Home assignment
As you have gathered, this workshop is about 5 different type of geometry.
1.
2.
3.
4.
5.
Hilbert's synthetic geometry
Birkhoff's metric geometry
Euclidean geometry
Compass-box geometry
Sulba-sutra geometry
To understand these five different types of geometry we need to understand that there are at
least two types of mathematics:
axiomatic math and
empirical math.
Axiomatic math
(also called formal math)
Empirical math
Begins from axioms or postulates
Begins from facts and observations
Started by Hilbert and Russell around 1900
Traditional (sulba sutra, Rhind papyrus)
(both first studied foundations of geometry
and analysed “errors in Euclid”).
Rejects empirical proof
Accepts empirical proof
The first two types of geometry (Hilbert's and Birkhoff's) are examples of axiomatic math.
Compass box geometry and sulba-sutra are examples of empirical math.
Additionally, there is a little known third type of math: religious math. “Euclidean” geometry
(or more properly Egyptian mystery geometry) is an example of religious math. Plato
advocated its teaching to arouse the soul. The church taught it (from around 13th c. to its
priests to teach them a special way of reasoning, in support of the Christian theology of
reason.
Which geometry does the school text teach?
Aim of the workshop is to help you to understand: what geometry IS actually taught in
the school text? And also to help you to decide: what geometry OUGHT be taught in
schools?
To this end, you must have a basic understanding of all the above 5 types of geometry.
The home assignment was intended to bring out these features.
(Brief answers to home assignment.)
We also need to understand the key differences between axiomatic math and empirical math.
A brief history of geometry teaching
We have all heard of the sulba sutra, From time to time the issue crops up in India that the
“Pythagorean” theorem was anticipated in the sulba sutra. But what is the correct situation?
Egyptian geometry also used a cord, as did many other traditional geometries. But the details
are lost and the sulba sutra is the oldest existing treatise on that. In India people wee
traditionally taught geometry using a rope or rajju, and we can still see masons using it for
construction.
In contrast, in Europe, “Euclidean” geometry first became known starting 1125 CE from
Latin translations of Arabic books, and the church soon adopted it to teach reasoning to its
priests in support of its newly invented “Christian theology of reason”.
“Euclidean” geometry, which came to Europe from Muslims, was also known to Muslims in
India and taught in some madrassas. Abul Fazl, Akbar's biographer, records it as part of his
talim. But the book Elements, which Euclid supposedly wrote, was not translated into
Sanskrit until around 1725, because it was thought to be more concerned with Muslim
religion than practical value. Thus, the Pythagorean theorem” was easily proved in one step
in Indian books, such as the Yuktibhasa, and did not require 47 intermediate propositions as
in “Euclid”. In 1725, Pandit Jagannath was commissioned by Sawai Jai Singh (author of
various jantar mantars in Delhi, Jaipur, and Ujjain) to translate “Euclid” from Farsi into
Sanskrit, which he did under the title Rekha ganita.
Mission schools in India used to teach “Euclidean” geometry, and a century later this
missionary education became compulsory under the university colonial education system
which came to India in 1857. At this time Cambridge used to teach what was believed to be
the “exact” Euclid. Later (at the end of the 19th c.) Cambridge changed it to teaching the
propositions in the same order, but not exactly the same proofs. We imitated that.
There the matter remained until the launch of sputnik by Soviet Union in 1957. This led to
panic in the US, since the launch of the sputnik demonstrated that the Soviet Union had the
capacity for Inter Continental Ballistic Missiles which could carry large “atom bombs”
(thermonuclear devices) and drop them directly on the US. Policy makers in the US decided
that they were lagging behind in science and technology. The first effect was a revision of
STEM (science, technology, engineering, mathematics) education.
The revised syllabus for geometry was put up the School Mathematics Study Group. This
came to be known as the “new math”. We adopted the SMSG, and that is what we teach to
this day except that we teach it in combination with compass-box geometry.
In this workshop we want to review these decisions, check if they were correct, and decide
what math to teach.
Note: The Carter administration in the US also introduced another type of math called “new
new math”, based on a philosophy called constructivism. This was fiercely contested by
critics as “no-right answer” math. We will not discuss it, since this was never introduced in
India.
Alternative math curriculum
Aim of teaching math. The first question is this: why teach math? Most people will agree
that we teach math for its use in science, engineering and commerce. [Some will assert that
math is done for aesthetics. This relates to Plato's spiritual beliefs about math and music. The
vast majority of students love music and find math ugly.] In practice, students not interested
in these subjects drop math after class X.
