“The foundations of mathematics”

Frank Ramsey's paper “The foundations of mathematics” (Ramsey, 1926) was presented to the London Mathematical Society on November 12, 1925, and so was likely written during the fall of 1924 after Ramsey had read the proofs for the second edition of Principia Mathematica. In the paper Ramsey makes repeated references to PM, citing the first volume of the new edition, and many points in the essay are best read as responses to the second edition. Indeed, the point for which the paper is most well known, the separation of the “mathematical” paradoxes from those due to semantics or “linguistics”, and the proposal to solve the former within the simple theory of types, misses the fact that Ramsey also proposes a new ramified theory of types as part of the solution of the semantic paradoxes. This modification of the type theory of PM turns attention to the very issues that concerned Russell in Appendix B. It is actually Ramsey's theory that should be given the joking name of the “ramsified theory of types” and which is here named “Ramsey's ramified simple types” or “RS-types”.

Ramsey introduces the paper as follows.

This chapter presents the notation and logical system of Principia Mathematica, taking as illustrations many formulas that will appear in the second edition and the manuscripts, and so will prepare the reader who is new to the work. While the syntax of PM is not presented in that work explicitly, and is not up to contemporary standards in logic in certain respects, it is easy to distinguish primitive from defined expressions, and follow the introduction of defined symbols into the system. Symbols are then used in a uniform way after they are introduced. Russell's notation evolved from 1900 to 1910, starting with his adoption of Peano's symbolism, and then developed through his collaboration with Whitehead. The second edition of PM uses the notation of the first, with the exception of the new Sheffer stroke, and so the changes are primarily matters of logical theory. This chapter will also review the distinctive logical doctrines of the first edition of PM, particularly the theory of types, the theory of definite descriptions, and the “no-class theory of classes”, as necessary prerequisites for understanding the additions of the second edition.

Primitive symbols and punctuation

The basic symbols include the following. Some are primitive and some are defined:

✻ Pronounced “star”; indicates a chapter or section (“number”) and one of three kinds of sentence, either an axiom, a theorem, or a definition. “✻20 General Theory of Classes” is a number, with ✻20·01 its first definition, and ✻20·1 its first theorem.

Reconstructing the original manuscript

The “Hierarchy of propositions and functions” (HPF) consists of 53 leaves, all but one with writing on only one side. The item is in file RA 230.031400. The leaves are all numbered (foliated) with only one exception, from 1 to 78. There are some duplicate numberings and some numbers out of order at the end of the manuscript. Almost all of the 25 missing folios reappear renumbered in the manuscripts for the Introduction and Appendix A. The numbers are written in the upper right of each page, underlined following Russell's practice, and they are reproduced here as italic. There are very few deletions, except for the last few pages. Some entire pages were deleted with a single line, or large ‘X’ through the whole page, and that is indicated where it happens.

At one point this HPF manuscript was the entire draft of the new material for the second edition. Russell then decided to split the new material into an introduction and an appendix. Long sections of the manuscript were moved to become parts of the Introduction to the second edition, and the core of Appendix A. Several folios were added at the beginning of each of those two. The rest of the material in the HPF manuscript was revised to differing extents and turns up in the published version there and in the Introduction, Appendix A, and Appendix B.

Russell begins the Introduction to the second edition by writing that the body of the text has not been changed for the new edition because any new propositions would disrupt the elaborate system of numbered theorems. Changes, he says, will be proposed in the Introduction, but not worked into the system. Appendix A does have a number (✻8), and numbered theorems, but the other changes are not formulated so technically. Thus even the definitions of the standard connectives in terms of the Sheffer stroke remain unnumbered. More serious for the reader, however, is the lack of precision in the discussion of extensionality and the abandonment of the axiom of reducibility. The informal approach to the discussion of the changes leaves it unclear which are minor, and which central, and for all, exactly what is being proposed. Following the opening excuse for not modifying the technical body of the work, Russell goes on to present a seemingly disjointed list of changes which are “desirable”, beginning with what he describes as “the most definite improvement resulting from work in mathematical logic during the past fourteen years…”. This improvement turns out to be the reduction of the propositional connectives to one, the Sheffer stroke “p|q” for “not both p and q.” It is true that the foundation for the logic circuits of contemporary computer chips is the thousands of connections executing operations on current that model this very stroke, now called the “nand” connective, so, in a way, this use of the Sheffer stroke in “logic” is the greatest legacy of the second edition of PM.

Editorial note

This manuscript is item 111-01-01 A in the Carnap archive at the University of Pittsburgh, and is published here by their permission. The manuscript consists of 35 leaves foliated by Russell from 1 to 35. “Russell” is written by Carnap in the upper left corner of the first leaf. Russell used underlining for emphasizing what was set in print as capitalized section heads and italics. These underlined words are set in italics below. Some numbers are not in numerical order, but rather follow the order in which they are discussed in the “summary” of a given chapter. Russell adds comments and picks out theorems in a manner that reveals that he was also reading through the symbolic portion of each chapter as he composed the list. The selection of definitions is partial, but self contained, with no evidence of Russell slipping, and using symbols which were not defined earlier in the list.

Whitehead and Russell generally define a notion as a relation, and then present a more familiar form as a theorem, e.g. D'R. He says “It is easier, though less correct, to give the latter forms as Dfs. This plan is adopted forthwith.” (p.5) Notice that ✻56, the last number of the paperback reprint of PM (1962), is completed on page 7. Carnap's Abriss uses this notation and follows definitions into Volume II.

