Principia Mathematica is modern and insightful

Principia Mathematica by Whitehead and Russell was published back successful 1910 -- and yet it sounds for illustration a modern matter connected programming languages. I person recovered Principia rather engaging and difficult to put away. Principia discusses, pinch awesome insight, specified modern topics as extensionality/intensionality, referential transparency, type. It contains possibly the first mentioning of `domain', `alpha renaming' and `type' successful the modern sense. Its `incomplete symbols' -- the ones that only make consciousness successful a discourse -- expect continuations and control operators. It insightfully observes that the notions of free and bound variables, substitution, abstraction, and exertion all come from linguistics. I could not thief but consciousness that Principia already contained lambda-calculus. It besides seems that Russell and Whitehead anticipated intuitionism, for example, erstwhile insisting on separate notations for 'any' vs. `all' (although admitting the equivalence of these notions successful their theory).

The full Principia is very large: It is said that the book is famous for taking a 1000 pages to beryllium that 1+1=2. As the preface stresses, the proofs are excruciatingly elaborate truthful to region the chance of an unstated premise being utilized successful a proof. The extremity of Principia was to put guardant a group of very basal notions, and show that they and they unsocial are capable for the full Mathematics. If Principia were to beryllium published today, each the proofs would be relegated to a Supplement (or a theorem prover). What important are the basal notions and the group up -- astir of which is explained successful the Preface and Chapter 1.

These pursuing are a fewer notes taken while reference Chapter 1 of Principia, pinch respective comments very kindly fixed by Jacques Carette.

Version

The existent type is 1.3, August 2026

References

Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Cambridge: University Press, 1910-
<http://name.umdl.umich.edu/AAT3201.0001.001>
The afloat scanned text, galore acknowledgment to The University of Michigan Historical Mathematics Collection

Linsky, Bernard. The Notation successful 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/>

On p12, the book states that definitions are simply typographic conveniences. On the different hand, definitions are of astir importance, because they show the intent.

…the definitions are not portion of our subject, but are, strictly speaking, specified typographical conveniences.… In spite of the fact that definitions are theoretically superfluous, it is nevertheless existent that they often convey much important information than is contained successful the propositions successful which they are used. … The postulation of definitions embodies our prime of subjects and our judgement arsenic to what is astir important. Secondly, … the definition contains an study of a communal idea, and whitethorn therefore express a notable advance.

Propositional functions: anticipation of lambda-calculus

Page 15 introduces ``propositional functions'', what is now known as lambda-terms. See for yourself, from the moving illustration connected the page.

"x is hurt" [called ambiguous] really makes nary assertion astatine all, till we person settled who x is. Yet owing to the personality retained by the ambiguous adaptable x, it is an ambiguous illustration from the collection of propositions arrived astatine by giving each imaginable determinations to x successful "x is hurt" which output a proposition, existent aliases false.

The authors then present the notation for that ``propositional function'': "\hat{x} is hurt". Although "x is hurt" and "y is hurt" occurring successful the aforesaid discourse tin beryllium distinguished, ``"\hat{x} is hurt" and "\hat{y} is hurt" convey nary favoritism of meaning at all.'' The paragraph concludes: ``More generally, φx is an ambiguous value of the propositional usability φ\hat{x}, and erstwhile a definite signification a is substituted for x, φa is an unambiguous value of φ\hat{x}.'' Here we person it: free variables, bound variables, substitution and alpha-equivalence.

The taxable of variables comes up again, connected p17, successful the chat of quantified formulas:

The awesome "(x).φx" [in modern notation, ∀x.φ(x)] denotes 1 definite proposition, and location is no favoritism successful meaning betwixt "(x).φx" and "(y).φy" when they hap successful the aforesaid context. … The awesome "(x).φx" has immoderate affinity to the symbol ∫abφ(x) dx since successful neither lawsuit is the look a function of x. … The x which occurs successful "(x).φx" or "(∃x).φx" is called (following Peano) an "apparent variable".

The page past goes connected to present the conception of a adaptable scope.

What Principia calls `apparent variable' is bound adaptable in modern terminology; `real variable' is now called free variable. The illustration of a definite integral to exemplify bound variables and alpha-equivalence is striking. It besides shows that lambda calculus has a agelong pedigree. I couldn't thief but respect the Leibniz insight.

p18 and p19 of Principia deals pinch what we now call schematic variables and schematic assertions, of the form ⊢ f x.

