of classes and relations from the ultimate logical premisses. In the
course of this process, undertaken for the first time with the rigour
of mathematicians, some contradictions have become apparent.
That first discovered is known as Burali-Forti’s contradiction,[1] and
consists in the proof that there both is and is not a greatest infinite
ordinal number. But these contradictions do not depend upon
any theory of number, for Russell’s contradiction[2] does not involve
number in any form. This contradiction arises from considering
the class possessing as members all classes which are not members
of themselves. Call this class w; then to say that x is a w is
equivalent to saying that x is not an x. Accordingly, to say that w
is a w is equivalent to saying that w is not a w. An analogous
contradiction can be found for relations. It follows that a careful
scrutiny of the very idea of classes and relations is required.
Note that classes are here required in extension, so that the class of
human beings and the class of rational featherless bipeds are
identical; similarly for relations, which are to be determined by the
entities related. Now a class in respect to its components is many.
In what sense then can it be one? This problem of “the one and the
many” has been discussed continuously by the philosophers.[3] All
the contradictions can be avoided, and yet the use of classes and
relations can be preserved as required by mathematics, and indeed
by common sense, by a theory which denies to a class—or relation—existence
or being in any sense in which the entities composing it—or
related by it—exist. Thus, to say that a pen is an entity and the
class of pens is an entity is merely a play upon the word “entity”;
the second sense of “entity” (if any) is indeed derived from the
first, but has a more complex signification. Consider an incomplete
proposition, incomplete in the sense that some entity which ought
to be involved in it is represented by an undetermined x, which may
stand for any entity. Call it a propositional function; and, if φx
be a propositional function, the undetermined variable x is the
argument. Two propositional functions φx and ψx are “extensionally
identical” if any determination of x in φx which converts
φx into a true proposition also converts ψx into a true proposition,
and conversely for ψ and φ. Now consider a propositional function
Fχ in which the variable argument χ is itself a propositional function.
If Fχ is true when, and only when, χ is determined to be either φ or
some other propositional function extensionally equivalent to φ,
then the proposition Fφ is of the form which is ordinarily recognized
as being about the class determined by φx taken in extension—that
is, the class of entities for which φx is a true proposition when x is
determined to be any one of them. A similar theory holds for relations
which arise from the consideration of propositional functions with
two or more variable arguments. It is then possible to define
by a parallel elaboration what is meant by classes of classes,
classes of relations, relations between classes, and so on. Accordingly,
the number of a class of relations can be defined, or of a class
of classes, and so on. This theory[4] is in effect a theory of the use
of classes and relations, and does not decide the philosophic question
as to the sense (if any) in which a class in extension is one entity.
It does indeed deny that it is an entity in the sense in which one of
its members is an entity. Accordingly, it is a fallacy for any
determination of x to consider “x is an x” or “x is not an x” as
having the meaning of propositions. Note that for any determination
of x, “x is an x” and “x is not an x,” are neither of them
fallacies but are both meaningless, according to this theory. Thus
Russell’s contradiction vanishes, and an examination of the other
contradictions shows that they vanish also.
Applied Mathematics.—The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications. For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting. It is only recently that the succession of processes which is involved in any act of counting has been seen to be irrelevant to the idea of number. Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. A legend of the Council of Nicea[5] illustrates this point: “When the Bishops took their places on their thrones, they were 318; when they rose up to be called over, it appeared that they were 319; so that they never could make the number come right, and whenever they approached the last of the series, he immediately turned into the likeness of his next neighbour.” Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analogous considerations. In the first place, some physical process of addition is presupposed, involving some inductively inferred law of permanence during that process. Thus in the theory of masses we must know that two pounds of lead when put together will counterbalance in the scales two pounds of sugar, or a pound of lead and a pound of sugar. Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of real numbers. Indeed, mathematicians now reserve “continuity” as the term for the latter kind of continuity; the mere property of having an infinite number of terms between any two terms is called “compactness.” The compactness of the series of rational numbers is consistent with quasi-gaps in it—that is, with the possible absence of limits to classes in it. Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. In calculations the latter hypothesis is made because of its mathematical simplicity. But, the assumption has certainly no a priori grounds in its favour, and it is not very easy to see how to base it upon experience. For example, if it should turn out that the mass of a body is to be estimated by counting the number of corpuscles (whatever they may be) which go to form it, then a body with an irrational measure of mass is intrinsically impossible. Similarly, the continuity of space apparently rests upon sheer assumption unsupported by any a priori or experimental grounds. Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity.
In one sense there is no science of applied mathematics. When once the fixed conditions which any hypothetical group of entities are to satisfy have been precisely formulated, the deduction of the further propositions, which also will hold respecting them, can proceed in complete independence of the question as to whether or no any such group of entities can be found in the world of phenomena. Thus rational mechanics, based on the Newtonian Laws, viewed as mathematics is independent of its supposed application, and hydrodynamics remains a coherent and respected science though it is extremely improbable that any perfect fluid exists in the physical world. But this unbendingly logical point of view cannot be the last word upon the matter. For no one can doubt the essential difference between characteristic treatises upon “pure” and “applied” mathematics. The difference is a difference in method. In pure mathematics the hypotheses which a set of entities are to satisfy are given, and a group of interesting deductions are sought. In “applied mathematics” the “deductions” are given in the shape of the experimental evidence of natural science, and the hypotheses from which the “deductions” can be deduced are sought. Accordingly, every treatise on applied mathematics, properly so-called, is directed to the criticism of the “laws” from which the reasoning starts, or to a suggestion of results which experiment may hope to find. Thus if it calculates the result of some experiment, it is not the experimentalist’s well-attested results which are on their trial, but the basis of the calculation. Newton’s Hypotheses non fingo was a proud boast, but it rests upon an entire misconception of the capacities of the mind of man in dealing with external nature.
- ↑ “Una questione sui numeri transfiniti,” Rend. del circolo mat. di Palermo, vol. xi. (1897); and Russell, loc. cit., ch. xxxviii.
- ↑ Cf. Russell, loc. cit., ch. x.
- ↑ Cf. Pragmatism: a New Name for some Old Ways of Thinking (1907).
- ↑ Due to Bertrand Russell, cf. “Mathematical Logic as based on the Theory of Types,” Amer. Journ. of Math. vol. xxx. (1908). It is more fully explained by him, with later simplifications, in Principia mathematica (Cambridge).
- ↑ Cf. Stanley’s Eastern Church, Lecture v.
