Ideas from 'Grundlagen der Arithmetik (Foundations)' by Gottlob Frege [1884], by Theme Structure

[found in 'The Foundations of Arithmetic (Austin)' by Frege,Gottlob (ed/tr Austin,J.L.) [Blackwell 1980,0-631-12694-5]].

green numbers give full details    |     back to texts     |     expand these ideas

---

1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual

13876

The syntactic category is primary, and the ontological category is derivative [Wright,C]

1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis

8415	Never lose sight of the distinction between concept and object

1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis

9841	Frege was the first to give linguistic answers to non-linguistic questions [Dummett]

9840	Frege initiated linguistic philosophy, studying number through the sense of sentences [Dummett]

1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis

15948

Frege developed formal systems to avoid unnoticed assumptions [Lavine]

2. Reason / A. Nature of Reason / 3. Pure Reason

10804

Thoughts have a natural order, to which human thinking is drawn [Yablo]

2. Reason / A. Nature of Reason / 5. Objectivity

9832	Frege sees no 'intersubjective' category, between objective and subjective [Dummett]

8414	Keep the psychological and subjective separate from the logical and objective

2. Reason / D. Definition / 7. Contextual Definition

9844	Originally Frege liked contextual definitions, but later preferred them fully explicit [Dummett]

9822	Nothing should be defined in terms of that to which it is conceptually prior [Dummett]

2. Reason / E. Argument / 6. Conclusive Proof

17495

Proof aims to remove doubts, but also to show the interdependence of truths

2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes

8632	You can't transfer external properties unchanged to apply to ideas

3. Truth / B. Truthmakers / 5. What Makes Truths / c. States of affairs make truths

13881

We need to grasp not number-objects, but the states of affairs which make number statements true [Wright,C]

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL

9154	Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Burge]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set

9157	The null set is only defensible if it is the extension of an empty concept [Burge]

9835	It is because a concept can be empty that there is such a thing as the empty class [Dummett]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes

9854	We can introduce new objects, as equivalence classes of objects already known [Dummett]

9883	Frege introduced the standard device, of defining logical objects with equivalence classes [Dummett]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V

18104

Frege, unlike Russell, has infinite individuals because numbers are individuals [Bostock]

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets

9834	A class is, for Frege, the extension of a concept [Dummett]

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

8645	Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"

5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic

16891

Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Burge]

16906

The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Jeshion]

5. Theory of Logic / G. Quantification / 6. Plural Quantification

14236

Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley]

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

22294

We can show that a concept is consistent by producing something which falls under it

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

17624

To understand axioms you must grasp their logical power and priority [Burge]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

8640	We cannot define numbers from the idea of a series, because numbers must precede that

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero

9838	Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Dummett]

9564	For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Chihara]

10551

If objects exist because they fall under a concept, 0 is the object under which no objects fall [Dummett]

8653	Nought is the number belonging to the concept 'not identical with itself'

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One

8654	One is the Number which belongs to the concept "identical with 0"

8636	We can say 'a and b are F' if F is 'wise', but not if it is 'one'

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units

8641	You can abstract concepts from the moon, but the number one is not among them

9989	Units can be equal without being identical [Tait]

17429

Frege says only concepts which isolate and avoid arbitrary division can give units [Koslicki]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts

17427

Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]

17437

Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki]

17438

Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki]

17426

A concept creating a unit must isolate and unify what falls under it

17428

Frege says counting is determining what number belongs to a given concept [Koslicki]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation

15916

Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Lavine]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

10034

The number of natural numbers is not a natural number [George/Velleman]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers

16883

Arithmetical statements can't be axioms, because they are provable [Burge]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers

10625

Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright]

13871

Frege claims that numbers are objects, as opposed to them being Fregean concepts [Wright,C]

13872

Numbers are second-level, ascribing properties to concepts rather than to objects [Wright,C]

9816	For Frege, successor was a relation, not a function [Dummett]

9953	Numbers are more than just 'second-level concepts', since existence is also one [George/Velleman]

9954	"Number of x's such that ..x.." is a functional expression, yielding a name when completed [George/Velleman]

17636

A cardinal number may be defined as a class of similar classes [Russell]

10139

Frege gives an incoherent account of extensions resulting from abstraction [Fine,K]

10028

For Frege the number of F's is a collection of first-level concepts [George/Velleman]

10029

Numbers need to be objects, to define the extension of the concept of each successor to n [George/Velleman]

9973	The number of F's is the extension of the second level concept 'is equipollent with F' [Tait]

16500

Frege showed that numbers attach to concepts, not to objects [Wiggins]

