Wittgenstein a Church­Turingova
teze
Jakub Güttner, guttner@fit.vutbr.cz, leden 2002
Abstrakt
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je absurdní.
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filosofie matematLN\ "t]HQt VH SUDYLGO\ V\QWD[H D VpPDQWLND
Abstract
This essay looks at Ludwig Wittgenstein's objections to Turing's version of the Church
Thesis. It contains a brief introduction to Church and Turing Theses together with an
identification of the philosophical aspects of Turing's approach and their connection
with the Mechanist Thesis. It also contains an explanation of the crux of
Wittgenstein's objections, a discussion on rule­following and an analysis of the
concept of computation. Turing claimed that a machine can think, while Wittgenstein
tries to show that such claim is absurd.
Keywords
Wittgenstein, Turing Thesis, Church Thesis, Mechanist Thesis, Artificial Intelligence,
Filosophy of Mathematics, Rule­following, Syntax and Semantics
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Ludwig Wittgenstein se ve svém díle mimo jiné zabýval filosofií matHPDWLN\ D RGYiçLO VH
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Churchova teze
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Wittgenstein: $KRM $ORQ]R -VHP WH
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Second Number Class".
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integers by identifying it with the notion of a recursive function of positive integers
(or of a lambda-definable function of SRVLWLYH LQWHJHUV $ HVN\ 'HILQXMPH SRMHP
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GHILQRYiQ VQDG MHQ LQWXLWLYQ MDNR SRVWXS NWHU QHS"HVDKXMH QDH OLGVNp YSR HWQt
schopnosti. Však také koOHJD *|GHO SURWHVWRYDO çH MH YHOPL QHXVSRNRMLYp definovat
HIHNWLYQ Y\ tVOLWHOQp IXQNFH MDNR QMDNRX W"tGX DQLç E\FK QHMG"tY XNi]DO çH
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práci "On Computable Numbers With an Application To the
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Turingova teze
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Turing: We may compare a man in the process of computing a real number to a
machine which is only capable of a finite number of conditions which will be called
m-configurations. The machine is supplied with a "tape"... running through it and
divided into sections... each capable of bearing a "symbol"... All effective numbertheoretic
functions (viz. algorithms) can be encoded in binary terms, and these
binary-encoded functions are Turing machine computable.
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Mechanistická teze
Proces lidského myšlení lze simulovat strojem.
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inteligence?
Problém zastavení -- Entscheidungsproblem
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Dç N 0HFKDQLVWLFNp WH]L 3RVORXFKHM GiO D SRVX
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Wittgensteinova kritika
Normativnost matematiky
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Mathematics: 'i VH Y*EHF "tFL çH SR tWDFt VWURM SR tWi" 3"HGVWDY VL çH SR ítací stroj
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VSRMHQt D GRGiYi MLP SRW"HEQp NRQFHSW\
Pravidla a opice
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Wittgenstein: 3UiY VH FK\VWiP FHO WHQ DUJXPHQW REUiWLW SURWL 7XULQJRYP VWURM*P
tak poslouchej dál. Dostal jsem se k SULQFLSX "t]HQt VH SUDYLGO\ 6QDG VH PQRX
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Wittgenstein: 7H
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nápad?
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QD UWQH QD ]HP REUD]HF ´#­" a druhý pak vedle do písku vyryje "@­@­@­@­@­".
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chování?
