Die Grundlagen der Arithmetik (The Foundations of Arithmetic) is a book by Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
, published in 1884, in which he investigates the philosophical foundations of arithmetic. In a tour de force of literary and philosophical merit, Frege demolished other theories of number and developed his own theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
The book was not well received and was not read widely when it was published. It did, however, draw the attentions of Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
and Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
, who were both heavily influenced by Frege's philosophy.
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 which 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 the order of ideas, whereas psychology studies certain thought processes in individual minds.
Frege greatly appreciates the work of Kant
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
. 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 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.
This criticism is probably unfair to Kant. Kant's example of counting points or fingers in the introduction to the Critique is actually meant merely as an illustration of the fact that arithmetic is constructive in nature. He does not mean that we literally need to count dots in order to represent large numbers. His point is only that in arithmetic, we construct magnitudes. So, for example, to add 1,115,677 to 4,322,899, we construct the sum according to the rules we have defined for addition, which include carrying, etc. But when we are doing this what we are doing is proceeding according to some method by which we can construct a sum. Now each of the rules we define in this method is reducible to simpler rules, until ultimately, if we wish, we can reduce the whole thing to the pure intuition of time, i.e., the counting of successive instants of time. The reduction, of course, does not need to be actually carried out. It is enough if it is possible in principle. Kant's discussion of algebra makes this clear. Although he is not talking specifically about arithmetic here, this discussion is obviously relevant to it:
"But mathematics does not merely construct magnitudes (quanta), as in geometry, but also mere magnitude (quantitataem), as in algebra, where it entirely abstracts from the constitution of the object that is to be thought in accordance with such a concept of magnitude. In this case it chooses a certain notation for all construction of magnitudes in general (numbers), as well as addition, subtraction, extraction of roots, etc., and, after it has also designated the general concept of quantities in accordance with their different relations, it then exhibits all the procedures through which magnitude is generated and altered in accordance with certain rules in intuition; where one magnitude is to be divided by another, it places their symbols together in accordance with the form of notation for division, and thereby achieves by a symbolic construction equally well what geometry does by an ostensive or geometrical construction (of the objects themselves), which discursive cognition could never achieve by means of mere concepts."
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 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
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
), 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.