The Foundations of Arithmetic
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
Criticisms of predecessors
Criticisms of 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.
of Mill
of 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.
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. Numbers are objects not in time and space. As numbers are not used as predicates for the external world, Frege argues numbers are, in fact, second order predicates- predicates of predicates. Frege explains that number modifies concepts. These concepts, it is important to note, are objective just as the numbers are. Evidence is provided in that the same and exact sense perception can lead to varying concepts, and thereby to varying true numerical statements. For example the same sense data can be seen as 1 football team or 11 players. It is clear that the number modifies the concept (objectively) and not the object in the external world.
Frege's definition of a number
Frege argues that numbers belong to concepts, not to objects. The reason is that the very same object can purport many different numbers, and this is based on the concept by which we analyze the object. So 1 classroom may be seen as 20 students. Frege concludes that every number must have a concept to which it belongs. The number 0 is defined as belonging to the concept "X is different from itself", as there are no objects in the extension of this concept. The number 1 is defined as belonging to the concept "X is 0", the number 2 "X is either 0 or 1" and so forth.
External Links
http://www.ac-nancy-metz.fr/enseign/philo/textesph/Frege.pdf Original German text (maintained by Alain Blachair, Académie de Nancy-Metz)
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