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Appendix A Reformulation of Example 9
APPENDIX
A REFORMULATION OF EXAMPLE 9 The system of Example 9 formalizes in a rough way what we do in elementary algebra. Indeed the identities arrived at by removing parentheses, multiplying, long division with remainder, factoring, etc. are elementary propositions in that system. Naturally one can pass on to fractions, etc. by making appropriate changes. Thus the system is a counter example for the contention that the notion of formal system is confined to abstruse logical matters; likewise for the idea that logic must be presupposed before one can operate a system of elementary character. To be sure the system does not reproduce with complete adequacy the actual processes of elementary instruction; but there is enough similarity to show that elementary algebra is essentially a study of formal systems which are neither more complicated than, nor dependent on, those of logic itself. However the reasoning in elementary algebra involves the usc of inequalities as well as equalities. Thus the passage from to
a.c
=
b.c
a=b
requires the additional hypothesis c =F 0; and the neglect of this proviso leads to serious fallacies 1. 1 For a case where a fallacy of this general nature occurs in an otherwise reputable book, see G. W. Pierce, Electric Oscillations and Electric Waves. 1922, pp. 57 -58 § 59 "Condition that makes the transient term zero". Since a sine wave and its derivative can never vanish simultaneously it follows that the transient term, in the case considered by Pierce, cannot be zero under any circumstances. The fallacy arises from equating to zero the numerator of a certain fraction and forgetting that then the denominator also vanishes. (This fallacy was also noted independently by D. C. Bourgin when we were graduate students together).
71
APPENDIX
Now of course it can be shown that the system of Example 9 is resolvable, and therefore negation can be constructively defined in the meta-theory. But not only is this foreign to elementary alge bra, but it is thoroughly unsatisfactory from other points of view. Accordingly we consider here a system in which inequalities can also be derived as elementary propositions. This system involves some other changes in Example 9; if we omit everything related to inequality we may regard this system as a revision of that example. The constituents of the primitive frame as grouped in a manner suggested by Chapter VII; this is more in accord with tradition, and I now regard it as more significant than the grouping of Chapter IV. Constituents in parentheses are variants, some of which will be discussed in the comment following the formulation. I. PRIMITIVE IDEAS
A. Primitive Terms: 0, 1, x (other variables, i). B. Operations. Binary operations on a and b: a Unary operations on a: - a, a-I. C.
+ b,
a.b (or ob).
Predicates. 1. Morphological categories: (V); I; C; P. 2. Theoretical binary predicates on a and b:
a
=
b; a =J:. b.
II. POSTULATES or Primitive Propositions. A. Rules of Formation. (0. Variables - x is in V.) 1. Integers - I.
a. 1 is in I.
b. If a is in I, then a + 1 is in I. (c. If a is in I and a = b, then b is in I).
72
APPENDIX
2. Constants -
a. b. (c. d. e. f.
C.
°If isa inis inC. I, then a is in C.
If a and b are in C, then a + b is in C). If a and b are in C, then a. b is in C. If a is in C, then - a is in C. If a is in I, then a-I is in C. (f" If a is in C and a =f:. 0, then a-I is in C). (g. If a is in C and a = b, then b is in C). 3. Polynomials - P. a. If a is in C, then a is in P. b. x is in P (or if a is in V, then a is in P). c. If a and b are in P, then a + b is in P. d. If a and b are in P, then a.b is in P. e. If a is in P, then - a is in P.
D. Axiom Schemes. 1. For all a, b, c in P:
a. a + b = b + a. b. a + (b + c) = (a + b) + c. c. a + 0= a. d. a + (- a) = O. e. c.b = b .c. f. a. (b.«) = (a. b). c. g. a.l = a. h. a.(b + c) = (a.b) + (o .c).
2. For all a in I: a. a.a-1 = 1. b. a =f:. o. c. a-I =f:. o. C. Rules
0/ Procedure.
1. For all a, b, c in P:
a' If a = b, then b = a. b. If a = band b = c, then a = c.
