philosopher who played a crucial role in the emergence of modern logic
and analytic philosophy. Frege's logical works were revolutionary, and
are often taken to represent the fundamental break between
contemporary approaches and the older, Aristotelian tradition. He
invented modern quantificational logic, and created the first fully
axiomatic system for logic, which was complete in its treatment of
propositional and first-order logic, and also represented the first
treatment of higher-order logic. In the philosophy of mathematics, he
was one of the most ardent proponents of logicism, the thesis that
mathematical truths are logical truths, and presented influential
criticisms of rival views such as psychologism and formalism. His
theory of meaning, especially his distinction between the sense and
reference of linguistic expressions, was groundbreaking in semantics
and the philosophy of language. He had a profound and direct influence
on such thinkers as Russell, Carnap and Wittgenstein. Frege is often
called the founder of modern logic, and he is sometimes even heralded
as the founder of analytic philosophy.
1. Life and Works
Frege was born on November 8, 1848 in the coastal city of Wismar in
Northern Germany. His full christened name was Friedrich Ludwig
Gottlob Frege. Little is known about his youth. His father, Karl
Alexander Frege, and his mother, Auguste (Bialloblotzsky) Frege, both
worked at a girl's private school founded in part by Karl. Both were
also principals of the school at various points: Karl held the
position until his death 1866, when Auguste took over until her death
in 1878. The German writer Arnold Frege, born in Wismar in 1852, may
have been Frege's younger brother, but this has not been confirmed.
Frege probably lived in Wismar until 1869; in the years from 1864-1869
he is known to have studied at the Gymnasium in Wismar.
In Spring 1869, Frege began studies at the University of Jena. There,
he studied chemistry, philosophy and mathematics, and must have
solidly impressed Ernst Abbe in mathematics, who later become of
Frege's benefactors. After four semesters, Frege transferred to the
University of Göttingen, where he studied mathematics and physics, as
well as philosophy of religion under Hermann Lotze. (Lotze is
sometimes thought to have had a profound impact on Frege's
philosophical views.) In late 1873, Frege finished his doctoral
dissertation, under the guidance of Ernst Schering, entitled Über eine
geometrische Darstellung der imaginären Gebilde in der Ebene ("On a
Geometrical Representation of Imaginary Figures in a Plane"), and
received his Ph.D.
In 1874, with the recommendation of Ernst Abbe, Frege received a
lectureship at the University of Jena, where he stayed the rest of his
intellectual life. His position was unsalaried during his first five
years, and he was supported by his mother. Frege's
Habilitationsschrift, entitled Rechnungsmethoden, die auf eine
Erweiterung des Grössenbegriffes gründen ("Methods of Calculation
Based upon An Amplification of the Concept of Magnitude,"), was
included with the material submitted to obtain the position. It
involves the theory of complex mathematical functions, and contains
seeds of Frege's advances in logic and the philosophy of mathematics.
Frege had a heavy teaching load during his first few years at Jena.
However, he still had time to work on his first major work in logic,
which was published in 1879 under the title Begriffsschrift, eine der
arithmetischen nachgebildete Formelsprache des reinen Denkens
("Concept-Script: A Formula Language for Pure Thought Modeled on That
of Arithmetic"). Therein, Frege presented for the first time his
invention of a new method for the construction of a logical language.
Upon the publication of the Begriffsschrift, he was promoted to
ausserordentlicher Professor, his first salaried position. However,
the book was not well-reviewed by Frege's contemporaries, who
apparently found its two-dimensional logical notation difficult to
comprehend, and failed to see its advantages over previous approaches,
such as that of Boole.
Sometime after the publication of the Begriffsschrift, Frege was
married to Margaret Lieseburg (1856-1905). They had at least two
children, who unfortunately died young. Years later they adopted a
son, Alfred. However, little else is known about Frege's family life.
Frege had aimed to use the logical language of the Begriffsschrift to
carry out his logicist program of attempting to show that all of the
basic truths of arithmetic could be derived from purely logical
axioms. However, on the advice of Carl Stumpf, and given the poor
reception of the Begriffsschrift, Frege decided to write a work in
which he would describe his logicist views informally in ordinary
language, and argue against rival views. The result was his Die
Grundlagen der Arithmetik ("The Foundations of Arithmetic"), published
in 1884. However, this work seems to have been virtually ignored by
most of Frege's contemporaries.
Soon thereafter, Frege began working on his attempt to derive the
basic laws of arithmetic within his logical language. However, his
work was interrupted by changes to his views. In the late 1880s and
early 1890s Frege developed new and interesting theories regarding the
nature of language, functions and concepts, and philosophical logic,
including a novel theory of meaning based on the distinction between
sense and reference. These views were published in influential
articles such as "Funktion und Begriff" ("Function and Concept",
1891), "Über Sinn und Bedeutung" ("On Sense and Reference", 1892) and
"Über Begriff und Gegenstand" ("On Concept and Object", 1892). This
maturation of Frege's semantic and philosophical views lead to changes
in his logical language, forcing him to abandon an almost completed
draft of his work in logic and the foundations of mathematics.
However, in 1893, Frege finally finished a revised volume, employing a
slightly revised logical system. This was his magnum opus,
Grundgesetze der Arithmetik ("Basic Laws of Arithmetic"), volume I. In
the first volume, Frege presented his new logical language, and
proceeded to use it to define the natural numbers and their
properties. His aim was to make this the first of a three volume work;
in the second and third, he would move on to the definition of real
numbers, and the demonstration of their properties.
Again, however, Frege's work was unfavorably reviewed by his
contemporaries. Nevertheless, he was promoted once again in 1894, now
to the position of Honorary Ordinary Professor. It is likely that
Frege was offered a position as full Professor, but turned it down to
avoid taking on additional administrative duties. His new position was
unsalaried, but he was able to support himself and his family with a
stipend from the Carl Zeiss Stiftung, a foundation that gave money to
the University of Jena, and with which Ernst Abbe was intimately
involved.
Because of the unfavorable reception of his earlier works, Frege was
forced to arrange to have volume II of the Grundgesetze published at
his own expense. It was not until 1902 that Frege was able to make
such arrangements. However, while the volume was already in the
publication process, Frege received a letter from Bertrand Russell,
informing him that it was possible to prove a contradiction in the
logical system of the first volume of the Grundgesetze, which included
a naive calculus for classes. For more information, see the article on
"Russell's Paradox". Frege was, in his own words, "thunderstruck". He
was forced to quickly prepare an appendix in response. For the next
couple years, he continued to do important work. A series of articles
entitled "Über die Grundlagen der Geometrie," ("On the Foundations of
Geometry") was published between 1903 and 1906, representing Frege's
side of a debate with David Hilbert over the nature of geometry and
the proper construction and understanding of axiomatic systems within
mathematics.
However, around 1906, probably due to some combination of poor health,
the early loss of his wife in 1905, frustration with his failure to
find an adequate solution to Russell's paradox, and disappointment
over the continued poor reception of his work, Frege seems to have
lost his intellectual steam. He produced very little work between 1906
and his retirement in 1918. However, he continued to influence others
during this period. Russell had included an appendix on Frege in his
1903 Principles of Mathematics. It is from this that Frege came be to
be a bit wider known, including to an Austrian student studying
engineering in Manchester, England, named Ludwig Wittgenstein.
