19th Century Logic between Philosophy and Mathematics
Volker Peckhaus
Institut für Philosophie
der Universität Erlangen-Nürnberg
Bismarckstr. 1, D-91054 Erlangen
E-mail: [EMAIL PROTECTED]
From:
http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/texte/logic_phil_math.html
1 Introduction
Doubt could be expressed that a special section on late 19th
century mathematics, or, more specifically, on Victorian mathematics, was an
appropriate place for a lecture on 19th century logic. Most 19th century
scholars would have been of the opinion that philosophers are responsible for
research on logic. On the other hand, the history of late 19th century logic
indicates clearly a very dynamic development instigated not by philosophers,
but by mathematicians. The central feature of this development was the
emergence of what has been called the "new logic'', "mathematical logic'',
"symbolic logic'', or, since 1904, "logistics''. This new logic came from Great
Britain, and was created by mathematicians in the second half of the 19th
century, finally becoming a mathematical subdiscipline in the early 20th
century. This development is, thus, at the heart of Victorian mathematics.
The 100th anniversary of the death of Charles L. Dodgson, i. e.,
Lewis Carroll (1832-1898) was the motivation behind this special section on
late 19th century mathematics given at the Commonwealth University in Ottawa
which has just commemorated its 150th anniversary. Carroll's well-known books
on logic, The Game of Logic of 1887 and Symbolic Logic of 1896 of which a
fourth edition had already appeared in 1897, were written "to be of real
service to the young, and to be taken up, in High Schools and in private
families, as a valuable addition of their stock of healthful mental
recreations'' (Carroll 1896, xiv). They were meant "to popularize this
fascinating subject,'' as Carroll wrote in the preface of the fourth edition of
Symbolic Logic (ibid.). But, astonishingly enough, in both books there is no
definition of the term "logic''. Given the broad scope of these books the title
"Symbolic Logic'' of the second book should at least have been explained.
Maybe the idea of symbolic logic was so widely spread at the end of
the 19th century in Great Britain that Carroll regarded a definition as simply
unnecessary. Some further observations support this thesis. They concern a
remarkable interest by the general public in symbolic logic, after the death of
the creator of the algebra of logic, George Boole, in 1864.
Recalling some standard 19th century definitions of logic as, e.g.,
the art and science of reasoning (Whately) or the doctrine giving the normative
rules of correct reasoning (Herbart), it should not be forgotten that
mathematical or symbolic logic was not set up from nothing. It arose from the
old philosophical collective discipline logic. The standard presentations of
the history of logic ignore the relationship between the philosophical and
mathematical side of its development; they sometimes even deny that there has
been any development of philosophical logic at all. Take for example William
and Martha Kneale's programme in their eminent The Development of Logic. They
wrote (1962, iii): "But our primary purpose has been to record the first
appearances of these ideas which seem to us most important in the logic of our
own day,'' and these are the ideas leading to mathematical logic.
Another example is J. M. Bochenski's assessment of "modern
classical logic'' which he scheduled between the 16th and the 19th century. It
was for him a noncreative period in logic which can therefore justly be ignored
in the problem history of logic (1956, 14). According to Bochenski classical
logic was only a decadent form of this science, a dead period in its
development (ibid., 20).
Such assessments show that the authors adhered to the predominant
views on logic of our time, i. e. actual systems of mathematical or symbolic
logic. As a consequence, they have not been able to give reasons for the final
divorce between philosophical and mathematical logic, because they have ignored
the seed from which mathematical logic has emerged. Following Bochenski's view
Carl B. Boyer presented a consistent periodization of the development of logic
(Boyer 1968, 633): "The history of logic may be divided, with some slight
degree of oversimplification, into three stages: (1) Greek logic, (2)
Scholastic logic, and (3) mathematical logic.'' Note Boyer's "slight degree of
oversimplification'' which enabled him to skip 400 years of logical development
and ignore the fact that Kant's transcendental logic, Hegel's metaphysics and
Mill's inductive logic were called "logic'', too.
In discussing the relationship between the philosophical and the
mathematical development of logic, at least the following questions will be
answered:
1.. What were the reasons for the philosophers' lack of interest
in formal logic?
2.. What were the reasons for the mathematicians' interest in
logic?
