ABSTRACT
Wehavenowseenexamplesofhowbeliefsystemscanberepresentedby statementsinthefirst-orderlanguageofarithmetic,L1Ar,ontheonehand, andbystatementsinthesecond-orderlanguageofrealnumbers,L.?Real,on theotherhand.Incomputationalhydraulicsweneedtoalternatebetween statementsmadeinfirst-orderlanguages,astheonlykindsofstatementsthat ourdigitalmachinescanaccommodate,andstatementsmadeinhigherorderlanguages,whicharebestadaptedtoourown,humanunderstanding. Wethushaveaverybasicneedtopassbetweenstatementsexpressedinthe onelanguageandstatementsexpressedintheother,andwecallsuch passagesthataremadebetweendifferentlanguagestranslations,denoting thembythesymbol®.Thus,symbolically:
(StatementsmadeinL1Ar)~®-+(StatementsmadeinL2Real)
Wehavealsoalreadyseenhowthe'principleofeconomyofthought'leads ustoprefertheshortestpossiblestatements,andsothosecomposedofthe leastnumberofthemostsimply-compoundedterms;andthus,inparticular andultimatelyofabsolutenecessity,wealwaysseektousestatementswitha finitenumberofterms,witheachtermbeingitselfcomposedfromafinite setofatomicsymbols.Wemayalreadyanticipate,however-andindeedwe shallshowgenerallyinChapter4-thatifwewishtomakeanexact translation,inasensethatthestatementintheonelanguagecomesto expressexactlythesamesetofbeliefsasastatementintheotherlanguage (sothatnootherbeliefwhatsoeverneedstobeaddedorsubtractedin passingfromtheonestatementtotheother)thenastatementof.finite lengthintheonelanguagewillalmostalwaystranslateintoastatementof infinitelengthintheother.Thusinourmathematicallanguages,justasin ournaturallanguages,wecanhardlyevermakeanexacttranslationbetween statementsmadeindifferentlanguages.Since,however,neitherwenorour computerscanprovideanyexplicitrepresentationofstatementsofinfinite length,wehaveinalmostallcasestomakedowithapproximatetranslations
if we are to make any workable translations at all.* As we may also anticipate - and as we shall come to see more clearly in Chapter 4 - the process of approximation itself corresponds to the introduction of one or more further beliefs, so that the approximate translation comes to extend the belief system that underlies the original representation. Thus, in particular, the statement in L1Ar - the numerical scheme that approximately translates the original statement in ~Real (the partial differential equation) always says more than does this original statement This situation then naturally leads to the potential inherent in all translation, whereby traduttore traditore-the translator becomes a traitor or betrayer - in that the extension may itself introduce a belief that subverts the original belief system markedly, and indeed one that may come to contradict this original belief system in its entirety. As we shall see in Chapter 4, the flrst, subversive situation leads to all manner of errors (e.g. various kinds of truncation errors, which may be interpreted as amplillcation errors, phase errors, etc.) while the second situation, involving contradiction, appears to be inextricably linked to the phenomenon that we shall come to refer to as numerical instability.
