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10 Postulational Mathematics

Axiomatic Mathematical Systems An axiomatic system begins with definitions of terms plus a set of axioms. One

constructs valid deductive arguments using the definitions and axioms as premises. If the axioms are true, then every conclusion derived from them must be true as well.

One statement of the Pythagorean theorem is: In any triangle, the sum of the squares of the lengths of two adjacent sides is equal to the square

of the length of the third side, if and only if the angle between the adjacent sides will be a right angle. Egyptians and Babylonians found many triangles the lengths of whose sides agreed with these two statements. They seemed to have believed that these statements were true because in every case that they tried, a triangle that satisfied one side of the biconditional also satisfied the other. They did not know that the statements were universally and necessarily true. The Pythagoreans proved they were.

The earliest axiomatic mathematics is the 13 books of Euclid's Elements. The Elements begins with 23 definitions, five "postulates," and five "common notions." The postulates and common notions were considered to be "self-evident truths" – so obviously true as to need no proof. If the postulates and common notions are true, then every conclusion derived from them by valid deductive argument must be true.

Euclid's five postulates of plane geometry were: P1. A straight line may be drawn between any two points. P2. Any straight line may be extended indefinitely. P3. A circle may be drawn with any point as centre and any radius. P4. All right angles are equal. P5. If two straight lines lying in a plane are crossed by another straight line, and if the

sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.

A Problem with Euclid's Postulates Postulates P1 through P4 are easy to understand.

They may be "self-evident." However, postulate 5 is much more complicated than the others. In the diagram, the lines labelled L1 and L2 are the original "two straight lines lying in a plane." Line L3 is "another straight line" that crosses them. Angles A and B are the "internal angles on one side" of line L3. The postulate says that if the sum of angles A+B is less than

L1

L2

L3

A

B

AXIOMATIC MATHEMATICAL SYSTEMS 141

twice the size of a right angle, then if we extend lines L1 and L2 indefinitely, they'll meet somewhere off to the right. The definition of "parallel lines" says "Parallel straight lines are straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction."

Even when the meaning of the postulate is clear, its truth is not self-evident. In the diagram, L1 and L2 are clearly inclined toward each other. But what if the sum of angles A+B were only very slightly less than two right angles, so that the lines would have to be drawn millions of miles long before they met? Would they meet?

For two thousand years, geometers tried to derive the parallel postulate from the other postulates. They failed, but they did prove that the parallel postulate was logically equivalent to other statements. Some statements that are equivalent to P5 are: E1. If a straight line intersects one of two parallel lines, it will intersect the other. E2. Straight lines parallel to the same straight line are parallel to each other. E3. Two straight lines that intersect one another cannot be parallel to the same line. E4. Given a line L and a point P in a plane, where P is not on L, there is one and only

one line through P which is parallel to L.

If we could prove any of these, then (by equivalence) we would have proved pos-tulate P5. They failed to find a proof using direct methods. They tried Indirect Proof. If one derives a contradiction from the assumption that postulate P5 is false, then one will have proved that the postulate is true.

Occasionally somebody would claim to have proved a contradiction, but either the "proof" was invalid or the "contradiction" was not really a contradiction.

If there is no contradiction, we could invent new plane geometries by replacing Euclid's fifth postulate with its denial. Lobachevskian geometry replaces postulate E4 with "Given a line L and a point P in a plane, where P is not on L, there is more than one line through P which is parallel to L." Riemannian geometry uses the state-ment "Given a line L and a point P in a plane, where P is not on L, there is no line through P which is parallel to L."

Each of these postulates was proved to be logically independent of the other postulates. A statement is logically independent of other statements if it can be false when the other statements are all true. That entails that none of them can be proved by deductive argument using the other postulates as premises. It was also shown that if Euclidean geometry is consistent1 then Riemannian geometry and Lobachevskian geometry are consistent too. By all purely mathematical and logical standards, Riemannian and Lobachevskian geometries are "just as good" as Euclid's.

1 That is, if all of its postulates can be true on some interpretation, so it must be impossible to derive any

contradiction from them by any valid logical argument. (The meaning of "on some interpretation," is explained below.)

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A geometry (one of the many different geometries) came to be seen as one particular set of undefined terms and basic postulates. The postulates specify the relations between the terms.

Why "undefined terms"? Euclid's definitions of "point," "line," "surface," and so on are not very clear anyway. What does it mean to define "point" as "that which has no part," or "line" as "breadthless length"? These definitions require us to go on to define "part" and "breadth" and "length." The words used in those definitions require definition also. We're always going to have undefined terms. Why not stop at "point" and "line," and leave them undefined?

