Conceptual Notation and Related Articles

152 Pages·1972·58.56 MB·English

by  Gottlob Frege

GOTTLOB FREGE
CONCEPTUAL
NOTATION
AND RELATED ARTICLES
Translated aud edited
with a Bíography and lutroducíion by
TERRELL WARD BYNUM
OXFORD
AT THE CLARENDON PRESS
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PRESS i<;72 GOTTLOB FREGE
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ONiversitY
PREFACE
The present volumc is the culmination of a project begun in 1962
at the University of Delaware where 1 first became acquainted
with Frege's philosophy. After reading several of Frcge's works, I
attenipted to learn more about the man himself, only to discover that
very little was known about him and only a few of hís forty or more
publications had been translated into English. I was surprised to
discover that the book in which Frcge was said to have rcvolutionized
logic, the Be^riffsschrift, was( except for Geach's translation of most
of Chapter I) untranslated and unavailable at major universities even
in the original Germán. My curiosity had been aroused, however,
and I managed to acquire a microfilm of the Begriffsschrift, so I
began transiating parts of it. The more I learned about Frege, the
more interested 1 became in his Ufe and work.
During the year 1963-4 a Fulbright Fellowship in England pro-
vided the time and wherewithal to seek out information on Frege
and find some of his lesser-known works in European libraries.
The summer of 1965 proved to be the most fruitful and exciting
time for the project. A grant from the Danforth Foundation and one
from the Robert Wethcrill Trainer Memorial Fund of Chester,
Pennsylvania, made it possible to go to Europe. My wife and I visited
the Frege Archives in Münster, Germany, and received permission
from Professor Hans Hermes to read some of Frege's unpublished
works and letters. At Fribourg, Switzerland, we met Ignacio Ange-
lelli, who generously provided information and biographical data
which he had gathered over several years. In London I talked with
Rudolf Carnap about his experiences as a student in Frege s courses
at Jena University. At Bertrand Russelfs home in Wales I discussed
with him his discovery of Russell's Paradox and his subsequent cor-
respondence with Frege. Conversing with Russell over tea and read
ing Frege's original letters at Russell's desk are among the most
memorable events of the project for me.
By the end of the summer of 1965 I realized that a Frege renaj^s-
sance' was under way in América and Europe. Frege s phdosophy
was being studied avidly, and some of his works, mcludmg the
Begriffsschrifu were to appear in English. I was disappomted to learn
PREFACE ix
V"' PREFACE
that my translation would not be the first into print; but my proiect ledgcmcnts are also diic to Gcorg Olms Verlagsbuchhandiung and
was well advanced, and I resolved to finish it. A Fregc biography Harvard University Press for permission to publish my translation
now would be even more timely, and a second translation could be of the Be^rijfsschrift.
beneficial in brmging out difTerences of interpretation. It is impossible to cxpress fully my gratitude lo my wife, Aliñe,
The first draft of the present volunte was written at Princeton whose encouragement, translatíng skills, and crilical and secretarial
Universtty dunng 1965 and 1966 on a grant from the Danforth eíTorts wcre indispensable to the success of this project.
ountoion. In completmg the translations, I benefited especially
Albany, New York
from my wife, Aliñe, who aiso shared
WbLgraphV^^'^'"^''"®' endiess, job of compiiing the 20 March 1972
The volume was prepared with two major objectives in mind; (1)
to serve as an mtroduction to Frege's eariy philosophy, and (2) to
rsTnTntr H r' rosearch. The student using the book
™ssir nü n" 'he more teehnical
íirarks p ^"holar could forgo the more elementary
lopirallv ^ ^ themselves, rather íhan appearing chrono-
two latp' u" 'hat makes them easier to understand—
aim of tíip^fi'^ T respectively the justification and the
readerwilirfnTsf" Begriffsschrift itself. The
essav and fhe hetween the biographical
on its own tu ' ® fa'roduction, since each was written to stand
on Its own without reference to the other
man^^^Znlp"!*'''' from the help of
wh^eenLousí ™ ""'«hted to my friend Ignacio Angelelli
2c ours'er to profit from his wide knowledge and
hébedTn wa 7' A. Thomas who has
Sed topTf too subtie to describe. I am in-
ÍaSlwfv bef 777^"^ Radolf Carnap, and 1 regret that they
fuTtfpre^r S'^titude in print. I am grate-
thl manu crb: ; "Ú E- Jordán for reading
I have benefitpH^i criticisms and suggestíons.
Fidaríans Church Dagfinn
Günther 'pat ' "'teiithal, Montgomery Furth, William Kneale,
tbtrod, ced R^omrb. who first
and wispc i°t °f'o" been a source of encouragement
n
that^ri H f t° the late John R. Parsons and sorry
The cltnd" 7V P°'trait of Frege on the Frontispiece.
nrenarino provided valuable advice and assistance in
oOIf g«rreeaatt sseerrvviicde in helpmg to prepare athned ISnadnedxy o Sfa Ñvaagmee sS.hu Afcelktn owwa-s
V, . .'íM
tt
CONTENTS
ON THE LIFE AND WORK OF GOTTLOB FREGE
§1. Introduction 1
§2. Family Background 2
§3. Boyhood 2
§4. University Studies 3
4
§5. Habilitation
§6. Between Habilitation and Conceptual Notation* 6
§7. Conceptual Notation 10
§8. Reception of Conceptual Notation 15
§9. Between Conceptual Notation and Foundations ofA rithmetic 20
§10. Foundations of Arithmetic 24
§11. Reception oí Foundations of Arithmetic 28
§12. Between Foundations of Arithmetic and Basic Laws of
Arithmetic I 30
§13. Basic Laws of Arithmetic I 34
§14. Reception of Basic Laws of Arithmetic I. Frege and Peano 37
§15. Between Basic Laws of Arithmetic I and Russell's Paradox 41
§16. Russell's Paradox: Frege and Russell 46
§17. After Russell's Paradox 50
55
editor's introduction
translations of frege's works
On the Scientific Justification of a Conceptual Notation 83
On the Aim of the "Conceptual Notation" 90
Conceptual Notation 101
Applications of the "Conceptual Notation 204
* 'Conceptual notation' is the English translation of Frege's term 'Begriffs-
schrift' In the present volume, when 'conceptual notation' appears in italics, it
refers tó Frege's book; when it is enclosed in double quotation marks,i t denotes
Frege's symbolic language; when neither occurs, it means simply a language
like Frege's.
xii
CONTENTS
appendix i: reviews of the Conceptual Noiation by
fregeos CONTEMPORARIES 209 ON THE LIFE AND WORK OF
appendix II: evaluations of frege's Conceptual GOTTLOB FREGE
Notation by present-day scholars 236
§1. INTRODUCTION
A frece bibliography, 1873-19661
239
During the past twenly ycars, Gottlob Frege has become one of
The works of Frege
240
the most studied modcrn philosophers. Because of the present strotig
Secondary Sources 248 emphasis ¡n philosophy upoti logic and litiguistic a.ialys,s the works
Additional Sources 270 of Frege are of primary interest, since he (1) revolutiomzed logtc and
senrantics,( 2)¡ nitiated the "linguistic turn" in philosophy.( 3) riiadc
índex 289 a major contribution to the philosophy of mathemat.cs and (4)
significantly influenccd leading scholars hke Peano, Husserl, Rtissell,
in the bibhography are rcferred to in the text by their serial rtuntl't''®'
t Items Wittgcnstein, Carnap, and Church.
Given thcse facts, it is astonishing that no major biography of
Frege was cver published; and it is tragic that the present biographical
sketch was made only after the Second World War,i n which valuable
records and documents were destroyed (including a biography of
Frege by his adoptcd son Alfred).i Even before the war, the amount
of inaterial avaiiable would have been small. Now, relevant informa-
tion is scarce indeed; and the following account liad to be pieced
together from biographical fragments gathered from many places.
Very little is knowti about Frege's personality and prívate life, so
the prLent elTort must deal, for the most part, w.th hts works and
the main events in his career. A sketchy account, at least, can be
given of his person and eharaeter. A deseription from W.ttgenstem
Ld an extatit photograph3 y.yeal that "Frege was a
with a short beard," who bounced around the room when he talked .
