Who is LEJ Brouwer in Philosophy?
Luitzen Egbertus Jan Brouwer, better known as LEJ Brouwer, was a Dutch mathematician and philosopher who lived between 1881 and 1966. He is considered one of the main representatives of the movement known as Intuitionism, which had a great influence on the philosophy of mathematics of the XNUMXth century. XX.
Intuitionism in the Philosophy of Mathematics
Intuitionism is a philosophical current that defends the idea that mathematics is not based on absolute and objective truths, but rather on intuitions and mental constructions. According to Brouwer, mathematical objects do not exist independently of the human mind, being created through intuitive mental processes.
For Brouwer, mathematics is not a deductive science, as defended by mathematical formalism, but rather a constructive science, in which mathematical objects are constructed through mental processes. He believed that mathematical intuition was the basis for creating new mathematical concepts and theories.
Brouwer's Contributions to the Philosophy of Mathematics
Brouwer made several important contributions to the philosophy of mathematics. One of his main ideas was the notion of “constructivity”, which refers to the ability to construct mathematical objects through mental processes. He argued that mathematical construction was an active and creative process, in which the mathematician actively participates in the creation of mathematical objects.
Furthermore, Brouwer also proposed the principle of “exclusion of the excluded middle”, which states that a mathematical proposition can only be true or false, with no possibility of a third option. This idea was a criticism of the principle of excluded middle, defended by mathematical formalism, which states that a mathematical proposition is true or false, without the possibility of a third option.
Brouwer's influence on the Philosophy of Mathematics
Brouwer's ideas had a great influence on the philosophy of mathematics in the XNUMXth century. His intuitionism was a response to mathematical formalism, which dominated the philosophy of mathematics at the time. Brouwer argued that mathematics should not be reduced to a set of formal rules, but rather be seen as a creative and intuitive activity.
Furthermore, Brouwer's ideas also influenced the development of set theory, particularly through the work of his student, Arend Heyting. Heyting developed intuitionistic logic, which is a non-classical logic based on Brouwer's ideas. This logic was an alternative to classical logic, which is based on the principle of excluded middle.
Criticisms of Brouwer's Intuitionism
Brouwer's intuitionism has also received criticism from other philosophers and mathematicians. One of the main criticisms is that intuitionism is not capable of accounting for certain mathematical results, such as Gödel's completeness theorem. This theorem, proven by Kurt Gödel, states that there are mathematical propositions that are true, but that cannot be proven within a formal system.
Another criticism of intuitionism is that it is not capable of explaining the objectivity of mathematics. According to its critics, if mathematics is just a mental construction, as defended by Brouwer, then it would be subjective and dependent on the individual intuitions of each mathematician.
Brouwer's Legacy in the Philosophy of Mathematics
Even with the criticism he received, Brouwer's legacy in the philosophy of mathematics is undeniable. His ideas influenced not only the philosophy of mathematics, but also logic and set theory. His intuitionism brought a new perspective to the understanding of mathematics, highlighting the importance of intuition and constructiveness in mathematical activity.
Furthermore, Brouwer's work also paved the way for the development of other philosophical currents in the philosophy of mathematics, such as constructivism and finitism. These currents also emphasize the importance of construction and intuition in mathematics, but in different ways.
Conclusion
In summary, LEJ Brouwer was a Dutch mathematician and philosopher who played a fundamental role in the development of XNUMXth century philosophy of mathematics. His intuitionism brought a new perspective to the understanding of mathematics, highlighting the importance of intuition and constructiveness in mathematical activity. Despite the criticism he received, Brouwer's legacy in the philosophy of mathematics is undeniable, influencing not only the philosophy of mathematics, but also logic and set theory.
