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Chapter

9

Chapter III: Figure and Ground

What  is the sound of one hand clapping?  – Zen Koan 

9.1 Abstract

Thedistinctionbetweenfigureandgroundinartiscomparedtothedistinctionbetweentheoremsandnontheorems in formal systems. Thequestion “Does afigureneccessarily contain the sameinformationasitsground”leadstothedistinctionbetweenrecursivelyenumerablesetsandrecursive

sets.GEB pp. ix 

9.2 Primesvs. Composites

9.3 The tq-System

AXIOMSCHEMA:xt-q x isanaxiom,wheneverx isahyphen-string. RULEOFINFERENCE:

Suppose

that

x,

y,and

z

are

all

hyphen-strings.

And

suppose

that

xtyq z

is

an

old

theorem.

Then,

xty-q zx isanewtheorem.

9.4 Capturing Compositeness

RULE:Supposex,y,andzarehyphen-strings. If x-ty-q z isatheorem,thenCz isatheorem.

9.5 Illegally Characteriziing Primes

PROPOSEDRULE:Supposex isahyphen-string. If  Cx isnot  atheorem,thenPx isatheorem.Why isthis“illegal”?

9.6 Figure and Ground

Applications to art. How important is recognizable inbothmath, art, andmusic to figure and

gound?

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9.7 Figure and Ground in Music

Whatmusic isrecursive?

9.8 Recursively Enumberable Setsvs. Recursive Sets

Sowhat isrecursive? It isdifferentthanthetoolforthinkingwe introducedawhileago,butstillconnected. Tounderstandwhat recursivemeans, imagineyouhaveacollectionof  things (a set),andyouwanttodetermineoutof allthingsintheuniverse,whichonesaremembersof thatoriginalcollection(set). If there isanalgorithm  orasetof  intstructionswhichcandeterminemembership

of aset infinitetime,thentheset isrecursive.Thehottamale isthat(GEBpp. 72)

There exist formal systemswhosenegative space (setof non-theorems) isnot thepositivespace(setof theorems)of anyformalsystem.

Whichstatedmoreformallysays

Thereexistrecursivelyenumerablesetswhicharenotrecursive

andfinallythat

Thereexistformalsystemsforwhichthere isnotypographicaldecisionprocedure.

9.9 Primes as Figure Rather than Ground

AXIOMSCHEMA:xyDNDxwherexandyarehyphen-strings. RULE:If xDNDy isatheorem,thenso isxDNDxy. RULE:If  - - D N Dz isatheorem,so iszDF - -. RULE:If zDFx isatheoremandalsox- DNDzisatheorem,thenzDFx-  isatheorem. RULE:If z-DFz isatheoremand,thenPz-  isatheorem. 

AXIOM:

P -  

9.10 Study Questions

Morespecifics...

9.10.1

Primes

vs.

Composites

in

the

tq- systems

1. Don’t do it, butwould you have trouble listingwhich of  the first 30 positive integers areprimenumbersandwhicharecompositenumbers?

2. Definethetq-system,bynamingthe(a)legalsymbols,(b)axiom(s),and(c)ruleof produc

tion,

all

in

symbolic

form.

3. Hofstadterclaimsthatthetq-system issupposedtocapturemultiplication.Does it? Translate theaxioms toarithmetic statements. Translate the three theorems shownonp.65 into

arithmeticstatements.

4. Whyistheruleof  inferencefromof thetq-systemarithmeticallyreasonable?

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9.10.2

Capturing

Compositeness

and

Illegally

Characterizing

Primes

1. Hofstadter implicitly introduces a new symbol into the tq-system in order to state a new

RULE.What is the symbol,andwhat is the complete setof allowed symbols forwhat thenewC-system?Areyouof theopinionthatvariablesoughttobe included inthe list?

2. 

.

What

is

the

arithmetic

meaning

of 

the

new

rule,

and

how

does

it

capture

compositeness?Whydoesthenewrulehavethetwohyphens in it? Wouldn’txtyqzwork justaswellto

expresscompositeness? (Thinkof aspecialcase)

9.10.3

Figure

and

Ground

1. Commenton theSmokeSignaldrawing,p.702,as it relates towhatwevedone so far.  Try

toseeamessagethatsays“Cecin’estpasunmessage”(meaning“This isnotamessage”).Youmightalsocomparethisfigurewiththeoneonp.494.

2. Consider thedefinitionsof “cursivelydrawable” (by theway,whatwouldbe“cursivelyundrawable”or“uncursivelydrawable”?) and“recursive”onp.67. DrawaVenndiagramthat

shows

both

the

relationship

between

the

two

and

the

meaning

of 

the

statement

on

p.68

“Thereexistcursivelydrawablefiguresthatarenotrecursive”

3. WhatpuzzledoesFig. 17solve?Doyouseethat itsolves it?

4. ExplainFigure18asyouunderstand it toyour roommate,playmate, soulmate, shipmate,orevenarandompasserby.This isveryimportant!

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Gödel, Escher, Bach: A Mental Space Odyssey

Summer 2007 

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