Lacan and Gödel

Richard Klein 

http://www.lacan.com/thesymptom/?p=54

In “Science and Truth” there is a reference to Gödel’s theorem of incompleteness (Lacan,196! "61#$ Gödel %as a mathematical lo&ician %ho in'ented his theorem in 191$ It isapplied to formal s)stems and asserts that those containin&” minimum of arithmetic areincomplete and inconsistent$ The arithmetic the) contain is lo&ical accordin& to *eano’ sa+ioms$

Lacan is appl)in& this theorem to the su-ect of science$ I %ill tr) to sho% that this is thepoint in the Écrits at %hich a formalisation of ps)choanal)sis e&ins as a s)stem %hichcontains a minimum of arithmetic$

Gödel’s proof his conclusion is left for %homsoe'er desires to %or. throu&h it$ /nl) theconclusion is stated as follo%s! In an) formal lan&ua&e 0 there e+ists a statement S suchthat if 0 is consistent, neither S nor its ne&ati'e can e pro'ed in 0$ The propositions of 0cannot e pro'ed ) reference to 0$ It does not mean that S is false ut undecidale$ Thea+ioms of 0 are not -ust incomplete ut incompletale since the addition of an a+iom doesnot loc. the emer&ence of another statement S’ that cannot e decided$ is theorem ofincompleteness entails that the consistenc) of 0 cannot e pro'ed ) an) means %ithin 0$

Let 0 e the /ther %hich is the concept of the unconscia2us$ In “Science and Truth”, Lacan isnot appl)in& the theorem to the concept of the unconscious ut to the su-ect of theunconscious$ 3e'ertheless, Gödel’s theorem asserts that there is a lac. in the /ther, thatthe /ther is incomplete and inconsistent, %hich is %h) Lacan %rites it as the arred /ther!0

0 statement S in the field of the /ther cannot e &uaranteed as true$ Ta.e a construction oran) form of interpretation %hich attempts to complete the /ther and ma.e it consistent,that is, to fill out the lac. in the /ther$ Such an intelpretation is neither true nor false, utundecidale4 it is a proposition of the /ther and cannot e pro'ed ) reference to the /ther$5hate'er fresh .no%led&e follo%s in the %a.e of an interpretation is not an indication of itstruth since this .no%led&e is also a proposition of the /ther$ 3either the anal)sand’s )es’nor his no’ are si&ns of the truth or falsit) of an interpretation$ 7reud sa)s in “8onstructionsin 0nal)sis” that %hat is important is %hat comes indirectl) (7reud, 19d#$ The anal)sandsa)s “no” and some%here else sa)s “)es$” That %ould e an interpretation that .eeps thesu-ect di'ided et%een the true and the false, %hich does not suture the su-ect$

The su-ect of science is ein& made the focus of lo&ic$ :odem lo&ic sutures the su-ect ofscience$ Gödel’s theorem, sa)s Lacan, demonstrates that the suture has failed$;

Lo&ic ma.es a decision on %hat is true and false$ 0ristotelian lo&ic ma.es it in naturallan&ua&e$ :odern lo&ic creates an artificial lan&ua&e, that is, a formal s)stem, in %hich thedecision is made$

The su-ect of natura< lan&ua&e is descried ) Lacan in “Science and Truth” as thespea.in& suh-ect of lin&uistics in %hich the su-ect is determined as meanin& in a atter) ofsi&nifiers (Lacan, 196! "6=#$ This is not the su-ect of science$ The su-ect inps)choanal)sis, ho%e'er, is the su-ect of science (Lacan, 196! ""#$ The su-ect of

 

science cannot e, then, a su-ect of natural lan&ua&e$ It seems to me that this can eta.en as the point at %hich a formalisation is e&innin&$

:odern lo&ic attempts to re'eal the structure of science ostensil)$ Lacan sa)s it sutures,not science, ut the su-ect of science$ Science does not sa) true or false4 the su-ect does$In a philosoph) of science called lo&ical empiricism theoretical terms are made dependenton oser'ation terms$ The truth of the oser'ation terms must e &uaranteed in theoreticalterms$ The su-ect of science must al%a)s e true$ The su-ect %ho sa)s “no” andsome%here else sa)s )es’ is di'ided et%een the true and the false (see :iller, 199>#$Suturin& this di'ision ma.es the su-ect true$ Gödel’s theorem confIrms the e+istence of thedi'ision$ The su-ect is a lo&ical inconsistenc)$ The fIrst step in this formalisation assertsthat the su-ect is undecidale, %hich is an indication that the su-ect contains arithmetic$

