• I •
1. Constants and variables
Every scientific theory is a system of sentences which are accepted as true and which may be called LAWS or ASSERTED STATEMENTS or, for short, simply STATEMENTS. In mathematics, these statements follow one another in a definite order according to certain principles which will be discussed in detail in Chapter VI, and they are, as a rule, accompanied by considerations intended to establish their validity. Considerations of this kind are referred to as PROOFS, and the statements established by them are called THEOREMS.
Among the terms and symbols occurring in mathematical theorems and proofs we distinguish CONSTANTS and VARIABLES.
In arithmetic, for instance, we encounter such constants as “number”, “zero” (“0”), “one” (“1“), “sum” (“+”), and many others.1 Each of these terms has a well-determined meaning which remains unchanged throughout the course of the considerations.
As variables we employ, as a rule, single letters, e.g. in arithmetic the small letters of the English alphabet: “a”, “b”, “c”, …, “x”, “y”, “z”. As opposed to the constants, the variables do not possess any meaning by themselves, Thus, the question:
does zero have such and such a property?
e.g.:
is zero an integer?
can be answered in the affirmative or in the negative; the answer may be true or false, but at any rate it is meaningful. A question concerning x, on the other hand, for example the question:
is x an integer?
cannot be answered meaningfully.
In some textbooks of elementary mathematics, particularly the less recent ones, one does occasionally come across formulations which convey the impression that it is possible to attribute an independent meaning to variables. Thus it is said that the symbols “x”, “y”, … also denote certain numbers or quantities, not “constant numbers” however (which are denoted by constants like “0”, “1”, … ), but the so-called “variable numbers” or rather “variable quantities”. Statements of this kind have their source in a gross misunderstanding. The “variable number” x could not possibly have any specified property, for instance, it could be neither positive nor negative nor equal to zero; or rather, the properties of such a number would change from case to case, that is to say, the number would sometimes be positive, sometimes negative, and sometimes equal to zero. But entities of such a kind we do not find in our world at all; their existence would contradict the fundamental laws of thought. The classification of the symbols into constants and variables, therefore, does not have any analogue in the form of a similar classification of the numbers.
2. Expressions containing variables—sentential and designatory functions
In view of the fact that variables do not have a meaning by themselves, such phrases as:
x is an integer
are not sentences, although they have the grammatical form of sentences; they do not express a definite assertion and can be neither confirmed nor refuted. From the expression:
x is an integer
we only obtain a sentence when we replace “x” in it by a constant denoting a definite number; thus, for instance, if “x” is replaced by the symbol “1”, the result is a true sentence, whereas a false sentence arises on replacing “x” by “”. An expression of this kind, which contains variables and, on replacement of these variables by constants, becomes a sentence, is called a SENTENTIAL FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term “function” with a different meaning. More often the word “CONDITION” is employed in this sense; and sentential functions and sentences which are composed entirely of mathematical symbols (and not of words of everyday language), such as:
x + y = 5,
are usually referred to by mathematicians as FORMULAS. In place of “sentential function” we shall sometimes simply say “sentence”—but only in cases where there is no danger of any mis-understanding.
The role of the variables in a sentential function has sometimes been compared very adequately with that of the blanks left in a questionnaire; just as the questionnaire acquires a definite content only after the blanks have been filled in, a sentential function becomes a sentence only after constants have been inserted in place of the variables. The result of the replacement of the variables in a sentential function by constants—equal constants taking the place of equal variables—may lead to a true sentence; in that case, the things denoted by those constants are said to SATISFY the given sentential function. For example, the numbers 1, 2 and 2 satisfy the sentential function:
x < 3,
but the numbers 3, 4 and 4 do not.
Besides the sentential functions there are some further expressions containing variables that merit our attention, namely, the so-called DESIGNATORY or DESCRIPTIVE FUNCTIONS. They are expressions which, on replacement of the variables by constants, turn into designations (“descriptions”) of things. For example, the expression:
2x + 1
is a designatory function, because we obtain the designation of a certain number (e.g., the number 5), if in it we replace the variable “x” by an arbitrary numerical constant, that is, by a constant denoting a number (e.g., “2”).
Among the designatory functions occurring in arithmetic, we have, in particular, all the so-called algebraic expressions which are composed of variables, numerical constants and symbols of the four fundamental arithmetical operations, such as:
Algebraic equations, on the other hand, that is to say, formulas consisting of two algebraic expressions connected by the symbol “=”, are sentential functions. As far as equations are concerned, a special terminology has become customary in mathematics; thus, the variables occurring in an equation are referred to as the unknowns, and the numbers satisfying the equation are called the roots of the equation. E.g., in the equation:
x2 + 6 = 5x
the variable “x” is the unknown, while the numbers 2 and 3 are roots of the equation.
Of the variables “x”, “y”, … employed in arithmetic it is said that they STAND FOR DESIGNATIONS OF NUMBERS or that numbers are VALUES of these variables. Thereby approximately the following is meant: a sentential function containing the symbols “x”, “y”, … becomes a sentence, if these symbols are replaced by such constants as designate numbers (and not by expressions designating operations on numbers, relations between numbers or even things outside the field of arithmetic like geometrical configurations, animals, plants, etc.). Likewise, the variables occurring in geometry stand for designations of points and geometrical figures. The designatory functions which we meet in arithmetic can also be said to stand for designations of numbers. Sometimes it is simply said that the symbols “x”, “y”, … themselves, as well as the designatory functions made up out of them, denote numbers or are designations of numbers, but this is then a merely abbreviative terminology.
3. Formation of sentences by means of variables—universal and existential sentences
Apart from the replacement of variables by constants there is still another way in which sentences can be obtained from sentential functions. Let us consider the formula:
x + y = y + x.
It is a sentential function containing the two variables “x” and “y” that is satisfied by any arbitrary pair of numbers; if we put any numerical constants in place of “x” and “y”, we always obtain a true formula. We express this fact briefly in the following manner:
for any numbers x and y, x + y = y + x.
The expression just obtained is already a genuine sentence and, moreover, a true sentence; we recognize in it one of the fundamental laws of arithmetic, the so-called commutative law of addition. The most important theorems of mathematics are formulated similarly, namely, all so-called UNIVERSAL SENTENCES, or SENTENCES OF A UNIVERSAL CHARACTER, which assert that arbitrary things of a certain category (e.g., in the case of arithmetic, arbitrary numbers) have such and such a property. It has to be noticed that in the formulation of universal sentences the phrase “for any things (or numbers) x, y, …” is often omitted and has to be inserted mentally; thus, for instance, the commutative law of addition may simply be given in the following form:
x + y = y + x.
This has become a well accepted usage, to which we shall generally adhere in the course of our further considerations.
Let us now consider the sentential function:
x > y + 1.
This formula fails to be satisfied by every pair of numbers; if, for instance, “3” is put in place of “x” and “4” in place of “y”, the false sentence:
3 > 4 + 1
is obtained. Therefore, if one says:
for any numbers x and y, x > y + 1,
one does undoubtedly state a meaningful, though obviously false, sentence. There are, on the other hand, pairs of numbers which satisfy the sentential function under consideration; if, for example, “x” and “y” are replaced by “4” and “2”, respectively, the result is the true formula:
4 > 2 + 1.
This situation is expressed briefly by the following phrase:
for some numbers x and y, x > y + 1,
or, using a more frequently employed form:
there are numbers x and y such that x > y + 1.
The expressions just given are true sentences; they are examples of EXISTENTIAL SENTENCES, or SENTENCES OF AN EXISTENTIAL CHARACTER, stating the existence of things (e.g., numbers) with a certain property.
With the help of the methods just described we can obtain sentences from any given sentential function; but it depends on the content of the sentential function whether we arrive at a true or a false sentence. The following example may serve as a further illustration. The formula:
x = x + 1
is satisfied by no number; hence, no matter whether the words “for any number x” or “there is a number x such that” are prefixed, the resulting sentence will be false.
In contradistinction to sentences of a universal or existential character we may denote sentences not containing any variables, such as:
3 + 2 = 2 + 3,
as SINGULAR SENTENCES. This classification is not at all exhaustive, since there are many sentences which cannot be counted among any of the three categories mentioned. An example is represented by the following statement:
for any numbers x and y there is a number z such that
x = y + z.
Sentences of this type are sometimes called CONDITIONALLY EXISTENTIAL SENTENCES (as opposed to the existential sentences considered before, which may also be called ABSOLUTELY EXISTENTIAL SENTENCES); they state the existence of numbers having a certain property, but on condition that certain other numbers exist.
4. Universal and existential quantifiers; free and bound variables
Phrases like:
for any x, y, …
and
there are x, y, … such that
are called QUANTIFIERS; the former is said to be a UNIVERSAL, the latter an EXISTENTIAL QUANTIFIER. Quantifiers are also known as OPERATORS; there are, however, expressions counted likewise among operators, which are different from quantifiers. In the preceding section we tried to explain the meaning of both quantifiers. In order to emphasize their significance it may be pointed out that, only by the explicit or implicit employment of operators, can an expression containing variables occur as a sentence, that is, as the statement of a well-determined assertion. Without the help of operators, the usage of variables in the formulation of mathematical theorems would be excluded.
