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Single Idea 18142

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem ]

Full Idea

Frege inferred from the Julius Caesar problem that even though Hume's Principle sufficed as a single axiom for deducing the arithmetic of the finite cardinal numbers, still it does not explain our ordinary understanding of those numbers.

Gist of Idea

One-one correlations imply normal arithmetic, but don't explain our concept of a number

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 9.A.2

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.267

Related Idea

Idea 18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]

The 358 ideas from Gottlob Frege

22270 Frege changed philosophy by extending logic's ability to check the grounds of thinking [Potter on Frege]
8939 We should not describe human laws of thought, but how to correctly track truth [Frege, by Fisher]
17745 For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki]
7728 Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Frege, by Weiner]
7622 In 1879 Frege developed second order logic [Frege, by Putnam]
7729 Frege replaced Aristotle's subject/predicate form with function/argument form [Frege, by Weiner]
9950 A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) [Frege, by George/Velleman]
9991 For Frege the variable ranges over all objects [Frege, by Tait]
10536 Frege's domain for variables is all objects, but modern interpretations first fix the domain [Dummett on Frege]
7730 Frege introduced quantifiers for generality [Frege, by Weiner]
7742 Frege reduced most quantifiers to 'everything' combined with 'not' [Frege, by McCullogh]
13824 Proof theory began with Frege's definition of derivability [Frege, by Prawitz]
13609 Frege produced axioms for logic, though that does not now seem the natural basis for logic [Frege, by Kaplan]
17855 It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C]
10607 Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P]
11008 Existence is not a first-order property, but the instantiation of a property [Frege, by Read]
22280 Frege's account was top-down and decompositional, not bottom-up and compositional [Frege, by Potter]
7741 The predicate 'exists' is actually a natural language expression for a quantifier [Frege, by Weiner]
4971 I don't use 'subject' and 'predicate' in my way of representing a judgement [Frege]
16881 The laws of logic are boundless, so we want the few whose power contains the others [Frege]
18265 We don't judge by combining subject and concept; we get a concept by splitting up a judgement [Frege]
10535 Frege's 'objects' are both the referents of proper names, and what predicates are true or false of [Frege, by Dummett]
18995 Frege mistakenly takes existence to be a property of concepts, instead of being about things [Frege, by Yablo]
10317 It is unclear whether Frege included qualities among his abstract objects [Frege, by Hale]
9949 There is the concept, the object falling under it, and the extension (a set, which is also an object) [Frege, by George/Velleman]
9167 Frege felt that meanings must be public, so they are abstractions rather than mental entities [Frege, by Putnam]
4973 As I understand it, a concept is the meaning of a grammatical predicate [Frege]
4974 For all the multiplicity of languages, mankind has a common stock of thoughts [Frege]
4975 A thought can be split in many ways, so that different parts appear as subject or predicate [Frege]
9839 Frege equated the concepts under which an object falls with its properties [Frege, by Dummett]
14238 A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class [Frege]
18806 Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt]
9947 Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman]
9948 Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman]
4028 Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege]
18899 Frege takes the existence of horses to be part of their concept [Frege, by Sommers]
10319 An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale]
4972 I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege]
8488 A concept is a function whose value is always a truth-value [Frege]
8487 Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege]
8489 The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege]
8490 First-level functions have objects as arguments; second-level functions take functions as arguments [Frege]
8491 The Ontological Argument fallaciously treats existence as a first-level concept [Frege]
8492 Relations are functions with two arguments [Frege]
10623 Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright]
9975 Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege]
13733 Frege considered definite descriptions to be genuine singular terms [Frege, by Fitting/Mendelsohn]
9190 A concept is a function mapping objects onto truth-values, if they fall under the concept [Frege, by Dummett]
13665 Frege took the study of concepts to be part of logic [Frege, by Shapiro]
9874 Contradiction arises from Frege's substitutional account of second-order quantification [Dummett on Frege]
18252 Real numbers are ratios of quantities, such as lengths or masses [Frege]
18271 We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege]
18165 My Basic Law V is a law of pure logic [Frege]
9889 Real numbers are ratios of quantities [Frege, by Dummett]
10553 A number is a class of classes of the same cardinality [Frege, by Dummett]
10020 Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege]
13886 Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C]
9886 Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
9890 The modern account of real numbers detaches a ratio from its geometrical origins [Frege]
9891 The first demand of logic is of a sharp boundary [Frege]
10019 Only what is logically complex can be defined; what is simple must be pointed to [Frege]
9845 We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege]
9887 Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett]
8751 Only applicability raises arithmetic from a game to a science [Frege]
11846 If we abstract the difference between two houses, they don't become the same house [Frege]
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]
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]
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]
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]
17432 Frege's universe comes already divided into objects [Frege, by Koslicki]
12153 Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry on Frege]
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]
