Scientific American - 18 September 1920 - New Conceps of the Past Century the Change in Outlook...
Transcript of Scientific American - 18 September 1920 - New Conceps of the Past Century the Change in Outlook...
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276 September 18, ]9?0SCIENTIFIC AMERICAN
New Concepts of the Past Century‘The Change
in Outlook Since Classical Days, Which‘ Makes Non-Euclidean Geometry a Possibility
- '. J‘
HE scienceof geometryhas undergonea revolutionof which the outsider is not informed. The clas
sical geometerregarded his sciencefrom a narrow viewpoint, as the study of a certain set of observed phe
nomena—thoseof the space about us, consideredas anentity in itself and divorced from everything in it. Itis clear that some things about, that space are not asthey appear (optical illusions), and that other thingsabout it are true but by no means apparent (the sumof-squares property of a right triangle, the formalinfor surface and volume of a sphere, etc.). Whilemany things about space are “obvious,” these need inthe one case disproof and in the other discovery andproof. With all their love of mental processes fortheir own sake, it is then not surprising that theGreeks should have set themselvesthe task of provingby logical process the properties of space, which aless thoughtful folk would have regarded as a subjectonly for observational and experimental determination.
But, abstract or concrete, the logical structure musthave a starting point. It has been pointed out in thesecolumns that the simplest words cannot be defined interms of anything still simpler. In any logical struclure they must therefore stand undefined. Equally,the simplest facts cannot be proved in the absenceof simpler facts to cite as authority. If we are to have a
logical structure of any sort, we must begin by layingdown certain terms which we shall not attempt todefine, and certain statements\vhich \ve shall not tryto prove.
The classical geometersensedthe difficulty of defining his first terms. But besupposed that he had met it when hedefined these in words free of technicalsignificance. “A point is that which hasposition without size" seemed to him anadequate definition, because "position"and “size” are words of the ordinarylanguage with which we may"all be assumed familiar. But today we feel that“position” and “size" represent ideas thatare not necessarily more fundamentalthan those of “line" and “point," andthat such a definition begs the question.We get nowhere by replacing the nudefined terms “point” and “line" and“plane," which really everybody understands, by other undefined terms whichnobody understands any better.
In handling the facts that it was notconvenientto prove, the classical geometercame closer to modern practice. He laiddown at the beginning a few statementswhich he called “axioms," and which heconsidered to be so self-evident that demonstrationwas superfluous. That the term “self-evident" leftroom for a vast amount of ambiguity appears to haveescaped him altogether. His axioms were axiomssolely because they were obviously true.
'
Laying the Foundation
The modern geometer meets these dlfiiculties fromanother quarter. In the first place he is always insearch of the utmost possible generality, for he hasfound this to be his most eifective tool, enabling him asit does to make a single general statement take theplace and do the work of many particular statements.
The classical geometer attained generality of a sort,for all his statements were of am] point or line orplane. But the modern geometer, confronted with arelation that holds among points or between points
and lines, at once goes to speculating whether therearenot other elements among or betweenwhich it holds.
The classical geometerisn't interested in this question
at all, because he is seeking the absolute truth- aboutthe points and lines and planes which he sees as the
elements of space; to him it is actually an object so
to circumscribe his statements that they may by nopossibility refer to anything other than these elements.
Whereas the modern geometerfeels that his primary
concern is with the fabric of logical propositions that
he is building up, and _not at all with the elements
about which those propositions revolve.
It is of obvious value if the mathematician can laydown a proposition true of points, lines and planes.
But he would muchrather lay down a proposition true
at once of these and of numerous other things: for
such a proposition will group more phenomenaunder
W
Euclidean geometry.
By the Einstein Prize Essay Editor
a single principle. He feels that on pure scientificgrounds there is quite as much interest in any one setof elementsto which his proposition applies as there isin any other; that i f any person is to confinehis attention to the set that stands for the physicistfs space,that person ought to be the physicist, not the geometer.If he has produced a tool which the physicist can use,the physicist is welcome to use i t; but the geometercannot understand why, on that ground, he should beaskedto confinehis attention to the materials on whichthe ph_vsiclstemploys that tool.
It will be alleged that points and lines and planeslie in the mathen1atician’sdomain, and that the otherthings to which his propositions may apply may not:10lie—and especially that if he will not name them inadvance he cannot expect that they will so lie. But
the mathematician will not admit this. If mathematics is definedon narrow grounds as the scienceof number, even the point and line and plane may be excludedfrom its field. If any wider definition be sought—andof course one must be—there is just one definition that
the mathematician will accept: Dr. Keyser's statement
that “mathematics is the art or science of rigorousthinking.”