3.. What did "logic reform'' mean in the 19th century? Were the
systems of mathematical logic initially regarded as contributions to a reform
of logic?
4.. Was mathematical logic regarded as art, as science or as
both?
This paper focuses not only on the situation in Britain, but also
on the development in Germany. This needs some justification in a symposium on
Victorian mathematics. British logicians regarded Germany as the logical
paragon. John Venn can be regarded as a chief witness. He deplored, in the
second edition of his Symbolic Logic of 1894, the lack of a traditions in logic
in Great Britain which caused problems in creating the collection of books on
logic for the Cambridge University Library (1894, 533):
At the time when I commenced the serious study of Symbolic Logic
many of the most important works which bore on the subject were not to be found
in any of those great libraries in this country to which one naturally refers
in the first place, and could therefore only be obtained by purchase from
abroad. [...] I suppose that the almost entire abandonment of Logic as a
serious academic study, for so many years in this country at least, had
prevented the formation of those private professorial libraries, the frequent
appearance of which in the market has kept the second-hand booksellers' shops
in Germany so well supplied with works on this subject.
It should be stressed, however, that when speaking of German logic
Venn wasn't referring to contemporary German logical sytems, but to the great
18th century rationalistic precursors of the British algebra of logic beginning
with Gottfried Wilhelm Leibniz and ending with the Swiss, Johann Heinrich
Lambert.
In the following sections surveys are given of the philosophical
and mathematical contexts in which the new logic emerged in Great Britain and
Germany. The strange collaboration of mathematics and philosophy in promoting
the new systems of logic will be discussed, and finally answers to the four
questions already posed will be given.
2 Contexts
2.1 The Philosophical Context in Great Britain
The development of the new logic started in 1847, completely
independent of earlier anticipations, e.g. by the German rationalist Gottfried
Wilhelm Leibniz (1646-1716) and his followers (cf. Peckhaus 1994a; 1997, ch.
5). In that year the British mathematician George Boole (1815-1864) published
his pamphlet The Mathematical Analysis of Logic (Boole 1847). Boole mentioned
that it was the struggle for priority concerning the quantification of the
predicate between the Edinburgh philosopher William Hamilton (1788-1856) and
the London mathematician Augustus De Morgan (1806-1871) which encouraged this
study. Hence, he referred to a startling philosophical discussion which
indicated a vivid interest in formal logic in Great Britain. This interest was,
however, a new interest, not even 20 years old. One can even say that neglect
of formal logic could be regarded as a characteristic feature of British
philosophy up to 1826 when Richard Whately (1787-1863) published his Elements
of Logic.1In his preface Whately added an extensive report on the languishing
research and education in formal logic in England. He complained (1826, xv)
that only very few students of the University of Oxford became good logicians
and that
by far the greater part pass through the University without
knowing any thing of all of it; I do not mean that they have not learned by
rote a string of technical terms; but that they understand absolutely nothing
whatever of the principles of the Science.
Thomas Lindsay, the translator of Friedrich Ueberweg's important
System der Logik und Geschichte der logischen Lehren (1857, translation 1871),
was very critical of the scientific qualities of Whately's book, but he,
nevertheless, emphasized its outstanding contribution for the renaissance of
formal logic in Great Britain (Lindsay 1871, 557):
Before the appearance of this work, the study of the science had
fallen into universal neglect. It was scarcely taught in the universities, and
there was hardly a text-book of any value whatever to be put into the hands of
the students.
One year after the publication of Whately's book, George Bentham's
An Outline of a New System of Logic appeared (1827) which was to serve as a
commentary to Whately. Bentham's book was critically discussed by William
Hamilton in a review article published in the Edinburgh Review (1833). With the
help of this review Hamilton founded his reputation as the "first logical name
in Britain, it may be in the world.''2 Hamilton propagated a revival of the
Aristotelian scholastic formal logic without, however, one-sidedly preferring
the syllogism. His logical conception was focused on a revision of the standard
forms by quantifying the predicates of judgements.3 The controversy about
priority arose, when De Morgan, in a lecture "On the Structure of the
Syllogism'' (De Morgan 1846) given to the Cambridge Philosophical Society on
9th November 1846, also proposed quantifying predicates. None had any priority,
of course. Application of the diagrammatic methods of the syllogism proposed e.
g., by the 18th century mathematicians and philosophers Leonard Euler,
Gottfried Ploucquet, and Johann Heinrich Lambert, presupposed quantification of
the predicate. The German psychologistic logician Friedrich Eduard Beneke
(1798-1854) suggested quantifying the predicate in his books on logic of 1839
and 1842, the latter of which he sent to Hamilton. In the context of this paper
it is irrelevant to solve the priority question. It is, however, important that
a dispute of this extent arose at all. It indicates there was new interest in
research on formal logic.
