1874 | Rechnungsmethoden (dissertation) |
Ch.6 | p.68 | 9831 | Geometry appeals to intuition as the source of its axioms |
p.2 | p.279 | 18256 | Quantity is inconceivable without the idea of addition |
1879 | Begriffsschrift |
p.17 | 8939 | We should not describe human laws of thought, but how to correctly track truth |
p.19 | 7742 | Frege reduced most quantifiers to 'everything' combined with 'not' |
p.23 | 9950 | A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) |
p.23 | 17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B |
p.31 | 7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) |
p.37 | 7729 | Frege replaced Aristotle's subject/predicate form with function/argument form |
p.44 | 7730 | Frege introduced quantifiers for generality |
p.59 | 9991 | For Frege the variable ranges over all objects |
p.118 | 10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. |
p.124 | 7622 | In 1879 Frege developed second order logic |
p.126 | 11008 | Existence is not a first-order property, but the instantiation of a property |
p.133 | 7741 | The predicate 'exists' is actually a natural language expression for a quantifier |
p.191 | 13609 | Frege produced axioms for logic, though that does not now seem the natural basis for logic |
p.207 | 13824 | Proof theory began with Frege's definition of derivability |
§03 | p.12 | 4971 | I don't use 'subject' and 'predicate' in my way of representing a judgement |
§13 | p.29 | 16881 | The laws of logic are boundless, so we want the few whose power contains the others |
1881 | Boole calculus and the Concept script |
p.17 | p.17 | 18265 | We don't judge by combining subject and concept; we get a concept by splitting up a judgement |
1884 | Grundlagen der Arithmetik (Foundations) |
p.4 | 10803 | Frege himself abstracts away from tone and color |
p.7 | 10625 | Frege had a motive to treat numbers as objects, but not a justification |
p.7 | 13871 | Frege claims that numbers are objects, as opposed to them being Fregean concepts |
p.8 | 16022 | The idea of a criterion of identity was introduced by Frege |
p.10 | 13872 | Numbers are second-level, ascribing properties to concepts rather than to objects |
p.11 | 9154 | Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence |
p.11 | 10309 | Frege says singular terms denote objects, numerals are singular terms, so numbers exist |
p.11 | 13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities |
p.13 | 13875 | Frege's platonism proposes that objects are what singular terms refer to |
p.13 | 13876 | Syntactic category which is primary and ontological category derivative. |
p.13 | 9816 | For Frege, successor was a relation, not a function |
p.15 | 13878 | Concepts are, precisely, the references of predicates |
p.17 | 9945 | Logicism shows that no empirical truths are needed to justify arithmetic |
p.18 | 10642 | Second-order quantifiers are committed to concepts, as first-order commits to objects |
p.23 | 9951 | It appears that numbers are adjectives, but they don't apply to a single object |
p.24 | 9952 | Numerical adjectives are of the same second-level type as the existential quantifier |
p.25 | 13879 | For Frege, ontological questions are to be settled by reference to syntactic structures |
p.25 | 9953 | Numbers are more than just 'second-level concepts', since existence is also one |
p.25 | 13881 | We need to grasp not number-objects, but the states of affairs which make number statements true |
p.26 | 9157 | The null set is only defensible if it is the extension of an empty concept |
p.26 | 9158 | For Frege a priori knowledge derives from general principles, so numbers can't be primitive |
p.27 | 9954 | "Number of x's such that ..x.." is a functional expression, yielding a name when completed |
p.29 | 10139 | Frege gives an incoherent account of extensions resulting from abstraction |
p.30 | 10028 | For Frege the number of F's is a collection of first-level concepts |
p.30 | 9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' |
p.33 | 10029 | Numbers need to be objects, to define the extension of the concept of each successor to n |
p.35 | 10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor |
p.39 | 10033 | Why should the existence of pure logic entail the existence of objects? |
p.41 | 10034 | The number of natural numbers is not a natural number |