It is clearLm eisting records and Frege's wntmgs that he was a
. . VI» íitems ¡n the Bibliography of the present
^ Listed (1935) m the numbers) as among the documents in the
volume are rcferred to by (gj. Germany. However,P rofessor Hans
Frege Archives at the University biography is unknown to him and
Hcrmes at Münster has mformed me bmgrap y
must have been lost during the Second World War.
® Included in S22. . ^ frontispiece of the present volume is based
The portrait which appea provided from the Archives of
upon that photograph, which g indebted to Professor
Jena University by Protessor . • n^otograph's existence and enabling
Ignazio Angelelli for mformmg me of the pnotogr p
me to obtain it.
8243501
I
• • ''V*' '- :. V'v
'i ':X
2 UFE ANDWORKOFGOTTLOB FRECE
LIFE AND WORK OF GOTTLOB FREGE 3
He was an effect¡vra''ndtonrc''^ remarkable energy and persistcnce.
wliole pcriod in Wismar (a ílourishing commercial town, with a
explaining very complex materia"' He h'^^H í'"''
populalion of about 12,000, and one of ihe best harbours on the
could cut to the heart of a nmM logical insighl which Baltic). Frcge's family, like most familics in the región, was Lutheran.
polemicist. When aroused he always a formidable His father died in 1866 whilc Fregc was still at the Gptj/iasi'um in
bis only major interests were hhZ v ^ ñongue/' Apparently, Wismar, whicii he attended for five years (1864-9) under the head-
. Throughout bis lífe Fr^r mastcrs Krain (father and son). He passed his graduation examina-
just a boy, bis fatber died I P'^Sued by tragedy. When he was
tion (A/?in/r) in the spring of 1869 and immediately entcred Jena
ied young, and then bis wifí> marricd, bis children
University.
formal system in which to derive^^f^h ^^^"ty-five years developing a
This is all that is known at present about Frege's early life. There
that ene of bis assumptions can Je may have been many more details about this pcriod in tlic biography
of Frege that his adopted son Alfred gave to Professor Heinrich
bis monumental achievementc ^ ín spite
w 1 e he was alive; and bis death' Jo-?^ '"oceived little recognilion Scholz for the Fregc Archives in Münster in the 1930s.'
he seholarly world. It i onlvt ^nnotmed by
has arisen. and bis woJk^are h ^ ^ " general interest in his
§4. UNIVERSITY STUDIES
s betng Widely rcad and discussed.
Again, unfortunately, almost nothing is known about Fregc s life
as a university student, except the courses and professors he had. It is
Friedrich Ludwi' ^■^CKground olear, howcver, that he impressed his teachers as a bright, assiduous
MecMenburg on^Líuic Sf Germany (in young man p but it is unknown, for example, where or how he lived,
or what induced him to move to Goltingen University after two years
F"mS"í,í" '■ «Sí;
at Jena.-
Frege spent four semesters at Jena, from the spring of 1869 to the
Hamburo c was a businec' Goiihb Emanuel
winter of 1870. His professors were Geuther in chemistry; Fischer in
blotzky (9_i87of girls' school His m ,t founder philosophy; and Abbe, Schaeffer, and Snell in mathematics.^ During
Hter princ¡n!i Frege was B ^"£<^510 Biallo- his five semesters at Gottingen (Easter 1871-December 1873) he
Folish extra r ° school He was a teacher and studied philosophy of religión under Lotze; physics under Weber and
in the area S'°n- aiso ñame may indícate
Riecke; and mathematics under Clebsch, Schering, and Voss.-^
Arnold Freee ^ ^^'®oburg región sinr^ ° peoples who * See S253 and footnote ! to §1 above.
P-£e's brotheS " « W'" century A.a ^ This was the descriptlon of Frege as a student included in an officíal memor
ándum from Professor Cari Snell to Professor Ernst Haeckel at Jena University.
io52, may have been
^^"pcrLrpt Ernst Abbe. who had gone to both Jena and Gottingen himself, and
who donated millions of dollars (through the Cari Zeiss Síi/íiws) to Jena ín an
Almost nothing is ®°^HOO0
eíTort to make it as good as Gottingen, persuaded Frege to go to Gottingen in
order to get the best education possible, and then return to Jena to teach. Abbe
Was always impressed with Frege's great ability in logic and mathematics.
■--«..a.,,,, ••"-PParentlyspentthis = From a Lebenslauf submittcd by Frege to Gottingen University on 12
August 1873. r 11 •
* The courses which Frege took at Gottingen were the following:
Summcr 1871- Anaíytic Geometry with Professor CIcbsch, Images of Sur-
15 «n 2 August 1873. faces with Professor Clebsch, Functions of Complex Variables with
Professor Schering, Philosopliy of Religión with Professor Lotze.
"itrom the Frankfurt Goethe Winter 1871-2: Experimental Physics with Professor Weber.
[cont. onp.4
* work OF GOTTLOB FREGE LIFE AND WORK OF GOTTLOB FREGE 5
and which she was then dirccting, was suíTicient lo support herself and
he'sf h"' of complex
her son.
with Scherins in r.Ktti ' n- ^ Abbe in Jcna° and
Anothcr document which Fregc submitted was the article "Methods
a functional calculas coma' ' whlch dcvclops
of Calculaíion Bascd upon an Amplification of the Concept of
te logic,' was apparentlv '"s great contribulions
Magnitude", whicii was his Hahilitationsschrift? In March 1874
tion. composed at Gottingen bcfoce bis gradua-
Professor Ernst Abbe submitted to the University officials a report
on this paper.'* The report shows that the Jena mathematics faculty
"'etrical disscrtation, On a Geo-
were most impressed by Fregc's Schrift. Abbe found it to be erudite,
granted the decree of Dn^t l Z « Plane,^ and was
clear, and to the point; bearing the mark of truc originality and
December of that year ° Philosophy in Góttingen on 12 extraordinary creatlve powcrsj and containing novel points of view,
sagacious explanations, and surprising relations among widely
separated regions of analysis. He speculated that it might contain the
seeds of a comprehensive new point of view which, when fully
§5. HABILITATION
As soon as he received his Ph n t? developed, would atlain a durable significance for mathematical
Erast Abbe) applicd for a tP i- ' sponsored by analysis. Thus, Abbc's report was a truly impressive recommendation
several documenís he submitt^^ Position at Jena. Among the and, as it lurned out, a remarkable piece of foresight.
n
that she would support hím i statement from his mother Since Frege's documenís were acceptable, he was admitted to an
salaried.2 At that time she'w P^^^ion he sought was un- oral examination on 18 April 1874.-'^ The questions asked were very
the income from the girls' school^ ''"t apparently
general ones related to topics in Frege's Habilitatiomschrift. At the
end the committee was satisfied that Frege had mastered liis subject
Sum™ had founded
matter and achieved defensible views; but they were less impressed
Wimer'l872-3'^rforr' Professor W b
than they liad been by tlie Habilitationsschrifí itself, since Frege's
maticfli TI, '0<-ir-hour Dhvsirc uu . Webcr.
■^ieche, Mathe- answers were "neither quickwitted ñor ñuenf.®
Having sustained his oral examination, Frege was recommended
Schering. Selected
by the committee for acceptance into the University. They requested
speed in accepting him, because Professor Snell was not well, and
Frege was willing to take ovar part of Snell's duties. On 6 May he was
admitted as a lecturer {Privatdozent), provided that he would sustain
pús is Fz"?SS"' ^ prcsent section.
a public disputation and present a satisfactory trial lecture. By 18
" "S virtually unstudied^F °f ¡nteresf aíd P''"''^"'"^S"ires griinden.
Pmfessor An8elel|¡wl:^°rt™Mely, this dro,' ™Pwtance; but, thus far, May these requirements had been fulfilled;' and Frege had become
--.w^ovi rtngeie]irsrpnr:„f 7^ ^nis circumctn^ uui, mus lai,
a member in good standing of Jena's mathematics faculty.®
SlxpsT'ely" there is
=» See footnote 6 of §4, and also F2 of the Bibliography below.
• Apparently sent to Professor Ernst Haeckel on 30 March 1874.
pber eme geometrillZn "^e
® Given by Professors Abbe, Fortlage, Geuther, Haeckel, E. Schmid, M.
