Talk:Metamathematics

obsolete term?

The term 'metamathematics' is obsolete? Hmm... Maybe I've been reading way too many old texts on math, but I've certainly come across it more often than this article seems to state. Just because mathematical logic has more google hits means nothing in this circumstance. Anyway, metamathematics does not have to be mathematical, does it? But mathematical logic does. I would think this means that mathematical logic is actually a subset of metamathematics. Eric Herboso 02:00, 18 Feb 2005 (UTC)

Yes, metamathematics does have to be mathematical. It is (in my not-at-all-humble opinion) a subset of the intersection of mathematics and logic. 86.132.220.143 (talk) 20:25, 25 November 2017 (UTC)

Googlefight

Is using Googlefight in the way the article does really necessary and/or encyclopedic? --Aioth 15:21, 13 January 2006 (UTC)

Proposed merge

There seems to be no difference between metalogic and metamathematics. At least, all the examples in metalogic are or should be in metamathematics, and everything in metamathematics fits the definitions in metalogic. If metalogic were to discuss the meta-theory of non-mathematical logic, that could be different, but that would seem to require a formal theory of non-mathematical logic — which would make that aspect mathematical logic. — Arthur Rubin | (talk) 19:55, 14 January 2008 (UTC)

Oppose strongly the suggested merger :The above is quite incorrect: for example metalogic also includes many-valued logic, MV-logic algebras, \{}Lukasiewicz logics ,LM-logic, etc., that most ( e.g. >> 99.999 percent of today's mathematicians would not consider as metamathematics; thus one can study metacategories and multicategories without many-valued logics. The two entries should not be merged, It simply means that the metalogic authors' suggestions are not based on mathematical reality as published today. Furthermore, metalogic may also include various quantum logics that most mathematicians quite correctly would not regard as metamathematics, but as logical foundations of quantum physics theories that can also be considered under metalogics, but not under metamathematics. Nu 16:37, 2 March 2009 (UTC)Bci2Nu 16:37, 2 March 2009 (UTC)WP:Mathematics Project{WP: Physics Project}; 3rd March 2009 [UTC.]

The people I know who are doing what is described as "metalogic" consider themselves (and are called by others) logicians, not metalogicians – as for example those in Category:American logicians – and the field they are working in is known as "(mathematical) logic", not "metalogic". The latter term does not even occur in the AMS Mathematics Subject Classification. I don't think these logicians are particularly concerned with the "truths-of-logic", whatever that may be. (The term is not explained.)  --Lambiam 23:26, 14 January 2008 (UTC)

• Metalogic and metamathematics do not seem to be the same subject, though there appears to be some overlap in the matters considered and studied. Metamathematics appears to study many of the foundational problems in formal logic and seems geared towards system analysis. Metalogic seems much more narrowly focused on the advantages and shortcomings of formal language, rather than taking the systemic approach. Vassyana (talk) 15:03, 15 January 2008 (UTC)

Thinking about it, metalogic seems just to be what is commonly known in mathematical circles as model theory and proof theory. Although Britannica (online) has an article on it, the article metalogic has no secondary sources (the current reference being a primary source), and the article at present has nothing distinguishing it from other better known concepts in mathematical logic. I think, perhaps, I should withdraw the merge request in favor of changing it to a quasi-disambiguation page, unless sources can be provided that people doing it use the term. — Arthur Rubin | (talk) 17:38, 15 January 2008 (UTC)

Assuming (I can't find an explicit statement to the effect) that the author of Metalogic: An Introduction to the Metatheory of Standard First-Order Logic equates metalogic with "the metatheory of logic", the definition offered is:

"The metatheory of logic is the theory of those formal languages and systems that for one reason or another matter to the logician."

There is a chapter with the title "Metalogic" in Hao Wang's From Mathematics to Philosophy, p. 166.[1] It defines the notion as follows:

"Metalogic may be defined as the study of the syntax and the semantics of formal languages and formal systems."

