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

Robert Lindauer rlindauer at gmail.com
Tue Aug 3 14:20:00 EDT 2010

On Tue, Aug 3, 2010 at 9:06 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> Aquinas and Spinoza may be examples of philosophically careful
> theology, but they do not match up to mathematics.  There work is not
> deductive.

Spinoza's work is certainly deductive as is Aquinas, they carefully
explain their definitions and rules of inference, axioms and theorems
and show why their theorems follow from their

 There are too many undefined terms, loose inferences,
> extraneous hypotheses.

Compare euclid's definition "A point is that which has no parts" and
spinoza's definitional system and you'll see that both have the same
problem.  It's truth that much mathematics is carried out purely
formally but nevertheless when you attach that formalism to "ordinary
language" in any way you end up with the same kind of issues you
describe.  This shouldn't be surprising since theology is clearly not
mathematics and has a subject matter beyond the pure formalism and so
is required to attach its definitions to something real.

Like in philosophy generally, the path from
> premise to conclusion is not deductive but goes through a number
> inferences that are not justified by soundness but by seeming
> plausible.

Again, compare Euclid's elements or Cantor's Contributions or
Dedekind's "essays on the theory of numbers" ...

I think what you should be saying here is -some mathematics- is
rigorous and formal but in the end is based in the same kind of loose
talk that you find in philosophy or theology.

A rigorous definition of "empty" or "set" is currently unavailable,
there is no proof of any of the axioms of ZFC, nor is there a
demonstration of the soundness of modus ponens or LEM.

On these mathematical topics, there's lots of "seemingly plausible
talk", unless you're aware of someplace I might find a rigorous
demonstration of the ZF(C) axioms (hopefully without itself being an
axiomatic system in need of rigorous defense).

Many times I've heard the term "self evident" used with regard to the
axioms, in much the same way I've heard it used in other practices,
generally not Theology any more though, since Theology's practice is
more mature, and having tried several systems of formal theology I
think theologists generally have come to the conclusion that the
axiomatization of a system of thinking doesn't provide any support for


> Monroe
> On Mon, Aug 2, 2010 at 4:47 PM, Robert Lindauer <rlindauer at gmail.com> wrote:
>> On Mon, Aug 2, 2010 at 12:29 PM, Monroe Eskew <meskew at math.uci.edu> wrote:
>>> Vaughan,
>>> Are there writings that one might call "formal theology"?  Are there
>>> theological theorems and open problems?  I once read an essay by a
>>> Catholic Cardinal that was much more philosophically careful and
>>> sophisticated than typical religious talk, but I have never seen
>>> something like you describe--theology which differs from mathematics
>>> only in subject matter.
>>> Best,
>>> Monroe
>> Monroe:
>> There are of course many formal theological systems dating from both
>> in western and eastern thought:
>> The summa theologica:
>> http://www.newadvent.org/summa/
>> Spinoza's Ethics:
>> http://www.marxists.org/reference/subject/philosophy/works/ne/spinoza.htm
>> And Patanjali's "How to know God" (more practical so perhaps less
>> obviously formal, but certainly intended as a step-by-step guide for
>> achieving a certain result, but more like an applied
>> mathematics-for-engineering style, as long as you're comparing).
>> While modern theologists tend not to formalize -in the same way as
>> mathematicians- their arguments, that they are formalizable as axioms,
>> definitions, inference rules and theorems is generally thought to be
>> self-evident, for those theologists who like this kind of approach.
>> There are good reasons for why many theologists have tended away from
>> mathematical-formalisms: a recognition of the effectiveness of a
>> dialectical critique of any axiomatic approach, the consequent
>> wistfulness for transcendental theology.
>> I hope the parallel opposing attitudes in mathematical practice are
>> clear enough.
>> (Perhaps the thought that it is sufficient to produce arguments that
>> are formalizable in theory even if no such proof is produced in
>> reality, is something they have in common as disciplinary activities).
>> As for "open theological problems":
>> I think there are many many such things, including foundational ones,
>> again very parallel to those found in foundational-mathematics:
>> a) is there any such thing as "God"? (formalists versus realists)
>> b) does the word "God" have a single meaning? (irrealism versus realism)
>> c) what is the nature of God's infinitude (power, knowledge, goodness,
>> etc.) "Can God make a rock he can not destroy?"
>> d) how can humans have relations with God?
>> While philosophers have been more likely to study these questions,
>> they are in the end, theological questions just as
>> i) "Are there any such things as numbers?"
>> ii) "What constitutes being a number?" or
>> iii) "what is the nature of mathematical infinity?" or
>> iv) "how can finite, inconsistent beings have absolute knowledge of
>> mathematical truths?"
>> are, in the end, both philosophical and mathematical questions.
>> A good place to look into current lively topics is here:
>> http://www.press.uillinois.edu/journals/ajtp.html
>> Best,
>> Robbie Lindauer
>> ps - because Theology is also meant to be practical, there are much
>> more interesting things done in other methods , for instance, in
>> Liberation Theology,which because generally dialectical in nature tend
>> to resist formalism in the same way that a survey of "all the truths
>> of mathematics" would also resist formalization.

It gives.

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