Wittgenstein studied the work of Frege and Russell closely, and in
1911, he wrote to both of them concerning his own solution to
Russell's paradox. Frege invited him to Jena to discuss his views.
Wittgenstein did so in late 1911. The two engaged in a philosophical
debate, and while Wittgenstein reported that Frege "wiped the floor"
with him, Frege was sufficiently impressed with Wittgenstein that he
suggested that he go to Cambridge to study with Russell–a suggestion
that had profound importance for the history of philosophy. Moreover,
Rudolf Carnap was one of Frege's students from 1910 to 1913, and
doubtlessly Frege had significant influence on Carnap's interest in
logic and semantics and his subsequent intellectual development and
successes.
After his retirement in 1918, Frege moved to Bad Kleinen, near Wismar,
and managed to publish a number of important articles, "Der Gedanke"
("The Thought", 1918), "Der Verneinung" ("Negation", 1918), and
"Gedankengefüge" ("Compound Thoughts", 1923). However, these were not
wholly new works, but later drafts of works he had initiated in the
1890s. In 1924, a year before his death, Frege finally returned to the
attempt to understand the foundations of arithmetic. However, by this
time, he had completely given up on his logicism, concluding that the
paradoxes of class or set theory made it impossible. He instead
attempted to develop a new theory of the nature of arithmetic based on
Kantian pure intuitions of space. However, he was not able to write
much or publish anything about his new theory. Frege died on July 26,
1925 at the age of 76.
At the time of his death, Frege's own works were still not very widely
known. He did not live to see the profound impact he would have on the
emergence of analytic philosophy, nor to see his brand of logic–due to
the championship of Russell–virtually wholly supersede earlier forms
of logic. However, in bequeathing his unpublished work to his adopted
son, Alfred, he wrote prophetically, "I believe there are things here
which will one day be prized much more highly than they are now. Take
care that nothing gets lost." Alfred later gave Frege's papers to
Heinrich Scholz of the University of Münster for safekeeping.
Unfortunately, however, they were destroyed in an Allied bombing raid
on March 25, 1945. Although Scholz had made copies of some of the more
important pieces, a good portion of Frege's unpublished works were
lost.
Although he was a fierce, sometimes even satirical, polemicist, Frege
himself was a quiet, reserved man. He was right-wing in his political
views, and like many conservatives of his generation in Germany, he is
known to have been distrustful of foreigners and rather anti-semitic.
Himself Lutheran, Frege seems to have wanted to see all Jews expelled
from Germany, or at least deprived of certain political rights. This
distasteful feature of Frege's personality has gravely disappointed
some of Frege's intellectual progeny.
2. Contributions to Logic
Trained as a mathematician, Frege's interests in logic grew out of his
interests in the foundations of arithmetic. Early in his career, Frege
became convinced that the truths of arithmetic are logical, analytic
truths, agreeing with Leibniz, and disagreeing with Kant, who thought
that arithmetical knowledge was grounded in "pure intuition", as well
as more empiricist thinkers such as J. S. Mill, who thought that
arithmetic was grounded in observation. In other words, Frege
subscribed to logicism. His logicism was modest in one sense, but very
ambitious in others. Frege's logicism was limited to arithmetic;
unlike other important historical logicists, such as Russell, Frege
did not think that geometry was a branch of logic. However, Frege's
logicism was very ambitious in another regard, as he believed that one
could prove all of the truths of arithmetic deductively from a limited
number of logical axioms. Indeed, Frege himself set out to demonstrate
all of the basic laws of arithmetic within his own system of logic.
Frege concurred with Leibniz that natural language was unsuited to
such a task. Thus, Frege sought to create a language that would
combine the tasks of what Leibniz called a "calculus ratiocinator" and
"lingua characterica", that is, a logically perspicuous language in
which logical relations and possible inferences would be clear and
unambiguous. Frege's own term for such a language, "Begriffsschrift"
was likely borrowed from a paper on Leibniz's ideas written by Adolf
Trendelenburg. Although there had been attempts to fashion at least
the core of such a language made by Boole and others working in the
Leibnizian tradition, Frege found their work unsuitable for a number
of reasons. Boole's logic used some of the same signs used in
mathematics, except with different logical meanings. Frege found this
unacceptable for a language which was to be used to demonstrate
mathematical truths, because the signs would be ambiguous. Boole's
logic, though innovative in some respects, was weak in others. It was
divided into a "primary logic" and "secondary logic", bifurcating its
propositional and categorical elements, and could not deal adequately
with multiple generalities. It analyzed propositions in terms of
subject and predicate concepts, which Frege found to be imprecise and
antiquated.
Frege saw the formulae of mathematics as the paradigm of clear,
unambiguous writing. Frege's brand of logical language was modeled
upon the international language of arithmetic, and it replaced the
subject/predicate style of logical analysis with the notions of
function and argument. In mathematics, an equation such as "f(x) = x2
+ 1″ states that f is a function that takes x as argument and yields
as value the result of multiplying x by itself and adding one. In
order to make his logical language suitable for purposes other than
arithmetic, Frege expanded the notion of function to allow arguments
and values other than numbers. He defined a concept (Begriff) as a
function that has a truth-value, either of the abstract objects the
True or the False, as its value for any object as argument. See below
for more on Frege's understanding of concepts, functions and objects.
The concept being human is understood as a function that has the True
as value for any argument that is human, and the False as value for
anything else. Suppose that "H( )" stands for this concept, and "a" is
a constant for Aristotle, and "b" is a constant for the city of
Boston. Then "H(a)" stands for the True, while "H(b)" stands for the
False. In Frege's terminology, an object for which a concept has the
True as value is said to "fall under" the concept.
The values of such concepts could then be used as arguments to other
functions. In his own logical systems, Frege introduced signs standing
for the negation and conditional functions. His own logical notation
was two-dimensional. However, let us instead replace Frege's own
notation with more contemporary notation. For Frege, the conditional
function, "-->">" is understood as a function the value of which is
the False if its first argument is the True and the second argument is
anything other than the True, and is the True otherwise. Therefore,
"H(b) "> H(a)" stands for the True, while "H(a) -->H(b)" stands for
the False. The negation sign "~" stands for a function whose value is
the True for every argument except the True, for which its value is
the False. Conjunction and disjunction signs could then be defined
from the negation and conditional signs. Frege also introduced an
identity sign, standing for a function whose value is the True if the
two arguments are the same object, and the False otherwise, and a
sign, which he called "the horizontal", "—", that stands for a
function that has the True as value for the True as argument, and has
the False as value for any other argument.
Variables and quantifiers are used to express generalities. Frege
understands quantifiers as "second-level concepts". The distinction
between levels of functions involves what kind of arguments the
functions take. In Frege's view, unlike objects, all functions are
"unsaturated" insofar as they require arguments to yield values. But
different sorts of functions require different sorts of arguments.
Functions that take objects as argument, such as those referred to by
"( ) + ( )" or "H( )", are called first-level functions. Functions
that take first-level functions as argument are called second-level
functions. The quantifier, "For all x(…x…)", is understood as standing
for a function that takes a first-level function as argument, and
yields the True as value if the argument-function has the True as
value for all values of x, and has the False as value otherwise. Thus,
"For all xH(x)" stands for the False, since the concept H( ) does not
have the True as value for all arguments. However, "For all x[H(x)
-->H(x)]" stands for True, since the complex concept H( ) -->H( ) does
have the True as value for all arguments. The existential quantifier,
now written "There exists x(…x…)" is defined as "~For all x~(…x…)".
Those familiar with modern predicate logic will recognize the
parallels between it and Frege's logic. Frege is often credited with
having founded predicate logic. However, Frege's logic is in some ways
different from modern predicate logic. As we have seen, a sign such as
"H( )" is a sign for a function in the strictest sense, as are the
conditional and negation connectives. Frege's conditional is not, like
the modern connective, something that flanks statements to form a
statement. Rather, it flanks terms for truth-values to form a term for
a truth-value. Frege's "H(b) -->H(a)" is simply a name for the True,
by itself it does not assert anything. Therefore, Frege introduces a
sign he called the "judgment stroke", |-, used to assert that what
follows it stands for the True. Thus, while "H(b) -->H(a)" is simply a
term for a truth-value, "|- H(b) -->H(a)" asserts that this
truth-value is the True, or in this case, that if Boston is human,
then Aristotle is human. Moreover, Frege's logical system was
second-order. In addition to quantifiers ranging over objects, it also
contained quantifiers ranging over first-level functions. Thus, "|-For
all xThere exists F[F(x)]" asserts that every object falls under at
least one concept.
Frege's logic took the form of an axiomatic system. In fact, Frege was
the first to take a fully axiomatic approach to logic, and the first
even to suggest that inference rules ought to be explicitly formulated
and distinguished from axioms. He began with a limited number of fixed
axioms, introduced explicit inference rules, and aimed to derive all
other logical truths (including, for him, the truths of arithmetic)
from them. Frege's first logical system, that of the 1879
Begriffsschrift, had nine axioms (one of which was not independent),
one explicit inference rule, and also employed a second and third
inference rule implicitly. It represented the first axiomatization of
logic, and was complete in its treatment of both propositional logic
and first-order quantified logic. Unlike Frege's later system, the
system of the Begriffsschrift was fully consistent. (It has since been
proven impossible to devise a system for higher-order logic with a
finite number of axioms that is both complete and consistent.)
In order to make deduction easier, in the 1893 logical system of the
Grundgesetze, Frege used fewer axioms and more inference rules: seven
and twelve, respectively, this time leaving nothing implicit. The
Grundgesetze also expanded upon the system of the Begriffsschrift by
adding axioms governing what Frege called the "value-ranges"
(Werthverlaüfe) of functions, understood as objects corresponding to
the complete argument-value mappings generated by functions. In the
case of concepts, their value-ranges were identified with their
extensions. While Frege did sometimes also refer to the extensions of
concepts as "classes", he did not conceive of such classes as
aggregates or collections. They were simply understood as objects
corresponding to the complete argument-value mappings generated by
concepts considered as functions. Frege then introduced two axioms
dealing with these value-ranges. Most infamous was his Basic Law V,
which asserts that the truth-value of the value-range of function F
being identical to the value-range of function G is the same as the
truth-value of F and G having the same value for every argument. If
one conceives of value-ranges as argument-value mappings, then this
certainly seems to be a plausible hypothesis. However, from it, it is
possible to prove a strong theorem of class membership: that for any
object x, that object is in the extension of concept F if and only if
the value of F for x as argument is the True. Given that value-ranges
themselves are taken to be objects, if the concept in question is that
of being a extension of a concept not included in itself, one can
conclude that the extension of this concept is in itself just in case
it is not. Therefore, the logical system of the Grundgesetze was
inconsistent due to Russell's Paradox. See the entry on Russell's
Paradox for more details. However, the core of the system of the
Grundgesetze, that is, the system minus the axioms governing
value-ranges, is consistent and, like the system of the
Begriffsschrift, is complete in its treatment of propositional logic
and first-order predicate logic.
Given the extent to which it is taken granted today, it can be
difficult to fully appreciate the truly innovative and radical
approach Frege took to logic. Frege was the first to attempt to
transcribe the old statements of categorical logic in a language
employing variables, quantifiers and truth-functions. Frege was the
first to understand a statement such as "all students are hardworking"
as saying roughly the same as, "for all values of x, if x is a
student, then x is hardworking". This made it possible to capture the
logical connection between statements such as "either all students are
hardworking or all students are intelligent" and "all students are
either hardworking or intelligent" (for example, that the first
implies the second). In earlier logical systems such as that of Boole,
in which the propositional and quantificational elements were
bifurcated, the connection was wholly lost. Moreover, Frege's logical
system was the first to be able to capture statements of multiple
generality, such as "every person loves some city" by using multiple
quantifiers in the same logical formula. This too was impossible in
all earlier logical systems. Indeed, Frege's "firsts" in logic are
almost too numerous to list. We have seen here that he invented modern
quantification theory, presented the first complete axiomatization of
propositional and first-order "predicate" logic (the latter of which
he invented outright), attempted the first formulation of higher-order
logic, presented the first coherent and full analysis of variables and
functions, first showed it possible to reduce all truth-functions to
negation and the conditional, and made the first clear distinction
between axioms and inference rules in a formal system. As we shall
see, he also made advances in the logic of mathematics. It is small
wonder that he is often heralded as the founder of modern logic.
On Frege's "philosophy of logic", logic is made true by a realm of
logical entities. Logical functions, value-ranges, and the
truth-values the True and the False, are thought to be objectively
real entities, existing apart from the material and mental worlds. (As
we shall see below, Frege was also committed to other logical entities
such as senses and thoughts.) Logical axioms are true because they
express true thoughts about these entities. Thus, Frege denied the
popular view that logic is without content and without metaphysical
commitment. Frege was also a harsh critic of psychologism in logic:
the view that logical truths are truths about psychology. While Frege
believed that logic might prescribe laws about how people should
think, logic is not the science of how people do think. Logical truths
would remain true even if no one believed them nor used them in their
reasoning. If humans were genetically designed to use regularly the
so-called "inference rule" of affirming the consequent, etc., this
would not make it logically valid. What is true or false, valid of
invalid, does not depend on anyone's psychology or anyone's beliefs.
To think otherwise is to confuse something's being true with
something's being-taken-to-be-true.
3. Contributions to the Philosophy of Mathematics
Frege was an ardent proponent of logicism, the view that the truths of
arithmetic are logical truths. Perhaps his most important
contributions to the philosophy of mathematics were his arguments for
this view. He also presented significant criticisms against rival
views. We have seen that Frege was a harsh critic of psychologism in
logic. He thought similarly about psychologism in mathematics. Numbers
cannot be equated with anyone's mental images, nor truths of
mathematics with psychological truths. Mathematical truths are
objective, not subjective. Frege was also a critic of Mill's view that
arithmetical truths are empirical truths, based on observation. Frege
pointed out that it is not just observable things that can be counted,
and that mathematical truths seem to apply also to these things. On
Mill's view, numbers must be taken to be conglomerations of objects.
Frege rejects this view for a number of reasons. Firstly, is one
conglomeration of two things the same as a different conglomeration of
two things, and if not, in what sense are they equal? Secondly, a
conglomeration can be seen as made up of a different number of things,
depending on how the parts are counted. One deck of cards contains
fifty two cards, but each card consists of a multitude of atoms. There
is no one uniquely determined "number" of the whole conglomeration. He
also reiterated the arguments of others: that mathematical truths seem
apodictic and knowable a priori. He also argued against the Kantian
view that arithmetic truths are based on the pure intuition of the
succession of time. His main argument against this view, however, was
simply his own work in which he showed that truths about the nature of
succession and sequence can be proven purely from the axioms of logic.
