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.
Q. 0 Why do we teach math?
A. 0
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 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.
1(b)
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.