What math do we teach? The next question is how do we teach math? The simple answer is
that we imitate the West. On the doctrine articulated by Macaulay, the West is
“immeasurably superior” in science therefore to acquire it, we must imitate the West. This
Western math is also called formal math or axiomatic math. It is different from the math was
traditionally taught as in Bhaskar (Lilavati) or Mahavira (Ganita Sara Sangraha) or Sridhar
(Patiganita) or Aryabhata (Aryabhatiya) or the sulba sutra or even the Rhind papyrus dating
back to 3500 years.
Blind imitation of the West is like rote learning because it is done without understanding.
Therefore, we need to understand the real nature of axiomatic math and how it differs from
traditional math.
The coloniser left, but has left behind the colonial education system which still shapes the
minds of our children by teaching blind imitation of the West. This is education for slavery.
To achieve freedom, to decolonise, we need to decide on our own what math we teach:
whether to teach axiomatic math or traditional math.
Be warned the process of deciding is not easy because it requires real knowledge, and
colonial education instills ignorance. If it were easy, the whole country would not have been
enslaved for so long. But freedom requires effort especially when it is the mind which is
captured, and the chains are not visible. In this workshop we will focus on geometry: textbook geometry vs geometry of the sulba sutra.
So, what is the difference between axiomatic and traditional math? Axiomatic math begins
with postulates and uses solely deductive proofs. This is NOT the same as the traditional
math of Traditional math does not reject deductive proofs, but it accepts also empirical
proofs. (E.g. from Aryabhata). But axiomatic math or formal math rejects empirical proofs as
inferior and fallible (rope vs snake).
By implication, deductive proofs are regarded as superior and infallible. Traditional math,
done for practical value, accepts approximation, Western math claims to be exact and eternal
truth. These differences will become clearer as we go along.
Clarification: (Decolonisation is not mere advocacy of the traditional.)
I am not suggesting traditional math just because it is traditional. I am suggesting that
we critically compare the two types of math and choose what is best. Was such a critical
comparison done when Western universities came to India in 1857? No. Was it done
when India became independent in 1947? No. We adopted Western math, without a
critical comparison. At least today, we should carry out such a critical comparison. I
will only facilitate the comparison by explaining things: you will make the ultimate
decision.
For a critical comparison, we need to ask several questions and satisfy ourselves about the
answers.
Q. 1 Does axiomatic math help to achieve practical value?
A. 1 It does not. Why not? There are several reasons.
1. The NCERT class 9 text promotes axiomatic math by deprecating practical
math. Suggests that practical math is inferior and axiomatic math is something
superior. This is also what “classic” histories of math also say. Rouse Ball
quote.
2. It is claimed that the superiority of axiomatic math relates to the use of
deductive reasoning. However, various other cultures too, such as Indian
culture, certainly accepted deductive inference and used it. All systems of
Indian philosophy (except Lokayata) accept inference.
3. E.g Indian proof of Pythagorean theorem involves a process of deduction.
Why is this rejected? Because it involves the empirical. (Explanation of how it
involves the empirical.) The empirical is considered inferior and fallible hence
rejected by the West on philosophical grounds. In contrast, deduction is
considered infallible.
4. That is, the unique feature of the Western/formal math is NOT the mere use of
reason but (a) the avoidance of the empirical (“pure deductive proofs”), on
the belief that (b) deductive proofs are infallible or less fallible than empirical
proofs. There is a dangerous doublespeak here: clearly note that “the use of
reason” does not mean mere use of reason as others did. It actually means “the
avoidance of the empirical”. The NCERT text never clarifies this point. But
it proceeds to condition and indoctrinate children into this avoidance of the
empirical from an early age. Thus, the NCERT 6th standard text defines a point
as something invisible and inaccessible to the senses hence something nonempirical.
5. But deductive reasoning without an empirical base is extremely unreliable,
and can go absolutely haywire. It does NOT lead to valid knowledge or
approximately valid knowledge. Case of horny rabbit.
6. Lokayata argument of wolf's paws, and Sherlock Holmes “The priory school”
(horse shoes which leave the mark of a cow. Sherlock Holmes is renowned for
deduction, but he always kept facts in the forefronts: “It is a capital mistake
to theorize before one has data. Insensibly one begins to twist facts to suit
theories, instead of theories to suit facts.”] Therefore, the vast majority of
theorems proved deductively may be of no practical use. To determine
whether or not they are of practical use we must check the axioms of
mathematics empirically against factual data.