This interest represented only one side of the effect released by
Whately's book. Another line of research stood in the direct tradition of
Humean empiricism and the philosophy of inductive sciences: the inductive logic
of John Stuart Mill (1806-1873), Alexander Bain (1818-1903) and others. Boole's
logic was in clear opposition to inductive logic. It was Boole's follower
William Stanley Jevons (1835-1882; cf. Jevons 1877-1878) who made this
opposition explicit.
Boole referred to the controversy between Hamilton and De Morgan,
but this influence should not be overemphasized. In his main work on the Laws
of Thought (1854) Boole went back to the logic of Aristotle by quoting from the
Greek original. This can be interpreted as indicating that the influence of
contemporary philosophical discussion was not as important as his own words
might suggest. In writing a book on logic he was doing philosophy, and it was
thus a matter of course that he related his results to the philosophical
discussion of his time. This does not mean, of course, that his thoughts were
really influenced by this discussion.
2.2 The Philosophical Context in Germany
It seems clear that, in regard to the 18th century dichotomy
between German and British philosophy represented by the philosophies of Kant
and Hume, Hamilton and Boole stood on the Kantian side. There are some
analogies with the situation in Germany, where philosophical discussion on
logic after Hegel's death was determined by the Kantian influence. In the
preface to the second edition of his Kritik der reinen Vernunft of 1787,
Immanuel Kant (1723-1804) wrote that logic has followed the safe course of a
science since earliest times. For Kant this was evident because of the fact
that logic had been prohibited from taking any step backwards from the time of
Aristotle. But he regarded it as curious that logic hadn't taken a step forward
either (B VIII). Thus, logic seemed to be closed and complete. Formal logic, in
Kant's terminology the analytical part of general logic, did not play a
prominent rôle in Kant's system of transcendental philosophy. In any case it
was a negative touchstone of truth, as he stressed (B 84). Georg Wilhelm
Friedrich Hegel (1770-1831) went further in denying any relevance of formal
logic for philosophy (Hegel 1812/13, I, Introduction, XV-XVII). Referring to
Kant, he maintained that from the fact that logic hadn't changed since
Aristotle one could infer that it needed a complete rebuilding (ibid., XV).
Hegel created a variant of logic as the foundational science of his
philosophical system, defining it as "the science of the pure idea, i.e., the
idea in the abstract element of reasoning '' (1830, 27). Hegelian logic thus
coincides with metaphysics (ibid., 34).
This was the situation when after Hegel's death philosophical
discussion on logic in Germany started. This discussion on logic reform stood
under the label of "the logical question'', a term coined by the
Neo-Aristotelian Adolf Trendelenburg (1802-1872). In 1842 he published a paper
entitled "Zur Geschichte von Hegel's Logik und dialektischer Methode'' with the
subtitle "Die logische Frage in Hegel's Systeme''. But what is the logical
question according to Trendelenburg? He formulated this question explicitly
towards the end of his article: "Is Hegel's dialectical method of pure
reasoning a scientific procedure?'' (1842, 414). In answering this question in
the negative, he provided the occasion of rethinking the status of formal logic
within a theory of human knowledge without, however, proposing a return to the
old (scholastic) formal logic. In consequence the term "the logical question''
was subsequently used in a less specific way. Georg Leonard Rabus, the early
chronicler of the discussion on logic reform, wrote that the logical question
emerged from doubts concerning the justification of formal logic (1880, 1).