73
APPENDIX
c. If a = b, then a + c = b + c. d. If a = b then c .c = b.c. (e. If a.b = and a then b
°
* 0,
=
2. For all a, b, c in P: a. If a b, then b a. b. If a band b = c, then a c. c. If a then a + c b + c. (d. If a. c b. c, then a b). e. If a*O and b then
* * * b, *
*
*0
*
* *
u.b *0.
3. For all a, b in 0: (a. If a then a.a- l = (b. If a then a-I
1).
4. For a in P, b in 0: a. If a then ax + b
* 0.
* 0, *° * 0,
* 0).
0).
We now consider some comments on this system. In these comments references to the primitive frame are alway to II, and therefore 'II' is unexpressed. 1. Without assuming any of the postulates in parentheses it can be shown that the system is adequate for the representation of polynomials in one indeterminate with rational coefficients. Furthermore, it is a constructive metatheorem that for any two such polynomials a and b, one and only one of the alternatives a=b holds. Thus these two alternatives are the negations of one another in the usual sense. It is a corollary of this that, when everything involving inequality is left out, the system is resolvable, and the metatheoretic inequality in the truncated system coincides with that formulated here.
*
b' is regarded as the negation of 'a = b', then the 2. If 'a rules 02 follow from the corresponding rules 01, and vice versa, by the ordinary logic of negation. This is the point of view of elementary algebra. For this and the metatheorem of the preceding
74
APPENDIX
paragraph it follows that the rules in parentheses (in Cl and 2) are superfluous. Nevertheless, it seems desirable to state them in order to give an adequate account of inequality on the elementary level. We shall see below that if B2 and C3b are dropped, we need C2d 1. It is evident that Cle is precisely the type of inference mentioned in the introduction to this Appendix. 3. The parenthesized postulates in Al and A2 do not affect the adequacy of the system for representing polynomials; but they do affect the content of the classes I and C. Thus if Alc is omitted, I is the class of terms 1, 1 + I, (1 + 1) + 1, ... ; but such terms as (1 + 1) + (- 1), 1 + (1 + 1), (1 + 1) . (1 + ltl , all of which are equal to elements I, do not belong to I; furthermore, if A2gc, are also denied. the first two of there terms do not belong to C. If A2f' is denied, then the inverse a-I is not defined except for a E I. This is an unnatural state of affairs. The only objection to these postulates is that, while morphological, they contain a theoretical premise. But there is no reason for insisting that morphological postulates must be pure. It was pointed out in Chapter VII that the ultimate in formalization is the completely formalized system; the whole distinction between morphology and theoretics is nothing but a concession to practical convenience. As a matter of fact, little has to be done to the system to completely formalize it, irrespective of whether these postulates are assumed or not. The only changes are matters of terminology - Ala, A2a, and A3b become axioms, all the rest become rules of procedure. 4. If we admit the postulates Ale, A2f' (in place of A2f) and A2g, then A2c becomes proveable and B2ac can be replaced by C3ab. Of the latter C3b becomes a consequence of C2d (put a-I 1 Note that if we did not wish to exhibit the parallelism between CI and C2 it would be more convenient to state C2d in the form
If a. b =1= 0, then a =1= 0 Likewise Cle is equivalent to the rule. If a. c = b. c and c
"* O. then a = b.
APPENDIX
75
for c, n for c, and b = 0). (The postulates C2d is then no longer superfluous. ) 5. The postulate B2b corresponds to the algebraic requirement that the characteristic be zero. For a modular field we must replace B2 by something else. We must also drop A2f, but we can use A2f'. 6. The postulate C4 states that x is an indeterminate in the sense of modern algebra. This was done in Example 9 in order to have a connection with Example 5. To represent elementary algebra one should the replace C4 by a substitution rule. In such a system inequalities c =F 0 would only be derivable for constants, and the metatheorem in negation would only apply to equations n =F 0 where a is a constant (i.e., in C or equal to a member of C). 7. It is possible to modify the system in various ways so as
to deal with several variables, algebraic (as opposed to transcend-
ental) extensions, etc. This can be done by modifications of C4. In the case of transcendental extensions with several variables we need to formulate the (morphological) predicate 'x does not occur in a'. This will not be gone into further.