p.43 | 9973 | The number of F's is the extension of the second level concept 'is equipollent with F' |
p.44 | 16500 | Frege showed that numbers attach to concepts, not to objects |
p.44 | 9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one |
p.48 | 11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish |
p.49 | 11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' |
p.54 | 7731 | How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? |
p.55 | 15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted |
p.57 | 7736 | A concept is a non-psychological one-place function asserting something of an object |
p.59 | 7737 | Identities refer to objects, so numbers must be objects |
p.59 | 9990 | Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts |
p.64 | 9631 | Formalism fails to recognise types of symbols, and also meta-games |
p.66 | 8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? |
p.66 | 7738 | Zero is defined using 'is not self-identical', and one by using the concept of zero |
p.66 | 10831 | Frege only managed to prove that arithmetic was analytic with a logic that included set-theory |
p.76 | 11100 | Frege's algorithm of identity is the law of putting equals for equals |
p.77 | 9832 | Frege sees no 'intersubjective' category, between objective and subjective |
p.78 | 10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) |
p.87 | 10606 | Frege treats properties as a kind of function, and maybe a property is its characteristic function |
p.91 | 9834 | A class is, for Frege, the extension of a concept |
p.92 | 9835 | It is because a concept can be empty that there is such a thing as the empty class |
p.96 | 9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person |
p.111 | 13887 | Frege started with contextual definition, but then switched to explicit extensional definition |
p.114 | 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 |
p.118 | 7739 | Arithmetic is analytic |
p.123 | 10010 | Frege's belief in logicism and in numerical objects seem uncomfortable together |
p.125 | 9844 | Originally Frege liked contextual definitions, but later preferred them fully explicit |
p.136 | 13897 | Each number, except 0, is the number of the concept of all of its predecessors |
p.162 | 9853 | Identity between objects is not a consequence of identity, but part of what 'identity' means |
p.166 | 8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers |
p.167 | 9855 | Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects |
p.167 | 9854 | We can introduce new objects, as equivalence classes of objects already known |
p.168 | 9856 | Frege's account of cardinals fails in modern set theory, so they are now defined differently |
p.171 | 8785 | For Frege, objects just are what singular terms refer to |
p.190 | 17442 | Frege thinks number is fundamentally bound up with one-one correspondence |
p.200 | 9870 | Early Frege takes the extensions of concepts for granted |
p.204 | 9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined |
p.246 | 5658 | Numbers are definable in terms of mapping items which fall under concepts |
p.252 | 15948 | Frege developed formal systems to avoid unnoticed assumptions |
p.257 | 10278 | Without concepts we would not have any objects |
p.277 | 17636 | A cardinal number may be defined as a class of similar classes |
p.281 | 9902 | Frege's incorrect view is that a number is an equivalence class |
p.305 | 10526 | Fregean abstraction creates concepts which are equivalences between initial items |
p.305 | 10525 | Frege put the idea of abstraction on a rigorous footing |
p.325 | 16883 | Arithmetical statements can't be axioms, because they are provable |
p.354 | 17623 | To understand a thought you must understand its logical structure |
p.355 | 17814 | The natural number n is the set of n-membered sets |
p.356 | 17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition |
p.357 | 17819 | A set doesn't have a fixed number, because the elements can be seen in different ways |
p.358 | 17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too |
p.361 | 16891 | Despite Gödel, Frege's epistemic ordering of all the truths is still plausible |
p.371 | 16896 | If numbers can be derived from logic, then set theory is superfluous |
p.407 | 17430 | Fregean concepts have precise boundaries and universal applicability |
p.408 | 17431 | Vagueness is incomplete definition |
