Symbols — the Meat of Mathematics

Let us take a gentle look at algebra now. (The present article is derived from a similar article by Tobias Dantzig). It is an expository article only.

Algebra, in  the broad sense in which the term is used today, deals with operations upon symbolic forms. In this capacity, it not only permeates all of mathematics, but encroaches upon the domain of formal logic and even of metaphysics. Furthermore, when so construed, algebra is as old as man’s faculty to deal with general propositions; as old as his ability to discriminate between “some” and “any”.

Here, however, we are interested in algebra in a much more restricted sense, that part of general algebra which is very properly called the theory of equations. It is in this narrower sense that the term algebra was used at the outset. The word is of Arabic origin. Al is the Arabic title “the”, and gebar is the verb “to set”, to  restitute. To this day the word algebrista is used in Spain to  designate a bone-setter, a sort of chiropractor.

It is generally true that algebra in its development in individual countries passed successively through three stages: the rhetorical, the syncopated, and the symbolic. Rhetorical algebra is characterized by the complete absence of any symbols, except, of course, that the words themselves are being used in their symbolic sense. To this day, rhetorical algebra is used in such a statement as “the sum is independent of the order of the terms”, which in symbols would be designated by a+b=b+a.

Syncopated algebra, of which the Egyptian is a typical example, is a further development of rhetorical. Certain words of frequent use are gradually abbreviated. Eventually, these abbreviations become contracted to the point where their origin has been forgotten, so that the symbols have no obvious connection with the operation which they represent. The syncopation has become a symbol.

The history of the symbols + and – may illustrate the point. In mediaval Europe, the latter was long denoted by the full word “minus”, then by the first letter “m” duly superscribed. Eventually the letter itself was dropped, leaving the superscript only. The sign “plus” passed through a similar metamorphosis.

The turning point in the history of algebra was an essay written late in the sixteenth century by a Frenchman, Viete, who  wrote under the Latin name Franciscus Vieta. His great achievement appears, simple enough to us today. It is summed up in  the following passage from this work:

In this we are aided by an artifice which permits us to distinguish given magnitudes from those which are unknown or sought, and this by means of a symbolism which is permanent in nature and clear to understand — for instance, by denoting the unknown magnitudes by A or any other vowels, while the given magnitudes are designated by B,C, G or other consonants.

This vowel-consonant notation had a short existence. Within half a century of Vieta’s death appeared Descartes’s Geometrie, in which the first letters of the alphabet were used for given quantities, the last for  those unknown. The Cartesian notation not only displaced the Vietan, but has survived to this day.

But, while few of Vieta’s proposals were carried out in letter, they certainly were adopted in spirit. The systematic use of letters for undetermined but constant magnitudes, the “logistica speciosa” as he called it, which has played such a dominant role in the development of mathematics, was the great achievement of Vieta.

The lay mind may find it difficult to estimate the achievement of Vieta at its true value. Is not the literal notation a mere formality after all, a convenient shorthand at best? There is, no doubt, economy in writing


but does it really convey more to the mind than the verbal form of the same identity: the square of the sum of two numbers equals the sum of the squares of the numbers, augmented by twice their product?

Again, the literal notation had the fate of all very successful innovations. The universal of these makes it difficult to conceive of a time when inferior methods were in vogue. Today formulae in which letters represent general magnitudes are almost as familiar as common script, and our ability to handle symbols is regarded by many almost as a natural endowment of any intelligent man; but it is natural only because it has become a fixed habit of our minds. In the days of Vieta this notation constituted a radical departure from the tradition of ages.

Wherein lies the power of this symbolism?

First of all, the letter liberated algebra from the slavery of the word. And, by this, I do not mean merely that without the literal notation any general statement would become a mere flow of verbiage, subject to all the ambiguities and misinterpretations of human speech. This is important enough; but, what is still more important is that the letter is free from the taboos which have attached to words through centuries of use. The A of Vieta or our present “x” has an existence independent of the concrete object which it is assumed to represent. The symbol has a meaning which transcends the objects symbolized: that is why it is not a mere formality.

In the second place, the letter is susceptible of operations which enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is the power of transformations that lifts algebra above the level of a convenient shorthand.