Frege was also an opponent of formalism, the view that arithmetic can
be understood as the study of uninterpreted formal systems. While
Frege's logical language represented a kind of formal system, he
insisted that his formal system was important only because of what its
signs represent and its propositions mean. The signs themselves,
independently of what they mean, are unimportant. To suggest that
mathematics is the study simply of the formal system, is, in Frege's
eyes, to confuse the sign and thing signified. To suggest that
arithmetic is the study of formal systems also suggests, absurdly,
that the formula "5 + 7 = 12″, written in Arabic numerals, is not the
same truth as the formula, "V + VII = XII", written in Roman numerals.
Frege suggests also that this confusion would have the absurd result
that numbers simply are the numerals, the signs on the page, and that
we should be able to study their properties with a microscope.
Frege suggests that rival views are often the result of attempting to
understand the meaning of number terms in the wrong way, for example,
in attempting to understand their meaning independently of the
contexts in which they appear in sentences. If we are simply asked to
consider what "two" means independently of the context of a sentence,
we are likely to simply imagine the numeral "2″, or perhaps some
conglomeration of two things. Thus, in the Grundlagen, Frege espouses
his famous context principle, to "never ask for the meaning of a word
in isolation, but only in the context of a proposition." The
Grundlagen is an earlier work, written before Frege had made the
distinction between sense and reference (see below). It is an active
matter of debate and discussion to what extent and how this principle
coheres with Frege's later theory of meaning, but what is clear is
that it plays an important role in his own philosophy of mathematics
as described in the Grundlagen.
According to Frege, if we look at the contexts in which number words
usually occur in a proposition, they appear as part of a sentence
about a concept, specifically, as part of an expression that tells us
how many times a certain concept is instantiated. Consider, for
example, "I have six cards in my hand" or "There are 11 members of
congress from Wisconsin." These propositions seem to tell us how many
times the concepts of being a card in my hand and being a member of
congress from Wisconsin are instantiated. Thus, Frege concludes that
statements about numbers are statements about concepts. This insight
was very important for Frege's case for logicism, as Frege was able to
show that it is possible to define what it means for a concept to be
instantiated a certain number of times purely logically by making use
of quantifiers and identity. To say that the concept F is instantiated
zero times is to say that there are no objects that instantiate F, or,
equivalently, that everything does not instantiate F. To say that F is
instantiated one time is to say there is an object x that instantiates
F, and that for all objects y, either y does not instantiate F or y is
x. To say that F is instantiated twice is to say that there are two
objects, x and y, each of which instantiates F, but which are not the
same as each other, and for all z, either z does not instantiate F, or
z is x or z is y. One could then consider numbers as "second-level
concepts", or concepts of concepts, which can be defined in purely
logical terms. (For more on the distinction of levels of concepts, see
above.)
Frege, however, does not leave his analysis of numbers there.
Understanding number-claims as involving second-level concepts does
give us some insight into the nature of numbers, but it cannot be left
at this. Mathematics requires that numbers be treated as objects, and
that we be able to provide a definition of the number "two"
simpliciter, without having to speak of two Fs. For this purpose,
Frege appeals to his theory of the value-ranges of concepts. On the
notion of a value-range, see above. We saw above that we can gain some
understanding of number claims as involving second-level concepts, or
concepts of concepts. In order to find a definition of numbers as
objects, Frege treats them instead as value-ranges of value-ranges.
Exactly, however, are they to be understood?
Frege notes that we have an understanding of what it means to say that
there are the same number of Fs as there are Gs. It is to say that
there is a one-one mapping between the objects that instantiate F and
the objects instantiating G, i.e. that there is some function f from
entities that instantiate F onto entities that instantiate G such that
there is a different F for every G, and a different G for every F,
with none left over. (In this, Frege's views on the nature of
cardinality were in part anticipated by Cantor.) However, we must bear
in mind that the propositions:
(1) There are equally many Fs as there are Gs.
(2) The number of Fs = the number of Gs
must obviously have the same truth-value, as they seem to express the
same fact. We must, therefore, look for a way of understanding the
phrase "the number of Fs" that occurs in (2) that makes clear how and
why the whole proposition will be true or false for the same reason as
(1) is true or false. Frege's suggestion is that "the number of Fs"
means the same as "the value-range of the concept being a value-range
of a concept instantiated equally many times as F." This means that
the number of Fs is a certain value-range, containing value-ranges,
and in particular, all those value-ranges that have as many members as
there are Fs. Then (2) is understood as saying the same as "the
value-range of the concept being a value-range of a concept
instantiated equally many times as F = the value-range of the concept
being a value-range of a concept instantiated equally many times as
G", which will be true if and only if there are equally many Fs as Gs,
i.e. if every value-range of a concept instantiated equally many times
as F is also a value-range of a concept instantiated equally many
times as G.
To give some examples, if there are zero Fs, then the number of Fs,
i.e. zero, is the value-range consisting of all value-ranges with no
members. Recall that for Frege, classes are identified with
value-ranges of concepts. (See above.) To rephrase the same point in
terms of classes, zero is the class of all classes with no members.
Since there is only one such class, zero is the class containing only
the empty class. If there is one F, then the number of Fs, i.e. one,
is the class consisting of all classes with one member (the extensions
of concepts instantiated once). Here we can see the connection with
the understanding of number expressions as being statements about
concepts. Rather than understanding zero as the concept a concept has
just in case it is not instantiated, zero is understood as the
value-range consisting of value-ranges of concepts that are not
instantiated. Rather than understanding one as the concept a concept
has just in case it is instantiated by a unique object, it is
understood as the value-range consisting of value-ranges of concepts
instantiated by unique objects. This allows us to understand numbers
as abstract objects, and provide a clear definition of the meaning of
number signs in arithmetic such as "1″, "2″, "3″, etc.
Some of Frege's most brilliant work came in providing definitions of
the natural numbers in his logical language, and in proving some of
their properties therein. After laying out the basic laws of logic,
and defining axioms governing the truth-functions and value-ranges,
etc., Frege begins by defining a relation that holds between two
value-ranges just in case they are the value-ranges of concepts
instantiated equally many times. This relation holds between
value-ranges just in case they are the same size, i.e. just in case
there is one-one correspondence between the entities that fall under
their concepts. Using this, he then defines a function that takes a
value-range as argument and yields as value the value-range consisting
of all value-ranges the same size as it. The number zero is then
defined as the value-range consisting of all value-ranges the same
size as the value-range of the concept being non-self-identical. Since
this concept is not instantiated, zero is defined as the value-range
of all value-ranges with no members, as described above. There is only
one such number zero. Since this is true, then the concept of being
identical to zero is instantiated once. Frege then uses this to define
one. One is defined as the value-range of all value-ranges equal in
size to the value-range of the concept being identical to zero. Having
defined one is this way, Frege is able to define two. He has already
defined one and zero; they are each unique, but different from each
other. Therefore, two can be defined as the value-range of all
value-ranges equal in size to the value-range of the concept being
identical to zero or identical to one. Frege is able to define all
natural numbers in this way, and indeed, prove that there are
infinitely many of them. Each natural number can be defined in terms
of the previous one: for each natural number n, its successor (n + 1)
can be defined as the value-range of all value-ranges equal in size to
the value-range of the concept of being identical to one of the
numbers between zero and n.
