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.
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.
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. (Forgot to give, will
give now.)
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.
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.
Further, (b) it is a SECOND FALSE
STORY that the book Elements contains proofs based purely on reason.
The very first proposition of the Elements contains an empirical
proof as does its fourth proposition (SAS). 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.
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.
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.
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.
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
.
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).
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.
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.
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.
You never did this. We do so now.
Q. 1 (a) What exactly is axiomatic math?
(b) Does axiomatic math have practical value?
A. 1 (a)
The text states the story that the unique feature of axiomatic math is the use of deductive inference.
This story is false: for example deductive inference was widely accepted in Indian culture and in the Indian proof of the Pythagorean proposition.
However, such non-Western proofs are declared as inferior since they involve empirical elements.
So, the unique feature of axiomatic math is actually avoidance of the empirical (avoidance of facts) (not merely the use of deduction).
On some wrong religious belief that this avoidance of facts leads to infallible and exact knowledge.
However, if facts are avoided, any nonsense can be proved as a mathematical theorem as in the case of the rabbit with two horns.
A method of proof which enables one to PROVE any nonsense dogma as ”reasonable” greatly suited the church and its theology of reason. While this was of political value to the church, it is of no political or practical value to us. Hence, we should NOT imitate this method of proof glorified by the church.
To hide its real motives for glorifying this method of proof (by avoiding facts) , the church attributed this method of proof to “Euclid” and a book he supposedly wrote called the Elements.
Your geometry text begins with that story of Euclid. But this story is false. There is no evidence that Euclid existed and no pure deductive proofs (which completely avoid the empirical) are found in the book Elements. This was admitted by both Russell and Hilbert around 1900.
There is no way to empirically
check the axioms underlying axiomatic math. E.g. if a point is
declared invisible, we cannot empirically check that a unique line
passes through any two points. The axioms of formal math are pure
metaphysics.
This metaphysics is actually a
metaphysics of infinity. There are an infinity of points on a
line, so it would take an eternity of time to check empirically that
any two points on a line determine the same line.
It is the avoidance of the
empirical which results in an infinite regress. If a word is
defined using other words, those other words cannot be defined
empirically. Hence, a point, line, plane etc. are declared to be
undefined. To cover up the avoidance of the empirical, it is not
explained that this problem arises solely from avoidance of the
empirical.
There is no unique metaphysics of
infinity. Specifically, Indians used non-Archimedean arithmetic
instead of formal real numbers.
In practice we blindly accept the axioms laid down by Western authorities: Russell, Hilbert, Birkhoff, von Neuman, Godel, Zermelo etc.
Thus, axiomatic math teaches us to
distrust commonsense and trust the metaphysics of infinity laid down
by Western authority. It subtly teaches a slave mentality:
that Western authorities are more reliable than the empirical.
In fact that metaphysics of infinity is related to church doctrines of eternity.
Axiomatic math does NOT have
practical value, since the metaphysics of infinity can NEVER be used
in practice. In practice we must always work with finite decimal
expansions.
For example, to send a rocket to
moon, we use a computer program to calculate the trajectory. But a
computer has a finite memory (howsoever large) so it can never work
with formal reals, but works instead with floating point numbers.
Practical value comes from efficient techniques of calculation, not from metaphysical proofs.