p.409 | 17432 | Frege's universe comes already divided into objects |
p.418 | 17437 | Non-arbitrary division means that what falls under the concept cannot be divided into more of the same |
p.424 | 17438 | Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage |
p.504 | 10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall |
p.504 | 10550 | Frege establishes abstract objects independently from concrete ones, by falling under a concept |
p.947 | 16905 | Arithmetic must be based on logic, because of its total generality |
p.947 | 16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries |
Intro | p.-9 | 8619 | To learn something, you must know that you don't know |
Intro | p.-9 | 8620 | Thought is the same everywhere, and the laws of thought do not vary |
Intro | p.-7 | 8621 | Mental states are irrelevant to mathematics, because they are vague and fluctuating |
Intro | p.-5 | 8622 | Psychological accounts of concepts are subjective, and ultimately destroy truth |
Intro p.x | p.-3 | 8415 | Never lose sight of the distinction between concept and object |
Intro p.x | p.-3 | 8414 | Keep the psychological and subjective separate from the logical and objective |
§02 | p.2 | 8623 | Proof reveals the interdependence of truths, as well as showing their certainty |
§02 | p.18 | 17495 | Proof aims to remove doubts, but also to show the interdependence of truths |
§02 | p.190 | 17443 | Many of us find Frege's claim that truths depend on one another an obscure idea |
§02 | p.944 | 16903 | Justifications show the ordering of truths, and the foundation is what is self-evident |
§03 | p.4 | 9352 | An a priori truth is one derived from general laws which do not require proof |
§03 | p.5 | 9370 | A statement is analytic if substitution of synonyms can make it a logical truth |
§03 | p.108 | 8743 | Frege considered analyticity to be an epistemic concept |
§03 | p.359 | 16889 | A truth is a priori if it can be proved entirely from general unproven laws |
§03 n | p.4 | 8624 | Induction is merely psychological, with a principle that it can actually establish laws |
§10 | p.16 | 8626 | In science one observation can create high probability, while a thousand might prove nothing |
§13 | p.360 | 16890 | Frege's problem is explaining the particularity of numbers by general laws |
§18 | p.25 | 8630 | Individual numbers are best derived from the number one, and increase by one |
§24 | p.31 | 8632 | You can't transfer external properties unchanged to apply to ideas |
§25 | p.33 | 8633 | There is no physical difference between two boots and one pair of boots |
§26 | p.35 | 8634 | The equator is imaginary, but not fictitious; thought is needed to recognise it |
§26 | p.382 | 16900 | Intuitions cannot be communicated |
§26,85 | p.480 | 10539 | Frege refers to 'concrete' objects, but they are no different in principle from abstract ones |
§27 | p.38 | 8635 | Numbers are not physical, and not ideas - they are objective and non-sensible |
§29 | p.40 | 8636 | We can say 'a and b are F' if F is 'wise', but not if it is 'one' |
§30 | p.41 | 8637 | The number 'one' can't be a property, if any object can be viewed as one or not one |
§34 | p.58 | 9988 | If we abstract 'from' two cats, the units are not black or white, or cats |
§41 | p.53 | 8639 | If numbers are supposed to be patterns, each number can have many patterns |
§42 | p.54 | 8640 | We cannot define numbers from the idea of a series, because numbers must precede that |
§44 | p.57 | 8641 | You can abstract concepts from the moon, but the number one is not among them |
§46 | p.41 | 17460 | A statement of number contains a predication about a concept |
§46 | p.43 | 11029 | 'Exactly ten gallons' may not mean ten things instantiate 'gallon' |
§46 | p.125 | 14236 | Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed |
§47 | p.61 | 8642 | Abstraction from things produces concepts, and numbers are in the concepts |
§53 | p.65 | 8643 | Affirmation of existence is just denial of zero |
§53 | p.65 | 8644 | Because existence is a property of concepts the ontological argument for God fails |
§54 | p.59 | 9989 | Units can be equal without being identical |
§54 | p.403 | 17426 | A concept creating a unit must isolate and unify what falls under it |
§54 | p.405 | 17428 | Frege says counting is determining what number belongs to a given concept |
