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Ideas of Gottlob Frege, by Text

[German, 1848 - 1925, Led a quiet and studious life as Professor at the University of Jena.]

1874 Rechnungsmethoden (dissertation)
Ch.6 p.68 Geometry appeals to intuition as the source of its axioms
p.2 p.279 Quantity is inconceivable without the idea of addition
1879 Begriffsschrift
p.1 Frege changed philosophy by extending logic's ability to check the grounds of thinking [Potter]
p.17 We should not describe human laws of thought, but how to correctly track truth [Fisher]
p.19 Frege reduced most quantifiers to 'everything' combined with 'not' [McCullogh]
p.23 A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) [George/Velleman]
p.23 For Frege, 'All A's are B's' means that the concept A implies the concept B [Walicki]
p.29 Frege's account was top-down and decompositional, not bottom-up and compositional [Potter]
p.31 Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Weiner]
p.37 Frege replaced Aristotle's subject/predicate form with function/argument form [Weiner]
p.44 Frege introduced quantifiers for generality [Weiner]
p.59 For Frege the variable ranges over all objects [Tait]
p.118 Frege's logic has a hierarchy of object, property, property-of-property etc. [Smith,P]
p.124 In 1879 Frege developed second order logic [Putnam]
p.126 Existence is not a first-order property, but the instantiation of a property [Read]
p.133 The predicate 'exists' is actually a natural language expression for a quantifier [Weiner]
p.161 It may be possible to define induction in terms of the ancestral relation [Wright,C]
p.191 Frege produced axioms for logic, though that does not now seem the natural basis for logic [Kaplan]
p.207 Proof theory began with Frege's definition of derivability [Prawitz]
p.475 Frege's domain for variables is all objects, but modern interpretations first fix the domain [Dummett]
§03 p.12 I don't use 'subject' and 'predicate' in my way of representing a judgement
§13 p.29 The laws of logic are boundless, so we want the few whose power contains the others
1881 Boole calculus and the Concept script
p.17 p.17 We don't judge by combining subject and concept; we get a concept by splitting up a judgement
1884 Grundlagen der Arithmetik (Foundations)
p.-15 Frege tried to explain synthetic a priori truths by expanding the concept of analyticity [Katz]
p.-14 Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz]
p.-1 Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C]
p.3 If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen]
p.4 Frege himself abstracts away from tone and color [Yablo]
p.7 Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright]
p.7 Frege claims that numbers are objects, as opposed to them being Fregean concepts [Wright,C]
p.8 The idea of a criterion of identity was introduced by Frege [Noonan]
p.10 Numbers are second-level, ascribing properties to concepts rather than to objects [Wright,C]
p.11 Numbers seem to be objects because they exactly fit the inference patterns for identities
p.11 Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Hale]
p.11 Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Burge]
p.13 The syntactic category is primary, and the ontological category is derivative [Wright,C]
p.13 Frege's platonism proposes that objects are what singular terms refer to [Wright,C]
p.13 For Frege, successor was a relation, not a function [Dummett]
p.15 Concepts are, precisely, the references of predicates [Wright,C]
p.17 Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman]
p.18 Second-order quantifiers are committed to concepts, as first-order commits to objects [Linnebo]
p.23 It appears that numbers are adjectives, but they don't apply to a single object [George/Velleman]
p.24 Numerical adjectives are of the same second-level type as the existential quantifier [George/Velleman]
p.25 For Frege, ontological questions are to be settled by reference to syntactic structures [Wright,C]
p.25 We need to grasp not number-objects, but the states of affairs which make number statements true [Wright,C]
p.25 Numbers are more than just 'second-level concepts', since existence is also one [George/Velleman]
p.26 The null set is only defensible if it is the extension of an empty concept [Burge]
p.26 For Frege a priori knowledge derives from general principles, so numbers can't be primitive
p.27 "Number of x's such that ..x.." is a functional expression, yielding a name when completed [George/Velleman]
p.29 Frege gives an incoherent account of extensions resulting from abstraction [Fine,K]
p.30 For Frege the number of F's is a collection of first-level concepts [George/Velleman]
p.30 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [George/Velleman]
p.33 Numbers need to be objects, to define the extension of the concept of each successor to n [George/Velleman]
p.35 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [George/Velleman]
p.39 Why should the existence of pure logic entail the existence of objects? [George/Velleman]
p.41 The number of natural numbers is not a natural number [George/Velleman]
p.43 The number of F's is the extension of the second level concept 'is equipollent with F' [Tait]
