Author Archives: vinaire

I am originally from India. I am settled in United States since 1969. I love mathematics, philosophy and clarity in thinking.

Euclid’s ELEMENTS

November 27, 2011 – 8:00 PM

Reference: Mathematical proof

Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek “axios” meaning “something worthy”), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

.

EUCLID’S AXIOMS:

I.      Things which are equal to the same, or to equals, are equal to each other.

Thus, if there be three things, and if the first, and the second, be each equal to the third, we infer by this axiom that the first is equal to the second. This axiom relates to all kinds of magnitude. The same is true of Axioms ii., iii., iv., v., vi., vii., ix.; but viii., x., xi., xii., are strictly geometrical.

II.    If equals be added to equals the sums will be equal.

III.  If equals be taken from equals the remainders will be equal.

IV.   If equals be added to unequals the sums will be unequal.

V.     If equals be taken from unequals the remainders will be unequal.

VI.   The doubles of equal magnitudes are equal.

VII.   The halves of equal magnitudes are equal.

VIII.  Magnitudes that can be made to coincide are equal.

The placing of one geometrical magnitude on another, such as a line on a line, a triangle on a triangle, or a circle on a circle, &c., is called superposition. The superposition employed in Geometry is only mental, that is, we conceive one magnitude placed on the other; and then, if we can prove that they coincide, we infer, by the present axiom, that they are equal.  Superposition involves the following principle, of which, without explicitly stating it, Euclid makes frequent use:—“Any figure may be transferred from one position to another without change of form or size.”

IX.   The whole is greater than its part.

This axiom is included in the following, which is a fuller statement:—

IX’.  The whole is equal to the sum of all its parts.

X.    Two right lines cannot enclose a space.

This is equivalent to the statement, “If two right lines have two points common to both, they coincide in direction,” that is, they form but one line, and this holds true even when one of the points is at infinity.

XI.  All right angles are equal to one another.

This can be proven by superposition.

XII.   If two right lines meet a third line, so as to make the sum of the two interior angles on the same side less than two right angles, these lines being produced shall meet at some finite distance.

.

NOTE 1: Some of the definitions that I like, though different from Euclid’s, are as follows:

  1. A point is a location in space. It does not extend in any direction, therefore, it is said to have no dimensions.
  2. A line is a path traced in space by a moving point. If a point moves without changing its direction it will describe a straight line. A straight line extends in one direction, therefore, it is said to have one dimension.  If the moving point continually changes its direction it will describe a curve; hence it follows that only one straight line can be drawn between two points.
  3. A surface is a path traced in space by a line that is moving in a direction that goes across the line.  If a straight line moves without changing its direction it will describe a plane. A surface extends in two directions, therefore, it is said to have two dimensions.
  4. An angle is the difference between the directions represented by two straight lines. For example the difference (angle) between the directions east and north is 90 degrees. The difference (angle) between two parallel lines (same direction) is zero.

NOTE 2: The perception of “straight” is always in the direction in which light  propagates, regardless of any curvature.

.

Posted in Mathematics | Comments (0)

Learning from a barefoot movement

November 24, 2011 – 4:48 PM

VERY, VERY  INSPIRING !!  LISTEN CAREFULLY !!!

In Rajasthan, India, an extraordinary school teaches rural women and men — many of them illiterate — to become solar engineers, artisans, dentists and doctors in their own villages. It’s called the Barefoot College, and its founder, Bunker Roy, explains how it works.

.

Why you should listen to him:

Development projects the world over run into one crucial point: For a project to live on, it needs to be organic, owned and sustained by those it serves. In 1972,  Sanjit “Bunker” Roy founded the Barefoot College, in the village of Tilonia in Rajasthan, India, with just this mission: to provide basic services and solutions in rural communities with the objective of making them self-sufficient. These “barefoot solutions” can be broadly categorized into solar energy, water, education, health care, rural handicrafts, people’s action, communication, women’s empowerment and wasteland development. The Barefoot College education program, for instance, teaches literacy and also skills, encouraging learning-by-doing. (Literacy is only part of it.)  Bunker’s organization has also successfully trained grandmothers from Africa and the Himalayan region to be solar engineers so they can bring electricity to their remote villages.

As he says, Barefoot College is “a place of learning and unlearning: where the teacher is the learner and the learner is the teacher.”

.

Posted in Education | Comments (4)

Additional Math Concepts

November 24, 2011 – 11:50 AM

These basics are not presented in as rigorous a manner as the axioms, but they serve to clarify and build upon the axioms.

 (1) A unit is anything that can be grasped as an entity.

In mathematics, the most fundamental idea is that of a unit. A unit is commonly represented by the number “1” (one).

(2) A unit is counted one at a time.

Counting provides natural numbers that are commonly represented by the numbers, 1, 2, 3, 4, 5, and so on.

(3) All numbers are referenced from 0 (zero).

Any quantity present is relative to no quantity. Therefore, all numbers are referenced from the idea of “no units.” This concept is commonly represented by the numeral “0” (zero).

(4)  Therefore, the number n is fully defined by “0 + n”.

Therefore, the number 1 is fully defined by “0+1”; the number 2 is fully defined by “0+2”; the number 3 is fully defined by “0+3”, and so on.

(5) The number “0 + n” is abbreviated as n, or as the positive integer +n.

The number “0+1” is abbreviated as +1; the number “0+2” is abbreviated as +2; the number “0+3” is abbreviated as +3, and so on.  The numbers +1, +2, +3, +4, +5, etc. are called positive integers.

(6) If a number is n, then the next number is “n + 1”.

The next number is obtained by counting one more. This gives us the basic function of adding. Addition is represented by the sign “+”. Thus, the next number after 1 is “1+1” written as 2; the next number is “2+1” written as 3; the next number is “3+1” written as 4, and so on. One may keep on counting forward without limit.

 (7) If a number is n, then the previous number is “n – 1”. 

The previous number is obtained by counting one less. This function of taking away (subtracting) is the opposite of adding. Subtraction is represented by the sign “–”. Thus, the number previous to 3 is “3–1” or 2; the number previous to 2 is “2–1” or 1; the number previous to 1 is “1–1” or 0.

(8) The counts previous to 0 (zero) account for units that are missing.

As mentioned in (3) above, 0 (zero) represents the reference point of “no units”. The number previous to 0 is, 0–1; the number previous to 0–1 is 0–2; the number previous to 0–2 is 0–3, and so on. These counts define units that are missing. One may thus keep on counting backward without limit.

(9) A missing number is fully defined by “0 – n”.

“0–1”, “0–2”, “0–3”, etc., provide a count of units that are missing. An example would be a count of the money that one owes.

(10) The number “0 – n” is abbreviated as the negative integer –n.

The number “0–1” is abbreviated as –1; the number “0–2” is abbreviated as –2; the number “0–3” is abbreviated as –3, and so on. The numbers “–1, –2, –3, –4, –5, etc.” are called negative integers.

(11) The reference point zero (0) is neither positive nor negative.

Zero (0) is simply the reference point for quantities that are present, as well as for the quantities that are missing.

.

Posted in Mathematics | Comments (12)

False Predictions

November 24, 2011 – 8:08 AM

FALSE PREDICTIONS from http://www.mhealthtalk.com/2011/09/false-barriers/

These quotes show that it’s risky to say that something can’t or won’t be done, especially when technology is concerned and it’s the right thing to do.

  • PHONOGRAPH – “The phonograph has no commercial value at all.” (Thomas Edison)

  • TELEGRAPH – “I watched his countenance closely, to see if he was not deranged … and I was assured by other senators after he left the room that they had no confidence in it.” (U.S. Senator Smith of Indiana, after witnessing a demonstration of Samuel Morse’s telegraph, 1842)

  • TELEPHONE – “Well-informed people know it is impossible to transmit the voice over wires and that were it possible to do so, the thing would be of no practical value.” (Boston Post, on the telephone, 1865)

  • TELEPHONE – “This telephone has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us.” (Western Union internal memo, 1876)

  • TELEPHONE – “The Americans think we need of the telephone, but we do not. We have plenty of messenger boys.” (Sir William Preece, chief engineer of Britain’s Post Office, 1876)

  • ELECTRICITY – “Fooling around with alternating current is just a waste of time. Nobody will use it, ever.” (Thomas Edison, 1889)

  • CARS – “The horse is here to stay, but the automobile is only a novelty–a fad.” (President of the Michigan Savings Bank, speaking to Henry Ford’s lawyer, Horace Rackham. Rackham ignored the advice, invested $5000 in Ford stock, and sold it later for $12.5 million.)

  • PLANES – “Heavier-than-air flying machines are fantasy. Simple laws of physics make them impossible.” (Lord Kelvin, president, British Royal Society, 1895)

  • INVENTION – “Everything that can be invented has been invented.” (Charles H. Duell, commissioner of the US Patent Office, recommending that his office should be abolished, 1899)

  • PLANES – “Man will not fly for 50 years.” (Wilbur Wright, to brother Orville after a disappointing flying experiment in 1901. Their first successful flight was in 1903.)

  • PLANES – “There will never be a bigger plane built.” (A Boeing engineer, after the first flight of the 247, a twin-engine plane that holds ten people)

  • RADIO – “The wireless music box has no imaginable commercial value. Who would pay for a message sent to nobody in particular?” (David Sarnoff’s associates responding to his urgings for investment in radio, 1912)

  • TANKS – “Caterpillar land ships are idiotic and useless. Those officers and men are wasting their time and are not pulling their proper weight in the war.” (Fourth Lord of the British Admiralty, regarding the introduction of tanks in war, 1915)

  • TANKS – “The idea that cavalry will be replaced by these iron coaches is absurd. It is little short of treasonous.” (ADC to Field Marshal Haig, at tank demonstration, 1916)

  • MOVIES – “Who the hell wants to hear actors talk?” (H. M. Warner, Warner Brothers, 1927)

  • NUCLEAR – “There is not the slightest indication that nuclear energy will ever be obtainable. It would mean that the atom would have to be shattered at will.” (Albert Einstein, 1932)

  • NUCLEAR – “That is the biggest fool thing we have ever done. The bomb will never go off, and I speak as an expert in explosives.” (Admiral William Leahy, when President Truman asked for his opinion on the project to build an atomic bomb)

  • SPACE – “A rocket will never be able to leave the earth’s atmosphere.” (New York Times, 1936)

  • COMPUTERS – “I think there is a world market for about five computers.” (Thomas J. Watson Jr., chairman of IBM, 1943)

  • TELEVISION – “Television won’t last because people will soon get tired of staring at a plywood box every night.” (Darryl Zanuck, Movie Producer, 20th Century Fox, 1946)

  • TELEVISION – “The problem with television is that the people must sit and keep their eyes glued on a screen; The average American family hasn’t time for it.” (New York Times, 1949)

  • SPACE – “Space travel is bunk.” (Sir Harold Spencer Jones, Astronomer Royal of the UK, 1957, two weeks before Sputnik orbited the Earth)

  • COPIERS – “The world potential market for copying machines is 5000 at most.” (IBM to the founders of Xerox as it turned down their proposal, 1959)

  • SPACE – “There is practically no chance communications space satellites will be used to provide better telephone, telegraph, television, or radio service inside the United States.” (T. Craven, FCC Commissioner, 1961)

  • MUSIC – “Guitar music is on the way out.” (Decca Records, declining to record a new group called The Beatles, 1962)

  • COMPUTERS – “There is no reason for any individual to have a computer in their home.” (Kenneth Olson, founder of Digital Equipment Corporation, 1977)

  • COMPUTERS – “So we went to Atari and said, ‘Hey, we’ve got this amazing thing, even built with some of your parts, and what do you think about funding us? Or we’ll give it to you. We just want to do it. Pay our salary, we’ll come work for you.’ And they said, ‘No.’ So then we went to Hewlett-Packard, and they said, ‘Hey, we don’t need you. You haven’t gone through college yet.‘” (Steve Jobs, founder of Apple)

