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Frege’s theory of arithmetic

Frege, by virtue of his work creating predicate logic, is one of the founders of modern (mathematical) logic. His view was that mathematics is reducible to logic. His major works published with the goal of doing this are these:

• in 1879 -- Begriffsschrift, defining his “axiomatic-deductive” predicate calculus for the ultimate purpose of proving the basic truths of arithmetic "by means of pure logic."
• in 1884 -- Die Grundlagen der Arithmetik, using his predicate calculus to present an axiomatic theory of arithmetic.
• in 1893/1903 -- Die Grundgesetze der Arithmetik, presenting formal proofs of number theory from an intuitive collection of axioms.

As we have seen, Boole’s developed his algebra of logic as a means by which deduction becomes calculation. Frege's predicate calculus in the Begriffsschrift stood Boole’s purpose on its head:

• Frege’s goal: “to establish ... that arithmetic could be reduced to logic” and, thus, to create a logic by means of which calculation becomes deduction
• Frege’s program: to develop arithmetic as an axiomatic system such that all the axioms were truths of logic

Driven by “an over-ruling passion to demonstrate his position conclusively” and not “content with the usual informal mathematical standard of rigour,” Frege’s exposition in Grundgesetze is characterized by a great degree by precision and explicitness. He himself tells us why this is so:

[T]he fundamental propositions of arithmetic should be proved…with the utmost rigour; for only if every gap in the chain of deductions is eliminated with the greatest care can we say with certainty upon what primitive truths the proof depend.

Frege gave the following reason for developing his logic as an axiomatic system:

Because we cannot enumerate all of the boundless number of laws that can be established, we can obtain completeness only by a search for those [laws] which, potentially, imply all the others.

Frege also commented on the role of proof in mathematics:

The aim of proof is not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another.

Frege identified as the kernel of his system the axioms (laws) of his logic that potentially imply all the other laws. His statements above imply that he thought his system to be complete and his axioms to be independent. He did not, however, provide a precise definition of completeness and independence nor did he attempt a proof that his system was complete and his axioms independent.

Early in his first book on the foundations of arithmetic, Frege established his purpose:

[I]t is above all $Number$ which has to be either defined or recognized as indefinable. This is the point which the present work is meant to settle.

Frege began the introduction of numbers into his logic by defining what is meant by saying that two $Numbers$ are equal:

two concepts $F$ and $G$ are equal if the things that fall under them can be put into one-one correspondence

From this he arrives at the notion that “a $Number$ is a set of concepts”:

• §72 the $Number$ that belongs to the concept $F$ is the extension of the concept “equal to the concept $F$”