In everyday language it is not customary (though quite possible) to use variables, and quantifiers are also, for this reason, not in use. There are, however, certain words in general usage which exhibit a very close connection with quantifiers, namely, such words as “every”, “all”, “a certain”, “some”. The connection becomes obvious when we observe that expressions like:
all men are mortal
or
some men are wise
have about the same meaning as the following sentences, formulated with the help of quantifiers:
for any x, if x is a man, then x is mortal
and
there is an x, such that x is both a man and wise,
respectively.
For the sake of brevity, the quantifiers are sometimes replaced by symbolic expressions. We can, for instance, agree to write in place of:
for any things (or numbers) x, y, …
and
there exist things (or numbers) x, y, … such that
the following symbolic expressions:
respectively (with the understanding that the sentential functions following the quantifiers are put in parentheses). According to this agreement, the statement which was given at the end of the preceding section as an example of a conditionally existential sentence, for instance, assumes the following form:
A sentential function in which the variables “x”, “y”, “z”, … occur automatically becomes a sentence as soon as one prefixes to it one or several operators containing all those variables. If, however, some of the variables do not occur in the operators, the expression in question remains a sentential function, without becoming a sentence. For example, the formula:
x = y + z
changes into a sentence if preceded by one of the phrases:
for any numbers x, y and z;
there are numbers x, y and z such that;
for any numbers x and y, there is a number z such that;
and so on. But if we merely prefix the quantifier:
there is a number z such that or
we do not yet arrive at a sentence; the expression obtained, namely:
is, however, undoubtedly a sentential function, for it immediately becomes a sentence when we substitute some constants in the place of “x” and “y” and leave “z” unaltered, or else, when we prefix another suitable quantifier, e.g.:
for any numbers x and y or
It is seen from this that, among the variables which may occur in a sentential function, two different kinds can be distinguished. The occurrence of variables of the first kind—they will be called FREE or REAL VARIABLES—is the decisive factor in determining that the expression under consideration is a sentential function and not a sentence; in order to effect the change from a sentential function to a sentence it is necessary to replace these variables by constants or else to put operators in front of the sentential function that contain those free variables. The remaining, so-called BOUND or APPARENT VARIABLES, however, are not to be changed in such a transformation. In the above sentential function (II), for instance, “x” and “y” are free variables, and the symbol “z” occurs twice as a bound variable; on the other hand, the expression (I) is a sentence, and thus contains bound variables only.
*It depends entirely upon the structure of the sentential function, namely, upon the presence and position of the operators, whether any particular variable occurring in it is free or bound. This may be best seen by means of a concrete example. Let us, for instance, consider the following sentential function:
(III) for any number x, if x = 0 or y ≠ 0, then
there exists a number z such that x = y · z.
This function begins with a universal quantifier containing the variable “x”, and therefore the latter, which occurs three times in this function, occurs at all these places as a bound variable; at the first place it makes up part of the quantifier, while at the other two places it is, as we say, BOUND BY THE QUANTIFIER. The situation is similar with respect to the variable “z”. For, although the initial quantifier of (III) does not contain this variable, we can, nevertheless, recognize a certain sentential function forming a part of (III) which opens with an existential quantifier containing the variable “z”; this is the function:
(IV) there exists a number z such that x = y · z.
Both places at which the variable “z” occurs in (III) belong to the partial function (IV) just stated. It is for this reason that we say that “z” occurs everywhere in (III) as a bound variable; at the first place it makes up part of the existential quantifier, and at the second place it is bound by that quantifier. As for the variable “y” also occurring in (III), we see that there is no quantifier in (III) containing this variable, and therefore it occurs in (III) twice as a free variable.
The fact that quantifiers bind variables—that is, that they change free into bound variables in the sentential functions which follow them—constitutes a very essential property of quantifiers. Several other expressions are known which have an analogous property; with some of them we shall become acquainted later (in Sections 20 and 22), while some others—such as, for instance, the integral sign—play an important role in higher mathematics. The term “operator” is the general term used to denote all expressions having this property.*
5. The importance of variables in mathematics
As we have seen in Section 3 variables play a leading role in the formulation of mathematical theorems. From what has been said it does not follow, however, that it would be impossible in principle to formulate the latter without the use of variables. But in practice it would scarcely be feasible to do without them, since even comparatively simple sentences would assume a complicated and obscure form. As an illustration let us consider the following theorem of arithmetic:
for any numbers x and y, x3 − y3 = (x − y) · (x2 + xy + y2).
Without the use of variables, this theorem would look as follows:
the difference of the third powers of any two numbers is equal to the product of the difference of these numbers and a sum of three terms, the first of which is the square of the first number, the second the product of the two numbers, and the third the square of the second number.
