New Mo dels of the Real-Number Line Recent developments in mathernatical logic reveal tltat tltere are a number of alternative wa)rs of defining the continttLlm, or connected nurnber system., to include all tlt,e rr:al, rtLLmbers by Lynn Arthur Stccn -f firtuallv all of mallrematics and tonic ideals-has beeu tl-re most enduring tempted to ans$,er this question by pos- \/ rrr,r"h'nf science is based on tlre and popular philosophy of mathernatics, tulatiug the a prioli existence in the hu- V abstlact concept of the leal-nrrrn- since it plovides mathematics with the man mind of a kernel of intuitive mathe- ber line: the continuum, or connected (scientiffc) discipline of confolming to matical and geometrical tluth. The con- number system, that includes all the real son-re kind of pelceived reality together siderable influence of Kant's philosophy nurlbels-zelo, positive and negative with the freedom to escape the bonds reinfolced the wideiy held view that the integels, rational numbers (flactions) of empilicism. axioms of mathematics should be self- and irrational numbers (such as r)-but N{ost wolking mathematicians, at evident truths. Euclidean geometry be- that excludes the "unreal," or complex, least those not directly engaged in prob- came the archetype of mathematics, numbers-explessions containing the Iems of mathematical logic, tend to be since it was a beautiful, useful theot'y imagirraly numbel y=1. tlre leal-num- Platonists in the sense that they talk and created by logical deduction from cer'- bel corrtinuum not only plovides the nat- act as if the abstract objects they study tain (nearly) self-evident axioms. In- ural setting for all the operations of have some kind of enduling ideal ex- deed, Kant hin-iself had asselted that the alithmetic and calculus but also selves istence. Standing in extieme opposition geometrical intuition of Euclidean space as our only intellectual model of time to the Platonists are the Formalists, rvho and of the time continuum r'vas one of and (one-dimensional) space. The plop- naintain that the essence of mathemat- the a priori characteristics of the human erties of the continuum were organized ics is not in its meaning but in its folm. mirrd. into a coherent axiomatic framework The problem of the natule of the real- Thus it came as a considerable shock duling the 19th century and have been number line is viewed quite difelently to the i.ntellectual community of the accepted and promulgated with great by adherents to these two schools of early 19th century to learn of the dis- conviction by most contempolary n'rath- thought. A Formalist is likely to say that covery of non-Euclidean geometries, in ematicians. the real-number iine is whatever rve de- r'vhich one of Euclid's axiorns (the paral- Yet in spite of the present unanimity ffne it to be; if we have several com- lel postulate) did not hold. One signifi- of opinion concerning the exact struc- peting defrnitions, then'"ve rvill have sev- cant consequence of this discovery rvas tule of the continuum several significant eral difielent real-number lines and that philosophels and mathematicians alternative systems have been discov- mathemp,tics will be enriched by their began to regard axioms not as self-evi- eled during the past 10 years. Although presence. A Platonist, on the other hand, dent truths but rather as arbitlary rules, none of these new models reflects any would be inclined to rvonder rvhich of subject only to the requiren-rent of con- Iogical flarv in the 19th-century theory, the different models replesented the sistency. For the axiorns to be great the their very existence shorvs quite clear'ly "real" real-number abstlaction of the theory delived from them had to be both that the epistemological foundation of space and time continuum, beautiful and useful, but to be rnathe- mathematical analysis is far from settled. Like beauty, mathematics exists in matics the theoly had only to be con- the eye of the beholder'. Unlike beauty, sistent. Ilthough mathcmatics is sometimes hor,vever, mathemdics enjoys an unpar- Yet in spite of the widespread 20th- I r called a science, it is usually distin- alleled r.rorldwide reputation for objec- century consensus on consistency as the guished from science by its relative in- tivity. If you labor to discover the plop- sole criterion of mathematical truth, the dependence flom empirical considera- erties of the real-number continuum and vast majolity of mathematicians and sci- tions. The intellectual models of science iI I do likewise, we shall leach the same entists maintain a Platonic vierv of the are judged by their ability to explain the conclusions. Science has a similar objec- continuum. Nearly everyone who stud- observed ploperties of the universe, tivity. But whereas the