Bíbliography, pi t?- • ® ^^rstelhmg der •
and important material. Uis^I Schmidt, and Strasburger.
® From a report on Frege's examination to the University officials submitted
' From a re "ncluded in F'52 contain interesting by Professor Haeckel on the afternoon of 18 April 1874.
' The public disputation was held on Saturday, 16 May 1874 at 10 a.m.; and
SSlPiSS-ss!"--
the trial lecture was given at noon on the followíng Monday.
® Fregc worked most of his Ufe on problems in mathematics and philosophy;
and it is likely that he discussed his work with fellow faculty members in those
two disciplines. Mathematics and philosophy faculty members who were at Jena
« UFE AND WORK OF GOTTLOB FRECE
LIFE AND WORK OF GOTTLOB FRECE 7
SS. BETWEEN HAB.L.TATION AND CONCEPTUAL
spondcd with many Europcan scholars, incliiding such important
people as Husserl, Peano, Russell, Hilbert, Lowenheim, and Wittgen-
notation
Thus, in the summer semester nf tr stcin.= He met Lukasicwicz, Wittgcnstein, and Vailati( and, no doubt,
career. His Ufe durine fortv 'i'S teaching many others). He pubÜshed 40 works during his lifelime, including
and hermit-like as is commLi secludcd 4 books, 24 articles, S revicws and 4 commcnls and remarks.^ He
the Deutsche Akademie der N t ^o^ay. He was a member of aiso wroic a scorc of other works which did not appear bcfore his
matemático di PaJermo death, but have recenlly been published in one volumc/^
n The happicst period of Fregc's profcssional career appears to have
^chaft für Medicin uitd Natu ° ^^25), the Jenaische Gescll-
the Deutsche t° '917), and been the beginning, wliile he was a lecturer {Privaidozent). He was
read several papers at meetine. '^97 to 1925). He
three years—1899-1901_he w organizations.' For 2 The following is a list of thosc who are known to have had corrcspondence
with Frcec Whercvcr possible. the date of the corrcspondence and the number of
Johannes Thomae) of the L7™ Treasurer (along with
extant pagcs (if any) are includcd. Many of thcsc Icltcrs contain intercsting and
important technical disciissions which oiight to be published.
"^n g^n'^ation. He corre-
at JenaR "unng F^ege^s forty.four year, ,h. , Avenarius 1882 I1 ppaaggee
Ballue 1895-7 66
Mathematics: Couturat 1899-1906 1122
Darmstaedler 1913 and 1919 22
KarírT^'^''^^®)
Dingler 1917 66
Falchenberg 1890 11
Hilbert 1895-1903 2222
Hónigswaid 1925 55
Huntington ? 33
Husserl 1891-1906 1100
Jourdain 1902-14 2299
^oben Karl Hermán Í '^0 8)
Klein 1881 11
Philosophy: Haussner (1905-31) Knoch 1893 22
Arnold'ioV^GÍr/p'^^:^^; 74-85) Koebner 1891 22
Korselt 1903 55
August Julius Lowenheim ? nnoonnee
Mayor 1896-8 44
Hermann Guido (1871-6)
Pasch 1894-1903 1100
(1873-5) Peano 1894—1903 2200
Russell 1902-12 5511
Scheibc 1919 11
Schlomilch 1881 II
Ulrici 1881 11
Vailati 1904-6 33
Wittgcnstein 1910 or 1911 none
An unknown number of Frege letters were lost during the Secoiid World War.
These include the large corrcspondence with Lowenheim mentioned in S253.
Wittgcnstein mcntions his corrcspondence with Frege in the quotation included
in S22; but so far there is no trace of Ihem. Perhaps the Second World War
^S:S!ar^~
claimed them as wcll.
(Most of the extant letters, and perhaps all of them, are collected at the Frege
Archives in Münster, Germany.)
® See the Bibliography below.
