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Single Idea 9816

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers ]

Full Idea

Frege was operating with a successor relation, rather than a successor function.

Gist of Idea

For Frege, successor was a relation, not a function

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.2

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.13

A Reaction

That is, succession is a given fact, not a construction. 4 may be the successor of 3 in natural numbers, but not in rational or real numbers, so we can't take the relation for granted.

The 174 ideas from 'Grundlagen der Arithmetik (Foundations)'

15948 Frege developed formal systems to avoid unnoticed assumptions [Frege, by Lavine]
10804 Thoughts have a natural order, to which human thinking is drawn [Frege, by Yablo]
9832 Frege sees no 'intersubjective' category, between objective and subjective [Dummett on Frege]
13876 The syntactic category is primary, and the ontological category is derivative [Frege, by Wright,C]
9841 Frege was the first to give linguistic answers to non-linguistic questions [Frege, by Dummett]
9844 Originally Frege liked contextual definitions, but later preferred them fully explicit [Frege, by Dummett]
13881 We need to grasp not number-objects, but the states of affairs which make number statements true [Frege, by Wright,C]
9154 Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Frege, by Burge]
9157 The null set is only defensible if it is the extension of an empty concept [Frege, by Burge]
9835 It is because a concept can be empty that there is such a thing as the empty class [Frege, by Dummett]
9854 We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett]
18104 Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock]
9834 A class is, for Frege, the extension of a concept [Frege, by Dummett]
16891 Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Frege, by Burge]
16906 The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Frege, by Jeshion]
9838 Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett]
9564 For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara]
10551 If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett]
17427 Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Frege, by Koslicki]
17437 Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki]
17438 Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki]
15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
10034 The number of natural numbers is not a natural number [Frege, by George/Velleman]
16883 Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
10625 Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege]
13871 Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C]
13872 Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C]
9816 For Frege, successor was a relation, not a function [Frege, by Dummett]
9953 Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman]
9954 "Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman]
17636 A cardinal number may be defined as a class of similar classes [Frege, by Russell]
10139 Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege]
10028 For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman]
10029 Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman]
9973 The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait]
16500 Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins]
9990 Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait]
7738 Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner]
23456 Frege said logical predication implies classes, which are arithmetical objects [Frege, by Morris,M]
13887 Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C]
13897 Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C]
9856 Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege]
9902 Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege]
17814 The natural number n is the set of n-membered sets [Frege, by Yourgrau]
17819 A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege]
9956 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman]
13527 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS]
22292 Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter]
17442 Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck]
11030 The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege]
10030 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman]
8690 From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend]
10219 Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro]
13889 Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C]
18142 One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock]
17820 If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege]
16896 If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
13874 Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege]
13875 Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C]
7731 How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner]
7737 Identities refer to objects, so numbers must be objects [Frege, by Weiner]
17816 Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau]
11031 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege]
7739 Arithmetic is analytic [Frege, by Weiner]
9951 It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman]
9952 Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman]
9945 Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman]
16905 Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion]
8782 Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Frege, by Hale/Wright]
13608 Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Frege, by Bostock]
5658 Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton]
10831 Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege]
13864 Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege]
10033 Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege]
10010 Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
9631 Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR]
9875 Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett]
8911 If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen on Frege]
17431 Vagueness is incomplete definition [Frege, by Koslicki]
13879 For Frege, ontological questions are to be settled by reference to syntactic structures [Frege, by Wright,C]
10642 Second-order quantifiers are committed to concepts, as first-order commits to objects [Frege, by Linnebo]
10309 Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Frege, by Hale]
10550 Frege establishes abstract objects independently from concrete ones, by falling under a concept [Frege, by Dummett]
8785 For Frege, objects just are what singular terms refer to [Frege, by Hale/Wright]
10278 Without concepts we would not have any objects [Frege, by Shapiro]
17432 Frege's universe comes already divided into objects [Frege, by Koslicki]
10803 Frege himself abstracts away from tone and color [Yablo on Frege]
16022 The idea of a criterion of identity was introduced by Frege [Frege, by Noonan]
11100 Frege's algorithm of identity is the law of putting equals for equals [Frege, by Quine]
12153 Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry on Frege]