Although this discussion was clearly connected to formal logic, the
so-called reform did not concern formal logic. The reason was provided by the
Neo-Kantian Wilhelm Windelband who wrote in a brilliant survey on 19th century
logic (1904, 164):
It is in the nature of things that in this enterprise [i.e. the
reform of logic] the lower degree of fruitfulness and developability power was
on the side of formal logic. Reflection on the rules of the correct progress of
thinking, the technique of correct thinking, had indeed been brought to
perfection by former philosophy, presupposing a naive world view. What
Aristotle had created in a stroke of genius, was decorated with the finest
filigree work in Antiquity and the Middle Ages: an art of proving and
disproving which culminated in a theory of reasoning, and after this
constructing the doctrines of judgements and concepts. Once one has accepted
the foundations, the safely assembled building cannot be shaken: it can only be
refined here and there and perhaps adapted to new scientific requirements.
Windelband was very critical of English mathematical logic. Its
quantification of the predicate allows the correct presentation of extensions
in judgements, but it "drops hopelessly" the vivid sense of all judgements,
which tend to claim or deny a material relationship between subject or
predicate. It is "a logic of the conference table'', which cannot be used in
the vivid life of science, a "logical sport'' which has, however, its merits in
exercising the final acumen (ibid., 166-167).
The philosophical reform efforts concerned primarily two areas:
1.. the problem of a foundation of logic which itself was
approached by psychological and physiological means, leading to new discussion
on the question of priority between logic and psychology, and to various forms
of psychologism and anti-psychologism (cf. Rath 1994, Kusch 1995);
2.. the problem of logical applications focusing interest on the
methodological part of traditional logic. The reform of applied logic attempted
to bring philosophy in touch with the stormy development of mathematics and
sciences of the time.
Both reform procedures had a destructive effect on the shape of
logic and philosophy. The struggle with psychologism led to the departure of
psychology (especially in its new, experimental form) from the body of
philosophy at the beginning of the 20th century. Psychology became a new,
autonomous scientific discipline. The debate on methodology emerged with the
creation of the philosophy of science which was separated from the body of
logic. The philosopher's ignorance of the development of formal logic caused a
third departure: Part of formal logic was taken from the domain of the
competence of philosophy and incorporated into mathematics where it was
instrumentalized for foundational tasks.
2.3 The Mathematical Context in Great Britain
As mentioned earlier, the influence of the philosophical discussion
on logic in Great Britain on the emergence of the new logic should not be
overemphasized. Of greater importance were mathematical influences. Most of the
new logicians can be related to the so-called "Cambridge Network'' (Cannon
1978, 29-71), i. e. the movement which aimed at reforming British science and
mathematics which started at Cambridge. One of the roots of this movement was
the foundation of the Analytical Society in 1812 (cf. Enros 1983) by Charles
Babbage (1791-1871), George Peacock (1791-1858) and John Herschel (1792-1871).
In regard to mathematics Joan L. Richards called this act a "convenient
starting date for the nineteenth-century chapter of British mathematical
development'' (Richards 1988, 13). One of the first achievements of the
Analytical Society was a revision of the Cambridge Tripos by adopting the
Leibnitian notation for the calculus and abandoning the customary Newtonian
theory of fluxions: "the principles of pure D-ism in opposition to the Dot-age
of the University'' as Babbage wrote in his memoirs (Babbage 1864, 29). It may
be assumed that this successful movement triggered off by a change in notation
might have stimulated a new or at least revived interest in operating with
symbols. This new research on the calculus had parallels in innovative
approaches to algebra which were motivated by the reception of Laplacian
analysis. Firstly the development of symbolical algebra has to be mentioned. It
was codified by George Peacock in his Treatise on Algebra (1830) and further
propagated in his famous report for the British Association for the Advancement
of Science (Peacock 1834, especially 198-207). Peacock started by drawing a
distinction between arithmetical and symbolical algebra, which was, however,
still based on the common restrictive understanding of arithmetic as the
doctrine of quantity. A generalization of Peacock's concept can be seen in
Duncan F. Gregory's (1813-1844) "calculus of operations''. Gregory was most
interested in operations with symbols. He defined symbolical algebra as "the
science which treats of the combination of operations defined not by their
nature, that is by what they are or what they do, but by the laws of
combinations to which they are subject'' (1840, 208). In his much praised paper
"On a General Method in Analysis'' (1844) Boole made the calculus of operations
the basic methodological tool for analysis. However in following Gregory, he
went further, proposing more applications. He cited Gregory who wrote that a
symbol is defined algebraically "when its laws of combination are given; and
that a symbol represents a given operation when the laws of combination of the
latter are the same as those of the former'' (Gregory 1842, 153-154). It is
possible that a symbol for an arbitrary operation can be applied to the same
operation (ibid., 154). It is thus necessary to distinguish between
arithmetical algebra and symbolical algebra which has to take into account
symbolical, but non-arithmetical fields of application. As an example Gregory
mentioned the symbols a and +a. They are isomorphic in arithmetic, but in
geometry they need to be interpreted differently. a can refer to a point marked
by a line whereas the combination of the signs + and a additionally expresses
the direction of the line. Therefore symbolical algebra has to distinguish
between the symbols a and +a. Gregory deplored the fact that the unequivocity
of notation didn't prevail as a result of the persistence of mathematical
practice. Clear notation was only advantageous, and Gregory thought that our
minds would be "more free from prejudice, if we never used in the general
science symbols to which definite meanings had been appropriated in the
particular science'' (ibid., 158).