In the Begriffsschrift, Frege had already been able to prove certain
results regarding series and sequences, and was able to define the
ancestral of a relation. To understand the ancestral of a relation,
consider the example of the relation of being the child of. A person x
bears this relation to y just in case x is y's child. However, x falls
in the ancestral of this relation with respect to y just in case x is
the child of y, or is the child of y's child, or is the child of y's
child's child, etc. Frege was able to define the ancestral of
relations logically even in his early work. He put this to use in the
Grundgesetze to define the natural numbers. We have seen how the
notion of successorship can be defined for Frege, i.e. the relation n
+ 1 bears to n. The natural numbers can be defined as the value-range
of all value-ranges that fall under the ancestral of the successor
relation with respect to zero. The natural numbers then consist of
zero, the successor of zero (one), the successor of the successor of
zero (two), and so on ad infinitum. Frege was then able to use this
definition of the natural numbers to provide a logical analysis of
mathematical induction, and prove that mathematical induction can be
used validly to demonstrate the properties of the natural numbers, an
extremely important result for making good on his logicist ambitions.
Frege could then use mathematical induction to prove some of the basic
laws of the natural numbers. Frege next turned his logicist method to
an analysis of integers (including negative numbers) and then to the
real numbers, defining them using the natural numbers and certain
relations holding between them. We need not dwell on the details of
this work here.
Frege's approach to providing a logical analysis of cardinality, the
natural numbers, infinity and mathematical induction were
groundbreaking, and have had a lasting importance within mathematical
logic. Indeed, prior to 1902, it must have seemed to him that he had
been completely successful in showing that the basic laws of
arithmetic could be understood purely as logical truths. However, as
we have seen, Frege's definition of numbers heavily involves the
notion of classes or value-ranges, but his logical treatment of them
is shown to be impossible due to Russell's paradox. This presents a
serious problem for Frege's logicist approach. Another heavy blow came
after Frege's death. In 1931, Kurt Gödel discovered his famous
incompleteness proof to the effect that there can be no consistent
formal system with a finite number of axioms in which it is possible
to derive all of the truths of arithmetic. This presents a serious
blow to more ambitious forms of logicism, such as Frege's, which aimed
to provide precisely the sort of system Gödel showed impossible.
Nevertheless, it cannot be denied that Frege's work in the philosophy
of mathematics was important and insightful.
4. The Theory of Sense and Reference
Frege's influential theory of meaning, the theory of sense (Sinn) and
reference (Bedeutung) was first outlined, albeit briefly, in his
article, "Funktion und Begriff" of 1891, and was expanded and
explained in greater detail in perhaps his most famous work, "Über
Sinn und Bedeutung" of 1892. In "Funktion und Begriff", the
distinction between the sense and reference of signs in language is
first made in regard to mathematical equations. During Frege's time,
there was a widespread dispute among mathematicians as to how the
sign, "=", should be understood. If we consider an equation such as,
"4 x 2 = 11 – 3″, a number of Frege's contemporaries, for a variety of
reasons, were wary of viewing this as an expression of an identity,
or, in this case, as the claim that 4 x 2 and 11 – 3 are one and the
same thing. Instead, they posited some weaker form of "equality" such
that the numbers 4 x 2 and 11 – 3 would be said to be equal in number
or equal in magnitude without thereby constituting one and the same
thing. In opposition to the view that "=" signifies identity, such
thinkers would point out that 4 x 2 and 11 – 3 cannot in all ways be
thought to be the same. The former is a product, the latter a
difference, etc.
In his mature period, however, Frege was an ardent opponent of this
view, and argued in favor of understanding "=" as identity proper,
accusing rival views of confusing form and content. He argues instead
that expressions such as "4 x 2″ and "11 – 3″ can be understood as
standing for one and the same thing, the number eight, but that this
single entity is determined or presented differently by the two
expressions. Thus, he makes a distinction between the actual number a
mathematical expression such as "4 x 2″ stands for, and the way in
which that number is determined or picked out. The former he called
the reference (Bedeutung) of the expression, and the latter was called
the sense (Sinn) of the expression. In Fregean terminology, an
expression is said to express its sense, and denote or refer to its
reference.
The distinction between reference and sense was expanded, primarily in
"Über Sinn und Bedeutung" as holding not only for mathematical
expressions, but for all linguistic expressions (whether the language
in question is natural language or a formal language). One of his
primary examples therein involves the expressions "the morning star"
and "the evening star". Both of these expressions refer to the planet
Venus, yet they obviously denote Venus in virtue of different
properties that it has. Thus, Frege claims that these two expressions
have the same reference but different senses. The reference of an
expression is the actual thing corresponding to it, in the case of
"the morning star", the reference is the planet Venus itself. The
sense of an expression, however, is the "mode of presentation" or
cognitive content associated with the expression in virtue of which
the reference is picked out.
Frege puts the distinction to work in solving a puzzle concerning
identity claims. If we consider the two claims:
(1) the morning star = the morning star
(2) the morning star = the evening star
The first appears to be a trivial case of the law of self-identity,
knowable a priori, while the second seems to be something that was
discovered a posteriori by astronomers. However, if "the morning star"
means the same thing as "the evening star", then the two statements
themselves would also seem to have the same meaning, both involving a
thing's relation of identity to itself. However, it then becomes to
difficult to explain why (2) seems informative while (1) does not.
Frege's response to this puzzle, given the distinction between sense
and reference, should be apparent. Because the reference of "the
evening star" and "the morning star" is the same, both statements are
true in virtue of the same object's relation of identity to itself.
However, because the senses of these expressions are different–in (1)
the object is presented the same way twice, and in (2) it is presented
in two different ways–it is informative to learn of (2). While the
truth of an identity statement involves only the references of the
component expressions, the informativity of such statements involves
additionally the way in which those references are determined, i.e.
the senses of the component expressions.
So far we have only considered the distinction as it applies to
expressions that name some object (including abstract objects, such as
numbers). For Frege, the distinction applies also to other sorts of
expressions and even whole sentences or propositions. If the
sense/reference distinction can be applied to whole propositions, it
stands to reason that the reference of the whole proposition depends
on the references of the parts and the sense of the proposition
depends of the senses of the parts. (At some points, Frege even
suggests that the sense of a whole proposition is composed of the
senses of the component expressions.) In the example considered in the
previous paragraph, it was seen that the truth-value of the identity
claim depends on the references of the component expressions, while
the informativity of what was understood by the identity claim depends
on the senses. For this and other reasons, Frege concluded that the
reference of an entire proposition is its truth-value, either the True
or the False. The sense of a complete proposition is what it is we
understand when we understand a proposition, which Frege calls "a
thought" (Gedanke). Just as the sense of a name of an object
determines how that object is presented, the sense of a proposition
determines a method of determination for a truth-value. The
propositions, "2 + 4 = 6″ and "the Earth rotates", both have the True
as their references, though this is in virtue of very different
conditions holding in the two cases, just as "the morning star" and
"the evening star" refer to Venus in virtue of different properties.
