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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.
P1.
P2.
P3.
P4.
P5.
Euclid's five postulates of plane geometry were:
A straight line may be drawn between any two points.
Any straight line may be extended indefinitely.
A circle may be drawn with any point as centre and any radius.
All right angles are equal.
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
L1
L3
Postulates P1 through P4 are easy to understand.
A
They may be "self-evident." However, postulate 5 is
much more complicated than the others. In the L2
B
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
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 postulate 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 statement "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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10 POSTULATIONAL MATHEMATICS
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 mathematician 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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10 POSTULATIONAL MATHEMATICS
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 "oneway" 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 propositional logic derivations, except that it doesn't mean anything.6 The goal is to derive the
theorem7 MUIIU.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
4
5
6
7
MI
MII
MIIII
MIIIIU
MUIU
MUIUUIU
MUIIU
Axiom
1, Rule II
2, Rule II
3, Rule I
4, Rule III
5, Rule II
6, Rule IV
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.
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.
A former student asked (about this sentence) "Then what's the point?" The point emerges in the section on
"Interpretation of Formal Systems," below.
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
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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 interpretation, ---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 interpretation, 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 elements). 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 elements 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.
2.
Write out all the permutations of four elements a, b, c and d. How many permutations are there?
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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10 POSTULATIONAL MATHEMATICS
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
2
3
4
5
6
7
8
9
1
2
6
120
720
5040
40320
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
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.
149
◦
A
B
C
D
E
F
A
A
B
C
D
E
F
B
B
A
D
C
F
E
C
C
E
A
F
B
D
D
D
F
B
E
A
C
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 concentrate 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
E
E
C
F
A
D
B
F
F
D
E
B
C
A
150
2.
3.
10 POSTULATIONAL MATHEMATICS
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.
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.
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.
◦
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
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An operation is commutative if and only if X◦Y = Y◦X for every possible choice of X and Y.
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PROPERTIES OF A POSTULATIONAL SYSTEM
151
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.
2.
3.
4.
5.
6.
7.
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.
Every ordered pair of objects (e.g., A and F in that order are an ordered pair) in the set
produces exactly one "product."
Every "product" is a member of the original set of objects.
Every member of the set of objects appears exactly once in each row and each column.
There is a column that looks exactly like the guide column.
There is a row that looks exactly like the guide row.
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.
2.
10
11
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.
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
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?
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
3.
4.
10 POSTULATIONAL MATHEMATICS
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.
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 corresponds 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
13
I've called it ◦, so that we don't confuse it with any of the usual symbols for arithmetic operators.
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.
154
5.
4.
10 POSTULATIONAL MATHEMATICS
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?
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 “handson” 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 numbers 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