Boole adopted this criticism almost word for word. In his
Mathematical Analysis of Logic of 1847 he claimed that the reception of
symbolic algebra and its principles was delayed by the fact that in most
interpretations of mathematical symbols the idea of quantity was involved. He
felt that these connotations of quantitative relationships were the result of
the context of the emergence of mathematical symbolism, and not of a universal
principle of mathematics (Boole 1847, 3-4). Boole read the principle of the
permanence of equivalent forms as a principle of independence from
interpretation in an "algebra of symbols''. In order to obtain further
affirmation, he tried to free the principle from the idea of quantity by
applying the algebra of symbols to another field, the field of logic. As far as
logic is concerned this implied that only the principles of a "true Calculus''
should be presupposed. This calculus is characterized as a "method resting upon
the employment of Symbols, whose laws of combination are known and general, and
whose results admit of a consistent interpretation'' (ibid., 4). He stressed
(ibid.):
It is upon the foundation of this general principle, that I
purpose to establish the Calculus of Logic, and that I claim for it a place
among the acknowledged forms of Mathematical Analysis, regardless that in its
objects and in its instruments it must at present stand alone.
Boole expressed logical propositions in symbols whose laws of
combination are based on the mental acts represented by them. Thus he attempted
to establish a psychological foundation of logic, mediated, however, by
language. The central mental act in Boole's early logic is the act of election
used for building classes. Man is able to separate objects from an arbitrary
collection which belong to given classes, in order to distinguish them from
others. The symbolic representation of these mental operations follows certain
laws of combination which are similar to those of symbolic algebra. Logical
theorems can thus be proven like mathematical theorems. Boole's opinion has of
course consequences for the place of logic in philosophy: "On the principle of
a true classification, we ought no longer to associate Logic and Metaphysics,
but Logic and Mathematics'' (ibid., 13).
Although Boole's logical considerations became increasingly
philosophical with time, aiming at the psychological and epistemological
foundations of logic itself, his initial interest was not to reform logic but
to reform mathematics. He wanted to establish an abstract view on mathematical
operations without regard to the objects of these operations. When claiming "a
place among the acknowledged forms of Mathematical Analysis'' (1847, 4) for the
calculus of logic, he didn't simply want to include logic in traditional
mathematics. The superordinate discipline was a new mathematics. This is
expressed in Boole's writing: "It is not of the essence of mathematics to be
conversant with the ideas of number and quantity'' (1854, 12).
2.4 The Mathematical Context in Germany
The results of this examination of the British situation at the
time when the new logic emerged-a reform of mathematics, with initially a lack
of interest in a reform of logic, by establishing an abstract view on
mathematics which focused not on mathematical objects, but on symbolic
operations with arbitrary objects-these results could be transferred to the
situation in Germany without any problem.
The most important representative of the German algebra of logic
was the mathematician Ernst Schröder (1841-1902) who was regarded as having
completed the Boolean period in logic (cf. Bochenski 1956, 314). In his first
pamphlet on logic, Der Operationskreis des Logikkalkuls (1877), he presented a
critical revision of Boole's logic of classes, stressing the idea of the
duality between logical addition and logical multiplication introduced by
William Stanley Jevons in 1864. In 1890 Schröder started on the large project,
his monumental Vorlesungen über die Algebra der Logik (1890, 1891, 1895, 1905)
which remained unfinished although it increased to three volumes with four
parts, of which one appeared only posthumously. Contemporaries regarded the
first volume alone as completing the algebra of logic (cf. Wernicke 1891, 196).
