Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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VII. | RELATIVITY OF STANDARDS OFMATHEMATICAL RIGOR |

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Dictionary of the History of Ideas | ||

#### RELATIVITY OF STANDARDS OF

MATHEMATICAL RIGOR

From a broad standpoint, rigor in any field of en-

deavor and particularly in scientific fields, means the

adherence to procedures that have been generally

accepted as leading to correct conclusions. Thus the

statement, “It has been rigorously established that

Norsemen reached the shores of North America before

Columbus,” can be taken to mean that documentary,

archaeological, or other evidence has been produced

which conforms to the standards of acceptance set by

the group to which the statement is addressed. Such

a group might be a society of professional historians,

in which case the term “rigorously established” implies

that the evidence offered as a basis for the assertion

conforms to the standards set by professional historians;

for example, the evidence might be documentary ma

terials whose validity is considered acceptable by such

a group. If addressed to a group of anthropologists,

the evidence might consist of archaeological materials

whose authenticity meets the standards recognized as

acceptable by this group.

Moreover, it is possible that the statement is accept-

able to a group of historians, but not to professional

anthropologists; or even to one group of historians and

not to another. An example of the latter kind could

concern the validity of a certain alleged miracle, which

might be established quite rigorously according to the

standards of a group of church historians, but not to

those of a lay historical group.

Furthermore, rigor is not just a function of the group

involved, but of time. Standards of rigor notoriously

change with the passage of time. What would have

been considered rigorous by scientists of the year 1800

would certainly not meet the standards set by the

professional scientists of 1900. On the other hand,

standards of rigor do not necessarily become more

stringent with time, since cultures rise and fall, and

standards set by one culture may be forgotten and have

to be recreated or replaced by succeeding cultures. The

classical case of this kind may be found in connection

with the decay of the Hellenic culture and the gradual

ascendency of its successors.

It may be expected, too, that standards of rigor will

sometimes become the subject of profound discussion

amongst members of a group concerned. Examples of

this kind are frequently brought to public attention

in connection with the marketing of new drugs, where

the standards of rigor governing pretesting are fre-

quently bitterly debated between manufacturer's

chemists and those of government agencies. It is prob-

ably not generally realized that similar instances occur

even in mathematics, a field popularly known as the

“most exact” of the sciences; and in which no motives

of a pecuniary nature becloud the issues as is often

the case when commercial interests are involved. A

classic story in mathematical circles relates that one

of the contemporaries of David Hilbert, late professor

of mathematics at the University of Göttingen, Ger-

many, exclaimed upon reading a short and elegant

proof that Hilbert had given, “This is theology, not

mathematics!”—indicating an opinion that the proof

did not conform to accepted mathematical standards.

And this same Hilbert, who was one of the leading

mathematicians of the first third of the twentieth cen-

tury, became engaged in a prolonged debate with the

famous Dutch mathematician, L. E. J. Brouwer, over

what constitutes rigorous methods of proof in mathe-

matics (see below). Such debates are not of rare occur-

rence, and have occurred frequently throughout the

history of mathematics.

The development of the concept of rigor in mathe-

matics provides a most instructive and revealing story,

which can be told without becoming involved in eso-

teric technicalities and which has meaningful parallels

in other fields of learning. As one of the oldest sciences,

and especially one in which the concept of rigor has

achieved mature formulations, mathematics has tradi-

tionally been most concerned with standards of rigor;

and the stages through which mathematical rigor has

passed, with attention to cultural influences (internal

and external), give a superb example of the evolution

of a concept (rigor), which in spite of the paucity of

ancient documents, can be observed virtually from its

inception to the present.

Presumably such a concept as rigor was at first only

intuitive, not a consciously realized ideal. The Sume-

rian-Babylonian mathematics was the earliest for which

historical records have been found, although it was not

a separate “discipline” such as it became in the later

European cultures. In it a number of mathematical

formulas and procedures which later became standard

were developed, as well as a system of numerals almost

as sophisticated as our present-day decimal system.

Methods for solving algebraic equations had also been

developed along with a number of geometric formulas.

