Did David Hilbert ever express his philosophical thoughts in writing?
Yes, David Hilbert did express his philosophical thoughts in writing. In addition to his groundbreaking work in mathematics, Hilbert was also deeply interested in philosophy and made significant contributions to the field of epistemology, which is the study of knowledge.
Hilbert's most famous philosophical statement is probably his assertion that mathematics is a game played according to certain formal rules, rather than a reflection of some kind of Platonic ideal. This idea is encapsulated in his famous statement "We must know, we will know," which he used to express his belief in the power of formal axiomatic systems to uncover the truth about mathematical objects.
Hilbert also wrote extensively on the nature of mathematical reasoning and the role of intuition in mathematical discovery. He believed that mathematical concepts and methods are ultimately grounded in our intuition about space and time, and that this intuition can be refined and made more precise through careful study and reflection.
Finally, Hilbert was deeply interested in the foundations of mathematics and spent much of his career working on problems related to the consistency and completeness of formal axiomatic systems. His work on these topics led to the development of the field of metamathematics and had a profound impact on the development of modern mathematical logic.