Schröder's opinion concerning the question as to the end to which
logic is studied (cf. Peckhaus 1991, 1994b) can be drawn from an
autobiographical note, published in 1901 (and written in the third person), the
year before his death. It contains Schröder's own survey of his scientific aims
and results. Schröder divided his scientific production into three fields:
1.. A number of papers dealing with some of the current problems
of his science.
2.. Studies concerned with creating an "absolute algebra,'' i.
e., a general theory of connections. Schröder stressed that such studies
represent his "very own object of research'' of which only little was published
at that time.
3.. Work on the reform and development of logic.
Schröder wrote (1901) that his aim was
to design logic as a calculating discipline, especially to give
access to the exact handling of relative concepts, and, from then on, by
emancipation from the routine claims of spoken language, to withdraw any
fertile soil from "cliché'' in the field of philosophy as well. This should
prepare the ground for a scientific universal language that, widely differing
from linguistic efforts like Volapük [a universal language like Esperanto very
popular in Germany at that time], looks more like a sign language than like a
sound language.
Schröder's own division of his fields of research shows that he
didn't consider himself a logician: His "very own object of research'' was
"absolute algebra,'' and in respect to its basic problems and fundamental
assumptions similar to modern abstract or universal algebra. What was the
connection between logic and algebra in Schröder's research? From the passages
quoted one could assume that they belong to two separate fields of research,
but this is not the case. They were intertwined in the framework of his
heuristic idea of a general science. In his autobiographical note he stressed
(1901):
The disposition for schematizing, and the aspiration to condense
practice to theory advised Schröder to prepare physics by perfecting
mathematics. This required deepening-as of mechanics and geometry-above all of
arithmetic, and subsequently he became by the time aware of the necessity for a
reform of the source of all these disciplines, logic.
Schröder's universal claim becomes obvious. His scientific efforts
served to provide the requirements to found physics as the science of material
nature by "deepening the foundations,'' to quote a famous metaphor later used
by David Hilbert (1918, 407) in order to illustrate the objectives of his
axiomatic programme. Schröder regarded the formal part of logic that can be
formed as a "calculating logic,'' using a symbolical notation, as a model of
formal algebra that is called "absolute'' in its last state of development.
But what is "formal algebra''? The theory of formal algebra "in the
narrowest sense of the word'' includes "those investigations on the laws of
algebraic operations [ ...] that refer to nothing but general numbers in an
unlimited number field without making any presuppositions concerning its
nature'' (1873, 233). Formal algebra therefore prepares "studies on the most
varied number systems and calculating operations that might be invented for
particular purposes'' (ibid.).
It has to be stressed that Schröder wrote his early considerations
on formal algebra and logic without any knowledge of the results of his British
predecessors. His sources were the textbooks of Martin Ohm, Hermann Günther
Graß mann, Hermann Hankel and Robert Graß mann. These sources show that
Schröder was a representative of the tradition of German combinatorial algebra
and algebraic analysis (cf. Peckhaus 1997, ch. 6).
Like the British tradition, but independent of it, the German
algebra of logic was connected to new trends in algebra. It differed from its
British counterpart in its combinatorial approach. In both traditions, algebra
of logic was invented within the enterprise to reform basic notions of
mathematics which led to the emergence of structural abstract mathematics. The
algebraists wanted to design algebra as "pan-mathematics'', i. e. as a general
discipline embracing all mathematical disciplines as special cases. The
independent attempts in Great Britain and Germany were combined when Schröder
learned about the existence of Boole's logic in late 1873, early 1874. Finally
he enriched the Boolean class logic by adopting Charles S. Peirce's theory of
quantification and adding a logic of relatives according to the model of Peirce
and De Morgan.
The main interest of the new logicians was to utilize logic for
mathematical and scientific purposes, and it was only in a second step, but
nevertheless an indispensable consequence of the attempted applications, that
the reform of logic came into the view. What has been said of the
representatives of the algebra of logic also holds for the proponents of
competing logical systems such as Gottlob Frege or Giuseppe Peano. They wanted
to use logic in their quest for mathematical rigour, something questioned by
the stormy development in mathematics.