§54 | p.406 | 17429 | Frege says only concepts which isolate and avoid arbitrary division can give units |
§55? | p.123 | 10013 | Numerical statements have first-order logical form, so must refer to objects |
§56 | p.68 | 9046 | Our definition will not tell us whether or not Julius Caesar is a number |
§57 | p.69 | 9999 | For science, we can translate adjectival numbers into noun form |
§57 | p.69 | 8645 | Convert "Jupiter has four moons" into "the number of Jupiter's moons is four" |
§60 | p.71 | 8646 | Words in isolation seem to have ideas as meanings, but words have meaning in propositions |
§60 | p.126 | 9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line |
§61 | p.72 | 8648 | Ideas are not spatial, and don't have distances between them |
§61 | p.72 | 8647 | Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates |
§62 | p.111 | 9840 | Frege initiated linguistic philosophy, studying number through the sense of sentences |
§64 | p.33 | 9822 | Nothing should be defined in terms of that to which it is conceptually prior |
§64 | p.75 | 10556 | We create new abstract concepts by carving up the content in a different way |
§64 | p.193 | 17445 | Parallelism is intuitive, so it is more fundamental than sameness of direction |
§64-68 | p.232 | 9882 | You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables |
§64-68 | p.232 | 9881 | From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements |
§64-68 | p.233 | 9883 | Frege introduced the standard device, of defining logical objects with equivalence classes |
§66 n | p.77 | 8651 | A concept is a possible predicate of a singular judgement |
§68 | p.79 | 18181 | The Number for F is the extension of 'equal to F' (or maybe just F itself) |
§68 n | p.80 | 8652 | Numbers are objects, because they can take the definite article, and can't be plurals |
§74 | p.87 | 8653 | Nought is the number belonging to the concept 'not identical with itself' |
§77 | p.90 | 8654 | One is the Number which belongs to the concept "identical with 0" |
§79 | p.36 | 10032 | 'Ancestral' relations are derived by iterating back from a given relation |
§87 | p.99 | 8655 | Arithmetic is analytic and a priori, and thus it is part of logic |
§87 | p.99 | 8656 | The laws of number are not laws of nature, but are laws of the laws of nature |
§90 | p.102 | 8657 | Mathematicians just accept self-evidence, whether it is logical or intuitive |
4 | p.355 | 17624 | To understand axioms you must grasp their logical power and priority |
55-57 | p.118 | 18103 | Numbers are objects because they partake in identity statements |
p.x | p.-3 | 7732 | Never ask for the meaning of a word in isolation, but only in the context of a proposition |
1890 | works |
p.15 | 7725 | 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate |
p.21 | 3307 | Frege put forward an ontological argument for the existence of numbers |
p.22 | 13455 | Frege did not think of himself as working with sets |
p.33 | 7307 | A thought is not psychological, but a condition of the world that makes a sentence true |
p.44 | 13473 | Frege thinks there is an independent logical order of the truths, which we must try to discover |
p.55 | 3318 | Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' |
p.56 | 7309 | Frege's 'sense' is the strict and literal meaning, stripped of tone |
p.59 | 3319 | Frege gives a functional account of predication so that we can dispense with predicates |
p.67 | 6076 | For Frege, predicates are names of functions that map objects onto the True and False |
p.67 | 7312 | 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness |
p.91 | 3328 | Frege proposed a realist concept of a set, as the extension of a predicate or concept or function |
p.104 | 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 |
p.116 | 7316 | Analytic truths are those that can be demonstrated using only logic and definitions |
p.119 | 5816 | Frege said concepts were abstract entities, not mental entities |
p.207 | 9871 | Frege always, and fatally, neglected the domain of quantification |
p.246 | 5657 | Frege's logic showed that there is no concept of being |
p.317 | 16880 | Frege aimed to discover the logical foundations which justify arithmetical judgements |
p.320 | 16882 | The building blocks contain the whole contents of a discipline |
p.337 | 16885 | To understand a thought, understand its inferential connections to other thoughts |