An even more essential significance, from the standpoint of the economy of thought, attaches to variables as far as mathematical proofs are concerned. This fact will be readily confirmed by the reader if he attempts to eliminate the variables in any of the proofs which he will meet in the course of our further considerations. And it should be pointed out that these proofs are much simpler than the average considerations to be found in the various fields of higher mathematics; attempts at carrying the latter through without the help of variables would meet with very considerable difficulties. It may be added that it is to the introduction of variables that we are indebted for the development of so fertile a method for the solution of mathematical problems as the method of equations. Without exaggeration it can be said that the invention of variables constitutes a turning point in the history of mathematics; with these symbols man acquired a tool that prepared the way for the tremendous development of the mathematical science and for the solidification of its logical foundations.2
Exercises
1. Which among the following expressions are sentential functions, and which are designatory functions:
(a) x is divisible by 3,
(b) the sum of the numbers x and 2,
(c) y2 − z2,
(d) y2 = z2,
(e) x + 2 < y + 3,
(f) (x + 3) − (y + 5),
(g) the mother of x and z,
(h) x is the mother of z ?
2. Give examples of sentential and designatory functions from the field of geometry.
3. The sentential functions which are encountered in arithmetic and which contain only one variable (which may, however, occur at several different places in the given sentential function) can be divided into three categories: (i) functions satisfied by every number; (ii) functions not satisfied by any number; (iii) functions satisfied by some numbers, and not satisfied by others.
To which of these categories do the following sentential functions belong:
(a) x + 2 = 5 + x,
(b) x2 = 49,
(c) (y + 2) · (y − 2) < y2,
(d) y + 24 > 36,
(e) z = 0 or z < 0 or z > 0,
(f) z + 24 > z + 36 ?
4. Give examples of universal, absolutely existential and conditionally existential theorems from the fields of arithmetic and geometry.
5. By writing quantifiers containing the variables “x” and “y” in front of the sentential function:
x > y
it is possible to obtain various sentences from it, for instance:
for any numbers x and y, x > y;
for any number x, there exists a number y such that x > y;
there is a number y such that, for any number x, x > y.
Formulate them all (there are six altogether) and determine which of them are true.
6. Do the same as in Exercise 5 for the following sentential functions:
x + y2 > 1
and
x is the father of y
(assuming that the variables “x” and “y” in the latter stand for names of human beings).
7. State a sentence of everyday language that has the same meaning as:
for every x, if x is a dog, then x has a good sense of smell
and that contains no quantifier or variables.
8. Replace the sentence:
some snakes are poisonous
by one which has the same meaning but is formulated with the help of quantifiers and variables.
9. Differentiate, in the following expressions, between the free and bound variables:
(a) x is divisible by y;
(b) for any x, x − y = x + (−y),
(c) if x < y, then there is a number z such that x < y and y < z;
(d) for any number y, if y > 0, then there is a number z such that x = y·z;
(e) if x = y2 and y > 0, then, for any number z, x > −z2;
(f) if there exists a number y such that x > y2, then, for any number z, x > −z2.
Formulate the above expressions by replacing the quantifiers by the symbols introduced in Section 4.
*10. If, in the sentential function (e) of the preceding exercise, we replace the variable “z” in both places by “y”, we obtain an expression in which “y” occurs in some places as a free and in others as a bound variable; in what places and why?
(In view of some difficulties in operating with expressions in which the same variable occurs both bound and free, some logicians prefer to avoid the use of such expressions altogether and not to treat them as sentential functions.)
*11. Try to state quite generally under which conditions a variable occurs at a certain place of a given sentential function as a free or as a bound variable.
12. Which numbers satisfy the sentential function:
there is a number y such that x = y2,
and which satisfy:
there is a number y such that x · y = 1 ?
1 By “arithmetic” we shall here understand that part of mathematics which is concerned with the investigation of the general properties of numbers, relations between numbers and operations on numbers. In place of the word “arithmetic” the term “algebra” is frequently used, particularly in high-school mathematics. We have given preference to the term “arithmetic” because, in higher mathematics, the term “algebra” is reserved for the much more special theory of algebraic equations. (In recent years the term “algebra” has obtained a wider meaning, which is, however, still different from that of “arithmetic”.)—The term “number” will here always be used with that meaning which is normally attached to the term “real number” in mathematics; that is to say, it will cover integers and fractions, rational and irrational, positive and negative numbers, but not imaginary or complex numbers.
2 Variables were already used in ancient times by Greek mathematicians and logicians,—though only in special circumstances and in rare cases. At the beginning of the 17th century, mainly under the influence of the work of the French mathematician F. VIETA (1540–1603), people began to work systematically with variables and to employ them consistently in mathematical considerations. Only at the end of the 19th century, however, due to the introduction of the notion of a quantifier, was the role of variables in scientific language and especially in the formulation of mathematical theorems fully recognized; this was largely the merit of the outstanding American logician and philosopher CH. S. PEIRCE (1839–1914).