3 Accepting the New Logic
Although created by mathematicians, the new logic was widely
ignored by fellow mathematicians. In Germany Schröder was only known as the
algebraist of logic, and regarded as rather exotic. George Boole was respected
by British mathematicians, but his ideas concerning an algebraical
representation of the laws of thought received very little published reaction.
He shared this fate with Augustus De Morgan, the second major figure of
symbolic logic at that time. In 1864, Samuel Neil, the early chronicler of
British mid 19th century logic, expressed his thoughts about the reasons for
this negligible reception: "De Morgan is esteemed crotchety, and perhaps
formalizes too much. Boole demands high mathematic culture to follow and to
profit from'' (1864, 161). One should add that the ones who had this culture
were usually not interested in logic.
The situation changed after George Boole's death in 1864. In the
following comments only some ideas concerning the reasons for this new interest
are hinted at. In particular the rôles of William Stanley Jevons and Alexander
Bain are stressed which exemplify "the strange collaboration of mathematics and
philosophy in promoting the new systems of logic'' mentioned in the
introduction.
3.1 William Stanley Jevons
A broader international reception of Boole's logic began when
William Stanley Jevons made it the starting point for his influential
Principles of Science of 1874. He used his own version of the Boolean calculus
introduced in his Pure Logic of 1864. Among his revisions were the introduction
of a simple symbolical representation of negation and the definition of logical
addition as inclusive "or''. He also changed the philosophy of symbolism (1864,
5):
The forms of my system may, in fact, be reached by divesting his
[Boole's] of a mathematical dress, which, to say the least, is not essential to
it. The system being restored to its proper simplicity, it may be inferred, not
that Logic is a part of Mathematics, as is almost implied in Professor Boole's
writings, but that the Mathematics are rather derivatives of Logic. All the
interesting analogies or samenesses of logical and mathematical reasoning which
may be pointed out, are surely reversed by making Logic the dependent of
Mathematics.
Jevons' interesting considerations on the relationship between
mathematics and logic representing an early logicistic attitude will not be
discussed. Similar ideas can be found not only in Gottlob Frege's work, but
also in that of Hermann Rudolf Lotze and Ernst Schröder. In the context of this
paper, it is relevant that Jevons abandoned mathematical symbolism in logic, an
attitude which was later taken up by John Venn. Jevons attempted to free logic
from the semblance of being a special mathematical discipline. He used the
symbolic notation only as a means of expressing general truths. Logic became a
tool for studying science, a new language providing symbols and structures. The
change in notation brought the new logic closer to the philosophical discourse
of the time. The reconciliation was supported by the fact that Jevons
formulated his Principles of Science as a rejoinder to John Stuart Mill's A
System of Logic of 1843, at that time the dominating work on logic and the
philosophy of science in Great Britain. Although Mill called his logic A System
of Logic Ratiocinative and Inductive, the deductive parts played only a minor
rôle, used only to show that all inferences, all proofs and the discovery of
truths consisted of inductions and their interpretations. Mill claimed to have
shown "that all our knowledge, not intuitive, comes to us exclusively from that
source'' (Mill 1843, Bk. II, ch. I, § 1). Mill concluded that the question as
to what induction is, is the most important question of the science of logic,
"the question which includes all others.'' As a result the logic of induction
covers by far the largest part of this work, a subject which we would today
regard as belonging to the philosophy of science.
Jevons defined induction as a simple inverse application of
deduction. He began a direct argument with Mill in a series of papers entitled
"Mill's Philosophy Tested'' (1877/78). This discourse proved that symbolic
logic could be of importance not only for mathematics, but also for philosophy.
Another effect of the attention caused by Jevons was that British
algebra of logic was able to cross the Channel. In 1877, Louis Liard
(1846-1917), at that time professor at the Faculté de lettres at Bordeaux and a
friend of Jevons, published two papers on the logical systems of Jevons and
Boole (Liard 1877a, 1877b). In 1878 he added a booklet entitled Les logiciens
anglais contemporaines which ran into five editions until 1907, and was
translated into German in 1880. Although Herman Ulrici had published a first
German review of Boole's Laws of Thought as early as 1855, the knowledge of
British symbolic logic was conveyed primarily by Alois Riehl, then professor at
the University of Graz, in Astria. He published a widely read paper "Die
englische Logik der Gegenwart'' ("English contemporary logic'') in 1877 which
reported mainly Jevons' logic and utilized it in a current German controversy
on the possibility of scientific philosophy.
