Abstraction 3

ABSTRACTION • Abstraction is the thought process wherein ideas are distanced from concrete objects. It is also the process of reducing the information content of a concept in order to retain only that information which is relevant to a particular purpose. Abstraction is an essential part of any intellectual activity. Its importance is derived from the ability of the mind to ignore irrelevant details and forming new conceptual objects and from the use of names to reference new objects.

• The importance of abstraction goes almost entirely unrecognized in everyday life. It is only when we think and think philosophically, that we recognize that thinking itself can only proceed at an abstract level.

• Many people believe that sticking to concrete objects is a safer and a more efficient way of thinking. The goal of this lecture is to show the futility of this opinion. In fact, we will show that without some level of abstraction our experience of the world would be a buzzing confusion. It would be all sensation and no perception, or all perception and no proposition, or all description and no explanation.

ABSTRACTION •

This is a football. It is made of leather.

It is supposed to be played with by the kicking. It has two colors. It is made by a certain manufacturer.

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Let us ignore:

✓ ✓ ✓ ✓ ✓

• • •

The material it is made of.

The purpose.

Its different colors.

Its manufacturer Etc.

What remains?

It has a spherical shape.

Therefore we can utilize and apply all our knowledge about spheres, for example: finding its volume, finding its surface area etc,

SYMMETRY

Further Examples of Abstraction • Abstracting an emotional state to happiness (for example) reduces the amount of information conveyed about the emotional state.

• Another example of abstraction is programming and building up software for a particular purpose.

• The grammar of a language is a further example of abstraction.

• The rules of a game, such as chess is another example of abstraction.

• Abstraction is used in the arts as any object or image, which has been distilled from the real world.

• However, mathematics seems to be the best field to illustrate abstraction, its levels and its importance.

IN FACT

MATHEMATICS IS A WEAPON OF MASS ABSTRACTION FOR THE PURPOSE OF UNDERSTANDING NATURE AND FOR DEVELOPING ITSELF. ______________________________ ABSTRACTION IN MATHEMATICS IS THE PROCESS OF EXTRACTING THE UNDERLYING ESSENCE OF A CONCEPT, REMOVING ITS DEPENDENCE ON REAL WORLD OBJECTS WITH WHICH IT IS CONNECTED OR MIGHT HAVE BEEN CONNECTED. LO! MATHEMATICS TURNS OUT TO HAVE AN INCREDIBLY WIDER

MATHEMATICS : A WEAPON OF MASS ABSTRACTIONS FOR UNDERSTANDING NATURE AND ITSELF • Mathematics is badly taught at most high school and even at the university level.

• Mathematics is usually presented as a collection of dull tricks for solving “canned” problems, devoid of historical perspective, of motivation of human touch and endeavor.

• NOW, let us catch a short glimpse of the process of abstraction of the concept of number as it was distanced from concrete objects by ancient people at some thousands years ago.

• Later on in the lecture, we will start to climb a ladder of higher level of abstraction from which we perceive for a much longer period of time a wider view onto a further scenario of abstraction towards the notion of Group Theory.

LACK OF ABSTRACT NOTION OF NUMBERS AMONG THE ANCIENTS •

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Modern society extensively use the concept of number, as well as, the word “number” in our every daily life. As a result, we tend to take the ideas of numbers for granted. Many of us do not realize its history, the labor of birth of new ideas.

In almost all civilizations, rudimentary counting is nearly as old as speech. When people, in the early Mesopotamian civilization (namely around 8000 to 3500 BC) wanted to express “five oxen”, for example, they would draw, on a clay tablet, a symbol for an ox five times. This practice indicates that those ancient people, had not yet, grasped the abstract notion of numbers.

Those people DID see five oxen. Yes, but they were not aware of the “FIVENESS” in the scene of the five oxen. In other words, those people did not yet discover that there is something in common among “five oxen”, “five trees”, “five apples”, and “five dogs,” …, etc.

For them, numbers were still tightly “locked” with things. Numbers did not have an independent conceptual existence.

