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 1980,0631126945]].
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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 nonlinguistic questions

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 numberobjects, 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 selfevidence [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

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
8636

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

8654

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

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

Nonarbitrary 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 onetoone correspondence replaces wellordering, 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 secondlevel, 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 'secondlevel 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 firstlevel 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 selfidentical', and one by using the concept of zero [Weiner]

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 nmembered sets [Yourgrau]

17819

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

17820

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

17460

A statement of number contains a predication about a concept

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 firstorder 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 firstlevel concepts) of the concept 'equinumerous with F' [George/Velleman]

13527

Frege's cardinals (equivalences of oneone correspondences) is not permissible in ZFC

22292

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

17442

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

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
10030

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

11030

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

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 11 [Wright,C]

18142

Oneone correlations imply normal arithmetic, but don't explain our concept of a number

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
8652

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

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 nonsensible

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
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 secondlevel type as the existential quantifier [George/Velleman]

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

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
7739

Arithmetic is analytic [Weiner]

9945

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

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]

16905

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

8655

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

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 settheory [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 metagames [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

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 nonmental, 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 / 9. Vagueness / d. Vagueness as linguistic
17431

Vagueness is incomplete definition [Koslicki]

7. Existence / D. Theories of Reality / 10. 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 / 10. Ontological Commitment / c. Commitment of predicates
10642

Secondorder quantifiers are committed to concepts, as firstorder 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 coordinates

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. SelfEvidence
8657

Mathematicians just accept selfevidence, 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

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 selfevident [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 nonpsychological oneplace 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]

10525

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

10526

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

10556

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

9881

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

9882

You can't simultaneously fix the truthconditions of a sentence and the domain of its variables [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. Society / E. State Functions / 4. Education / a. 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
8644

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

22286

Existence is not a firstlevel concept (of God), but a secondlevel property of concepts [Potter]
