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Principia Mathematica

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✱9 with a new Appendix B and Appendix C. PM was conceived as a sequel to Russell's 1903 The Principles of Mathematics, but as PM states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."

For Isaac Newton's book containing basic laws of physics, see Philosophiæ Naturalis Principia Mathematica.

PM, according to its introduction, had three aims: (1) to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions, axioms, and inference rules; (2) to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; (3) to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox.[3]


This third aim motivated the adoption of the theory of types in PM. The theory of types adopts grammatical restrictions on formulas that rules out the unrestricted comprehension of classes, properties, and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of PM.


PM sparked interest in symbolic logic and advanced the subject, popularizing it and demonstrating its power.[4] The Modern Library placed PM 23rd in their list of the top 100 English-language nonfiction books of the twentieth century.[5]

Scope of foundations laid[edit]

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.


A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

(1) Elementary propositions.

(2) Elementary propositions of functions.

(3) Assertion: introduces the notions of "truth" and "falsity".

(4) Assertion of a propositional function.

(5) Negation: "If p is any proposition, the proposition "not-p", or "p is false," will be represented by "~p" ".

(6) Disjunction: "If p and q are any propositions, the proposition "p or q, i.e., "either p is true or q is true," where the alternatives are to be not mutually exclusive, will be represented by "p ∨ q" ".

(cf. section B)

whether a contradiction could be derived from the axioms (the question of ), and

inconsistency

whether there exists a which could neither be proven nor disproven in the system (the question of completeness).

mathematical statement

Contents[edit]

Part I Mathematical logic. Volume I ✱1 to ✱43[edit]

This section describes the propositional and predicate calculus, and gives the basic properties of classes, relations, and types.

Part II Prolegomena to cardinal arithmetic. Volume I ✱50 to ✱97[edit]

This part covers various properties of relations, especially those needed for cardinal arithmetic.

Part III Cardinal arithmetic. Volume II ✱100 to ✱126[edit]

This covers the definition and basic properties of cardinals. A cardinal is defined to be an equivalence class of similar classes (as opposed to ZFC, where a cardinal is a special sort of von Neumann ordinal). Each type has its own collection of cardinals associated with it, and there is a considerable amount of bookkeeping necessary for comparing cardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite cardinals. ✱120.03 is the Axiom of infinity.

Part IV Relation-arithmetic. Volume II ✱150 to ✱186[edit]

A "relation-number" is an equivalence class of isomorphic relations. PM defines analogues of addition, multiplication, and exponentiation for arbitrary relations. The addition and multiplication is similar to the usual definition of addition and multiplication of ordinals in ZFC, though the definition of exponentiation of relations in PM is not equivalent to the usual one used in ZFC.

Part V Series. Volume II ✱200 to ✱234 and volume III ✱250 to ✱276[edit]

This covers series, which is PM's term for what is now called a totally ordered set. In particular it covers complete series, continuous functions between series with the order topology (though of course they do not use this terminology), well-ordered series, and series without "gaps" (those with a member strictly between any two given members).

Part VI Quantity. Volume III ✱300 to ✱375[edit]

This section constructs the ring of integers, the fields of rational and real numbers, and "vector-families", which are related to what are now called torsors over abelian groups.

The system of propositional logic and predicate calculus in PM is essentially the same as that used now, except that the notation and terminology has changed.

The most obvious difference between PM and set theory is that in PM all objects belong to one of a number of disjoint types. This means that everything gets duplicated for each (infinite) type: for example, each type has its own ordinals, cardinals, real numbers, and so on. This results in a lot of bookkeeping to relate the various types with each other.

In ZFC functions are normally coded as sets of ordered pairs. In PM functions are treated rather differently. First of all, "function" means "propositional function", something taking values true or false. Second, functions are not determined by their values: it is possible to have several different functions all taking the same values (for example, one might regard 2x+2 and 2(x+1) as different functions on grounds that the computer programs for evaluating them are different). The functions in ZFC given by sets of ordered pairs correspond to what PM call "matrices", and the more general functions in PM are coded by quantifying over some variables. In particular PM distinguishes between functions defined using quantification and functions not defined using quantification, whereas ZFC does not make this distinction.

PM has no analogue of the , though this is of little practical importance as this axiom is used very little in mathematics outside set theory.

axiom of replacement

PM emphasizes relations as a fundamental concept, whereas in modern mathematical practice it is functions rather than relations that are treated as more fundamental; for example, category theory emphasizes morphisms or functions rather than relations. (However, there is an analogue of categories called that models relations rather than functions, and is quite similar to the type system of PM.)

allegories

In PM, cardinals are defined as classes of similar classes, whereas in ZFC cardinals are special ordinals. In PM there is a different collection of cardinals for each type with some complicated machinery for moving cardinals between types, whereas in ZFC there is only 1 sort of cardinal. Since PM does not have any equivalent of the axiom of replacement, it is unable to prove the existence of cardinals greater than ℵω.

