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

Irving ianellis at iupui.edu
Wed Aug 4 14:47:46 EDT 2010

Vaughn Pratt wrote: "The mathematician's gut instinct is to dismiss
theology as lacking the precision of argument needed for reliable

But are we in fact being mathematical in any recognizable sense when we
go by gut instinct rather than logic? What is, or should be, at stake
here is analyzing the logical structure of arguments for the existence
of God, and looking for informal or formal fallacies in these proofs --
or alleged proofs. By doing so, we should be well able to avoid "gut

As an historical aside, let us recall that Kronecker, and many others,
dismissed Cantor's transfinite sets as mere "theology" (see,e.g. F. A.
Medvedev, "Kantorovskaya teoriya mnozhestv i teologiya",
Istoriko-matematicheskie Issledovaniya 29 (1985), 209–240; J. W.
Dauben, "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the
Infinite", Journal of the History of Ideas 38 (1977), 85–108;  C.
Ferrandi, "La discussione del concetto di infinito tra matematica,
filosofia e teologia. Georg Cantor e la Scolastica cristiana",
Angelicum 82/1 (2005) 187–214).

I would propose nevertheless that there may perhaps be a residual
Pythagorean-Platonistic theology in any thinking that treats mathematical
objects or structures as having some kind of platonistic ideal existence.
But then we may also want to carfully distinguish thinking philosophically
about mathematics and thinking philosophically about doing mathematics on
the one hand, in contrast with doing mathematics, or writing proofs of
theorems, on the other.

We are all probably also familiar with the folklore regarding Euler's
mathematical "proof" of the existence of God, directed at an allegedly
confounded Denis Diderot:

"Monsieur, (a + b^n)/n = x, therefore God exists. Please reply!"

Turning serious, and supposedly incumbent upon me by reason of my
connection with the Peirce Edition, I would point out that in
"A Neglected Argument for the Reality of God", first published in
the Hibbert Journal 7 (1908), 90-112, and reprinted in Charles Hartshorne &
Paul Weiss (eds.), Collected Papers of Charles Peirce, vol. VI, Charles
Peirce, apropos his argument for the *reality*, rather than the
*existence*, of God as an ens necessarium, wrote: " 
 there is a reason,
an interpretation, a logic in the course of scientific advance, and
this indisputably proves to him who has perceptions of rational or
significant relations, that man's mind must have been attuned to the
truth of things in order to discover what he has discovered. It is the
very bedrock of logical truth."  A copy of the text of the "Neglected
Argument" article is available online at:

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
URL: http://www.irvinganellis.info

More information about the FOM mailing list