Most surprising among the latter was the famous “Py-

thagorean theorem,” relating the square on the hypot-

enuse of a right triangle to the squares on the other

two sides—traditionally attributed to the Pythagorean

school which flourished over a millennium later in the

Greek culture. Such materials presumably imply the

development of some kind of standards according to

which these algebraic and geometric ideas became

admissible for those uses (usually commercial) to which

they were put. The nature of these standards is un-

known, but there is no evidence as yet available that

they were as advanced as the methods that developed

in the later Hellenic civilization. They were probably

of an intuitive, traditional nature, although they could

also have embraced certain pragmatic and diagram-

matic tests. For example, if an ancient Sumerian “Ein-

stein” were faced with a problem involving the deter-

mination of the quantity of material needed to erect

a certain structure, he might have found a formula for

the purpose. Then presumably this formula would not

have gained acceptance by his contemporaries without

his first convincing them in some way of its validity.

It can be surmised that this would have been accom-

plished merely by showing that the method

“worked”—that is, that it gave the desired amount of

material, or a reasonable approximation thereto. Or,

if the problem were of a geometric character, demon-

stration of the validity of the formula might have been

accomplished by certain visual methods consisting of

counting pebble arrangements, or of geometric pat-

terns displaying “obvious” properties. This is con-

jecture of course; but since the earliest methods used

by their Greek successors consisted of just such tests

for validity, and since there were cultural contacts

between the later Babylonians and the early Greeks,

it seems a not improbable hypothesis (Neugebauer

[1957], Ch. II).

The course of Greek mathematics, thanks to the

extant traces of the unusual intellectual atmosphere in

which it developed, is somewhat less conjectural. Spe-

cifically, its development within a philosophical milieu

influential in both the Greek and succeeding cultures

resulted in the preservation of more important written

records. Moreover, the circumstances of its evolution

contain suggestions of the manner in which cultural

influences, both environmental and intrinsic, promoted

its development toward increased rigor. This first be-

comes noticeable in the Pythagorean school of the sixth

and fifth centuries B.C. The geographical location of

this school was Croton, in the southeastern section of

Italy. In nearby Elea, the Eleatic school of philosophy

was centered, and one of its foremost exponents,

Parmenides, was apparently associated for a time with

the Pythagorean school of mathematics. Usually, when

two cultural entities meet and mingle, diffusion of ideas

from each to the other occurs. In this case, the cultural

entities were the Pythagorean school of mathematics

and the Eleatic system of philosophy. The cosmological

system conceived by Parmenides was evidently influ-

enced by Pythagorean points of view; on the other

hand, the Pythagoreans could have become acquainted

with the dialectic of the Eleatics, one of whose features

was indirect argument (Szabó, 1964).

If such was the case, we have here one of the earliest

examples of concepts external to mathematics com-

bining with intrinsic mathematical needs to produce

a method promoting greater rigor of proof. Up to this

time, Pythagorean methods of proof had not advanced

much further than the primitive diagrammatic methods

termed “visual.” By arranging objects, such as pebbles,

in simple geometrical arrays, a number of elementary

formulas had been discovered by direct observation.

In other instances, the use of superposition—moving

one geometric configuration into coincidence with

another—was employed. Some have conjectured that

the first proofs of the Pythagorean Theorem were

accomplished in this way. While such methods were

well adapted to the discovery of simple arithmetic and

geometric facts, they were not as conclusive as the

deductive methods which came into use and which

were possibly influenced by adaptation of the Eleatic

dialectic to mathematical proofs. The previous primi-

tive methods could never have sufficed to prove certain

visual methods, as, for example, the existence of in-

commensurable line segments; i.e., line segments for

which there exists no common unit of measurement,

such as the side and diagonal of a square. We can

conjecture that the Pythagoreans began to suspect the

existence of incommensurable segments and realized

the inadequacy of their traditional proof methods. If

such was the case, there was thereby set up an intrinsic

motivation to find a more rigorous type of reasoning.

It appears likely that the proof of incommensura-

bility of the side of a square with its diagonal was one

of the earliest, if not the earliest, to make appeal to

the dialectic method. For this proof, it was necessary

to show that there cannot exist any unit of length, no

matter how small, that will exactly measure both the

side of a square and its diagonal. A geometric fact of

this kind cannot be handled by visual methods, since

the stipulation “no matter how small” places it beyond

the range of human perception. However, if the as-

sumption that by using some sufficiently small unit of

length, both the side and diagonal of a square can be

exactly measured, can be shown to lead to contra-

diction, then one may conclude that such a unit of

length cannot possibly exist; i.e., that the side and

diagonal are incommensurable. (Later, the basis for this

type of argument was formulated by Aristotle in the

Law of Contradiction: “Contradiction is impossible”

or more explicitly, “No proposition can be both true

and false”; and the Law of the Excluded Middle:

“Every proposition is either true or false.” Thus, the

proposition that there exists a common unit of measure

for the side and diagonal of a square is either true or

false; and since its truth is untenable, having been

shown to imply contradiction, it must be false.)

Such arguments are called “indirect” forms of

proof—later called “reductio ad absurdum.” They

produce a conviction not attainable by visual argu-

ments, which are always open to the objection that

they cover only particular cases and may be the result

of illusory perceptions. Consequently they soon became

standard in Greek mathematics, not to be matched in

quality of rigor by visual methods. Indeed, it soon

became the rule that no longer was a mathematical

formula or method to be accepted because it always

seemed to work in particular cases (in Plato's dialogues,

Socrates frequently rejects a definition of a concept

like justice by enumeration of particular cases falling

under it, and demands an essential or universal prop-

erty); it must be proved by a logical argument such

as that of the indirect type. Rigorous proof came to

be synonymous with proof by logic.

Although not all historians agree on the details of

the above interpretation of the available historical

literature, the evidence strongly implies that as a result

of a need internal to mathematics combined with the

existence of a philosophical dialectic in the culture

external to mathematics, greater rigor was achieved

in Greek mathematics. Moreover, this was possibly not

the only case in which mathematical rigor was in-

debted to influences external to mathematics. For ex-

ample, Zeno of Elea, a pupil of Parmenides, had been

led by his work on the extension of his master's philos-

ophy to a series of paradoxes which were ultimately

recognized to be of fundamental importance to mathe-

matics, in that they raised questions concerning the

continuous character of the straight line. For instance,

if the line is made up of points, and a point has no

length, then how can a line have length? Zeno also

argued that motion in a straight line would be impossi-

ble since an object could never get from one point

to another. Again, historians differ in their opinions

regarding whether Zeno's work was influential or not

in the development of ancient mathematical thought,

but it may have been in the effort to get around such

difficulties that mathematicians came to realize that

the vague intuitive conceptions on which geometry had

been based must be replaced by an explicit set of

assumptions which embodied the intuitive “facts” on

which proofs could be based. The fourth-century (B.C.)

mathematician Eudoxus was most prominently iden-

tified with this accomplishment, and it is generally

agreed that a considerable part of Euclid's *Elements*

stems directly from Eudoxus' work. In the *Elements*

the basic assumptions are called “axioms” and “postu-

lates,” and the proofs display the mature form of which

the indirect method was the first example. These

proofs, ultimately called proofs by logical deduction,

demonstrate that by “taking thought” alone, one can

establish the validity of an assertion covering infinitely

many special cases. Another type of reasoning, impor-

tant heuristically (as a method of discovery), by “anal-

ysis,” used the device of first assuming the truth of the

assertion to be proved in order to ascertain its conse-

quences; if these consequences consisted of basic as-

sumptions (axioms or postulates) or previously proved

assertions, then it was sometimes possible to reverse

the process by showing that the consequences had the

desired assertion as one of their consequences.

The Greek philosopher Aristotle made a noteworthy

study of logical deduction, arriving at general frame-

works for the methods involved which were applicable

to all fields of study, not just to geometry. He proposed

a general definition of a demonstrative science which

became a model for centuries of later scientific work.

According to this definition, a demonstrative science

should consist of a collection of basic assumptions, and

of the theorems which these assumptions imply (Beth,

various forms of logical deduction set forth in Aris-

totle's study of argumentative methods.

So far as rigor is concerned, little further significant

progress was made until the nineteenth century, when

a combination of circumstances, bearing a curious

resemblance to those which seem to have brought

about an increase in rigor during the Greek era came

into play. These circumstances developed in the fol-

lowing manner.