I agree that this covers the areas of model theory and proof theory within mathematical logic, and the article should indeed point that out.  --Lambiam 21:06, 15 January 2008 (UTC)

Oppose. Metalogic straddles mathematics and philosophy. Just doing a quick Google search, UC Berkeley offers Metalogic as a course in the Philosophy Dept. UC Irvine offers it as a joint course of the Logic & Philosophy of Science and the Philosophy Departments. Compare Philosophy with Metaphilosophy. Chet Ubetcha (talk) 02:42, 23 January 2008 (UTC)

• Comment. Well, UCI is not really known for having a first-class logic or philosophy department, but that's beside the point. I attempted to withdraw the merge request above, but others have said it should be merged somewhere, so it seems improper just to delete the request. — Arthur Rubin | (talk) 02:51, 23 January 2008 (UTC)
• Comment. Both those courses are based on the book Metalogic, not necessarily on the subject. Odd, isn't it. — Arthur Rubin | (talk) 02:54, 23 January 2008 (UTC)

We are not the first to argue this. The players were Tarski versus Carnap, who were so important that I don't have to pipe their names to get the links to work. Chet Ubetcha (talk) 03:10, 23 January 2008 (UTC)

• off-topic remark I have to take exception to Arthur's comment about UCI. I don't know about their philosophy department in general, but in philosophy of math specifically it is extremely strong (that's where Penelope Maddy is), and they have a whole department of "Logic and Philosphy of Science", which I believe is the first of its kind. In math logic, on the other hand, they have Matthew Foreman, so not hurting on that score either. --Trovatore (talk) 03:17, 24 January 2008 (UTC)

Oppose. Metamathematics is not the same as metalogic. Furthermore, metamathematics is different enough (and important in its own right by enough) to warrant its own article in wikipedia. By the way, how many official votes do we need before we can take that merge tag down? — Eric Herboso 09:41, 6 February 2008 (UTC)

Question If the two topics are different enough to warrant distinct articles why is the distinction not made clear?

• Comment Neither article is very long, detailed or clear. They seem to refer to each other, I'd say that the topic is almost circular. I have no doubt that the two subjects are different and that if you had two experts, one from each field, they could tell you how the subjects are related but that they look at different ASPECTS of the theory. I'm a math person myself, and it seems to me that 'metamathematics' is a specific field of study within the larger on context of 'metalogic'. I think that if you want to keep the two articles disjoint then you should probably ADD enough context in both articles to warrant a distinction and disjoint articles; otherwise they should be merged and put into correct context of one another. Jka02 (talk) 15:50, 22 March 2008 (UTC)

Oppose. Shouldn't we prove that mathematics = logic before merging the articles at the metalevel? Pouring oil on burning waters. RJBotting 19:05, 16 June 2008 (UTC)

What exactly is metamathematics?

Hibbert spoke about metamathematics and mathematics, about synthesizing complete systems by using rules to manipulate arbitrary symbols, and using these rules to manipulate the symbols such that their completeness is shown. My question is, what qualifies mathematics and what qualifies metamathematics? Is the statement "x is x" mathematical, or metamathematical? What about "'x' is 'x'"? That is to say, are the rules that define the manner in which the symbols manipulated mathematical or metamathematical? How can mathematics be derived from metamathematical rules while metamathematics is defined as being about mathematics? Shouldn't it be the other way around if any formal mathematical system is derived from rules that lie in the domain of metamathematics?

NOTE: I assume you meant "Hilbert" not "Hibbert" (?!), as one does not recognize any "Hibbert" as any kind of authority in metamathematics, or metalogics for that matter.

Furthermore, metamathematics is just one field of science, whereas metalogics are many distinct fields, different from metamathematics in both methodology and fundamental concepts.

Turing

Shouldn't there be mention of Alan Turing as a key contributor to metamathematics? His work on decidability, which gave rise to the notion of the Turing machine, was prompted by the Hilbert programme -- The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900. —Preceding unsigned comment added by 81.151.101.2 (talk) 20:54, 10 March 2009 (UTC)

Yes, the Entscheidungsproblem should be mentioned, as well as the problems from Hilbert's problems that had a foundational aspect. — Carl (CBM · talk) 21:58, 10 March 2009 (UTC)

Trimmed

I trimmed the article some today. User:Bci2 expanded it some, and some of the new content was good, but much of it seemed to make the article hard to read (to my eye) or to diverge into ruminations about the difference between metamathematics, logic, mathematics, and metalogic. I do think that we can expand this article quite a bit, and I added a "milestones" section to start that. I also found a nice article on JStor about Tarski's use of the term metamathematics.