Euclid's definitions were scrapped. The postulates were sanitized to use undefined terms. For example, Euclid's first postulate says, "A straight line may be drawn between any two points," which is equivalent to "Given any two points, there is at least one straight line that contains them." To keep this abstract, we shouldn't talk about "straight line," because people may interpret that to mean the path of a ray of light or the streak of ink left by a pen following a ruler. Abstract geometry is not about the paths of rays of light or streaks of ink. Using an undefined word like prosfo would be better. Similarly with "point," where we might say clamit instead. The relation "contains" is tied to the intuitive interpretation of "point" and "line" so we use another undefined word (e.g., gragga). Now we can restate the postulate as "Given any two clamits there is at least one prosfo such that both clamits have a gragga relation to that prosfo." Better yet, we can replace the verbal formulation with a symbolic one, like: ∀x∀y((Cx ∧ Cy) ⊃ ∃z(Dz ∧ (Gxz ∧ Gyz))). The parallels postulate E4 can be replaced with "Given any prosfo P1 and a clamit C1 that does not have the gragga relation to P1, there is one and only one prosfo P2 that has the gragga relation to C1 such that, no matter how much we nichkak2 P1 and P2, there will never be a clamit that has the relation gragga to both P1 and P2."

What does it mean to say that something is a clamit? What does it mean to say that some x has the property C or that there is a G relation between two things? At this level of abstraction, it doesn't matter. We can say that the theorems of a geometry are true of any things, properties, operations and relations that satisfy the postulates. If we can interpret the postulates so that they state truths about some set of things, then the theorems will also be true of that set of things on that interpretation. The mathe-matician can concentrate on deriving theorems without caring about what sort of things she is dealing with. She can also focus on the properties of the system, in abstraction from questions about the things the system describes.

David Hilbert created such an abstract axiomatic geometry. In the process, he showed that some of Euclid's proofs were invalid. Euclid had assumed things that were not "contained" in the premises (axioms and postulates) he was supposed to be using. 2 I cannot say "extend" because that is likely to be interpreted.

POSTULATIONAL SYSTEMS 143

One can interpret Hilbert's abstract geometry in terms of Euclidean lines and points. On that interpretation it is a cleaned-up and corrected version of Euclid's geometry. One can also interpret it differently (e.g., if we interpreted clamit to mean line and prosfo to mean point) to give different geometries. Similar abstraction and interpretation could be done with the non-Euclidean geometries. We can create all sorts of abstract "geometries" unrelated to anything that had been done before.

Postulational Systems Pure (or abstract) mathematics is the construction and investigation of

postulational systems.

A postulational system consists of a set or sets of undefined terms, predicates, relations and operations, and a set of unproved basic statements that establish relationships between these terms. These basic statements are just assumed as

postulates of the system.

The basic statements don't have to be true or even meaningful (they may contain terms that have no meaning other than what the postulates assert about them).

Consider symbolic logic as a pure postulational system. Give up the idea that the A's and B's "stand for" statements. Just accept them as the things we manipulate with our rules. Connectives and operators would not be defined in terms of truth-values T and F. Any kind of symbolic constants (e.g., and ) can be used in the tables that define them. We don't have to be limited to just two possible values – we can construct alternative "logics" using as many values as we like. We just specify the formation rules and transformation rules of a system, and that will be a logic.

Formation rules are rules for constructing well-formed formulas (WFFs). We could distinguish A, B, C, … as simple WFFs, and then describe how to make compound WFFs out of simple WFFs plus connectives. The value of a compound WFF would be a function of the values of its atomic components, but we wouldn't specify that these values are truth-values.

The transformation rules specify allowable transformations on WFFs, describing what WFF you can get by transforming one or more other WFFs. To state the transformation rules, we'd create a symbolic shorthand using a special kind of variable to represent the form of a WFF (like our p, q, r, …) and rules about when a WFF is a "substitution instance" of a form. Our derivation rules are transformation rules.3 We would also state some postulates or specify a form whose substitution instances are postulates (such as ~(p ∧ ~p)). The postulates are just postulated in the system. We don't commit ourselves to saying that they are "true." 3 Although we'd have some problem stating rules CP and IP in such an abstract logic.

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Here are a couple of pure postulational systems from Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.4

Hofstadter's MIU System Hofstadter created the MIU system to illustrate the idea of postulational systems.