During the period which followed the Greek decline,

mathematics underwent an extensive development and

evolution in both symbolic and conceptual content. In

arithmetic, the remarkable number system of the

Sumerian-Babylonian culture evolved essentially into

the decimal system used today. Although the numerals

used by the Babylonians were cumbersome (due, per-

haps, to the necessity of having to adapt them to the

use of the stylus and baked clay media), their place

value system in which the “value” of a single digit

depended on its position (“place”) within the numeral

was the same as that used in the decimal system. (It

lacked a true zero, but this was clearly evolving by

the end of the Babylonian era.) However, the symbolic

algebra which we now use was a product of the later

European cultures. And (in the seventeenth century)

it was the imposition of this algebra on the geometry

bequeathed by the Greeks which resulted in analytic

geometry, and enabled Newton and Leibniz to crystal-

lize their ideas on the calculus. Although Newton and

Leibniz are popularly credited with creating the cal-

culus, what they essentially did was to synthesize, in

symbolic form, concepts that had been developed by

a host of predecessors going back to the Greeks (Boyer,

1949; Rosenthal, 1951; Bochner, 1966). This achieve-

ment was a breakthrough whose motivation lay at

least as much in the search for a medium in which to

express natural laws as in the desire to bolster the

purely symbolic aspects of mathematics: in short, in a

combination of cultural and intrinsic mathematical

stresses.

However, the success of the symbolic machinery set

up by Newton and Leibniz was so great that it went

beyond the conceptual background; symbols and oper-

ations with them were created for which no one could

give a satisfactory meaning, although results achieved

with them generally justified their invention. They

passed the pragmatic test but flunked the conceptual.

As a result, that vaunted rigor for which mathematics

had been praised from the time of the Greeks was now

lacking, and there ensued a field day for philosophical

critics (such as the renowned Bishop Berkeley, who

called Newton's infinitesimals “the ghosts of departed

quantities”), not to mention the uncomfortable feeling

that the mathematical defenders of the new calculus

could not escape.

Actually, this lack of conceptual justification was not

a new phenomenon in mathematics in those areas

where the conditions laid down by Aristotle for a

demonstrative science had not been met. Consider the

ordinary arithmetic of the integers, for example; no

satisfactory conceptual background had ever been fur-

nished for it. But this caused little concern and there

is little evidence that anyone was aware of the lack

until quite recent times. True, some qualms were expe-

rienced by the introduction of negative numbers, which

for centuries had been toyed with but rejected as “ficti-

tious,” even after their use became common in the

seventeenth century. The conceptual basis for the

nonnegative integers was purely intuitive, but they had

been in use for untold centuries and had achieved

cultural acceptability—that is, as meeting the demands

of the rigor of the day. But the extension to negative

numbers was purely formal—a symbolic achievement

embodying such operational rules as the laws of signs,

but otherwise having no conceptual justification.

Moreover, no axiomatic basis satisfying Aristotle's con-

ditions was given for them until the late nineteenth

and early twentieth centuries (Landau, 1951).

A similar situation prevailed concerning complex

numbers of the form *a + bi* (where *i* stands for the

“imaginary” √-1) encountered in elementary prob-

lems such as the solution of quadratic and cubic equa-

tions (*a* and *b* being “real” numbers). These numbers

and arithmetical operations with them were success-

fully carried out for several centuries, although a satis-

factory conceptual background was not provided until

the twentieth century. Intuitive bases of a geometric

nature did develop for them much earlier, but by that

time geometry was coming to be no longer accepted

as a basis for numerical theories.

Thus the introduction of a new symbolic apparatus

like the calculus should “logically” not have caused

such concern, so long as it passed the pragmatic

test—which it certainly did. Of course it lacked the

long traditional background possessed by the natural

numbers, but this was also true, possibly to a lesser

extent, in the case of the negative integers and the

complex numbers. However, an idea had become

prominent which, although not strictly new in mathe-

matics, had nevertheless not caused much concern

since Eudoxus devised his theory of proportion. This

was the concept of the infinite. It intruded into all the

basic conceptions offered as an explanation of the new

calculus, and occurred in two opposing forms; the

“infinitely small” and the “infinitely great.” Attempts

at clarifying the basic concepts of the calculus, such

as that of the derivative of a function, made appeal

quite unconvincing (even, one sometimes suspects, to

those who devised them). And although the axiomatic

method of the Greeks enjoyed quite a vogue at the

time (Leibniz had used it in arguments of a political

and military nature), notably in social and philo-

sophical theories (as, for instance, by Spinoza), there

seems to have been little effort to use it as a means

for giving the calculus a firm basis.

Although appeal to the axiomatic method had to

await the latter part of the nineteenth century, certain

notable contributions to the rigorous development of

the calculus were made earlier. Chief among these was

that of A. Cauchy, whose *Cours d'analyse*... (1821)

gave the basic ideas of the calculus a quite rigorous

treatment, making no appeal to such vague notions as

“little zeroes.” In other fields of mathematics, the

realization was growing that the axiomatic method

offered an acceptable path to greater rigor. This was

helped by the accompanying realization that the num-

ber systems which had achieved mathematical accept-

ance either through tradition or by special needs were

not the only ones that could be devised. Similarly, the

geometry of Euclid was not the only type of geometry

that provided a consistent description of physical

space. The result of such considerations was the incep-

tion of new algebras and geometries, all rigorously

defined by means of the axiomatic method in the

Aristotelian tradition. Although these developments

had many consequences, the one of greatest importance

for present purposes was the casting into prominence

of the problem of consistency. How could one be

assured that all these algebraic and geometric systems,

frequently mutually incompatible (as, e.g., Euclidean

and non-Euclidean geometries), were within them-

selves consistent systems? For certainly if a mathe-

matical system harbored contradiction, then it could

not have been rigorously developed. In this way, rigor

and consistency began to be associated; that which is

rigorously structured ought not to be inconsistent, and

systems that turn out to be consistent must ipso facto

be the result of rigorous formulation.

In contrast, the calculus was still based on fuzzy

notions of a number system which, in addition to the

ordinary integers and fractions, contained irrationals

such as √2, π , etc. This number system, known techni-

cally as the real number system, had grown as new

accretions were needed. With the introduction of ana-

lytic geometry, it had been given a more satisfactory

intuitive background through association with the

straight line. By selecting an arbitrary point *P* on some

fixed line *L* as a representative of zero, the points to

one side of *P* were associated with the positive real

numbers in increasing order, and those to the other

side with negative numbers, each negative number

being the same distance from *P* as its positive counter-

part on the other side of *P* (Figure 1). It became intui-

tively evident that to each point of *L* corresponded

a unique real number in this manner, and that in

problems of the calculus appeal could be made to this

linear structure, considered as equivalent to the system

of real numbers. Proofs were given which made use

of this geometric concept and it gradually became clear

that the amount of geometric intuition employed in

the proofs of theorems of the calculus was exceeding

the limits imposed by new standards of rigor. This was

made all the more evident by the fact that many of

the geometric facts used to substantiate numerical

statements were frequently the same “facts” that had

seemed so evident to the Greeks that they had never

been adequately established in geometry. In short, they

had no firm basis either numerically or geometrically.

This unsatisfactory state of affairs became all the

more pronounced as the calculus gradually grew into

what is now termed classical analysis, which embodied

not only the advanced ideas owing to the successors

of Leibniz and Newton, but also a theory whose foun-

dation was the system of complex numbers geometri-

cally represented by the points of a Euclidean plane.

This growth of analysis was not just an internal evolu-

tion, influenced only by mathematical considerations,

but was in large measure due to the needs of physical

theories (Bochner, 1966). Of great importance was the

work of a French mathematical physicist Baron Joseph

Fourier (1768-1830), who was not a professional

“pure” mathematician—but was one who, in the opin-

ion of one historian (Bell [1945], p. 292), “had almost

a contempt for mathematics except as a drudge of the

sciences.” Being quite uninhibited by such qualms as

would have (and did—see Bell [1937], pp. 197-98)

beset a pure mathematician, he proceeded to set up

mathematical tools whose chief virtue was apparently

that they were suited to the needs of such studies as

the theory of heat. In particular they involved infinite

processes which had little rigorous foundation and

which stretched to its limits that geometric intuition

upon which mathematical analysts were wont to rely.

As so often happens, mathematicians found them-

selves confronted, much as in the case of the basic

notions of the calculus (which had by now been essen-

tially cleared up by Cauchy), with new symbolic and

operational apparatuses which could not be ignored.

It was not just that they seemed to prove their worth

only compensating feature, they might well have been

left to the whimsies of physics—but they rapidly

offered ways in which to treat purely mathematical

problems as well as suggestions for new concepts or

expansions of already existing concepts (such as that

of function). And to accept them meant, again, to find

a rigorous foundation for them.