The main difficulty I see with expanding this article is the disuse of the term "metamathematics" by many contemporary mathematical logicians. It may be hard to document more contemporary mathematical viewpoints, but we want to ensure that we don't write an article from the point of view of 1960 either. — Carl (CBM · talk) 21:54, 10 March 2009 (UTC)

Reply on your update

You also need a non-circular definition of metamathematics!

The present definition is not circular: the study of mathematics itself using mathematical methods. Could you explain what circularity you see? — Carl (CBM · talk) 13:03, 12 March 2009 (UTC)

Proposal to add the metatheory category

This is to propose adding the metatheory category to this article.
I just undid my last revision because it seemed that someone wasn't very happy about it.
Thanks for your attention.
Maurice Carbonaro (talk) 12:06, 10 November 2009 (UTC)

Metalogic

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

I'm proposing that Metalogic be merged into Metamathematics, although there is some justification for the reverse merge. As far as I can tell, there's been no discussion since it was proposed in 2009, nor any distinction given between Metalogic and Metamathematics. If we can't distinguish them, they should both be in the same article. — Arthur Rubin (talk) 14:01, 20 November 2012 (UTC)

I see I proposed this before. If both articles were to include the distinction (if any), I would withdraw the merge request, although both articles are basically stubs at this point. Perhaps an article "Foundations of logic" which includes both? — Arthur Rubin (talk) 14:04, 20 November 2012 (UTC)

Please ignore the tags, to some extent. After refreshing my memory of the previous discussions, I think Metalogic and Metamathematics should be merged somewhere. They have a lot in common, but the name of the merged article would be the subject of heated debate, probably only of interest to those on the metalogic side. — Arthur Rubin (talk) 14:09, 20 November 2012 (UTC)

Perhaps they should both be merged into foundations of mathematics, and then split on a different basis if enough can be found to be worthy of a split. Foundations of mathematics is a long article, but neither of these is much more than a stub. — Arthur Rubin (talk) 14:13, 20 November 2012 (UTC)

• Oppose I strongly oppose this repeated proposal, and I do question the motivation. I believe it is an excuse to delete valid material. There does exist a hypothetical universe, where I would support the creation of an article Metalogic and metamathematics, however, the political environment here is not it. Arthur originally believed and claimed that it is a neologism, which just shows where he is at, in terms of understanding the issue. I even showed him a university course schedule where Metalogic is still currently being taught. So this proposal is completely inappropriate, and I would like the issue closed for good this time.Greg Bard (talk) 14:57, 20 November 2012 (UTC)

  I pointed out the course was about the book (or at least drew its title from the book), rather than necessarily being about the topic. It still seems that "metamathematics" and "metalogic" are never used in the same context, so it is difficult to assert that they are different. If you (after 4 years) have been unable to describe the difference, isn't it possible that there isn't one.

  As far as I'm concerned, this article (metamathematics) is defective; since I can't think of sources for improvement, it seems a merge somewhere is in order. "Metalogic" may be worse, consisting primarily of examples of topics within the field of "metalogic", with no unifying theme.