objectivity of ied calculus in the past 50 yeals was ex- rvhereas those of mathematics ale science is a rather plausible consequence pected either to have a clear intuitive judged (by mathematicians) according of its conformity with reality, the objec- (Platonic) image of the real-number line to their consistency and beauty and (by tivity of mathematics is harder to ex- or to believe that all its properties rvere scientists) according to their utility. Pla- plain. Why should the ideal mathemati- consequences of some 10 to 15 suPPos- to's description of .n"rathematics as the cirl universe in your mind be the same as edly self-evident truths that describe discovery of the properties of objects in the one in my mind? what is calleri formally a "complete or'- an ideal universe-the universe of Plir- Trvo centulies ago Immanuel Kant at- deled field" and informally the "real- number system" lsce illustration on tlis finitesimals played a key role in the de- niz had r.vrought, squirmed out of the pagel. velopment of the definitions and nota- dilemma by describing infinitesimals as A cynic might vierv this ploglam as tions of calculus. Indeed, during the "fictions, but useful fictions." N{ean- indoctrination in Platonism, since its l8th and l9th centuries the name for while Bishop Berkeley scorned Nerv- clear purpose is to convince students of what we now call simply "calculus" rvas ton's inffnitesimals (or fluxions, as Nerv- something their plofessors have accept- "calculus of inffnitesimals." What Nerv- ton called them) as "ghosts of departed ed, namely that thele is a unique real- ton and Leibniz did was to shorv horv quantities." number line with certain self-evident one could calculate with infinitesimals But mathematics flourisl.red in spite of properties, and th:rt mathematicians and obtain reliable lesults, results that the philosophers, arvakened by the cal- have succeeded, by listing the axioms of moreover could not be obtained by any culus of inffr.ritesimals as it had not been a complete oldeled field, in capturing other method. since the glories of Athens and Alexan- the essence of this line in a dozen or so The calculus of infinitesimals u'as re- dria. The Platonic image of the real- sentences. Thus mathematicians norv ceived with plofound skepticisrn by number line r.vas as yet only a vague talk about /he real-number line just as many philosophers rvho, follorving Aris- ideal, and calculus developed more as a mathematicians and philosophels of the totle, abhon'ed the absolute inffnite. In descriptive science than as a deductive 18th century talked about f/i.e geon.retry. his Plrysics Alistotle distinguished be- logical system. It rvas not until the 19th Of course, the existence of different trveen the potential infinite and the ab- century that mathematicians begau to geometries does not necessitate the ex- solute infinite, acceptirrg the former but echo the philosophers'skepticism; it rvas istence of different real-rrun'rber lines. rejecting the latter as untenable, or be- only then, as the axioms fol the real- Nonetheless, in I931 Kurt Gtidel shorvecl yond the firm glasp of the human mind. numbel system rvere distilled fi'om the that in any mathematical system suffi- (In taking this position Alistotle rvas great unolganized mass of mystical ciently large to contain arithmetic, there merely reflecting the r,videspread Greek properties, that it became clear that the will alrvays be ur.rdecidable sentences: mistlust of the infinite, most popularly existence of infinitesimals r.vas incon- statements about the system that can be illustrated by the paradores of Zeno.) sistent r'vith these axioms. neither proved nor disproved by Iogical The philosophel Leibniz, somervhat cha- This inconsistency is an immediate deduction fi'om the axioms lsee "Ciidel's grined at rvhat the mathematicinn Leib- eonsequence of one of the most char'- Proof," by Elnest Nagel and James R. Ner'vman; ScmNrrrrc AnrinrceN, lune, 1956]. Gridel's nor.v farnous "ur.rdecid- ability theorem" implies that in the Pla- 1. Addition and Multiplication tonic universe of ideal mathematical ob- lf xandyare real numbers,ihen so are x+ y and xy, jects thele are many-in fact, infinitely many-objects that satisfy the axioms for 2. Associativity the real-number line, since each unde- lf w, x and y are real numbers,then (w + x) + y = w + (x + y) and (wx)y = w(xy). cidable proposition about the leal-num- belline may be true in one ideal model 3. Commutativity ll x and y arereal numbers,thenx+y = y + x aad xy = yx. and false in another. In 1963, more than 30 yeals after 4. Distributivity Godel proved that diffelerrt models for lf w, x and y are real nr,rnrbers,then vt(x+ y) = wx + wy. the real-number axioms must exist, Paul J. Cohen of Stanfold University actuirlly 5.