Before the introduction of literal notation, it was possible to speak of individual expressions only; each expression, such as 2x+3, 3x-5;

x^{2}+4x+7; 3x^{2}-4x+5, had an individuality all its own and had to be handled on its own merits. The literal notation made it possible to pass from the individual to the collective, from the “some” to the “any” and the “all”. The linear form ax+b, the quadratic form ax^{2}+bx+c, each of these forms is regarded now as a single species. It is this that made possible the general theory of functions, which is the basis of all applied mathematics.

But, the most important contribution of the logistica speciosa, and the one that concerns us most in this study, is the role it played in the formation of the generalized number concept.

As long as one deals with numerical equations, such as

x+4=0; 2x=8 and x^{2}=9, call this equation I

x+0=4; 2x=5; x^{2}=7, call this equation II

one can content himself (as most mediavel algebraists did) with the statement that the first group of equations is possible, while the second is impossible.

But, when one considers literal equations of  the same types:

x+b=a; bx=a; x^{n}=a

the very indeterminateness of the data compels one to give an indicated or symbolic solution to the problem:

x=a-b; x=a/b; x= (a)^{1/n}.

In vain, after this, will one stipulate that the expression a-b has a meaning only if a is greater than b, that a/b is meaningless when a is not a multiple of b, and that a^{1/n} is not a number unless a is a perfect nth power. The very act of writing down the meaningless has given it meaning; and, it is not easy to deny the existence of something that has received a name.

Moreover, with the reservation that a>b, that a is a multiple of b, that a is perfect nth power, rules are devised for operating on such symbols as a-b; a/b; a^{1/n}. But, sooner or later the very fact that there is nothing on the face of these symbols to indicate whether a legitimate or an illegitimate case is before us, will suggest that there is no contradiction involved in operating on  these symbolic beings as if they bona fide numbers. And from this there is but one step to recognizing these symbolic beings as numbers “in extenso”.

What distinguishes modern arithmetic from that of the pre-Vieta period is the changed attitude towards the “impossible”. Up to the seventh century the algebraists invested this term with an absolute sense. Committed to natural numbers as the exclusive field for all arithmetic operations, they regarded possibility, or restricted possiblity, as an intrinsic property of these operations.

Thus, the direct operations of arithmetic — addition (a+b), multiplication (ab), potentiation a^{n} — were omni-possible; whereas, the inverse operations — subtraction (a-b), division a/b, extraction or roots a^{1/n} — were possible only under restricted conditions. The pre-Vieta algebraists were satisfied with stating these facts, but were incapable of a closer analysis of the problem.

Thus, the direct operations of arithmetic are omnipossible because they are but a succession of iterations, a step-by-step penetration into the sequence of natural numbers, which is assumed a priori unlimited. Drop this assumption, restrict the field of the operand to a finite collection (say to the first 1000 numbers), and operations such as 925+125 or 67 x 15 become impossible and the corresponding operations meaningless.

Or, let us assume that the field is restricted to odd numbers only. Multiplication is still omni-possible, for the product of any two odd numbers is odd. However, in such a restricted field addition is an altogether impossible operation, because the sum of any two odd numbers is never an odd number.

Yet, again, if the field were restricted to prime numbers, multiplication would be impossible, for the simple reason that the product of two primes is never a prime; while, addition would be possible only in such rare cases as when one of the two terms is 2, the other being the smaller of a couple of twin-primes like


Other examples could be adduced, but even these few will suffice to bring out the relative nature of  the words possible, impossible, and meaningless. And, once this relativity is recognized, it is natural to inquire whether through a proper extension of  the restricted field the inverse operations of arithmetic may not be rendered as omni-possible as the direct are.

To accomplish this with respect to subtraction it is sufficient to adjoin to  the sequence of natural numbers zero and the negative integers. The field so created is called the general integer field.

Similarly, the adjunction of positive and negative fractions to this integer field will render division omni-possible.

The numbers thus created — the integers, and the fractions, positive and negative, and the number zero — constitute the rational domain. It supersedes the natural domain of integer arithmetic. The four fundamental operations, which heretofore applied to integers only, are now by analogy extended to these generalized numbers.

All this can be accomplished without a contradiction. And, what is more, with a single reservation which we shall take up presently, the sum, the difference, the product, and the quotient of  any two rational numbers are themselves rational numbers. This very important fact is often paraphrased into the statement: the rational domain is closed with respect to the fundamental operations of arithmetic.

