Ever since David Hilbert introduced his programme, a lot of work has gone into examining the question of what conclusions can and cannot be algorithmically extracted from a set of axioms. But very little has been said about whether it is possible to construct a complete set of axioms from an already well-defined body of knowledge. I am starting to suspect that incompleteness lurks here as well.
The problem is, there will always be assumptions which escape notice because they are so blatantly obvious. This is a difficulty which is well known to all aficionados of detective fiction.
Euclid gave five axioms and another five common notions. But are they complete? For instance, why don’t we have an axiom telling us that it is possible for the human mind to conceive of geometric objects? But, that is psychology, and not mathematics, you say? Fine. Then why don’t we have an axiom telling us that the idealists are wrong?
GÖDEL GOING DOWN
Ever since David Hilbert introduced his programme, a lot of work has gone into examining the question of what conclusions can and cannot be algorithmically extracted from a set of axioms. But very little has been said about whether it is possible to construct a complete set of axioms from an already well-defined body of knowledge. I am starting to suspect that incompleteness lurks here as well.
The problem is, there will always be assumptions which escape notice because they are so blatantly obvious. This is a difficulty which is well known to all aficionados of detective fiction.
Euclid gave five axioms and another five common notions. But are they complete? For instance, why don’t we have an axiom telling us that it is possible for the human mind to conceive of geometric objects? But, that is psychology, and not mathematics, you say? Fine. Then why don’t we have an axiom telling us that the idealists are wrong?