[FOM] Concerning proof, truth, and certainty in mathematics

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Tue Aug 3 05:22:06 EDT 2010

Monroe Eskew wrote:
> Are there writings that one might call "formal theology"?  Are there
> theological theorems and open problems? 

Here are two candidates:

Stephen D. Unwin,
The Probability of God
Crown Forum; 2003

This contains real theorems, though not fully formalized in the sense
of automated theorem proving.

An approximation -- to the extent that the logical writing
of Leibniz, say, approximates formal logic -- is the
Summa Theologica by Thomas Aquinas.

Certainly part of it could be completely formalized;
an example might be the following, taken from Chapter C5 of
the theoretical physics FAQ at 

<begin quote>

It looks as if Aquinas was the first writer anticipating quantum theory
and the Pauli exclusion principle. Replace 'angel' by 'electron' and he
sounds surprisingly modern; in modern terms, angels are Fermions,
according to Thomas Aquinas.

An English translation of the ''Summa Theologica'' is available online.
Part I (http://www.newadvent.org/summa/1.htm)
contains the chapter on angels.
The sections 50-53 on their substance relate to their physical
properties and hence are of scientific interest.

There he discusses the properties of a point particle from
a logical point of view. His 'angels' are not the winged creatures
we might imagine them to be, but incorruptible, indivisible,
extended objects, ''form without matter'', with quite precise

Two angels cannot be in the same place, but they have
virtual (sic!) positions, and can be in an extended place:
    ''So the entire body to which he is applied by his power,
    corresponds as one place to him.''
They may go from one place to another with or without being observable
in between:
    ''But an angel's substance is not subject to place as contained
    thereby, but is above it as containing it: hence it is under
    his control to apply himself to a place just as he wills,
    either through or without the intervening place.''
Their number roughly matches those of the number of electrons:
    ''Hence it must be said that the angels, even inasmuch as they are
    immaterial substances, exist in exceeding great number, far beyond
    all material multitude.''
(With ''angel'' interpreted as ''electron'', ''immaterial'' could thus
be interpreted as zero baryon number.)

Like early chemists hiding their scientific insights in an alchemist
guise, Aquinas might have phrased his speculations in terms of notions
acceptable to his clerical collegues...

<end quote>

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