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Creative mathematics and Gödel

``Creative mathematics'' sounds like an oxymoron. Mathematics is about those truths that are logically determined by assumptions. These assumptions are called axioms in formal mathematics. The classic example of a logical deduction is the syllogism about Socrates.

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
If all men share a common property and Socrates is a man then he must have that property. Logic at this level is self evident. But whether or not an object has a particular property such as being mortal is not always so easy to decide even if we know it is logically determined by our assumptions. Gödel proved that there are outcomes logically determined by a set of axioms that cannot be proved from those axioms. Without realizing it he established the creative nature of mathematics with important implications for the creative evolution of consciousness.

This section begins with a practical example of the kind of problem that Gödel proved could not be decided. This is intended to make Gödel's abstract mathematical result concrete and to show its relevance to human experience. Next is a brief history and statement of Gödel's result and an explanation of how the result relates to practical problems. The following section applies this mathematics to the evolution of consciousness.

Most of us have been frustrated by a computer that suddenly stops responding. Gödel's result implies that there can exist no general method to determine if the computer will eventually start responding or if you must reboot and loose the work you were doing. For some cases one can figure this out, but no method will work for every case. This is true even though one can determine exactly what the computer will do at any time. Its fate is determined by the computer design and the programs it is running. Of course real computers can act erratically because of a hardware failure, but the assumption here is that the hardware is working correctly.

The inability to predict if a computer will ever do something is at the root of mathematical creativity. The existence of unsolvable problems that have a logically determined outcome was discovered by Kurt Gödel in the 1930s. Around 1900 Hilbert, a famous mathematician, posed a challenge to the mathematical community. He presented a list of problems that included finding a method for deciding all well posed mathematical problems. Thirty years later Gödel proved this was impossible[8]. He showed that any mathematical system that was strong enough to include the primitive recursive functions could not prove its own consistency unless it was inconsistent. The primitive recursive functions are those that can be defined with elementary induction described later in this section. They are powerful enough to model the execution of all possible computer programs. A formal system is inconsistent if one can derive a statement and the negation of the same statement from the axioms of the system. A formal system that included the following axioms would be inconsistent.

All men are mortal.
Socrates is a man.
Socrates is not mortal.

The problem of knowing if a computer will ever accept more user input is equivalent to the problem of determining if a formal mathematical system is consistent1. For any problem of one type one can construct a problem of the other type such that the two problems have the same answer. The computer will eventually accept input if and only if the formal system is inconsistent. This is possible because a formal mathematical system is a precise set of rules for deducing theorems. One can think of it as a computer program for generating theorems. So all one has to do to see if the system is consistent is to generate every theorem and check each one against all previous theorems to see if the new theorem is the logical negation of any previous theorem. If one finds such a contradiction the process accepts user input. If no contradiction is found the process will run forever without ever accepting input.

What is the relevance of Gödel's result? To gain insight into this we need to understand a little about the hierarchy of mathematical truth developed as a consequence of Gödel's result. This starts with understanding laws of induction beginning with elementary induction. A mathematical system must have this level of induction for Gödel's proof to work. Using elementary induction one can prove that something is true for every integer without testing every case. The first step is to prove the property is true of the number 0. The next step is to prove that if it is true for any number $n$ it must be true for $n+1$. If one can show this than one knows how to prove the property for any integer. Start at 0 and work up to the desired number by iterating the second step. If we can prove the property holds for any integer, it must be true for all integers. The Appendix gives an example of proof by induction.

One way to extend induction is to operate not on the integers but on properties of integers. One can go higher and consider induction on methods for generating properties2. One can always go to higher levels of abstraction. There is no finite way to characterize this hierarchy. Nor is there any single path of development that can fully explore it. Limiting culture to a single mathematical system inevitably limits the power of mathematics to an infinitesimal fragment of what it could be. Mathematicians do not necessarily think that these very high levels of induction are that important. With the rarest of exceptions practical mathematics gets along quite well with a fragment of the power of induction of ZFC (defined in Section 5).

Godel's result was a shock to the mathematical community at the time and the result has yet to be fully digested. There can be no general method for determining mathematical truth. Any method that follows a single path may continue to make progress into an unbounded future, but all that path will accomplish over infinity is fully embodied in a single finite higher level axiom of induction. Every single or finite path process has such a limiting axiom that the process can never discover.

The only way to explore the full richness of this hierarchy is with an ever increasing number of schools of mathematics following mutually inconsistent mathematical systems. The is no way to know which school is true although some schools will prove to be false, but the total number of active schools must increase without limit. One can easily prove this. By exploring an ever increasing number of mathematical systems one can explore all possible formal mathematical systems. Of course one can be more selective than that, but it is clear that exploring every possible system is a way to consider every true axiom of induction.

Gödel's result has fundamentally altered the technical development of mathematics. New fields like Recursive Function Theory are founded upon it. But it has not altered the view that mathematical truth is absolute. Higher levels of induction are not needed for existing science and technology or normal mathematics. Is this proof of their irrelevance or a limitation of the level of evolutionary development that humans have reached? That is the subject of the next section.


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