Kurt Gödel
Kurt Gödel was a German mathematician who wrote a paper which sent the most powerful school of philosophy of his day (the Vienna school, also known as the logical positivists) crashing to the ground. His work inspired Alan Turning to discover the Halting Problem.
Here is a quick capsule of the conclusion of his Incompleteness Theorem: In any formal system capable of representing the arithmetic of natural numbers there exist statements which the system cannot prove as true or false.
In order to accomplish this proof, Gödel invented a new branch of mathematics, which is now called metamathematics. Very little has been done with this branch of metamathematics (perhaps because it dashed the hopes of so many scientists and mathematicians). Ernest Nagel and James R. Newman suggested in "Gödel's Proof" that the theorem implied a limitation on the power of computers. Douglas Hofstadter in a foreword to a recent edition of that book suggests it implied the opposite:
<blockquote><i>Gödel's great stroke of genius...was to realize that numbers are a universal medium for embedding patterns of any sort, and that for that reason, statements seemingly about numbers alone can in fact encode statements about other universes of discourse. In other words Gödel saw beyond the surface level of number theory, realizing that numbers could represent any kind of structure.
— Douglas R. Hofstadter
Gödel's metamathematics applies to any system for which certain properties are true of any system which has analogues of three notions ("formulas," "proof arrays," and "provable formulas." If these analogues can be defined in the system which Gödel called <i>PM</i> (the system Russell and Whitehead defined in Principia Mathematica ), then his conclusions apply to that system. This suggests rather widespread applicability of metamathematical conclusions. Not "any kind of structure" as Hofstadter says, but a wide variety. Including all in which emergence has been observed.
Gödel himself offered the opinion that a wide variety of systems are covered:
<blockquote><i>The method of proof just explained can clearly be applied to any formal system that, first, when interpreted as representing a system of notions and propositions, has at its disposal sufficient means of expression to define the notions occurring in the argument above (in particular, the notion "provable formula") and in which, second, every provable formula is true in the interpretation considered.</i>
— Kurt Gödel in "On formally undecideable propositions"
Some relevant questions are: Are there other kinds of systems which we do not usually think of as formal systems which have at their disposal sufficient means of expression to define the notions occurring in the argument? Are there ontological systems (i.e., systems of things which exist in the real world) for which Gödel's proof works as well?
I will try to show the answer to both these questions is yes. If it is, it will be possible to show a large number of propositions to be true which will meet Kant's requirements for the synthetic a priori. Specifically, I think it will be possible to prove that a large number of emergent phenomena are, in fact, necessary.
If this is the case, the usefulness of metamathematics will have been demonstrated using a single theorem. I suspect this will open the floodgates of inquiry into other theorems which might be proven under metamathematics. It is not unreasonable to assume that such additional theorems will also deal with emergence. Perhaps even the four Candidate Laws For Coevolving Systems (or some close facsimile) will be proven my such means.
See also The Goedel Connection and Investigations.
— Scotus - 27 Apr 2003