Thus the growth of mathematical analysis brought

mathematics to face much the same types of problems

as had confronted the ancient Greeks and which were

solved by such innovations as Eudoxus' theory of pro-

portion. In particular, it was necessary to replace the

largely intuitive conception of the structure of the real

number system by a precisely formulated axiomatic

system which would serve as a satisfactory base upon

which to found analysis. Such a foundation would, one

hoped, not only settle once and for all whether the

types of series, and functions related thereto, “worked”

in applications because of accidental circumstances or

whether they could be shown, by logical deduction

from the new foundation, to be mathematically sound.

The solution of the latter problem was found, as

anyone familiar with the way in which mathematics

evolved would expect, by several independent investi-

gators (Meray, Dedekind, Weierstrass, G. Cantor).

Although their solutions were not precisely the same,

they turned out to be equivalent (in the sense that each

could be derived from the others). And one now re-

joiced in the feeling that the one apparently remaining

insecure part of mathematics had been given a secure

foundation; and mathematics could resume its course

presumably assured of having once again achieved a

rigor safe from all criticism.

But this feeling of security was not to last long. As

usually happens when mathematics makes a great ad-

vance, new insights are achieved regarding concepts

which had long been taken for granted. A mathe-

matician of ancient Greece, for instance, knew per-

fectly well that a line joining a point exterior to a circle

to a point interior to the circle would have to intersect

the periphery of the circle; it was self-evident, and

needed no justification. Nevertheless, the time arrived

(during the nineteenth century) when it was forced

upon one that justification really was necessary if the

demands of modern rigor were to be met. Similarly,

one had no qualms in speaking of a “collection” of

numbers or geometric entities; for instance, no one

would object to speaking of the collection—the term

“set” is more in vogue today—of numbers that were

solutions of an algebraic equation. Correspondingly,

one might speak of the set of all even numbers, or

the set of all odd numbers. True, the latter sets each

contain infinitely many numbers, whereas the number

of solutions of an algebraic equation is finite. Never-

theless, the use of the words “set” and “collection”

was felt to be the same as their use in the physical

world. To speak of a collection of people or a set of

chairs is an ordinary usage of the natural language.

And although mathematics had become increasingly

symbolized over the centuries, employment of the

natural language (as in the statements of the axioms

and theorems of Euclidean geometry) continued to be

acceptable.

However, this apparently innocent use of the notion

of a collection turned out to be another case of a

concept borrowed from the general culture and put

to use in mathematics in ways never before dreamed

of. Not only was it used to define such a basic notion

as number (theretofore taken for granted, but whose

extension to numbers for infinite sets plainly demanded

definition), but it lay at the heart of the foundation

of mathematical analysis. It was inevitable that a study

of the concept for its own sake would become neces-

sary, and this was finally undertaken by the German

mathematician Georg Cantor during the latter half of

the nineteenth century. Symptomatic of the lack of

interest or concern generally felt by mathematicians

of his time, however, was the fact that most of Cantor's

contemporaries at first considered his researches as

neither mathematically justified nor even “good”

mathematics. Some of his colleagues considered that

Cantor was transgressing the bounds of what could be

called “mathematics.” Fortunately Cantor persisted in

his researches, and not only did they lead to a full-

blown theory of great inherent interest, but its appli-

cations to such problems as those bequeathed by Four-

ier proved unexpectedly fruitful. By the end of the

century, his ideas were coming to be generally ac-

cepted, and the Theory of Sets was well on the way

to becoming an established mathematical discipline.

About the same time, the German mathematician

and logician G. Frege was turning his attention to the

problem of furnishing a rigorous foundation for the

arithmetic of integers. He was convinced that all

mathematics could be derived from logic and thus

rendered free of all criticism regarding its lack of rigor.

In showing this, he did not hesitate to use the notion

of set, which he apparently felt to be itself rooted in

logic. From a somewhat different point of view, both

Dedekind and the Italian logician Peano (ca. 1890)

gave an axiomatic foundation for the system of natural

numbers from which, again using set theory, the real

number system could be derived.

As a result of these researches, the mathematical

world came to consider, around the turn of the century,

that mathematics had at last been placed on a rigorous

foundation, and that all criticism of the foundations

feeling were the words of the renowned French math-

ematician, Henri Poincaré, in an address at the Inter-

national Congress of Mathematicians of 1900: “We

believe that we no longer appeal to intuition in our

reasoning.... Now, in analysis today, if we take the

pains to be rigorous, there are only syllogisms or ap-

peals to the intuition of pure number that could possi-

bly deceive us. It may be said that today absolute rigor

is attained” (Bell [1945], Ch. 13; also see Poincaré

[1946], pp. 210-22).

