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Single Idea 9886

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number ]

Full Idea

The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity.

Gist of Idea

Cardinals say how many, and reals give measurements compared to a unit quantity

Source

Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.246

A Reaction

We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals.

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]