Those ancient people were not as fortunate as our modern school children, who have well trained teachers to enlighten the paths of children by using sophisticated audio-visual aids. But those people had none; they had to stumble on something that would ignite their minds (Or the minds of some of them) in such a way that illuminates for them the concept of one-to-one correspondence in the difficult and long way. Evidently, after hundreds or even thousands of years they seem that they did stumble. Anyway, it was a conceptual leap in abstraction.

THE INITIAL HISTORICAL STEPS IN ABSTRACTIONS • There are no historical records of the actual development of that intellectual leap.

• However, Mesopotamians after they developed their civilization, began to distance the notion of integers from objects counted.

• Slowly and gradually the “small” integers, say from one to ten or twelve were given names and symbols to be drawn on clay tablet. They had some difficulties with zero.

• Later, Mesopotamians developed a system for representing all (positive) integers using the names and symbols for those “small” numbers. Though their system possessed deficiencies, which were later perfected by themselves and by other civilizations.

• We, in this lecture, will not elaborate further except to mention that when an advanced stage of development was reached, numbers begin to “live” at the conceptual level in the minds of people as if they were as concrete objects as oxen, dogs, trees, and apples.

• Consequently, numbers began to be manipulated, factorized, and classified into classes such as even, odd numbers, primes, perfect squares, cubes. Tables of additions, multiplications, inverses were constructed, …, etc.

LET US MAKE THAT BIG TIME JUMP OF SEVERAL THOUSAND YEARS.

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Thus, at the beginning of the nineteenth century, we have inherited a well developed systems of numbers and efficient manipulations and symbolization.

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We had also inherited many unsolved problems from the field of numbers as well as from other fields of knowledge. Those unsolved problems motivated research.

FIRST INITIAL STEP TOWARDS ABSTRACTION OF NUMBERS • The system of numbers is a set with two operations usually called addition and multiplication each satisfying certain conditions and inter-operation relations.

• As a first initial step towards abstraction, let us, retain the operation of addition and ignore that of multiplication.

• Thus, after this initial “stripping”, we are left with a set possessing a single operation.

• LET US PEEP DEEPER ONTO THIS STRIPPING.

THE SYSTEM OF INTEGERS I The system of integers is a set of numbers with a machine (similar to a vending machine) which has a pair of openings, through each an integer is inserted; the sum of that pair of numbers is obtained at the outlet.

The system satisfies the following:

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There exists in the set of numbers a “privileged number” 0 whose sum with any number is the latter number.

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For any number there is a number whose sum with the former number is 0.

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The sum of any triple of numbers is independent of the way they are inserted.

That is (x+y)+z =x+(y+z) for any triple.

8

5

+ 13

THE SYSTEM OF INTEGERS II The system of integers is a set of numbers with a machine (similar to a vending machine) which has a pair of openings, through each an integer is inserted; the sum of that pair of numbers is obtained at the outlet.

The system satisfies the following:

•

There exists in the set of numbers a “privileged number” 0 whose sum with any number is the latter number.

•

For any number there is a number whose sum with the former number is 0.

•

The sum of any triple of numbers is independent of the way they are inserted.

That is (x+y)+z =x+(y+z) for any triple.

0

x

+ x

THE SYSTEM OF INTEGERS III The system of integers is a set of numbers with a machine (similar to a vending machine) which has a pair of openings, through each an integer is There is a number inserted; the sum of that pair of numbers is obtained at the outlet.

The system satisfies the following:

•

There exists in the set of numbers a “privileged number” 0 whose sum with any number is the latter number.

•

For any number there is a number whose sum with the former number is 0.

•

The sum of any triple of numbers is independent of the way they are inserted.

That is (x+y)+z =x+(y+z) for any triple

x

+ 0

• Abstractions and generalizations are carried by men and some women for the sake of developing and corroborating knowledge in order to understand and manipulate nature. • Let us have a glimpse onto the tragic life of a young man who, more than anyone else, made great strides of abstractions of the notion of numbers.

EVARISTE GALOIS (1811-1832)

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Galois’ life was a tragic one. His father committed suicide. Galois himself regularly failed his examinations in school, as well as, he failed to gain admittance to the Ecole Polytechnique. He suffered the additional misfortune of having his research work (on solution of algebraic equations of degree greater than four) not only ignored but also completely misplaced. He was politically active against the royalty. As a result he was imprisoned in 1831 after proposing a toast interpreted as a threat to the King. In 1832, Galois was shot to death in a duel with a rival over a coquette.

