Principia Mathematica: the first 100 years

Principia Mathematica: The first 100 years

Alasdair Urquhart

McMaster University

Victoria Day 2010

Alasdair Urquhart (McMaster University) Principia Mathematica: The first 100 years Victoria Day 2010 1 / 36

Queen Victoria 1819-1901


Russell’s nightmare

He was in the Cambridge University Library about two hundred years later,watching an assistant going round with a bucket, in which he was puttingbooks which he decided should be destroyed as not worth keeping. Theassistant picked up the only copy still in existence of PrincipiaMathematica, and stood hesitating. . . . At this stage Russell woke up.


How important was PM for 20th century logic?

Three possible directions from which we can approach this question.1 How important was PM as a basic treatise on logic?2 How important was it as a basic foundational framework?3 To what extent did it stimulate research in mathematical logic?


PM as a treatise on logic

Erwin Schrodinger

“I don’t believe that Russell and Whitehead had read it themselves.”

On the other hand . . .


W.V. Quine

“This is the book that has meant the most to me.”

The fact is, from about 1910 to 1930, PM was perhaps the most basicreference on symbolic logic, and not only Quine, but Bernays, Carnap,Church, Godel, C.I. Lewis, Emil Post and J. Barkley Rosser all started fromit (though they may have diverged considerably from it).


The appearance of PM began the relative eclipse of the previouslydominant tradition of algebraic logic; Schroder’s three-volume treatise wasput in the shade. This was in spite of the fact that Principia is rather heavilyindebted to the algebraic tradition, as the sections on relation algebrashow.


C.I. Lewis

C.I. Lewis’s remarks in his Survey of Symbolic Logic (1918) that his aim isto ease the transition between algebraic logic (with which he assumedstudents were familiar), and the work of Peano, Whitehead and Russell,“these most difficult and technical of treatises, in a new notation,developed by methods which are entirely novel to him, and bristling withlogical-metaphysical difficulties.”


John Crossley Stephen Kleene

A conversation from 1974

Crossley: What did you do, Steve, when you first started logic, you didn’thave books, did you?Kleene: Didn’t have books?


Gerald Sacks

Sacks: Well, he had Principia (laughter). Let’s see, was there a book byLewis on model theory?Kleene: Well, I never read Principia; of course I thumbed it a little bit.Rosser, I guess, started in logic that way, but I learned logic by learningChurch’s system, which was subsequently proved inconsistent. Out of thissystem we abstracted λ-definability. It was only after I got my degree that Ireally began to read much of the literature. Hilbert-Ackermann wasaround, and the first volume of Hilbert-Bernays.


Principia Mathematica: Decline and Fall

0

10

20

30

40

50

60

70

80

90

100

55

36

50

37

33

38

19

39

30

40

17

41

25

42

25

43

20

44

25

45

13

46

22

47

29

48

9

49

15

50

0

51

This chart shows the percentage of papers in the Journal of SymbolicLogic from 1936 to 1951 that cite Principia Mathematica in their list ofreferences.


PM as a foundation for mathematics

Louis Couturat

Russell to Couturat 21 August 1906:“Sad to say, it will be a long time before our work on the second volume isfinished. We are thinking of making it into an independent book, which weshall call Principia Mathematica.”


The failure of logicism

Principia Mathematica nowhere claims to show that mathematics isderivable from pure logic; in the Preface, Whitehead and Russell state astheir aim the more modest goal of the “mathematical treatment of theprinciples of mathematics.”This is a stark contrast with Russell’s bold claims in the Principles ofMathematics.


The Principles of Mathematics 1903

“Pure Mathematics is the class of all propositions of the form “pimplies q,” where p and q are propositions containing one ormore variables, the same in the two propositions, and neither pnor q contains any constants except logical constants.”


Two problems with this definition:1 Russell has forgotten to say that the propositions must be true;2 The restriction to implications is unnecessary.

Simplifying, we get: “Pure Mathematics is the class of all true propositionscontaining no constants except logical constants.”


Frank Ramsey

“It is really obvious that not all such propositions are propositions ofmathematics or symbolic logic. Take for example ‘Any two things differ inat least thirty ways’; this is a completely general proposition, it could beexpressed as an implication involving only logical constants and variables,and it may well be true. But as a mathematical or logical truth no one couldregard it.” – Ramsey 1925


Ramsey’s example could perhaps be dismissed as trivial. However, theaxiom of infinity is also one of those propositions that can be expressed inpurely logical terms (given the resources of higher order logic), and yet itdoes not appear to be a logical truth, at least by modern standards. This isvery bad news for Russell’s original logicist enterprise.


“That there are infinite classes is so evident that it will scarcely be denied.Since, however, it is capable of formal proof, it may be well to prove it” –Principles of Mathematics, p. 357.


Plato’s proof of the axiom of infinity

“A very simple proof is that suggested in the Parmenides, which is asfollows. Let it be granted that there is a number 1. Then 1 is, or has Being,and therefore there is Being. But 1 and Being are two: hence there is anumber 2; and so on” (Principles of Mathematics Ch. 43).


Henri Poincare

As late as 1906, in his reply to Poincare, Russell maintains that the Axiomof Infinity is a logical truth.

“By taking propositions into account, we can manufacture ℵ0

entities. E.g. put

p0 . = . a = u, pn+1 . = . pn = u;

it is not hard to prove that the successive p’s are all different, andthat there are therefore at least ℵ0 entities.”


With the adoption of the theory of types, the axiom of infinity was animmediate casualty.

Even simple propositions of arithmetic, such as 2 , 3, are unprovablewithout the axiom of infinity (1 , 0 and 2 , 0 are proved in Principia as∗101.22 and ∗101.34). This means that such propositions hold onlyhypothetically in PM.


An Astounding Discovery


If logicism is understood as the translation of mathematics into logicalsymbolism, then the translation of a proposition should have roughly thesame meaning as the original. However, the translation of 2 , 3, more orless literal in the original version of logicism, is now the hypothetical

Infin ax . ⊃ . 2 , 3,

which does not appear to have the same meaning as the original.


PM as a foundation for mathematics

Nicolas Bourbaki’s choice of set theory as the basis for his reconstructionof mathematics certainly had a considerable influence in making thelanguage of sets into the natural vernacular of contemporarymathematicians. Nevertheless, in the early years of the 20th century, typetheory in the style of Whitehead and Russell was a strong contender as afoundational framework.


The basic position of PM as a foundational framework is made clear inGodel’s great incompleteness paper of 1931. Not only is PrincipiaMathematica explicitly mentioned in the title of the paper, theincompleteness result is carried out for a streamlined and rigorous versionof the simple theory of types.


The early 1930s is the period when PM began to lose its status as a textfor learning basic logic. The same seems to be true for its status as afoundational scheme. Godel mentions Zermelo’s axioms for set theory inhis 1931 paper, but PM remains his basic framework. However, his latermajor work, the consistency proof for the axiom of choice and thegeneralized continuum hypothesis uses the Zermelo-Fraenkel version ofset theory.


Footnote 48a: “the true reason for the incompleteness inherent in allformal systems of mathematics is that the formation of ever higher typescan be continued into the transfinite . . . , while in any formal system atmost denumerably many of them are available”


Tarski 1931:

1 There is a definition of truth for languages of finite order;2 There is no definition of truth for languages of infinite order.

So, Godel’s almost casual introduction of transfinite types did not seemnatural to Tarski in 1931.


Tarski 1936: “In writing the present article [the 1931 paper on truth] I hadin mind only formalized languages possessing a structure which is inharmony with the theory of semantical categories and especially with itsbasic principles. . . . Today I can no longer defend decisively the view I thentook of this question.”


Defenders of the type-theoretical faith

Alonzo Church


Alan Turing

“One tends to feel that Russell’s type theory was largely anticipated byprehistoric man” – Turing 1948.


Type theory is alive and well in programming languages!


PM and the development of logic

Jacques Herbrand

Jacques Herbrand found inspiration for his work in proof theory in one ofthe oddest and most idiosyncratic passages in PM, namely ∗9. PrincipiaMathematica contains not one, but two, foundations for the theory ofquantification. It is the first, nonstandard foundation given in ∗9 thatinspired Herbrand.


Adele and Kurt Godel

Herbrand’s work in proof theory involved a radical reworking of ideas fromPrincipia. Godel’s work on the continuum hypothesis involved an evenmore radical reworking of the ramified type hierarchy, by extending it to alltransfinite types.


∗54.43 on your very own T-shirt!


Principia Mathematica: the first 100 years

Documents

Transcript of Principia Mathematica: the first 100 years