The immediate concern of this science is the means
of rigorous thinking-undefined terms and definitions,
axioms and propositions. Its collateral concern is thethings to which these may apply, the things which may
be thought about rlgorously—everything. But now the
mathematic-ian‘sdomain is so vastly extendedthat it
HEN the Einstein essays begin to appear in print, it will befound that many of them will have somethingto say about the
non-Euclidean character of the geometry of the Einstein fourdimcnsional lime-spacemanifold. This sounds like a very large order
but it is not half so bad as it sounds.will not be able, in the space allotted, to give this phase of ihe subject
anything like the background which it ought to have.backgroundmay properly be supplied in advance ralhcr than as an afterthought; so theEinstein Prize Essay Editor has prepared theaccompany
ing material which. with anotherarticle in a later issue, aims to make it
plain just what the mathematicianis getting at when he talks of non
becomesmore than ever important for him to attainthe utmost generality in all his pronouncements.
One barrier to such generalization is the very name“geometry," with the restricted significance which itsderivation and long usage carry. The geometertherefore must have it distinctly understood that for him“geometry” means simply the processof deducing a setof propositions from a set of undefinedprimitive termsand axioms; and that when he speaks of "a geometry" he means some particular set of propositions sodeduced, together with the axioms, etc., on whichthey are based. If you take a new set of axioms youget a new geometry.
The geometer will, if you insist, go on ‘calling hisundefined terms by the familiar names "point,” “line."“plane." But you must distinctly understand that thisis a concessionto usage, and that you are not for a
momentto restrict the application of his statementsinany way. He would much prefer, however, to be allowed new names for his elements, to say “We startwith three elements of different sorts, which we assume to exist, and to which we attach the namesA, B and 0-or if you prefer, primary, secondary andtertiary elements—-oryet again, names possessing no
intrinsic significance at all, such as ching, chang andchung." He will then lay down whatever statementshe requires to serve the purposesof the ancient axioms,
all of these referring to some one or more of his elements. Then he is ready for the serious business of proving that, all his hypothesesbeing granted, his elements A, B and C, or I, II and III, or ching, chang andchung, are subject to this and that and the otherpropositions.
The competingessays,however,
It seemsthat such
We have asked this memberof the slafi to try to
make good on the boast, which he recently incorporated into a review of
Einsleinian liieralurc. that he triesto be a human being. We hope thathehas .succeeded.—THi-1EDITOR.
The Role of Geometry
The objection will he urged that the mathemauc-iawho does all this usurps the place of the logician. Alittle reflection will show this not to
be the case. Th
logician in fact occupies the same position with reference to the geometerthat the geometer occupies withreference to the physicist, the chemist, the arithmetician, the engineer, or anybody else whose primary interest lies with some particular set of elements twhich the geometer’ssystem applies. The mathematician is the tool-maker of all science, but he does nomake his own tools-—thesethe logician supplies. Theloglcian in turn never descends to the actual practiceof rigorous thinking, save as he must necessarily dthis in laying down the general procedures whichgovern rigorous thinking. He is interested in processes,not in their application. He tells us that if proposition is true its conversemay be true or false oambiguous,but its contrapositive is always true, whileits negative is always false. But he never, from particular proposition “It A is B then C is D,“ drawsthe particular contrapositive inference “If C is not Dthen A is not B." That is the mathematicians business
The mathematician is the quantity-production maof science. In his absence,the worker in each narrowerfield where the elements under discussion take par
ticular concrete forms could work out, for himself, thpropositions of the logical structure that applies tthose elements. But it would then be found that thengineer had duplicated the work of the physicist, an
so for many other cases: for the wholetrend of modern science is toward showing that the same background of principles lies at the root of all things. Sthe mathematician develops the.fabric opropositions that follows from this, thaand the other group of assumptions, anddoes this without in the.least concerninhimself as to the nature of the elementof which these propositions may be trueHe knows only that they are true for anelements of which his assumptio\-s arirue, and that is all he needs to knowWhenever the worker in some particularfield finds that a certain group of th1.:e0meter‘sassumptions are true for hielements, the geometry of those elementis ready at hand for him to use.
Now it is all right purposely to avoidknowing what it is that we are talkingabout, so that the names of these thingsshall constitute mere blank forms whichmay be filled in, when and if we wishby the namesof any things in the universe
of which our “axioms” turn out to be true. But whatabout these axioms themselves? When we lay themdown, in ignorance of the identity of the elements towhich they may eventually apply, they cannot by anypossibility be “self-evident.” We may, at pleasureaccept as self-evident a statement about points andlines and planes; or one about electrons, centimeterand seconds; or one about integers, fractions, and irrational numbers; or one about any other concretthing or things whatever. But we cannot accept aself-evident a statement about chlngs, changs andchungs. So we must baseour “axioms” on someotherground than this; and our modern geometer has hisground ready and waiting. He accepts his axioms onthe ground that it pleases him to do so. To avoid alsuggestion that they are supposedto be self-evident, oreven necessarily true, he drops the term “axiom" and
substitutes for it the more colorless word "postulate.A postulate is merely something that we agree toaccept, for the time being, as a basis of further argument. If it turns out to be true, or if we can findcircumstances under which and elements to which itapplies, any conclusions which we deduce from it bytrustworthy processes are valid within the same limitations. And the propositions which tell us that, ifour postulates are true, such and such conclusions aretrue—tl1ey, too are valid, but without any reservationat all!
What May We Take for Granted?
But how doesthe geometerknow what postulates tolay down? One is tempted to say that he is at libertyto postulate anything that he pleases, and investigate
(Continuedonpage388')
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286September 18, 1920SCIENTIFIC AMERICAN
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New Concepts of the Past Century(Conlinuedfrompage276)
the results; and that whether or not hispostulate ever he realized, the propositions that he deducesfrom it, being true,are of scientific interest. Actually, however, it is not quite as simple as all that.If it were sufiicient to make a single postulate it would be as simple as all that;but it turns out that this is not sufficientany more than it is sufficient to have asingle undefined term. We must haveseveral postulates; and they must besuch, as a whole, that a geometry flowsout of them. The requirements are three.
In the first place, the system of postulates must be “cuteg0rical" or completethem must be enough of them, and theymust cover enough ground, for the support of a complete system of geometry.In practice the test for this is direct. If \\'e got to a point in the building up of ageometry where we could not provewhether a certain thing was one wayalways, or always the other way. or sometimes one way and sometimes the other,we should concludethat we neededan additioflal postulate covering this grounddirectly or indirectly. And we shouldmake that postulate——becauseit is precisely the things that we can't provcwhich, i n practical work, we agree to assume.
In the second place. the system of pos
tulates must be (-onslstent—no one ormore of them may lead, individually orcollectively, to consequencesthat contradict the results of any other or others.If in the course of building up a geometrywe find we have proved two propositionsthat dcny one another, we search out theimplied contradiction in our postulatesand remedy it.
Finally, the postulates ought to be independent. It should not be possible toprove any one of them as a consequenceof the others. If this property fails, thegcolnetr_vdoes not fail with_ it: but it isseriously disfigured by the superfluity of assumptions, and one of them should beeliminated. If we are to assumeanythingunnecessarily. we muy as well assumethewhole geometry and he done with it.
The gcometer‘sbusiness then is to drawup a set of postulates. This he may doon any basis wliutevcr. They may be suggested to him by the behavior of
points,lines and planes, or by some other concrete phenomenu: they may with equalpropriety be the product of an inventiveimagination. On proceeding to deducetheir consequences,he will discover andremedy any luck of cutegoricity or consistenceor independencewhich his original system of postulates may havelucked. In the end he will have so large11body of propositions without contradiction or failure that he will conclude thepropriety of his postulates to have beenestablished, and the geometry based onthem to be a vulid one.
And What is It All About?Is this geometry ever realized? Strict
ly it is not the ;.:cometer’sbusinessto askor answer this question. But researciidevelops two viewpoints. There is alwaysthe hian who indulges in the pursuit of facts for their sake alone, and equally the
man who wants to see his new facts leadto something else. One great mathematician is quoted as enunciating a newtheory of surpassing mathematical beautywith the climacteric remark “And. thankGod, no one will ever be able to find anyuse for it!" An equally distinguishedcontemporary, on being interrogated concerning possible applications for one of hismost abstruse theorems, replied thathe knew no present use for it; but thatlong experience had made him confidentthat the mathematician would never develop any tool, however remote from immediate utility, for which the delvers inother fields would not presently findsome use.
