irrefragable proof.
When the church taught us the “superiority” of this way of doing math was actually teaching us that by imitating the church we become superior.
Normal vs formal reason
Many people parrot that mathematics is a way to teach reasoning. However, they failed to understand the double speak about reason.
The church understanding of reason is different from the everyday understanding.
Normal vs formal reason (contd)
Normal reasoning proceeds from facts.
The church method of reasoning begins from non-factual assumptions called axioms or postulates. In the church method of (formal) reasoning, the use of facts is prohibited.
Normal vs formal reason (contd)
This is also the case in formal mathematical proof. In such a proof one is not allowed to use facts at any point. That is, normal everyday reasoning which uses facts is not to be confused with formal reasoning without facts.
The relevance for us today is this: math is difficult because we teach formal math. Formal math is metaphysics, not mere abstraction.
Invisible points are metaphysics not abstraction.
No pure deductive proofs in the Elements
Further, it is a SECOND FALSE STORY that the book Elements contains proofs based purely on deductive reason.
The very first proposition of the Elements contains an empirical proof as does its fourth proposition (SAS).
No pure deductive proofs in the Elements (contd)
The proof of the “Pythagorean theorem” is also empirical since based on SAS. It is prolix because very easy empirical proofs are available in other traditions.
Eventually, after 700 years it was admitted that the story about deductive proofs in the Elements is false.
However, the myth was still accepted as true that there was a Euclid who intended to give “superior” deductive proofs which are superior.
Hilbert’s synthetic geometry
Accordingly, various people tried to remedy matters. Hilbert invented synthetic geometry which corrected Euclid.
In synthetic geometry there is no notion of distance. Therefore, it is said that instruments of this geometry are unmarked straight-edge and the collapsible compasses. This ensures that distances cannot be picked and carried.
Hilbert’s synthetic geometry (contd)
Therefore, also, the NCERT school text says that “ideally” a ruler should be unmarked.
Hilbert also replaced the original term equality with the term congruence.
Hilbert’s synthetic geometry (contd)
In order to get a “Pythagorean theorem” in synthetic geometry we are led to the absurdity that although distances are not defined, length is not defined, area is defined
It is this effort which led to the birth of formal mathematics.
Birkhoff’s metric geometry
Synthetic geometry was invented to explain the prolixity of the Elements. But it lacks practical value.
Therefore, Birkhoff, in 1930, gave an axiomatic metric geometry. But, this two trivialises Elements and the proof of the Pythagorean theorem
Deductive vs empirical proofs
It is a THIRD FALSE STORY, part of church theology, that proofs based on reason alone are infallible. Everyone accepts that empirical proofs are fallible But proofs based on reason alone are also fallible.
Fallibility of deduction
Thus, one may mistake an invalid deductive proof for a valid one far more easily than one may mistake a rope for a snake.
For 7 centuries, all Western scholars foolishly thought that the proofs in the Elements were models of deductive proofs.
Fallibility of deduction (contd)
Reason divorced from facts can easily lead to any nonsense. Example of the horned rabbit. (1) All animals have horns, (2) A rabbit is an animal (3) Therefore a rabbit has horns. The conclusion is nonsense because it starts from a non-fact (all animals have horns). But there I no room for facts in axiomatic math
Fallibility of deduction (contd)
Even logic is not unique (as in the theological belief that logic binds God). Empirical proof is needed for the belief that only 2-valued logic applies to the real world. (Buddhist logic, quantum mechanics).
Fallibility of deduction (contd)
All schools of Indian philosophy (except Lokayata) accepted reason, but they also accepted empirical proofs (or प्रत्यक्ष प्रमाणor facts). Rejecting empirical proofs as inferior teaches students students that all Indian thought is inferior.
Various types of geometry
When it was finally admitted (at the beginning of the 20th c.) that there are no deductive proofs in the Elements, many attempts were made to save the story. This led to different types of geometry. We must understand those to decide which geometry to teach.
5 types of geometry.
Because we believe in blindly copying everything Western, we have copied everything resulting in a confused hotch-potch
Hence, your school text mixes up incompatible types of geometry: synthetic geometry (which does not define distance) with empirical compass-box geometry which does.
3 types of math: empirical, axiomatic, religious.
Compass box geometry and sulba sutra geometry are empirical.
Hilbert's synthetic geometry and Birkhoff's metric geometry are axiomatic.,
Greek geometry (as in Plato) is
religious.
Our purpose for teaching math
For its practical value for everyday commerce, science, and engineering,
NOT for aesthetic or religious reasons.
Therefore, we must decide which type of math and geometry to teach.
We can do this only after we understand the various type of math/geometry and compare them thoroughly.