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Frege's logical revolution: how he created the foundations of modern mathematics
In the early 20th century, the German logician Gottlob Frege's innovations in mathematics and logic changed people's understanding of formal logic and laid the foundation for modern mathematics. As one of the founders of analytic philosophy, his work had a profound impact not only on mathematics and logic, but also on the philosophy of language and philosophy of science.
Logicism proposed by Frege, that mathematics can be reduced to logic, became one of the core concepts of mathematical philosophy in the 20th century.
Frege elaborated on the externality of mathematics in his work "The Foundations of Arithmetic", arguing that mathematical objects exist in reality regardless of the mental state of the mathematician or logician. His views challenged the psychologism of the time, which held that mathematical concepts and their truths originated from mental processes. This view contributed to the development of realism in mathematics, which in turn influenced later mathematicians and philosophers.
Frege's logical revolution also introduced new logical forms, especially his predicate logic. In his book "Conceptual Notation", he conducted a more precise grammatical analysis of various types of sentences for the first time, which greatly expanded the scope of logical reasoning and surpassed ancient Aristotelian logic. This new logical tool enables complex logical problems and multiple general problems to be expressed and explained more clearly.
Frege's views on language led to a wide-ranging discussion of semantics, reference, and meaning, a discussion that charted the future of the philosophy of language.
In the philosophy of language, Frege's in-depth study of meaning had an important influence on subsequent philosophers. He distinguished between the concepts of "meaning" and "reference" and explored how language conveys meaning in "On Sense and Reference." This view not only promoted the rise of language analysis, but also influenced many well-known thinkers, such as Russell and Wittgenstein.
In contemporary philosophy, Frege's theory is also important in logical positivism. The Vienna Circle combined his ideas with empiricism, was skeptical of metaphysical issues, and argued that only verifiable statements have meaning, which provided fresh perspectives and methods for analytic philosophy.
Frege's teachings are not limited to mathematics and logic, but his ideas have also penetrated into every corner of scientific philosophy. Along with the reforms in mathematics and logic, the concept of science also underwent changes, allowing philosophers to examine the cornerstones of scientific practice and theory in a more rigorous way. His students and followers, such as Russell, promoted this change and expanded the application of liberal logic to many disciplines of science.
Frege's theory demonstrated the logical structure of mathematics and its central role in understanding the world, thus influencing the thinking of a group of later mathematicians.
With the revolution in logic and mathematics, many subsequent philosophers such as Wittgenstein and Kunn also developed on this basis and formed their own unique philosophical systems. In his early work "Tractatus Logico-Philosophicus" and later in "Philosophical Investigations", Wittgenstein made important criticisms and reinterpretations of Frege's logic, showing his continued concern for analytic philosophy.
In the face of philosophical developments since the 20th century, Frege's discoveries still inspire scholars to explore mathematics and logic. His ideas were not just a technical breakthrough, but a theoretical shift that allowed philosophers to fundamentally reexamine the structure of meaning and knowledge. In the ongoing academic debate, can Frege be regarded as the true father of analytic philosophy?
