[FOM] Charles Peirce on proofs for number theory

Irving ianellis at iupui.edu
Sat Jan 9 17:15:41 EST 2010

In the manuscript "Logical Studies of the Theory of Numbers" (ca. 
1890), Peirce asks whether there is an algorithm for finding solutions 
to equations in number theory. He also asks whether there is an 
algorithm for determining if there are proofs in number theory. "The 
object of the present investigation," he writes (p. 55 of Nathan Houser 
(ed.), Writings of Charles S. Peirce: A Chronological Edition, vol. 8: 
1890-1892 (Bloomington/Indianapolis: Indiana University Press, 2010), 
pp. 55-56), "is to analyze carefully the logic of the theory of 
numbers. I especially desire to clear up the question of whether there 
can be fundamentally different ways of proving a theorem from given 
premises; and the law of reciprocity seems to be instructive in this 
respect. I also wish to know whether there is not a regular method of 
proof in higher arithmetic, so that we can see in advance precisely how 
a given proposition is to be demonstrated." He thus seems to 
anticipate, in a more general way, David Hilbert's Tenth Problem, posed 
at the International Congress of Mathematicians in 1900, of determining 
whether there is an algorithm for solutions to Diophantine equations. 
Peirce proposes translating these equations into Boolean algebra, but 
does not show how to use that to solve equations.

Like much of his work, Peirce failed to follow through with this 
project. The manuscript was found interpolated into an unsent letter to 
an unknown correspondent.

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

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