The basics of arithmetic

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The basics of arithmetic ( 1884 ) with the subtitle “A logical-mathematical investigation into the concept of number” is one of Gottlob Frege's main works .

The subject of the book is the explanation of the term " number " or " number ". Using this basic concept, Frege outlines his logistic program, that is, the return of mathematics to logic . There are both remarks on the theoretical background of this program and a draft of a practical implementation. In contrast to the conceptual writing that appeared five years earlier, Frege dispenses with a formal notation , theorems and definition are expressed in natural language, in the case of proofs only the idea of ​​proof is suggested. A fully formalized implementation of the logistic program with the notation of the conceptual writing can be found later in “ The Basic Laws of Arithmetic ” (1893 and 1903). Frege's approach to orienting his research on observations about natural language usage had a great influence on the later Analytical Philosophy .

The theory of logicism

The program of logicism, as Frege presents in the fundamentals, is essentially based on his anti- psychologism and the conviction that the theorems of arithmetic are analytical .

Anti-psychologism

For Frege, mathematical objects are representational, but abstract. He opposes the conception that they only exist in thought: “ So the number is also something objective. [...] So, by objectivity, I understand independence from our perception, looking at and imagining [...]. “(§ 26) This also means that in mathematical proofs one should not fall back on intuition or view . I.e. Frege cannot accept the fact that “ the mathematician is satisfied when every transition to a new judgment is evident as correct without asking about the nature of this enlightenment, whether it is logical or intuitive ” (§ 90). According to Frege, every step in a proof must be covered by a final rule and the permissible rules must be specified in advance, a requirement that can ultimately only be implemented in a formal system such as Frege's terminology: “ The requirement is therefore irrefutable, all Avoid jumps in the conclusion. [...] In order to avoid these evils, I came up with my 'terminology' ”. (§ 91)

Analyticity of arithmetic

Frege opposes Kant's view that the theorems of arithmetic represent synthetic judgments a priori , that is, sentences that are not purely conceptual and yet experience-independent truths. (But he accepts this assessment in the case of geometry : “ By calling the geometrical truths synthetic and a priori, he has revealed their true essence ” (§ 89)).

John Stuart Mills , who considers the propositions of arithmetic to be a posteriori , that is, dependent on experience, is still further from Kant's view (cf. § 7).

Frege, on the other hand, believes that arithmetic is analytical ; That is, that it is possible to trace your sentences back to purely logical truths. “ It is now a matter of finding the evidence and tracing it back to the original truths. If one comes across only the general laws of logic and definitions in this way, one has an analytical truth ”(§ 3). His view is based primarily on the fact that he succeeded in reducing the inference rule of complete induction , which is only used in mathematics and therefore seems to have a non-logical character, to a definition : the inference from n to (n + 1), which is usually held to be a peculiar mathematical one, to be demonstrated as based on the general logical inference. "(§ 108) This definition, which, by the way, must be regarded as one of the outstanding achievements of the mathematician Frege, was already formulated in the terminology; in the" Fundamentals "it can be found in § 79.

The propositions of arithmetic can therefore be traced back to definitions, but these are far more complex than those that appear in Kant's examples of analytical propositions, so Kant had apparently underestimated the creative power of the definition: “ The more fruitful conceptual definitions draw boundary lines that do not yet exist were given. What can be deduced from them cannot be overlooked from the outset [...]. These inferences expand our knowledge and, according to Kant, should therefore be considered synthetic; nevertheless they can be proved purely logically and are therefore analytical ”(§ 88).

The practice of logicism: definition of number

About the first half of Frege's essay (§ 5 - § 44) deals with his criticism of philosophers and mathematicians who have tried to define the concept of number. In the second half (§ 45 - § 109) he presents his own definition and tries to show that it is free from the difficulties mentioned above.

Frege's criticism of his predecessors

Frege examines different views on the nature of the propositions of arithmetic, that of numbers and that of unit. He derives from these aporias and thus shows that they are untenable. For example, he proves that Leibniz 's proof for “2 + 2 = 4”, which apparently is based only on definitions, is flawed (§ 6).

He also criticizes Ernst Schröder's view , according to which numbers arise from collections of things through abstraction , in that all other properties of things, apart from their number, are disregarded (§ 21). A collection of things could not be the basis of the numbers, because different numbers can be assigned to the same collection: “[I can] understand the Iliad as 1 poem, as 24 chants or as a large number of verses. [...] In the same way, an object to which I can ascribe different numbers with the same right is not the actual bearer of a number ” (§ 22).

