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Chapter

11

Chapter IV:Consistency,

Completeness,andGeometry

I have obtained some results which are “repugnant  to the nature of a straight-line.” – Saccheri 

11.1 Abstract

TheprecedingDialogueisexplicatedtotheextentitispossibleatthisstage.Thisleadsbacktothequestionof howandwhensymbols inaformalsystemacquiremeaning. Thehistoryof Euclidean

andnon-Euclideangeometryisgiven,asanillustrationof theelusivenotionof “undefinedterms”.This leads to ideas about the consistency of  different andpossibly “rival” geometries. Through

thisdiscussion thenotion of undefined terms is clarified, and the relation of undefined terms to

perceptionandthoughtprocesses isconsidered.GEB pp. ix 

11.2 Questions

1. Givesomeexamplesof complexisomorphisms.WhatmachinaryisHofstadter(DRH)talking

about?

2. WhydoesDRHkeepapologizingabouthisuseof theterm“isomorphism”?

3. Giveanexampleof whenawordcanhavemultiplemeanings. Whenyousaysomething,how

doyoucommunicatethe intendedmeaning?

4. DRHasksthatweassumethateveryreaderof Englishusesmoreor lessthesame“isomorphism” insuckingmeaning fromthemarksonpaper.Whatdoeshemeanbythis? Inwhatwayscouldtworeadersof Englishusedifferentisomorphismsinextractingmeaningfromthe

same

piece

of 

writing?

Hint:

Think

about

the

conceptual

web

you

attach

to

even

very

simple

words.Asanexercisepickanynounfromthedialogueandcreateyourownpersonalconceptweb attached to that noun. For example if  you picked the noun “present”write it in themiddleof ablankpieceof paperanddrawtwo linestoseparateareasof thepaperandthen

write“now”at theendof one lineand“gift”at theother line’send. Branchoff of eachof these concepts (“SantaClause”, “MyBirthday”, etc.) and give yourself  a point for every

conceptyouattachtoyour initialword. Identify levelsof meaning if you’rekeen.

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5. What’s the problemwith interpretingmathematical objects?  In the case of  themodified

pq-system? In thecaseof Euclideangeometry? What’swrongwithour interpretationof a

“straight line”?

6. Drawpicturesof Euclid’sfivepostulates.Dotheyseem likeasafetruthstoassume?

7.

Do

you

think

mathematics

is

the

same

all

possible

worlds? 

Consider

carefully

what

youconsidertobemathematics.

8. Doyouthinklogicisthesameinallpossibleworlds?Whatthenconstitutesapossibleworld?

9. Pushingpossibleworldsaside,What logicdoesouruniverseobey? Consider if ouruniverseobeysThe

Law

of 

the

Excluded

MiddlewhichstatesthateitherPornot-Pmustbetrue,

there isnoroombetweentrueandfalse.Whatdoesquantummechanicssayaboutthelogicof theuniverse?

10.  GEB pp. 102 AnswerDRH’squestionabouttherecordandtherecord-player.Howcanthey

bothpasstheirrespectivetests?

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Gödel, Escher, Bach: A Mental Space Odyssey

Summer 2007 

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