In what sense is Principia mathematica of Russell and Whitehead a metatheory rather than an object theory?

Dorais wrote at https://mathoverflow.net/q/159818:

note that Gödel's Incompleteness Theorems were originally meta-meta-theorems: Gödel proved that the formal system of Principia Mathematica (PM) was incomplete and PM was intended by Russell and Whitehead as the foundation of all mathematics, i.e., the ultimate meta-theory

Wikipedia: "A metatheory or meta-theory is a theory whose subject matter is some theory." In what sense are theories the subject matter of principia mathematica?

  • 2
    Have you read PM?
    – MmmHmm
    Nov 15, 2016 at 15:19
  • 2
    Principia Mathematica was conceived before the object/meta distinction was made, and aimed at presenting a "universal" logical shell that encompasses all of mathematics, if not all of science, a rigorized version of natural language, which of course requires no metalanguage because its subject matter is everything thinkable. Approaching Principia "from outside" can only be done after rejecting its principles.
    – Conifold
    Nov 16, 2016 at 2:04
  • @Conifold: Can you explain in more detail what you meant by, "[a]pproaching Principia "from outside" can only be done after rejecting its principles"?
    – user13627
    Jul 24, 2017 at 14:29
  • @user170039 It means that one has to exit the Principia and treat it as just one among many formal logical systems. Then it can be subjected to external meta-theoretical analysis. This is what Gödel did in the original proof of incompleteness. But that meant rejecting the One Logic of Principia, as Russell complained. To logicists logic was more than merely formal.
    – Conifold
    Jul 24, 2017 at 21:22

2 Answers 2


The "standard" view, shared by most commentators, has been synthesized by Kurt Gödel :

It is to be regretted that this first comprehensive and thoroughgoing presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in *1 - *21 of Principia) that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism [emphasis added]. (Kurt Gödel, "Russell’s Mathematical Logic", Reprinted from The Philosophy of Bertrand Russell, Paul A. Schilpp (editor), 1944).

For a similar view, see :

  • Alasdair Urquhart, "The Theory of Types", The Cambridge Companion to Bertrand Russell (Nicholas Griffin, ed.), 2003.

Contra the "standard" view, see :

  • 1
    You may also mention Boyce's book Logic for Programming: New Evidence in Support of Principia Mathematica.
    – user13627
    Jul 18, 2017 at 13:46
  • Recently I have had a discussion with user21820 regarding Boyce's paper (see here and here).
    – user13627
    Jul 21, 2017 at 14:01
  • The sentence that seemed puzzling to me (and user21820 also said it in this room) that, "The system of Principia resists Gödel’s technique of arithmetisation and thus provides a viable classical theory of arithmetic." If you think that both of us have interpreted the main idea of the paper mistakenly, I would like if you could share your opinions in the said room. I think that will be enriching (at least for me).
    – user13627
    Jul 21, 2017 at 14:02
  • @user170039 - I've not studied Boyce's paper (and I've not seen the new book) but I'm not sure that Boyce's point of view is "cirerct". We have "modern rewriting" of Principia that fully satisfy the modern approach; see e.g. Peter Andrews, An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. We can call them Principia's system, if we want, but they are not principia (the book). The critique of Godel is still valid; at the same time, G's work shows that it is quite straghtforward to fix Principia. 1/2 Jul 21, 2017 at 14:30
  • 1
    @MauroALLEGRANZA: For the record, Stephen Boyce is a crank, which is obvious from his lot of crank 'papers' on arΧiv, one of which claims that first-order arithmetic is inconsistent but is so convoluted! He also put "University of Sydney" at the bottom of another crank 'paper' that claims that "the correctness of classical arithmetic implies that the metatheory of the formalist account of arithmetic is false", but it seems there is no such person related to sydney.edu.au.
    – user21820
    Dec 13, 2018 at 8:40

When one uses nickels to calibrate a digital scale, he assumes five nickels together weigh 25 grams. The question that follows is this: what kind of instrument can guarantee the precision and accuracy of a nickel's weight? Eventually, he will trace the standard of a gram to the IPK, then he discovers that the definition of a gram depends on temperature, length, atmospheric pressure and the purity of water. Next he hopes that those instruments that guarantee the precision and accuracy of T, L, P do not use components that are weight sensitive, and the definitions of T, L, P standard units do not depend on weight.

If a theory cannot stand on its own feet, it has no right to talk about other theories. Gödel used numbering to gauge PM without first explaining what numbers are. Gödel was a platonist, W&R were not; that was why Russell said that he and Gödel "never arrived at common premises from which to argue."(See Autobiography)

PM stands on its own feet. Although it was not intended to be a theory about other theories, you can use it to gauge other theories whenever it is applicable. In 2016, in this world of formalists, PM is very applicable.

In the department of philosophy and mathematics new branches sprout as prolifically as lianas in the rain forest and there is a pullulation of techniques, terms and symbols producing a heterogeneous crop full of chaff that conceals a few kernels of wisdom.

--Hilton, Alice Mary. Logic, Computing Machines, and Automation. Cleveland and New York: Meridian Books, 1964

  • George, +1 Do you have a citation for this claim: "Although it was not intended to be a theory about other theories..."? It would be helpful as it gets directly at the OPs questions. Also, per the OP, if not intended as metatheory would R&W describe their PM as an "object theory"?
    – MmmHmm
    Nov 15, 2016 at 16:41
  • 1
    PM 1st ed Preface said it was originally intended to be the second volume of the Principles. In the Principles preface, it states that the present work has two objectives: 1. to show that pure mathematics is deduced from a very small number of principles; 2. to explain some fundamental concepts. Nov 15, 2016 at 16:50
  • I seriously doubt Russell would use such concept as "meta-theory" because Russell's work makes things simple rather than taking the problems upstairs. Nov 15, 2016 at 16:53
  • George, thanks. I agree, my aim is to get the OP's question answered. As you know Russell very well (and I presume have also actually read PM) - do you think R&W would consider PM as an "object theory"?
    – MmmHmm
    Nov 15, 2016 at 16:57
  • 1
    Would it be possible to expand slightly on "PM stands on its own feet?" The version I was taught was that Godel proved that, if PM were to ever try to stand on its own feet, it would trip and fall. If, as you say, PM is very applicable for formalists today, I assume there's some verbiage to be had to explain how people can reach such disparate stances. I would find value in some explanation of the apparent agreement to disagree.
    – Cort Ammon
    Nov 15, 2016 at 20:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .