QuoteRef: fregG_1884

topics > all references > ThesaHelp: references e-f

references e-f
mathematics as a formal system
natural language as a system
analytic truth
empirical truth
what is a number
number and arithmetic as part of language
program module
discrete vs. continuous
abstraction by name
definition by example
meaning by language as a whole
meaning of words
meaning without reference
number as the extension of a class of equinumerous classes
object and value equivalence
task scheduling
infinity and infinitesimal
mathematical proof
philosophy of mathematics
science as mathematics
objects as a set of attributes
knowledge as interrelated facts
abstraction by common attributes
limitations of formalism


Frege, G., Die Grundlagen der Arithmetik, Breslau, 1884. Google

Other Reference

translated by J.L. Austin, Foundations of Arithmetic, Basil Blackwell 1953. notes from Northwestern University Press edition 1980 119 pages.


for a discussion see: Dummett, M., Frege: Philosophy of Mathematics, Cambridge Massachusetts: Harvard University Press, 1991.

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

Related Topics up

ThesaHelp: references e-f (168 items)
Topic: mathematics as a formal system (30 items)
Topic: natural language as a system (43 items)
Topic: analytic truth (51 items)
Topic: empirical truth (44 items)
Topic: what is a number (55 items)
Topic: number and arithmetic as part of language (30 items)
Group: program module   (10 topics, 334 quotes)
Topic: discrete vs. continuous (47 items)
Topic: abstraction (62 items)
Topic: religion (43 items)
Topic: entities (20 items)
Topic: abstraction by name (29 items)
Topic: definition by example (26 items)
Topic: meaning by language as a whole (26 items)
Topic: meaning of words (21 items)
Topic: meaning without reference (31 items)
Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: object and value equivalence (60 items)
Topic: recognition (50 items)
Group: sets   (7 topics, 148 quotes)
Topic: task scheduling (49 items)
Topic: infinity and infinitesimal (37 items)
Topic: mathematical proof (23 items)
Group: philosophy of mathematics   (11 topics, 330 quotes)
Topic: science as mathematics (26 items)
Topic: objects as a set of attributes (39 items)
Topic: knowledge as interrelated facts (23 items)
Topic: abstraction by common attributes (19 items)
Topic: elements (18 items)
Topic: limitations of formalism (92 items)
Group: formalism   (9 topics, 473 quotes)

Collected barberCB 10/90 3/94
Copyright © 2002-2008 by C. Bradford Barber. All rights reserved.
Thesa is a trademark of C. Bradford Barber.