The only formation rule of the MIU system is: a WFF is any string containing only the letters M, I, and U. There are four transformation rules, and all of them are "one-way" rules (like implicational rules). The rules are: Rule I: Given a WFF whose last letter is I, you can add a U at the end. For example,

from MMI you can get MMIU. Rule II: Given a WFF of the form Mx,5 you may repeat x to get Mxx. From MIU you

can get MIUIU, from MUM you get MUMUM and from MU you can get MUU, etc. Rule III: If III occurs in a WFF, you can replace the III with U. So, from UMIIIMU you

get UMUMU and from MIIII you get MIU or MUI. From MIII you can make MU. Note: these are "one-way" rules, so you cannot use Rule III to get MIII from MU.

Rule IV: If UU occurs in a WFF, you can drop the UU. So from UUU you can get U. From MUUIII you can get MIII.

There is one postulate (initial premise or axiom) in this system. It is MI. Hofstadter sets his readers a problem that he calls "the MU-puzzle." Starting from that one postulate as a given WFF, can you derive MU?

Here's an example of a derivation in the MIU system. It resembles our proposi-tional logic derivations, except that it doesn't mean anything.6 The goal is to derive the theorem7 MUIIU. (1) MI Axiom (2) MII 1, Rule II (3) MIIII 2, Rule II (4) MIIIIU 3, Rule I (5) MUIU 4, Rule III (6) MUIUUIU 5, Rule II (7) MUIIU 6, Rule IV

4 Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (A Metaphorical Fugue on Minds and

Machines in the Spirit of Lewis Carroll) (New York: Basic Books, 1979). This is an amazing, wonderful book. Gödel was a mathematician; Escher was a graphic artist/printmaker/painter; Bach was Johann Sebastian Bach. The book is about mathematics, graphic art, music, philosophy, minds and brains, meanings, computers and artificial intelligence, and lots more.

5 x here stands for any of M, I, or U or any string of them. It works something like our p, q, and r forms. That is, Mx is not a WFF – it is the form of a WFF. Mx is any WFF whose first letter is M.

6 A former student asked (about this sentence) "Then what's the point?" The point emerges in the section on "Interpretation of Formal Systems," below.

7 Any WFF that can be derived just from the basic axiom(s) or postulate(s) of a system is a theorem of that system. Remember this definition!!

POSTULATIONAL SYSTEMS 145

Exercise on the MIU System Play with the MIU system. Derive some theorems. Try to derive MU. If you cannot

derive MU, try to prove that it is impossible. The proof of impossibility cannot be done within the MIU system; you have to "step outside the system" and do your reasoning in traditional logic or math.

Hofstadter's pq- System WFFs in the pq- system contain just three kinds of symbols: p, q, and - (the

hyphen). There are infinitely many axioms. Hofstadter defines an axiom-schema, which is the form of an axiom. Every WFF of that form is an axiom of the system.

Definition: xp-qx- is an axiom, where x consists only of hyphens. Note that x is not part of a WFF, but is used to stand for a string of hyphens. The same string of hyphens (i.e., the same number of hyphens in the string) replaces both xs.

From this definition, we can see that -p-q-- is an axiom (substituting a single hyphen for x). So are --p-q--- and ---p-q---- and ----p-q-----. Any axiom of the system is also a theorem of the system.

There is only one transformation rule (Hofstadter calls it a "rule of production" or "production rule") in this system. Rule: Suppose x, y, and z stand for particular strings of hyphens. Suppose that xpyqz is a theorem. Then xpy-qz- is a theorem. If x is '--' and y is '---' and z is '-', then the rule tells us: If --p---q- is a theorem (is it?), then --p----q-- is a theorem.

What about formation rules? A derivation must start with an axiom or a theorem. If it begins with a theorem, that theorem will have to have been derived from an axiom or a theorem. Ultimately we arrive at premises that are axioms only. No axiom contains more than one or less than one p or q. Our transformation rule does not allow us to add a p or a q. So every WFF must contain exactly one p and exactly one q. By similar argument, since the transformation rule never permits derivation of a WFF that has fewer hyphens than the WFF it's derived from, and the simplest axiom is -p-q--, so every WFF in a derivation will begin with a hyphen, and the p and q will be separated by at least one hyphen, and the WFF will end with a hyphen. The axiom schema and the transformation rule determine the only allowable formulas in derivations. We don't need explicit formation rules.

Exercise on the pq- System Play around with the system. Get an interesting axiom using the schema and use

the transformation rule to derive theorems.

Interpretation of Formal Systems Now we get to the point of all this. MI is an axiom (postulate) of the MIU system.