9990	Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Tait]

7738	Zero is defined using 'is not self-identical', and one by using the concept of zero [Weiner]

23456

Frege said logical predication implies classes, which are arithmetical objects [Morris,M]

13887

Frege started with contextual definition, but then switched to explicit extensional definition [Wright,C]

13897

Each number, except 0, is the number of the concept of all of its predecessors [Wright,C]

9856	Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett]

9902	Frege's incorrect view is that a number is an equivalence class [Benacerraf]

17814

The natural number n is the set of n-membered sets [Yourgrau]

17819

A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau]

17460

A statement of number contains a predication about a concept

17820

If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau]

16890

Frege's problem is explaining the particularity of numbers by general laws [Burge]

8630	Individual numbers are best derived from the number one, and increase by one

11029

'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt]

10013

Numerical statements have first-order logical form, so must refer to objects [Hodes]

18181

The Number for F is the extension of 'equal to F' (or maybe just F itself)

18103

Numbers are objects because they partake in identity statements [Bostock]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle

9956	'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [George/Velleman]

13527

Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Wolf,RS]

22292

Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Potter]

17442

Frege thinks number is fundamentally bound up with one-one correspondence [Heck]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem

11030

The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt]

10030

'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [George/Velleman]

8690	From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Friend]

10219

Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Shapiro]

13889

Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Wright,C]

18142

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

9046	Our definition will not tell us whether or not Julius Caesar is a number

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory

16896

If numbers can be derived from logic, then set theory is superfluous [Burge]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

8639	If numbers are supposed to be patterns, each number can have many patterns

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

13874

Numbers seem to be objects because they exactly fit the inference patterns for identities

13875

Frege's platonism proposes that objects are what singular terms refer to [Wright,C]

7731	How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner]

7737	Identities refer to objects, so numbers must be objects [Weiner]

8635	Numbers are not physical, and not ideas - they are objective and non-sensible

8652	Numbers are objects, because they can take the definite article, and can't be plurals

6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics

17816

Frege's logicism aimed at removing the reliance of arithmetic on intuition [Yourgrau]

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

8633	There is no physical difference between two boots and one pair of boots

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival

11031

'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt]

8637	The number 'one' can't be a property, if any object can be viewed as one or not one

9999	For science, we can translate adjectival numbers into noun form

9951	It appears that numbers are adjectives, but they don't apply to a single object [George/Velleman]

9952	Numerical adjectives are of the same second-level type as the existential quantifier [George/Velleman]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

9945	Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman]

16905

Arithmetic must be based on logic, because of its total generality [Jeshion]

8782	Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright]

13608

Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]

5658	Numbers are definable in terms of mapping items which fall under concepts [Scruton]

8655	Arithmetic is analytic and a priori, and thus it is part of logic

7739	Arithmetic is analytic [Weiner]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

10831

Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine]

13864

Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C]

10033

Why should the existence of pure logic entail the existence of objects? [George/Velleman]

10010

Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes]

6. Mathematics / C. Sources of Mathematics / 7. Formalism

9631	Formalism fails to recognise types of symbols, and also meta-games [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

9875	Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Dummett]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism

8642	Abstraction from things produces concepts, and numbers are in the concepts

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism

8621	Mental states are irrelevant to mathematics, because they are vague and fluctuating

7. Existence / A. Nature of Existence / 1. Nature of Existence

8643	Affirmation of existence is just denial of zero

7. Existence / A. Nature of Existence / 4. Abstract Existence

8911	If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen]

8634	The equator is imaginary, but not fictitious; thought is needed to recognise it

7. Existence / C. Structure of Existence / 4. Ontological Dependence

17443

Many of us find Frege's claim that truths depend on one another an obscure idea [Heck]

17445

Parallelism is intuitive, so it is more fundamental than sameness of direction [Heck]

7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete

10539

Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Dummett]

7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic

17431

Vagueness is incomplete definition [Koslicki]

7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment

13879

For Frege, ontological questions are to be settled by reference to syntactic structures [Wright,C]

7. Existence / D. Theories of Reality / 11. Ontological Commitment / c. Commitment of predicates

10642

Second-order quantifiers are committed to concepts, as first-order commits to objects [Linnebo]

8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation

10032

'Ancestral' relations are derived by iterating back from a given relation [George/Velleman]

8. Modes of Existence / B. Properties / 1. Nature of Properties

10606

Frege treats properties as a kind of function, and maybe a property is its characteristic function [Smith,P]

9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta

8647	Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates

9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta

10309

Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Hale]

