Louisville Process Theology Network

Freeman Dyson on Inexhaustible Knowledge

Apr 02, 2009

Page 225, from ?“The Scientist As Rebel?” by Freeman Dyson.


?“Another reason why I believe science to be inexhaustible is G?¶del?’s theorem. The mathematician Kurt Gobel discovered and proved the theorem in 1931. The theorem says that given any finite set of rules for doing mathematics, there are undecidable statements; mathematical statements that cannot either be proved or disproved by using these rules.

G?¶del gave examples of undecidable statements that cannot be proved true or false using the normal rules of logic and arithmetic. His theorem implies that pure mathematics is inexhaustible.

No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. Now I claim that because of G?¶del?’s theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include rules for doing mathematics, so that G?¶del?’s theorem applies to them.

The theorem implies that even within the domain of the basic equations of physics, our knowledge will always be incomplete?… I rejoiced in the fact that science is inexhaustible, and hoped that non-scientists would rejoice too.

Science has three advancing frontier, which will remain open. There is the mathematical frontier, which will always be open thanks to G?¶del. There is the complexity frontier, which will always be open because we are investigating objects of ever increasing complexity; molecules, cells, animals, brains, human beings, societies.

And, there is the geographical frontier, which will always be open because our unexplored universe is expanding in space and time. My hope and belief is that there will never come a time when we shall say, ?‘We are done.?’?”



From Solomon Feferman?’s letter to ?“The New York Times?” in 2004; Professor of Mathematics and Philosophy, Stanford University.

?“Whether or not the kind of inexhaustibility of mathematics discovered by G?¶del is relevant to the application of mathematics in science, there is a different kind of inexhaustibility which is more significant for practice: no matter which axiomatic system S is taken to underlay one?’s work at any given stage in the development of mathematics and science, there is a potential infinity of propositions that can be demonstrated in S, and at any moment, only a finite number of them have been established.

Experience shows that significant progress at each such point depends to an enormous extent on creative ingenuity in the exploitation of accepted principles rather than essentially new principles. One can join Freeman Dyson in rejoicing in that kind of inexhaustibility as well.?”