Is it true? Is it false? The theorems of the system are derived according to the

146 10 POSTULATIONAL MATHEMATICS

transformation rules from the axiom MI. In a sense, then, they are conclusions of valid arguments using MI as a premise. Are they true or false? M and I and U were never defined. We don't know what these symbols denote or connote. They are undefined terms or primitives of the system.

Are the axioms of the pq- system true or false? We don't know what p or q or – stand for, so we don't know what (if anything) is stated by "---p-q----." The theorems can be validly (i.e., according to the transformation rules of the system) derived from the axioms, but are the axioms true or false?

WFFs in these systems don't make statements until we interpret the system.

One system is considered to be an interpretation of another if and only if one can map one system onto the other in such a way that each part8 of one system

corresponds to one part of the other, where "correspond" means that the two parts play similar roles in their respective systems. When such a mapping exists, we say that one system is an interpretation of the other. Mathematicians say that the two

systems are isomorphic, or that there is an isomorphism between them.

Hofstadter proposes that we consider the isomorphism or interpretation given by the mapping: p ↔ plus; q ↔ equals; - ↔ one; -- ↔ two; --- ↔ three; etc. On this inter-pretation, ---p-q---- maps to "three plus one equals four." It and all of the axioms and theorems are true on this interpretation.

Another interpretation is: p ↔ horse; q ↔ happy; - ↔ apple. Then -p-q-- translates as "apple horse apple happy apple apple." On this interpretation, axioms and theorems are no more true than non-axioms and non-theorems. Hofstadter says, "A horse might enjoy 'happy happy happy apple horse' (mapped onto qqq-p) just as much as any interpreted theorem."

A problem with interpretation is that one might "read too much into" the original system, based on knowledge of the system to which it is mapped. For example, on the "plus, equals, one, two, …" interpretation, someone might think that --p--p--q------ should be a theorem because "two plus two plus two equals six" is true. That would be a mistake. --p--p--q------ is not a theorem. It's not even a WFF in the system.

Is "plus, equals, one, two, …" the interpretation of the pq- system? Hofstadter gives the mapping: p ↔ equals; q ↔ taken from; - ↔ one; -- ↔ two; etc. On this interpretation, --p---q----- means "two equals three taken from five." This is another interpretation. Again, all the axioms and every theorem are true on this interpretation. Which is the real meaning of the string? Is there even any sense in asking about the real meaning?

8 "Part" includes the objects that are components of the two systems as well as the relations between and

operations on those objects.

PROPERTIES OF A POSTULATIONAL SYSTEM 147

Properties of a Postulational System Algebra is usually taught as if it were just symbolic arithmetic. That is, one learns

to use letters as variables to stand for numbers. Modern algebra is more abstract. The things the variables stand for don't have to be numbers. We can explore arithmetic operations and relations without thinking about numeric interpretations. The freedom that results from seeing mathematics as the construction of postulational systems allows us to construct algebras.

An algebraic structure or algebraic system (or an algebra, where one can describe many different algebras) is a system consisting of a set of elements or objects, together with operations on and relations between those objects. Ordinary arithmetic on integers is one algebra. Changing the set of objects to rational numbers gives a different (but similar) algebra. Defining different arithmetic operators or relations gives other algebras. Mathematicians explore the properties of various abstract algebras, including the properties that many different algebras share.

Although an algebraic system can be freely invented with no particular interpre-tation, it is easiest to construct an algebraic system based on something. We can then illustrate how one explores the properties of the system that results.

Because permutations are important in probability theory later in this course, we'll explore an algebra of permutations.

Permutations The order in which the members of an ordered pair are listed is important. We can

extend the notion of an ordered pair and talk about ordered triples (a list of three elements, where the order of the elements matters) or ordered quadruples (four ele-ments). In general, we can discuss ordered n-tuples (ordered lists of n elements).

Since order matters, two ordered lists are distinct if the elements are listed in different orders. Each distinct way of ordering the n elements of an n-tuple is called a permutation of those elements. "Permutation" is also the word for the operation of permuting the elements of an ordered n-tuple into a different ordering.

How many permutations of two elements a and b can we distinguish? The ele-ments can be put into ordered pairs as a, b or as b, a . If there are three elements a, b and c, they can be ordered in six distinct ways, so there are six permutations of three elements: a, b, c , a, c, b , b, a, c , b, c, a , c, a, b , and c, b, a .

Exercise on Permutations 1. Write out all the permutations of four elements a, b, c and d. How many permu-

tations are there? 2. Write your answer from question 1 into the following table. Is there a pattern?

