[FOM] Concerning proof, truth, and certainty in mathematics
pratt at cs.stanford.edu
Mon Aug 2 13:18:38 EDT 2010
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
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
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
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.
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