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

#### Reference

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.

#### Notes

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

Quotations
 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

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)
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Topic: number and arithmetic as part of language (30 items)
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Topic: discrete vs. continuous (47 items)
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Topic: object and value equivalence (60 items)
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Topic: limitations of formalism (92 items)
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