9853 Identity between objects is not a consequence of identity, but part of what 'identity' means [Frege, by Dummett]
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]
10034 The number of natural numbers is not a natural number [Frege, by George/Velleman]
9631 Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR]
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]
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]
10606 Frege treats properties as a kind of function, and maybe a property is its characteristic function [Frege, by Smith,P]
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]
16883 Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
16896 If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
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]
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]
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]
9844 Originally Frege liked contextual definitions, but later preferred them fully explicit [Frege, by Dummett]
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]
15948 Frege developed formal systems to avoid unnoticed assumptions [Frege, by Lavine]
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]
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]
9835 It is because a concept can be empty that there is such a thing as the empty class [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]
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]
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]
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]
8621 Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege]
8620 Thought is the same everywhere, and the laws of thought do not vary [Frege]
8622 Psychological accounts of concepts are subjective, and ultimately destroy truth [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]
16903 Justifications show the ordering of truths, and the foundation is what is self-evident [Frege, by Jeshion]
17443 Many of us find Frege's claim that truths depend on one another an obscure idea [Heck on Frege]
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]
14236 Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley on 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]
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]
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]
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]
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]
10556 We create new abstract concepts by carving up the content in a different way [Frege]
17445 Parallelism is intuitive, so it is more fundamental than sameness of direction [Frege, by Heck]
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]
11052 Psychological logic can't distinguish justification from causes of a belief [Frege]
11219 Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta]
16863 Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege]
16862 The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege]
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
16867 Logic not only proves things, but also reveals logical relations between them [Frege]
16865 'Theorems' are both proved, and used in proofs [Frege]
16866 Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
16868 The essence of mathematics is the kernel of primitive truths on which it rests [Frege]
16870 Axioms are truths which cannot be doubted, and for which no proof is needed [Frege]
16871 A truth can be an axiom in one system and not in another [Frege]
16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]
16873 Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege]
16872 A thought is the sense expressed by a sentence, and is what we prove [Frege]
16874 The parts of a thought map onto the parts of a sentence [Frege]
16876 We need definitions to cram retrievable sense into a signed receptacle [Frege]
16875 We use signs to mark receptacles for complex senses [Frege]
16877 A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege]
16878 We must be clear about every premise and every law used in a proof [Frege]
16879 A sign won't gain sense just from being used in sentences with familiar components [Frege]
9388 Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege]
16886 The truth of an axiom must be independently recognisable [Frege]
9831 Geometry appeals to intuition as the source of its axioms [Frege]
18256 Quantity is inconceivable without the idea of addition [Frege]
9821 A definition need not capture the sense of an expression - just get the reference right [Frege, by Dummett]
17446 Counting rests on one-one correspondence, of numerals to objects [Frege]
9577 The naïve view of number is that it is like a heap of things, or maybe a property of a heap [Frege]
9580 Our concepts recognise existing relations, they don't change them [Frege]
9578 If objects are just presentation, we get increasing abstraction by ignoring their properties [Frege]
9579 Disregarding properties of two cats still leaves different objects, but what is now the difference? [Frege]
9581 Many people have the same thought, which is the component, not the private presentation [Frege]
9582 Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege]
9583 Psychological logicians are concerned with sense of words, but mathematicians study the reference [Frege]
9584 Identity baffles psychologists, since A and B must be presented differently to identify them [Frege]
9585 Since every definition is an equation, one cannot define equality itself [Frege]
9586 In a number-statement, something is predicated of a concept [Frege]
9587 How do you find the right level of inattention; you eliminate too many or too few characteristics [Frege]
9588 Number-abstraction somehow makes things identical without changing them! [Frege]
9589 Numbers are not real like the sea, but (crucially) they are still objective [Frege]
10533 We can't get a semantics from nouns and predicates referring to the same thing [Frege, by Dummett]
4893 Frege was asking how identities could be informative [Frege, by Perry]
18752 'The concept "horse"' denotes a concept, yet seems also to denote an object [Frege, by McGee]
9462 Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first [Frege, by Jacquette]
18936 Frege moved from extensional to intensional semantics when he added the idea of 'sense' [Frege, by Sawyer]
8187 Frege was strongly in favour of taking truth to attach to propositions [Frege, by Dummett]
18937 If sentences have a 'sense', empty name sentences can be understood that way [Frege, by Sawyer]
