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

[Vaughan Pratt]

On 8/1/2010 4:23 PM, Harvey Friedman wrote:
> In practice, both the method of professional mathematical interaction
> and publication, as well as the laying down of the accepted axioms is
> already enough to prevent any dispute about correctness of proofs from
> lasting very long. This is exactly the opposite situation in all other
> intellectual endeavors.
 > [...]
> For the largest compendium of actual formally perfect proofs, look up
> MIZAR.

Indeed.  As a rebuttal to Epstein, Mizar is hard to beat.

> On Jul 31, 2010, at 9:30 PM, ARF (Richard L. Epstein) wrote:
>> If mathematics is a body of truths, with certain knowledge of them,
>> then such knowledge is the same
>> kind as theological knowledge.
>
> This is false. Mathematical knowledge is vastly different than what
> transpires in theology.

Is the distinction so clear cut?  Why couldn't St. Augustine's proof of 
the existence of God be rendered in MIZAR?  Or if not MIZAR then in de 
Bruijn's AUTOMATH?

Harvey's response brings to mind Gordan's initial response to Hilbert's 
finiteness theorem, "Das ist nicht Mathematik. Das ist Theologie." 
But after the general acceptance of Hilbert's nonconstructive proof, 
Gordan allowed that theology had its merits after all.

Theology has in common with mathematics the axiomatic method.  For 
example the Nicene creed of 325 AD sets out those postulates of 
Christianity which are to be taken on faith, and from which further 
inferences may be drawn.  Theological heresy is the substitution of 
other axioms inconsistent with the established dogma.

And yet there is something about theology that we all recognize as 
"vastly different" from mathematics.  What is the essence of that 
difference if not in the use of axioms and deductive reasoning?

The mathematician's gut instinct is to dismiss theology as lacking the 
precision of argument needed for reliable mathematics.  But this is 
almost surely based on encounters with the amateur theologians 
encountered in the pulpit, on blogs, etc.  To judge theology that way 
instead of analyzing the reasoning in the proceedings of a modern 
theological conference is like judging our understanding of global 
warming from Conservapedia's take on it instead of say the ninth edition 
of Donald C. Ahren's "Meteorology Today."

I would think the essential difference is in the choice of concepts. 
Whereas mathematics is about the sorts of abstractions that arise in 
geometry, analysis, and combinatorics, theology is more about divinity 
and spirituality.  The latter are certainly not central concepts of 
mathematics and, to many if not most mathematicians, of no mathematical 
interest at all.

So whereas I would agree with HF that theology is different from 
mathematics by virtue of its focus on mathematically uninteresting 
concepts, I would agree with ARF that mathematics as "a body of truths, 
with certain knowledge of them" shares that trait with theology to the 
extent that their respective professionals both strive for precision of 
argument from axioms to conclusions.

The agreements and disagreements can be summarized with respect to the 
following two links,

    Mathematics <---> certain knowledge <---> theology

along with their composite from mathematics directly to theology.

ARF and HF would appear to be in agreement on the absence of the 
composite link, but differ on which component of the composite to break: 
Richard denies the first while affirming the second, Harvey does the 
opposite.

I argued above for the simultaneous possibility of both links, and 
therefore that of their composite.  I would take ARF's and HF's 
arguments as evidence against the necessity of either link, with the 
common element of those arguments appearing to be that one of the two 
links holds yet the composite fails, whence the other link must fail.

Vaughan Pratt