The Foundations of Arithmetic
Die Grundlagen der Arithmetik — Frege's logicist account of the nature of number
Tradition: Analytic philosophy / logicism / philosophy of mathematics
Arithmetic is reducible to logic — numbers are not subjective ideas but objective abstract entities accessible to reason
The Foundations of Arithmetic is one of the founding works of analytic philosophy and the first systematic statement of logicism — the doctrine that the truths of arithmetic are reducible to truths of pure logic. Frege critically surveys earlier views of number (Mill's empiricism, Kant's synthetic-a-priori position, the formalist view) and develops his own account: number-statements predicate something of concepts; numbers themselves are abstract objects definable in terms of equinumerosity of concepts. Russell's 1902 letter pointing out the paradox in Frege's formal system (Russell's paradox) shattered the technical execution of the logicist program, but the Foundations' philosophical achievements — the sense-reference distinction, the context principle, the analytic-philosophical method — remained foundational.
Author
Editions cited
- The Foundations of Arithmetic (J. L. Austin, Oxford, 1980)
- Die Grundlagen der Arithmetik (Christian Thiel, Hamburg, 1986)
- Translations from the Philosophical Writings of Gottlob Frege (Peter Geach & Max Black, Blackwell, 3rd ed. 1980)
School Embodiments
The Foundations of Arithmetic is one of the founding texts of analytic philosophy. Russell, Wittgenstein, Carnap, and the entire analytic tradition acknowledge Frege as a major source.
"In the enquiry that follows, I have kept to three fundamental principles: always to separate sharply the psychological from the logical; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object." (Foundations, Introduction)
Frege's position that arithmetic is analytic and derivable from pure logic is a paradigm of modern philosophical rationalism — the truths of mathematics are knowable a priori without empirical input.
"In arithmetic we are not concerned with objects which we come to know as something alien from without through the medium of the senses." (Foundations §105)
Frege's realism about numbers as abstract objects — neither subjective ideas nor physical patterns — is the modern philosophical descendant of Platonic realism about mathematical entities.
"The laws of number are not laws of nature, but laws of the laws of nature." (Foundations §87)
Frege's logical analysis of language shaped the Vienna Circle's programmatic application of logical method to philosophy.
"Words have meaning only in the context of a sentence." (Foundations, Introduction, the "context principle")
Frege's robust anti-psychologism — the objectivity of logical and mathematical truth — shaped the realist strand of analytic philosophy.
"Number is something objective." (Foundations §27)
Frege engaged the German idealist tradition critically; his "third realm" of objective thoughts has some affinity with Hegelian objective spirit, though Frege would not have accepted the comparison.
"Thoughts are neither things in the external world nor ideas." (Frege, "Thought" — consonant with Foundations)
Analytic-philosophical tradition.
Internal Tensions
Russell's paradox destroyed the formal program of the Foundations; Frege's appendix to the Grundgesetze volume II (1903) acknowledged the catastrophe but could not resolve it. The philosophical achievements survived independently of the formal failure, and the twenty-first-century neo-logicism (Crispin Wright, Bob Hale) has attempted to recover the program with new principles.
I. Time
Numbers are timeless abstract objects. Real time in physical reality is the standard nineteenth-century framework.
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II. Space
Numbers are not in space. Standard treatment of physical space.
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III. Matter
Material objects are real but not the subject of arithmetic. Standard scientific realism.
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IV. Observer
The Fregean observer is the rational agent capable of grasping objective thoughts. The "third realm" of objective thoughts is accessible to all rational minds.
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V. Energy
Not engaged.
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VI. Information
Numbers and thoughts are substantival abstract informational entities. Personal information not philosophically privileged.
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How The Foundations of Arithmetic resolves each dilemma
51 resolved positions across 4 dimensions, including 6 distinctive where the majority of schools go the other way · 6 unaligned.
Each dimension is sorted so minority positions come first. Mainstream positions are folded into an expandable list.
Time · 9 dilemmas · 3 distinctive
Persistence, the future, and the direction of becoming.
6 mainstream positions
Matter · 7 dilemmas, all mainstream
Observer · 37 dilemmas · 3 distinctive
Mind, agency, and the knower's relation to the known.