In "Über Sinn und Bedeutung", Frege limits his discussion of the
sense/reference distinction to "complete expressions" such as names
purporting to pick out some object and whole propositions. However, in
other works, Frege makes it quite clear that the distinction can also
be applied to "incomplete expressions", which include functional
expressions and grammatical predicates. These expressions are
incomplete in the sense that they contain an "empty space", which,
when filled, yields either a complex name referring to an object, or a
complete proposition. Thus, the incomplete expression "the square root
of ( )" contains a blank spot, which, when completed by an expression
referring to a number, yields a complex expression also referring to a
number, e.g., "the square root of sixteen". The incomplete expression,
"( ) is a planet" contains an empty place, which, when filled with a
name, yields a complete proposition. According to Frege, the
references of these incomplete expressions are not objects but
functions. Objects (Gegenstände), in Frege's terminology, are
self-standing, complete entities, while functions are essentially
incomplete, or as Frege says, "unsaturated" (ungesättigt) in that they
must take something else as argument in order to yield a value. The
reference of the expression "square root of ( )" is thus a function,
which takes numbers as arguments and yields numbers as values. The
situation may appear somewhat different in the case of grammatical
predicates. However, because Frege holds that complete propositions,
like names, have objects as their references, and in particular, the
truth-values the True or the False, he is able to treat predicates
also as having functions as their references. In particular, they are
functions mapping objects onto truth-values. The expression, "( ) is a
planet" has as its reference a function that yields as value the True
when saturated by an object such as Saturn or Venus, but the False
when saturated by a person or the number three. Frege calls such a
function of one argument place that yields the True or False for every
possible argument a "concept" (Begriff), and calls similar functions
of more than one argument place (such as that denoted by "( ) > ( )",
which is doubly in need of saturation), "relations".
It is clear that functions are to be understood as the references of
incomplete expressions, but what of the senses of such expressions?
Here, Frege tells us relatively little save that they exist. There is
some amount of controversy among interpreters of Frege as to how they
should be understood. It suffices here to note that just as the same
object (e.g. the planet Venus), can be presented in different ways, so
also can a function be presented in different ways. While "identity",
as Frege uses the term, is a relation holding only between objects,
Frege believes that there is a relation similar to identity that holds
between functions just in case they always share the same value for
every argument. Since all and only those things that have hearts have
kidneys, strictly speaking, the concepts denoted by the expressions "(
) has a heart", and "( ) has a kidney" are one and the same. Clearly,
however, these expressions do not present that concept in the same
way. For Frege, these expressions would have different senses but the
same reference. Frege also tells us that it is the incomplete nature
of these senses that provides the "glue" holding together the thoughts
of which they form a part.
Frege also uses the distinction to solve what appears to be a
difficulty with Leibniz's law with regard to identity. This law was
stated by Leibniz as, "those things are the same of which one can be
substituted for another without loss of truth," a sentiment with which
Frege was in full agreement. As Frege understands this, it means that
if two expressions have the same reference, they should be able to
replace each other within any proposition without changing the
truth-value of that proposition. Normally, this poses no problem. The
inference from:
(3) The morning star is a planet.
to the conclusion:
(4) The evening star is a planet.
in virtue of (2) above and Leibniz's law is unproblematically valid.
However, there seem to be some serious counterexamples to this
principle. We know for example that "the morning star" and "the
evening star" have the same customary reference. However, it is not
always true that they can replace one another without changing the
truth of a sentence. For example, if we consider the propositions:
(5) Gottlob believes that the morning star is a planet.
(6) Gottlob believes that the evening star is a planet.
If we assume that Gottlob does not know that the morning star is the
same heavenly body as the evening star, (5) may be true while (6)
false or vice versa.
Frege meets this challenge to Leibniz's law by making a distinction
between what he calls the primary and secondary references of
expressions. Frege suggests that when expressions appear in certain
unusual contexts, they have as their references what is customarily
their senses. In such cases, the expressions are said to have their
secondary references. Typically, such cases involve what Frege calls
"indirect speech" or "oratio obliqua", as in the case of statements of
beliefs, thoughts, desires and other so-called "propositional
attitudes", such as the examples of (5) and (6). However, expressions
also have their secondary references (for reasons which should already
be apparent) in contexts such as "it is informative that…" or "… is
analytically true".
Let us consider the examples of (5) and (6) more closely. To Frege's
mind, these statements do not deal directly with the morning star and
the evening star itself. Rather, they involve a relation between a
believer and a thought believed. Thoughts, as we have seen, are the
senses of complete propositions. Beliefs depend for their make-up on
how certain objects and concepts are presented, not only on the
objects and concepts themselves. The truth of belief claims,
therefore, will depend not on the customary references of the
component expressions of the stated belief, but their senses. Since
the truth-value of the whole belief claim is the reference of that
belief claim, and the reference of any proposition, for Frege, depends
on the references of its component expressions, we are lead to the
conclusion that the typical senses of expressions that appear in
oratio obliqua are in fact the references of those expressions when
they appear in that context. Such contexts can be referred to as
"oblique contexts", contexts in which the reference of an expression
is shifted from its customary reference to its customary sense.
In this way, Frege is able to actually retain his commitment in
Leibniz's law. The expressions "the morning star" and "the evening
star" have the same primary reference, and in any non-oblique context,
they can replace each other without changing the truth-value of the
proposition. However, since the senses of these expressions are not
the same, they cannot replace each other in oblique contexts, because
in such contexts, their references are non-identical.
Frege ascribes to senses and thoughts objective existence. In his
mind, they are objects every bit as real as tables and chairs. Their
existence is not dependent on language or the mind. Instead, they are
said to exist in a timeless "third realm" of sense, existing apart
from both the mental and the physical. Frege concludes this because,
although senses are obviously not physical entities, their existence
likewise does not depend on any one person's psychology. A thought,
for example, has a truth-value regardless of whether or not anyone
believes it and even whether or not anyone has grasped it at all.
Moreover, senses are interpersonal. Different people are able to grasp
the same senses and same thoughts and communicate them, and it is even
possible for expressions in different languages to express the same
sense or thought. Frege concludes that they are abstract objects,
incapable of full causal interaction with the physical world. They are
actual only in the very limited sense that they can have an effect on
those who grasp them, but are themselves incapable of being changed or
acted upon. They are neither created by our uses of language or acts
of thinking, nor destroyed by their cessation.
Unfortunately, Frege does not tell us very much about exactly how
these abstract objects pick out or present their references. Exactly
what is it that makes a sense a "way of determining" or "mode of
presenting" a reference? In the wake of Russell's theory of
descriptions, a Fregean sense is often interpreted as a set of
descriptive information or criteria that picks out its reference in
virtue of the reference alone satisfying or fitting that descriptive
information. In giving examples, Frege implies that a person might
attach to the name "Aristotle" the sense the pupil of Plato and
teacher of Alexander the Great. This sense picks out Aristotle the
person because he alone matches this description. Here, care must be
taken to avoid misunderstanding. The sense of the name "Aristotle" is
not the words "the pupil of Plato and teacher of Alexander the Great";
to repeat, senses are not linguistic items. It is rather that the
sense consists in some set of descriptive information, and this
information is best described by a descriptive phrase of this form.
