athematics plays an important role in the human exploration of reality both in respect to its own kind of reality and also as a heuristic tool for other kinds of investigation. Often seen as the paradigm of rationality, mathematics has been shown by Gödel, in one of the greatest discoveries of the twentieth century, to be startlingly open in its logical structure. The nine scientists and philosophers coming together in Castel Gandolfo, a small town southeast of Rome overlooking Lake Albano, take an intellectual, experiencebased approach to nonphysical reality in their exploration of the significance of mathematics. The topics engaging them emerge from the following sets of observations
and questions:

The majority of mathematicians see their subject as discovery and not mere invention. The implication is that mathematical entities exist in a noetic realm to which the human mind has access. The Mandelbrot Set (with its endlessly proliferating complexity deriving from a deceptively simple definition) did not come into existence when Benoit Mandelbrot first began to consider its definition, but he found it. Many will see his discovery as being an encounter with a noetic reality lying beyond
the merely material.


Evolutionary survival seems to require little more than simple arithmetic and a
little Euclidean geometry. Whence then has come the human ability to explore noncommutative algebras and to prove Fermat’s last theorem? An adequate evolutionary anthropology seems to require a richer context than afforded by conventional neoDarwinism.


It is a technique of discovery in fundamental physics to seek theories whose equations are endowed with the unmistakable character of mathematical beauty, since only theories of this kind have proved to have the longterm fruitfulness that persuades us of their verisimilitudinous character. How should we understand this “unreasonable effectiveness,” as Nobel laureate E. P. Wigner’s famously put it, of abstract mathematics in physical science? Theologically, the deep rational transparency and rational beauty of the universe can be understood as reflecting the rationality of the universe’s Creator.


Gödel showed that axiomatised mathematical systems including the integers are either incomplete (not all stateable results are proveable results) or inconsistent. What does this imply for the possibility of the formulation of grand unified theories in physics? Apophaticism, it appears, is not restricted to theology. 
The gathering in the Alban Hills, under the aegis of the John Templeton Foundation in partnership with the Centre for Advanced Research in Theological and Religious Studies, Cambridge, and the Vatican Observatory, begins a conversation to be continued at Cambridge University.
