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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:<ref>O’Connor and Robertson (2002)</ref>
 
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:<ref>O’Connor and Robertson (2002)</ref>
* 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 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 1884 -- ''Die Grundlagen der Arithmetik'', using his predicate calculus to present an axiomatic theory of arithmetic.
* in 1893, the first volume of ''Die Grundgesetze der Arithmetik'', containing an intuitive collection of axioms and formal proofs of number theory.
+
* 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:<ref>Gillies pp. 74-75</ref>
 
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:<ref>Gillies pp. 74-75</ref>
Line 16: Line 16:
 
Frege identified as the ''kernel'' of his system the axioms (laws) of his logic that potentially imply all the other laws. His statement above implies that he thought his system to be complete, though he did not provide either a precise definition of completeness or a proof that his system was actually complete.
 
Frege identified as the ''kernel'' of his system the axioms (laws) of his logic that potentially imply all the other laws. His statement above implies that he thought his system to be complete, though he did not provide either a precise definition of completeness or a proof that his system was actually complete.
  
Central to all of this work was a distinction that Frege was developing, but only finally published in 1892, namely that every concept, mathematical or otherwise, had two important, entirely distinct aspects:<ref>Frege (1892)</ref>
+
Frege began the introduction of numbers into his logic by defining what is meant by saying that two $Numbers$ are equal:<ref>Frege (1884) cited in Gillies p. 46</ref>
 +
::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”:
 +
::the $Number$ that belongs to the concept $F$ is the extension of the concept “equal to the concept $F$”
 +
 
 +
Frege then continues as follows:<ref>Frege (1884) cited in Gillies p. 47-48</ref>
 +
* he defines the expression
 +
::::“$n$ is a $Number$”
 +
:to mean
 +
::::“there exists a concept such that $n$ is the $Number$ that belongs to it.”
 +
* he defines the $Number$ $0$ as
 +
::::“the $Number$ that belongs to the concept “not identical with itself”
 +
* 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 $m$
 +
* 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 $0$
 +
* he then proves several propositions regarding the $Successor$ relation
 +
::* the $Successor$ relation is 1-1
 +
::* every $Number$ except $0$ is a $Successor$
 +
* finally, he gives a sketch of a proof that
 +
:::every $Number$ has a $Successor$
 +
:using definitions of $series$ and $following$ $in$ $a$ $series$ from his earlier work of 1879.
 +
 
 +
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:<ref>Frege (1892)</ref><ref>Gillies p. 83</ref>
 
# ''Sinn'': a “meaning” or “sense” or “connotation”
 
# ''Sinn'': a “meaning” or “sense” or “connotation”
 
# ''Bedeutung'': an “extension” or “reference” or “denotation”
 
# ''Bedeutung'': an “extension” or “reference” or “denotation”
  
This distinction of Frege's is the basis of what Gödel (very much later) characterized as the ''dichotomic conception'':<ref>Ferreiros pp. 18-19</ref>
+
This distinction of Frege's is the basis of what Gödel (many years later) characterized as the ''dichotomic conception'':<ref>Ferreiros pp. 18-19</ref>
::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 $V$ (a universe of discourse) into two classes: the class $\{ x : P(x) \}$ and its absolute complement, the class $\{ x : \neg P(x) \}$.
+
::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.
This standpoint is based on two key assumptions:
+
: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 existence of a ''Universal Set'', $V$ -- what we have seen as Dedekind’s ''Gedankenwelt''
# an unrestricted principle of ''Comprehension'' as a basic law of thought: Given any well-defined property (formalized as an open, unquantified sentence $Φ(x)$) there exists the set $S = \{ x: Φ(x)\}$.
+
# the unrestricted principle of ''Comprehension'' -- ''any'' well-defined property determines, a set.
Just one of assumptions (1) and (2) suffices for “naïve” set theory:
+
For “naïve” set theory, these two assumptions are equivalent and either one of them suffices to derive the other:
* deriving the ''Universal Set'' from the principle of ''Comprehension'': replace $Φ(x)$ by a truism, such as the property $x = x$.
+
* to derive ''Universal Set'' from ''Comprehension'':
* deriving ''Comprehension'' from the ''Universal Set'': establish the existence of an all-encompassing domain $V$ given as a set; in order to establish the principle of comprehension, the key idea is that, since $V$ is assumed to be a set, any part of it should again be a set. Hence, since a well-defined concept $P(x)$ defines a subset of $V$, the set $\{ x: P(x) \}$ exists!
+
::::replace $Φ(x)$ by a truism, such as the property $x = x$.
To these two, one adds Dedekind's principle of ''Extensionality'' (Dedekind 1888, 345).
+
* 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’s ''Grundgesetze'' made an implicit appeal to such an unrestricted Comprehension principle according to which every predicate (concept/property) defines a set.<ref>Azzano p. 10</ref> This is in contrast to the use of restricted predicates found in Cantor's early theory of sets.
+
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.<ref>Azzano p. 10</ref><ref>Ferreiros pp. 18-19. Ferreiros notes (with surprise) that, in spite of its importance to naive set theory, the unrestricted principle of Comprehension was almost nowhere stated clearly before it was proved to be contradictory!</ref>
 