The property of being the pupil of Plato and teacher of Alexander is
unique to Aristotle, and thus, it may be in virtue of associating this
information with the name "Aristotle" that this name may be used to
refer to Aristotle. As certain commentators have noted, it is not even
necessary that the sense of the name be expressible by some
descriptive phrase, because the descriptive information or properties
in virtue of which the reference is determined may not be directly
nameable in any natural language.
From this standpoint, it is easy to understand how there might be
senses that do not pick out any reference. Names such as "Romulus" or
"Odysseus", and phrases such as "the least rapidly converging series"
or "the present King of France" express senses, insofar as they lay
out criteria that things would have to satisfy if they were to be the
references of these expressions. However, there are no things which do
in fact satisfy these criteria. Therefore, these expressions are
meaningful, but do not have references. Because the sense of a whole
proposition is determined by the senses of the parts, and the
reference of a whole proposition is determined by the parts, Frege
claims that propositions in which such expressions appear are able to
express thoughts, but are neither true nor false, because no
references are determined for them.
This interpretation of the nature of senses makes Frege a forerunner
to what has since been come to be known as the "descriptivist" theory
of meaning and reference in the philosophy of language. The view that
the sense of a proper name such as "Aristotle" could be descriptive
information as simple as the pupil of Plato and teacher of Alexander
the Great, however, has been harshly criticized by many philosophers,
and perhaps most notably by Saul Kripke. Kripke points out that this
would make a claim such as "Aristotle taught Alexander" seem to be a
necessary and analytic truth, which it does not appear to be.
Moreover, he claims that many of us seem to be able to use a name to
refer to an individual even if we are unaware of any properties
uniquely held by that individual. For example, many of us don't know
enough about the physicist Richard Feynman to be able to identify a
property differentiating him from other prominent physicists such as
Murray Gell-Mann, but we still seem to be able to refer to Feynman
with the name "Feynman". John Searle, Michael Dummett and others,
however, have proposed ways of expanding or altering Frege's notion of
a sense to circumvent Kripke's worries. This has lead to a very
important debate in the philosophy of language, which, unfortunately,
we cannot fully discuss here.
5. References and Further Reading
a. Frege's Own Works
* "Antwort auf die Ferienplauderei des Herrn Thomae."
Jahresbericht der Deutschen Mathematiker-Vereinigung 15 (1906):
586-90. Translated as "Reply to Thomae's Holiday Causerie." In
Collected Papers on Mathematics, Logic and Philosophy [CP], 341-5.
Translated by M. Black, V. Dudman, P. Geach, H. Kaal, E.-H. W. Kluge,
B. McGuinness and R. H. Stoothoff. New York: Basil Blackwell, 1984.
* "Über Begriff und Gegenstand." Vierteljahrsschrift für
wissenschaftliche Philosophie 16 (1892): 192-205. Translated as "On
Concept and Object." In >CP 182-94. Also in The Frege Reader [FR],
181-93. Edited by Michael Beaney. Oxford: Blackwell, 1997. And In
Translations from the Philosophical Writings of Gottlob Frege [TPW],
42-55. 3d ed. Edited by Peter Geach and Max Black. Oxford: Blackwell,
1980.
* Begriffsschrift, eine der arithmetischen nachgebildete
Formelsprache des reinen Denkens. Halle: L. Nebert, 1879. Translated
as Begriffsschrift, a Formula Language, Modeled upon that of
Arithmetic, for Pure Thought. In From Frege to Gödel, edited by Jean
van Heijenoort. Cambridge, MA: Harvard University Press, 1967. Also as
Conceptual Notation and Related Articles. Edited and translated by
Terrell W. Bynum. London: Oxford University Press, 1972.
* "Über die Begriffsschrift des Herrn Peano und meine eigene."
Verhandlungen der Königlich Sächsischen Gesellschaft der
Wissenschaften zu Leipzig 48 (1897): 362-8. Translated as "On Mr.
Peano's Conceptual Notation and My Own." In CP 234-48.
* "Über formale Theorien der Arithmetik." Sitzungsberichte der
Jenaischen Gesellschaft für Medizin und Naturwissenschaft 19 (1885):
94-104. Translated as "On Formal Theories of Arithmetic." In CP
112-21.
* Funktion und Begriff. Jena: Hermann Pohle, 1891. Translated as
"Function and Concept." In CP 137-56, TPW 21-41 and FR 130-48.
* "Der Gedanke." Beträge zur Philosophie des deutschen Idealismus
1 (1918-9): 58-77. Translated as "Thoughts." In CP 351-72. Also as
part I of Logical Investigations [LI], edited by P. T. Geach. Oxford:
Blackwell, 1977. And as "Thought." In FR 325-45.
* "Gedankengefüge." Beträge zur Philosophie des deutschen
Idealismus 3 (1923): 36-51. Translated as "Compound Thoughts." In CP
390-406, and as part III of LI.
* Über eine geometrische Darstellung der imaginären Gebilde in der
Ebene. Ph. D. Dissertation: University of Göttingen, 1873. Translated
as "On a Geometrical Representation of Imaginary Forms in the Plane."
In CP 1-55.
* Grundgesetze der Arithmetik. 2 vols. Jena: Hermann Pohle,
1893-1903. Translated in part as The Basic Laws of Arithmetic:
Exposition of the System. Edited and translated by Montgomery Furth.
Berkeley: University of California Press, 1964.
* "Über die Grundlagen der Geometrie." Jahresbericht der Deutschen
Mathematiker-Vereinigung 12 (1903): 319-24, 368-75, 15 (1906):
293-309, 377-403, 423-30. Translated as "On the Foundations of
Geometry." In CP 273-340. Also as On the Foundations of Geometry and
Formal Theories of Arithmetic. Translated by Eike-Henner W. Kluge. New
York: Yale University Press, 1971.
* Die Grundlagen der Arithmetik, eine logisch mathematische
Untersuchung über den Begriff der Zahl. Breslau: W. Koebner, 1884.
Translated as The Foundations of Arithmetic: A Logico-Mathematical
Enquiry into the Concept of Number. 2d ed. Translated by J. L. Austin.
Oxford: Blackwell, 1953.
* "Kritische Beleuchtung einiger Punkte in E. Schröders
Vorlesungen über die Algebra der Logik." Archiv für systematsche
Philosophie 1 (1895): 433-56. Translated as "A Critical Elucidation of
Some Points in E. Schröder, Vorlesungen über die Algebra der Logik."
In CP 210-28, and TPW 86-106.
* Nachgelassene Schriften. Hamburg: Felix Meiner, 1969. Translated
as Posthumous Writings. Translated by Peter Long and Roger White.
Chicago: University of Chicago Press, 1979.
* "Le nombre entier." Revue de Métaphysique et de Morale 3 (1895):
73-8. Translated as "Whole Numbers." In CP 229-33.
* Rechnungsmethoden, die auf eine Erweiterung des Grössenbegriffes
gründen. Habilitationsschrift: University of Jena, 1874. Translated as
"Methods of Calculation based on an Extension of the Concept of
Quantity." In CP 56-92.
* Review of Zur Lehre vom Transfiniten, by Georg Cantor.
Zeitschrift für Philosophie und philosophische Kritik 100 (1892):
269-72. Translated in CP 178-181.
* Review of Philosophie der Arithmetik, by Edmund Husserl.
Zeitschrift für Philosophie und philosophische Kritik 103 (1894):
313-32. Translated in CP 195-209.