7. But there is no way to empirically check the axioms at the base of formal
mathematics. They are metaphysics on Popper's criterion. [Popper's criterion
explained. Formal mathematical theorems as tautologies. Metaphysics as nonphysics etc. ]
8. In practice, we accept the axioms of formal math solely on the strength of
Western authority (Hilbert's axioms, Peano's axioms, axioms of set theory etc.)
Thus children are taught to trust Western authority over their senses. Children
are told that they do formal mathematics to find the reasons why certain
theorems are true: but the reasons are traced to axioms and the axioms to
the beliefs of those in authority. Thus, the only reasons they ultimately
learn is blind trust in Western authority,
9. In fact, there is no other way but to accept axioms on authority: the axioms of
math involve a metaphysics of infinity. For example, there are an infinity of
points on a line so even if points were visible we cannot check Hilbert's
postulate that any two points determine a line for such empirical verification
will take an eternity of time.
10. This metaphysics of infinity has no practical value, and is never used in
practice: for example, to calculate the trajectory of a rocket (to send a man to
the moon) both NASA and ISRO used numerical calculation on a computer,
which can only deal with a finite set of floating point numbers (not formal real
numbers).
11. The evasion of the empirical precipitates an infinite regress: one word can
only be defined in terms of others. The chain (of using one word to define
another) can never terminate at an empirical point. Explaining an invisible
point to a child is like explaining the color white to a blind man. This
phenomenon of an infinite regress is pointed out in the class IX text. But it is
not explained that appeal to the empirical can resolve this regress. It is not
explained that, therefore, this regress is peculiar to formal math and its evasion
of the empirical in the name of infallibility. That is, it is NOT explained that
this infinite regress arises from avoidance of the empirical. The example used
in the NCERT text is that of “Euclid's” axioms “a point is that which has no
part”, but the term part is not defined. Students are then told that a point is an
undefined concept. Thus, a point is both invisible (as taught in class 6) and
undefined (as taught in class 9). How to escape from this pedagogical trap?
We will see that later.
12. The belief in a point as invisible is often confounded with Platonic idealism.
However, Plato advocated geometry for religious purposes. Plato rejected
empirical concerns as inferior to his spiritual concerns. There is religiosity in
his belief that the empirical is inferior. (Case of Horus.) E.g., the word math
derives from Plato's religious belief in mathesis meaning learning by arousing
the soul and making it recall its innate knowledge;
13. But the church exploited this religiosity and developed it in a different
direction. In fact, if there is no empirical control on the hypothesis, deductive
reasoning can be used to conclude anything that one wants. The way this was
exploited is little known.
1. Just because this sort of deductive reasoning (divorced from facts) can
be used to prove anything one want, it greatly suited the church. (Note
that this political value to the church is also some form of practical
value, but it is not the same as practical value for science, engineering
and commerce, or practical value for us.) It is practical value for the
coloniser, not the colonised, for the master, not the slave.
2. The fact is that Western culture was dominated by the church which
ruled the West for centuries. The church glorified evasion of the
empirical, since empirical facts went against its doctrines like virgin
birth. It gave this evasion a grand name: calling it metaphysics or
something higher than physics. Plato's religiosity was changed to
church religiosity.
3. The church used Euclid's Elements to teach metaphysical deductive
reasoning to its priests. This aspect that “Euclid” was part of the
church syllabus (and therefore of use to the church) is not revealed to
students.
14. There is no unique understanding of infinity. An alternative metaphysics of
infinity may be used as in the non-Archimedean arithmetic of Brahmagupta's
avyakt numbers to do calculus. This requires an alternative philosophy of math
as in the sulba sutra assertion of math as inexact and impermanent. This
philosophy is better suited to deliver practical value.
Summary of day 1
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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.
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3 types of math: empirical, axiomatic, religious.
o Compass box geometry and sulba sutra geometry are empirical.
o Hilbert's synthetic geometry and Birkhoff's metric geometry are axiomatic.,
o Greek geometry (as in Plato) is religious.
Q. 0 Why do we teach math?
A. 0
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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)
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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
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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 nonArchimedean 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)
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Axiomatic math does NOT have practical value, since the metaphysics of infinity can
NEVER be used in practice.
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.
Q. 2 (Where exactly is the practical value of math?) Math does have practical value. But
traditional math too had practical value which persisted for millennia before the advent of
formal math. That is, for our purposes we should not accept a package deal. We need to
understand: exactly what aspect of math results in practical value? Formal math is
characterised by formal proof, and traditional math by efficient techniques of calculation. So,
what gives practical value, efficient techniques of CALCULATION, or formal proof?
1. It is a myth that formal math has any practical value. Traditional math had practical
value which pre-dates formal math by thousands of years. Does formal math add to
that practical value, or does it just get in the way.
1. Arithmetic was used for commerce from long before the advent of formal
math. For practical applications the focus is on efficient CALCULATION, not
PROOF.
1. People continue to use practical arithmetic exactly the same old way
even after the advent of formal math in the 20th c.
2. Most people do not have any knowledge of Peano's axioms or
Whitehead and Russell's 378 page proof of 1+1=2 from a metaphysics
of infinity. This knowledge is irrelevant for any practical applications:
the axioms and proof add nothing to the practical value of arithmetic
for commerce. Imagine someone going through those 378 pages and
more while carrying out a commercial transaction.
3. Practical value came from efficient algorithms (for calculation), not
proofs. Indian techniques of arithmetic were more efficient at
calculation than arithmetic with Roman techniques. Hence the West
gave up “Roman numerals” and adopted “Arabic numerals”. The West
imported Indian techniques of arithmetic in 3 phases. First Gerbert
(Pope Sylvester), from Cordoba, 10th c. He first used the term Roman
numerals which showed he did not understand the system. Then
Florentine merchants from Africa, 12th c., then Jesuits in the 16th c.
from Cochin 16th c. The West imported and then taught Indian
techniques of arithmetic for practical value in calculations, not proofs.
2. Likewise today we have adopted computers which calculate super-efficiently.
Computers deliver practical value. But they cannot use formal math. In fact, a
computer cannot handle the metaphysics of infinity at the root of formal math
as already noted.
3. A third case we can consider is that of geometry.
1. First the claim that Greeks did something special is a complete
falsehood. Pure deductive proofs completely avoiding the empirical
did not exist before the 20th c. SAS, and its use for the proof of the
“Pythagorean” proposition.
2. An important corollary: This shows that deductive proofs are fallible.
A wrong deductive proof may be mistaken for a valid proof. Exactly as
a rope may be mistaken for a snake. However, errors in deductive
proofs can be more persistent than errors of observation.
3. Let us grant the religious Western belief that deductive proofs are
infallible, like the pope. Nevertheless, a chain is as weak as its weakest
link. Therefore, the story that Greeks did something special is wholly
false.
4. Second, the ability to prove the Pythagorean proposition is less important than
the ability to perform calculations with it.
5.
6.
7.
8.
1. Mere proofs whether empirical or metaphysical do not provide the
ability to calculate. In particular, it is no use knowing the proposition
unless one has some way to calculate the diagonal. That needs square
roots. Manava sulba sutra
2. The square root algorithm. (Aryabhata, GSS, PT, and as taught) Its
non-termination for
.
3. Key issue: Interpretation of this non-termination, savisesa (quote from
sulba sutra) vs formal reals. In practice, mathematics is NOT exact as
wrongly believed in the West,
4. While choosing the interpretation, bear in mind the objective of
teaching math. We are doing it for its practical applications for science
and engineering where approximations are acceptable, and no
possibility of exact knowledge.
5. Sidelight: The very word “surd” shows that the West had no
understanding of square roots until the Toledo translations.
Another kind of “Pythagorean calculation” is today taught as trigonometry.
1. This involves CALCULATING the sides when the diagonal and its
angle with one side is given.
2. This is a practical problem which arises in the context of longitude
determination.
3. For the practical problem we immediately find the limitation that the
“Pythagorean theorem” is NOT the general proposition it is made out
to be, but is invalid knowledge anywhere on the earth.
4. This last fact was known to Bhaskar 1, though the West learnt about
non-Euclidean geometry very late.
5. The calculation involves the trigonometric functions (which are
actually circular functions).
6. The very name sine shows that these functions were not known in the
West until the 12th c. Toledo translations.
7. They were not properly understood in the West because they again
involve an infinite series. Descartes on ratios of curved and straight
lines.
The story is the same for most other math. We can easily understand this
historically. Most mathematics of practical value (starting from arithmetic)
developed in the non-West for its practical value and was transmitted to the
West also for its practical value. Later, the West added a metaphysics of
infinity, and returned the same math as “superior” through colonial education.
Case of calculus.
There is a difference in the case of geometry which the West imported from
Egypt via Greeks and Arabs. This got linked with church dogmas and delinked
from its original spiritual significance. These dogmas were and are used for
political domination: that is practical value of another kind. Thus, the
metaphysics of infinity is tied to church dogmas of eternity: the formal reals
are a model for church notions of linear time. But most people find this too
hard to understand.
Because of these linkages to political power, Western scholars will not easily
abandon the metaphysics of infinity used in formal mathematics for though it
has no practical value, for science etc. it has political value since it helps
dominate others through claims of superiority. It is exactly like racism, in
enabling moral justification for domination by a false claim of superiority.
9. We need to pay close attention to the fact that colonial education came as
church education, and these myths are important to assert
Christian/racial/colonial superiority. To reiterate, this is the practical value of
another kind. Not practical value for application to science, engineering and
commerce but political value. We accept the metaphysical postulates of
axiomatic math on grounds of Western authority, not on any other grounds.
This makes us dependent on Western authority. It alienates people from their
own tradition.
10. The accompanying history of Greeks and their use of deductive reason is
completely false and originated in Crusading times when when Western priesthistorians paid no attention to evidence. Thus, Euclid was asserted to be from
Megara and a contemporary of Plato: this wild speculation was rejected after 5
centuries. There is still no evidence for Euclid and plenty of counter-evidence.
But when asked for evidence people do not provide it, there stock reaction is
to express outrage, the way the church expressed outrage at atheists.
Indoctrinated people guess that 20% of the myth is false: but it is 100% false.
The fact is that every aspect of the myth is false. The book Elements has no
pure deductive proofs. There is no evidence to connect Aristotle to the
syllogism. The church concocted these myths to support the church dogmas of
reason (as in Christian rational theology.
Q. 3. (Pedagogical problem) Irrespective of which kind of math is better, can formal
math at all be taught in schools? If we do try to teach it, what compromises are
required? What is it that is actually taught? What exactly is the skill that the children
eventually learn?
3. Thus, on the understanding developed above, formal math involves (a) avoidance of
the empirical, and (b) the resulting metaphysics is a metaphysics of infinity (c) and we
believe it because we have blind faith in Western authority.
1. The issue with mathematics is not abstraction but abstraction based on
metaphysics. Children understand abstractions like “dog”, based on physics
not metaphysics.
2. A point is invisible, a line too is invisible, since infinitesimally thin, and so is
a plane. How can students discriminate between these three different kinds of
ghostly and invisible entities? They cannot.
3. Likewise a line is said to extend indefinitely in both directions. On the surface
of the earth, such a line may be a loxodrome (curved line) not a straight line.
So, first we need a plane which extends indefinitely in two directions! So,
now, more than two directions are required.
4. So an attempt is made to teach empirical correlates. A physical dot, a physical
line segment (but not a line), etc.
5. This is deceptive: a line is “extended indefinitely”. What is the guarantee that
this can be done? What is the guarantee that this will result in a straight line,
not a loxodrome.
6. The key thing that children learn is to deprecate the empirical and distrust
commonsense. They are taught that the dot is erroneous and that a “real point”
is invisible. How can children be expected to understand such nonsense? The
other thing they learn is that math involves something beyond their
understanding, and in this matter they must trust Western authority. (This was
a typical church teaching, to teach faith in church authority; colonial education
modified it to teach faith in Western authority.)
7. Some practical value has to be taught. Hence, students are taught two different
types of geometry side by side: axiomatic synthetic geometry and empirical
compass box geometry. The difference and contradictions between these two
types of geometry are never explained. But there are stray confusing remarks:
“ideally a ruler should be unmarked”. This is a clear reference to Hilbert's
synthetic geometry in which distance is not defined. It is confusing because
the student naturally wonders: should we erase the markings on the ruler?
(Likewise in Maharashtra texts it is stated that “superposition is not a proper
procedure”.)
8. Likewise, consider the teaching of congruence. This term “congruence” was
absent in the Elements and introduced in Hilbert's synthetic geometry. Most
people cannot differentiate between congruence and equality. If congruence is
established by superposition, putting one triangle on top of another (equality)
then this permits empirical processes, so distances can be measured. If
superposition is not a valid process and is disallowed, why is measurement in
compass-box geometry allowed. In Birkhoff's metric approach, congruence is
equality. As soon as this empirical process of proof is accepted, the whole
story of Euclid and his special deductive proofs collapses. Congruence is
actually a synthetic notion: Hilbert's program was to do geometry without
distance. But this project is not explained and the word congruence is used in a
confusing way.
9. Likewise, this synthetic project is immediately contradicted by defining a
straight line as the shortest distance between two points. What sort of
definition of distance is involved: an axiomatic metric definition? An
empirical metric definition? Or no definition (synthetic geometry)?
10. The formal definition of a point is confused with the idealistic. [Hilbert's
remark on table and beer mugs.] Children confuse Platonic geometry with
synthetic.
11. The simple fact is that pedagogy at the school level MUST involve the
empirical. There is no other way to teach children. Therefore, a project of
teaching formal math or avoidance of the empirical to school students is
doomed to failure. The only other option is to teach everything: formal set
theory, its philosophy, formal real numbers etc. This is admittedly not
possible. Formal set theory is so complicated that even most formal
mathematicians too do not learn it. Teaching avoidance of the empirical by
coy references to the empirical only confuses the students. If students are
confused it is squarely the fault of the textbook writer and of the Western
mathematics community which defines the subject to its political advantage.
4. Axiomatic geometry: Synthetic and metric geometry.
1. As we saw above: school texts teach two incompatible types of geometry side
by side: synthetic geometry (without distance) and empirical compass-box
geometry.
2. What is the use of teaching geometry without distance? This has no practical
value. [Not even in general relativity; ref Kosambi.] But it has political value.
Hilbert's project was invented to save the tottering story of “Euclid and the
Elements”, a story in which the West is deeply invested. The collapse of the
story, if openly admitted, would make the West look excessively foolish. For
seven centuries their priest-scholars did not understand the elementary
“errors” in the purported pure deductive proofs in the Elements.
3. What actually failed, and failed miserably was the Crusading re-interpretation
of the Elements. Though it is a book about Egyptian mystery geometry,
intended to arouse the soul, it was reinterpreted during the Crusades as a book
about reasoning in support of Christian rational theology and against the
original notion of soul championed by Egyptian mystery geometry.
4. Hilbert's synthetic geometry was invented to save the myth. To save the myth,
it was not enough to offer the excuse that Euclid intended deductive proofs but
made errors. This is a classic case of defending a failed theory by
accumulating hypotheses. The mere add on story of intentions gone wrong still
did not fit the apparent prolixity of the Elements. Hilbert was smart enough to
understand (what most people don't) that metric notions would trivialise the
Elements. Synthetic geometry makes the theorems harder to prove, and serves
to explains the apparent prolixity of the Elements (on the Crusading
interpretation of it as a book about deductive reasoning; it is not prolix if
regarded as a book about Egyptian mystery geometry).
5. Synthetic geometry uses the notion of “congruence” introduced by Hilbert,
and not the original term equality. That is, it prohibits the comparison of
figures by suerposition or putting one figure on top of another. Note that this is
the same process used to measure length: we superpose a ruler on a line
segment.
6. Because, however, the “Pythagorean theorem” must be proved, Hilbert's
synthetic geometry does define areas. To reiterate, it does NOT define length
but defines area! Because formal math is all about blind acceptance of
Western authority we are expected to applaud this stupidity on the strength of
Hilbert's authority.
7. The proof of the “Pythagorean theorem” becomes very easy with Birkhoff's
metric postulates. So why not use them? Both Hilbert and Birkhoff were
authorities. As with any aspect of axiomatic math, the decision rests with the
“mathematics community” a community based in the West.
8. Instead of going by the social authority of the “mathematics community” we
should ask what is the practical value for us? Let the “mathematics
community” tell us exactly what that practical value is. For something like
synthetic geometry may be of political value to the Western math community,
and hence acceptable to it, but of no practical value to us. If no explanation
has been given for why synthetic geometry is useful to us, we abandon it as
useless.
Q. 5. Thus it is clear that we should teach empirical geometry in schools. But which
empirical geometry should we teach? Why teach only the geometry of the compass-box
(or instrument box or geometry box)?
5. The compass box is useful only for diagrams drawn on paper. It is assumed that the
paper is not rough hand-made paper but is smooth machine made paper.
1. It cannot be used to draw diagrams on the ground (in fact the process of
levelling the ground is not explained to students.) Thus it is of no use to
measure land areas.
2. The compass box has redundant features: set squares and dividers which are
never used. The vast majority of students do not understand their use.
3. The compass box has a protractor whose construction is not explained. A
common childish doubt: does the size of the protractor matter? If not why not?
The answer to the question involves the properties of a circle and the ratio of
its circumference and diameter today called as the number π.
4. A notion of distance is needed even to define a circle (circles are not defined
in Hilbert's synthetic geometry). Distance is easily defined empirically.
5. But additionally a method of measuring the circumference of a circle (a curved
line) is also needed to define a degree and construct a protractor.
6. What is a degree? What is it that is divided into 90 equal parts? A degree is
defined as the length of the arc relative to the circumference.
7. What is a radian? Most students are unclear about it. A radian is the length of
the arc relative to the radius. With an an angle defined as the arc, there is no
difficulty in understanding either radian or degree.
8. An angle is defined in the text as a pair of rays. This is a bad definition since it
only works for angles up to 2π
9. An angle is better defined as the length of a curved arc which can be greater
than 2π.
10. Thus, it is clear that a compass box should contain a string or a measuring
tape. But a string can replace the entire compass box especially for practical
purposes like land measurement.
11. Therefore, the empirical geometry we should teach is string geometry.
Q 6. What new features would students learn by teaching string geometry?
6. The greatest new features is conceptual clarity, since
(a) we teach only one kind of geometry not a mixture of two incompatible geometries
(b) we teach geometry empirically
(c) we allow the use a string to measure the length of curved lines.
1.
2.
3.
4.
5.
Practical measurement of lengths and areas using a cord.
The radian measure of angles.
The historical origin of the degree measure of angles in astronomy.
Similar triangles and the arithmetic rule of three.
The calculation of π by three methods.
1. Empirical method
2. The octagon doubling method.
1. Corollary: why the ratio of circumference to radius is constant.
3. Monte Carlo method
6. The Manava sulba sutra method of stating the Pythagorean proposition
7. The square root algorithm. Its difficulty, the origin of the term surd.
8. Why it does not terminate for the case of 2.
9. Meaning of savisesa.
10. Is there any place where the Pythagorean proposition holds exactly? No
escape from approximation. How to handle approximations and noonuniqueness using zeroism.
11. The second Pythagorean calculation: trigonometry
12. Circular functions defined using a circle (else no way to understand their
relation to pi)
13. Origin of the term sine.
14. The measurement of real life angles
15. The calculation of intermediate sine values: similar triangles rule of and linear
interpolation.
16. The measurement of tree heights.
17. Measuring the height of a hill.
18. Measuring the radius of the earth.
What are the new things you have learnt?
Conceptual clarity:
1.
2.
3.
4.
That there are several systems of geometry which are mutually incompatible.
That your text book jumbles them up, resulting in conceptual confusion.
That there is no virtue in avoiding the empirical.
Better and easy to understand (empirical) definitions of point, line segment, distance
etc.
5. A string is needed as part of the compass box, and it can replace the entire compass
box.
Angles
1. Conceptual clarity about angles: that an angle is the length of a curved arc rather than
a pair of straight lines.
2. Why a protractor is circular in shape?
3. Why its size does not matter in measuring an angle.
4. What is a degree? What is its historical origin?
5. What is a radian?
6. How to convert from degrees to radians and vice versa
7. What is the definition of the number pi.
8. How its value is calculated. Three methods: empirical, Monte Carlo method and
octagon method.
9. How to measure real life angles.
Pythagorean proposition: that calculation is more important than theorem
1. That the theorem is not a universal truth: it does not hold on the surface of the earth.
(Longitude problem.)
2. That we do not actually know of any place where it holds EXACTLY.
3. Better form of stating the proposition. Doing the actual calculation involves square
roots.
4. Square root calculation as non-terminating and inexact: how to deal with
approximations in the natural way of zeroism.
Trigonometry
1. That the Pythagorean proposition can be stated in another form, which involves
another calculation corresponding to trigonometry.
2. That you are taught the values of these circular functions (trigonometric ratios) only
for some values of the angle, and that this does not enable you to solve real life
problems.
3. You learnt how to calculate sine and cosine values not only for some for intermediate
values by linear interpolation: this is the beginning of the calculus.
History
1. That the story of Euclid and his deductive proofs is faulty when examined critically.
2. That you know something only when you have yourself critically examined it.