Metamathematics has a reasonably clear connotation, enough to have a Wikipedia page, with Gödel, Tarski, and Turing playing leading roles; Kleene's book (Introduction to Metamathematics (Amazon link)); Chaitin's article ("Meta-mathematics and the foundations of mathematics." EATCS Bulletin, June 2002, vol. 77, pp. 167-179); etc. My question is:

Q. Is there an identifiable meta-metamathematics, a scholarly study of metamathemaics, perhaps in the philosophical (rather than mathematical) literature? Or does all the literature essentially "devolve" to metamathemathics, without an identifiable line that can be drawn between metamathematics and meta-metamathematics?

Citations to possibly-meta-metamathematical studies would be appreciated! Thanks!

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    $\begingroup$ This might be a naive question but don't metamathematical theorems, once you formalize them, become mathematical and therefore meta-metamathematical theorems are simply metamathematical? For example, you can prove theorems like "the theory T proves that the theory S proves...etc.", which you might consider meta-meta-meta-...-mathematical, but isn't this just metamathematics? $\endgroup$ – Burak Jun 28 '14 at 0:58
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    $\begingroup$ Shouldn't this question be asked on the meta site? :-) $\endgroup$ – Asaf Karagila Jun 28 '14 at 1:30
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    $\begingroup$ @Asaf I agree with your comment, but doesn't it belong on the meta-meta-site? $\endgroup$ – Andreas Blass Jun 28 '14 at 1:37
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    $\begingroup$ @Andreas: I believe this was discussed before; on the StackExchange network, the meta operator is idempotent. $\endgroup$ – Asaf Karagila Jun 28 '14 at 1:43
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    $\begingroup$ @JosephO'Rourke: Is the four-color theorem "just a theorem of mathematics?" Is the Pythagorean theorem "just a theorem of mathematics"? Is there some relevant sense in which these differ from Godel's theorems? $\endgroup$ – Steven Landsburg Jun 28 '14 at 1:51

My opinion is that there is no crisp distinction between mathematics, metamathematics and meta-metamathematics, and the subjects thoroughly blend one into another in such a way that prevents any coherent distinction.

Furthermore, even the categorization of particular topics as mathematics or metamathematics has changed radically over time, and many topics that were formerly considered metamathematics are now just mathematics. For example, the ultrapower construction was born in metamathematics, but is now widely seen as a fundamental mathematical construction. The method of forcing was initially used only for relative consistency proofs, but is now saturated with a mixture of infinite combinatorics, ideals, Boolean algebras, topology, transfinite limits, and so on. Computability theory was born in purely philosophical speculation about what it means for a human to undertake a computable procedure, but gave birth to complexity theory and other extremely applied mathematical topics. Is the polynomial time hierarchy regarded today as metamathematics? I don't think so, but it is a part of complexity theory, which is a part of computability theory, which is traditionally considered metamathematics. The study of large cardinals is tied to fundamental issues in logic, such as definability and constructiblity, but also involves at its core essentially mathematical questions about infinite combinatorics, measure theory, complex systems of embeddings and so on. Where does the mathematics end and the metamathematics begin? It is all wrapped up together.

The term metamathematics has traditionally included the entire subjects of model theory, set theory, proof theory and computability theory, but I think this kind of usage of the term is simply no longer accurate, since huge parts of these subjects are now more mathematical than metamathematical. I think that the term "metamathematics" may have made more sense as a unifying umbrella term in an earlier age, when many mathematicians were simply less familiar with these subjects than is the case today.

Consider my work with Benedikt Löwe on the modal logic of forcing. The main theorem is that the ZFC provably valid principles of forcing are exactly those in the modal theory known as S4.2. Now, the principles under consideration, the principles of forcing, can themselves surely be considered as metamathematical, as they concern how truth varies in the generic multiverse, the Kripke model of possible worlds consisting of the set-theoretic universe in the context of all its forcing extensions. Since the principles are thus metamathematics, and we are proving theorems about which principles are provably valid, one could consider this to be solidly a case of meta-metamathematics. But if you look at the paper, I think you will mainly find just plain old mathematics, with detailed inductions and finite partial order combinatorics and some infinite combinatorics and forcing iterations, mixed in with some modal logic, which is essentially finite combinatorics. This example therefore illustrates my point that there is really no coherent distinction into mathematics/metamathematics/meta-metamathematics.

  • $\begingroup$ I have learned a great deal from this and the other responses and comments---Thanks! $\endgroup$ – Joseph O'Rourke Jun 28 '14 at 12:46
  • $\begingroup$ It is clear that there are mathematical statements that are not metamathematical and vice versa. For example, the statement `ZFC is consistent' is purely metamathematical, while its translation Con(ZFC) via Godel numbering is a mathematical (in fact, arithmetical) statement. Is not the distinction a valid and useful one? $\endgroup$ – Jesse Elliott Oct 13 '14 at 9:47
  • $\begingroup$ @JesseElliott I believe that the distinction is not as clear as your remark suggests. For example, I believe that many people would say that both of those statements are meta-mathematical, and others would say that those are the same statement, with the same meaning, even though they are said differently. $\endgroup$ – Joel David Hamkins Oct 13 '14 at 11:45
  • $\begingroup$ I'm not sure I agree. Con(ZFC) is a (probably true) statement to the effect that a certain complicated Diophantine equation has no integer solutions, expressed as a statement of PA. `ZFC is consistent' is the metamathematical statement that ZFC is consistent. They happen to be equivalent. Con(ZFC) is actually many statements, since it depends on what Godel numbering assignment you choose. So if they are all the same statement, then you are implying that all of the Diophantine equations you can get by different Godel number assignments are all identical, which is definitely not the case. $\endgroup$ – Jesse Elliott Oct 14 '14 at 7:59

This is has been considered in philosophy perhaps even more so than mathematical logic. I would consider Lindstroem's theorem (roughly, first order logic is the strongest system in some sense for which one can have a compactness theorem and a downard Lowenheim-Skolem theorem) as a result in meta-meta-mathematics, because it talks about several systems of possible logics and provides an interesting result. Beyond that level may be distinctions that can be appreciated by practitioners, but from my non-professional and non-philosophical perspective, it is a form of hair-splitting to be practiced by people other than me. I don't see how higher levels will benefit my or anyone elses studies, and I've done more than an average amount of reading in mathematical logic.


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