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YODVWQ Y 3\WKDJRURY YW MGH
Pokud chceme mluvit o aplikaci pravidel, chování sledovaného objektu musí být
FKiSiQR MDNR QRUPDWLYQt $ WtP Pt"tP N VDPpPX MiGUX VYp QiPLWN\ 3R tWiQt MH
WRWLç VRX iVWt PQRçLQ\ QRUPDWLYQtFK NRQFHSW* ]DWtPFR 7XULQJRY\ VWURMH QLNROL
3RXKp SURGXNRYiQt VSUiYQFK YVOHGN* QHVWD t N WRPX DE\FKRP PRKOL "tFL çH QNGR
QHER QFR SR tWi
Turing: $ SUR QH" 9çG\( SUiY Y 7XULQJRY WHVWX MGH R WR çH Pi VWURM ]D ~NRO
produkovat stejné výstuS\ MDNp E\FKRP R HNiYDOL RG ORYND 8GiORVWL NWHUp VH
RGHKUiYDMt QNGH ´]D SOHQWRX QHMVRX SRGVWDWQp
Wittgenstein: -i WYUGtP çH Y S"tSDG SURYiGQt YSR WX SRGVWDWQp MVRX .G\ç VH W
]HSWiP MDN VH YSR WHP GRVSOR NH VSUiYQpPX YVOHGNX MDNRX PL QDEídneš
RGSRY
"
Turing: 2GSRY
 Pi SRGOH P GY iVWL SUYQt ] QLFK MH SRVORXSQRVW GXHYQtFK VWDY*
SR Wi"H L I\]LFNFK VWDY* VWURMH YL] SDUDJUDI\  D  D GUXKi ]DKUQXMH SRSLV
DOJRULWPX L SURJUDPX NWHU E\O N YSR WX SRXçLW SDUDJUDI  D GiO
Wittgenstein: -HQçH P\ S"HFH GLVNXWXMHPH R YSR WX D Mi PiP ]D WR çH ]G*YRGQQt
MHKR VSUiYQRVWL E\ VH POR VNOiGDW ] SRVORXSQRVWL SRXçLWFK SUDYLGHO -- matematika
MH S"HFH ]DORçHQi SUiY QD MHMLFK SRXçtYiQt D SUiY WDWR SUDYLGOD ]DUX XMt çH
výsledek bude správný! -DN P*çH EW VSUiYQRVW ]DUX HQD NG\ç YHGH N YVOHGNX MHQ
VH]QDP MDNFKVL VWDY* P\VOL"
Turing: 2 WR PL SUiY MGH 0*M VWURM WRWLç SURYiGt YSR HW PHFKDQLFNP ]S*VREHP
3URJUDP NWHU GR QM YORçtP REVDKXMH MHGQRWOLYi HOHPHQWiUQt SUDYLGOD SRGOH
kterých sH YSR HW "tGt 9LPQL VL çH POXYtP R SUDYLGOHFK -- NDçGp ] QLFK MH
VRX iVWt VWDQGDUGQtKR SRVWXSX YSR WX '*OHçLWp S"LWRP MH çH NDçGp ] WFKWR
SUDYLGHO MH QDWROLN MHGQRGXFKp çH YODVWQ QHPi çiGQ NRJQLWLYQt Y]QDP QHER
obsah. Aby ho stroj provedl, nemust VH VQDçLW KR QHMG"tY SRFKRSLW 7R SUDYLGOR MH
SURVW WDN ]iNODGQt çH KR P*çH DSOLNRYDW LVW PHFKDQLFN\
Pravidla versus popisy, syntaxe versus sémantika
Wittgenstein: %XGX SRNUD RYDW ]D WHEH 'RQDOG .QXWK Y\VYWOXMH çH DOJRULWPXV MH
PQRçLQD SUDYLGHO QHER SRN\Q* SUR ]tVNiQt SRçDGRYDQpKR YVWXSX ] GDQpKR YVWXSX
$OJRULWPXV VH Y\]QD XMH WtP çH YHFKQ\ QHMDVQRVWL PXVt EW Y\ORX HQ\ SUDYLGOD
PXVt SRSLVRYDW RSHUDFH NWHUp MVRX WDN MHGQRGXFKp D GRE"H GHILQRYDQp çH MH P*çH
vykonávat i stroj.
Turing: Krásni GHILQLFH 1H]EYi QHç V Qt SOQ VRXKODVLW
Wittgenstein: 1H]DUD]LOR W WYU]HQt çH ´SUDYLGOD SRSLVXMt RSHUDFH" 3UDYLGOR VH S"HFH
Y\]QD XMH WtP çH QLF nepopisuje, DOH VWDQRYXMH ]S*VRE SRXçLWt QMDNFK NRQFHSW*
8]QiYiP çH MHGQRWOLYp NURN\ WYpKR VWURMH se chovají jako popisy operací, ale na
UR]GtO RG WHEH P WR QHYHGH N 0HFKDQLVWLFNp WH]L DOH N ]DP\OHQt QDG WtP MHVWOL VH
P*çHPH RGYiçLW MH QD]YDW jednoduchá pravidla.