It is doubtful if Poincaré could have been aware,

at the time he uttered these words, that contradiction

had already been discovered in the theory of sets

(communication between various national mathe-

matical groups was rather poor at that time). In the

unrestricted use of set-theoretic methods in the realm

of the infinite, contradiction had been, and was being

found.

The earliest attempt to meet the situation was to

call again upon the axiomatic method for help. The

first set of axioms for the theory of sets was given in

1908 by the German mathematician Ernst Zermelo.

Thus the apparently innocent notion of set, universally

used in common discourse, and having come into

mathematics because of the use of the natural language,

became the central concept of a mature mathematical

theory, deserving of axiomatic foundation in the same

way that geometry had been axiomatized by the

Greeks. And much as the Greeks succeeded in avoiding

the difficulties posed by the discovery of incommensu-

rable magnitudes, so did the axiom system of Zermelo

promise to avoid the contradictions to which the un-

restricted notion of set had led. Unfortunately there

was no guarantee that it would suffice to avoid all

possible contradictions; that is, there appeared no way

of proving Zermelo's system consistent, even though

the axioms in themselves seemed to restrict the theory

of sets sufficiently to avoid contradiction. One could

no longer assert, consequently, that mathematics had

attained that absolute rigor which Poincaré had cited.

Concurrently with the axiomatization of the theory

of sets, other approaches were made to the problem

of giving mathematics a rigorous foundation, and for

a time three distinct “schools of thought” emerged

(Wilder [1965], Chs. 8-11). One of these, associated

with the name of Bertrand Russell, but actually pre-

sented in the monumental *Principia Mathematica*

(1910-13) of A. N. Whitehead and B. Russell, followed

a path based conceptually on Frege's ideas and sym-

bolically upon Peano's work. The central thesis of the

Whitehead-Russell doctrine was again that mathe-

matics could be founded on logic. But it developed

that in order to build a secure theory of number, free

from contradiction, axioms had to be introduced which

had not only never been part of classical logic, but

were obviously framed solely to suit the needs of

mathematics. Moreover, they did not have the charac-

ter of universality that one might expect of an axiom

of logic, but were clearly manufactured to meet a

special situation. Consequently, although the White-

head-Russell “school” acquired a sizable following for

a time, it had only a limited life. Nevertheless, the

central theme—that mathematics is derivable from

logic—persisted, and the *Principia Mathematica* has

continued to be a source of both inspiration and sym-

bolic modes for workers in the foundations of mathe-

matics and logic.

In particular, the so-called “Formalist School,”

starting under the leadership of the great German

mathematician David Hilbert, adopted a symbolism

obviously inspired by that of the *Principia Mathe-
matica.* However, the motivating philosophy of this

school was not that mathematics is derivable from

logic, but that all of mathematics could be formulated

in a symbolic framework which, although formally

meaningless, could be interpreted by mathematical

concepts and shown to be consistent. More specifically,

it was Hilbert's idea to set up certain axioms using

symbols alone and no words from the natural language,

along with a set of rules which, although not an in-

trinsic part of the symbolic system and couched in the

natural language, would specify how theorems could

be derived from the axioms. The object of this program

was to show that a symbolic system could be set up

which would, when interpreted by mathematical con-

cepts, give all of mathematics, and which could be

shown would never give the formula for a contra-

diction. In this way, it was hoped that absolute rigor

could be established.

Meanwhile a distinctly different and radical ap-

proach to the problem of rigor was being promoted

by the Dutch mathematician L. E. J. Brouwer (who

was influenced by the ideas of the nineteenth-century

mathematician L. Kronecker). Brouwer maintained

that mathematical concepts are intuitively given and

that language and symbolism are necessary only for

communicating these concepts. The intuition basic to

mathematics, according to Brouwer, is that of stepwise

progression as in the passage of time, conceived as one

instant following another; for mathematics, the basic

intuition gives the sequence of natural numbers: 1, 2,

3,.... All mathematics must be constructed on the

basis of this sequence. In particular, “existence” of a

mathematical concept depends upon such a con-

struction; appeal to the logical Law of the Excluded

Middle to prove the existence of a mathematical entity,

involving showing that assumption of nonexistence

leads to contradiction, is not acceptable, for example.