If we wish, however, we may inquire(Continucdonpage288)
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288 September 18, 1920SCIENTIFIC AMERICAN
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‘New Concepts of the Past Century
1 (Continuedfrompage286)with perfect propriety, from the side lines,whether a given geometry is ever realized. We may learn that so far as has
i yet beendiscoveredthere are no elementsfor which all its postulates are verified,and that there is therefore no realizationknown. On the other hand, we may morelikely find that many different sets of elements are such that the postulates canbe interpreted as applying to them, and
i that we therefore have numerous realiza, tions of the geometry. As a human being the geometermay be interested in allthis, but as a geometer it really makes
ilittie difference to him.i When we look at space about us, we
see it, for some reason grounded in the; psychological history of the human race,
‘
as made up i n the small of points, whichgo to make up lines, which in turn constitute planes. Or we can start at the
i other end and break space down first intoplanes, then into lines, finally into points.Our perceptions and conceptions of thesepoints, lines and planes are very definite
Eindeed; it seems indeed, as the Greeksi thought, that certain things about them‘
ore self-evident. If we wish to take theseself-evident properties of point. line andplane, and combine with them enoughadditional hair-splitting specifications toassure the modern geometerthat we have
‘really a categorical system of as21iniptions, we shall have the basis of :1 per
, fectly good system of geonictry. This \vill‘be what we unavoidably think of its the
absolute truth with regard to the spaceabout us; but you mustn't say so in thepresenceof the geomcter. It will also bewhat we call the Euclidean geometry. Ithas been satisfactory in the last degree,because _notonl_vspace, but pretty muchevery other system of two or three elements beuring nn_v relations to one on
1other can be made, by employing as ameans of interpretation the Descartenn
‘scheme of plotting. to fit into the frame‘work of Euclidean geometry. But it is‘ not the only thing in the world of concep, tunl possibilities, and it begins to appear
that it may not even be the only thing inthe world of cold hard fact that surrounds
i us. I had hopedto makeclear in this placethe reason for these statementsand something of their significance: but after re
peatedboiling down I find thnt these introductory remarks occupy all the spacei that can be given to the subject in it single, issue, so must postpone the conclusion of ithe mutter till another date. This pre
liminnry discussion will have fulfilled itsmission if it has made the lil_\‘ll‘|ilIlunderstand why Bertrand Russell, eminentmathematician, was able to say: “Mathematics is the science in which _v0uneverknow what you are talking about, or
] whether what you say is true.”
Running the Gauntlet of QualityProduction
(Continuedfrompage.978)
sandth of an inch, the tolerance on theiiurdness is five points on the scleroscope,which is the closest test commercially
1practicable. Altogether, there are, on this‘single part, :1 total of 34 inspection operations, and since many of them must be
repeated on each cam. the total numberof inspection points is 140. On till the
- parts named there is one-hundredpercent inspection; that is, every one is inspected. On pnrts where such great accuracy is not required, the inspections aremade on five or ten per cent of the parts,and unless trouble is discovered the balance are passed. Because of this fact, itis impossible to say exactly how manyinspections of parts have been made foreach car assembled. The number will beabove 20,000and below 25,000.
But if inspection of finished parts isnecessary, we must not forget that the
foundation of the quality of any productmust be the materials that enter into it.and in its methodsof selecting materialsand handling them the American quantity
.('ll0S9I1 to fit :1
production system inns \von :1 long advantage over the European hand-work system. These methods hnve now reached itpoint where nothing is left to chance,and where the exact properties of everypiece of metal and of any other materialare accurately calculated for the duty itis to do.
The automobile factory of this articlepurchases 18 fundamentally differentkinds of steels. under 45 different set.of specifications, in order to meet thewide range of the needs of a single car.
It uses, for the same purpose, seven dif ferent brnsses, six different bronzes, andso on down the list, each material being
yery definite function.Among the non-metallic goods it purchases leather, rubber. wood, glass, felt,fabric, asbestos, cork, gasket materials,bakelite and fibers. The inspection of these materials on their arrival at theplnnt is only the first step in the processof control.
Many of the materials undergo processes which produce fundamental Cll8Il2‘P.\'during manufacture. The specificationsfor the steel alloys, for example, are sovaried under heat treatment and carbonizing that it is possible for the factoryexperts to tell front a single particle of the finished niatcrinl from which one of more than a hundred working parts itwas broken. I