GoKawiilPrincipia Mathematica is modern and insightful
Introduction
Principia Mathematica by Whitehead and Russell was published back in 1910 -- and yet it reads like a modern text on programming languages. I have found Principia quite engaging and hard to put away. Principia discusses, with great insight, such modern topics as extensionality/intensionality, referential transparency, type. It contains perhaps the first mentioning of `domain', `alpha renaming' and `type' in the modern sense. Its `incomplete symbols' -- the ones that only make sense in a context -- anticipate continuations and control operators. It insightfully observes that the notions of free and bound variables, substitution, abstraction, and application all come from linguistics. I could not help but feel that Principia already contained lambda-calculus. It also seems that Russell and Whitehead anticipated intuitionism, for example, when insisting on separate notations for 'any' vs. `all' (although admitting the equivalence of these notions in their theory). The whole Principia is very large: It is said that the book is famous for taking a thousand pages to prove that 1+1=2 . As the preface stresses, the proofs are excruciatingly detailed so to remove the chance of an unstated premise being used in a proof. The goal of Principia was to put forward a set of very basic notions, and show that they and they alone are sufficient for the whole Mathematics. If Principia were to be published today, all the proofs would be relegated to a Supplement (or a theorem prover). What important are the basic notions and the set up -- most of which is explained in the Preface and Chapter 1. These following are a few notes taken while reading Chapter 1 of Principia, with several comments very kindly given by Jacques Carette. Version The current version is 1.3, August 2026 References The current version is 1.3, August 2026 Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Cambridge: University Press, 1910-
<http://name.umdl.umich.edu/AAT3201.0001.001>
The full scanned text, many thanks to The University of Michigan Historical Mathematics Collection Linsky, Bernard. The Notation in Principia Mathematica
The Stanford Encyclopedia of Philosophy (Summer 2026 Edition), Edward N. Zalta & Uri 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 literature of intensions and extensions, and what is now called `referential transparency': ``if p≡q we shall have f(p)≡f(q) ''. Here f(p) is a proposition that includes another proposition p . In modern terms, we would call f a context and denote by C[] , and say that if p≡q then C[p]≡C[q] , which is the familiar statement of a referential transparent context. The page then shows an example of a non-referentially transparent context ``A believes p'': a proposition whose meaning varies when p is substituted with equivalent propositions. The example betrays the origin of this concept, from linguistics, specifically, from the work of Frege (who is mentioned in a footnote). The book states that ``mathematics is always concerned with extensions rather than intensions.'' (again borrowing Frege terms, but in English translation.)
Definitions: a mere typographic convenience of most importance
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