Brouwer called the resulting philosophy “Intuition-

numbers does not exist, since it cannot be built up from

the natural numbers without using certain axioms of

the theory of sets which are not constructive and hence

are not admissible to the Intuitionist. The contra-

dictions encountered in the “orthodox” mathematics

are due not to the use of the infinite per se, but to

the “unjustified” extension of the laws of logic from

the finite to the infinite. By using constructive methods

only, these contradictions are avoided.

While the Intuitionist contention that their methods

yielded an absolutely rigorous mathematics was appar-

ently correct, unfortunately only a portion of the

mathematics which had been built up during the pre-

ceding three centuries was attainable by these methods.

Acceptance of the Intuitionist path to absolute rigor

meant, then, giving up concepts which had not only

proved their usefulness but had become firmly im-

bedded in the culture. It is not surprising, therefore,

that Intuitionism attracted few converts, and that the

major part of the mathematical community looked for

another way out of the difficulties posed by the contra-

dictions.

Later attempts to establish an absolutely rigorous

mathematics, employing chiefly the methods of formal

axiomatics, have revealed that such a concept as abso-

lute rigor is apparently an ideal toward which to strive,

but one that is in practice unattainable except in cer-

tain limited domains. It is in much the same category

as such an intuitively conceived abstraction as abso-

lutely perfect linear measurement; no matter how

much more precise measuring instruments are made,

it is in practice unattainable. This does not imply that

certain restricted portions of mathematics are not rig-

orously founded; quite the contrary. It applies chiefly

to those parts of mathematics in which the (infinite)

theory of sets is employed. Moreover, logic itself has

been revealed as only an intuitive cultural construct

which gives rise to the same kind of problems and

variations as mathematics when subjected to formal

symbolic analysis (Beth, 1959).

In the natural sciences such as physics, chemistry,

and zoology, at least in their experimental aspects, the

amount of rigor attainable is dependent upon technical

factors such as measuring devices, and will increase

as the related technology becomes more precise. Simi-

lar conclusions hold in the social sciences. Both cate-

gories of science—natural and social—tend toward

greater mathematization as they develop; and so long

as the portions of mathematics which they employ can

be shown to be rigorous, they will not be affected by

the types of difficulty still encountered in the parts of

mathematics dependent upon general set theory.

It must be recognized, too, that a sizable group of

mathematicians of Platonistic persuasion take the view

that mathematics simply has not yet advanced far

enough to be able to cope with such vexing questions

as arise in modern set theory; that the “truth” con-

cerning these is still a matter for investigation and that

their rigorous solutions are still attainable. The situa-

tion is much like that of a natural scientist who believes

that the “laws” of nature as presently formulated are

only an approximation to the true situation which

exists. Whether this “true” situation will ever be dis-

covered, or even whether it can be formulated in

linguistic or mathematical terms if it does exist, he

cannot say. Similarly, the mathematician who feels that

rigorous mathematical truth does exist must admit, in

the present state of knowledge, that it may never be

possible to attain it.

## *BIBLIOGRAPHY*

E. T. Bell, *The Development of Mathematics* (New York,

1945); idem, *Men of Mathematics* (New York, 1937). E. W.

Beth, *The Foundations of Mathematics* (Amsterdam, 1959).

S. Bochner, *The Role of Mathematics in the Rise of Science*

(Princeton, 1966). C. B. Boyer, *The History of the Calculus
and its Conceptual Development* (New York, 1949). E. G.

H. Landau,

*Foundations of Analysis*(New York, 1951). O.

Neugebauer,

*The Exact Sciences in Antiquity*(Providence,

1957). H. Poincaré,

*The Foundations of Science*(Lancaster,

Pa., 1946). A. Rosenthal, “The History of Calculus,”

*Ameri-*

can Mathematical Monthly,58 (1951), 75-86. A. Szabo, “The

can Mathematical Monthly,

Transformation of Mathematics into Deductive Science and

the Beginnings of its Foundation on Definitions and

Axioms,”

*Scripta Mathematica,*27 (1964), Part I, 24-48A,

Part II, 113-39. R. L. Wilder,

*Introduction to the Founda-*

tions of Mathematics,2nd ed. (New York, 1965).

tions of Mathematics,

RAYMOND L. WILDER

[See also Axiomatization; Continuity; Infinity; Number;Pythagorean...; Relativity.]

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