# In what sense is Principia mathematica of Russell and Whitehead a metatheory?

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?

• Have you read PM? – Mr. Kennedy Nov 15 '16 at 15:19
• 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 '16 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 '17 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 '17 at 21:22

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 :

• You may also mention Boyce's book Logic for Programming: New Evidence in Support of Principia Mathematica. – user13627 Jul 18 '17 at 13:46
• Recently I have had a discussion with user21820 regarding Boyce's paper (see here and here). – user13627 Jul 21 '17 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 '17 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 – Mauro ALLEGRANZA Jul 21 '17 at 14:30
• For sure (for me, at least) Principia is a sytem and not a meta-system. As said, there are (in the book) no issue about consistency, completeness, and so on. We can find them, for the first time, into Hilbert & Ackermann's 1928 textbook. 2/2 – Mauro ALLEGRANZA Jul 21 '17 at 14:33

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"? – Mr. Kennedy Nov 15 '16 at 16:41
• 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. – George Chen Nov 15 '16 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. – George Chen Nov 15 '16 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"? – Mr. Kennedy Nov 15 '16 at 16:57
• 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 '16 at 20:37