The Growth of Mathematical Knowledge: An Open World View

1 The closed world view In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: " The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past " (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege's idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because " in mathematics we must always strive after a system that is complete in itself " (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because " in history we have development; a system is static " , but he sticks to the view that " science only comes to fruition in a system " because " only through a system can we achieve complete clarity and order " (Frege 1979, p. 242). He even goes so far as saying that " no science can be so enveloped in obscurity as mathematics, if it fails to construct a system " (Frege 1979, p. 242). By 'system' Frege means 'axiomatic system'. In his view, in mathematics we cannot rest content with the fact that " we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction " , that is, to find " what the primitive truths are " , because " only in this way can a system be constructed " (Frege 1979, p. 205). The primitive truths are the principles of the axiomatic system. Frege's stress on the role of systems also determines his views on the growth of mathematical knowledge. According to him, building up a body of mathematical