One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. “He doesn’t think concretely.” Consider by contrast the Indian mathematician Ramanujan, who was intimately familiar with properties of many numbers, some of them huge. That way of thinking represents a world antipodal to that of Grothendieck. “He really never worked on examples,” Mumford observed. “I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieck in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It’s just very strange. That’s the way his mind worked.
Rene Descartes was a French philosopher, mathematician, and scientist who is often referred to as the father of modern western philosophy. He is best known for his contributions to the fields of mathematics and philosophy, including his famous statement "I think, therefore I am."
Descartes made significant contributions to the development of analytical geometry, which uses algebraic methods to solve geometric problems. He also developed the Cartesian coordinate system, which is still used in mathematics and science today. Additionally, Descartes' work in calculus and optics laid the foundation for future advancements in these fields.
Descartes' philosophical ideas, particularly his emphasis on skepticism and the importance of reason, had a profound impact on the development of modern western philosophy. He is known for his method of doubt, which encourages individuals to question everything they believe to be true in order to arrive at certain knowledge.
Descartes' statement "I think, therefore I am" is a philosophical argument that he used to prove his own existence. He argued that the very act of thinking and doubting one's own existence is proof that one exists. This statement is often seen as a starting point for Descartes' philosophical ideas.
Descartes' contributions to mathematics and philosophy have had a lasting impact on modern science. His emphasis on reason and skepticism laid the groundwork for the scientific method, which is still used today. His work in optics and calculus also paved the way for future scientific advancements in these fields.