* These are edited by H. Mermes, et al., and published (1969) by Félix Memer
in Hamburg, Germany. See F45.
ját
8 l ife and work of gottlob frece
LIFE AND WORK OF GOTTLOB FRECE
For Frcge it was especially important to get the fundamental
malhemalical sludents at points absolutely pcrfect—to make complelely olear the root
DtónfthTs b ° Ive" ""T"' f"'"--
concepls and basic assumptions upon which a discipline is built.
courses and published fcle extra-heavy load of
Indeed, this was the chief aim of all his major works and the root of
a little book {Conceptual Notar Seomelry, and
most of his criticisni of other scholars.
later scholars would come to r/"" °" matlicmatics that
This desire for clarily and correctness at the fundamental level,
From the hpainn; V as monumental, together with a seemingly insignificant revicw which Frege published
particularly in 1878 when n unusually large, during his first year of teaching,® may have bcen the origin of his idea
volunteered to taL over n. 7^?^ i"- ^regc
for a project that shaped his career and consumed most of his vorking
carry fais own load besides so doT'' mathematics and
life—namely, proving that ariíhmetic is a branch of logic. The book
that Frege rcviewed was H. Secger's The Elemenís of Anthmetic,^°
gapmthemathematicalteacbina ' a" intolerable
could be acquired a yearTr J°'^^""^sThomac
which was meant to explain to students the fundamentáis of arith-
him to sacrifice much of extra teaching ccrtainly forced metic. Frege's main criticisms are that the basic laws of arithmetic
making great strides in logic and jusí when lie was
are left unproved and the fundamental concepts are poorly defined:
for the sake of the stu7etrm7k* designcd his lectures Aftcr somc parlially unfortunatc cxplanations of the calculating opera-
tions and thcir symbols, some propositions are prescnted in the second and
avoiding unnecessary comniexitv Tf ^ thoughtful and third chapters undcr the title of "the fundamental theorcms and the most
appreciated by the conseípnt;^ ^ his courses were much essential transformation formulas". Thcse propositions, which actually
seems to have been true from T ^^o attended them. This
form the foundation of the wholc of arithmetic, are lumped together
end. Rudolph Carnap, who atte^deíF
without proof; while, later, theorems of a much more limitcd importance
who is perhaps Frege's ereatP./ 7 ® courses in 1910-13, and are distinguished with particular ñames and proved in detail. .. . The
inspiration I received from im' ^ í"eports: "the most fruitful amplification of concepts which is so highly important for arithmetic,
those in the fields of philosonhT!'7^ not come from and is often the source of great confusión for the student, Icaves much to
ft her from the lectures of Fr. ">^'11=0,31105 proper, but be desircd. .. . The result of all these deficiencies will be that the student
fields".' °f F'ege on the borderlands between those will nierely memorize the laws of arithmetic and become accustomed to
being satisficd with words he does not understand.^^
Frege was not only an pff..,-
upon those who attended his lectur demanding Perhaps it was his disappointment with Seeger's book that gave
Umverstty offieials in Januarv mo W''work for the Frege the idea of wriling a olear, exact presentation of the funda
courses were little suited to niLe 1 reported that Frege's
mental notions of arithmetic. The very things that he finds lacking in
I i s just an exercise fo fu! "for whom Seeger's book—olear definitions of fundamental arithmetical
by yutue of the great clSld?" ' concepts and proofs of the basic laws of arithmetic—become the
major aims of his whole logistic programme. Conceptual Notation,^^
intr f ° "looghtfulness of °f his expression and by
the first of four volumes devoted to the project, was completed four
'"toduce aspiring listeners to the d «1 is particularly fit to
years after the appearance of the review in questíon enough time
leftu 7 repeatedlv^'"! ®"^'''''"'"'hcn""'<=al
to conceive a plan and publish a little book about it.
Iverv f J appeTredt "PP^^'-ndy to hear
y undamental point."8 be absolutely perfect on We shall probably never learn whether or not this actually was the
path that led Frege to his logistic programme. From this point on.
Pr®o fFersnom" D5rí ' ...1 • . «"-"ten by Abbc on 10 I
Unívepity ArchiSrr °/ '<="-. ProfesTor « am indebted to ® F3
" H SecRcr Die Elemente der Arithmeíik (Schwerin i. M.: A. Hildebrand,
" Also from 81^°' this docu^^ "^"eeldli. and the Jena
1874). ' " From F3. "F 7.
, ' .wiu.;
"om A34, p. 4.