Nothing matters too much, or Wright is wrong (Q2701944)

The paper centrally deals with the question whether ``Hume's Principle'' provides a ``\textit{philosophically interesting} rather than merely formal foundation for arithmetic''. The author's ideas are expressed with extensive references to \textit{C. Wright} [Frege's conception of numbers as objects. Aberdeen Univ. Press (1983; Zbl 0524.03005)] and \textit{G. Boolos} [Logic, logic, and logic. Harvard Univ. Press, Cambridge (1998; Zbl 0955.03008)], where Hume's Principle (HP) is expressed as (a modification of a (fatal) principle of Frege): NEWLINE\[NEWLINE\# X = \# Y \leftrightarrow X\approx Y \tag{HP}NEWLINE\]NEWLINE \(X\) and \(Y\) are second-order variables ranging over Fregean concepts (properties; e.g., to be a subclass), and \(X \approx Y\) abbreviates the statement in second-order logic that there is a one-one correspondence between the \(X\)s and the \(Y\)s (between the objects falling under the concept \(X\) and those falling under the concept \(Y\)). \(\#\) is a total function from concept to object, a particular object, the \textit{number of} \(X\). HP is now regarded as a principle from which, together with second-order logic, second-order Peano arithmetic is derivable, essentially in Frege's way, using appropriate definitions of zero, successor, and finite number. NEWLINENEWLINENEWLINEHP entails, according to Wright, that there is a partition of concepts into equivalence classes, in which two concepts belong to the same class if and only if they are equinumerous. If there are only \(k\) objects, \(k\) a finite number, then, since there are \(k + 1\) natural numbers \(\leq k\), there will be \(k + 1\) equivalence classes, viz. a class containing each concept under which zero objects fall, a class containing each concept under which exactly one object falls, ..., and a class containing each concept under which all \(k\) objects fall. Thus, if there are only \(k\) objects, there is no function mapping concepts to objects that takes nonequinumerous concepts to different objects, for there won't be enough objects around to serve as values of the function, since \(k + 1\) are needed. So if HP holds, there must be infinitely many objects. NEWLINENEWLINENEWLINEThe author objects to HP as naturally being regarded as \textit{foundational} for arithmetic: ``the problem is the way in which Hume's Principle generates an infinity of numbers, generating new numbers to count the numbers already there with a tail-biting circularity which is lucky enough to avoid paradox so long as we stick with cardinals.'' ``My feeling is that the infinity of the number series ... should rather be seen as one with the Cantorean idea of a completable infinity. The reason why there are infinitely many natural numbers ought to be the same sort of reason as the reason why \(V_{\omega}\) is a set. But I shall make no attempt here to say just what sort of reason that is.'' NEWLINENEWLINENEWLINEReviewer's comments. The remarks of the author raise the problem of how to understand the \textit{nature of a foundation}. Consider the fundamental problem of how, in mathematics, to \textit{interpret} an axiom of infinity, formulated with the \textit{intended meaning} that a set exists (like the set of natural numbers) which is infinite. The axiom is but a finite string of symbols (even when the whole theory, where it occurs, is taken into account -- every formal theory is finitely representable). How can such a finite string be interpreted as describing an infinite object -- if not the interpretation process itself (beyond the axiom) is what accounts for the infinity. This problem has developed in two directions. \textbf{First}, towards \textit{language} as a whole of entangled description-interpretation processes, explicity recognizing the impossibility of completely describing the interpretations in the language where they occur. The idea of mathematicians \textit{sharing} a language when they communicate about infinity, and infinities, is philosophically satisfactory for understanding the otherwise stubborn problem with ``intended interpretations''. \textbf{Second}, the problem has developed in a direction of \textit{inaccessibility}, conceived as mathematically objectifiable by axiomatization. Recalling the author's point, any such \textit{mathematically expressible foundation} is bound to be (philosophically) incomplete: the axiomatizability of inaccessibility (the set of natural numbers is inaccessible) implies a form of accessibility. Only \textit{partial} aspects of the full inaccessibility of the infinite from the finite are axiomatizable.