  • COMPUTERS – “640 K [of computer memory] ought to be enough for anybody.” (Bill Gates, founder and CEO of Microsoft, 1981)

  • COMPUTERS – “We see a corporate market of maybe 15,000 PCs a year by 1990.” (DataQuest, 1984)

  • COMPUTERS – “By 1990 75-80 percent of IBM compatible computers will be sold with OS/2.” (Bill Gates, founder and CEO of Microsoft, 1988)

  • COMPUTERS – “I predict that the last mainframe will be unplugged on March 15, 1996.” (Stewart Alsop, InfoWorld columnist, 1991)

  • INTERNET – “I predict the Internet will soon go spectacularly supernova and in 1996 catastrophically collapse.” (Robert Metcalfe, founder of 3Com and inventor of Ethernet, 1995)

  • NEW BUSINESSES – “A cookie store is a bad idea. Besides, market research and focus groups confirm that America wants soft, not chewy, cookies.” (Investor rejection letter to Debby Fields, founder of Mrs. Fields’ Cookies)

  • NEW BUSINESSES – “The concept is interesting and well-informed, but in order to earn better than a ‘C’ the idea must be feasible.” (Yale professor’s comments on a term paper submitted by Fred Smith for an overnight delivery system. Two years later, Smith founded Federal Express.)

.

Posted in General | Comments (3)

Fundamentals, Consistency and Breakthroughs

November 21, 2011 – 11:22 PM

Breakthroughs occur when deeper consistencies are found in the fundamentals of a subject. A breakthrough is always preceded by the discovery of some inconsistency. It then leads to the breakthrough and a deeper consistency.

For a long time, mathematics was based on numerals that did not include the zero. Look at the Roman numerals. There is no zero there. This is because a number was supposed to represent “something” only. For a long time it was inconceivable that the idea of “nothing” had any use in mathematics.

We know that the next number is arrived at by adding one to the number. There is no limit to the next number. It keeps on going for ever. We also know that the previous number is determined by subtracting one from the number. However, at one time the previous number stopped at one, because counting started from one. Nobody seemed to see any utility in seeking a number previous to one.

It was not until the 4th century BC that somebody took a hard look at the inconsistency of not going beyond one for the previous number. This new number was perceived as “void” or “nothing.” If a person spent all his money, he had no money left. This was consistent with the idea of previous number. But this was resisted simply because counting started from one, and “nothing” could not be counted. The moment that resistance was overcome, breakthroughs started to occur in the subject of mathematics. Zero could now be used as a reference point from which to measure. The biggest breakthrough came with the positional notation for the numbering system. Suddenly, it became much simpler to write numbers (compare current numbering system to the Roman system), and mathematics started to progress by leaps and bounds.

Inconsistencies at the fundamental level are hard to accept as inconsistencies because the popular consensus endorses it as the norm. Thus, it took several centuries before the possibility of numbers previous to zero could even be considered. The idea of negative numbers was resisted in Europe as late as the 17th century AD, even though they were known to facilitate the representation of debt. Finally, with the formal introduction of negative numbers later in 17th century, the “previous” numbers could also be extended without limit. A much deeper consistency was achieved at the fundamental level of mathematics. This led to incredible breakthroughs.

There are many such examples of overcoming mental resistance at the fundamental level for the sake of greater consistency. And each time it happened it was followed by wonderful breakthroughs. Here is one more example. We know that larger and larger magnitudes may be represented by the idea of a unit, such as, a planet, a galaxy, a cluster of galaxies, and a universe. Is there a limit in magnitude to which a unit is bound? The expanding universe seems to indicate that the answer is no. Now looking in the opposite direction of smaller and smaller magnitudes, is their another limit to which a unit is bound? The discovery of the irrational number seems to indicate that the answer is no again. So, the mathematical logic points to the possibility of discovering smaller and smaller particles endlessly. We don’t know for certain yet. We are simply looking at a deeper consistency.

In summary, breakthroughs occur when one seeks a deeper consistency in the fundamentals of a subject, even when this requires going against the general consensus.

.

Posted in KHTK | Comments (11)