In PM ordinals are treated as equivalence classes of well-ordered sets, and as with cardinals there is a different collection of ordinals for each type. In ZFC there is only one collection of ordinals, usually defined as . One strange quirk of PM is that they do not have an ordinal corresponding to 1, which causes numerous unnecessary complications in their theorems. The definition of ordinal exponentiation αβ in PM is not equivalent to the usual definition in ZFC and has some rather undesirable properties: for example, it is not continuous in β and is not well ordered (so is not even an ordinal).

von Neumann ordinals

The constructions of the integers, rationals and real numbers in ZFC have been streamlined considerably over time since the constructions in PM.

This section compares the system in PM with the usual mathematical foundations of ZFC. The system of PM is roughly comparable in strength with Zermelo set theory (or more precisely a version of it where the axiom of separation has all quantifiers bounded).

A 54-page introduction by Russell describing the changes they would have made had they had more time and energy. The main change he suggests is the removal of the controversial axiom of reducibility, though he admits that he knows no satisfactory substitute for it. He also seems more favorable to the idea that a function should be determined by its values (as is usual in modern mathematical practice).

Appendix A, numbered as *8, 15 pages, about the Sheffer stroke.

Appendix B, numbered as *89, discussing induction without the axiom of reducibility.

Appendix C, 8 pages, discussing propositional functions.

An 8-page list of definitions at the end, giving a much-needed index to the 500 or so notations used.

Apart from corrections of misprints, the main text of PM is unchanged between the first and second editions. The main text in Volumes 1 and 2 was reset, so that it occupies fewer pages in each. In the second edition, Volume 3 was not reset, being photographically reprinted with the same page numbering; corrections were still made. The total number of pages (excluding the endpapers) in the first edition is 1,996; in the second, 2,000. Volume 1 has five new additions:


In 1962, Cambridge University Press published a shortened paperback edition containing parts of the second edition of Volume 1: the new introduction (and the old), the main text up to *56, and Appendices A and C..

Legacy[edit]

Andrew D. Irvine says that PM sparked interest in symbolic logic and advanced the subject by popularizing it; it showcased the powers and capacities of symbolic logic; and it showed how advances in philosophy of mathematics and symbolic logic could go hand-in-hand with tremendous fruitfulness.[4] PM was in part brought about by an interest in logicism, the view on which all mathematical truths are logical truths. Though flawed, PM would be influential in several later advances in meta-logic, including Gödel's incompleteness theorems.


The logical notation in PM was not widely adopted, possibly because its foundations are often considered a form of Zermelo–Fraenkel set theory.


Scholarly, historical, and philosophical interest in PM is great and ongoing, and mathematicians continue to work with PM, whether for the historical reason of understanding the text or its authors, or for furthering insight into the formalizations of math and logic.


The Modern Library placed PM 23rd in their list of the top 100 English-language nonfiction books of the twentieth century.[5]

Axiomatic set theory

Boolean algebra

– first computational demonstration of theorems in PM

Information Processing Language

Introduction to Mathematical Philosophy

(2001) [1972]. A Mathematical Introduction to Logic (2nd ed.). San Diego, California: Academic Press. ISBN 0-12-238452-0.

Enderton, Herbert B.

(1944). "Russell's Mathematical Logic". In Schilpp, Paul Arthur (ed.). The Philosophy of Bertrand Russell. The Library of Living Philosophers. Vol. 5 (1st ed.). Chicago: Northwestern University Press. pp. 123–153. LCCN 44006786. OCLC 2007378. OL 6467049M.

Gödel, Kurt

Gödel, Kurt (1990). ; et al. (eds.). Collected Works, Volume II, Publications 1938–1974. New York: Oxford University Press. ISBN 0-19-503972-6.

Feferman, Solomon

(2000). The Search for Mathematical Roots 1870–1940. Princeton, New Jersey: Princeton University Press. ISBN 0-691-05857-1.

Grattan-Guinness, Ivor

(2004) [1940]. A Mathematician's Apology. Cambridge: Cambridge University Press. ISBN 978-0-521-42706-7.

Hardy, G. H.

(1952). Introduction to Metamathematics (6th reprint ed.). Amsterdam, New York: North-Holland Publishing Company. LCCN 53001848. OCLC 9296141.

Kleene, Stephen Cole

(1986). Bollobás, Béla (ed.). Littlewood's Miscellany. Cambridge: Cambridge University Press. ISBN 0-521-33058-0. MR 0872858.

Littlewood, J. E.

(1967). From Frege to Gödel: A Source book in Mathematical Logic, 1879–1931 (3rd printing ed.). Cambridge, Massachusetts: Harvard University Press. ISBN 0-674-32449-8.

van Heijenoort, Jean

; Desmond, William Jr., eds. (2008). Handbook of Whiteheadian Process Thought, Volume 1. Heusenstamm: Ontos Verlag. ISBN 978-3-938793-92-3. Retrieved 13 September 2023 – via Academia.edu.

Weber, Michel

(2009). Major Works: Selected Philosophical Writings. New York: HarperCollins. ISBN 978-0-06-155024-9.

Wittgenstein, Ludwig

Stanford Encyclopedia of Philosophy

Principia Mathematica

in a more modern notation (Metamath)

Proposition ✱54.43