more on this theme
|
more from this text
Single Idea 9831
[filed under theme 6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
]
Full Idea
The elements of all geometrical constructions are intuitions, and geometry appeals to intuition as the source of its axioms.
Gist of Idea
Geometry appeals to intuition as the source of its axioms
Source
Gottlob Frege (Rechnungsmethoden (dissertation) [1874], Ch.6), quoted by Michael Dummett - Frege philosophy of mathematics
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.68
A Reaction
Very early Frege, but he stuck to this view, while firmly rejecting intuition as a source of arithmetic. Frege would have known well that Euclid's assumption about parallels had been challenged.
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 naïve 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]
|