BETWEEN THE FINITARY AND THE IDEAL

Within contemporary philosophy of mathematics there is a trend focussing on how mathematics is done and how it evolves, rather than how it should be done or how it should evolve. This fact is somewhat contrary to the philosophy of mathematics in the 20th century, which to a large extent was dominated by views developed during the so-called foundational crisis in the beginning of that very century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justication and consistency of mathematics. Mathematicians and philosophers like Gottlob Frege (1848{1925), Bertrand Russell (1872{1970), David Hilbert (1862{1943), Kurt Godel (1906{1978) and others were very successful in their development of logic from around 1890 to, say, 1940 and they had a huge impact on the phi-losophy of mathematics of those days. Most probably it was Hilbert's program, rise and fall, which was the most single in uential factor of the foundational studies until 1960's. Among the results was the widespread conception that the proper { if not the only { approach to philosophy of mathematics was throug