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Single Idea 8782

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism ]

Full Idea

Since Frege's defence of his thesis that the laws of arithmetic are analytic depended upon a realm of independently existing objects - the finite cardinal numbers and the real numbers - his view amounted to a Platonist version of logicism.

Gist of Idea

Frege offered a Platonist version of logicism, committed to cardinal and real numbers

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Logicism in the 21st Century 1

Book Ref

'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.166

A Reaction

Nice to have this spelled out. Along with Gödel, Frege is the most distinguished Platonist since the great man. Frege has lots of modern fans, but I would have thought that this makes his position a non-starter. Alternatives are needed.

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]