The Foundations of Arithmetic
Title page of the original 1884 edition
|Original title||Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl|
|Translator||J. L. Austin|
|Subject||Philosophy of Mathematics|
|Pages||119 (original German)|
The Foundations of Arithmetic (German: Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other theories of number and develops his own theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism. The book was not well received and was not read widely when it was published. It did, however, draw the attentions of Bertrand Russell and Ludwig Wittgenstein, who were both heavily influenced by Frege's philosophy. An English translation was published (Oxford, 1950) by J. L. Austin, with a second edition in 1960.
Criticisms of predecessors
Psychologistic accounts of mathematics
Frege objects to any account of mathematics based on psychologism, that is the view that math and numbers are relative to the subjective thoughts of the people who think of them. According to Frege, psychological accounts appeal to what is subjective, while mathematics is purely objective: mathematics are completely independent from human thought. Mathematical entities, according to Frege, have objective properties regardless of humans thinking of them: it is not possible to think of mathematical statements as something that evolved naturally through human history and evolution. He sees a fundamental distinction between logic (and its extension, according to Frege, math) and psychology. Logic explains necessary facts, whereas psychology studies certain thought processes in individual minds.
Frege greatly appreciates the work of Immanuel Kant. He criticizes him mainly on the grounds that numerical statements are not synthetic-a priori, but rather analytic-a priori. Kant claims that 7+5=12 is an unprovable synthetic statement. No matter how much we analyze the idea of 7+5 we will not find there the idea of 12. We must arrive at the idea of 12 by application to objects in the intuition. Kant points out that this becomes all the more clear with bigger numbers. Frege, on this point precisely, argues towards the opposite direction. Kant wrongly assumes that in a proposition containing "big" numbers we must count points or some such thing to assert their truth value. Frege argues that without ever having any intuition toward any of the numbers in the following equation: 654,768+436,382=1,091,150 we nevertheless can assert it is true. This is provided as evidence that such a proposition is analytic. While Frege agrees that geometry is indeed synthetic a priori, arithmetic must be analytic.
Frege roundly criticizes the empiricism of John Stuart Mill. He claims that Mill's idea that numbers correspond to the various ways of splitting collections of objects into subcollections is inconsistent with confidence in calculations involving large numbers. He also denies that Mill's philosophy deals adequately with the concept of zero. He goes on to argue that the operation of addition cannot be understood as referring to physical quantities, and that Mill's confusion on this point is a symptom of a larger problem of confounding the applications of arithmetic for arithmetic itself.
Development of Frege's own view of a number
Frege makes a distinction between particular numerical statements such as 1+1=2, and general statements such as a+b=b+a. The latter are statements true of numbers just as well as the former. Therefore, it is necessary to ask for a definition of the concept of number itself. Frege investigates the possibility that number is determined in external things. He demonstrates how numbers function in natural language just as adjectives. "This desk has 5 drawers" is similar in form to "This desk has green drawers". The drawers being green is an objective fact, grounded in the external world. But this is not the case with 5. Frege argues that each drawer is on its own green, but not every drawer is 5. Frege urges us to remember that from this it does not follow that numbers may be subjective. Indeed, numbers are similar to colors at least in that both are wholly objective. Frege tells us that we can convert number statements where number words appear adjectivally (e.g., 'there are four horses') into statements where number terms appear as singular terms ('the number of horses is four'). Frege recommends such translations because he takes numbers to be objects. It makes no sense to ask whether any objects fall under 4. After Frege gives some reasons for thinking that numbers are objects, he concludes that statements of numbers are assertions about concepts.
Frege takes this observation to be the fundamental thought of Grundlagen. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept horse in the barn. Frege attempts to explain our grasp of numbers through a contextual definition of the cardinality operation ('the number of...', or ). He attempts to construct the content of a judgment involving numerical identity by relying on Hume's principle (which states that the number of Fs equals the number of Gs if and only if F and G are equinumerous, i.e. in one-one correspondence). He rejects this definition because it doesn't fix the truth value of identity statements when a singular term not of the form 'the number of Fs' flanks the identity sign. Frege goes on to give an explicit definition of number in terms of extensions of concepts, but expresses some hesitation.
Frege's definition of a number
Frege argues that numbers are objects and assert something about a concept. Frege defines numbers as extensions of concepts. 'The number of F's' is defined as the extension of the concept G is a concept that is equinumerous to F. The concept in question leads to an equivalence class of all concepts that have the number of F (including F). Frege defines 0 as the extension of the concept being non self-identical. So, the number of this concept is the extension of the concept of all concepts that have no objects falling under them. The number 1 is the extension of being identical with 0.
The book was fundamental in the development of two main disciplines, the foundations of mathematics and philosophy. Although Bertrand Russell later found a major flaw in Frege's work (this flaw is known as Russell's paradox, which is resolved by axiomatic set theory), the book was influential in subsequent developments, such as Principia Mathematica. The book can also be considered the starting point in analytic philosophy, since it revolves mainly around the analysis of language, with the goal of clarifying the concept of number. Frege's views on mathematics are also a starting point on the philosophy of mathematics, since it introduces an innovative account on the epistemology of numbers and math in general, known as logicism.
The text is divided into five chapters, which are further divided into certain headings or topics (phrased as questions or statements), and then these into 109 sections.
- Views of certain writers on the nature of arithmetical propositions.
- Are numerical formulae provable? (§5-8)
- Are the law of arithmetic inductive truths (§9-11)
- Are the laws of arithmetic synthetic a priori or analytic? (§12-17)
- Views of certain writers on the concept of Number. (§18-20)
- Is Number a property of external things? (§21-25)
- Is number something subjective? (§26-27)
- The set theory of Number (§28)
- Views on unity and one.
- Does the word "one" express a property of objects? (§29-33)
- Are units identical with one another? (§34-39)
- Attempts to overcome the difficulty. (§40-44)
- Solution of the difficulty. (§45-54)
- The Concept of Number
- Every individual number is a self-subsistent object. (§55-61)
- To obtain the concept of Number, we must fix the sense of a numerical identity. (§62-69)
- Our definition completed and its worth proved (§71-83)
- Infinite Numbers. (§84-86)
- Conclusion (§87-91)
- Other numbers. (§92-109)
- Gottlob Frege (1884). Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: Verlage Wilhelm Koebner.
- Frege 1960.
- Frege, §27.
- Frege, §12: "But an intuition in this [Kant's] sense cannot serve as ground of our knowledge of the laws of arithmetic."
- Frege, §5: "Kant declares [statements such as 2 + 3 = 5] to be unprovable and synthetic, but hesitates to call them axioms because they are not general and because the number of them is infinite. Hankel justifiably calls this conception of infinitely numerous unprovable primitive truths incongruous and paradoxical."
- Frege, §14: "The fact that [denying the parallel postulate] is possible shows that the axioms of geometry are independent of one another and of the primitive laws of logic, and consequently are synthetic. Can the same be said of the fundamental propositions of the science of number? Here, we have only to try denying any one of them, and complete confusion ensues."
- Frege 1960, p. 9-12.
- Shapiro 2000, p. 96: "Frege's Foundations of Arithmetic contains a sustained, bitter assault on Mill's account of arithmetic"
- Frege 1960, p. 10: "If the definition of each individual number did really assert a special physical fact, then we should never be able to sufficiently admire, for his knowledge of nature, a man who calculates with nine-figure numbers."
- Shapiro 2000, p. 98: "Frege also takes Mill to task concerning large numbers."
- Frege 1960, p. 11: "[...] the number 0 would be a puzzle; for up to now no one, I take it, has ever seen or touched 0 pebbles."
- Frege, §22: "Is it not in totally different senses that we speak of a tree having 1000 leaves and again as having green leaves? The green colour we ascribe to each single leaf, but not the number 1000."
- Frege, §57: "For example, the proposition 'Jupiter has four moons' can be converted into 'the number of Jupiter's moons is four'"
- Frege, §63: "Hume long ago expressed such a means: 'When two numbers are so combined as that one has always a unit answering to every unit of the other, we pronounce them equal'"
- Boolos 1998, p. 154: "Frege defines 0 as the number of the concept: being non-self-identical. Since everything is self-identical, no object falls under this concept. Frege defines 1 as the number of the concept being identical with the number zero. 0 and 0 alone falls under this latter concept."
- The Foundations of Arithmetic. A logico-mathematical enquiry into the concept of number (Oxford: Basil Blackwell, 1950) by Gottlob Frege, translated by J. L. Austin.
- Frege, Gottlob (1960). The foundations of arithmetic; a logico-mathematical enquiry into the concept of number. Translated by Austin, J. L. (2nd ed.). Evanston, Illinois: Northwestern University Press. ISBN 0810106051. OCLC 650.
- Boolos, George (1998). "Chapter 9: Gottlob Frege and the Foundations of Arithmetic". Logic, logic, and logic. Edited by Richard C. Jeffrey, introduction by John P. Burgess. Cambridge, Mass: Harvard University Press. ISBN 9780674537675. OCLC 37509971.
- On Frege's criticism of Mill: Shapiro, Stewart (2000). Thinking about mathematics: the philosophy of mathematics. New York: Oxford University Press. pp. 95–98. ISBN 9780192893062. OCLC 43864339.
- Free, full-text German edition
- Stanford Encyclopedia of Philosophy: "Frege's Theorem and Foundations for Arithmetic" by Edward Zalta.
- Nechaev, V.I. (2001) , "Number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Peter Suber, Geometry and Arithmetic are Synthetic, 2002.