p.44 Frege showed that numbers attach to concepts, not to objects [Wiggins]
p.44 Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait]
p.48 The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt]
p.49 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt]
p.54 How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner]
p.55 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Lavine]
p.57 A concept is a non-psychological one-place function asserting something of an object [Weiner]
p.59 Identities refer to objects, so numbers must be objects [Weiner]
p.59 Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Tait]
p.64 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Wolf,RS]
p.64 Formalism fails to recognise types of symbols, and also meta-games [Brown,JR]
p.66 Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine]
p.66 Zero is defined using 'is not self-identical', and one by using the concept of zero [Weiner]
p.66 From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Friend]
p.76 Frege's algorithm of identity is the law of putting equals for equals [Quine]
p.77 Frege sees no 'intersubjective' category, between objective and subjective [Dummett]
p.78 Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Shapiro]
p.87 Frege treats properties as a kind of function, and maybe a property is its characteristic function [Smith,P]
p.91 Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry]
p.91 A class is, for Frege, the extension of a concept [Dummett]
p.92 It is because a concept can be empty that there is such a thing as the empty class [Dummett]
p.96 Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Dummett]
p.109 Frege said logical predication implies classes, which are arithmetical objects [Morris,M]
p.111 Frege started with contextual definition, but then switched to explicit extensional definition [Wright,C]
p.112 Frege was the first to give linguistic answers to non-linguistic questions [Dummett]
p.114 Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Wright,C]
p.118 Arithmetic is analytic [Weiner]
p.123 Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes]
p.125 Originally Frege liked contextual definitions, but later preferred them fully explicit [Dummett]
p.127 Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Potter]
p.135 Frege, unlike Russell, has infinite individuals because numbers are individuals [Bostock]
p.136 Each number, except 0, is the number of the concept of all of its predecessors [Wright,C]
p.162 Identity between objects is not a consequence of identity, but part of what 'identity' means [Dummett]
p.166 Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright]
p.167 We can introduce new objects, as equivalence classes of objects already known [Dummett]
p.167 Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Dummett]
p.168 Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett]
p.171 For Frege, objects just are what singular terms refer to [Hale/Wright]
p.190 Frege thinks number is fundamentally bound up with one-one correspondence [Heck]
p.191 Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]
p.200 Early Frege takes the extensions of concepts for granted [Dummett]
p.204 For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Chihara]
p.223 Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Dummett]
p.246 Numbers are definable in terms of mapping items which fall under concepts [Scruton]
p.246 Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo]
p.252 Frege developed formal systems to avoid unnoticed assumptions [Lavine]
p.257 Without concepts we would not have any objects [Shapiro]
p.258 Thoughts have a natural order, to which human thinking is drawn [Yablo]
p.267 One-one correlations imply normal arithmetic, but don't explain our concept of a number [Bostock]
p.277 A cardinal number may be defined as a class of similar classes [Russell]
p.281 Frege's incorrect view is that a number is an equivalence class [Benacerraf]
p.305 Frege put the idea of abstraction on a rigorous footing [Fine,K]
p.305 Fregean abstraction creates concepts which are equivalences between initial items [Fine,K]
p.325 Arithmetical statements can't be axioms, because they are provable [Burge]
p.354 To understand a thought you must understand its logical structure [Burge]
p.355 The natural number n is the set of n-membered sets [Yourgrau]
p.356 Frege's logicism aimed at removing the reliance of arithmetic on intuition [Yourgrau]
p.357 A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau]
p.358 If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau]
p.361 Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Burge]
p.371 If numbers can be derived from logic, then set theory is superfluous [Burge]
p.405 Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]
p.407 Fregean concepts have precise boundaries and universal applicability [Koslicki]
p.408 Vagueness is incomplete definition [Koslicki]
p.409 Frege's universe comes already divided into objects [Koslicki]
p.418 Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki]
p.424 Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki]