Frege also investigates attempts to create numbers from “units”, where a unit is characterized by its undividedness. He points out that one cannot, in principle, assume that the unit is indivisible: “But there are cases where one cannot avoid thinking about the separability, where even the conclusion is based on the composition of the unit, e.g. B. with the task: a day has 24 hours, how many hours have 3 days. " (§ 33)

In addition, the attempt to obtain numbers from units is faced with the difficulty of determining whether these units are equal or unequal. Hobbes represents e.g. B. the opinion that these units must be equal to each other ( Hume sees it similarly , § 34). But if these units are really the same, then apparently no more distinction can be made between them; the question then arises as to how one should talk about several things at all. For this reason, Descartes , for example, takes the opposite view, namely that the units must be different from one another (§ 35). Jevons takes the same view, he even goes so far as to claim that in the expression "1 + 1" the two ones are different from each other (§ 36). According to Frege, the equation “1 = 1” would then be wrong, and Jevons' view is therefore untenable (§ 36). Frege sums up the discussion as follows: “If we want to create the number through the combination of different objects, we get a cluster in which the objects are contained with the very properties by which they differ, and that is not that Number. If, on the other hand, we want to form the number by combining like elements, then this always merges into one and we never come to a majority ” (§ 39).

Attempts to analyze numbers by units also suffer from the fundamental difficulty that they can hardly be applied to one and even less to zero (§ 44).

Frege's own analysis

Frege's starting point for solving the difficulties is the recognition “ that the number given contains a statement about a concept ” (§ 46). Frege defines “term” in § 70 as follows: “ If we separate“ the earth ”in the sentence“ The earth has more mass than the moon ”, we get the term“ more mass than the moon ”. “So a concept is the same thing that is called a predicate in modern logic . According to Frege, numbers are now assigned to concepts rather than objects. This solves the problem that the same object (e.g. the Iliad) can be given different numbers (1 poem, 24 chants). The object is then described by different terms and these, not the object itself, are assigned the respective number. The number zero then also no longer causes problems: “ When I say 'Venus has 0 moons', there is no moon [...] about which something could be said, but the term 'Venus moon' becomes one Attribute, namely that of not dealing with anything ”(§ 46). Frege points out that similar considerations can already be found in Spinoza (§ 49).

Frege can now also resolve the difficulties related to "unity". For him, a unit is a concept that designates a thing, the parts of which are no longer assigned the same concept (§ 54). An example of a unit is the term “syllable”: parts of a syllable are not syllables. In contrast to this, parts of red things can also be red, so the term “red” is not a unit. Concepts that refer to numbers must always be units. “ Unity in relation to a finite number can only be such a concept that clearly delimits what falls under it and does not allow any arbitrary division. "(§ 54).

Frege is now going to define the numbers as independent objects, as they are used in mathematics. This definition is also applicable to the natural language number statements, only a sentence like “Jupiter has four moons” has to be transformed to “The number of Jupiter's moons is (the) four” (§ 58).

Frege's first attempt at definition consists in defining “the number that belongs to the term F = the number that belongs to the term G” as “between F and G there is a one-to-one association ”; in Frege's terminology: the two terms are “equal” (cf. his definition in § 72). The realization of this connection he wrote to Hume, in the literature, this set is also sometimes called " Hume's principle " ( Hume's principle called). Hume's principle, however, is not a definition, since, according to him, it cannot be decided whether the number to which F is assigned is identical to an arbitrary thing or not; this can only be decided if the other thing is also a number: “ [If If we] hereby introduce the expression 'the number which belongs to the concept F', then we only have a meaning for the equation if both sides have the form just mentioned. According to such a definition, we could not judge whether an equation is true or false if only one side has this form ”(§ 107).

Frege therefore defines as follows: " The number that belongs to the term F is the scope of the term 'equal to the term F' ". However, Frege is aware that he has not yet defined the “scope of a term”. Only in the basic laws of arithmetic will he introduce the scope of terms (axiomatically). In any case, Frege understands a conceptual scope to mean what we today call a set . The scope of the term “equal to the term F” is the set of all terms that are equal to F.

From this definition Hume's principle can be derived (§ 73). Frege defines the number 0 (§ 74) and proves some of its properties (§ 75). He then determines what it means for two numbers to follow one another (§ 79). From this definition he can deduce that there are infinitely many numbers (§ 81 ff.).

On the consistency of Frege's suggestion

In the later work, Basic Laws of Arithmetic , the practical implementation of the logistic program, as indicated in the Fundamentals , is specified and expanded in various ways. First, all evidence is strictly formal. Second, the theorems proven cover many more parts of arithmetic. Thirdly, the scope of terms is introduced axiomatically and thus the actual foundation of the program is created (strictly speaking, Frege introduces "value curves" of functions, which are, however, equivalent to the scope of terms). However, this foundation turns out to be fragile: Bertrand Russell discovers a contradiction here (“ Russell's Paradox ”) and Frege's life's work collapses. Russell himself should reserve the right to present the first implementation of the logistic program together with Alfred North Whitehead in the Principia Mathematica (1910).

The inconsistency in the basic laws also affects the foundations , because this work also makes use of the ultimately contradicting characterization of the scope of terms. Modern Frege research (including George Boolos ) has found that at least Hume's principle is consistent and that the logicist program can be implemented on its basis using Frege's theorem , which led to the formation of neo-logicism.

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