Turing: -i E\FK "HNO MHQ WR çH .QXWK SRSLVXMH DOJRULWP\ MDNR ]YOiWQt W"tGX IXQNFt
(SURWRçH ]REUD]XMt YVWXS QD YVWXS S"L HPç S"LGiYi GYD SRçDGDYN\ -- aby se
DOJRULWPXV GDO VSHFLILNRYDW MDNR VRXERU SUDYLGHO D DE\ WDWR SUDYLGOD POD ]KUXED
VWHMQRX WULYLiOQt VORçLWRVW $ Mi GRGiYiP çH 7XULQJ*Y VWURM VH S"L VYp LQQRVWL "tGt
SUiY WDNRYmi pravidly.
Wittgenstein: $ S"HGSRNOiGi çH NH Y]QLNX XPOp LQWHOLJHQFH MH W"HED SRVWDYLW VWURM
NWHU VH X t WtP çH SRFKRSt QMDNi SUDYLGOD D SDN MH SRXçtYi 2Wi]ND ]Qt ]GD P*çH
být chápání vybudováno na zvládnutí "syntaktických instrukcí", kterými se "tGt MDN
ORYN WDN VWURM $XWRPDWLFN\ S"HGSRNOiGi çH WY*M VWURM GHPRQVWUXMH VYRX
VFKRSQRVW "tGLW VH MHGQRWOLYPL ´EH]REVDçQPL SRGSUDYLGO\ VNU]H WHQt WLVN D
Y\PD]iYiQt V\PERO* QD SiVFH -VRX EH]REVDçQi SURWRçH MH VWURM PXVt SURYiGW EH]
jakékoli inWHOLJHQFH D VFKRSQRVWL ]SUDFRYiYDW LQIRUPDFH Y QP REVDçHQp DOH
VRX DVQ MVRX WR SRGSUDYLGOD DE\ VH MLPL PRKO "tGLW
-i PiP DOH QiPLWNX MDN P*çH POXYLW R ´EH]REVDçQpP SUDYLGOH" -H WR SUDYLGOR
NWHUp QLF QH"tNi"
Turing: 6QDçtP VH WtP QD]QD LW çH VWURM provádí dedukci, která je zcela mechanická.
Wittgenstein: $OH YçG\( WR MH ~SOQ VFHVWQp 6OX XMH GY QHVOX LWHOQp YFL -- logickou
LQIHUHQFL D PDQLSXODFL VH V\PERO\ 0*çX SRURYQiYDW WYDU QHER YHOLNRVW
EH]REVDçQFK V\PERO* DOH QHPRKX SURKOiVLW çH QMaký takový symbol vyplývá z
MLQpKR 3RFKRSLW çH p implikuje q ]QDPHQi YGW MDN MH NRQFHSWXiOQt Y]WDK PH]L
obsahem p a q UR]XPW WRPX çH q vyplývá z p .G\ç QNGR QD SR tWDFtP VWURML
QiKRGQ ]Pi NQH WOD tWND  [  D QNGH VH REMHYt  QHGi VH "tFL çH E\ WHQ
VWURM QFR VSR tWDO 6WURM SRX]H PDQLSXOXMH VH ]QDN\ D SRNXG ]QDN*P QHS"L"DGtPH
QMDN VpPDQWLFN Y]QDP S*MGH MHQ R V\QWDNWLFNRX KUX
Turing: 'RE"H DOH WD EH]REVDçQi SUDYLGOD
Wittgenstein: 3URVtP W MHGQRGXFKRVW SUDYLGOD VH QHP*çH SOpVW V otázkou jeho
VpPDQWLFNpKR REVDKX 3RNXG UR]ORçtP SRFKRSHQt VORçLWpKR SUREOpPX GR SRFKRSHQt
MHKR MHGQRGXFKFK iVWt MH W"HED Y\"HLW RWi]NX WRKR MDNi MH MHMLFK VNXWH Qi
VpPDQWLND 5R]GOHQtP QD PHQt SUDYLGOD ]QDPHQi MHQ WR çH VWRMtP S"HG
