32. Why a Given Function requires Arguments of a Certain Type

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

Why a Given Function requires Arguments of a Certain Type



The considerations so far adduced in favor of the view that a function cannot significantly have as argument anything defined in terms of the function itself have been more or less indirect. But a direct consideration of the kinds of functions which have functions as arguments and the kinds of functions which have arguments other than functions will show, if we are not mistaken, that not only is it impossible for a function φz^ to have itself or anything derived from it as argument, but that, if yz^ is another function such that there are arguments a with which both "φa" and "ya" are significant, then yz^ and anything derived from it cannot significantly be argument to φz^. This arises from the fact that a function is essentially an ambiguity, and that, if it is to occur in a definite proposition, it must occur in such a way that the ambiguity has disappeared, and a wholly unambiguous statement has resulted. A few illustrations will make this clear. Thus "(x) . φx," which we have already considered, is a function of φx; as soon as φx^ is assigned, we have a definite proposition, wholly free from ambiguity. But it is obvious that we cannot substitute for the function something which is not a function: "(x) . φx" means "φx in all cases," and depends for its significance upon the fact that there are "cases" of φx, i.e. upon the ambiguity which is characteristic of a function. This instance illustrates the fact that, when a function can occur significantly as argument, something which is not a function cannot occur significantly as argument. But conversely, when something which is not a function can occur significantly as argument, a function cannot occur significantly. Take, e.g. "x is a man," and consider "φx^ is a man." Here there is nothing to eliminate the ambiguity which constitutes φx^; there is thus nothing definite which is said to be a man. A function, in fact, is not a definite object, which could be or not be a man; it is a mere ambiguity awaiting determination, and in order that it may occur significantly it must receive the necessary determination, which it obviously does not receive if it is merely substituted for something determinate in a proposition*. This argument does not, however, apply directly as against such a statement as "{(x) . φx} is a man." Common sense would pronounce such a statement to be meaningless, but it cannot be condemned on the ground of ambiguity in its subject. We need here a new objection, namely the following: A proposition is not a single entity, but a relation of several; hence a statement in which a proposition appears as subject will only be significant if it can be reduced to a statement about the terms which appear in the proposition. A proposition, like such phrases as "the so-and-so," where grammatically it appears as subject, must be broken up into its constituents if we are to find the true subject or subjects*². But in such a statement as "p is a man," where p is a proposition, this is not possible. Hence "{(x). φx)} is a man" is meaningless.

* Note that statements concerning the significance of a phrase containing "φx^" concern the symbol "φz^," and therefore do not fall under the rule that the elimination of the functional ambiguity is necessary to significance. Significance is a property of signs.

*² C.f. Chapter III

Script: Otto Binder Pencils and Inks: George Papp © Respective copyright/trademark holders.

This small patch of ‘mental driver’ is deceptively simple, but serves as a giant step towards comprehension of some aspects of the system that we have, until now largely had to accept on face value. Speaking for myself, the rules regarding the use of functions, and the broad restrictions of the vicious-circle principle seemed to be very restrictive and cumbersome obstacles that will arise at every turn. This post goes very far to shrinking that field by providing important specific arguments to guide us in using propositions, and seeing more specifically how functions fit into useful calculations.

Still this post is mainly perceived in the abstract, and our comfort with its ambiguity at this point may be indicative of our confidence in Principia Mathematica. To study this system calls for a leap of faith, for even as the rational perception of value may take hold early, there is so much that must be absorbed, with no clear application beyond ambiguity, in sight. This is a trip into the shaman mind. Very likely anyone reading this work is interested from a philosophical field of study. I suspect few mathematicians in the post war era ever bothered to pursue the intensional aspects of the system, instead seeking some arcane advantage by perusing an obscure text that everyone knew of, but few had actually read.

 

Two world wars changed the field of mathematics radically; from a few Cambridge professors corresponding with their counterparts in Berlin or Rome with papers of pure and universal mathematics. War, industry and big profits changed the Mathematicscape in every way imaginable, so that by 1950, the mind of the mathematician was so far removed from that resembling Bertrand Russell’s struggling to finish developing our system from 1905-1910, as to be seen as completely divergent in both process and intent.

Script Archie Goodwin Pencils and Inks John Severin © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.