10550

Frege establishes abstract objects independently from concrete ones, by falling under a concept [Dummett]

9. Objects / A. Existence of Objects / 3. Objects in Thought

8785	For Frege, objects just are what singular terms refer to [Hale/Wright]

10278

Without concepts we would not have any objects [Shapiro]

9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation

17432

Frege's universe comes already divided into objects [Koslicki]

9. Objects / F. Identity among Objects / 1. Concept of Identity

16022

The idea of a criterion of identity was introduced by Frege [Noonan]

11100

Frege's algorithm of identity is the law of putting equals for equals [Quine]

9. Objects / F. Identity among Objects / 3. Relative Identity

12153

Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry]

9. Objects / F. Identity among Objects / 6. Identity between Objects

9853	Identity between objects is not a consequence of identity, but part of what 'identity' means [Dummett]

11. Knowledge Aims / A. Knowledge / 2. Understanding

17623

To understand a thought you must understand its logical structure [Burge]

12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori

9158	For Frege a priori knowledge derives from general principles, so numbers can't be primitive

12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence

8657	Mathematicians just accept self-evidence, whether it is logical or intuitive

12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities

9352	An a priori truth is one derived from general laws which do not require proof

16889

A truth is a priori if it can be proved entirely from general unproven laws

12. Knowledge Sources / A. A Priori Knowledge / 8. A Priori as Analytic

2514	Frege tried to explain synthetic a priori truths by expanding the concept of analyticity [Katz]

12. Knowledge Sources / E. Direct Knowledge / 2. Intuition

16900

Intuitions cannot be communicated [Burge]

13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / d. Rational foundations

16903

Justifications show the ordering of truths, and the foundation is what is self-evident [Jeshion]

14. Science / C. Induction / 1. Induction

8624	Induction is merely psychological, with a principle that it can actually establish laws

8626	In science one observation can create high probability, while a thousand might prove nothing

15. Nature of Minds / A. Nature of Mind / 1. Mind / c. Features of mind

8648	Ideas are not spatial, and don't have distances between them

18. Thought / A. Modes of Thought / 1. Thought

8620	Thought is the same everywhere, and the laws of thought do not vary

18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts

9870	Early Frege takes the extensions of concepts for granted [Dummett]

18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts

13878

Concepts are, precisely, the references of predicates [Wright,C]

7736	A concept is a non-psychological one-place function asserting something of an object [Weiner]

17430

Fregean concepts have precise boundaries and universal applicability [Koslicki]

8622	Psychological accounts of concepts are subjective, and ultimately destroy truth

18. Thought / D. Concepts / 5. Concepts and Language / b. Concepts are linguistic

8651	A concept is a possible predicate of a singular judgement

18. Thought / E. Abstraction / 1. Abstract Thought

9846	Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett]

18. Thought / E. Abstraction / 2. Abstracta by Selection

9976	Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait]

18. Thought / E. Abstraction / 3. Abstracta by Ignoring

10803

Frege himself abstracts away from tone and color [Yablo]

9988	If we abstract 'from' two cats, the units are not black or white, or cats [Tait]

18. Thought / E. Abstraction / 7. Abstracta by Equivalence

9855	Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Dummett]

10802

Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo]

10526

Fregean abstraction creates concepts which are equivalences between initial items [Fine,K]

10525

Frege put the idea of abstraction on a rigorous footing [Fine,K]

10556

We create new abstract concepts by carving up the content in a different way

9882	You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett]

9881	From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Dummett]

19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning

8646	Words in isolation seem to have ideas as meanings, but words have meaning in propositions

7732	Never ask for the meaning of a word in isolation, but only in the context of a proposition

19. Language / E. Analyticity / 1. Analytic Propositions

9370	A statement is analytic if substitution of synonyms can make it a logical truth [Boghossian]

8743	Frege considered analyticity to be an epistemic concept [Shapiro]

19. Language / E. Analyticity / 2. Analytic Truths

20295

All analytic truths can become logical truths, by substituting definitions or synonyms [Rey]

19. Language / E. Analyticity / 4. Analytic/Synthetic Critique

2515	Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz]

25. Social Practice / E. Policies / 5. Education / b. Education principles

8619	To learn something, you must know that you don't know

26. Natural Theory / D. Laws of Nature / 6. Laws as Numerical

8656	The laws of number are not laws of nature, but are laws of the laws of nature

28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique

22286

Existence is not a first-level concept (of God), but a second-level property of concepts [Potter]

8644	Because existence is a property of concepts the ontological argument for God fails