problémem pochopit SUiY WDWR SUDYLGOD
-HGLQ ]S*VRE MDN RGVWUDQLW QRUPDWLYLWX NWHURX E\ PO VWURM SRFKRSLW MH GHILQRYDW
pravidla jako popis DNFH NWHUi VH Pi RGHKUiW Y P\VOL SR Wi"H 7tP VH DOH ]WUiFt
PDWHPDWLND NWHUi MH S"tVQ QHNDX]iOQt 0DWHPDWLND VSRMXMH WYU]HQt QD ]iNODG
DVRFLDFt D SHYQ GDQFK SUDYLGHO QH DNFt
Turing: 3DN WHG\ P*çHPH DOJRULWP\ FKiSDW MDNR QMDNp DNFH NWHUp PDQLSXOXMt VH
V\PERO\ D SUREOpP MH Y\"HHQ
Wittgenstein: $ VSROX V WtP MH Y\"HHQD L RWi]ND MHVWOL P*çH VWURM WDWR SUDYLGOD
pochopiW 3URWRçH SDN QHPDMt QLF VSROH QpKR V PDWHPDWLNRX QHO]H MH DQL FKiSDW
MDNR QFR FR E\ VH GDOR SRXçtW S"L YSR WX 9H]PL VL W"HED SUDYLGOR ´*272 67(3 ,
IF INPUT=0" .G\E\FK S"LVWRXSLO QD WY*M QiYUK WR SUDYLGOR E\ Y SRGVWDW ]QOR ´
$.7,98-( 3(6289$& 0(&+$1,6086 D PXVt X]QDW çH L NG\E\V GRNRQDOH
SRFKRSLO MDN WDNRY PHFKDQLVPXV IXQJXMH PDWHPDWLNX E\V ] QM QHY\VWDYO
Turing: $OH W\ I\]LFNp DNFH S"HFH S"HVQ RGSRYtGDMt MHGQRWOLYP SUDYLGO*P QRUPDWLYQt
matematiky, ne?
Wittgenstein: Ne tak zhurta. OdSRYtGDMt WR DQR $OH UR]KRGQ QH ]WOHVXMt. Pokud jsi
VFKRSHQ QMDNRX VRXVWDYX ]DNyGRYDW GR MLQp VRXVWDY\ MGH R QMDNRX VXEVWLWXFL DOH
Y]QDP Wp SUYQt VRXVWDY\ VH ]WUiFt .G\ç VL MDEOND KUXN\ D EDQiQ\ R]QD tP tVO\
P*çH EW ]DNyGRYiQt SURYHGHQR ]FHOD S"HVQ DOH Y RNDPçLNX NG\ WDNRYRX VpULL
tVHO S"HGORçtP QMDNpPX ORYNX Xç Y Qt QHQt DQL EDUYD DQL FKX(
$QR PRKX QMDNi PDWHPDWLFNi SUDYLGOD ]DNyGRYDW GR RSHUDFt D PRKX
]NRQVWUXRYDW VWURM NWHU EXGH W\WR RSHUDFH WXS SURYiGW 1HP*çH VH DOH X LW nic o
PDWHPDWLFH SURWRçH SUDFXMH SRX]H VH V\PERO\ NWHUp ]WUDWLO\ VYRX S*YRGQt
VpPDQWLNX 9VOHGHN P*çH EW VSUiYQ DOH QHMVHP VFKRSHQ GRNi]DW çH MH VSUiYQ
SURWRçH SRVWXS NWHU N QPX YHGO E\O PHFKDQLFN D QH PDWHPDWLFN
Pokud by si naopak pravidla VYRX VpPDQWLNX XFKRYDOD MH W"HED DE\ VH MLPL "tGLO
QNGR NGR MH VFKRSHQ MHMLFK VNXWH Q REVDK SRFKRSLW $OH 7XULQJ*Y VWURM SR tWi V
WtP çH RQ\ ]iNODGQt LQVWUXNFH QHY\çDGXMt çiGQp NRJQLWLYQt VFKRSQRVWL DQL
LQWHOLJHQFL 7R E\ PXVHO VWURM EW Xç ViP R VRE LQWHOLJHQWQt D Y SUDYpP VORYD
VP\VOX SUDYLGO*P UR]XPW.