Describe the pattern in a conjecture about the relation between the number of

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elements and the number of permutations. Your conjecture should be one that would hold for all collections (sets) of elements of any finite size. Can you prove that your conjecture is universally true?

Number of Elements Number of Permutations 1 1 2 2 3 6 4 5 120 6 720 7 5040 8 40320 9 362880

Permutations as Operations We develop a notation. Let's say that (2 1) specifies a permutation of an ordered

pair that puts the first element into the second place and the second element into the first place. Let's call that permutation B. So B = (2 1). When we apply the permutation B to the ordered pair a, b we get b, a . We symbolize the operation of applying the B-permutation to the ordered pair a, b as B( a, b ). We say B( a, b ) = b, a , pronounced "B of a, b equals b, a ." Another permutation on ordered pairs would be (1 2). Call it A, so A = (1 2). This is the null permutation or identity permutation. It leaves the order unchanged. A( a, b ) = a, b .

This is an example of "thingification." We started with ordered pairs and treated them as things. Then we treated operations on ordered pairs as things, and gave the operations names, calling them "A" and "B."

Working on triples is more interesting. Call the identity permutation A, again. So A = (1 2 3). Let B = (1 3 2), C = (2 1 3), D = (2 3 1), E = (3 1 2) and F = (3 2 1). If we apply the F permutation to a, b, c we get c, b, a . If we apply the F permutation to c, b, a we get a, b, c . So applying the F permutation twice is like applying the identity permutation. Also, D( a, b, c ) = b, c, a . D( b, c, a ) = c, a, b . So, by applying D twice to a, b, c we get c, a, b , which is the same thing we'd get by applying E to a, b, c . We can describe this as D(D( a, b, c ) = E( a, b, c ). This says that applying D to the result of applying D to a, b, c is the same as the result of applying E to a, b, c . We pronounce it "D of D of a, b, c equals E of a, b, c ."

What happens if you apply F to the result of applying B to a, b, c ? Applying B to a, b, c gives you a, c, b . Applying F to a, c, b gives b, c, a . This is just what you'd get from D( a, b, c ). So F(B( a, b, c )) = D( a, b, c ).

What if we didn't start with a, b, c ? What is the result of applying F to the result of applying B to b, a, c ? B( b, a, c ) = b, c, a . F( b, c, a ) = a, c, b . But a, c, b is just what you'd get from D( b, a, c ). Check it and see. What's more, it doesn't

PROPERTIES OF A POSTULATIONAL SYSTEM 149

matter which of the six ordered triples you start with: Applying F to the result of applying B to any triple is the same as applying D to it. If we limit our universe of discourse to just the six ordered triples (all possible permutations of a, b, c ), we can say: ∀x(F(B(x)) = D(x)). For any x (of the six triples), F of B of x is equal to D of x.

◦ A B C D E F A A B C D E F B B A E F C D C C D A B F E D D C F E A B E E F B A D C F F E D C B A

We saw that D(D( a, b, c ) = E( a, b, c ). You can check to see that the generalization ∀x(D(D(x)) = E(x)) is true for any (x) of the six triples. Again, it does not matter what triple you start with: applying D to the result of applying D to any triple will give the same result as just applying E to that triple.

Applying a permutation to the result of applying a permutation to a triple is an operation (as addition and multiplication are operations). We call the operation "composition" of permutations.

Since the effect of composition of permutations is independent of the particular triple we start with, so the triples can "drop out" of the discussion and we can concen-trate on the permutation operations (as things). We can say "B followed by F is the same as D" and "D followed by D is the same as E" and so on. Let's represent "is the same as" with the = symbol. Instead of "B followed by F" let's use B◦F. The ◦ symbol stands for the operation of composing two permutations (i.e., composition of two permutations). The ◦ symbol is an operator-symbol, like + or × in ordinary arithmetic. Now we can say B◦F = D and D◦D = E.

We summarize the statements about the results of all possible compositions of permutations on triples in a table, rather like a multiplication table. In the table above, we find B◦F by looking in the column on the left for B and then following that row across to the column with F at its top. We see that B◦F = D.

This is an algebraic system consisting of six elements A, B, C, D, E and F and the operation ◦. Its creation was motivated by the notion of permutations of ordered triples. Now that we've got it, we can study it on its own, unrelated to its origins.