The single but very important reservation is that of division by zero. This is equivalent to the solution of the equation x.0=a. If a is not zero, the equation is impossible, because we are compelled, in defining the number zero, to admit the identity a.0=0. There exists therefore no rational number which satisfies the equation x.0=a.

On the contrary, the equation x.0=a is satisfied for any rational value of x. Consequently, x is here an indeterminate quantity. Unless the problem that led to such equations provides some further information, we must regard 0/0 as the symbol of *any* rational number, and a/0 as the symbol of *no* rational number.

Elaborate though these considerations may seem, in symbols they reduce to the following succinct statement: if a, b and c are any rational numbers, and *a* is not zero, then there always exists a rational number x, and only one, which will satisfy the equation ax+b=c

This equation is called “linear”, and it is the simplest type in a great variety of  equations. Next to linear some quadratic, then cubic, quartic, quintic and generally algebraic equations of any degree, the degree n meaning the highest power of  the unknown x in ax^{n}+bx^{n-1}+cx^{n-2}+...+px+q=0

But even these do not exhaust the infinite variety of equations; exponential, trigonometric, logarithmic, circular, elliptic, etc., constitute a still vaster variety, usually classified under the all-embracing term transcendental.

Is the rational domain adequate to handle this infinite variety? This is emphatically not the case. We must anticipate an extension of  the number domain to greater and greater complexity. But this extension is not arbitrary; there is concealed in  the very mechanism of the generalizing scheme a guiding and unifying idea.

This idea is sometimes called the principle of permanence. It was first explicity formulated by the German mathematician, Hermann Hanckel, in 1867, but the germ of the idea was already contained in the writings of Sir William Rowan Hamilton, one of the most original and fruitful minds of the nineteenth century.

I shall formulate this principle as a definition:

A collection of symbols infinite in number shall be called a number field, and each individual element in it a number,

First. If among the  elements of the collection we can identify the sequence of natural numbers.

Second. If we can establish criteria of rank which will permit us to tell of any two elements whether they are equal, or if not equal, which is greater; these criteria reducing to the natural criteria when the two elements are natural numbers.

Third. If for any two elements of  the collection we can devise a scheme of addition and multiplication which will have the commutative, associative, and distributive properties of the natural operations bearing these names, and which will reduce to these natural operations when the two elements are natural numbers.

These very general considerations leave the question open as to how  the principle of permanence operates in special cases. Hamilton pointed the way by a method which he called algebraic pairing. We shall illustrate this on the natural numbers.

If a is a multiple of b, then the symbol a/b indicates the operation of division of a by b. Thus 9/3=3 means that the quotient of the indicated division is 3. Now, given two such indicated operations, is  there a way of  determining whether the results are equal, greater, or less, without actually performing the operations? Yes, we have the following:

Criteria of Rank. a/b=c/d if ad=bc

a/b > c/d if ad>bc

a/b < c/d if ad<bc

And we can even go further than that: without  performing the indicated operations we can devise rules for manipulating on these indicated quantities:

Addition: (a/b)+(c/d) = (ad+bc)/bd

Multiplication. (a/b).(c/d)= (ac)/(bd)

Now, let us not stipulate any more that a be a multiple of b. Let us consider a/b as the symbol of a new field of mathematical beings. These symbolic beings depend on two integers a and b written in proper order. We shall impose on this collection  of  couples the criteria of  rank mentioned above,i.e., we shall claim that, for instance:

(20/15)=(16/12) because  20 x 12 = 15 x 16

(4/3)>(5/4) because (4)( 4) >( 3)(5)

We shall define  the operations on these couples in accordance with the rules which, as we have shown above, are true for the case when a is a multiple of b, and c is a multiple of d, i.e., we shall for instance:

(2/3)+(4/5)=((2)( 5)+(3) ( 4))/((5)( 3))=22/15

We have now satisfied all the stipulations of the principle of  permanence.

1. The new field contains the natural numbers as a subfield, because we can write any natural number in the form of a couple:

1/1; 2/1; 3/1; 4/1, and so on and on.

2. The  new field criteria of  rank which reduce to  the natural criteria when a/b and c/d are natural numbers.

3. The new field has been provided with two  operations which have all the properties of addition and multiplication, to which they reduce  when a/b and c/d are natural numbers.

And, so these new beings satisfy all the stipulations of the principle. They have proved their right to be adjoined to the natural numbers, their right to be invested with the dignity of the same name “number”. They are therewith admitted, and the field of numbers comprising both old and new is christened the rational domain of numbers.

It would seem at first glance that  the principle of permanence leaves such a latitude in the choice of operations as to make the general number it postulates too general to be of much practical value. However, the stipulations that the natural sequence should be a part of  the field, and that  the fundamental operations should be commutative, associative and distributive (as the natural operations are), impose restrictions which, as we shall see, only very special fields can meet.

The position of arithmetic,as formulated in the principle of permanence, can be compared to the policy of a state bent on expansion, but desirous to  perpetuate the fundamental laws on which it grew strong. These two different objectives — expansion on the one hand, preservation of uniformity on  the other — will naturally influence the rules for admission of new states to  the Union.

Thus, the first point in the principle of  permanence corresponds to the pronouncement that the nucleus state shall set the tone of the Union. Next, the original state being an oligarchy in which every citizen has a rank, it imposes this requirement on the new states. This requirement corresponds to the second point of  the principle of superposition.

Finally, it stipulates that the laws of commingling between the citizens of each individual state admitted to the Union shall be of  a type which will permit unimpeded relations between citizens of that state and those of the nucleus state.

Of course, I do not want the reader to take this analogy literally. It is suggested in the hope  that it may invoke mental associations from a more familiar field, so that the principle of permanence may lose its seeming artificiality.

The considerations, which led up to the construction of the rational domain, were the first steps in a historical process called the arithmetization of  mathematics. This movement, which began with Weierstrass in the sixties of the 19th century, had for its object the separation of purely mathematical concepts, such as “number” and “correspondence” and “aggregate”, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.

These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of  the most careful circumspection in the choice of words, the meaning concealed behind these words may influence our reasoning. For  the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure forms of thought.

But, how can we avoid the use of human language? The answer is found in the word “symbol”. Only by using a symbolic language not  yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason —- only thus may we hope to build mathematics on the solid foundation of  logic.

Such is the platform of  this school, a school which was founded by the Italian Peano and whose most modern representatives were Bertrand Russell and Alfred North Whitehead. In the fundamental work of the latter men, the Principia Mathematica, they had endeavoured to reconstruct the whole foundation of modern mathematics, starting with clear-cut, fundamental assumptions and proceeding on principles of  logic.

I confess that I  am out  of sympathy with the extreme formalism of  the Peano-Russell school, that I have never acquired the taste for their methods of symbolic  logic, that my repeated efforts to master their involved symbolism have invariably resulted in helpless confusion and despair. This personal ineptitude has undoubtedly coloured my opinion — a powerful reason why I should not air my prejudices here.

Yet I  am certain that these prejudices have not caused me to underestimate the role of mathematical symbolism. To me, the tremendous importance of this symbolism lies not in these sterile attempts to banish intuition from the realm of  human thought, but in its unlimited power to aid intuition in creating  new forms of thought.

To recognize this, it is not necessary to master the intricate technical symbolism of modern mathematics. It is sufficient to contemplate the  more simple, yet much more subtle, symbolism of language. For, in so far as our language is capable of precise statements, it is but a systems of  symbols, a rhetorical algebra par excellence. Nouns and phrases are but symbols of classes of objects, verbs symbolize relations, and sentences are but propositions connecting these classes. Yet, while the word is the abstract symbol of a class, it has also the capacity to  invoke an image, a concrete picture of some representative element of the class. It is in this dual function of our language that we should seek the germs of the conflict which later arises between logic and intuition.

And what is true of words generally is particularly true of  those words which represent natural numbers. Because they have the  power to evoke in our minds images of concrete collections, they appear to us so rooted in firm reality as to be endowed with an absolute nature. Yet in the sense in which they are used in arithmetic, they are but a set of abstract symbols subject to a system of operational rules.

Once we recognize this symbolic nature of  the natural number, it loses its absolute character. Its intrinsic kinship with the wider domain of which it is the  nucleus becomes evident. At the same time, the successive extensions of the number concept become steps in an inevitable process of natural evolution, instead of the artificial and arbitrary legerdemain which they seem at first.

More later…






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