The Foundations of Arithmetic
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Die Grundlagen der Arithmetik (The Foundations of Arithmetic) is a book by Gottlob Frege, published in 1884, in which he investigates the philosophical foundations of arithmetic. In a tour de force of literary and philosophical merit, he demolishes other theories of number and develops his own view of a number. It 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 Wittgenstein who were both heavily influenced by Frege's philosophy.
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[edit] Criticisms of predecessors
[edit] Psychologistic accounts of mathematics
Frege objects to any psychological account of mathematics. Psychological accounts appeal to what is subjective, while mathematics are purely objective. Mathematics are completely independent from human thought. Mathematical entities have objective properties regardless of humans thinking of them. It is not possible to think of mathematical statements as something which evolved naturally through the human history and evolution.
[edit] Mill
[edit] Kant
Frege greatly appreciates the work of Kant. He criticizes him mainly on that numerical statements are not synthetic-a priori, but rather analytic-a priori. Kant claims that 7+5=12 is a 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 experienced world. 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 to that such a proposition is analytic. While Frege agrees that geometry is indeed synthetic a priori, arithmetic must be analytic.
[edit] 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 Gl. So, when I say that the number of horses in the barn is four, I am saying that four objects fall under the concept horse. Frege attempts to explain our grasp of numbers through a contextual definition of the cardinality operation ('the number of...', or Nx:Fx). 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 [or in one-one correspondence]). He rejects this definition because it doesn't fix the truth value of identity statments 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.
[edit] 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.
[edit] External links
- Original German text. Maintained by Alain Blachair, Académie de Nancy-Metz.
- Stanford Encyclopedia of Philosophy: "Frege's Logic, Theorem, and Foundations for Arithmetic," -- by Edward Zalta.