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Galois, on the eve of his death wrote a letter to a friend in which he described a new method of attack at a problem that was bothering the mathematical community. The method he hammered was brand new, it nothing but an abstraction of the system of integers which he called group.

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He must have felt that he has an important message to the world that he has to deliver. He did.

GROUPS BEGAN TO HAVE A CONCEPTUAL LIFE OF THEIR OWN • After the death of Galois, the abstract concept of groups was strongly developed during the nineteenth century by leading mathematicians among whom are: Cauchy, Cayley, Jordan, Sylow, Sophus Lie. • In little more than a century group theory had effected a remarkable unification of mathematics, revealing connections between parts of algebra and geometry that were long considered distinct and unrelated.

• Wherever groups disclosed themselves themselves or could be introduced, simplicity crystallized out of comparative disorder.

• Groups seem to crop up in all manner of different contexts. Once you look for them you find them everywhere.

• Group theory – as we will see later during the lecture – has also helped physicists, chemists and scientists to the basic structure of phenomenal world.

WHAT IS A GROUP ?

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• 1.

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1.

A group is an abstraction of the system of integers. HOW? First by ignoring the constituent set of integers and replacing it by ANY SET WHATSOEVER. Second by ignoring the addition machine and replacing it by a similarly built machine, through which a pair of elements of the NEW set is inserted at the openings. The machine operates on the pair by some methods or others (illustrated by the forthcoming examples). Thus, obtaining a single element at the outlet of the machine. Upon this scaffolding a structure is built that has some similarity to the system of integers. Thus we obtain: A group is a system consisting of a set (which we will call carrier set) and a machine (which let us name as *– machine). The system possesses the following properties: The “starring” of any triple of elements in the carrier set is independent of the way they are inserted. That is (x * y) * z = x * (y * z) for any triple of elements.

There is in the carrier set a “privileged element”, which let us denote by e. The result of “starring” e with any element x of the set is the element x itself. That is: x * e = x = e * x.

For any element in the carrier set there is an element whose “starring” with the former element is the “privileged element” e. That is for any element x there is an element y such that x * y = e = y * x.

*-

OPERATION MACHINE Y X STARRING OPERATION

X*Y

*-

OPERATION MACHINE

e

X STARRING OPERATION

X

*-

OPERATION MACHINE

There is an element

X

STARRING OPERATION

e

DEFINITION OF A GROUP IN STANDARD MATHEMATICAL TERMINOLOGY The standard parlance of group theory are: ➢ ➢ ➢ ➢

A binary operation satisfying the first property is said to be an “associative operation” The operation machine is called “binary operation”.

An element, which we described as “privileged” is called a “neutral element” or an “identity element”.

If the starring of an element with an element x is a neutral element then we call that element an “opposite of x” and we denoted it by opp(x).

• Thus, using the above terminologies we define: A group is a system consisting of a set with an associative binary operation, and possessing a neutral element, and each element in the set has an opposite element.

AUDIENCES ARE DEMANDING

“We want Examples!!!”

EXAMPLES • The Prototype Model: The group of integers. Usually denoted by Z. • The set of all even integers under addition. […, - 4, - 2, 0, 2 , 4, 6, …] ➢The neutral element is 0. ➢The opposite of x is –x, where x is an even integer

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• The non-zero rational numbers under multiplication.

➢The constituent set is the the set of all the rational numbers, [ for example, 5/7, 43/12 , - 15/7, and so on.] ➢The operation is the usual multiplication, ➢The neutral element is 1, ➢The opposite of x is 1/x where x is a non-zero rational number

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EXAMPLE CLOCK-GROUP Clock arithmetic

➢ The constituent set is:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.}

➢ The operation, which we denote by * is the reading of the clock addition; say x * y will mean the reading of the clock at y hours after x. ➢ The neutral element is 12. ➢ Opp(1) = 11, opp(2) = 10, …, opp(11) = 1, opp(12) = 12 .

