[FOM] Gödel, Darwin and creating mathematics

[Paul Budnik]

Gödel's Incompleteness Theorems suggests a philosophy of mathematical 
truth that is both objective and creative. Logically determined 
statements, like the consistency of a formal system, are objectively 
true or false even though there can be no general process for deciding 
this. Emil Post reached a similar conclusion over 60 years ago: "The 
conclusion is unescapable that even for such a fixed, well defined body 
of mathematical propositions [a formulation of the recursively 
enumerable sets], _mathematical thinking is, and must remain, 
essentially creative_." (from the published lecture "Recursively 
enumerable sets of positive integers and their decision problems" 
www.projecteuclid.org/euclid.bams/1183505800 ).

I discussed  objective mathematics in 
www.cs.nyu.edu/pipermail/fom/2010-May/014792.html This posting focuses 
on the creation of mathematics. My assumptions are that physical reality 
is always finite but could be potentially infinite and the laws of 
physics are recursive.

In the light of Gödel's work, these assumptions may seem to be an 
obstacle to the evolution of a mind capable of developing mathematics. 
Gödel established limits on what mathematics a recursive process can 
decide, but not on what it can explore. It is straight forward to write 
a single nondeterministic TM program to enumerate all the axiom systems 
definable in a formal language and to deduce all the theorems decidable 
in each of these systems. While this is not a practical way for us to 
create new mathematics, biological evolution did something a bit like 
this in evolving the mathematically capable mind. That process involved 
an immense diversity of life over billions of years.

Our sense of the innate truth of fragments of mathematics is an 
evolutionary legacy. It evolved because there are modes of thought that 
consistently lead to accurate conclusions. These modes of thought were 
refined and formalized through cultural evolution. This suggests two 
questions. How close are we to the limits of what is achievable with our 
evolutionary legacy? That is how far can we confidently extend our 
ability to decide questions about objective mathematics?  The second 
question is what happens when we approach the limits of what is 
achievable from that legacy?

I suspect the answer to the first question is that we are far from those 
limits in part because computers have not been used as a research tool 
for expanding the foundations of mathematics. They are used for 
automated theorem proving and proof verification, but not to explore and 
understand the ordinal hierarchy that is essential to expanding the 
logical power of mathematics. There is a combinatorial explosion of 
complexity in developing the ordinal hierarchy at the level of the 
recursive and countable admissible ordinals, where it is directly 
connected to recursive processes and can be investigated using computer 
experiments. Although currently the most powerful expansions of this 
hierarchy come from large cardinal axioms, I suspect that, eventually, 
more powerful results will be obtained through a mastery of the innate 
complexity of the ordinal hierarchy at these lower levels. At some point 
it becomes impossible to manage this complexity without the aid of 
computers. For more about this see my page on the ordinal calculator, 
www.mtnmath.com/ord .

Appel and Haken pioneered using computers to manage complexity in 
mathematics over 30 years ago in their proof of the four color theorem. 
They used the computer to deal with the combinatorial explosion of 
special cases that had to be considered. The complexity of their proof 
justified a careful process of acceptance, but the use of computers 
should have been recognized as a powerful new technique for dealing with 
complexity and not seen as a possible reason for rejecting the proof.

My answer to the second question is that we will eventually need to 
return to the nondeterministic search of biological evolution. This will 
require following an ever increasing number of paths as resources become 
available. Unlike biological evolution, this increasingly divergent 
search will be guided by a deep understanding of which paths may be 
worth pursuing.  Under my assumptions about physical reality, any 
approach limited to a fixed finite number of alternatives will run up 
against what I call a Gödel limit. Continual progress can be made 
defining more powerful axiom systems into an unbounded future but the 
entire sequence of correct results will be theorems provable from a more 
general axiom that will never be considered or explored.

Paul Budnik
www.mtnmath.com