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''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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
. 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
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 Psychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, laws, or entities. The word was coined by Johann ...
, 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 Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and ...
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.


Kant

Frege greatly appreciates the work of Immanuel Kant. He criticizes him mainly on the grounds that numerical statements are not
synthetic Synthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to: Science * Synthetic chemical or compound, produced by the process of chemical synthesis * 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.


Mill

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 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
. 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 with 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 Nx: Fx ). He attempts to construct the content of a judgment involving numerical identity by relying on
Hume's principle Hume's principle or HP says that the number of ''F''s is equal to the number of ''G''s if and only if there is a one-to-one correspondence (a bijection) between the ''F''s and the ''G''s. HP can be stated formally in systems of second-order logic. ...
(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.


Legacy

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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
), the book was influential in subsequent developments, such as ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
''. 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.


Editions

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See also

*
Basic Law V In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 ''Die Grund ...
* '' Begriffsschrift'' *
Context principle In the philosophy of language, the context principle is a form of semantic holism holding that a philosopher should "never ... ask for the meaning of a word in isolation, but only in the context of a proposition" (Frege 884/1980x). Analysis The c ...
* Foundationalism * Linguistic turn *
Psychologism dispute In logic, anti-psychologism (also logical objectivism or logical realism) is a theory about the nature of logical truth, that it does not depend upon the contents of human ideas but exists independent of human ideas. Overview The anti-psychologisti ...
*
Round square copula In metaphysics and the philosophy of language, the round square copula is a common example of the dual copula strategy used in reference to the problem of nonexistent objects as well as their relation to problems in modern philosophy of language. ...


References


Sources

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External links

* – Free, full-text German edition
''Die Grundlagen der Arithmetik''
at
archive.org The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
– Free, full-text German edition * Stanford Encyclopedia of Philosophy
"Frege's Theorem and Foundations for Arithmetic"
by Edward Zalta. * *
Peter Suber Peter Dain Suber (born November 8, 1951) is a philosopher specializing in the philosophy of law and open access to knowledge. He is a Senior Researcher at the Berkman Klein Center for Internet & Society, Director of the Harvard Office for Scholarl ...

"Geometry and Arithmetic are Synthetic"
2002. {{DEFAULTSORT:Foundations Of Arithmetic, The 1884 non-fiction books Books by Gottlob Frege Logic books Philosophy of mathematics literature