We said that our operator symbol ◦ was "like + or × in ordinary arithmetic." How much (or little) does it resemble those common arithmetic operators? For example, is our ◦ operator associative? In ordinary arithmetic, + is associative (for any numbers x and y and z, (x + y) + z = x + (y + z)). Is that true for our ◦ operator? That is, among the six "things" A, B, C, D, E and F described in our table, is it true that ∀x∀y∀z((x◦y) ◦ z = x ◦ (y ◦ z)). Now we're treating the operation ◦ as a thing!

Exercise on Permutation-pairs 1. Explore the conjecture that X◦(Y◦Z) always gives the same result as (X◦Y)◦Z

(where X, Y and Z are variables standing for permutations). Try B◦(B◦B), C◦(D◦D), F◦(E◦C), D◦(C◦D) and F◦(F◦E). Five successful tests do not prove that

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the conjecture is true – we'd either have to try all the 216 possible combinations or else derive a formal deductive proof – but it may encourage you to believe that the conjecture is true.

2. Check if the operation of permuting an ordered pair and then permuting the result (i.e., composition of permutations) is commutative.9 Go through the table and find all those pairs where X◦Y = Y◦X. Note any patterns (if any) that you discover.

3. We listed and named the two permutations of two elements. Make a "multiplication table" showing the product of pairs of permutations of ordered pairs.

Studying General Properties of a System After you have done #3 in the exercise, you have a table for the compositions of

permutations on ordered pairs. The table for the compositions of permutations on triples is given above. There are 24 different permutations of four elements. The table of compositions of permutations on ordered quadruples is called "Table of 'Products' of Permutations on Quadruples," below.

9 An operation is commutative if and only if X◦Y = Y◦X for every possible choice of X and Y.

◦ A B C D E F G H I J K L M N O P Q R S T U V W X A A B C D E F G H I J K L M N O P Q R S T U V W X B B C A L F J I D K E G H P M S N X U W R T Q O V C C A B H J E K L G F I D N P W M V T O U R X S Q D D J G E A H L K B I F C Q U M W O S V P X R T N E E I L A D K C F J B H G O X Q T M V R W N S P U F F K H B L G A J E C D I S V X R P Q U O M W N T G G D J K I A F C L H B E U W T Q R P M X S N V O H H F K J C L D I A G E B V R N S W O X M Q T U P I I L E G K B J A H D C F T O R X U N P V W M Q S J J G D C H I B E F A L K W Q V U N X T S P O M R K K H F I G C E B D L A J R S U V T M N Q O P X W L L E I F B D H G C K J A X T P O S W Q N V U R M M M N P O Q V T S U X R W A B D C E K H G I F L J N N P M W V X U O R Q T S C A H B J I L K G E D F O O X T Q M S W R N U V P E I A L D H F C J K G B P P M N S X L R W T V U P B C L A F G D I K J H E Q Q U W M O R P V X N S T D J E G A F K L B H C I R R S V U T P Q N O W M X K H I F G A B E D C J L S S V R X P W O U M T Q N F K B H L D J A E G I C T T O X R U M V P W S N Q I L G E K C A J H B F D U U W Q T R N X M S O P V G D K J I B C F L A E H V V R S N W T M X Q P O U H F J K C E I D A L B G W W Q U V N O S T P R X M J G C D H L E B F I K A X X T O P S U N Q V M W R L E F I B J G H C D A K

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It doesn't matter that the tables were originally made to show the result of successive permutations. We can treat them as defining the result ("product") of a binary operation (◦) on undefined mathematical objects. In the tables, the operator is indicated as ◦ (in the top left corner), to help you stop thinking of the table as showing composition of permutations. Think of it as just an operation called "◦" on things called "A," "B," and so on. The tables define the operations, just as truth tables defined the logical operators and addition and multiplication tables define those operations.

Mathematics studies the patterns that emerge among the tables. See if you can find any properties that all three tables have in common.

Table of "Products" of Permutations on Quadruples Some of the things all of these tables have in common are:

1. Each table deals with a certain set of objects. Every object is listed in the guide (top) row and guide (leftmost) column of the table.

2. Every ordered pair of objects (e.g., A and F in that order are an ordered pair) in the set produces exactly one "product."

3. Every "product" is a member of the original set of objects. 4. Every member of the set of objects appears exactly once in each row and each column. 5. There is a column that looks exactly like the guide column. 6. There is a row that looks exactly like the guide row. 7. The "products" have the associative property. That is, for all X, Y, and Z in the set,

X◦(Y◦Z) = (X◦Y)◦Z.

Exercise on Properties of Tables 1. Check that the table of compositions of permutations of pairs is associative.

Figure out and compare the following pairs of complex products: (A◦A)◦A and A◦(A◦A); (A◦A)◦B and A◦(A◦B); (A◦B)◦A and A◦(B◦A); (B◦A)◦A and B◦(A◦A); (A◦B)◦B and A◦(B◦B); (B◦A)◦B and B◦(A◦B); (B◦B)◦A and B◦(B◦A); (B◦B)◦B and B◦(B◦B). If, in all possible cases, the pairs of products are equal, then the composition operation has the associative property.

2. Composition of permutations of triples and of quadruples also have the associative property, but it would take a long time to check all the cases.10 Using the table of composition of permutations on quadruples, show that (F◦J)◦M = F◦(J◦M), (C◦E)◦L = C◦(E◦L), (R◦M)◦T = R◦(M◦T), and (X◦H)◦Q = X◦(H◦Q).11

10 Each case involves three (not necessarily different) permutations. There are 8 = 23 possible combinations of

three permutations of pairs; there are 63 = 216 possible combinations of three permutations of triples; there are 243 possible combinations of three permutations of quadruples. Do you understand why?

11 Checking less than all of the possible combinations does not prove that the operation is associative. You could check them all, but see previous footnote. Mathematicians would prove that it is associative by a deductive proof that would apply to permutations of any size of n-tuple.

152 10 POSTULATIONAL MATHEMATICS

3. Go through the table of composition of permutations of quadruples and find all those pairs where X◦Y = Y◦X. Note any patterns (if any) that you discover.

4. Is composition of permutations of triples commutative? Is composition of permutations of quadruples commutative? Give reasons for your answers.

Groups One of the simplest algebraic systems is called a group.

A group is an algebraic system consisting of a nonempty set (call it G) of elements and one binary operation12 that satisfies four postulates (see below). One

convention for describing an algebraic system is to put the set(s) and operator(s) inside angle-brackets, as G, ◦ .

The members (elements) of the set G (which we denote, below, by the small letters a, b, c, …) are undefined objects, and the operation ◦ is an undefined operation. The properties of the objects and the operation are given explicitly in the postulates.

The postulates that must be satisfied if G, ◦ is a group are:

G1: To every pair of elements a and b of G, given in the stated order, there corre-sponds a definite unique element of G, denoted by a◦b. This is called the law of closure for the operation ◦. This property of groups is called the "closure property." We say that a group is closed for an operation ◦ if and only if ◦ exhibits the closure property.

G2: For all a, b, and c in G, (a◦b)◦c = a◦(b◦c). This is the associative law for the operation ◦. It describes the associative property of groups.

G3: There exists an element i in G such that, for any a in G, a◦i = i◦a = a. The element i is called an identity element of the group. We can prove that a group will never contain more than one identity element.

G4: For each element a of G there is an element a of G such that a◦a = a ◦a = i. The element a is called the inverse of a. We can prove that an element a of a group possesses only one inverse element.

If the following postulate is also satisfied,13 the group is called a commutative or Abelian group.

G5: For all a, b in G, a◦b = b◦a. This is the commutative law for operation ◦.

12 I've called it ◦, so that we don't confuse it with any of the usual symbols for arithmetic operators. 13 That is, if the operation ◦ is commutative.

GROUPS 153

A group for which postulate G5 does not hold is called a non-Abelian group. If the set G of a group contains only a finite number of distinct elements, the group is called a finite group; otherwise it is called an infinite group.

Our three tables of compositions of permutations are examples of finite groups. Another way of saying this is to say that they are interpretations of the postulates that define a finite group. The identity element in each table is the element called A; that is, for every X in G, X◦A = A◦X = X, so our tables satisfy postulate G3. Postulate G4 requires that the identity element must appear at least once in every row, in which case the element in the guide row at the top of that column will be the inverse element for the element in the guide column at the left end of that row, and vice versa. Thus, in the big table, C is the inverse element for B (i.e., B = C) because B◦C = C◦B = A.

We can easily check that the table of composition of permutations on pairs represents an Abelian group, but the table of composition of permutations on triples does not (because, for example, D◦B = F, but B◦D = E, and C◦E = D, but E◦C = F, and E◦F = C, whereas F◦E = B. A glance at the table of composition of permutations on quadruples shows that B◦D = L, but D◦B = J, so it is not an Abelian group.

Exercise on Group Properties 1. Prove that a group can never have more than one identity element. 2. Prove that an element of a group cannot have more than one inverse. 3. Consider a set of two objects {E, O} (think of E as standing for "even" and O as

standing for "odd"). Make a table for an operation ◦ on {E, O}, where the "product" is odd or even depending on whether the sum of the two operands14 is odd or even. For example, the sum of two even numbers is an even number, so E◦E = E, and the sum of an even plus an odd number is odd, so E◦O = O. Does this table specify a group (i.e., does {E, O}, ◦ satisfy the definition of "group")? Is the table like (isomorphic to) any group we have already considered?

4. ("Clock arithmetic") Imagine a "clock" having the numbers 0, 1, 2, 3, 4, 5, 6 equally spaced on its face. To find the sum of any two numbers in that set of numbers, we start at the first number and move clockwise around the dial a number of spaces equal to the second number. Thus, the "sum" of 2+3 is found by starting at 2 and moving clockwise three places (to 3, then 4, then 5). The "sum" is 5. 4+5 is found by starting at the 4 position and moving through 5, 6, 0, 1, 2, giving a "sum" of 2. Write out the table of clock-arithmetic "sums" formed from the set {0, 1, 2, 3, 4, 5, 6}. Does this table define a group? Why or why not?

14 When you use an operation (like addition or multiplication or ◦) on one or more objects, the objects are called

the operands of the operation.

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5. Each of the following algebraic systems satisfies the definition of an infinite group. In each example, name the identity element and specify the inverse x for each element x. a) , + (i.e., the set of integers and the operation of ordinary arithmetic

addition). b) -, · (i.e., the set of rational numbers excluding 0 and the operation of

ordinary arithmetic multiplication). Why was it necessary to exclude 0? 4. Explain why the following algebraic systems are not infinite groups:

a) , · (i.e., the set of integers and the operation of ordinary arithmetic multiplication).

b) , + (i.e., the set of natural numbers with ordinary addition).

In this exercise we saw that parts of our ordinary arithmetic systems can be studied as interpretations of the abstract algebraic systems called groups. We can also dream up new "arithmetics" that are groups.

We can go on to study the abstraction itself. We can derive theorems from the postulates, and know that any system that satisfies the definition of a group will be such that all the theorems will be true for that system. Moses Richardson's Fundamentals of Mathematics gives a number of theorems and their proofs. Many books and articles in scientific journals are devoted to the study of groups.

Beyond this, mathematicians study more complicated abstract algebraic systems like rings, subrings, ideals, integral domains, ordered integral domains, and fields. These involve more than one operator and additional postulates and properties. One or the other system will describe almost any arithmetic system, including those that we invent with our new kinds of numbers. In studying such abstract systems, we can study our ordinary arithmetic at a level behind the mere numbers, and discover isomorphisms between arithmetic systems and other deductive structures.

11 Estimating Arithmetic

Numeracy

Most children enjoy their first encounters with arithmetic. Mastering the basic operations is “empowering” – it gives one power over numbers. After years of practicing these operations over and over – adding long columns of numbers or multiplying long strings of digits – it becomes tedious. Students come to associate mathematics with boring drill. Exploration of mathematical concepts bogs down because of the need to perform calculations. Calculators and computers permit one to study mathematical ideas and free one from long, error-prone computations. That’s good.

On the other hand, the use of calculators often leads people to be alienated from numbers. They can get numeric results quickly, but they often fail to understand what the numbers mean. When a student takes out a calculator to figure out what 12% of 200 is, or to figure what percentage 20 is of 1000, she demonstrates innumeracy. That’s bad.

To divide one integer by another integer and report the result to eight or ten decimal places is innumerate and ignorant. Such precision is utterly bogus and meaningless, but the calculator or computer gives results to that precision. People learn to trust the calculator, even when it seems to tell them that 42 is 0.251497005988% or 2.51497005988% of 167. Numerate people would notice that these answers are obviously wrong.

Numerate people know how to do manual calculations, and how to simplify the task of manual calculation. They should also know how to use calculators, without losing the “hands-on” familiarity and “feel” for numbers that comes from intimate acquaintance. To be numerate requires that one have a “vocabulary” of numbers against which one can compare other num-bers that one encounters. One should know (approximately) the population of one’s city, one’s province or state, one’s country, and the world. One should have a bunch of distances (how wide is North America? how far away is Europe? the moon? the sun? the nearest star?). Sizes of commonly-met things (cars, bricks, books, packages of 500 sheets of paper, your hand, your height, the height of one storey of a building) are useful for comparisons.

One should exercise this vocabulary frequently. When driving, estimate (mentally calculate) the distance traveled, the average speed, the distance remaining to the destination. While shopping, practice keeping a running total in one’s head. Exercising one’s “arithmetic muscles” leads to a familiar ease with numbers, just as frequent reading and talk improve

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