10424 A Fregean proper name has a sense determining an object, instead of a concept [Frege, by Sainsbury]
10510 Frege ascribes reference to incomplete expressions, as well as to singular terms [Frege, by Hale]
18772 We can treat designation by a few words as a proper name [Frege]
14075 Proper name in modal contexts refer obliquely, to their usual sense [Frege, by Gibbard]
22318 Frege failed to show when two sets of truth-conditions are equivalent [Frege, by Potter]
15597 Frege's Puzzle: from different semantics we infer different reference for two names with the same reference [Frege, by Fine,K]
17002 Frege's 'sense' is ambiguous, between the meaning of a designator, and how it fixes reference [Kripke on Frege]
18778 Every descriptive name has a sense, but may not have a reference [Frege]
7805 Frege started as anti-realist, but the sense/reference distinction led him to realism [Frege, by Benardete,JA]
9180 Holism says all language use is also a change in the rules of language [Frege, by Dummett]
15155 Expressions always give ways of thinking of referents, rather than the referents themselves [Frege, by Soames]
11126 'Sense' gives meaning to non-referring names, and to two expressions for one referent [Frege, by Margolis/Laurence]
8164 Frege was the first to construct a plausible theory of meaning [Frege, by Dummett]
9817 Earlier Frege focuses on content itself; later he became interested in understanding content [Frege, by Dummett]
8171 Frege divided the meaning of a sentence into sense, force and tone [Frege, by Dummett]
4954 Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined [Kripke on Frege]
7304 Frege explained meaning as sense, semantic value, reference, force and tone [Frege, by Miller,A]
18773 People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander' [Frege]
4976 The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense [Frege]
4977 In maths, there are phrases with a clear sense, but no actual reference [Frege]
4978 The meaning of a proper name is the designated object [Frege]
4979 We are driven from sense to reference by our desire for truth [Frege]
4980 The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false [Frege]
4981 The reference of a word should be understood as part of the reference of the sentence [Frege]
18940 It is a weakness of natural languages to contain non-denoting names [Frege]
18939 In a logically perfect language every well-formed proper name designates an object [Frege]
9545 Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Frege, by Chihara]
16379 Thoughts about myself are understood one way to me, and another when communicated [Frege]
8162 Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') [Frege, by Dummett]
9818 A thought is distinguished from other things by a capacity to be true or false [Frege, by Dummett]
9877 Late Frege saw his non-actual objective objects as exclusively thoughts and senses [Frege, by Dummett]
7740 There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts [Frege, by Weiner]
19466 The word 'true' seems to be unique and indefinable [Frege]
19465 There cannot be complete correspondence, because ideas and reality are quite different [Frege]
19467 A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences [Frege]
19468 The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets' [Frege]
19469 We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion) [Frege]
19470 Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist [Frege]
19471 A fact is a thought that is true [Frege]
19472 A sentence is only a thought if it is complete, and has a time-specification [Frege]
8446 We understand new propositions by constructing their sense from the words [Frege]
8447 In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference [Frege]
8448 Any object can have many different names, each with a distinct sense [Frege]
8449 Senses can't be subjective, because propositions would be private, and disagreement impossible [Frege]
18166 The loss of my Rule V seems to make foundations for arithmetic impossible [Frege]
18269 Logical objects are extensions of concepts, or ranges of values of functions [Frege]
18253 I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege]
16884 Basic truths of logic are not proved, but seen as true when they are understood [Frege, by Burge]
9871 Frege always, and fatally, neglected the domain of quantification [Dummett on Frege]
3331 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 on Frege]
6076 For Frege, predicates are names of functions that map objects onto the True and False [Frege, by McGinn]
3319 Frege gives a functional account of predication so that we can dispense with predicates [Frege, by Benardete,JA]
13473 Frege thinks there is an independent logical order of the truths, which we must try to discover [Frege, by Hart,WD]
3328 Frege proposed a realist concept of a set, as the extension of a predicate or concept or function [Frege, by Benardete,JA]
16895 The null set is indefensible, because it collects nothing [Frege, by Burge]
13455 Frege did not think of himself as working with sets [Frege, by Hart,WD]
5657 Frege's logic showed that there is no concept of being [Frege, by Scruton]
16880 Frege aimed to discover the logical foundations which justify arithmetical judgements [Frege, by Burge]
5816 Frege said concepts were abstract entities, not mental entities [Frege, by Putnam]
16882 The building blocks contain the whole contents of a discipline [Frege]
16894 An apriori truth is grounded in generality, which is universal quantification [Frege, by Burge]
16885 To understand a thought, understand its inferential connections to other thoughts [Frege, by Burge]
16887 Frege's concept of 'self-evident' makes no reference to minds [Frege, by Burge]
3318 Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' [Frege, by Benardete,JA]
7307 A thought is not psychological, but a condition of the world that makes a sentence true [Frege, by Miller,A]
7725 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate [Frege, by Weiner]
7309 Frege's 'sense' is the strict and literal meaning, stripped of tone [Frege, by Miller,A]
7312 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness [Frege, by Miller,A]
3307 Frege put forward an ontological argument for the existence of numbers [Frege, by Benardete,JA]
7316 Analytic truths are those that can be demonstrated using only logic and definitions [Frege, by Miller,A]
8689 Eventually Frege tried to found arithmetic in geometry instead of in logic [Frege, by Friend]
22317 Truth does not admit of more and less [Frege]
9179 Frege frequently expressed a contempt for language [Frege, by Dummett]