3.2 Alexander Bain
Finally a few words on Alexander Bain (1818-1903): This Scottish
philosopher was an adherent of Mill's logic. Bain's Logic, first published in
1870, had two parts, the first on deduction and the second on induction. He
made explicit that "Mr Mill's view of the relation of Deduction and Induction
is fully adopted'' (1870, I, iii). Obviously he shared the "[ ...] general
conviction that the utility of the purely Formal Logic is but small; and that
the rules of Induction should be exemplified even in the most limited course of
logical discipline'' (ibid., v). The minor rôle of deduction showed up in
Bain's definition " Deduction is the application or extension of Induction to
new cases '' (40).
Despite his reservations about deduction, Bain's Logic was quite
important for the reception of symbolic logic because of a chapter of 30 pages
entitled "Recent Additions to the Syllogism.'' In this chapter the
contributions of William Hamilton, Augustus De Morgan and George Boole were
introduced. Presumably many more people became acquainted with Boole's algebra
of logic through Bain's report than through Boole's own writings. One example
is Hugh MacColl (1837-1909), the pioneer of the calculus of propositions
(statements) and of modal logic. He created his ideas independently of Boole,
eventually realizing the existence of the Boolean calculus by means of Bain's
report. Even in the early parts of his series of papers "The Calculus of
Equivalent Statements'' he quoted from Bain's presentation when discussing
Boole's logic (MacColl 1877/78). In 1875 Bain's logic was translated into
French, in 1878 into Polish. Tadeusz Batóg and Roman Murawski (1996) have shown
that it was Bain's presentation which motivated the first Polish algebraist of
logic, Stanisaw Pi atkiewicz (1848-?) to begin his research on symbolic logic.
The remarkable collaboration of mathematics and philosophy can be
seen in the fact that a broader reception of symbolic logic commenced only when
its relevance for the philosophical discussion of the time came to the fore.
4 Conclusions
Finally, these are the answers to the initial questions:
1.. What were the reasons for the philosophers' lack of interest
in formal logic?
In Germany philosophers shared Kant's opinion that formal logic
was a completed field of knowledge. They were interested primarily in the
foundations and application of logic. In Great Britain there was hardly any
vivid logical tradition. Philosophy was predominated by empiricist conceptions.
New systems of formal logic therefore had difficulties in gaining a footing in
the philosophical discussion.
2.. What were the reasons for the mathematicians' interest in
logic?
Foundational problems and problems in grasping new mathematical
objects forced some mathematicians to look intuitively at the logical
foundations of their subject. The interest in formal logic was thus a result of
the dynamic development of late 19th century mathematics. One should not
assume, however, that this was a general interest. Most mathematicians did not
(and still do not) care about foundations.
3.. How did the mathematicians' logical activities fit into the
reform of logic conceptions of the time?
In Germany in the second half of the 19th century, Logic reform
meant overcoming the Hegelian identification of logic and metaphysics. In Great
Britain it meant enlarging the scope of the syllogism or elaborating the
philosophy of science. Mathematicians were initially interested in utilizing
logic for mathematical means, or they used it as a language for structuring and
symbolizing extra-mathematical fields. Applications were e. g. the foundation
of mathematics (Boole, Schröder, Frege), the foundation of physics (Schröder),
the preservation of rigour in mathematics (Peano), the theory of probabilities
(Boole, Venn), the philosophy of science (Jevons), the theory of human
relationships (Alexander Macfarlane), and juridical questions. The
mathematicians' preference for the organon aspect of formal logic seems to be
the point of deviation between mathematicians and the philosophers who were not
interested in elaborating logic as a tool.
4.. Was mathematical logic regarded as art or as science?
From the applicational interest it follows that it was mainly
regarded as an art. The scientific aspect grew, however, with the insight into
the power of logical calculi. Nevertheless, in an institutional sense the new
logic was established only in the beginning of the 20th century as an academic
subject, i. e. as an institutionalized domain of science.
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