Gottlob Frege

Gottlob Frege . German logician, mathematician and philosopher. In 1879 – 1918 professor at the University of Jena. His works opened a new stage in mathematical logic. Frege carried out, for the first time, the axiomatic structuring of the logic of statements and predicates, he established the principles of the theory of mathematical proof. He founded the part of logical semantics linked to the concepts of meaning and meaning of language expressions and to the relationship of designation (or naming). [1]


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  • 1 Biographical synthesis
  • 2 Influence on the development of mathematical logic
  • 3 Signs in language
  • 4 Relevant works
  • 5 References
  • 6 Sources

Biographical synthesis

Born on 8 November as as 1848 in Wismar , Germany today. The son of humble parents, he entered the University of Jena at the age of 21. At the age of 23, he moved to Göttingen with the aim of completing his studies in chemistry , physics , philosophy and mathematics . In 1873 he graduated in mathematics and returned to Jena as a teacher of that subject. For around 50 years he continued to teach until his death in 1925 .

Influence on the development of mathematical logic

Frege’s works laid the foundations for logical semantics. He invented many symbolic notations, such as quantifiers and variables, thus laying the foundations of modern mathematical logic. Alluding to the role of symbols in mathematics and logic, he indicated that there should be no sign without significance. He considered erroneous the purpose of mathematicians to make the private signs of significance the object of mathematics, to convert numbers into signs.

The main works of Frege – Calculation of concepts and Basic laws of arithmetic – were read by few of his contemporaries because readers were scared by their complex symbolism, however, his work had a great impact on the history of the foundation of mathematics in the first half of the 20th century . It was based largely Principia Mathematica of Russell and Whitehead . Frege was opposed to the subjectivist approach to logic. To explain the process of knowledge, he pointed out:

“We must conceive of knowledge as an activity that does not create what is known, but captures what already exists.”

He believed that sensory data, obtained from the outside world through the sense organs, were not necessarily the starting point of knowledge. He approached logical laws as the more general laws that prescribe how one should think, assuming that these laws were given once and for all and were immutable. Frege argued that the definition does not create the defined object and did not recognize the attempts of some mathematicians to approach the definition of an object as its creator. In his opinion, you couldn’t define everything.

Signs in language

Frege distinguishes two identity statements “a = a” and “a = b”. The identity relationship that appears in these statements cannot be between signs of objects or between objects. If the identity is between objects, the information you provide us (a = a) is not different from that you provide us (a = b). If the relationship is between names of objects, then we are not saying anything extralinguistic. Thus Frege solves this question by distinguishing in the expressions the reference and the sense. The reference is the object itself that we designate with a sign, the meaning expresses the way the object is given. That is, with (a = b) we express two different ways of referring to the same object.

The theory of meaning presented by Frege tells us that the signs signify the ways of giving the objects to which we refer with our words. Meaning is an approximation to the object itself.

For Frege our words refer to objects and also express ways of giving such objects, that is, they make sense. In this way, the reference of a sign is an object, if the object is sensible, the representation that I have is nothing more than an “internal image” constructed from the memory of the sensations that such object produced in me, and in this it is differentiates subjective representation from reference. The meaning of an expression is understood to the extent that you have a certain knowledge of the referent.

Frege rejects the thesis that words are signs of ideas. Only the speaker himself has access to these ideas, we use words as signs of these ideas to communicate them. According to Frege, meanings and concepts are private entities, to open up to a new Platonic paradigm: the realism of meaning, from which he defends that our words refer to objects in the world, they have reference and, also, meaning. The meaning is given by the knowledge that one has of the reference, without it follows that it is something subjective, with respect to this Frege says that

“Humanity has a common treasure of thoughts, which it transmits from one generation to another.”

That is, the meanings and meanings of words belong to communities of speakers and not to the minds of individuals; what is unique to speakers are their subjective representations, of which words are not signs.

Relevant works

  • Conceptual writing( 1879 )
  • The Foundations of Arithmetic( 1884 )
  • The Basic Laws of Arithmetic(two volumes) ( 1893 – 1903 )
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