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Single Idea 8782
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
]
Full Idea
Since Frege's defence of his thesis that the laws of arithmetic are analytic depended upon a realm of independently existing objects - the finite cardinal numbers and the real numbers - his view amounted to a Platonist version of logicism.
Gist of Idea
Frege offered a Platonist version of logicism, committed to cardinal and real numbers
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Logicism in the 21st Century 1
Book Ref
'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.166
A Reaction
Nice to have this spelled out. Along with Gödel, Frege is the most distinguished Platonist since the great man. Frege has lots of modern fans, but I would have thought that this makes his position a non-starter. Alternatives are needed.
The
358 ideas
from Gottlob Frege
7741
|
The predicate 'exists' is actually a natural language expression for a quantifier
[Frege, by Weiner]
|
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]
|
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]
|
9950
|
A quantifier is a second-level predicate (which explains how it contributes to truth-conditions)
[Frege, by George/Velleman]
|
13609
|
Frege produced axioms for logic, though that does not now seem the natural basis for logic
[Frege, by Kaplan]
|
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]
|
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]
|
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]
|
10317
|
It is unclear whether Frege included qualities among his abstract objects
[Frege, by Hale]
|
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]
|
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]
|
18899
|
Frege takes the existence of horses to be part of their concept
[Frege, by Sommers]
|
9948
|
Unlike objects, concepts are inherently incomplete
[Frege, by George/Velleman]
|
9947
|
Concepts are the ontological counterparts of predicative expressions
[Frege, by George/Velleman]
|
4028
|
Frege allows either too few properties (as extensions) or too many (as predicates)
[Mellor/Oliver on Frege]
|
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]
|
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]
|
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]
|
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]
|
10020
|
Frege's biggest error is in not accounting for the senses of number terms
[Hodes on Frege]
|
10553
|
A number is a class of classes of the same cardinality
[Frege, by Dummett]
|
9889
|
Real numbers are ratios of quantities
[Frege, by Dummett]
|
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]
|
10803
|
Frege himself abstracts away from tone and color
[Yablo 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]
|
9875
|
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
[Frege, by Dummett]
|
9631
|
Formalism fails to recognise types of symbols, and also meta-games
[Frege, by Brown,JR]
|
8911
|
If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental
[Rosen on Frege]
|
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]
|
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]
|
7739
|
Arithmetic is analytic
[Frege, by Weiner]
|
9945
|
Logicism shows that no empirical truths are needed to justify arithmetic
[Frege, by George/Velleman]
|
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]
|
16905
|
Arithmetic must be based on logic, because of its total generality
[Frege, by Jeshion]
|
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]
|
17431
|
Vagueness is incomplete definition
[Frege, by Koslicki]
|
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]
|
17820
|
If you can subdivide objects many ways for counting, you can do that to set-elements too
[Yourgrau on Frege]
|
16896
|
If numbers can be derived from logic, then set theory is superfluous
[Frege, by Burge]
|
11031
|
'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many'
[Rumfitt on Frege]
|
17816
|
Frege's logicism aimed at removing the reliance of arithmetic on intuition
[Frege, by Yourgrau]
|
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]
|
10606
|
Frege treats properties as a kind of function, and maybe a property is its characteristic function
[Frege, by Smith,P]
|
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]
|
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]
|
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]
|
12153
|
Geach denies Frege's view, that 'being the same F' splits into being the same and being F
[Perry on Frege]
|
9853
|
Identity between objects is not a consequence of identity, but part of what 'identity' means
[Frege, by Dummett]
|
17623
|
To understand a thought you must understand its logical structure
[Frege, by Burge]
|
9158
|
For Frege a priori knowledge derives from general principles, so numbers can't be primitive
[Frege]
|
17432
|
Frege's universe comes already divided into objects
[Frege, by Koslicki]
|
2514
|
Frege tried to explain synthetic a priori truths by expanding the concept of analyticity
[Frege, by Katz]
|
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]
|
13878
|
Concepts are, precisely, the references of predicates
[Frege, by Wright,C]
|
9870
|
Early Frege takes the extensions of concepts for granted
[Frege, by Dummett]
|
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]
|
9844
|
Originally Frege liked contextual definitions, but later preferred them fully explicit
[Frege, by Dummett]
|
9154
|
Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence
[Frege, by Burge]
|
13881
|
We need to grasp not number-objects, but the states of affairs which make number statements true
[Frege, by Wright,C]
|
15948
|
Frege developed formal systems to avoid unnoticed assumptions
[Frege, by Lavine]
|
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]
|
9854
|
We can introduce new objects, as equivalence classes of objects already known
[Frege, by Dummett]
|
9157
|
The null set is only defensible if it is the extension of an empty concept
[Frege, by Burge]
|
9835
|
It is because a concept can be empty that there is such a thing as the empty class