 
- - - - -
 
  
 
==Notes==
 
==Notes==
Line 40: Line 75:
 
==Primary sources==
 
==Primary sources==
  
* Frege, G. (1884). ''Die Grundlagen der Arithmetik'', K¨obner, Breslau.
+
* Frege, G. (1879). ''Begriffsschrift ...'', [“Conceptual Notation …”, English translation by T W Bynum, Oxford University Press, 1972].
  
* Frege, G. (1892) ''Uber Sinn und Bedeuting''.
+
* Frege, G. (1884). ''Die Grundlagen der Arithmetik'', [''The Foundations of Arithmetic'', English translation J L Austin, Basil Blackwell, 1968].
 +
 
 +
* 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==
 
==References==
 +
 +
* Azzano
 +
 +
* Ferreiros “Hilbert, Logicism, and Mathematical Existence”
 +
 +
* Gillies
 +
 +
* O’Connor and Robertson

Revision as of 14:45, 14 August 2015

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:[1]

  • 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:[2]

  • 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.[3]

Frege gave the following reason for developing his logic as he did:[4]

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 which, potentially, imply all the others.

Frege identified as the kernel of his system the axioms (laws) of his logic that potentially imply all the other laws. His statement above implies that he thought his system to be complete, though he did not provide either a precise definition of completeness or a proof that his system was actually complete.

Frege began the introduction of numbers into his logic by defining what is meant by saying that two $Numbers$ are equal:[5]

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”:

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

Frege then continues as follows:[6]

  • he defines the expression
“$n$ is a $Number$”
to mean
“there exists a concept such that $n$ is the $Number$ that belongs to it.”
  • he defines the $Number$ $0$ as
“the $Number$ that belongs to the concept “not identical with itself”
  • 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 $m$ * 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 $0$ * he then proves several propositions regarding the $Successor$ relation ::* the $Successor$ relation is 1-1 ::* every $Number$ except $0$ is a $Successor$ * finally, he gives a sketch of a proof that :::every $Number$ has a $Successor$ :using definitions of $series$ and $following$ $in$ $a$ $series$ from his earlier work of 1879. 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-00000006-QINU`"''"`UNIQ--ref-00000007-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-00000008-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.[10][11]

Notes

  1. O’Connor and Robertson (2002)
  2. Gillies pp. 74-75
  3. Azzano p. 12
  4. Frege (1879) p. 136 cited in Gillies p. 71
  5. Frege (1884) cited in Gillies p. 46
  6. Frege (1884) cited in Gillies p. 47-48
  7. Frege (1892)
  8. Gillies p. 83
  9. Ferreiros pp. 18-19
  10. Azzano p. 10
  11. Ferreiros pp. 18-19. Ferreiros notes (with surprise) that, in spite of its importance to naive set theory, the unrestricted principle of Comprehension was almost nowhere stated clearly before it was proved to be contradictory!

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. (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
How to Cite This Entry:
Whayes43. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whayes43&oldid=36641