Primarily at issue were the kinds of assumptions Whitehead and Russell needed to complete their project. Although Principia succeeded in providing detailed derivations of many major theorems in finite and transfinite arithmetic, set theory, and elementary measure theory, two axioms in particular were arguably non-logical in character: the axiom of infinity and the axiom of reducibility. The axiom of infinity in effect states that there exists an infinite number of objects. Arguably it makes the kind of assumption generally thought to be empirical rather than logical in nature. Although technically feasible, many critics concluded that the axiom was simply too ad hoc to be justified philosophically. In the minds of many, the issue of whether mathematics could be reduced to logic, or whether it could be reduced only to set theory, thus remained open.
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Primarily at issue were the kinds of assumptions Whitehead and Russell needed to complete their project. Although Principia succeeded in providing detailed derivations of many major theorems in finite and transfinite arithmetic, set theory, and elementary measure theory, two axioms in particular were arguably non-logical in character: the axiom of infinity and the axiom of reducibility.
The axiom of infinity in effect states that there exists an infinite number of objects. Arguably it makes the kind of assumption generally thought to be empirical rather than logical in nature. Although technically feasible, many critics concluded that the axiom was simply too ad hoc to be justified philosophically. In the minds of many, the issue of whether mathematics could be reduced to logic, or whether it could be reduced only to set theory, thus remained open.
In response, Whitehead and Russell argued that both axioms were defensible on inductive grounds. As they tell us in the Introduction to the first volume of Principia, self-evidence is never more than a part of the reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true if the axiom were false, and nothing which is probably false can be deduced from it.
If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false. And if the axiom itself is nearly indubitable, that merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a radically different kind. Infallibility is never attainable, and therefore some element of doubt should always attach to every axiom and to all its consequences.
In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premisses which were not previously known to require limitations. As Russell writes, Both Whitehead and I were disappointed that Principia Mathematica was only viewed from a philosophical standpoint.
People were interested in what was said about the contradictions and in the question whether ordinary mathematics had been validly deduced from purely logical premisses, but they were not interested in the mathematical techniques developed in the course of the work. I will give two illustrations: Mathematische Annalen published about ten years after the publication of Principia a long article giving some of the results which unknown to the author we had worked out in Part IV of our book.
This article fell into certain inaccuracies which we had avoided, but contained nothing valid which we had not already published. The author was obviously totally unaware that he had been anticipated.
The second example occurred when I was a colleague of Reichenbach at the University of California. I told him that this subject was fully treated in the third volume of the Principia. When I saw him a week later, he told me that he had verified this.
First, it popularized modern mathematical logic to an extent undreamt of by its authors. By using a notation superior to that used by Frege, Whitehead and Russell managed to convey the remarkable expressive power of modern predicate logic in a way that previous writers had been unable to achieve. Second, by exhibiting so clearly the deductive power of the new logic, Whitehead and Russell were able to show how powerful the idea of a modern formal system could be, thus opening up new work in what soon was to be called metalogic.
Third, Principia Mathematica re-affirmed clear and interesting connections between logicism and two of the main branches of traditional philosophy, namely metaphysics and epistemology, thereby initiating new and interesting work in both of these areas. Just as importantly, it initiated a tradition of common technical work in fields as diverse as philosophy, mathematics, linguistics, economics and computer science.
Today a lack of agreement remains over the ultimate philosophical contribution of Principia, with some authors holding that, with the appropriate modifications, logicism remains a feasible project. Others hold that the philosophical and technical underpinnings of the project remain too weak or too confused to be of great use to the logicist.
For more detailed discussion, readers should consult Quine a , Quine b , Landini , Landini , Linsky , Linsky , Hale and Wright , Burgess , Hintikka and Gandon There is also lack of agreement over the importance of the second edition of the book, which appeared in Volume 1 and Volumes 2 and 3.
The revisions were done by Russell, although Whitehead was given the opportunity to advise. In addition to the correction of minor errors throughout the original text, changes to the new edition included the inclusion of a new Introduction and three new appendices. The appendices discuss the theory of quantification, mathematical induction and the axiom of reducibility, and the principle of extensionality respectively.
The book itself was reset more compactly, making page references to the first edition obsolete. Russell continued to make corrections as late as for the printing, the year he and Mrs Whitehead finally began to receive royalties. Appendix B has been notoriously problematic. Urquhart Linsky provides helpful discussion, both of the Appendix itself and of the suggestion that by Russell may have been out of touch with recent developments in the quickly changing field of mathematical logic.
He also addresses the suggestion, made by some commentators, that Whitehead may have been opposed to the revisions, or at least indifferent to them, concluding that both charges are likely without foundation. Did you like this?
Whitehead & Russell: Principia Mathematica Vol. I, II, III pdf
PM —94, for the first edition: 1 Elementary propositions. Anything implied by a true elementary proposition is true. The revised theory is made difficult by the introduction of the Sheffer stroke " " to symbolise "incompatibility" i. In the revised theory, the Introduction presents the notion of "atomic proposition", a "datum" that "belongs to the philosophical part of logic". These have no parts that are propositions and do not contain the notions "all" or "some". For example: "this is red", or "this is earlier than that".
Principia Mathematica Volume I