  (And, for what it's worth, the previous discussion did not have a formal resolution, so writing "result=keep" is somewhat questionable; one could easily justify a delete of both articles on the grounds of notability.) — Arthur Rubin (talk) 15:39, 20 November 2012 (UTC)

Arthur, your statement is so full of falsehoods, presumptions, and wrong analysis, that it is hard to know where to begin, where to end, or if there is any hope that you will understand or respect the decisions of those who do. No. Neither the article, nor any courses taught are "about the book." Metalogic is the title of chapters of books, used in the titles of scholarly articles, and is a term well used in the literature. You need to face the fact that you are late to the party in that regard. It isn't up to me to prove that the terms are the same or different. If you are proposing to merge them, it's up to you to prove they are the same, as there are sources for the use of each, and you, yourself state explicitly that they are used in different contexts. You state your opinion that you see no "unifying theme" in the article metalogic. Being able to see the conceptual connections between things is a function of intelligence. I cannot help you there beyond the explicit and specific organization of the article metalogic itself. Furthermore, your claim that the previous discussion did not have a formal resolution is so disingenuous, as to call it question your integrity. There was a very clear consensus to oppose your proposal, by any reasonable standard. It wasn't "questionable" AT ALL. Greg Bard (talk) 06:02, 22 November 2012 (UTC)

• Oppose: I fail to understand the reasoning behind this proposal. Metalogic is NOT metamathematics, though they are definitely intertwined. Metamathematics has to do with proof theory and deals with how we describe and justify what we use as mathematical rules. Meanwhile, metalogic deals with how we use describe and justify what we use as logical rules of inference. These are separate issues. Sure, metalogic contains metamathematics within it, and so the terms are sometimes used interchangeably. But they refer to separate concepts. Logics need not necessarily use mathematical reasoning within them, and so there are some issues in the field of metalogic that would not be included in metamathematics. As one example, see this sentence in encyclopedia's entry for metamathematics: "Thus, these metatheorems, strictly speaking, do not belong to metamathematics and are therefore often said to belong to metalogic or to what is called set-theoretic predicate logic." — Eric Herboso 04:14, 22 November 2012 (UTC)

  Did you, perhaps read the heading of that entry. "From The Great Soviet Encyclopedia" .... Even if the entry were not ideologically based, it is translated from Russian. — Arthur Rubin (talk) 07:08, 22 November 2012 (UTC)

  The English translation of the Great Soviet Encyclopedia is quite good, as the wikipedia article on it shows, so your claim of possible mistranslation seems unlikely. As for your claim of ideological bias, should we also refuse to use the Academic American Encyclopedia, the New American Cyclopedia, or the New American Desk Encyclopedia? What about Encyclopaedia Britannica? After all, it has Great Britain right there in the name. Of course, I take the point that every encyclopedia has inherent bias of some kind. But I fail to understand how the ideology of the Soviet Union at that time would cause it to say something incorrect about metalogic and metamathematics. If you seriously think that the article is ideologically biased in this respect, does this mean you are suggesting that there was a Soviet conspiracy to undermine the field of metalogic through surreptitious encyclopedia entries? — Eric Herboso 00:43, 25 November 2012 (UTC)

Discussion

One should also note a statement in the (current) version of metalogic:

Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by mathematical logic in academia.

— Arthur Rubin (talk) 15:45, 20 November 2012 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Russell?

What contributions to metamathematics did Russell make? I ask because /Principia Mathematica/ has no discussion of such metamathematical topics as redundancy of axioms, decidability, completeness. Can anyone cite a work were Russell indicated an interest in any such matters? 86.132.220.143 (talk) 20:22, 25 November 2017 (UTC)

Error

I hope it's not impolite to say so, but whoever wrote "Gödel's incompleteness theorem: proof that given any finite number of axioms for Peano arithmetic, there will be true statements about that arithmetic that cannot be proved from those axioms." doesn't know what they're talking about. There is an axiomatization of PA that unavoidably has infinitely many axioms - the first-order axiomatization. And there is an axiomatization of PA that has finitely many axioms - the second-order axiomatization. Both are incomplete. But Gödel's incompleteness theorem isn't really about PA (2nd or 1st order), it is about a broad class of arithmetics. Shall I delete the offending sentence? 86.132.220.143 (talk) 20:35, 25 November 2017 (UTC)

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Gentzens' Cut-Theorem and Lorenzen?

I've read the German edition of this article and there is a reference to the philosopher and mathematican Paul Lorenzen and a work about Genzens' Hauptsatz. Would it be a good Idea to put this in the article, too? 2A02:908:424:9D60:0:0:0:4A03 (talk) 16:31, 25 August 2022 (UTC)