When we asseverate thing containing a existent variable, arsenic successful e.g. ⊢ x = x we are asserting any worth of the propositional function. When we asseverate thing containing an evident variable, arsenic in ⊢ (x).x = x [which is ⊢ ∀ x. x=x successful modern notation] we are asserting ... all values of the proposition usability successful question. It is plain that we tin only asseverate ``any value'' if all values are true; for otherwise, since the worth of the adaptable remains to beryllium determined, it mightiness beryllium truthful wished arsenic to springiness a mendacious proposition. Thus successful the supra instance, since we person ⊢ x = x we whitethorn infer ⊢ (x).x = x

The authors past spell connected to present what we now telephone generalization, of ∀-introduction. (Page 20 introduces the inverse, ∀-elimination, or, arsenic Principia puts it, ``what holds for all, holds for any''.)

Although a schematic look (for any) is balanced to the corresponding universally quantified look successful Principia's logic [which was later distilled to is now called First-Order Logic], the authors still wish to keep the 2 notions distinct.

The mean formulae of mathematics incorporate specified [real-variable] assertions; for example sin² x + cos² x = 1 does not asseverate this aliases that peculiar lawsuit of the formula, nor does it assert that the look holds for all imaginable values of x, although this is balanced to this second assertion; it simply asserts that the look holds, leaving x wholly undetermined; and it is able to do this legitimately, because nevertheless x is determined, a true proposition results.

On page 20, Principia says, aft describing ∃-introduction: ⊢ φy ⊂ (∃x).φx:

The supra proposition gives what is successful believe the only measurement of proving existence theorems: we ever person to find immoderate peculiar y for which φy holds, and hence to infer (∃x).φx. If we were to presume what is called the multiplicative axiom, aliases the equivalent axiom enunciated by Zermello, that would, successful an important class of cases, springiness an existence-theorem wherever nary peculiar lawsuit of truth tin beryllium found.

Thus, for Russell and Whitehead, ``the only measurement successful practice'' of proving existence theorems was to grounds a witness. They have, perhaps unconsciously, took up intuitionistic, aliases moreover constructivist view. And this was published successful 1910...

Jacques Carette noted that Brouwer was besides publishing astir that time. (Although it has to beryllium said that Brouwer writings of that time were hardly comprehensible to a mathematician. The intuitionistic vs. classical controversy has really started pinch Hermann Weyl.) Jacques has further noted that some aspects of that constructivism tin beryllium traced back Kronecker 30 years earlier.

Types

On p21, aft asserting a proposition (in modern notation)

the authors constitute ``this requires φ and ψ should beryllium functions which take arguments of the aforesaid type. (We shall explicate this request astatine a later stage).'' How contemporary! That was possibly the first usage of the connection `type' successful the consciousness now truthful communal successful programming.

Origin of set-membership

On p26, the authors statement that the awesome for group rank is really the Greek epsilon, the first missive of the connection ἐστί -- which, by a Russian analogue, I assume intends ``to be''. So x ∈ man virtually intends "x is simply a man". (I don't mean that Principia first projected that notation. It was already established.)

Descriptive functions

Page 33 is astir apt the first modern meaning of a usability arsenic a particular shape of a binary relation: immoderate binary narration R induces a usability R'y arsenic the unsocial x specified that xRy holds. No restriction connected R is imposed; however, later `domain' is introduced as a people of those y for which location exists only 1 x truthful that xRy holds. A one-to-many narration hence does specify a function, with the quiet domain.

Principia calls specified binary-relation--induced functions `descriptive functions' (now often called ``definite descriptions'). The sanction and the exposition follows the mentation of descriptions successful earthy languages that Russell developed 5 years prior (in his celebrated insubstantial ``On denoting'', Mind 14(4), 1905).

Jacques Carette noted that Principia anticipated the quality between ``definite description'' and ``explicit function'' backmost successful 1910, because there were already examples successful mathematics of these. ``Analytic continuation is 1 of those processes successful mathematics which is functional but not a function, arsenic it involves a definite magnitude of choice.''

References

Ludlow, Peter. Descriptions
The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.)
<https://plato.stanford.edu/archives/win2023/entries/descriptions/>