Principia Mathematica
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Principia Mathematica is modern and insightful
Introduction
Referential transparency, extensionality
Definitions: a mere typographic convenience of most importance
Propositional functions: anticipation of lambda-calculus
for any vs for all: a glimpse of Intuitionism
Intuitionistic view on existence
Types
Origin of set-membership
Descriptive functions
Introduction
Principia Mathematica by Whitehead and Russell was published<br>back in 1910 -- and yet it reads like a modern text on programming<br>languages. I have found Principia quite engaging and hard to put<br>away. Principia discusses, with great insight, such modern topics as<br>extensionality/intensionality, referential transparency, type. It<br>contains perhaps the first mentioning of `domain', `alpha renaming'<br>and `type' in the modern sense. Its `incomplete symbols' -- the ones<br>that only make sense in a context -- anticipate continuations and<br>control operators. It insightfully observes that the notions of free<br>and bound variables, substitution, abstraction, and application all<br>come from linguistics. I could not help but feel that Principia<br>already contained lambda-calculus. It also seems that Russell and<br>Whitehead anticipated intuitionism, for example, when insisting on<br>separate notations for 'any' vs. `all' (although admitting the<br>equivalence of these notions in their theory).<br>The whole Principia is very large: It is said that the book is famous<br>for taking a thousand pages to prove that 1+1=2. As the preface<br>stresses, the proofs are excruciatingly detailed so to remove the<br>chance of an unstated premise being used in a proof. The goal of<br>Principia was to put forward a set of very basic notions, and show<br>that they and they alone are sufficient for the whole Mathematics.<br>If Principia were to be published today, all the proofs would be<br>relegated to a Supplement (or a theorem prover). What important are<br>the basic notions and the set up -- most of which is explained in the<br>Preface and Chapter 1.<br>These following are a few notes taken while reading Chapter 1 of<br>Principia, with several comments very kindly given by Jacques Carette.
Version<br>The current version is 1.3, August 2026<br>References
Principia Mathematica by Alfred North Whitehead and Bertrand Russell.<br>Cambridge: University Press, 1910-
http://name.umdl.umich.edu/AAT3201.0001.001>
The full scanned text, many thanks to The University of Michigan<br>Historical Mathematics Collection
Linsky, Bernard. The Notation in Principia Mathematica
The Stanford<br>Encyclopedia of Philosophy (Summer 2026 Edition), Edward N. Zalta & Uri<br>Nodelman (eds.)
https://plato.stanford.edu/archives/sum2026/entries/pm-notation/>
Referential transparency, extensionality
Page 8 of Principia has perhaps the first mention in mathematical<br>literature of intensions and extensions, and what is now called<br>`referential transparency': ``if p≡q we shall have f(p)≡f(q)''.<br>Here f(p) is a proposition that includes another proposition p. In<br>modern terms, we would call f a context and denote by C[], and say<br>that if p≡q then C[p]≡C[q], which is the familiar statement of<br>a referential transparent context. The page then shows an<br>example of a non-referentially transparent context ``A believes p'': a<br>proposition whose meaning varies when p is substituted with<br>equivalent propositions. The example betrays the origin of this<br>concept, from linguistics, specifically, from the work of Frege (who<br>is mentioned in a footnote). The book states that ``mathematics is<br>always concerned with extensions rather than intensions.'' (again<br>borrowing Frege terms, but in English translation.)
Definitions: a mere typographic convenience of most importance
On p12, the book states that definitions are merely typographic<br>conveniences. On the other hand, definitions are of most importance,<br>because they show the intent.
…the definitions are not part of our subject, but are, strictly<br>speaking, mere typographical conveniences.… In spite of the<br>fact that definitions are theoretically superfluous, it is<br>nevertheless true that they often convey more important information<br>than is contained in the propositions in which they are used. …<br>The collection of definitions embodies our choice of subjects and<br>our judgement as to what is most important. Secondly, … the<br>definition contains an analysis of a common idea, and may therefore<br>express a notable advance.
Propositional functions: anticipation of lambda-calculus<br>Page 15 introduces ``propositional functions'', what is now known as<br>lambda-terms. See for yourself, from the running example on the page.
"x is hurt" [called ambiguous]<br>really makes no assertion at all, till we have settled who x is.<br>Yet owing to the individuality retained by the<br>ambiguous variable x, it is an ambiguous example from the collection<br>of propositions arrived at by giving all possible determinations to<br>x in "x is hurt" which yield a proposition, true or false.
The authors<br>then introduce the notation for that...