p.504 If objects exist because they fall under a concept, 0 is the object under which no objects fall [Dummett]
p.504 Frege establishes abstract objects independently from concrete ones, by falling under a concept [Dummett]
p.947 Arithmetic must be based on logic, because of its total generality [Jeshion]
p.947 The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Jeshion]
Intro p.-9 Thought is the same everywhere, and the laws of thought do not vary
Intro p.-9 To learn something, you must know that you don't know
Intro p.-7 Mental states are irrelevant to mathematics, because they are vague and fluctuating
Intro p.-5 Psychological accounts of concepts are subjective, and ultimately destroy truth
Intro p.x p.-3 Keep the psychological and subjective separate from the logical and objective
Intro p.x p.-3 Never lose sight of the distinction between concept and object
§005, 88 p.3 All analytic truths can become logical truths, by substituting definitions or synonyms [Rey]
§02 p.18 Proof aims to remove doubts, but also to show the interdependence of truths
§02 p.190 Many of us find Frege's claim that truths depend on one another an obscure idea [Heck]
§02 p.944 Justifications show the ordering of truths, and the foundation is what is self-evident [Jeshion]
§03 p.4 An a priori truth is one derived from general laws which do not require proof
§03 p.5 A statement is analytic if substitution of synonyms can make it a logical truth [Boghossian]
§03 p.108 Frege considered analyticity to be an epistemic concept [Shapiro]
§03 p.359 A truth is a priori if it can be proved entirely from general unproven laws
§03 n p.4 Induction is merely psychological, with a principle that it can actually establish laws
§053 p.60 Existence is not a first-level concept (of God), but a second-level property of concepts [Potter]
§095 p.129 We can show that a concept is consistent by producing something which falls under it
§10 p.16 In science one observation can create high probability, while a thousand might prove nothing
§13 p.360 Frege's problem is explaining the particularity of numbers by general laws [Burge]
§18 p.25 Individual numbers are best derived from the number one, and increase by one
§24 p.31 You can't transfer external properties unchanged to apply to ideas
§25 p.33 There is no physical difference between two boots and one pair of boots
§26 p.35 The equator is imaginary, but not fictitious; thought is needed to recognise it
§26 p.382 Intuitions cannot be communicated [Burge]
§26,85 p.480 Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Dummett]
§27 p.38 Numbers are not physical, and not ideas - they are objective and non-sensible
§29 p.40 We can say 'a and b are F' if F is 'wise', but not if it is 'one'
§30 p.41 The number 'one' can't be a property, if any object can be viewed as one or not one
§34 p.58 If we abstract 'from' two cats, the units are not black or white, or cats [Tait]
§41 p.53 If numbers are supposed to be patterns, each number can have many patterns
§42 p.54 We cannot define numbers from the idea of a series, because numbers must precede that
§44 p.57 You can abstract concepts from the moon, but the number one is not among them
§46 p.41 A statement of number contains a predication about a concept
§46 p.43 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt]
§46 p.125 Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley]
§47 p.61 Abstraction from things produces concepts, and numbers are in the concepts
§53 p.65 Affirmation of existence is just denial of zero
§53 p.65 Because existence is a property of concepts the ontological argument for God fails
§54 p.59 Units can be equal without being identical [Tait]
§54 p.403 A concept creating a unit must isolate and unify what falls under it
§54 p.405 Frege says counting is determining what number belongs to a given concept [Koslicki]
§54 p.406 Frege says only concepts which isolate and avoid arbitrary division can give units [Koslicki]
§55? p.123 Numerical statements have first-order logical form, so must refer to objects [Hodes]
§56 p.68 Our definition will not tell us whether or not Julius Caesar is a number
§57 p.69 For science, we can translate adjectival numbers into noun form
§57 p.69 Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
§60 p.71 Words in isolation seem to have ideas as meanings, but words have meaning in propositions
§60 p.126 Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett]
§61 p.72 Ideas are not spatial, and don't have distances between them
§61 p.72 Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates
§62 p.111 Frege initiated linguistic philosophy, studying number through the sense of sentences [Dummett]
§64 p.33 Nothing should be defined in terms of that to which it is conceptually prior [Dummett]
§64 p.75 We create new abstract concepts by carving up the content in a different way
§64 p.193 Parallelism is intuitive, so it is more fundamental than sameness of direction [Heck]
§64-68 p.232 You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett]
§64-68 p.232 From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Dummett]
§64-68 p.233 Frege introduced the standard device, of defining logical objects with equivalence classes [Dummett]
