8/18/2019 Scientific American - 18 September 1920 - New Conceps of the Past Century the Change in Outlook Since Classic…

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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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8/18/2019 Scientific American - 18 September 1920 - New Conceps of the Past Century the Change in Outlook Since Classic…

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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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8/18/2019 Scientific American - 18 September 1920 - New Conceps of the Past Century the Change in Outlook Since Classic…

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288   September   18,   1920SCIENTIFIC   AMERICAN

 The   service   is   tlzere   whenthe  metal   delivers it

OU   pay   for the   design-time   and   labor that go  to make a metaluseful.   Then   rust.   acid   action,   steam   wear,   salt   pitting   and

high   heats   limit   its service life and so   discount   your   investment.

 These   corrosive   forces have least   effect on   Monel   metal.   In   fact

such seasoned engineering   organizations   as  J.   G.   White,   Stoneand   Webster,   Dwight   Robinson   Co.,   etc.,   have   adopted   Monel   asstandard for anti-corrosive   service.

Monel   valve seats   and   stems   on   2001b. 125deg.   F.   superheat   linesatthe ShermanCreekstation, United Electric Light  and Power Co., New York,show no wearaftersix years’continuous service.

Monel   shaft   sleeves,   impellers   and  runners on  condensers   and condenserpumpsat theCos  Cob Plant of theN.  Y. N. H. 8:H. R. R. havestoodupfor yearsagainstthecorrosiveactionof pollutedsaltwater.

Monel   metal for   thermo-couples,   oxy-acetyline   bodies and otherheatexposedpartsis  giving an equallygood accountof itself in  exactingservice.(its  ratio of  expansionis practicallythatof  steel.)

In   chemical   process machinery,   Monel   is   specified   for filter clothor for parts directlyin contactwith activesolutions by thebestinformedconsulting engineers.

 THE INTERNATIONAL   NICKEL COMPANY43   Exchange Place, New  York

 The internationalNickelCompanyof Canada,Ltd., Toronto.Ontario

 ThenameMonelisgivento alineofmetalproductsproducedby The InternationalNickelCompanyfromanaturalnickelall0y—-67%nickel.28";copper,and!)% othermetals. These

fioductsincl udeMonelblocks,‘

onelrods, Monelcastings,Monelwire,Monelstripstock,Monelsheets,etc.   ThenameMonelindentifiesthe naturalnickelalloyas oduredby TheInternationalickelCompany.

‘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