This group is denoted by:

Z12

EXAMPLE


5

8

CLOCK ADDITION

1

EXAMPLE WORD GROUP •

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How an ordinary typewriter is used? We usually type letters in juxtaposition to form dictionary or non-dictionary words.

“Our” typewriter types words also, but in a generalized sense, (or abstract sense) that is: The words typed by “Our” typewriter are not necessarily dictionary words.

Capital and small letters may be used in the same word, for example: “gFtttuPrs.” However, a small letter next to the capital of the same letter, or vice versa, cancel each other for example: “FerRsuUmj” becomes the reduced word “Fesmj”. Therefore the carrier set is the set of all reduced words.

The typewriter space is considered to be a word. It is called the “empty word.”

EXAMPLE


wdoGH

CaTW

Word Juxtaposition

CaTdoGH

EXAMPLE

wtAc

CaTW

Word Juxtaposition

––

QUESTION •Work with a duet letter typewriter say “a and A” only.

•The words obtained are: …… , aaaaa, aaaaa, aaa, aa, a, –, A, AA, AAA, AAAA, AAAAA, ….

•Does this ring something in the minds of the audience?

EXAMPLE • The set of all permutations of a finite collection of objects coup[ed with the operation of applying one permutation after another forms in a natural manner a group. • It can be proved that any finite group has the same structure as a permutation group.

EXAMPLE RUBIK CUBE • • •

• •

The set of all sequences of moves on a Rubik's Cube forms a group. The binary operation is composition of moves.

This is a finite group.

What is the number of elements of the group of the cube?

We can solve this question by counting the permissible permutations of edges and the permissible permutations of corners.There are 8 corner cubelets and 12 edge cubelets.

Each corner cubelet can have three possible orientations ("twists") and each edge cubelet can have two possible orientations ("flips").

Order of Cube Group = (8!)(11!)(6^8)(16) This is a rather large number:

43,252,003,274,489,856,000 • •

(or a little over 43 quintillion). At the rate of 1,000,000 configurations per second it would take well over a million years to run through them all.

From this we can safely conclude that coupling random twiddling with patience is a poor strategy for solving the cube.

EXAMPLE EQUILATERAL TRIANGLE ROTATIONS I Stands for leaving the triangle in its position.

J Stands for rotating the triangle counterclockwise 120 degrees.

K Stands for rotating the triangle counterclockwise 240 degrees.

• The group of rotations of an equilateral triangle

➢ The set is {I, J, K}. ➢ The operation is the composition of rotations, denoted by ©. ➢ The neutral element is I. ➢ J©J = K; K©J = I, ➢ J©K = I; K©K = J. ➢ opp I = I; opp J = K; opp K = J.

• This group has the same structure as

z3.

DEMANDS OF THE AUDIENCE • Examples.

• Consequences and Their Relevancies.

• Different kinds of groups. [finite groups, order of a finite group, infinite groups, cyclic groups, commutative. non-commutative groups, .. Etc.]

• Classifications.

• Construction of new groups from already known groups [subgroups, multiplications of groups, killing normal subgroups, … Etc.]

• Applications inside Mathematics.

• Applications outside Mathematics.

• How remote is the abstract group from the prototype?

HOUSEHOLD CONSEQUENCES Given an abstract group with a binary operation, which let us denote by *. Let each of a, b, c be an element in the group.

1.

If each e and e` is a neutral element then e = e`. [Uniqueness of neutral element].

3.

Henceforth we will say “THE neutral element” instead of “A neutral element.”

If each b and c is an opposite element of a then b = c. [Uniqueness of opposite element] . Henceforth we will say “ THE opposite element of a.” instead of “ A opposite element of a.” opp(e) = e.

4.

5.

6.

7.

8.

opp(opp(a)) = a. opp(a * b) = opp(b) * opp(a).

If a * b = e then a = opp(b) and b = opp(a).

If a * c = b * c then a = b. [Right cancellation property].

If c * a = c * b then a = b. [Left cancellation property].

2.

PROOFS Proofs of theorems 1 and 3. e`= e`* e = e.

opp(e) = opp(e) * e = e.