A REFORMULATION OF EXAMPLE 9 The system of Example 9 formalizes in a rough way what we do in elementary algebra. Indeed the identities arrived at by removing parentheses, multiplying, long division with remainder, factoring, etc. are elementary propositions in that system. Naturally one can pass on to fractions, etc. by making appropriate changes. Thus the system is a counter example for the contention that the notion of formal system is confined to abstruse logical matters; likewise for the idea that logic must be presupposed before one can operate a system of elementary character. To be sure the system does not reproduce with complete adequacy the actual processes of elementary instruction; but there is enough similarity to show that elementary algebra is essentially a study of formal systems which are neither more complicated than, nor dependent on, those of logic itself. However the reasoning in elementary algebra involves the usc of inequalities as well as equalities. Thus the passage from to
a.c
=
b.c
a=b
requires the additional hypothesis c =F 0; and the neglect of this proviso leads to serious fallacies 1. 1 For a case where a fallacy of this general nature occurs in an otherwise reputable book, see G. W. Pierce, Electric Oscillations and Electric Waves. 1922, pp. 57 -58 § 59 "Condition that makes the transient term zero". Since a sine wave and its derivative can never vanish simultaneously it follows that the transient term, in the case considered by Pierce, cannot be zero under any circumstances. The fallacy arises from equating to zero the numerator of a certain fraction and forgetting that then the denominator also vanishes. (This fallacy was also noted independently by D. C. Bourgin when we were graduate students together).
71
APPENDIX
Now of course it can be shown that the system of Example 9 is resolvable, and therefore negation can be constructively defined in the meta-theory. But not only is this foreign to elementary alge bra, but it is thoroughly unsatisfactory from other points of view. Accordingly we consider here a system in which inequalities can also be derived as elementary propositions. This system involves some other changes in Example 9; if we omit everything related to inequality we may regard this system as a revision of that example. The constituents of the primitive frame as grouped in a manner suggested by Chapter VII; this is more in accord with tradition, and I now regard it as more significant than the grouping of Chapter IV. Constituents in parentheses are variants, some of which will be discussed in the comment following the formulation. I. PRIMITIVE IDEAS
A. Primitive Terms: 0, 1, x (other variables, i). B. Operations. Binary operations on a and b: a Unary operations on a: - a, a-I. C.
+ b,
a.b (or ob).
Predicates. 1. Morphological categories: (V); I; C; P. 2. Theoretical binary predicates on a and b:
a
=
b; a =J:. b.
II. POSTULATES or Primitive Propositions. A. Rules of Formation. (0. Variables - x is in V.) 1. Integers - I.
a. 1 is in I.
b. If a is in I, then a + 1 is in I. (c. If a is in I and a = b, then b is in I).
72
APPENDIX
2. Constants -
a. b. (c. d. e. f.
C.
°If isa inis inC. I, then a is in C.
If a and b are in C, then a + b is in C). If a and b are in C, then a. b is in C. If a is in C, then - a is in C. If a is in I, then a-I is in C. (f" If a is in C and a =f:. 0, then a-I is in C). (g. If a is in C and a = b, then b is in C). 3. Polynomials - P. a. If a is in C, then a is in P. b. x is in P (or if a is in V, then a is in P). c. If a and b are in P, then a + b is in P. d. If a and b are in P, then a.b is in P. e. If a is in P, then - a is in P.
D. Axiom Schemes. 1. For all a, b, c in P:
a. a + b = b + a. b. a + (b + c) = (a + b) + c. c. a + 0= a. d. a + (- a) = O. e. c.b = b .c. f. a. (b.«) = (a. b). c. g. a.l = a. h. a.(b + c) = (a.b) + (o .c).
2. For all a in I: a. a.a-1 = 1. b. a =f:. o. c. a-I =f:. o. C. Rules
0/ Procedure.
1. For all a, b, c in P:
a' If a = b, then b = a. b. If a = band b = c, then a = c.
