Did Peirce have Hilbert's 9th & 10th problems?

[irvanellis]

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.