§66 n p.77 A concept is a possible predicate of a singular judgement
§68 p.79 The Number for F is the extension of 'equal to F' (or maybe just F itself)
§68 n p.80 Numbers are objects, because they can take the definite article, and can't be plurals
§74 p.87 Nought is the number belonging to the concept 'not identical with itself'
§77 p.90 One is the Number which belongs to the concept "identical with 0"
§79 p.36 'Ancestral' relations are derived by iterating back from a given relation [George/Velleman]
§87 p.99 Arithmetic is analytic and a priori, and thus it is part of logic
§87 p.99 The laws of number are not laws of nature, but are laws of the laws of nature
§90 p.102 Mathematicians just accept self-evidence, whether it is logical or intuitive
4 p.355 To understand axioms you must grasp their logical power and priority [Burge]
55-57 p.118 Numbers are objects because they partake in identity statements [Bostock]
p.x p.-3 Never ask for the meaning of a word in isolation, but only in the context of a proposition
1890 works
p.15 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate [Weiner]
p.21 Frege put forward an ontological argument for the existence of numbers [Benardete,JA]
p.22 Frege did not think of himself as working with sets [Hart,WD]
p.33 A thought is not psychological, but a condition of the world that makes a sentence true [Miller,A]
p.44 Frege thinks there is an independent logical order of the truths, which we must try to discover [Hart,WD]
p.55 Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' [Benardete,JA]
p.56 Frege's 'sense' is the strict and literal meaning, stripped of tone [Miller,A]
p.59 Frege gives a functional account of predication so that we can dispense with predicates [Benardete,JA]
p.67 For Frege, predicates are names of functions that map objects onto the True and False [McGinn]
p.67 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness [Miller,A]
p.91 Frege proposed a realist concept of a set, as the extension of a predicate or concept or function [Benardete,JA]
p.104 If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA]
p.116 Analytic truths are those that can be demonstrated using only logic and definitions [Miller,A]
p.119 Frege said concepts were abstract entities, not mental entities [Putnam]
p.207 Frege always, and fatally, neglected the domain of quantification [Dummett]
p.246 Frege's logic showed that there is no concept of being [Scruton]
p.317 Frege aimed to discover the logical foundations which justify arithmetical judgements [Burge]
p.320 The building blocks contain the whole contents of a discipline
p.337 To understand a thought, understand its inferential connections to other thoughts [Burge]
p.337 Basic truths of logic are not proved, but seen as true when they are understood [Burge]
p.351 Frege's concept of 'self-evident' makes no reference to minds [Burge]
p.369 An apriori truth is grounded in generality, which is universal quantification [Burge]
p.371 The null set is indefensible, because it collects nothing [Burge]
3.4 p.66 Eventually Frege tried to found arithmetic in geometry instead of in logic [Friend]
CP 353 p.325 Truth does not admit of more and less
p.228 p.228 Frege frequently expressed a contempt for language [Dummett]
1891 Function and Concept
p.4 Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver]
p.17 Frege takes the existence of horses to be part of their concept [Sommers]
p.18 Frege thought traditional categories had psychological and linguistic impurities [Rumfitt]
p.20 Unlike objects, concepts are inherently incomplete [George/Velleman]
p.20 Concepts are the ontological counterparts of predicative expressions [George/Velleman]
Ch.2.II p.35 An assertion about the concept 'horse' must indirectly speak of an object [Hale]
p.14 p.29 I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false
p.30 p.30 A concept is a function whose value is always a truth-value
p.30 p.30 Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism
p.32 p.32 The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place
p.38 p.38 First-level functions have objects as arguments; second-level functions take functions as arguments
p.38 n p.38 The Ontological Argument fallaciously treats existence as a first-level concept
p.39 p.39 Relations are functions with two arguments
1892 On Concept and Object
p.16 Frege mistakenly takes existence to be a property of concepts, instead of being about things [Yablo]
p.21 There is the concept, the object falling under it, and the extension (a set, which is also an object) [George/Velleman]
p.33 It is unclear whether Frege included qualities among his abstract objects [Hale]
p.150 Frege felt that meanings must be public, so they are abstractions rather than mental entities [Putnam]
p.474 Frege's 'objects' are both the referents of proper names, and what predicates are true or false of [Dummett]
p.193 p.43 As I understand it, a concept is the meaning of a grammatical predicate
p.196n p.46 For all the multiplicity of languages, mankind has a common stock of thoughts
p.199 p.49 A thought can be split in many ways, so that different parts appear as subject or predicate