Turing: 7DNçH ]PHFKDQL]RYDW YSR HW "t]HQ SUDYLGO\ ]QDPHQi KR QDKUDGLW MLQP
PHFKDQLVPHP D QH KR QMDN ´]WOHVQLW 'REUi 7UYiP DOH QD Wp iVWL Pp WH]H NWHUi
XND]XMH çH UHNXU]LYQt IXQNFH VH YHOPL GRE"H KRGt N PHFKDQLFNp LPSOHPHQWDFL
Wittgenstein: 9 WRPWR ERG V WHERX VRXKODVtP D Y\VORYXML WL KOXERN REGLY
Turing: $ Mi VL MHW SURP\VOtP MHVWOL EXGX WUYDW L QD WRP DE\ RVWDWQt S"LMDOL WX
ILORVRILFNRX iVW Pp WH]H
=iYU
:LWWJHQVWHLQ XSR]RUXMH QD WR çH 7XULQJRYD SUiFH ´2Q &RPSXWDEOH 1XPEHUV Pi
GY iVWL -- PDWHPDWLFNRX D ILORVRILFNRX 9 SUYQt iVWL XNi]DO QHRW"HO SRKOHG QD
Y\PH]HQt W"tG\ HIHNWLYQ Y\ tVOLWHOQFK IXQNFt D Y\W\ LO VPU YHGRXFt N PHFKDQLFNp
LPSOHPHQWDFL SDUFLiOQ UHNXU]LYQtFK IXQNFt 9 GUXKp iVWL VH S"HVXQXO N GLVNXVL R
SRYD]H SR tWiQt D "t]HQt VH SUDYLGO\ S"L HPç SURVW"HGN\ MHKR DUJXPHQWDFH E\O\ YH VYp
SRGVWDW HSLVWHPRORJLFNp QH PDWHPDWLFNp
= WpWR GUXKp iVWL 7XULQJ RGYRGLO VYRX S"HGVWDYX X tFtFK VH VWURM* NWHUp ]YOiGDMt nová
SUDYLGOD D SDN MH SRXçtYDMt WDNçH MVRX VDP\ VFKRSQ\ "HLW VWiOH YtFH REHFQMtFK
SUREOpP* :LWWJHQVWHLQ WHQWR VPU XYDçRYiQt QDSDGi 7YUGt çH DUJXPHQWHP NWHU Pi
GY VWUiQN\ =DSUYp SRNXG Pi ]iNODGQt SRGSUDYLGOR QMDN Y]QDP QHP*çH VH MtP
"tGLW stroj bez kognitivních schopností. Zadruhé, pokud je podpravidlo zcela bez
Y]QDPX QHP*çH EW VWURMHP Y SUDYpP VORYD VP\VOX SRFKRSHQR D SDN SRXçtYiQR N
"HHQt SUREOpP*
9 WpWR SUiFL E\O\ REMDVQQ\ ]iNODG\ ]H NWHUFK 7XULQJRYD WH]H Y\FKi]t D MHKR SR]GMí
LQWHUSUHWDFH ILOR]RILFNFK Qi]RU* NWHUp Y WpWR WH]L XYHGO 6RX DVQ VH ]GH ]NRXPi
:LWWJHQVWHLQ*Y SRKOHG QD ILORVRILFNRX iVW WH]H KODYQt REODVW MHKR QiPLWHN D GLVNXVH
QDG SRYDKRX SUDYLGHO D MHMLFK PHFKDQLFNpKR SURYiGQt 6DPRWQp ]KRGQRFHQt YVOHGNX
tétR GLVNXVH SRQHFKiYiPH QD ODVNDYpP WHQi"L
Literatura
Shanker, S. G.: Wittgenstein versus Turing on the Nature of Church's Thesis, Notre
Dame Journal of Formal Logic vol. 28/4, 1987
Church, A.: The Constructive Second Number Class, Bulletin of the American
Mathematical Society, vol. 44, 1938
Davis, M.: Why Gödel Didn't Have Church's Thesis, Information and Control, vol. 54,
1982
Knuth, D.: Algorithms, Scientific American, vol. 234, 1977
Turing, A.: On Computable Numbers, with an application to the
Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42,
1939
Turing, A.: Intelligent Machinery (1948), Machine Intelligence 5, Edinburgh, 1969
Wang, H.: From Mathematics to Philosophy, Routledge & Kegan Paul, Londýn, 1974
Wittgenstein, L.: Philosophical Investigations, Bassil Blackwell, Oxford, 1973
Wittgenstein, L.: Philosophical Grammar, ed. Rush Rhees, Basil Blackwell, Oxford,
1974
Wittgenstein, L.: Lectures on the Foundations of Mathematics, Basil Blackwell, Oxford
1978
Wittgenstein, L.: Ludwig Wittgenstein and the Vienna Circle, rozhovory zaznamenané
Friedrichem Waismannem, Basil Blackwell, Oxford, 1979