9853 Identity between objects is not a consequence of identity, but part of what 'identity' means [Frege, by Dummett]
17623 To understand a thought you must understand its logical structure [Frege, by Burge]
2514 Frege tried to explain synthetic a priori truths by expanding the concept of analyticity [Frege, by Katz]
9158 For Frege a priori knowledge derives from general principles, so numbers can't be primitive [Frege]
10606 Frege treats properties as a kind of function, and maybe a property is its characteristic function [Frege, by Smith,P]
9870 Early Frege takes the extensions of concepts for granted [Frege, by Dummett]
13878 Concepts are, precisely, the references of predicates [Frege, by Wright,C]
7736 A concept is a non-psychological one-place function asserting something of an object [Frege, by Weiner]
17430 Fregean concepts have precise boundaries and universal applicability [Frege, by Koslicki]
9976 Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait on Frege]
9855 Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Frege, by Dummett]
10802 Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo on Frege]
10526 Fregean abstraction creates concepts which are equivalences between initial items [Frege, by Fine,K]
10525 Frege put the idea of abstraction on a rigorous footing [Frege, by Fine,K]
2515 Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz on Frege]
8619 To learn something, you must know that you don't know [Frege]
8622 Psychological accounts of concepts are subjective, and ultimately destroy truth [Frege]
8620 Thought is the same everywhere, and the laws of thought do not vary [Frege]
8621 Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege]
8415 Never lose sight of the distinction between concept and object [Frege]
8414 Keep the psychological and subjective separate from the logical and objective [Frege]
20295 All analytic truths can become logical truths, by substituting definitions or synonyms [Frege, by Rey]
17495 Proof aims to remove doubts, but also to show the interdependence of truths [Frege]
17443 Many of us find Frege's claim that truths depend on one another an obscure idea [Heck on Frege]
16903 Justifications show the ordering of truths, and the foundation is what is self-evident [Frege, by Jeshion]
9352 An a priori truth is one derived from general laws which do not require proof [Frege]
16889 A truth is a priori if it can be proved entirely from general unproven laws [Frege]
9370 A statement is analytic if substitution of synonyms can make it a logical truth [Frege, by Boghossian]
8743 Frege considered analyticity to be an epistemic concept [Frege, by Shapiro]
8624 Induction is merely psychological, with a principle that it can actually establish laws [Frege]
22286 Existence is not a first-level concept (of God), but a second-level property of concepts [Frege, by Potter]
22294 We can show that a concept is consistent by producing something which falls under it [Frege]
8626 In science one observation can create high probability, while a thousand might prove nothing [Frege]
16890 Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge]
8630 Individual numbers are best derived from the number one, and increase by one [Frege]
8632 You can't transfer external properties unchanged to apply to ideas [Frege]
8633 There is no physical difference between two boots and one pair of boots [Frege]
8634 The equator is imaginary, but not fictitious; thought is needed to recognise it [Frege]
16900 Intuitions cannot be communicated [Frege, by Burge]
10539 Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Frege, by Dummett]
8635 Numbers are not physical, and not ideas - they are objective and non-sensible [Frege]
8636 We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege]
8637 The number 'one' can't be a property, if any object can be viewed as one or not one [Frege]
9988 If we abstract 'from' two cats, the units are not black or white, or cats [Tait on Frege]
8639 If numbers are supposed to be patterns, each number can have many patterns [Frege]
8640 We cannot define numbers from the idea of a series, because numbers must precede that [Frege]
8641 You can abstract concepts from the moon, but the number one is not among them [Frege]
11029 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege]
17460 A statement of number contains a predication about a concept [Frege]
14236 Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley on Frege]
8642 Abstraction from things produces concepts, and numbers are in the concepts [Frege]
8643 Affirmation of existence is just denial of zero [Frege]
8644 Because existence is a property of concepts the ontological argument for God fails [Frege]
17426 A concept creating a unit must isolate and unify what falls under it [Frege]
17428 Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki]
9989 Units can be equal without being identical [Tait on Frege]
17429 Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki]
10013 Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes]
9046 Our definition will not tell us whether or not Julius Caesar is a number [Frege]
8645 Convert "Jupiter has four moons" into "the number of Jupiter's moons is four" [Frege]
9999 For science, we can translate adjectival numbers into noun form [Frege]
9846 Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett on Frege]
8646 Words in isolation seem to have ideas as meanings, but words have meaning in propositions [Frege]
8648 Ideas are not spatial, and don't have distances between them [Frege]
8647 Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates [Frege]
9840 Frege initiated linguistic philosophy, studying number through the sense of sentences [Frege, by Dummett]
9822 Nothing should be defined in terms of that to which it is conceptually prior [Frege, by Dummett]
17445 Parallelism is intuitive, so it is more fundamental than sameness of direction [Frege, by Heck]
10556 We create new abstract concepts by carving up the content in a different way [Frege]
9882 You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege]
9881 From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Frege, by Dummett]
9883 Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett]
8651 A concept is a possible predicate of a singular judgement [Frege]
18181 The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege]
8652 Numbers are objects, because they can take the definite article, and can't be plurals [Frege]
8653 Nought is the number belonging to the concept 'not identical with itself' [Frege]
8654 One is the Number which belongs to the concept "identical with 0" [Frege]
10032 'Ancestral' relations are derived by iterating back from a given relation [Frege, by George/Velleman]
8655 Arithmetic is analytic and a priori, and thus it is part of logic [Frege]
8656 The laws of number are not laws of nature, but are laws of the laws of nature [Frege]
8657 Mathematicians just accept self-evidence, whether it is logical or intuitive [Frege]
17624 To understand axioms you must grasp their logical power and priority [Frege, by Burge]
18103 Numbers are objects because they partake in identity statements [Frege, by Bostock]
7732 Never ask for the meaning of a word in isolation, but only in the context of a proposition [Frege]