In the clinic it is an empirical fact that the su-ect spea.s a natural lan&ua&e$ /n the otherhand$ the di'ision of the su-ect is not an empirical fact ut the effect of a reduction %hichma) ta.e a lon& time to accomplish (Lacan, 196! "#$ This reduction has to do %ith theshrin.a&e of .no%led&e since the su-ect is also descried ) Lacan as the result of there-ection of .no%led&e (Lacan, 196! "6#$ The reduction is the direction of the treatmentto a decompleted and inconsistent /ther$ the effect of %hich is the su-ect of science$

The su-ect is di'ided, Lacan ar&ues, et%een truth and .no%led&e (Lacan, 196! "6#$ If.no%led&e shrin.s, it is contin&ent$ In lo&ic truth is necessar)$ It is not, ho%e'er, thesu-ect thal is necessaril) true$ 0ccordin& to Gödel’s theorem, it is a lo&ical inconsistenc)$

There is a formalisation of oth ends of the fantas)4 e&innin& %ith the su-ect, arithmeticis introduced into the s)stem, and, therefore, Gödel’s theorem is asserted$ In the paper thatprecedes the one under consideration, Lacan e&ins ) statin& that the dri'e as constructed) 7reud is prohiited to ps)cholo&isin& thou&ht %hich supposes a moral in nature (Lacan,196>! "1#$ ere is a &ood reason for formalisation$ The asis of ps)cholo&isin& thou&ht isnatural lan&ua&e$ The dri'e cannot enter natural lan&ua&e$

In his introduction to the Foundarions of Arithmetic , 7re&e sa)s that his method &oesa&ainst ps)cholo&isin& thou&ht$ 7ormalisation constitutes a reduction that ma) ta.e a lon&time of ps)cholo&isin& thou&ht$ ?no%led&e shrin.s, and the su-ect encounters the truth ofthe dri'e$ The dri'e di'ides the su-ect and desire (Lacan, 196>! "#$ The truth of desireseems to account for the lo&ical inconsistenc) of the su-ect$ Such is the structure offantas), accordin& to Lacan$

0lso in the article %hich precedes “Science and Truth” is Lacan’s account of the point at%hich an anal)st is made (Lacan, 196>! ">#$ 0t the end of anal)sis the dri'e hassomethin& to do %ith the emer&ence of the desire of the anal)st$ This is not enlar&ed upon) Lacan here, thou&h else%here he states that it is also the desire of the anal)st %hich haseen operatin& in the accomplishment of the anal)sis$ It must e the desire to create alan&ua&e in %hich the su-ect can sa) the truth$ 7ormalisatn is the e+pression of therelation of the desire of the anal)st to the truth$ It seems to me that the desire of theanal)st is structured ) Gödel’s theorem, and the form of interpretation must e affected )it$ 5ithout it, there %ill e no concept of the sometimes lon& reduction to the di'ision of thesu-ect, to the point of a manque à savoir , a %ant2to2.no%, since the su-ect is a lac.outside .no%led&e$ It seems to e a reduction to the first a+iom of *eano! @ero is a numer$0 lac. in the foundations is -ust this @ero4 a painful emptiness that %ill ma.e the su-ectinconsistent et%een the true and the false, and ma.e it desire to find an /ther that iscomplete and consistent$

 

Gödel’s Incompleteness theorem

The openin& para&raph of ?urt Gödel’s 191 paper!

The de'elopment of mathematics in the direction of &reater precision has led to lar&e areasof it ein& formali@ed, so that proofs can e carried out accordin& to a fe% mechanical rules$The most comprehensi'e formal s)stems to date are, on the one hand, the PrincipiaMathematica of 5hitehead and Aussell and, on the other, the Bermelo27raen.el s)stem ofa+iomatic set theor)$ Coth s)stems are so e+tensi'e that all methods of proof used inmathematics toda) can e formali@ed in them4 i$e$, can e reduced to a fe% a+ioms andrules of inference It %ould seem reasonale, therefore, to surmise that these a+ioms andrules of inference are sufficient to decide all mathematical Duestions %hich can eformulated in the s)stem concerned$ In %hat follo%s it %ill e sho%n that this is not thecase, ut rather that, in oth of the cited s)stems, there e+ist relati'el) simple prolems oftheor) of ordinar) %hole numers %hich cannot e decided on the asis of the a+ioms$

?urt Gödel, “Eer unentscheidare SFt@e del Principia Mathematica und 'er%andterS)steme I”, Monatshefte für Mathematik und Physik  "! 129"$ 8omposite translation,cited in Aa)mond Smull)an, The Lady or the Tiger? , armonds%orth! *en&uin, 19", p$16$