* "Über Sinn und Bedeutung." Zeitschrift für Philosophie und
philosophische Kritik 100 (1892): 25-50. Translated as "On Sense and
Meaning." In CP 157-77. As "On Sinn and Bedeutung." In FR 151-71. And
as "On Sense and Reference." In TPW 56-78.
* "Über das Trägheitsgesetz." Zeitschrift für Philosophie und
philosophische Kritik 98 (1891): 145-61. Translated as "On the Law of
Inertia." In CP 123-36.
* "Die Unmöglichkeit der Thomaeschen formalen Arithmetik aus Neue
nachgewiesen." Jahresbericht der Deutschen Mathematiker-Vereinigung 17
(1908): 52-5. Translated as "Renewed Proof of the Impossibility of Mr.
Thomae's Formal Arithmetic." In CP 346-50.
* "Der Verneinung." Beträge zur Philosophie des deutschen
Idealismus 1 (1918-9): 143-57. Translated as "Negation." In CP 373-89,
part II of LI, and FR 346-61.
* "Was ist ein Funktion?" In Festschrift Ludwig Boltzmann gewidmet
zum sechzigsten Geburtstage, 656-66. Leipzig: Amrosius Barth, 1904.
Translated as "What is a Function?" In CP 285-92, and TPW 285-92.
* Wissenschaftlicher Briefwechsel. Hamburg: Felix Meiner, 1976.
Translated as Philosophical and Mathematical Correspondence.
Translated by Hans Kaal. Chicago: University of Chicago Press, 1980.
* Über die Zahlen des Herrn H. Schubert. Jena: Hermann Pohle,
1899. Translated as "On Mr. H. Schubert's Numbers." In CP 249-72.
b. Important Secondary Works
* Angelelli, Ignacio. Studies on Gottlob Frege and Traditional
Philosophy. Dordrecht: D. Reidel, 1967.
* Baker, G. P. and P. M. S. Hacker. Frege: Logical Excavations.
New York: Oxford University Press, 1984.
* Beaney, Michael. Frege: Making Sense. London: Duckworth, 1996.
* Beaney, Michael. Introduction to The Frege Reader, by Gottlob
Frege. Oxford: Blackwell, 1997.
* Bell, David. Frege's Theory of Judgment. New York: Oxford
University Press, 1979.
* Bynum, Terrell W. "On the Life and Work of Gottlob Frege. "
Introduction to Conceptual Notation and Related Articles, by Gottlob
Frege. London: Oxford University Press, 1972.
* Carl, Wolfgang. Frege's Theory of Sense and Reference.
Cambridge: Cambridge University Press, 1994.
* Carnap, Rudolph. Meaning and Necessity. 2d ed. Chicago:
University of Chicago Press, 1956.
* Church, Alonzo. "A Formulation of the Logic of Sense and
Denotation." In Structure, Method and Meaning: Essays in Honor of
Henry M. Sheffer, edited by P. Henle, H. Kallen and S. Langer, 3- 24.
New York: Liberal Arts Press, 1951.
* Currie, Gregory. Frege: An Introduction to His Philosophy.
Totowa, NJ: Barnes and Noble, 1982.
* Dummett, Michael. Frege: Philosophy of Language. 2d ed.
Cambridge, MA: Harvard University Press, 1981.
* Dummett, Michael. Frege: Philosophy of Mathematics. Cambridge,
MA: Harvard University Press, 1991.
* Dummett, Michael. Frege and Other Philosophers. Oxford: Oxford
University Press, 1991.
* Dummett, Michael. The Interpretation of Frege's Philosophy.
Cambridge, MA: Harvard University Press, 1981.
* Geach, Peter T. "Frege." In Three Philosophers, edited by G. E.
M. Anscombe and P. T. Geach, 127-62. Oxford: Oxford University Press,
1961.
* Gödel, Kurt. "On Formally Undecidable Propositions of Principia
Mathematica and Related Systems I." In From Frege to Gödel, edited by
Jan van Heijenoort, 596-616. Cambridge, MA: Harvard University Press,
1967. Originally published as "Über formal unentscheidbare Sätze der
Principia Mathematica und verwandter Systeme I." Monatshefte für
Mathematik und Physik 38 (1931): 173-98.
* Grossmann, Reinhardt. Reflections on Frege's Philosophy.
Evanston: Northwestern University Press, 1969.
* Haaparanta, Leila and Jaakko Hintikka, eds. Frege Synthesized.
Boston: D. Reidel, 1986.
* Kaplan, David. "Quantifying In." Synthese 19 (1968): 178-214.
* Klemke, E. D., ed. Essays on Frege. Urbana: University of
Illinois Press, 1968.
* Kluge, Eike-Henner W. The Metaphysics of Gottlob Frege. Boston:
Martinus Nijhoff, Boston, 1980.
* Kneale, William and Martha Kneale. The Development of Logic.
London: Oxford University Press, 1962.
* Kripke, Saul. Naming and Necessity. Cambridge, MA: Harvard
University Press, 1980. First published in Semantics of Natural
Languages. Edited by Donald Davidson and Gilbert Harman. Dordrecht: D.
Reidel, 1972.
* Linsky, Leonard. Oblique Contexts. Chicago: University of
Chicago Press, 1983.
* Resnik, Michael D. Frege and the Philosophy of Mathematics.
Ithaca: Cornell University Press, 1980.
* Ricketts, Thomas G., ed. The Cambridge Companion to Frege.
Cambridge: Cambridge University Press, forthcoming.
* Russell, Bertrand. "The Logical and Arithmetical Doctrines of
Frege." In The Principles of Mathematics, Appendix A. 1903. 2d. ed.
Reprint, New York: W. W. Norton & Company, 1996.
* Russell, Bertrand. "On Denoting." Mind 14 (1905): 479-93.
* Salmon, Nathan. Frege's Puzzle. Cambridge: MIT Press, 1986.
* Schirn. Matthias, ed. Logik und Mathematik: Frege Kolloquium
1993. Hawthorne: de Gruyter, 1995.
* Schirn. Matthias, ed. Studien zu Frege. 3 vols. Stuttgart-Bad
Cannstatt: Verlag-Holzboog, 1976.
* Searle, John R. Intentionality: An Essay in the Philosophy of
Mind. Cambridge: Cambridge University Press, 1983.
* Sluga, Hans. "Frege and the Rise of Analytic Philosophy."
Inquiry 18 (1975): 471-87.
* Sluga, Hans. Gottlob Frege. Boston: Routledge & Kegan Paul, 1980.
* Sluga, Hans. The Philosophy of Frege. 4 vols. New York: Garland
Publishing, 1993.
* Sternfeld, Robert. Frege's Logical Theory. Carbondale: Southern
Illinois University Press, 1966.
* Thiel, Christian. Sense and Reference in Frege's Logic.
Translated by T. J. Blakeley. Dordrecht: D. Reidel, 1968.
* Tichý, Pavel. The Foundations of Frege's Logic. New York: Walter
de Gruyter, 1988.
* Walker, Jeremy D. B. A Study of Frege. London: Oxford University
Press, 1965.
* Weiner, Joan. Frege in Perspective. Ithaca: Cornell University
Press, 1990.
* Wright, Crispin. Frege's Conception of Numbers as Objects.
Aberdeen: Aberdeen University Press, 1983.
* Wright, Crispin. Frege: Tradition and Influence. Oxford: Blackwell, 1984.
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