The abundance of manuscript material and correspondence relating to the second edition of Principia Mathematica in the Bertrand Russell Archives makes possible a reconstruction of the process of its composition. It is clear that the second edition is solely the work of Russell, and what is new is confined to the Introduction to the second edition and three appendices, with the rest of the first edition reset with corrections or simply reprinted. Frank Ramsey was involved with the new edition, reading manuscript material for the new material, and then the printer's proofs late in 1924.

From the manuscript material we can also learn more of the details of this story. The new material for the second edition was originally planned to take the form of a revised chapter of the Introduction to the first, and it survives as a manuscript of seventy odd leaves, called “The Hierarchy of Propositions and Functions.” (“HPF” in what follows.) This was the first draft, with its leaves dispersed and renumbered, later filled out with the rest of the material for the Introduction to the second edition and Appendices A and B. Appendix C appears to have been then added, with only the final draft surviving.

Appendix B, which is not reprinted in the paperback Principia Mathematica to ✻56, with the technical problems identified by Gödel (1944) and confirmed by Myhill's (1974) proof, was the subject of the most preliminary work.

At the end of the manuscript of the Introduction to the second edition, Russell introduces a list of works in mathematical logic with this:

The list has become an issue of discussion among commentators on the second edition because of what it reveals about Russell's knowledge of logical developments in 1924.

Ray Monk proposes a list of difficulties that Russell faced when beginning to work on a genuinely revised second edition, including this damning charge:

In 1929 Russell wrote a letter of reference which recommends Leon Chwistek over Alfred Tarski for a position at the University of Lwow. It raises the question of what Russell knew of Tarski's work in 1924. In their biography of Tarski, Feferman & Feferman (2004) endorse the view that it was “Ramsey and others in Russell's circle” who supplied the references for the list.

Editorial note

The manuscripts of the Introduction and Appendices A, B, and C are transcribed below with almost all the changes due to the publisher's “house style” imposed, as in the use of capitals in the title. More substantial changes which Russell made as corrections to the printer's proof sheets are indicated in footnotes as “‘…’ becomes ‘…’”. Russell struck out some material with a single line (reproduced here in the text as deletions) and sometimes added some material above the line, and that is indicated as well. These insertions, usually above the line with a caret to indicate their position, are inserted in the line with ‘⌊’ and ‘⌋’ around the inserted material, as follows:⌊insertions⌋. A sequence below such as “Given p and p|(q|r), we can infer ⌊r⌋ q” indicates that the letter q was deleted and changed to r probably immediately after it was first written. Footnotes in the manuscript are indicated by a line across the page but are presented here as they appear in print. The printed Introduction begins on page xiii, indicated below in parentheses. Subsequent page numbers are inserted in the text where the page begins in the printed version. The number in the manuscripts (“foliation”) is reproduced in italics in the upper right hand corner of the material from each page. When Russell reused a leaf, giving it a new number, this is indicated in a footnote.

The second edition of Alfred N. Whitehead and Bertrand Russell's Principia Mathematica was published by Cambridge University Press in three volumes between 1925 and 1927. It consists of a reprint of the first edition, which appeared between 1910 and 1913, with the addition of a new Introduction and three Appendices (A, B, and C) written by Russell alone, and a List of Definitions. The new material takes up only 66 pages, yet it proposed radical changes to the system of Principia Mathematica, some of which require fundamental rethinking of the nature of logic.

What Russell oddly introduces as the “…most definite improvement resulting from work in mathematical logic during the past fourteen years…” is the proposal to replace the familiar fundamental logical connectives “or” and “not” with the single “Sheffer stroke”, “not-both.” This technically trivial change is in fact carried out in a rigorous fashion and does not require any rewriting of the body of Principia Mathematica in order to be properly implemented. A second and genuinely fundamental change is the adoption of “extensionality” in the second edition. This requires that all propositional connectives are to be truth-functional, and that co-extensive propositional functions, true of the same arguments, are identified. Russell characterizes this doctrine as the result of two theses, that:

What this change amounts to, and how it fits with the details of the various traces of the non-extensional system of the first edition that are unaltered, such as the definition of identity, will be discussed below.

Induction and inductive classes

The topic of Appendix B is the principle of mathematical induction which, together with the definition of numbers as classes of equinumerous classes, forms the heart of the logicist account of arithmetic. Throughout the earlier history of mathematics, induction had been recognized as a method of proof distinctive of arithmetic. The great achievement of the logicists, in particular, of Frege, was to show how induction could be shown to follow from logical truths and definitions alone.

The principle of induction is used to show that a property holds of all numbers by first showing that it holds of 0 and then that if it holds for some arbitrary n then it holds for n + 1. Proofs “by induction” thus have two parts. The first, “basis step” for the induction, is to prove that the given property holds of 0. The second, “induction step”, proceeds by assuming that the property holds of an arbitrary number n, the so-called “inductive hypothesis”, and then showing that it holds for n + 1. These two together suffice for proving that the property holds of all numbers.

Induction is not at the heart of the development of mathematics in Principia Mathematica, nor even that of Frege's earlier Grundgesetze der Arithmetik (1893). By the time of Introduction to Mathematical Philosophy (IMP) in 1919, however, Russell had come to present it as a central part of what is now the standard account of how mathematics emerges from logic.