iii ;;Quote: calculation is not aggregative mechanical thought; only possible if mathematical notation has been thoroughly developed

iii ;;Quote: thought is in its essentials the same everywhere; but may be more pure, using words and numerals as aids, aspires to surpass all sciences

4 ;;Quote: an a posteriori truth depends on facts; an a priori truth depends on general, primitive laws that neither need nor admit to proof

4 ;;Quote: to demonstrate that arithmetic is analytic need to prove its fundamental propositions with utmost rigour; makes clear the primitive truths

21 ;;Quote: the truths of arithmetic govern all that is numerable; this includes everything thinkable; closely tied the laws of thought

29 ;;Quote: what number belongs to a pile of playing cards depends on how we chose to regard it; e.g., as packs

33 ;;Quote: number is entirely the creature of the mind; 1 house has many windows, 1 city has many houses

50 ;;Quote: number can not be defined as a collection of unit objects; if distinguished, can't do arithmetic; if not, they merge together

59 ;;Quote: a statement of number is an assertion about a concept; e.g., "Venus has 0 moons" concerns the concept "moon of Venus"

59 ;;Quote: concepts can change their properties, e.g., the number of inhabitant's of Germany

59+;;Quote: picking a time for "inhabitant of Germany", fixes the number for all eternity

60 ;;Quote: statements such as "All whales are mammals" and of number concern concepts instead of objects; an indefinite object is really a concept

61 ;;Quote: number applies to concepts that are abstracted from things; this explains number's wide range of applicability

62 ;;Quote: a thing is called one or single simply with respect to its existence; number applies to things with a common genus, e.g., coins

65 ;;Quote: the ontological argument for the existence of God fails because existence and oneness are properties of concepts

65+;;Quote: affirmation of existence is nothing but denial of the number nought

65 ;;Quote: oneness is a characteristic of the higher order concept of all unitary concepts; this differs from genus and species

66 ;;Quote: can not define numbers as belonging to a concept; e.g., is Caesar a number? What is the object 0?

70 ;;Quote: we can form no idea of our distance from the sun; yet we don't doubt the calculation nor avoid using it for further inferences

71 ;;Quote: only in a proposition have words really a meaning; even though mental pictures float before us all the while

71+;;Quote: we are led by our thought beyond the scope of our imagination

73 ;;Quote: to define Number need to fix the sense of numerical identity; of the number which belongs to the concept F and to the concept G

73 ;;Quote: define numeric identity by a one-one relationship; need to use identity to define numeric identity

73 ;;Quote: define numeric identity in the same way as defining direction from the extension of the concept of parallel lines

78 ;;Quote: for identity to be a useful concept need to be able to recognize two things as the same even though they differ

78+;;Quote: Frege assumes that the extension of a concept is known; basis for recognizing identity

83 ;;Quote: define a 1-1 correlation as a relation that assigns an object for concept G to each object in concept F and vice versa

143 ;;Quote: two concepts are equinumerous if there is a one-to-one correlation; 'n is a number' is a concept that is equinumerous to a concept F

145 ;;Quote: 0 is the number that applies to the concept 'unequal to itself'

87 ;;Quote: for logic and rigorous proof, a concept must have sharp limits, i.e., clear membership determination

88 ;;Quote: 0 is a number because any relation, including identity, is a 1-1 relation for concepts under which no object falls

89 ;;Quote: define successor as the Number belonging to a concept of one additional element

90 ;;Quote: define 1 as the Number belonging to the concept "identical with 0"; this concept only contains 0

90+;;Quote: Frege's definition of 1 does not presuppose, for its objective legitimacy, any matter of observed fact

93 ;;Quote: Frege's definition of successor is objectively definite; something independent of the laws that govern the movements of our attention

94 ;;Quote: Frege sketches the proof that every natural number has a successor and therefore forms an infinite sequence

153 ;;Quote: showed that arithmetic laws are analytic judgments (a priori)

153+;;Quote: arithmetic laws are the laws of the laws of nature

100 ;;Quote: a poor way to form concepts is through a list of characteristics; instead the elements of a definition are richly interconnected

100+;;Quote: Frege's concept of Number extends our knowledge even though it is analytic; like the seed produces the plant

102 ;;Quote: Frege did not claim to prove the analytic character of number; there may be a gap in his proof or a missing premise

156 ;;Quote: a proof must be a step-by-step procedure; devised a concept writing to reduce their length; conforms with rules

116 ;;Quote: define number by fixing the sense of numerical identity, i.e., the recognition statement for numbers