Proof of theorem 2. b = b * e = b * (a * opp(a)) = (b * a) * opp(a) = e * opp(a) = c = c * e = c * (a * opp(a)) = (c * a) * opp(a)

Proof of theorem 4.

For notational simplicity denote opp(a) by k. [therefore, k * a = e]

opp(opp(a)) = opp(k) = opp(k) * e = opp(k) * (k * a) = a = e * a = (opp(k) * k) * a

PROOFS •

Proof of 5 For notational simplicity we denote a * b by k and b * opp(k) by r.

Consider: opp(a * b) = opp(k) = e * opp(k) = [(opp(b) * b] * opp(k) = (opp(b) * [b * opp(k)] = opp(b) * r = opp(b) * (e * r) = opp(b) * [(opp(a) * a) * r] = opp(b) * {(opp(a) * [a * r]} = opp(b) * {(opp(a) * [a * (b * opp(k)]} = opp(b) * {(opp(a) * [(a * b) * opp(k)]} = opp(b) * {(opp(a) * [k * opp(k)]} = opp(b) * [opp(a) * e] = opp(b) * opp(a).

RELEVANCIES OF CONSEQUENCES • •

These theorems seem at first sight to be trivial.

But in the absence of abstraction, it would not occur to anyone to prove the uniqueness of, the already unique, 0. However, dealing with abstract groups with elements having the same structure as 0, there is a natural temptation to demand a a proof.

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The proof given establishes by a single stroke the uniqueness of the neutral element FOR EVERY GROUP. The same thing can be said for the uniqueness of opposite element and the for the following results:

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➢ – (–x) = x ➢ 1/(1/x) = x ➢ – (a+b) = (–b)+(–a) ➢ 1/(xy) = (1/x)(1/y) ➢–0=0 ➢ 1 /1 = 1

• Anything proved or constructed for an abstract group is true for all groups. This is one instance of the importance of abstraction.

DEMANDS OF THE AUDIENCE • Examples.

• Consequences and Relevancies.

• Different kinds of groups.

[finite groups, order of a finite group, infinite

groups, cyclic groups, commutative. non-commutative groups, .. Etc.]

• Construction of new groups from already known groups. [subgroups, product of groups, killing normal subgroups, … Etc.]

• • • •

Classifications.

Applications inside Mathematics.

Applications outside Mathematics.

How remote is the abstract group from the Prototype?

BACK TO SYMMETRIES I

BACK TO SYMMETRIES II • Platonic solids were well known from antiquity some thousand years before Plato.

• It can be proved that there are exactly five platonic solids.

• E – F + V = 2.

• The sphere has angular symmetry • There are exactly seventeen wallpaper symmetries.

• A crystal structure is composed of a unit cell, a set of atoms arranged in a particular way, which is periodically repeated in space. Using crystal symmetry one can prove that there are exactly thirty two different crystal structures. Isn't this a great result?

BACK TO SYMMETRY III •

Symmetry refers to special property of bodies or scientific laws or formulae that are left unchanged by permutations.

• Symmetries and groups are very much interrelated. Wherever you find a symmetries you find groups that describe them all.

• In a triangle with sides of lengths a, b, and c, we have the following symmetric formulae. ➢ P = (a + b + c) /2 ➢ A = P (P – a)(P – b)(P – c) ➢ r = (P – a)(P – b)(P – c) / P ➢ R = a b c /(4 √A).

P, A, r, and R are symmetric under the group of permutations, Each must stand for some intrinsic property of the triangle. In fact: P = half the perimeter of the triangle.

A = (area of the triangle)^2. r = (radius of the inscribed circle in the triangle)^2.

R = radius of the circumscribed circle for the triangle.

PERMUTATION GROUP ON THREE OBJECTS. IT IS THE SMALLEST NON-COMMUTATIVE GROUP.

Shuffling of the Music Cards

• • • • • •

The move “I” means let the cards as they are. (ab) means interchange card a with b leaving c fixed. (bc) means interchange card b with c leaving a fixed. (ca) means interchange card c with a leaving b fixed. (abc) means a goes to b, b goes to c and c goes to a. (acb) means a goes to c, c goes to b and b goes to a.

