Chance against constructibility
pp. 263-281
in: Jaakko Hintikka (ed), From Dedekind to Gödel, Berlin, Springer, 1995Abstract
Some time towards the end of the last century, mathematical existence acquired a new meaning that has since become the standard one. This change can be seen in for example David Hilbert's book on the foundations of geometry of 1899. In brief, the old sense was existence as something constructed, whereas the new one was existence as consistency. So, Hilbert considered a problem to possess a solution if a contradiction could be derived from the assumption that the problem is impossible. Such an indirect proof of existence need not give any indication of how to actually find a solution or even guarantee that one can be found.