p.337 | 16884 | Basic truths of logic are not proved, but seen as true when they are understood |
p.351 | 16887 | Frege's concept of 'self-evident' makes no reference to minds |
p.369 | 16894 | An apriori truth is grounded in generality, which is universal quantification |
p.371 | 16895 | The null set is indefensible, because it collects nothing |
3.4 | p.66 | 8689 | Eventually Frege tried to found arithmetic in geometry instead of in logic |
p.228 | p.228 | 9179 | Frege frequently expressed a contempt for language |
1891 | Function and Concept |
p.4 | 4028 | Frege allows either too few properties (as extensions) or too many (as predicates) |
p.20 | 9948 | Unlike objects, concepts are inherently incomplete |
p.20 | 9947 | Concepts are the ontological counterparts of predicative expressions |
Ch.2.II | p.35 | 10319 | An assertion about the concept 'horse' must indirectly speak of an object |
p.14 | p.29 | 4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false |
p.30 | p.30 | 8488 | A concept is a function whose value is always a truth-value |
p.30 | p.30 | 8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism |
p.32 | p.32 | 8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place |
p.38 | p.38 | 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments |
p.38 n | p.38 | 8491 | The Ontological Argument fallaciously treats existence as a first-level concept |
p.39 | p.39 | 8492 | Relations are functions with two arguments |
1892 | On Concept and Object |
p.16 | 18995 | Frege mistakenly takes existence to be a property of concepts, instead of being about things |
p.21 | 9949 | There is the concept, the object falling under it, and the extension (a set, which is also an object) |
p.33 | 10317 | It is unclear whether Frege included qualities among his abstract objects |
p.150 | 9167 | Frege felt that meanings must be public, so they are abstractions rather than mental entities |
p.474 | 10535 | Frege's 'objects' are both the referents of proper names, and what predicates are true or false of |
p.193 | p.43 | 4973 | As I understand it, a concept is the meaning of a grammatical predicate |
p.196n | p.46 | 4974 | For all the multiplicity of languages, mankind has a common stock of thoughts |
p.199 | p.49 | 4975 | A thought can be split in many ways, so that different parts appear as subject or predicate |
p.201 | p.98 | 9839 | Frege equated the concepts under which an object falls with its properties |
1892 | On Sense and Reference |
p.1 | 8187 | Frege was strongly in favour of taking truth to attach to propositions |
p.4 | 11126 | 'Sense' gives meaning to non-referring names, and to two expressions for one referent |
p.6 | 9462 | Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first |
p.10 | 8164 | Frege was the first to construct a plausible theory of meaning |
p.13 | 9817 | Earlier Frege focuses on content itself; later he became interested in understanding content |
p.17 | 15155 | Expressions always give ways of thinking of referents, rather than the referents themselves |
p.36 | 18752 | 'The concept "horse"' denotes a concept, yet seems also to denote an object |
p.41 | 8171 | Frege divided the meaning of a sentence into sense, force and tone |
p.46 | 10510 | Frege ascribes reference to incomplete expressions, as well as to singular terms |
p.59 | 4954 | Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined |
p.59 | 16402 | Frege's 'sense' is ambiguous, as the meaning of a designator, and how its reference is determined |
p.79 | 18772 | We can treat designation by a few words as a proper name |
p.91 | 18778 | Every descriptive name has a sense, but may not have a reference |
p.100 | 4893 | Frege was asking how identities could be informative |
p.154 | 18936 | Frege moved from extensional to intensional semantics when he added the idea of 'sense' |
p.203 | 14075 | Proper name in modal contexts refer obliquely, to their usual sense |
p.357 | 7805 | Frege started as anti-realist, but the sense/reference distinction led him to realism |
p.395 | 10424 | A Fregean proper name has a sense determining an object, instead of a concept |
p.472 | 10533 | We can't get a semantics from nouns and predicates referring to the same thing |
Pref | p.-8 | 7304 | Frege explained meaning as sense, semantic value, reference, force and tone |
note | p.79 | 18773 | People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander' |
p.27 | p.57 | 4976 | The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense |
p.28 | p.58 | 4977 | In maths, there are phrases with a clear sense, but no actual reference |
p.30 | p.60 | 4978 | The meaning of a proper name is the designated object |
p.33 | p.63 | 4979 | We are driven from sense to reference by our desire for truth |
p.34 | p.63 | 4980 | The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false |
p.35 | p.65 | 4981 | The reference of a word should be understood as part of the reference of the sentence |
p.40 | p.69 | 18940 | It is a weakness of natural languages to contain non-denoting names |
p.41 | p.70 | 18939 | In a logically perfect language every well-formed proper name designates an object |
1893 | Grundgesetze der Arithmetik 1 (Basic Laws) |
p.3 | 10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions |
p.44 | 9975 | Frege ignored Cantor's warning that a cardinal set is not just a concept-extension |
p.147 | 9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept |
p.177 | 13665 | Frege took the study of concepts to be part of logic |
p.250 | 13733 | Frege considered definite descriptions to be genuine singular terms |
§25 | p.217 | 9874 | Contradiction arises from Frege's substitutional account of second-order quantification |
III.1.73 | p.269 | 18252 | Real numbers are ratios of quantities, such as lengths or masses |
p.2 | p.122 | 18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear |
p.4 | p.6 | 18165 | My Basic Law V is a law of pure logic |
1894 | Review of Husserl's 'Phil of Arithmetic' |
p.32 | 9821 | A definition need not capture the sense of an expression - just get the reference right |
p.193 | 17446 | Counting rests on one-one correspondence, of numerals to objects |
p.323 | p.323 | 9577 | The naïve view of number is that it is like a heap of things, or maybe a property of a heap |
p.324 | p.324 | 9580 | Our concepts recognise existing relations, they don't change them |
p.324 | p.324 | 9578 | If objects are just presentation, we get increasing abstraction by ignoring their properties |
p.324 | p.324 | 9579 | Disregarding properties of two cats still leaves different objects, but what is now the difference? |
p.325 | p.325 | 9581 | Many people have the same thought, which is the component, not the private presentation |
p.326 | p.326 | 9583 | Psychological logicians are concerned with sense of words, but mathematicians study the reference |
p.326 | p.326 | 9582 | Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves |
p.327 | p.327 | 9585 | Since every definition is an equation, one cannot define equality itself |
p.327 | p.327 | 9584 | Identity baffles psychologists, since A and B must be presented differently to identify them |
p.328 | p.328 | 9586 | In a number-statement, something is predicated of a concept |
p.330 | p.330 | 9587 | How do you find the right level of inattention; you eliminate too many or too few characteristics |
p.332 | p.332 | 9588 | Number-abstraction somehow makes things identical without changing them! |
p.337 | p.337 | 9589 | Numbers are not real like the sea, but (crucially) they are still objective |
1895 | Elucidation of some points in E.Schröder |
p.212 | p.126 | 14238 | A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class |
1897 | Logic [1897] |
p.147 | 11052 | Psychological logic can't distinguish justification from causes of a belief |
1900 | On Euclidean Geometry |
183/168 | p.348 | 16886 | The truth of an axiom must be independently recognisable |
1902 | Letters to Russell |
1902.06.22 | p.127 | 18166 | The loss of my Rule V seems to make foundations for arithmetic impossible |
1902.07.28 | p.113 | 18269 | Logical objects are extensions of concepts, or ranges of values of functions |
1903.05.21 | p.270 | 18253 | I wish to go straight from cardinals to reals (as ratios), leaving out the rationals |
1903 | Grundgesetze der Arithmetik 2 (Basic Laws) |
p.109 | 13886 | Later Frege held that definitions must fix a function's value for every possible argument |
p.139 | 10020 | Frege's biggest error is in not accounting for the senses of number terms |
p.261 | 9889 | Real numbers are ratios of quantities |
p.510 | 10553 | A number is a class of classes of the same cardinality |