and then continues as follows by defining the expression

$n$ is a $Number$ :to mean ::::there exists a concept such that $n$ is the $Number$ that belongs to it. * §73 he draws this inference ::::the concept $F$ is equal to the concept $G$ :implies ::::the $Number$ belonging to the concept $F$ is identical to the $Number$ belonging to the concept $G$ * §74 he defines the $Number$ $0$ as ::::the $Number$ that belongs to the concept “not identical with itself” * §75 he immediately clarifies this, stating ::::Every concept under which no object falls is equal to every other concept under which no object falls, and to them alone. :and therefore ::::$0$ is the $Number$ which belongs to any such concept, and no object falls under any concept if the number which belongs to that concept is $0$. * §76 he defines the $Successor$ relation ::::$n$ follows in the series of $Numbers$ directly after $m$ :to mean ::::there exists a concept $F$ and an object falling under it, $x$, such that ::::::the $Number$ belonging to the concept $F$ is $n$ ::::and ::::::the $Number$ belonging to the concept “falling under $F$ but not equal to $x$” is $m$ * §77 he defines the $Number$ $1$ as ::::“the $Number$ belonging to the concept ‘identical with $0$’” :from which it follows that ::::$1$ is the $Number$ that follows directly after $0$ * §78-81 he proves or gives a proof sketch for several propositions regarding the $Successor$ relation, using definitions of $series$ and $following$ $in$ $a$ $series$ from his earlier work of 1879 ::* the $Successor$ relation is 1-1 ::* every $Number$ except $0$ is a $Successor$ ::* every $Number$ has a $Successor$ * §82 he outlines a proof that there is no last member in the series of $Numbers$ * §83 he provides a definition of finite Number, noting that no finite Number follows in the series of natural numbers after itself * §84 he notes that the $Number$ which belongs to the concept 'finite N$umber$' is an infinite $Number$. Central to all of this work was a distinction that Frege was developing, but only finally published in 1892 and incorporated in the ''Grundgesetze'', namely, that every concept, mathematical or otherwise, had two important, entirely distinct aspects:'"`UNIQ--ref-00000009-QINU`"''"`UNIQ--ref-0000000A-QINU`"' # ''Sinn'': a “meaning” or “sense” or “connotation” # ''Bedeutung'': an “extension” or “reference” or “denotation” This distinction of Frege's is the basis of what Gödel (many years later) characterized as the ''dichotomic conception'':'"`UNIQ--ref-0000000B-QINU`"' ::Any well-defined concept (property or predicate) $P(x)$ establishes a dichotomy of all things into those that are $P$s and those that are non-$P$s. :In other words, ::a concept partitions $V$ (the universe of discourse) into two classes: the class $\{ x : P(x) \}$ and its absolute complement, the class $\{ x : \neg P(x) \}$. Underlying this notion are two key assumptions: # the existence of a ''Universal Set'', $V$ -- what we have seen as Dedekind’s ''Gedankenwelt'' # the unrestricted principle of ''Comprehension'' -- ''any'' well-defined property determines, a set. For “naïve” set theory, these two assumptions are equivalent and either one of them suffices to derive the other: * to derive ''Universal Set'' from ''Comprehension'': ::::replace $Φ(x)$ by a truism, such as the property $x = x$. * to derive ''Comprehension'' from the ''Universal Set'': ::assume an all-encompassing set $V$, ::::note that any part of $V$ is also a set, ::::and that any well-defined concept $P(x)$ defines a subset of $V$, ::therefore the set $\{ x : P(x) \}$ exists! To these two assumptions, add Dedekind's principle of ''Extensionality''. Frege intended the ''Grundgesetze'' to be the implementation of his program to demonstrate “every proposition of arithmetic” to be ”a [derivative] law of logic.” In this work of 19 years duration, there was no explicit appeal to an ''unrestricted'' principle of Comprehension. Instead, Frege's theory of arithmetic appealed to Comprehension by virtue of its symbolism, according to which for any predicate $Φ(x)$ (concept or property) one can form an expression $S = \{ x : Φ(x)\}$ defining a set. Frege's theory assumes that (somehow) there is a mapping which associates an object (a set of objects) to every concept, but he does not present comprehension as an explicit assumption. All of this is in contrast to the use of restricted predicates in Cantor's early theory of sets.

Primary sources

• Frege, G. (1879). Begriffsschrift ..., [“Conceptual Notation …”, English translation by T W Bynum, Oxford University Press, 1972].

• Frege, G. (1884). Die Grundlagen der Arithmetik, [The Foundations of Arithmetic, English translation J L Austin, Basil Blackwell, 1968].

• Frege, G. (1884) ... [Dietz, A. "Frege: The Foundations of Arithmetic (1884)," A brief summary keyed to sections of the original text, URL: https://sites.google.com/site/philosophysummaries/personal/previous-classes/500/frege-the-foundations-of-arithmetic-1884, Accessed: 2015/08/14.]

• Frege, G. (1892) Uber Sinn und Bedeuting, [“On Sense and Reference,” Translations from the Philosophical Writings of Gottlob Frege, Geach and Black (eds.) Blackwell, 1960, pp. 56-78].

• Frege, G. (1893) Grundgesetze, [The Basic Laws of Arithmetic, English translation by M Firth, University of California, 1964].

References

• Azzano

• Ferreiros “Hilbert, Logicism, and Mathematical Existence”

• Gillies

• O’Connor and Robertson