p.201 p.98 Frege equated the concepts under which an object falls with its properties [Dummett]
1892 On Sense and Reference
p.1 Frege was strongly in favour of taking truth to attach to propositions [Dummett]
p.4 'Sense' gives meaning to non-referring names, and to two expressions for one referent [Margolis/Laurence]
p.6 Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first [Jacquette]
p.10 Frege was the first to construct a plausible theory of meaning [Dummett]
p.13 Earlier Frege focuses on content itself; later he became interested in understanding content [Dummett]
p.17 Expressions always give ways of thinking of referents, rather than the referents themselves [Soames]
p.34 Frege's Puzzle: from different semantics we infer different reference for two names with the same reference [Fine,K]
p.36 'The concept "horse"' denotes a concept, yet seems also to denote an object [McGee]
p.41 Frege divided the meaning of a sentence into sense, force and tone [Dummett]
p.46 Frege ascribes reference to incomplete expressions, as well as to singular terms [Hale]
p.59 Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined [Kripke]
p.59 Frege's 'sense' is ambiguous, between the meaning of a designator, and how it fixes reference [Kripke]
p.79 We can treat designation by a few words as a proper name
p.91 Every descriptive name has a sense, but may not have a reference
p.100 Frege was asking how identities could be informative [Perry]
p.154 Frege moved from extensional to intensional semantics when he added the idea of 'sense' [Sawyer]
p.154 If sentences have a 'sense', empty name sentences can be understood that way [Sawyer]
p.203 Proper name in modal contexts refer obliquely, to their usual sense [Gibbard]
p.248 Holism says all language use is also a change in the rules of language [Dummett]
p.335 Frege failed to show when two sets of truth-conditions are equivalent [Potter]
p.357 Frege started as anti-realist, but the sense/reference distinction led him to realism [Benardete,JA]
p.395 A Fregean proper name has a sense determining an object, instead of a concept [Sainsbury]
p.472 We can't get a semantics from nouns and predicates referring to the same thing [Dummett]
Pref p.-8 Frege explained meaning as sense, semantic value, reference, force and tone [Miller,A]
note p.79 People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander'
p.27 p.57 The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense
p.28 p.58 In maths, there are phrases with a clear sense, but no actual reference
p.30 p.60 The meaning of a proper name is the designated object
p.33 p.63 We are driven from sense to reference by our desire for truth
p.34 p.63 The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false
p.35 p.65 The reference of a word should be understood as part of the reference of the sentence
p.40 p.69 It is a weakness of natural languages to contain non-denoting names
p.41 p.70 In a logically perfect language every well-formed proper name designates an object
1893 Grundgesetze der Arithmetik 1 (Basic Laws)
p.3 Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Hale/Wright]
p.44 Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait]
p.147 A concept is a function mapping objects onto truth-values, if they fall under the concept [Dummett]
p.177 Frege took the study of concepts to be part of logic [Shapiro]
p.250 Frege considered definite descriptions to be genuine singular terms [Fitting/Mendelsohn]
§25 p.217 Contradiction arises from Frege's substitutional account of second-order quantification [Dummett]
III.1.73 p.269 Real numbers are ratios of quantities, such as lengths or masses
p.2 p.122 We can't prove everything, but we can spell out the unproved, so that foundations are clear
p.4 p.6 My Basic Law V is a law of pure logic
1894 Review of Husserl's 'Phil of Arithmetic'
p.32 A definition need not capture the sense of an expression - just get the reference right [Dummett]
p.193 Counting rests on one-one correspondence, of numerals to objects
p.323 p.323 The naïve view of number is that it is like a heap of things, or maybe a property of a heap
p.324 p.324 Our concepts recognise existing relations, they don't change them
p.324 p.324 Disregarding properties of two cats still leaves different objects, but what is now the difference?
p.324 p.324 If objects are just presentation, we get increasing abstraction by ignoring their properties
p.325 p.325 Many people have the same thought, which is the component, not the private presentation
p.326 p.326 Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves
p.326 p.326 Psychological logicians are concerned with sense of words, but mathematicians study the reference
p.327 p.327 Identity baffles psychologists, since A and B must be presented differently to identify them
p.327 p.327 Since every definition is an equation, one cannot define equality itself
p.328 p.328 In a number-statement, something is predicated of a concept
p.330 p.330 How do you find the right level of inattention; you eliminate too many or too few characteristics
p.332 p.332 Number-abstraction somehow makes things identical without changing them!
p.337 p.337 Numbers are not real like the sea, but (crucially) they are still objective
1895 Elucidation of some points in E.Schröder
p.212 p.126 A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class
1897 Logic [1897]
p.147 Psychological logic can't distinguish justification from causes of a belief
1900 On Euclidean Geometry
183/168 p.348 The truth of an axiom must be independently recognisable
1902 Letters to Russell
1902.06.22 p.127 The loss of my Rule V seems to make foundations for arithmetic impossible
1902.07.28 p.113 Logical objects are extensions of concepts, or ranges of values of functions
1903.05.21 p.270 I wish to go straight from cardinals to reals (as ratios), leaving out the rationals
1903 Grundgesetze der Arithmetik 2 (Basic Laws)
p.109 Later Frege held that definitions must fix a function's value for every possible argument [Wright,C]
p.139 Frege's biggest error is in not accounting for the senses of number terms [Hodes]
p.261 Real numbers are ratios of quantities [Dummett]
p.510 A number is a class of classes of the same cardinality [Dummett]
§157 p.246 Cardinals say how many, and reals give measurements compared to a unit quantity
§159 p.262 The modern account of real numbers detaches a ratio from its geometrical origins
§160 p.277 The first demand of logic is of a sharp boundary
§180 p.137 Only what is logically complex can be defined; what is simple must be pointed to
§66 p.268 We can't define a word by defining an expression containing it, as the remaining parts are a problem
§86-137 p.252 Formalism misunderstands applications, metatheory, and infinity [Dummett]
§91 p.147 Only applicability raises arithmetic from a game to a science
§99 p.174 If we abstract the difference between two houses, they don't become the same house
1910 Letters to Jourdain
p.43 p.43 We understand new propositions by constructing their sense from the words
p.43 p.43 In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference
p.44 p.44 Any object can have many different names, each with a distinct sense
p.44 p.44 Senses can't be subjective, because propositions would be private, and disagreement impossible
1914 Logic in Mathematics
p.4 Frege suggested that mathematics should only accept stipulative definitions [Gupta]
p.203 p.203 Does some mathematical reasoning (such as mathematical induction) not belong to logic?
p.203 p.203 The closest subject to logic is mathematics, which does little apart from drawing inferences
p.203 p.203 If principles are provable, they are theorems; if not, they are axioms
p.204 p.204 Tracing inference backwards closes in on a small set of axioms and postulates
p.204 p.204 Logic not only proves things, but also reveals logical relations between them
p.204 p.204 'Theorems' are both proved, and used in proofs
p.204-5 p.204 The essence of mathematics is the kernel of primitive truths on which it rests
p.205 p.205 Axioms are truths which cannot be doubted, and for which no proof is needed
p.205 p.205 To create order in mathematics we need a full system, guided by patterns of inference
p.205 p.205 A truth can be an axiom in one system and not in another
p.206 p.206 Thoughts are not subjective or psychological, because some thoughts are the same for us all
p.206 p.206 A thought is the sense expressed by a sentence, and is what we prove
p.207 p.207 The parts of a thought map onto the parts of a sentence
p.209 p.209 We use signs to mark receptacles for complex senses
p.209 p.209 We need definitions to cram retrievable sense into a signed receptacle
p.210 p.210 A 'constructive' (as opposed to 'analytic') definition creates a new sign
p.212 p.212 We must be clear about every premise and every law used in a proof
p.213 p.213 A sign won't gain sense just from being used in sentences with familiar components
p.229 p. Every concept must have a sharp boundary; we cannot allow an indeterminate third case
1918 The Thought: a Logical Enquiry
p.5 Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') [Dummett]
p.15 A thought is distinguished from other things by a capacity to be true or false [Dummett]
p.129 There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts [Weiner]
p.209 Thoughts about myself are understood one way to me, and another when communicated
p.225 Late Frege saw his non-actual objective objects as exclusively thoughts and senses [Dummett]
p.327 (60) p.327 The word 'true' seems to be unique and indefinable
p.327 (60) p.327 There cannot be complete correspondence, because ideas and reality are quite different
p.327-8 (61) p.328 A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences
p.328 (61) p.328 The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets'
p.329 (62) p.329 We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion)
p.337(69) p.337 Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist
p.342(74) p.342 A fact is a thought that is true
p.343(76) p.343 A sentence is only a thought if it is complete, and has a time-specification
1922 Sources of Knowledge of Mathematics
p.3 Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Chihara]