§157 | p.246 | 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity |
§159 | p.262 | 9890 | The modern account of real numbers detaches a ratio from its geometrical origins |
§160 | p.277 | 9891 | The first demand of logic is of a sharp boundary |
§180 | p.137 | 10019 | Only what is logically complex can be defined; what is simple must be pointed to |
§66 | p.268 | 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem |
§86-137 | p.252 | 9887 | Formalism misunderstands applications, metatheory, and infinity |
§91 | p.147 | 8751 | Only applicability raises arithmetic from a game to a science |
§99 | p.174 | 11846 | If we abstract the difference between two houses, they don't become the same house |
1910 | Letters to Jourdain |
p.43 | p.43 | 8447 | In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference |
p.43 | p.43 | 8446 | We understand new propositions by constructing their sense from the words |
p.44 | p.44 | 8449 | Senses can't be subjective, because propositions would be private, and disagreement impossible |
p.44 | p.44 | 8448 | Any object can have many different names, each with a distinct sense |
1914 | Logic in Mathematics |
p.4 | 11219 | Frege suggested that mathematics should only accept stipulative definitions |
p.203 | p.203 | 16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? |
p.203 | p.203 | 16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences |
p.203 | p.203 | 16864 | If principles are provable, they are theorems; if not, they are axioms |
p.204 | p.204 | 16866 | Tracing inference backwards closes in on a small set of axioms and postulates |
p.204 | p.204 | 16865 | 'Theorems' are both proved, and used in proofs |
p.204 | p.204 | 16867 | Logic not only proves things, but also reveals logical relations between them |
p.204-5 | p.204 | 16868 | The essence of mathematics is the kernel of primitive truths on which it rests |
p.205 | p.205 | 16870 | Axioms are truths which cannot be doubted, and for which no proof is needed |
p.205 | p.205 | 16871 | A truth can be an axiom in one system and not in another |
p.205 | p.205 | 16869 | To create order in mathematics we need a full system, guided by patterns of inference |
p.206 | p.206 | 16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all |
p.206 | p.206 | 16872 | A thought is the sense expressed by a sentence, and is what we prove |
p.207 | p.207 | 16874 | The parts of a thought map onto the parts of a sentence |
p.209 | p.209 | 16875 | We use signs to mark receptacles for complex senses |
p.209 | p.209 | 16876 | We need definitions to cram retrievable sense into a signed receptacle |
p.210 | p.210 | 16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign |
p.212 | p.212 | 16878 | We must be clear about every premise and every law used in a proof |
p.213 | p.213 | 16879 | A sign won't gain sense just from being used in sentences with familiar components |
p.229 | p. | 9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case |
1918 | The Thought: a Logical Enquiry |
p.5 | 8162 | Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') |
p.15 | 9818 | A thought is distinguished from other things by a capacity to be true or false |
p.129 | 7740 | There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts |
p.209 | 16379 | Thoughts about myself are understood one way to me, and another when communicated |
p.225 | 9877 | Late Frege saw his non-actual objective objects as exclusively thoughts and senses |
p.327 (60) | p.327 | 19466 | The word 'true' seems to be unique and indefinable |
p.327 (60) | p.327 | 19465 | There cannot be complete correspondence, because ideas and reality are quite different |
p.327-8 (61) | p.328 | 19467 | A 'thought' something for which the question of truth can arise; thoughts are senses of sentences |
p.328 (61) | p.328 | 19468 | The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets' |
p.329 (62) | p.329 | 19469 | We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion) |
p.337(69) | p.337 | 19470 | Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist |
p.342(74) | p.342 | 19471 | A fact is a thought that is true |
p.343(76) | p.343 | 19472 | A sentence is only a thought if it is complete, and has a time-specification |
1922 | Sources of Knowledge of Mathematics |
p.3 | 9545 | Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical |