Practical value of math

• Where exactly is the practical value of math?

• Traditional math had practical value which persisted for millennia before the advent of formal math.

• Traditional math characterized by efficient techniques of calculation.

  • Arithmetic used for commerce. Efficient Indian arithmetic adopted by Europe which rejected its inferior Roman arithmetic.

  • Not traditional math alone. Computers also do efficient arithmetic. They are welcome. But they cannot use any metaphysics of infinity (formal real numbers) because they have finite memory. They use floating point numbers.

• Axiomatic math added metaphysical proof.

  • Peano's axioms or Whitehead and Russell's 378 page proof of 1+1=2 do not add to practical value. Most people continue to do arithmetic without knowing these axioms.

E.g. for practical use of “Pythagorean proposition”, mere proof (metaphysical or empirical) is not enough. We need to be able to calculate with it.

1. To calculated diagonal from a knowledge of sides needs square roots. Manava sulba sutra

   1. The square root algorithm. (Aryabhata, and as taught) Its non-termination for .

   2. Sidelight: The very word “surd” shows that the West had no understanding of square roots until the 12^th c. Toledo translations.

2. 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 2^nd Pythagorean 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 12^th 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.

3. 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.

4. School text says we should imitative “Euclid” but such imitation gives nil practical value.

5. Because of linkages of education to political power, Western scholars will not easily abandon the metaphysics of infinity used in formal mathematics for though it has o practical value, for science etc. it has political value since it helps dominate others through claims of superiority. If we imitate that, it helps them, not us.
   
   

Pedagogical problem: Irrespective of which kind of math is better, formal math cannot be taught in schools.

1. 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. So an attempt is made to teach it using empirical correlates. A physical dot, a physical line segment (but not a line), etc.

4. 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.

5. 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. The other thing they learn is that math involves something beyond their understanding, and in this matter they must trust Western authority.

6. Some practical value has to be taught. Hence, students are taught four different types of geometry side by side: axiomatic synthetic geometry (Hilbert), axiomatic metric geometry and empirical compass box geometry. These are embedded in the story of a fourth type of geometry (Euclidean or religious geometry).

7. Teaching such a mixture causes confusion.

8. 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”.)

9. 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.

10. 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)?

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.

Thus it is clear that we should teach empirical geometry in schools. But which empirical geometry should we teach? The geometry of the compass-box or the sulba sutra (rajju ganita)?

1. 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.

2. 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.

3. The compass box has redundant features: set squares and dividers which are never used. The vast majority of students do not understand their use.

4. ANGLE Angle is defined using two straight lines but measured using a semi-circular protractor. Why?

5. The construction of the protractor 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.

1. 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.

2. 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.

3. 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

4. An angle is better defined as the length of a curved arc which can be greater than 2π.

5. 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.

6. Therefore, the empirical geometry we should teach is string geometry.

What are the new things you have learnt?

Conceptual clarity:

1. That there are several systems of geometry which are mutually incompatible.

2. That your text book jumbles them up, resulting in conceptual confusion.

3. That there is no virtue in avoiding the empirical.

4. 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.