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13876 | The syntactic category is primary, and the ontological category is derivative |

15948 | Frege developed formal systems to avoid unnoticed assumptions |

9840 | Frege initiated linguistic philosophy, studying number through the sense of sentences |

8415 | Never lose sight of the distinction between concept and object |

9832 | Frege sees no 'intersubjective' category, between objective and subjective |

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

9844 | Originally Frege liked contextual definitions, but later preferred them fully explicit |

9822 | Nothing should be defined in terms of that to which it is conceptually prior |

8623 | Proof reveals the interdependence of truths, as well as showing their certainty |

17495 | Proof aims to remove doubts, but also to show the interdependence of truths |

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

13881 | We need to grasp not number-objects, but the states of affairs which make number statements true |

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

9157 | The null set is only defensible if it is the extension of an empty concept |

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

9854 | We can introduce new objects, as equivalence classes of objects already known |

9883 | Frege introduced the standard device, of defining logical objects with equivalence classes |

9834 | A class is, for Frege, the extension of a concept |

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

16891 | Despite Gödel, Frege's epistemic ordering of all the truths is still plausible |

16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries |

14236 | Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed |

17624 | To understand axioms you must grasp their logical power and priority |

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

9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person |

9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined |

10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall |

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

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" |

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

9989 | Units can be equal without being identical |

17426 | A concept creating a unit must isolate and unify what falls under it |

17429 | Frege says only concepts which isolate and avoid arbitrary division can give units |

15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted |

17437 | Non-arbitrary division means that what falls under the concept cannot be divided into more of the same |

17438 | Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage |

17428 | Frege says counting is determining what number belongs to a given concept |

10034 | The number of natural numbers is not a natural number |

16883 | Arithmetical statements can't be axioms, because they are provable |

10625 | Frege had a motive to treat numbers as objects, but not a justification |

13871 | Frege claims that numbers are objects, as opposed to them being Fregean concepts |

13872 | Numbers are second-level, ascribing properties to concepts rather than to objects |

9816 | For Frege, successor was a relation, not a function |

9953 | Numbers are more than just 'second-level concepts', since existence is also one |

9954 | "Number of x's such that ..x.." is a functional expression, yielding a name when completed |

17636 | A cardinal number may be defined as a class of similar classes |

10139 | Frege gives an incoherent account of extensions resulting from abstraction |

10028 | For Frege the number of F's is a collection of first-level concepts |

10029 | Numbers need to be objects, to define the extension of the concept of each successor to n |

9973 | The number of F's is the extension of the second level concept 'is equipollent with F' |

16500 | Frege showed that numbers attach to concepts, not to objects |

9990 | Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts |

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

13887 | Frege started with contextual definition, but then switched to explicit extensional definition |

13897 | Each number, except 0, is the number of the concept of all of its predecessors |

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

9902 | Frege's incorrect view is that a number is an equivalence class |

17814 | The natural number n is the set of n-membered sets |

17819 | A set doesn't have a fixed number, because the elements can be seen in different ways |

17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too |

16890 | Frege's problem is explaining the particularity of numbers by general laws |

18103 | Numbers are objects because they partake in identity statements |

17460 | A statement of number contains a predication about a concept |

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' |

10013 | Numerical statements have first-order logical form, so must refer to objects |

18181 | The Number for F is the extension of 'equal to F' (or maybe just F itself) |

17442 | Frege thinks number is fundamentally bound up with one-one correspondence |

9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' |

10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor |

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

10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) |

13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 |

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

11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish |

16896 | If numbers can be derived from logic, then set theory is superfluous |

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

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 |

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

7737 | Identities refer to objects, so numbers must be objects |

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

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

17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition |

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

9951 | It appears that numbers are adjectives, but they don't apply to a single object |

9952 | Numerical adjectives are of the same second-level type as the existential quantifier |

11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' |

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 |

9945 | Logicism shows that no empirical truths are needed to justify arithmetic |

8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers |

5658 | Numbers are definable in terms of mapping items which fall under concepts |

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

7739 | Arithmetic is analytic |

16905 | Arithmetic must be based on logic, because of its total generality |

10831 | Frege only managed to prove that arithmetic was analytic with a logic that included set-theory |

10010 | Frege's belief in logicism and in numerical objects seem uncomfortable together |

10033 | Why should the existence of pure logic entail the existence of objects? |

9631 | Formalism fails to recognise types of symbols, and also meta-games |

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

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

8643 | Affirmation of existence is just denial of zero |

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

17443 | Many of us find Frege's claim that truths depend on one another an obscure idea |

17445 | Parallelism is intuitive, so it is more fundamental than sameness of direction |

10539 | Frege refers to 'concrete' objects, but they are no different in principle from abstract ones |

17431 | Vagueness is incomplete definition |

13879 | For Frege, ontological questions are to be settled by reference to syntactic structures |

10642 | Second-order quantifiers are committed to concepts, as first-order commits to objects |

10032 | 'Ancestral' relations are derived by iterating back from a given relation |

10606 | Frege treats properties as a kind of function, and maybe a property is its characteristic function |

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

10309 | Frege says singular terms denote objects, numerals are singular terms, so numbers exist |

10550 | Frege establishes abstract objects independently from concrete ones, by falling under a concept |

8785 | For Frege, objects just are what singular terms refer to |

10278 | Without concepts we would not have any objects |

17432 | Frege's universe comes already divided into objects |

16022 | The idea of a criterion of identity was introduced by Frege |

11100 | Frege's algorithm of identity is the law of putting equals for equals |

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

17623 | To understand a thought you must understand its logical structure |

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

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

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 |

16900 | Intuitions cannot be communicated |

16903 | Justifications show the ordering of truths, and the foundation is what is self-evident |

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 |

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

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

9870 | Early Frege takes the extensions of concepts for granted |

13878 | Concepts are, precisely, the references of predicates |

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

17430 | Fregean concepts have precise boundaries and universal applicability |

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

8651 | A concept is a possible predicate of a singular judgement |

9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line |

9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one |

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

10803 | Frege himself abstracts away from tone and color |

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

10525 | Frege put the idea of abstraction on a rigorous footing |

10526 | Fregean abstraction creates concepts which are equivalences between initial items |

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 |

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

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 |

9370 | A statement is analytic if substitution of synonyms can make it a logical truth |

8743 | Frege considered analyticity to be an epistemic concept |

20295 | All analytic truths can become logical truths, by substituting definitions or synonyms |

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

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

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