Frege: Fundamentals of Logic and ArithmeticJune 27, 2021
Frege’s project of reducing arithmetic to the logical involves both describing the numbers one by one and capturing the order of the numbers.
Frege, one by one, in his 1884 work on the foundations of arithmetic (Die Grundlagen der Arithmetik, eine logisch-matematische Untersuchung über den Begriff der Zahl) and in his work on the basic laws of arithmetic in which he published the first and only volume in 1893 (Die Grundgeseztze der Arithmetik). sayal; English cardinal) tried to reveal how numbers can be derived from pure logic.
However, in his work (Begriffsschrift) of 1879, he primarily tried to show that the concept of order in a series could be reduced to the concept of logical result in his effort to define numbers one by one. The idea of ideography, which Frege began to develop with this work, was also a turning point in the development of formal logic. Begriffsschrift allowed functions and variables to be represented in a formal language. Frege was trying to show that mathematics is based on logic and develops from logic. In order to show this, logic had to be taken much further than it was then. Neither Aristotle’s theory of syllogism nor Stoic propositional logic were sufficient to represent propositions and inferences. For example, the implications of Euclidean geometry could not be handled within classical logic.
Although classical logic made progress on logic invariants, the handling of propositions containing statements such as “all” and “some” was not detailed enough. The quantification logic developed by Frege completely solved all these problems and became a new beginning in the development of modern logic. In the logic developed by Frege, the most complex mathematical expressions could be represented, and the proofs could be handled with logical tools in a way that did not allow any heuristic element to leak in his own sense.
All advances in logic in the 20th century are based on Frege’s logic of quantification. Bertrand Russell’s (1872-1970) and Alfred North Whitehead (1861-1947) famous works Principia Mathematica (1910-1913), in which they discussed and formalized logical concepts, Bertrand Russell’s theory of definite descriptors was a turning point in the development of mathematical logic and calculus theory. Kurt Gödel’s (1906-1978) incompleteness theorems and Alfred Tarski’s (1901-1983) truth theory based on the separation of object language and metalanguage are based on the logic developed by Frege.
Frege expressed his purpose in his Begriffsschrift in the preface as follows:
…we divide all truths that require verification into those that can be justified by reason and those that have to be supported by the facts of experience. However, the fact that a proposition can be of the first kind is undoubtedly compatible with the fact that the proposition in question cannot appear in the human mind without the accompaniment of the senses. Therefore, the best method of proof, rather than the psychological source, is at the heart of this classification. Now, if I wish to address the question of which of these kinds the judgments of arithmetic belong to, I must first make certain how far I can go in arithmetic only with the support of the laws of thought that transcend all particulars, that is, by means of inferences alone. My first step will be to try to reduce the concept of order in a sequence to the concept of logical result and from there to the concept of number. I had to make every effort to ensure that the chains of inference were not gapless, to prevent anything visual from leaking out of sight. While I was trying to comply with this requirement as strictly as possible, I found the inadequacy of language as an obstacle… This lack of language led me to this idea of ideography (Begriffsschrift, p.1).
As understood from this excerpt, Frege aims to create a purely logical system for representing inferences. This system will not allow any visual element to seep into it. Frege’s aim in a wider context is to show that arithmetic is a sub-branch of logic. The true propositions of arithmetic can be proved as a theorem of logic and within the possibilities of a logical system.
In order to understand the change Frege achieved, we need to go into the details of his perspective on concepts, objects and quantifiers. First, let’s consider a proposition that expresses a number. Frege’s first observation about such propositions is that they say something “about concepts”. To say that an object is as many as any numerical number is essentially to say that there are a certain number of objects that fall under a certain concept. In other words, such propositions are in the form of “There are n Fs”. Therefore, the analysis of numerical propositions requires us to explain how concepts can be handled logically.