73
APPENDIX
c. If a = b, then a + c = b + c. d. If a = b then c .c = b.c. (e. If a.b = and a then b
°
* 0,
=
2. For all a, b, c in P: a. If a b, then b a. b. If a band b = c, then a c. c. If a then a + c b + c. (d. If a. c b. c, then a b). e. If a*O and b then
* * * b, *
*
*0
*
* *
u.b *0.
3. For all a, b in 0: (a. If a then a.a- l = (b. If a then a-I
1).
4. For a in P, b in 0: a. If a then ax + b
* 0.
* 0, *° * 0,
* 0).
0).
We now consider some comments on this system. In these comments references to the primitive frame are alway to II, and therefore 'II' is unexpressed. 1. Without assuming any of the postulates in parentheses it can be shown that the system is adequate for the representation of polynomials in one indeterminate with rational coefficients. Furthermore, it is a constructive metatheorem that for any two such polynomials a and b, one and only one of the alternatives a=b holds. Thus these two alternatives are the negations of one another in the usual sense. It is a corollary of this that, when everything involving inequality is left out, the system is resolvable, and the metatheoretic inequality in the truncated system coincides with that formulated here.
*
b' is regarded as the negation of 'a = b', then the 2. If 'a rules 02 follow from the corresponding rules 01, and vice versa, by the ordinary logic of negation. This is the point of view of elementary algebra. For this and the metatheorem of the preceding
74
APPENDIX
paragraph it follows that the rules in parentheses (in Cl and 2) are superfluous. Nevertheless, it seems desirable to state them in order to give an adequate account of inequality on the elementary level. We shall see below that if B2 and C3b are dropped, we need C2d 1. It is evident that Cle is precisely the type of inference mentioned in the introduction to this Appendix. 3. The parenthesized postulates in Al and A2 do not affect the adequacy of the system for representing polynomials; but they do affect the content of the classes I and C. Thus if Alc is omitted, I is the class of terms 1, 1 + I, (1 + 1) + 1, ... ; but such terms as (1 + 1) + (- 1), 1 + (1 + 1), (1 + 1) . (1 + ltl , all of which are equal to elements I, do not belong to I; furthermore, if A2gc, are also denied. the first two of there terms do not belong to C. If A2f' is denied, then the inverse a-I is not defined except for a E I. This is an unnatural state of affairs. The only objection to these postulates is that, while morphological, they contain a theoretical premise. But there is no reason for insisting that morphological postulates must be pure. It was pointed out in Chapter VII that the ultimate in formalization is the completely formalized system; the whole distinction between morphology and theoretics is nothing but a concession to practical convenience. As a matter of fact, little has to be done to the system to completely formalize it, irrespective of whether these postulates are assumed or not. The only changes are matters of terminology - Ala, A2a, and A3b become axioms, all the rest become rules of procedure. 4. If we admit the postulates Ale, A2f' (in place of A2f) and A2g, then A2c becomes proveable and B2ac can be replaced by C3ab. Of the latter C3b becomes a consequence of C2d (put a-I 1 Note that if we did not wish to exhibit the parallelism between CI and C2 it would be more convenient to state C2d in the form
If a. b =1= 0, then a =1= 0 Likewise Cle is equivalent to the rule. If a. c = b. c and c
"* O. then a = b.
APPENDIX
75
for c, n for c, and b = 0). (The postulates C2d is then no longer superfluous. ) 5. The postulate B2b corresponds to the algebraic requirement that the characteristic be zero. For a modular field we must replace B2 by something else. We must also drop A2f, but we can use A2f'. 6. The postulate C4 states that x is an indeterminate in the sense of modern algebra. This was done in Example 9 in order to have a connection with Example 5. To represent elementary algebra one should the replace C4 by a substitution rule. In such a system inequalities c =F 0 would only be derivable for constants, and the metatheorem in negation would only apply to equations n =F 0 where a is a constant (i.e., in C or equal to a member of C). 7. It is possible to modify the system in various ways so as
to deal with several variables, algebraic (as opposed to transcend-
ental) extensions, etc. This can be done by modifications of C4. In the case of transcendental extensions with several variables we need to formulate the (morphological) predicate 'x does not occur in a'. This will not be gone into further.