[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]
|
9838
|
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person
[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]
|
16883
|
Arithmetical statements can't be axioms, because they are provable
[Frege, by Burge]
|
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]
|
10034
|
The number of natural numbers is not a natural number
[Frege, by George/Velleman]
|
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]
|
15916
|
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
[Frege, by Lavine]
|
10625
|
Frege had a motive to treat numbers as objects, but not a justification
[Hale/Wright on Frege]
|
17636
|
A cardinal number may be defined as a class of similar classes
[Frege, by Russell]
|
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]
|
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]
|
8619
|
To learn something, you must know that you don't know
[Frege]
|
8622
|
Psychological accounts of concepts are subjective, and ultimately destroy truth
[Frege]
|
8620
|
Thought is the same everywhere, and the laws of thought do not vary
[Frege]
|
8621
|
Mental states are irrelevant to mathematics, because they are vague and fluctuating
[Frege]
|
8414
|
Keep the psychological and subjective separate from the logical and objective
[Frege]
|
8415
|
Never lose sight of the distinction between concept and object
[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]
|
17443
|
Many of us find Frege's claim that truths depend on one another an obscure idea
[Heck on Frege]
|
16903
|
Justifications show the ordering of truths, and the foundation is what is self-evident
[Frege, by Jeshion]
|
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]
|
9999
|
For science, we can translate adjectival numbers into noun form
[Frege]
|
8645
|
Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
[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]
|
8647
|
Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates
[Frege]
|
8648
|
Ideas are not spatial, and don't have distances between them
[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]
|
16866
|
Tracing inference backwards closes in on a small set of axioms and postulates
[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]
|
16868
|
The essence of mathematics is the kernel of primitive truths on which it rests
[Frege]
|
16871
|
A truth can be an axiom in one system and not in another
[Frege]
|
16870
|
Axioms are truths which cannot be doubted, and for which no proof is needed
[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]
|
17446
|
Counting rests on one-one correspondence, of numerals to objects
[Frege]
|
9821
|
A definition need not capture the sense of an expression - just get the reference right
[Frege, by Dummett]
|
9577
|
The naďve view of number is that it is like a heap of things, or maybe a property of a heap
[Frege]
|
9578
|
If objects are just presentation, we get increasing abstraction by ignoring their properties
[Frege]
|
9580
|
Our concepts recognise existing relations, they don't change them
[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]
|
18752
|
'The concept "horse"' denotes a concept, yet seems also to denote an object
[Frege, by McGee]
|
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]
|
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]
|
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]
|
18937
|
If sentences have a 'sense', empty name sentences can be understood that way
[Frege, by Sawyer]
|
14075
|
Proper name in modal contexts refer obliquely, to their usual sense
[Frege, by Gibbard]
|
10424
|
A Fregean proper name has a sense determining an object, instead of a concept
[Frege, by Sainsbury]
|
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]
|
15155
|
Expressions always give ways of thinking of referents, rather than the referents themselves
[Frege, by Soames]
|
9180
|
Holism says all language use is also a change in the rules of language
[Frege, by Dummett]
|
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]
|
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]
|
9877
|
Late Frege saw his non-actual objective objects as exclusively thoughts and senses
[Frege, by Dummett]
|
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]
|
16379
|
Thoughts about myself are understood one way to me, and another when communicated
[Frege]
|
7740
|
There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts
[Frege, by Weiner]
|
19465
|
There cannot be complete correspondence, because ideas and reality are quite different
[Frege]
|
19466
|
The word 'true' seems to be unique and indefinable
[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]
|
9871
|
Frege always, and fatally, neglected the domain of quantification
[Dummett on Frege]
|
16884
|
Basic truths of logic are not proved, but seen as true when they are understood
[Frege, by Burge]
|
3319
|
Frege gives a functional account of predication so that we can dispense with predicates
[Frege, by Benardete,JA]
|
6076
|
For Frege, predicates are names of functions that map objects onto the True and False
[Frege, by McGinn]
|
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]
|
3318
|
Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x'
[Frege, by Benardete,JA]
|
16887
|
Frege's concept of 'self-evident' makes no reference to minds
[Frege, by Burge]
|
16885
|
To understand a thought, understand its inferential connections to other thoughts
[Frege, by Burge]
|
16894
|
An apriori truth is grounded in generality, which is universal quantification
[Frege, by Burge]
|
3307
|
Frege put forward an ontological argument for the existence of numbers
[Frege, by Benardete,JA]
|
16882
|
The building blocks contain the whole contents of a discipline
[Frege]
|
5816
|
Frege said concepts were abstract entities, not mental entities
[Frege, by Putnam]
|
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]
|
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]
|
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]
|
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]
|
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]
|