I

(ab)

(bc)

(ab)

I

(bc)

(abc)

I

(ca)

(acb)

(abc)

(abc)

(bc)

(ac)

(ab) (acb)

(acb)

(ac)

(ab)

(bc)

(ca)

(acb) (abc)

(abc) (acb)

(ac)

(bc)

(ac)

(ab)

(ac)

I

(bc)

(ab)

I

I (abc)

A POSSIBLE QUESTION BY THE AUDIENCE.

ISN’T THIS A GAME PLAYED BY MATHEMATICIANS FOR THEIR OWN AMUSEMENTS OR THINKING-WHIMS ????

NO THIS IS THE VERY UNIMAGINATIVE IMPORTANCE OF ABSTRACTION

EMMY NOETHER 1882-1935 • •

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Traditionally people consider mathematicians to be men. This however is not entirely to be true.

One of those women mathematicians was the German-born Emmy Noether. Being a woman, she was refused to be admitted to take mathematics at the University where her father and brother were Mathematics university professors. However, she was later granted permission to audit classes. After struggling with the administration she obtained her doctorate, but was not allowed to teach. However, she helped her father in his teaching.

She made very profound contribution to group theory and its application to physics. So deep her results that the most prominent mathematicians Felix Klein and Hilbert invited by to Gottingen University. Many of the faculty did not want her because of her gender. At last she was allowed to teach on her own but was underpaid.

Noether was a warm person who inspired and cared deeply for her students.

In 1933 she moved to the United States and taught at Bryn Mawr College, until her death at 1935.

Albert Einstein wrote her obituary.

Our universe can be thought of as a space with some structure – constituted by the laws of physics and chemistry. We can ask what transformations of space and time will preserve a certain structure (like Newton’s Laws, or Maxwell Equations, or Einstein’s Formulae).

This leads us to groups of transformations of space and time that preserve desired properties.

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Something intriguing happens: the groups correspond in a natural and intimate way with physical conservation laws. The wonderful results proved by our lady friend Emmy Noether were: ➢ In classical physics each symmetry of the laws of physics gives rise to a conserved quantity. ➢ From the translational symmetries we get conservation of momentum. We see this whenever two billiard balls collide. ➢ From the rotational symmetries we get conservation of angular momentum. ➢ The laws of physics are also invariant under time evolution itself. Time evolution is a symmetry of space-time, given by: t goes to t + constant, x, y, and z remain invariant.

The corresponding conserved quantity is energy. Ultimately, this is because energy tells you how fast things are wiggling around as time passes! ➢ The fundamental postulate of the special theory of relativity, that the laws of physics take the same form with respect to one another at a fixed velocity is clearly another statement about the symmetry of physical law.

• Things get really interesting when we realize that given the group of transformation, each irreducible representation of the group very naturally corresponds to a different fundamental particle, with force carrying and force carrying particles neatly delineated from each other by the degree of the irreducible representation.

ABSTRACT GROUPS APPLIED

IMPORTANCE AND DIFFICULTIES OF ABSTRACTION • The importance of of abstraction are:

➢Revealing deep connections between different fields of knowledge. ➢Suggesting conjectures in one field from known results in related areas. ➢Applying techniques and methods in one field to prove results in related areas.

• However, to be fair and unbiased, abstraction is easily misunderstood mainly because highly abstract concepts are more difficult to learn and require a degree of mental maturity and experience before they can be assimilated.

BEGINNING & END • Many people believe that sticking to concrete objects is a much more safer and efficient way of thinking than abstraction. The goal of the lecture was to show the futility of this opinion. In fact, we will show that without some level of abstraction our experience of the world would be indeed a buzzing confusion. It would be all sensation and no perception, or all perception and no proposition, or all description and no explanation.

• The main illustration of my thesis originated from one main example of that field of supreme art of abstraction, namely, Mathematics. The example seemed at the beginning to be nothing more than a little abstract game that mathematicians were playing off by themselves to solve their own problems. But later it turned out that this “game” is possessing an unbelievable impact on the real world we live in.

• This is the incredible power of abstraction and mathematics.

• It was just ONE example, more could have been chosen.