"It is always possible to create new knowledge in science, because scientific inquiry is essentially inductive; on the other hand, it is never possible to create new knowledge in mathematics,...
"It is always possible to create new knowledge in science, because scientific inquiry is essentially inductive; on the other hand, it is never possible to create new knowledge in mathematics, because mathematical inquiry is purely deductive." Critically discuss the view expressed in the statement.
On the first point, inductive reasoning is based on observation, and since we can never know what we will observe, for all practical purposes, it would scientific enquiry is limited only by the range of possible observations. Of course, these are in turn constrained by technological limitations, our cognitive abilities, and perhaps by the disciplinary constraints of scientific inquiry. So practically speaking, of course there are limits to what we can learn scientifically. But philosophically speaking, the new knowledge available as a result of scientific inquiry is vast.
As far as mathematics, we again have to consider our own cognitive limitations, but we should also recognize that science, too, works from deductive reasoning to some extent, arguing deductively from conclusions reached inductively. So the conclusions we reach from deductive reasoning are not as limited as one might imagine. Applied specifically to mathematics, for all practical purposes, it really does not seem that this mode of enquiry is limited by deductive reasoning, and that it actually involves a great deal of inductive reasoning and experimentation. Attempts to encompass all mathematical expressions within a single system have failed. The limits to mathematics, like those of science, may have to do with our cognitive limitations.
Source: Chaitin, Gregory J. 1998. The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning.